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Nonparametric and parametric analyses of food demand in the United States
Lee, Hwang-Jaw, Ph.D.
The Ohio State University, 1990
Copyright ©1991 by Lee, Hwang-Jaw. All rights reserved.
UMI 300 N. ZeebRd. Ann Arbor, MI 48106
NONPARAMETRIC AND PARAMETRIC ANALYSES OF FOOD DEMAND IN THE UNITED STATES
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy In the Graduate School of The Ohio State University
By Hwang-Jaw Lee, B.A., M.B.A.
The Ohio State University 1990
Dissertation Committee Approved by Wen S. Chern, Ph.D. Joseph Havlicek Jr., Ph.D. Cameron S. Thraen, Ph.D. Wen S. Chern, Adviser Department of Agricultural Economics and Rural Sociology DEDICATION
To Hy Parents
II ACKNOWLEDGEMENTS Achieving goals In life Is often Impossible without others. Numerous Individuals helped me 1n my development process. I wish to express my gratitude for their assistance. I extend my sincere appreciation to Dr. Wen S. Chern, my advisory committee chairman. I have greatly benefltted from his guidance and encouragement during my studies in the United States. Furthermore, his advice, criticism, as well as many suggestions are all necessary and extremely helpful Inputs to the completion of this research. Sincere thanks are extended to my committee members, Dr. Joseph Havlicek Jr. and Dr. Cameron S. Thraen for their helpful comments and suggestions. I would like to express special thanks and gratitude to the Department of Agricultural Economics and Rural Sociology and the Ohio Agricultural Research and Development Center (OARDC) for the financial supports during my Ph.D studies at The Ohio State University. Appreciation is also extended to the Instruction and Research Computer Center (IRCC) at The OSU for providing computer facilities and services for the completion of this research. Special thanks are expressed to my parents, brothers, and sisters in Taiwan for providing the love and support throughout my childhood and adult years. Finally, I express my deepest appreciation, gratitude, as well as love to Grace Chen, for her understanding, patience and encouragement while I struggled to complete this dissertation.
Ill VITA
August 28, 1957 ...... Born - Tainan, Taiwan 1979 ...... B.S., National Chung-Hsing University, Taiwan 1979-1981 ...... Second Lieutenant, Military Service, R.O.C. Army 1981-1986 ...... Teaching Assistant, Department of Agricultural Marketing, National Chung-Hsing University, Taiwan 1983-1985 ...... M.B.A., Institute of Graduate Studies in Business Administration, Tunghai University, Taiwan 1986-1987 ...... Research Assistant, Department of Textile and Consumer Economics, University of Maryland 1987-1990 ...... Graduate Research Associate, Department of Agricultural Economics and Rural Sociology, The Ohio State University
PUBLICATIONS Hwang-Jaw Lee and Wen S. Chern, "Effects of Different Income Sources on Food Expenditures," Working Paper, Department of Agricultural Economics and Rural Sociology, The Ohio State University, September, 1990. Hwang-Jaw Lee and Wen S. Chern, "Can We Use Aggregate Subgroup Data from BLS's Consumer Expenditure Surveys in Demand Analysis?" Working Paper, Department of Agricultural Economics and Rural Sociology, The Ohio State University, January, 1990. Hwang-Jaw Lee and Wen S. Chern, "Nonlinear Regression for Estimating a Quadratic Expenditure System," Proceedings of Business and Economic Statistics Section. American Statistics Association, 1990.
IV Hwang-Jaw Lee, "Estimation of The Demand for Food at Home and Food Away from Home," paper presented at the Third Annual Graduate Research Forum at The Ohio State University, Columbus, OH, April 15, 1989. Wen S. Chern and Hwang-Jaw Lee, "Nonparametrlc and Parametric Analyses of Food at Home and Food Away from Home," Paper presented at the 1989 Annual Meeting of the American Agricultural Economics Association, Baton Rouge, Louisiana, July 28 - August 2, 1989. Wen S. Chern and Hwang-Jaw Lee, "Complete Demand System of Nondurable Goods and Services," Proceedings of the 35th Conference of American Council on Consumer Interests. 1989. Wen S. Chern and Hwang-Jaw Lee and Horacio Soberon-Ferrer, "Estimation of Complete Demand Systems: A Comparative Analysis of Aggregate vs. Household Level Data," paper presented at the Annual Meeting of the Allied Social Science Association (ASSA), New York City, December 28-30, 1988. Wen S. Chern, Soberon-Ferrer Horacio and Hwang-Jaw Lee, "Estimation of Price and Income Elasticities of Energy and Oil Demand in LDC's," Proceedings of Ninth International Conference of International Association of Energy Economists. 1987. Hwang-Jaw Lee, "Risk and Investment Decision," Journal of Taiwan Economy. No. 110, February, 1986. (in Chinese). Hwang-Jaw Lee, The Impacts of Business Marriage on Shareholders' Wealth. Master Thesis, Institute of Graduate Studies in Business Administration, Tunghai University, 1985. (in Chinese). Hwang-Jaw Lee, "Multiple Attribute Utility Theory and It's Applications for Consumer Purchasing Decision," Journal of Taiwan Economy. No. 99, March, 1985. (in Chinese). Hwang-Jaw Lee, "Dynamic Pricing Strategy -- An Application of Learning Curve and Diffusion Curve, Journal of Taiwan Economy. No. 95, November, 1984. (in Chinese). Hwang-Jaw Lee, An Economic Study on Direct Sale Marketing of Fruits and Vegetables in Taiwan. Technical Report, Department of Agricultural Marketing, National Chung-hsing University,June, 1983. (in Chinese). Hwang-Jaw Lee, "The Export Development of Canned Sea Foods in Taiwan," Quarterly Journal of Bank of Taiwan. Vol. 30, No.3, September, 1979. (in Chinese).
V FIELDS OF STUDY Major Field Agricultural Economics, Demand and Consumption Analysis .... Prof. Wen S. Chern Prof. Leroy J. Hushak Prof. Alan Randal1 Minor Field Econometrics and Statistics .... Prof. Stephen R. Cosslett Prof. Adrian C. Cameron Prof. Irigmar R. Prucha Prof. Joseph S. Verducci Prof. Mike Flinger Family Resource Management and Consumer Economics ..... Prof. Nancy M. Rudd Prof. Sherman Hanna Prof. Kathryn Stafford Prof. Rachel Dardis
VI TABLE OF CONTENTS DEDICATION ...... II ACKNOWLEDGEMENTS ...... Ill VITA ...... IV TABLE OF CONTENTS ...... VII LIST OF FIGURES ...... X LIST OF TABLES ...... XII CHAPTER PAGE I INTRODUCTION ...... 1 1. Statement of Problem ...... 2 2. Objectives of The Study ...... 6
II THEORETICAL AND EMPIRICAL CONSIDERATIONS ...... 7 1. The Neoclassical Consumer Demand Theory ...... 7 2. Empirical Considerations ...... 9 1) Data consistency „..... „...... 10 2) Functional f o r m ...... 12 3) Aggregation problem ...... 15 a) Aggregation over coranodities ...... 15 b) Aggregation across consumers ...... 17 4)Demographic variables ...... 19
III LITERATURE R E V I E W ...... 22 1. Single Equation Food Demand Studies ...... 23 2. Engel Relationships Studies ...... 25 3. Food Demand Elasticity Matrix Studies...... 28 4. Complete Demand System Approach ...... 31 5. Concluding Remarks ...... 35 IV DATA SOURCES AND DESCRIPTION...... 37 1. Data Sources ...... 37 2. Data Description...... 43 VII V METHODOLOGY ...... 54 1. Nonparametric Approach ...... 56 1) Testing of data consistency ...... 57 2) Testing of rank in the demand system ...... «... 59 3) Factor and clustering analysis for commodity grouping 64 2. Parametric Approach ...... 71 1) Estimating demand equations under two>stage budgeting 71 2) Testing of model specification ...... 73 3) Model e s t i m a t i o n...... 78 4) Test statistics on parameters restriction ...... 83 5) Computation of elasticities ...... <>...... 85 VI RESULTS OF NONPARAMETRIC ANALYSES ...... 90 1. Data Consistency Test Results .... 90 2. Results of Rank Test ...... 94 3. Results of Commodity Grouping ...... 100 4. Construction of The Basic Set of Six Food Groups .... 113 5. Concluding Remarks ...... 121
VII RESULTS FROM PARAMETRIC ESTIMATION ...... 124 1. Model Specification Test ...... 124 2. Specification of PIGLOG Demand Systems ...... 127 3. Estimation of The LA/AIDS for 19 Food Categories in One Stage ...... 132 1) Regression results ...... 134 2) Comparisons of key elasticities for 19 food categories ...... 141 3) A complete demand elasticity matrix for 19 food coranodities ...... 149 4. Estimated Results from The Two Stage Procedure ...... 157 5. Further Comparison of Elasticities ...... 177 6. Predictive Performance Evaluation ...... 185
VIII CONCLUSIONS ...... 198 1. Objective of Research ...... 198 2. Empirical Finding ...... 199 1) Nonparametric results ...... 200 2) Parametric results ...... 202 3) Evaluation of predictive performance ...... 205 3. Limitation and Further Study ...... 206 FOOTNOTES ...... 210
VIII APPENDICES A. Review on Complete Demand Systems ...... 213 B. Food Expenditure Categories ...... 224 C. Regression Results of LA/AIDS Dynamic Model II ...... 227 D. Regression Results of Full Model under Two-Stage Procedure ...... 230 E. Elasticities Formulas for The Full Model ...... 237 F. Uncompensated Price and Expenditure Elasticities Matrix for 19 Food Categories ...... 239 G. Compensated Price Elasticities Matrix for 19 Food Categories based on Dynamic Model I ...... 252 H. Compensated Price Elasticities Matrix for Aggregate Commodity Groups ...... 255 I. Graphs for Comparing Actual and Predicted Expenditures . 261
LIST OF REFERENCES ... 272
IX LIST OF FIGURES FIGURE PAGE 1. Aggregate Food Consumption Patterns by Month ...... 50 2. Consumption Patterns of Meats and Seafood by Month ...... 51 3. Consumption Patterns of Vegetables by Month ...... 52 4. Consumption Patterns of Fruits by Month ...... 53 5. Relations between Different Systems of Budget Share Equation ...... 77 6. Hierarchical Structure of Clustering Analysis ...... Ill 7. Relationships between Additive Demand Systems ...... 216 8. Comparison of Actual and Predict Cereal Expenditure by Month ...... 262 9. Comparison of Actual and Predict Bakery Products Expenditure by Month ...... 263 10. Comparison of Actual and Predict Beef Expenditure by Month ...... 263 11. Comparison of Actual and Predict Pork Expenditure by Month ...... 264 12. Comparison of Actual and Predict Other Meats Expenditure by Month ...... 264 13. Comparison of Actual and Predict Poultry Expenditure by Month ...... 265 14. Comparison of Actual and Predict Seafood Expenditure by Month ...... 265 15. Comparison of Actual and Predict Eggs Expenditure by Month ...... 266 17. Comparison of Actual and Predict Milk Expenditure by Month ...... 266
X 18. Comparison of Actual and Predict Dairy Products Expenditure by. Month ...... 267 19. Comparison of Actual and Predict Fresh Fruits Expenditure by Month ...... 267 20. Comparison of Actual and Predict Fresh Vegetables Expenditure by Month ...... 268 21. Comparison of Actual and Predict Processed Fruits Expenditure by Month ...... 268 22. Comparison of Actual and Predict Processed Vegetables Expenditure by. Month ...... 269 23. Comparison of Actual and Predict Sweets Expenditure by Month ...... 269 24. Comparison of Actual and Predict Oil Expenditure by. Month ...... 270 25. Comparison of Actual and Predict Nonalcoholic Beverage Expenditure by Month ...... 270 26. Comparison of Actual and Predict Miscellaneous Foods Expenditure by. Month ...... 271 27. Comparison of Actual and Predict Food Away from Home Expenditure by Month ...... 271
XI LIST OF TABLES TABLE PAGE 1. Survey of Food Demand Studies Using Single Equation ..... 24 2. Survey of Studies on Food Consumption Using Engel Relationship, Socio-economic and Demographic Factors .... 26 3. Survey of Studies Using A Food Demand Matrix ...... 30 4. Survey of Studies on Food Demand In Using Complete Demand System ...... 33 5. Sample Size by Year and Total ...... 42 6. Meekly Expenditure of Food Categories for Urban Consumer Units ...... 47 7. Sample Mean of Family Size and Age of Household Head .... 48 8. Descriptive Statistics of Budget Shares and Prices for 19 Food Categories ...... 49 9. Results of Nonparametric Test ...... 92 10. Results of Specification of Engel Function for 19 Food Categories ...... 97 11. Estimated Gorman Statistics ...... „...... 99 12. Results of Principal Component Factor Analysis ...... 101 13. Results of Factor Patterns ...... 103 14. Results of Cluster Listing ...... 105 15. Cluster S t r u c t u r e ...... 107 16. Inter-Cluster Structure ...... 108 17. Summary of Oblique Centroid Component Cluster Analysis .... 110 18. Results of Consistency (GARP) Testing for Selected Commodity Groupings ...... 112
XII 19. Comparison of Consistency (GARP) Test Results Between Cluster and BLS Groupings ...... 116 20. Results of Consistency (GARP) Testing for Six Food Groups . 118 21. Results of Nonparametric Rank Test for Six Food Groups .... 120 22. Results of Engel Function Specification Test for Six Food Groups .... 126 23. Regression Result of The LA/AIDS Dynamic Model 1 .... 135 24. Test Statistics for Testing LA/AIDS Specifications .... 140 25. Comparison of Uncompensated Own-Pr1ce Elasticities Estimated for Different LA/AIDS Specifications ...... 142 26. Compari son of Expenditure Elasticities Estimated for Different LA/AIDS Specifications ...... 144 27. Comparison of Estimated Elasticities for Family Sizes with Different LA/AIDS Specifications ...... 146 28. Comparison of Estimated Elasticities for Age with Different LA/AIDS Specifications ...... 147 29. Uncompensated Price and Expenditure Elasticities Matrix for 19 Food Categories Based on LA/AIDS Dynamic Model II .. 150 30. Compensated Price Elasticities Matrix Based on LA/AIDS Dynamic Model II ...... 153 31. Test Statistics Based on Objective Function for Testing Model Specifications ...... 159 32. Test Statistics Based on Likelihood Ratio for Testing Model Specifications ...... 160 33. Uncompensated Price and Expenditure Elasticities Matrix for Aggregate Groups in The First Stage ...... 161 34. Uncompensated Price and Expenditure Elasticities Matrix for Subgroup I ...... 163 35. Uncompensated Price and Expenditure Elasticities Matrix for Subgroup II ...... 165 36. Uncompensated Price and Expenditure Elasticities Matrix for Subgroup III ...... 166 37. Uncompensated Price and Expenditure Elasticities Matrix for Subgroup IV ...... 167 XIII 38. Uncompensated Price and Expenditure Elasticities Matrix for Subgroup V ...... 169 39. A Complete Estimated Elasticities Matrix from Full Model for 19 Food Categories under Two-Stage Procedure ...... 171 40. A Complete Estimated Elasticities Matrix from Translog Model for 19 Food Categories under Two-Stage Procedure .... 173 41. A Complete Estimated Elasticities Matrix from LA/AIDS Model for 19 Food Categories under Two-Stage Procedure .... 175 42. Comparisons of Estimated Uncompensated Own-Price Elasticities from One-Stage and Two-Stage Procedures .... 178 43. Comparisons of Estimated Expenditure Elasticities from One-Stage and Two-Stage Procedures ...... 180 44. Comparisons of Elasticity Estimates with Other Studies .... 182 45. Comparison of Average Information Inaccuracies for Different LA/AIDS Specifications ...... 187 46. Comparison of Root-Mean-Square Errors for Different LA/AIDS Specifications ...... 189 47. Comparison of Root-Mean-Square Percentage Errors for Different LA/AIDS Specifications ...... 191 48. Comparison of Accuracy Number on Turning Points for Different LA/AIDS Specifications ...... 194 49. Comparison of Slope Coefficients in The Merton Market Timing Test for Different LA/AIDS Specifications ...... 197 50. Theoretical Characteristics of Selected Complete Demand Systems ...... 221 51. 19 Food Expenditure Categories ...... 225 52. Regression Results of The Dynamic Model II ...... 228 53. Estimates of First Stage Structural Parameters Based on Full Model ...... 231 54. Estimates of Second Stage Structural Parameters for Group I Based on Full Model ...... 232 55. Estimates of Second Stage Structural Parameters for Group II Based on Full Model ...... 233
; XIV 56. Estimates of Second Stage Structural Parameters for Group III Based on Full Hodel ...... 234 57. Estimates of Second Stage Structural Parameters for Group IV Based on Full Model ...... 235 58. Estimates of Second Stage Structural Parameters for Group V Based on Full Model ...... 236 59. Uncompensated Price and Expenditure Elasticities Matrix for 19 Food Commodities Using Static Model I ...... 240 60. Uncompensated Price and Expenditure Elasticities Matrix for 19 Food Commodities Using Static Model II ...... 242 61. Uncompensated Price and Expenditure Elasticities Matrix for 19 Food Conmodiiies Using Static Model III ...... 244 62. Uncompensated Price and Expenditure Elasticities Matrix for 19 Food Consnodities Using Dynamic Model I ...... 246 63. Uncompensated Price and Expenditure Elasticities Matrix for 19 Food Commodities Using Dynamic Model III ...... 248 64. Uncompensated Price and Expenditure Elasticities Matrix for 19 Food Commodities Using Dynamic Model IV ...... 250 65. Compensated Price Elasticities Matrix for 19 Food Commodities Using Dynamic Model I ...... 253 66. Compensated Price Elasticities Matrix for Aggregate Commodity Groups in The First Stage ...... 256 67. Compensated Price Elasticities Matrix in The Second Stage, Group I ...... 257 68. Compensated Price Elasticities Matrix in The Second Stage, Group II ...... 258 69. Compensated Price Elasticities Matrix in The Second Stage, Group III ...... 259 70. Compensated Price Elasticities Matrix in The Second Stage, Group IV ...... 259 71. Compensated Price Elasticities Matrix In The Second Stage, Group V ...... 260
XV CHAPTER I INTRODUCTION Consumer demand for food is a critical component in the economic analysis of agricultural policies. Understanding the structure and pattern of food consumption is essential for designing and assessing food and agricultural programs and policies. For example, the estimated price and income elasticities of food demand provide important information for assessing the impacts of government price support and income maintenance programs. Other policy areas in which the estimated demand elasticities and projection of consumer demand could be useful include structural analysis and strategic planning by the food industry. Demographic and socio-economic characteristics of the U.S. population which may affect food consumption have changed significantly in recent years. The important changes include the declining rate of growth of population, the composition of age, and size of households, the increasing labor force participation of women, and the racial mix of the population. In addition, increasing nutrition and health concerns may have, as claimed by many researchers, changed food consumption patterns in the U.S.. These claims have been based on the fact that there have been substantial decreases in per capita consumption of red meat and eggs during the last two decades. These and other changes in food consumption patterns have major implications for the food industry, especially if these changes continue in the future.
1 The purpose of this dissertation research is to investigate and apply the most appropriate complete demand systems to analyzing the consumption and demand for food commodities in the United States. In addition, this study also attempts to narrow the gap between theoretical and empirical demand analysis using one of the most complete and consistent data bases available to date and more advanced econometric procedures than previously employed. Therefore, the empirical results of this study will be consistent with the underlying microeconomic theory and the estimated demand parameters will reflect accurately ood consumption behavior during the study period.
1.1 Statement of Problem Traditional consumer demand theory provides a framework for formulating a demand function for any commodities such as food. Basically, under given conditions, the quantity of a commodity demanded can be expressed as a function of prices and income as derived from the maximization of utility subject to budget constraint. However, many economists found that it is necessary to take into consideration the effects of many other demographic and socio-economic variables on food consumption as these data become available. For instance, Tomek (1977) indicated the need for economists to be attuned to changes in socio-demographic as well as economic factors. Specifically, important changes have been occurring in the size and age composition of U.S. households that may have an impact on food consumption levels and patterns. Many studies related to the demand for food are available in the literature. These studies can be classified into two groups. The first group used a single equation method for food (aggregate or subcategories) demand analysis. The other group used complete demand systems which either incorporated food as one of the expenditure items or dealt with a subgroup of food items only. The single-equation approach does not permit imposition of the restrictions that are implied in consumer demand theory. For this reason, the system-wide approach has gained its popularity in the literature of theoretical and applied demand analyses during the last two decades. The use of a complete demand system approach al1ows us to first test the constraints of microeconomic demand theory and then possibly impose these restrictions in parameter estimation. The data used in previous food demand studies relied primarily on either time-series data for food expenditure (aggregate or subcategories) from national income account or the cross-sectional household survey data which were available only for selected years. Due to the 1 imitation of data availability, there has been very 1ittle research on food consumption with pooling time-series and cross-sectional data. Also due to these data limitations, previous researchers usually must either define broad aggregate items as observation basis or use average expenditures as the analysis unit. These practices are often seen in food demand research using time-series data. Most studies based on time-series data could not shed much 1ight on the effects of social and demographic factors. Moreover, the problem of degrees of freedom often prevents one from using a more flexible functional form. All these limitations 1ikely affect the accuracy of estimated elasticities and the simulations using these empirical estimates. On the other hand, a study using cross-sectional data may have sufficient observations to apply a more flexible functional form and be able to detect the demographic effects on food consumption than that using the time-series data. Unfortunately, more often than not, the price data are not available, thus the price effects can not be examined. Further, such data sets could not be used to distinguish static versus dynamic behavior patterns. All these deficiencies result from a prevalent "gap between theory and empirical analysis" of consumer behavior (Houthakker, 1960). Fortunately, the BLS has been able to provide an excellent data basis in recent years. Specifically, it has been conducting a diary survey of consumer expenditures on frequently purchased iterns including detailed food commodity items yearly and continuously since 1980, These data bases can solve the problems of data 1 imitations discussed above. We can now conduct an analysis at a fairly disaggregate level of food commodity groupings which can alleviate the aggregation problem across commodities and provide more useful estimates of substitution patterns among food commodities than the previous aggregate models. We also can examine the effects of various demographic and household characteristics on food demand and consumption. Furthermore, many advanced econometric procedures developed in the econometrics 1iterature will be applied in this study. These developments of methodology permit us to explore a possibility of systematically bridging the gap. between the theoretical and empirical analysis of food demand. Among advanced methodologies related to this research include factor and cluster analyses, nonparametric testing on data consistency and rank condition of demand systems, flexible functional forms, translating methods, as well as nonlinear system estimation procedures. Specifically, we will firstly employ the nonparametric testing procedures to examine whether or not our data can be rationalized by a nonsatiated, continuous, concave, monotonic utility function. After testing data consistency, we will conduct the rank test in order to reduce the dimension of the commodity space. Two multivariate statistical techniques, factor and cluster analyses, are used to gauging data structure and grouping food items into "similar" commodity groups. In parametric analysis, we will draw appropriate inferences about demand elasticities over data space without prior constraints. Based on model specification test, we will select the best among various flexible functional forms such as the full demand system developed by Lewbel (1989) and its nested forms of indirect translog and almost ideal demand system (AIDS). In addition, we will use the translating procedures to incorporate the demographic effects into a complete demand system. Finally, five statistical measurements will use to evaluate the predictive performance of alternative model specifications. 1.2 Objectives of The Study Essentially, the primary objective of this study is to analyze empirically the food consumption pattern and demand structure using the integrated data bases and econometric procedures. In addition, important issues between theoretical and empirical demand analyses will be addressed. More specifically, the objectives of this study are summarized as follows: (1) To develop a methodology for implementing the neoclassical demand model for food commodities; (2) To construct monthly food expenditure data based on the survey data for more than seventy eight thousands households during the 1980-1986 sample period; (3) To employ nonparametric procedures for examining data consistency, data structure, and preference structure. (4) To find an appropriate flexible demand system for modeling consumer behavior and analyzing food consumption structure; (5) To apply both one-stage and two-stage budgeting procedures by means of flexible functional form for constructing a 19 by 20 demand elasticities matrix; (6) To empirically investigate the impacts of demographic or/and socio-economic factors on food consumption; (7) To simulate food demand/expenditure during the sample period; (8) To draw Implications on changing patterns of food consumption and demand in the United States. CHAPTER II THEORETICAL AND EMPIRICAL CONSIDERATIONS
Basically, this study 1s an applied and empirical research. A success of empirical work requires a solid theoretical foundation. This chapter presents a review of the neoclassical consumer demand theory and a discussion of several important issues for applying the theory to modeling food demand in the United States.
2.1 The Neoclassical Consumer Demand Theory Demand theory deals mainly with consumer choices. Any economic theory must be based on certain basic axioms or assumptions which form a logical conceptual foundation for the theory. The axioms themselves are accepted as basic truths within which the theory is to be developed (Simmons, 1974). Demand theory begins with a consumption set and a binary preference relation. Usually, a consumption set, X, is assumed to be the nonnegative orthant in Rn as well as a closed and convex set. Five axioms must be constructed for the existence of a utility function which represents the goal of consumers' desirability. These axioms are completeness, transitivity, reflexivity, continuity and strong monotonicity (see Debreu (1959), Green (1971), Deaton and Muellbauer (1980a), and Varian (1984)). If preference satisfies all these axioms, then there exists a continuous utility function U: X - R which
7 represents these preferences1. The consumer's choice problem is to allocate his Income within his budget constraint, M, into a bundle of market goods and services in the set X that maximizes U(x) under given market prices, P, which is an n-dimensional vector and p e Rn. If a local nonsatiation axiom is imposed for the preference structure then there exist some bundles x' in X with |x'-x| > e for any positive real number e, such that x' is preferred to x (Varian, 1984). The bundle which maximizes the utility subject to budget constraint can be solved by a Lagrange method. By means of the first-order and second-order conditions from a Lagrangian function we can obtain the well known Marshallian demand functions and the marginal utility of income2. Alternatively, an indirect utility function, V(P,M) corresponding to the maximum attainable utility function, given income and prices, can be derived by substituting Marshallian demand functions Into a direct utility function. The resultant indirect utility function can be shown to have the following properties: (a) V (P,M) is continuous at all P > 0, M > 0; (b) V (P,M) is nonincreasing in P; (c) V (P,M) is quasi-convex in P; (d) V (P»M) is homogenous of degree zero in (P»M). A more convenient way to obtain a demand function can be achieved through the indirect utility function by using Roy's Identity. Furthermore, the consumer's choice problem can also be solved by minimizing the expenditure of attaining a given level of utility. The resulting expenditure function, e(P,U), has five properties: (a) nondecreasing in P; (b) homogenous of degree one in P; (c) concave in P; (d) continuous in p, for p » 0; and (e) Shephard's Lemma, i.e., where they exist, the partial derivatives of the expenditure function with respect to prices are the Hicksian demand functions3. The expenditure function and Indirect utility function are intimately related. The relationships among utility functions (direct and indirect), expenditure function and demand functions (Marshallian and Hicksian) can be linked together to form a well known theory of duality in demand analysis (Deaton and Muellbauer, 1980a, pp. 38-41). The Marshallian and Hicksian demand functions derived from the neoclassical theory of consumer behavior should maintain four well-known properties: adding-up, homogeneity, symmetry, and negativity.
2.2 Empirical Consideration The preceding section described the basic neoclassical consumer theory. Under the axioms of preference structure, the demand functions which possess the four properties are derived from utility maximization subject to budget constraint. Note that the theory does not suggest any functional form for empirical studies. Two alternative approaches are usually adopted to link the theory with empirical work. One is that the theory is first assumed to be valid, then the functional forms of demand with the four properties imposed are chosen for empirical studies. Another approach Is that demand functions are specified which may or may not satisfy the properties of demand theory. However, these restrictions can be examined. Since the four properties listed above are sufficient to guarantee that the system is consistent with neoclassical consumer theory, tests of these properties are valid tests of the theory Itself. Each of these two approaches has its specific merits. The 10 advantage of the former approach of Imposing the restrictions can reduce the number of structural parameters to be estimated. However, the Imposition of restrictions on demand functions Implies that the neoclassical theory Is valid with respect to the particular set of data being used for estimation. This a priori assumption may or may not be true. The selection of these alternative approaches for empirical work often depends on the objectives of research. This section deals with several Important issues often raised when the neoclassical consumer theory 1s applied to empirical demand analysis. These are (1) the problem of data consistency; (2) the Issue of functional form which is still an open question in demand study; and (3) the two technical problems in aggregation and Incorporating demographic effects.
2.2.1 Data consistency Demand analysts often devote less attention to selecting appropriate data series than to the choice of econometric techniques, although data and methods are equally Important to the results (Manchester, 1990). After all, the rationality of consumer behavior 1s the basic premise In the neoclassical consumer theory. Based on this premise, a well-behaved utility function can be formulated. Then a system of demand functions can thus be derived from the utility maximization principle. Therefore, whether or not the data used are consistent with this maximizing behavior 1s critical 1n assessing the validity of estimated parameters of a demand system. Generally, two quite distinctive approaches can be used for testing whether a finite body of price and quantity data is consistent with the utility maximizing behavior. The first approach is a parametric test based on the Slutsky conditions. This approach assumes that demand behavior can be adequately described by some parametric family of functional forms; one can then estimate the parameters that best describe the data by various statistical techniques and test for the restrictions imposed by demand theory. This procedure suffers from the fact that the testing involves a joint hypothesis of the restrictions of utility maximization and the chosen functional form. The second approach, on the other hand, is a nonparametric test which stems from the revealed preference conditions. These conditions provide a complete list of restrictions imposed by a rational behavior. One attractive feature of nonparametric tests for data consistency is that they do not involve any estimation of demand parameters, and thus no assumption on functional form is necessary. Many previous studies rejected the utility maximization hypothesis by using the parametric approach. For instance, Christensen, et al. (1975) employed both the direct and indirect translog demand systems; and Gallant (1981) employed the Fourier flexible form to test consumer behavior using U.S. data for 1929-72 for three aggregate categories of consumer expenditures. Both studies rejected the hypothesis that restrictions such as symmetry implied by the theory of demand are valid. In contrast, Manser and McDonald (1988) used a nonparametric procedure to test the same data and found data to be consistent with rationality. Manser and McDonald suggested two possible explanations of their rejections. The aggregation to three commodities may be improper or unwarranted. Or perhaps, the flexible forms used In these two studies are still not flexible enough. The preceding discussion has an important implication for empirical demand study. Nonparametric testing is a useful pretest tool for the maintained behavioral hypothesis. If data used in the study is inconsistent with the assumed rationality, one should examine whether the inconsistencies are caused by measurement errors, or by cross- sectional heterogeneity, or by preference changes. Generally speaking, if a nonparametric test can not reject the hypothesis of data satisfying the rational behavior, then a parametric analysis with a specific functional form could be pursued with more confidence. Consequently, if the main purposes of demand study are to estimate parameters in the demand system and to make statistical inferences, researchers should examine and test data consistency before a parametric procedure is developed. This is because a parametric model can be correctly specified and be validly estimated only under the condition of rational behavior assumptions. In case of data failing to satisfy the premise of rationality, the estimated parameters of a demand system using these data are likely to be useless in policy applications such as conducting policy simulations or evaluating and computing the cost- benefit measures or other welfare comparisons.
2.2.2 Functional Fora Demand functions in a demand system are typically derived from the assumed utility function. However, a utility function is an Implicit function in the neoclassical consumer theory. Consequently, the choice of functional form is often quite arbitrary and 1s treated as an econometric problem of model specification. In fact, an Infinite number of possible functional forms for a demand function theoretically exist. There are three distinct approaches commonly used to derive an explicit functional form for empirical study. The first approach begins with specifying a direct utility function, and then derives the demand functions from the first order conditions of the constrained utility maximization. The well-known example using this approach is the Linear Expenditure System (LES) which is derived from a Stone-Geary utility function. In addition, the direct translog model (Christensen et al., 1975) and the direct addilog demand system (Houthakker, 1960) and CES-type demand system (Houthakker, 1965) are also be included in this category. The second approach uses duality and begins with specifying a functional form for the indirect utility function or expenditure function. The explicit demand functions are then derived through either Roy's identity in the case of indirect utility function, or Shephards' Lemma in the case of expenditure function. From a mathematic point of view, the duality theory rests on a theorem due to Minkowski (1911). i.e., every closed convex set in Rn can be characterized as the intersection of its supporting halfspace (Diewert,1974). Moreover, there are several practical and econometric advantages for using duality in empirical demand studies4. This approach has been popular even since duality theory was introduced. The resulting flexible demand systems include indirect translog demand system (Christensen et al.,1975), Almost Ideal Demand System (Deaton and Muellbauer,1980b), generalized 14 Leontiff demand system (Berndt et al.,1977), generalized Cobb-Douglas demand system (Diewert, 1973), quadratic expenditure system (Poliak and Wales, 1978,1980), non-linear preference demand model (Blundell and Ray, 1984), Miniflex translog demand system (Barnett, 1983, 1985), Mini flex generalized Leontiff demand system (Barnett, et al., 1985, 1987), normalized quadratic demand system (Diewert and Wales, 1988), and Fourier demand system (Gallant, 1981, 1984). The third approach is based on a straightforward specification of demand functions, imposing directly the properties of demand theory. The main advantage of this approach is that it does not require any assumption about the form of utility function. One shortcoming of this approach is that the data used for estimation are assumed to be characterized by the properties of consumer theory. The validity of this assumption can not be examined. The Rotterdam model (Thei1,1965) and the composite demand system (Huang and Haidacher, 1983), almost complete demand system (Helen, 1982,1983) are all specified under this approach. Conceptually, these three approaches are interrelated within the framework of duality. In choosing a functional form, one needs to consider the trade-off between elegance and practicality. In this specific context, one needs to weight the validity of imposing the theoretical restrictions on a demand system and the required characteristics of data for estimation. To a large extent, the choice may be determined by the objective of the research. In general, for an accurate representation of reality, an unrestricted functional form is desirable. The more restrictive the form is, the more likely the data may be forced to fit the form which could be unsuitable. A more detailed 15 literature review on the functional forms of demand system Is discussed In Appendix A.
2.2.3 Aggregate Problem Due to the limited availability of data and the difficulty 1n computation, a number of aggregation conditions often need to be fulfilled In order to formulate an appropriate demand system using aggregate observations. There are two specific aggregation problems to be dealt with in this section. They are the problems of aggregation over commodities and across consumers.
2.2.3.a Aggregation over commodities For empirical analysis, one of the most serious problems 1s a large number of commodities in the market place. For Instance, for a completely unrestricted model for n commodities, there are n(n+l) parameters to be estimated. This number may be too many to be manageable in empirical estimation. Of course, we can Impose the homogeneity, symmetry and adding-up conditions Into model specification. After imposing these restrictions, the number of parameters can reduce to n(n+l)]-l. This reduced number of parameters may still be too large to be estimated. Therefore, grouping commodities Into broad categories is usually needed in order to Increase the degrees of freedom in estimation. In practice, two alternatives strategies may be used to address this problem. One 1s to assume separability which would reduce the dimensionality of the estimation problem. Another approach 1s to use 16 composite commodity theorem which 1s based on the behavior of exogenous variables such as price rather than on the form of the utility function. Various forms of separability have been used 1n the literature5. Depending upon the degree of restrictiveness Imposed on a utility function, there are weak separability, quasi separability, strong separability, pearce separability, block-add1t1v1ty and Gossen additivity. The practical Importance of various separability assumptions lies in the fact that (a) separability provides a fundamental linkage between aggregation over goods and the maximization principles in economic theory, (b) separability provides a theoretical basis for partitioning the economic structure into sectors, and (c) separability provides a statistical hypothesis, permitting great simplification in estimation of large demand systems (Barnett and Choi, 1989). Therefore, the imposition of a separability not only directly reduces the number of parameters to be estimated, but also offers a piausible description of consumer behavior. The composite commodity theorem was developed by Hicks (1936) and Leontief (1936). The Hicks aggregation theorem states that if a group of prices move in parallel then the corresponding group of commodities can be treated as a single commodity. The essentially similar result was also proved by Leontief. The Leontief aggregation theorem asserts that if a group of commodities is always consumed in fixed proportions, then we can treat the bundle as a single commodity. Application of these two approaches in constructing commodity grouping for empirical analysis has been rather limited. This is because, 1n reality, there are no guidelines on which type of 17 separability 1s appropriate. Furthermore, given the degree of separability, it 1s not clear which commodities should compose a particular group (Young,1977). Moreover, the Imposition of a separability assumption on utility function may affect the accuracy of estimates. Blackorby, Primont, and Russell (1977) prove that the imposition of separability destroys the local flexibility property of a flexible functional form when separability 1s Imposed globally. On the other hand, the composite commodity theorem, from a practical standpoint, 1s overly restrictive and is difficult to achieve in the real world. As Green (1964) points out, there is no reason to suppose that in the absence of proportionality the group would be separable.
2.2.3.b Aggregation across consumers Neoclassical demand theory and restrictions are all related to a particular individual behavior. However, data available for empirical studies are usually aggregate macro data. Therefore, many previous empirical demand studies used aggregate macro data to estimate the demand relationships derived from the theory of individual behavior. Researchers generally assume that there exists a "representative" consumer whose behavior can reflect the average of population. The question is whether or not the behavior of this representative consumer is the same as the behavior of actual persons, and furthermore, whether or not the properties of individual utility can be carried over to form the "representative" utility function. One may approach this problem of aggregation through model specification with parametric restrictions. Specifically, in order to 18 achieve a consistency with exact aggregation, an individual demand function must be linear in total expenditure and prices. This is, of course, the well-known conditions for exact aggregation. An easy way to examine this exact aggregation condition is to check the form of Engle function. Muellbauer (1975,1976) provides further insights into this aggregation problem by restating the requirement for the exact aggregation as
Xi ® a (p) (p)Mb (1) where xj is quantity of i-th good demanded for h-th consumer, and Mh is total expenditure of h-th consumer. In order to show the individual household demand being consistent with utility maximization, Muellbauer points out that Eq. (1) can be derived from the following cost function:
Ch(p,Uh) = ab+Ub$ (p) (2 ) where Uh is the utility for the h-th household. Eq.(2) has the Gorman polar form. This form remains too restrictive to permit the use of average (or mean) income (or total expenditure) in the aggregate demand function. In order to relax this restriction, Muellbauer develops the condition under which the aggregate demand function reflects the behavior of a "representative consumer". This condition implies that the aggregate demand is a function of not only the mean income but also a representative level of income which can, In turn, be a function of income distribution and prices. Deaton and Muellbauer (1980a) show that if consumer preference belongs to a price independent generalized linear class (PIGL), then the aggregate demand can be represented as if it was 19 the outcomes by a rational representative consumer. Recently, Lau (1982) Incorporates Individual attributes Into the condition of exact aggregation and develops a more general condition, called the fundamental theorem of exact aggregation. Lau shows that an exact aggregation can be satisfied under the following conditions: (a) all the Individual demand functions must be sums of products of separate functions of price and of total expenditure and individual attributes; (b) aggregate demand functions depend only on general symmetric functions of individual Income and an Index of attributes. The empirical applications of this theorem Include those studies by Jorgenson, Lau and Stoker (1982), Jorgenson, Slesnick and Stoker (1988), and Nicol (1989).
2.2.4 Demographic Variables Demographic variables such as household size, age of household head, labor force participation of woman, and races have been showed to be important determinants of household consumption. Therefore, 1t would be important to Incorporate these demographic variables Into our analysis of food consumption and demand. Generally speaking, there are two different econometric procedures to incorporate demographic effects into a demand system (Derrick and Wolken, 1982). The first approach is called a pooling method which uses additional parameters for demographic variables in model specification to capture the effects of demographic variables on consumption. The manner in which various demographic groups differ 1n their consumption behavior is determined by directly analyzing the estimates of the demographic parameters. The second approach 1s an unpooled method 1n 20 which separate demand systems are determined for each of demographic groups. The demographic characteristics define membership In each group of consuming units. Variation In consumption across groups 1s determined by the specific parameters estimated for each group. Each approach has its own merits6. The advantages of the pooling technique are parameter parsimony as well as explicit measurement of demographic effects. However, the unpool1ng specification allows parameters 1n the demand system to be free to vary across the demographic groups, a less restrictive procedure than pooling. The extent to which these merits are important remain an empirical question. To incorporate demographic effects Into model specification, the oldest and most commonly used method 1s adult equivalent scales7. Two best-known and often employed equivalent scales techniques are translation and scaling procedures. Translation approach subtracts comodity specific Indices from quantities in the direct utility function. Specifically,
U(X) = U i x ^ . Xfj-dj,) (3) where d{ is a function of demographic variables. A commonly used function Is linear such as d, - £alrNP, where Nr are demographic variables. The d,'s may be Interpreted as committed quantities of commodities. The corresponding Indirect utility function associated with this specification can be expressed as V(p,M-Ephdh). Therefore, It 1s easy to see that the effects of demographic variables on consumption come through income effects. Alternatively, a scaling procedure was originally developed by Barten (1964). Under this procedure, the direct utility function can be 21 written as
u(X)=u (4) This specification is interpreted as commodity specific consumer equivalent scales. The resulting indirect utility function is of the form V(p1d1 pNdN, H). It is, therefore, obvious that the influence of demographic variables will be felt through prices. Recently, Lewbel (1985) extended and modified the model described by Gorman (1976) to develop a general method of incorporating demographic effects into a demand system. The general method is to introduce functions of demographic variables, prices and expenditures into cost function of a demand system. It permits complicated interactions of demographic variables with prices and income (or expenditure). The resulting demand equations with demographic effects using this approach are given explicitly as functions of the original demand equations and the modified functions of demographic variables. Therefore, this procedure is Interpreted as altering a household's technology as well as to encompass adult equivalent scale and related methods. The translation approach will be employed in this study to incorporate the effects of demographic variables such as family sizes and age of household head on demand for food in the United States. CHAPTER III LITERATURE REVIEW This chapter presents a review of the past studies In food demand and consumption analysis 1n the United states. It is noted that the relevant and useful literature is not limited to those in food area. The methodology used in this study is based on many previous researches and recent development in modeling complete demand systems. A review of complete demand system modeling is providing in Appendix A. Many studies related to the demand for food (aggregate, food at home, food away from home, or by subcategories) are available in the literature. These studies can be roughly classified into four groups on the basis of methodology. They are (a) using a single equation approach for selected food items; (b) estimating Engel relationships; (c) constructing a food demand elasticity matrix; and (d) using a complete system approach for a food group. The first two categories of study usually attempted to estimate demand parameters for individual commodities. In contrary, the main objective of the last two categories of study is to estimate a complete set of demand parameters including own-price, cross-price and income elasticities. We first review the studies based on single equation and Engel relationships, next on those the constructing food demand elasticities matrices, and then summarize the studies employing complete demand systems. This Chapter ends with some concluding remarks on the state of the art in demand modeling. 22 23 3.1 Single Equation Food Demand Studies In the 1950's and 1960's, considerable work was published on the demands for particular food products (Buse,1986). The objectives of a single equation approach to model food demand are usually to obtain consumer demand parameters for the food commodities (or groups) of interest, or to search for appropriate explanatory variables to interpret the differences of food consumption patterns, or to evaluate the performance of alternative functional forms and various statistical methods for estimating demand parameters. The most popular functional forms used in these food demand studies are linear, semi-logarithmic, double-logarithmic and Box-Cox general form. The explanatory variables are usually its own price, the prices of selected other foods, and disposable income. In addition, socioeconomic and demographic variables are sometimes incorporated into a demand function and justified as proxy variables for taste factors. Based on the alternative specifications of error structure, three methods were often used to estimate the demand parameters: (a) ordinary least squares or generalized least squares; (b) limited dependent variable estimation method such as the Tobit model and (c) maximum likelihood estimation. Table 1 presents a survey of recent studies of food demand in the U.S. using a single equation approach. 24 Table 1. Survey of Food Demand Studies in the U.S. Using Single Equation Method Study Data Commodity Functional Estimation Forms Method Chang 1935-1974 meat Box-Cox NMLE" (1977) (excluding 1942-1947) Blaylock and 1960 I - beef Box-Cox NMLE Smallwood 1979 IV (1983) Quarterly Wohlgenant 1947-1983 beef,pork, Fourier OLS (1985) (annual) poultry, fish Raunikar and 1977-1978 shell eggs Linear Tobit Huang Nationwide (1986) Food Consumption Survey Brown and 1978 I - six citrus Double-Log OLS Lee 1984 III juices Mixed (1986) (39 bimonthly Estimation data) Lee, Brown 1981 - 1982 orange juice Switching Two-Stage and Schwartz Panel data (two brands) Regression Heckman (1986) Estimation a/ NMLE represents Nonlinear Maximum Likelihood Estimation. 25 3.2 Engel Relationships Studies Demand analyses using cross-sectional data or other data without price variation information usually focus on Engel relationships or the impacts of specific socioeconomic and demographic factors on food consumption. This is because the cross-sectional data such as those obtained from a household food consumption survey has aboundant information about household characteristics, socioeconomic and demographic variables. Much of the incentive for doing this kind of study came from policy makers with responsibilities to formulate and establish effective welfare programs, and from food marketing administrators in charge of to identifying the target market and developing marketing strategy more efficiently. For instance, the estimation of the equivalent scales for welfare analysis has received much attention over the years. Recently, the proportion spent on food away from home has increased dramatically. This trend is partly attributed to demographic changes such as composition and size of the household, labor force participation, age structure of population and life styles. Therefore, many economists working in this area have attempted to analyze expenditures on food away from home. More detailed information on this category of previous study is summarized in Table 2. Table 2. Survey of Studies on Food Consumption in the U.S. Using Engel Relationship, Socioeconomic and Demographic Factors Study Data Commodify Functional Estimation Form Method Salathe & 1955 & 1965 Total food & Linear Nonlinear Buse HFCS* 5 food Least (1979) categories Squares Chavas 1972-73 t 17 food Linear OLS (1979) CEDS(BLS) categories Redman 1972-73 & FAFHC Linear OLS (1980) 1973-74 CEDS(BLS) Blaylock & 1965 Beef, pork, Box-Cox Nonlinear Green HFCS poultry, fish, Maximum (1980) and eggs Likelihood Price, 1972-73 65 types of Linear OLS Price & Washington fruits and West State vegetables (1980) Huang & Griffin Fresh beef, Spline Raunikar Consumer ground beef, Function OLS (1981) Panel beef roast, steak Smallwood & 1977-78 FAFH Linear OLS Blaylock NFCS(USDA) (1981) Tyrel1 & 1972-73 8 aggregate Linear OLS Mount CEIS(BLS)® commodities Logit (1982) Including food at home and FAFH Kinsey 1978 Panel FAFH Linear Tobit (1983) Study of Income Dynamic Haines 1977-78 FAFH Linear Tobit (1983) NFCS(USDA) Nyankori 1980 Total food & Linear OLS (1986) CEIS(BLS) 14 selected Spline food categories 27 Table 2 (continued) Capps & 1977-78 8 food Linear Tobit Pearson NFCS(USDA) categories (1986) Lee & 1977-78 FAFH Linear Switching Brown NFCS(USDA) Regression (1986) Price 1977-78 Total food & Double-Log Tobit (1986) NFCS(USDA) 28 food categories Chern & 1980-81 8 aggregate Box-Cox Maximum Ferrer CEIS(BLS) commodities Likelihood (1987) including food at home and FAFH McCracken 1977-78 FAFH Linear Tobit & Brandt NFCS(USDA) (1987) Kolodinsky 1980 FAFH Linear Tobit (1987) CEIS(BLS) Yang 1985-86 FAFH Linear Tobit (1988) CSFir OLS Basiotis & 1985-86 FAFH Linear Heckman Yang CSFII two-step (1989) a/ HFCS - Household Food Consumption Survey; b/ CEDS - Consumer Expenditure Diary Survey; c/ FAFH - Food Away From Home; d/ NFCS - Nationwide Food Consumption Survey; e/ CEIS - Consumer Expenditure Interview Survey; f/ CSFII- Continuing Survey of Food Intakes by Individuals. 28 3.3 Food Demand Elasticity Matrix Studies Three noteworthy works on constructing a complete disaggregate food demand matrix in the literature are due to Brandow (1961), George and King (1971), and Huang (1985). Brandow employed a synthetic approach to construct a complete structure of demand relationships for 24 foods and one nonfood commodity. The estimated parameters of demand matrix were constrained to be consistent with the utility maximization. In order to obtain a consistent demand matrix, Brandow adopted the assumption of want-independent (Frisch, 1959) and the previously estimated coefficient of money flexibility of -0.86, and used the direct-price elasticities, income elasticities, and expenditure share from a number of prior studies to derive the parameters of his food demand matrix. George and king used the similar approach to construct a demand matrix for 49 food commodities and one nonfood commodity. However, a main improvement in their study over the Brandow's is that the demand coefficients of commodities are obtained by means of using an uniform estimation procedure on the same sample observations rather than from other prior studies based on different data sources, various time periods, and alternative estimation methods. The data period used by George and King covered only the postwar period, which had relatively stable consumption pattern. The major limitations of both studies using a synthetic approach to construct a food demand matrix are related to the estimation procedure, derivation of cross-price elasticities, the problem of statistical reliability, and want-independent assumption. Recently, 29 Huang employed a novel approach to construct a disaggregate food demand matrix. In order to remedy the deficiencies In estimation procedure and testing of statistical reliability of estimates in the demand matrix form using this synthetic approach, Huang used a constrained maximum likelihood method to Incorporate the parametric restrictions derived from classical demand theory. He then applied these procedures to estimate a U.S. food demand system including 40 food items and one nonfood item, using annual observations for 1953-83. In addition, two more studies in this category appeared in the literature. One was due to Price and Mittelhammer (1979). They used prior information and annual observations for 1949-73, and then applied a mixed estimation technique to derive a demand matrix at farm level for 14 fresh fruits. Another study was done by Huang and Haidacher (1983). They used a constrained maximum likelihood method and parametric restrictions to derive a complete demand matrix for a composite food demand system covering 12 food categories and one nonfood sector using annual data for 1950-1981. These five studies using a complete demand matrix approach are summarized in the Table 3. 30 Table 3. Survey of Studies Using A Food Demand Matrix in the United States Study Data Commodity Method Brandow 1923 - 1958 24 food categories Synthetic (1961) annual data and one nonfood approach George & 1955 (6060 49 food categories Synthetic King Households)> and one nonfood approach (1971) 1965 (15101 Households) food consumption survey, and 1946-1968 Price & 1949 - 1973 14 fresh fruits Mixed Two- Mittlehammer annual data Stage Least (1979) Squares Huang & 1950 - 1981 12 food composite Constrained Haidacher annual data groups and one Maximum (1983) nonfood Likelihood Huang 1953 - 1983 40 food items and Constrained (1985) annual data one nonfood Maximum Likelihood 31 3.4 Complete Demand Systems Approach The last category of food demand studies used complete demand systems incorporating food (or separately food at home and food away from home) as one of the expenditure items or dealing with a subgroup of food items. The complete demand system approach has been widely employed in food demand analysis because it is based on a conceptual framework derived from the theory to deal with the interdependencies of demand for various commodities. Moreover, it also allows us to examine empirically and to have the flexibility to impose the theoretical constraints of utility maximization. These features can not be rigorously implemented in a single equation and Engel relations. The system approach also permits us to incorporate relevant socioeconomic and demographic factors into model specification which can not be accompl1 shed in the above mentioned demand matrix approach. However, the complete demand system is not without problems such as the difficulty in nonlinear estimation. Table 4 presents a survey of recent food demand studies using a complete system approach. Important findings from this review of previous models can be summarized as follows: (a) Linear expenditure system, linear approximate almost ideal demand system, and translog demand system are the three most often used models in estimating food demand; (b) The data used is these studies mainly come from three sources which are annual time series, consumer expenditure survey, and nationwide food consumption survey; (c) Food at home, food away from home and meats are three main categories analyzed in the literature due to their relative importance in food expenditure; (d) Estimation methods employed are either maximum likelihood estimation or iterative seemingly unrelated nonlinear regression method. Both estimation methods have been showed to yield identical results under given conditions; (e) Except those using the additive utility function, the previous studies analyzed usually had a small system for less than 5 commodities or groups of commodities, apparently due to the complexity of computation. In addition, some studies also incorporated habit and demographic factors into model specification. For instance, Manser (1976) and Menkhaus et al. (1985) used an extended translog function with habit formation to study meat consumption. Eales and Unnevehr (1988) and Moschini and Meilke (1989) adopted the first-difference LA/AIDS form to capture the dynamic effects on meat demand. Capps and Havllcek (1984) employed the translating techniques to include household size and the degree of urbanization in their S,-Branch system. Kokoski (1986) and Chern and Lee (1989) also used the translating method to incorporate family size into their quadratic expenditure system to analyze the effects of demographic factors on consumption. 33 Table 4. Survey of Studies on Food Demand in U.S. Using Complete Demand Systems Study Data Commodity Model Estimation Method Manser 1948 - 1972 Meats, fruits and Translog ITSUR (1976) annual data vegetables, cereal (TL) and bakery, and miscellaneous Christensen 1947 - 1971 Fish, beef, Translog ITSUR & Manser annual data poultry, and (1977) seafood Blackorby, 1946 - 1968 Meats(4 types) Generalized FILM Boyce & annual data vegetables( 6 S-branch Russell types), and (1978) fruits (6 types) Eastwood 1955 - 1978 Food at home, food ExtendED MLE & Craven annual data away from home, and LES (1981) 10 other categories Heien 1947 - 1979 14 food categories, Almost NTSLS* (1982) annual data service and Complete nondurable good Model Lanm 1960 I - Food at home, food Translog ITSUR (1982) 1980 III away from home, arid (quarterly other nondurable data) goods B1 an d fort 1 1948 - 1978 11 aggregated AIDS, LES FIML & Green annual data commodities (1983) including food Heien 1967 I - 14 aggregate Almost NTSLS (1983) 1979 III conmoditles Complete (quarterly including 5 Model data) food items Uohlgenant 1946 - 1968 Food and nonfood Fourier Nonlinear (1984) annual data TL, GL LS Capps & 1972 - 1974 Ground beef,steaks, SI-Branch FILM Havlicek CEDS(BLS) roasts, poultry, (1984) pork,variety meats, seafood 34 Table 4 (continued) Menkhaus, 1965 - 1981 Beef* pork and Translog ITSUR Clair and annual data chicken Hallingbye (1985) Blanciforti, 1948 - 1978 Meats* fruits and AIDS, LES FILM Green and annual data vegetables* cereal King and bakery* and (1986) miscellaneous Kokoski 1972-73 & Cereals, meats* QES FILM (1986) 1980-81 dairy, fruits and CEDS(BLS) vegetables* others Huang & 1977-78 8 food categories LES ML Raunikar NFCS(USDA) (1987) Craren & 1955 - 1978 11 aggregate LES-Leser Leser Haidacher annual data commodities LES-Powell Powell (1987) including food LES-Stone FMLE at home* food away from home Dahlgram 1950 - 1985 Beef* pork* Rotterdom Stepwide (1988) annual data chicken, other ML food and nonfood Eales & 1965 - 1985 Chicken, beef, LA/AIDS ITSUR Unnevehr annual data pork, non-meat (1988) food, non-food Heien & 1977 Steak, roast, AIDS ITSUR Pompel1 i Household ground beef (1988) Consumption Survey Chern I 1980 - 1986 8 aggregate QES* LES ITSUR Lee CEIS(BLS) commodities (1989) (mean including food expenditure at home and food data) away from home. Moschini 1967 I - Beef, pork, chicken LA/AIDS ITSUR & Meike 1987 IV fish (1989) (quarterly data) a/ NTSLS : nonlinear three-stage least squares. 35 3.5 Concluding Remarks The following two general remarks on this review of literature are appropriate: (1) Two important empirical issues, namely data consistency and aggregation problems, were given little attention in most of above mentioned food demand studies. Almost all previous food demand studies assumed that the data used (either time-series or cross section) are consistent with utility maximization. As discussed earlier, in the case where data violate the rational behavior due to measurement error or preference structure changes then the estimates of parameter using these inconsistent data sets are subject to specification errors. Aggregation problems in an empirical study include the aggregation across commodities related to separability condition, and the aggregation across consumers related to a "representative consumer". These aggregation problems were also often ignored or arbitrarily treated as though it is an appropriate assumption. These assumptions may not be appropriate without further testing. In addition, the use of annual time series data in food demand can not ref1ect the seasonal effects on consumption. Consequently, the cross-price elasticities may not be correctly estimated, especially for those food items having a strong seasonality as affected by food supply conditions. (2) Every approach to modeling food demand has its own merits. For instance, from theoretical and practical points of view, the single equation and Engel relation approaches have advantages in several practical aspects such as they can be designed to answer a particular policy question unique to a particular food commodity or food group. However, the demand matrix and complete demand system approaches have good properties from theoretical perspective and they can properly and effectively capture the Interrelatlonshlps among various food commodities. Therefore, the choice of which approach to use for an empirical food demand analysis may depend upon the objective of the study, the content of data, and the cost of estimation. CHAPTER IV DATA SOURCE AND DESCRIPTION This chapter deals chiefly with the data used in this study. The data sources and data base construction are reported first. Then the basic structure and pattern of food consumption in the U.S. during the sample period are presented and discussed. 4.1 Data Sources Three major data components of any demand analysis are expenditure (income or quantity), prices, and socioeconomic (and/or demographic) variables. The characteristics of available data series for these three components and how well they match up are important for accuracy of parameter estimates. The following three best known series of data on food expenditures are available for food demand studies in the U.S.. The first series is food expenditures by items published by the Economic Research Services (ERS) of U.S. Department of Agriculture. This series provides annual data reaching back to 1889. The second series is personal consumption expenditures, available from Bureau of Economic Analysis (BEA) of the U.S. Department of Coiranerce. Data from 1929 to a recent year are available. The third series is the Continuing Consumer Expenditure Survey (CES), conducted by the Bureau of Labor Statistics (BLS) of the U.S. Department of Labor. There are many distinctive 3 7 38 characteristics among these three data series in terms of the definition of expenditure and methodology of data collection. A detail discussion about the differences between these series is provided by Manchester (1987, 1990). The CES provides the most consistent set of data for the three components needed in a demand analysis. Thus this data set is selected for this study. These data are all provided by the BLS. Specifically, expenditure and demographic data are obtained directly from the BLS's Consumer Expenditure Surveys (CES) while the price data are the consumer price indexes (CPI) published by the BLS. The CPI has been computed using the expenditure weights obtained from the CES. Therefore, there are direct and consistent matches between the expenditure data and their corresponding CPI component for the categories of food used in this study. The sample periods in the study are from 1980 to 1986. The CES consists of two separate components which have been conducted continuously by the BLS. The first is the Diary or recordkeeping survey completed by each sample consumer unit (CU) for two consecutive one week period8. The second 1s the quarterly Interview panel survey in which each of sample consumer units reports information to an interviewer every three months for five consecutive quarters. Each survey (diary or interview) has its own questionnaire and sample. The only relationship between the two is that the samples are all chosen from the same frame at the same time. In this study, we use data (both expenditures of food categories and demographic characteristics of consumer units) obtained from the CES diary surveys. The diary survey is specifically designed for collecting 39 data about small, frequently purchased items, such as food purchases which are normally difficult to recal1 after a long period of time. In the diary survey, expenditures refer to transaction cost, including excise and sales taxes, for good and services acquired during the survey reference period. The full cost of each purchase is recorded, even though full payment may not have been made at the time of purchase. Note that expenditures incurred by members of the consumer unit while away from home overnight or longer are excluded from diary. Therefore, the food away from home expenditure does not include those items. In addition to expenditures, the diary survey also collects information about income sources such as food stamp and social welfare income, and demographic and household characteristics of members in each consumer unit. Data in the diary survey are collected from a nationwide probability sample of households designed to be representative of the national, noninstitutional population9. Besides the population residing in regular housing, persons residing in selected group quarters, such as college dormitories, are also represented. Weights are assigned to each consumer unit in the survey in order to provide estimates for the U.S. population10. Table 5 presents the sample sizes in the public-use tapes of the CES diary surveys for the years 1980 to 1986. The total number of observations in the sample periods exceeds seventy eight thousand. It is clearly difficult, if not impossible, to directly use these many observations for empirical estimation. Since there is a subsample for each month, we have ample flexibilities to construct a desirable data 40 base. Based on the original household data, we construct aggregate monthly time-series data to analyze U.S. consumer behavior for food commodities. The weighted average monthly time-series data are computed by using weighting factors, consumer unit weight 2111. Two aggregate levels of expenditures data are available in the BLS public-use tapes. They are the expenditure class level and the Universal Classification Code (UCC) level12. There are more than one hundred food expenditure items at the UCC level collected by BLS. Of course, it would be very difficult to manage these many expenditure components in an empirical study. Therefore, the expenditure class level is used as the basis for food grouping in this study. Specifically, the expenditures of food categories at the UCC level are aggregated into 18 expenditure components for food at home and one expenditure component for food away from home at the expenditure class level. These 19 food expenditure categories are: (1) cereal and cereal products, (2) bakery products, (3) beef, (4) pork, (5) other meats, (6) poultry, (7) fish and seafood, (8) eggs, (9) fresh milk and cream, (10) other dairy products, (11) fresh fruits, (12) fresh vegetables, (13) processed fruits, (14) processed vegetables, (15) sugar and other sweets, (16) nonalcoholic beverages, (17) fats and oils, (18) miscellaneous foods, and (19) food away from home. A more comprehensive description of the components included in each category is presented in Appendix B. In order to have more information on price variations than previous food demand studies, this study uses monthly data as the observation basis. Prices of food categories are obtained from the CPI Detailed Report which are published monthly by the BLS. The prices used in this study is seasonally unadjusted series of the monthly CPI-U, which is computed from U.S. city average, for all urban consumers. However, the use of monthly data may raise some problems in dealing with seasonal fluctuations found in the next section of data description. Note that if the seasonal fluctuations in the Independent variables (prices, Income and other socio-economic variables), fully account for the seasonal fluctuations in the corresponding dependent variable (expenditures), then no problem would exist for using monthly data. Gersonvitz and Mackinnon (1978) indicate that seasonal fluctuations may increase the precision of coefficient estimates by imparting additional variation to the independent variables. It 1s often the case, however, that when seasonally varying dependent variables are regressed on seasonally varying independent variables, the resulting residuals have a seasonal pattern. Facing with this situation, applied econometricians typically do one of the two things. Me either incorporate dumny variables (monthly in our case) to capture the effects of seasonality or employ seasonally adjusted data. In the premilimary examination of model specification, we attempted to employ the monthly dumny variables and the harmonic method reference to capture the seasonal effects on food consumption. However, the regression results show that the dumny variable approach overshoots and distorts the effects of prices and expenditure on consumption for some food groups while the harmonic method could not detect all major seasonal effects for some specific food commodities. In either of these approaches, we have to give up a lot of degrees of freedom for incorporating these seasonality variables in the complete demand systems. Consequently, we decided to use the seasonally adjusted data for parametric estimation of demand parameters in this study. The X-ll Table 5. Sample Size by Year and Total YEAR SAMPLE SIZE 1980 10,423 1981 10,547 1982 10,925 1983 10,791 1984 11,873 1985 11,618 1986 12,815 TOTAL 78,992 43 procedure (available in the computer software, SAS/ETS) which Is an adaptation of the Bureau of Census X-ll seasonal adjustment program, 1s used to seasonally adjust our original monthly unadjusted data series of expenditures and prices13. 4.2 Data Description Descriptive statistics are effective means of describing the general trends of the key variables used in an econometric analysis. Since the data used in the study are time-series, It 1s essential to understand these historical trends of the data series during the sample period. Table 6 shows that weighted average weekly expenditures for each of the 19 categories of food commodities for urban consumer units for 1980 and 1986. Table 6 reveals several interesting changes on food consumption pattern and trend during this period. These changing patterns are summarized as follows: (1) The total food expenditures (in current dollars) including food at home and away from home, have been increasing during the sample periods. The increase of total food expenditures from $48.41 in 1980 to $59.60 in 1986 represents a 23% increase during the sample period. Note that the expenditure of food away from home increased markedly more than the food at home during the same period (Food away from home in the diary survey did not include expenditures while out of town overnight or longer). (2) Even though the expenditure on food at home and away from home have been increasing, the share of these two components have changed 44 significantly. Specifically, the expenditure share of food at home has shown a downward trend, from 68.3% in 1980 to 63.3% in 1986. However, the proportion spent on food away from home has been rising, from 31.2% in 1980 to 36.7% in 1986. This trend can be partly attributed to the changes in demographic variables such as composition and size of the consumer unit, women labor force participation, age structure of population and life styles. (3) The composition of expenditure structure on food at home has also had substantial changes. The shares of total expenditure for two broad commodity groups, meats and eggs, as well as dairy products, have been declining. For instance, the expenditure shares for meat and eggs decreased from 23.8% to 18.2% during the sample period. On the contrary, the shares of other food commodities have been increasing while the cereal and bakery product, and fruits and vegetables still remain to have a stable share of expenditure on food at home. (4) Considering individual commodity expenditures, the decreases of expenditure on beef, pork, other meats, and eggs are clearly notable. Especially for beef, its expenditure decreased from $4.58 in 1980 to $3.63 in 1986. On the other hand, the increases in the expenditures of cereal and bakery products, fish and seafood, fruits and vegetables, as well as miscellaneous foods were substantial. For the miscellaneous foods categories, the average expenditure per consumer unit Increased from $2.85 in 1980 to $4.53 in. 1986. It is interesting to note that the food items with an increase in expenditure appear to be considered as healthful food such as fish and seafood with lower cholesterol, cereals with high fiber, fresh fruits 45 and vegetables with nutritional benefits, as well as convenience goods such as frozen prepared meals and other prepared food Items Included 1n the miscellaneous foods category. However, decreases In expenditure occurred 1n the red meats commodity group. These decreasing trends may signify concerns about the effects of cholesterol, saturated fat and other factors related to health risk associated with the consumption of red meats. These changes in expenditure pattern if truly related to health concern and time value of consumer unit should have Important implications for public policy and food industry. In addition, the last two columns in Table 6 presents the total sample means and shares of each category for the 1980-1986 period. Figures 1-4 illustrate the trends and changes of consumption patterns for some of these food categories over the sample periods. In general, these figures reveal that the expenditures (not seasonally adjusted) of various categories have substantial monthly fluctuations. These seasonal oscillations are particularly visible for fresh vegetables and fresh fruits. These changes in consumption pattern may result from relative prices, income, seasonal factors, shifts in the demographic structure of household, tastes as well as preferences. Table 7 presents the mean statistics for family size and age of household head. The average family size and age of household head during the sample period are 2.57 and 45.71, respectively. Generally speaking, the trend of family size has been gradually decreasing whereas age of household has been increasing. The changes in these two demographic variables will likely affect the food consumption pattern. Finally, summary statistics of two important variables, budget shares and prices of commodities, for estimating structural parameters in the demand system are reported 1n Table 8. The sample mean of budget shares for food away from home Is the 1argest among all food categories, while small budget shares are observed for egg, processed vegetables, fat and oil. The coefficient of variation (CV) Is used to measure variation of the data series for the budget shares and prices of 19 food commodities during the sample period. The values of CV indicate that the variations of budget shares for bakery and dairy products are fairly small, while beef, pork, and eggs have a relatively large variation. The results also Indicate that the variations of the prices of beef, milk, and dairy products are very small. On the contrary, fresh fruits and fresh vegetables have a large variation in prices during sample period. The low variation of budget shares and price may cause unstable estimates of parameters in the parametric analysis. Beef appears to be the most troublesome category to deal with because of the extremely low CV in its prices. 47 Table 6. Weekly Expenditures of Food Commodities for Urban Consumer Units 1980 1986 Sample Average ______(1980-19861 ITEMS Expenditure Share Expenditure Share Expenditure Share (dollars) (%) (dollars) (%) (dollars) (%) Food , Total 47.49 100 61.04 100 55.61 100 Food at home 32.70 68.9 38.60 63.2 36.29 65.3 Cereal and Bakery 4.18 8.8 5.43 8.9 4.88 8.8 cereal 1.33 2.8 1.83 3.0 1.56 2.8 bakery products 2.85 6.0 3.60 5.9 3.32 5.9 Meats and Eggs 11.17 23.5 11.03 18.1 11.30 20.3 beef 4.30 9.1 3.73 6.1 4.06 7.3 pork 2.33 4.9 2.25 3.7 2.30 4.1 other meats 1.48 3.1 1.53 2.5 1.57 2.9 poultry 1.47 3.1 1.65 2.7 1.56 2.8 fish & seafood 0.97 2.0 1.30 2.1 1.16 2.1 eggs 0.62 1.3 0.57 1.0 0.63 1.1 Dairy products 4.40 9.3 4.93 8.1 4.80 8.6 fresh milk 2.29 4.9 2.38 3.9 2.40 4.4 other dairy 2.11 4.4 2.55 4.2 2.40 4.3 Fruits and Veg. 4.94 10.4 6.33 10.4 5.88 10.5 fresh fruits 1.48 3.1 2.03 3.3 1.79 3.2 fresh vegetables 1.42 3.0 1.84 3.0 1.74 3.1 processed fruits 1.14 2.4 1.42 2.3 1.35 2.4 processed veg. 0.90 1.9 1.04 1.7 1.00 1.8 Other food at home 7.99 16.8 10.90 17.9 9.46 17.0 sugar and sweets 1.23 2.6 1.46 2.4 1.34 2.4 fat and oils 0.94 2.0 1.01 1.7 1.00 1.8 misc. foods 2.85 5.9 4.67 7.6 3.84 6.9 nonalcoholic ber. 2.97 6.3 3.76 6.2 3.28 5.9 Food away from home 14.79 31.1 22.44 36.8 19.32 34.7 Data sources: BLS's CES Diary Survey 1980-1986. 48 Table 7. Sample Mean of Family Size and Age of Household Head Year Family Size Age of Household Head 1980 2.60 45.75 1981 2.58 44.61 1982 2.52 45.21 1983 2.55 45.69 1984 2.60 46.65 1985 2.57 46.03 1986 2.60 46.03 Average 2.57 45.71 49 Table 8. Descriptive Statistics of Budget Share and Prices for 19 Food Commodities MEAN® C Vb Commodities U? 9? M, ?i Cereal 2.80 156.95 5.99 8.44 Bakery 5.96 152.22 4.37 11.37 Beef 7.39 272.71 16.89 1.84 Pork 4.16 247.36 11.28 9.22 Other meats 2.84 264.67 9.43 3.48 Poultry 2.81 207.12 7.98 7.90 Seafood 2.08 381.33 10.32 9.02 Eggs 1.15 184.17 14.85 10.84 Milk 4.35 135.58 8.26 3.03 Dairy product 4.32 146.99 4.72 5.73 Fresh fruits 3.21 315.33 6.23 12.32 Fresh veg. 3.12 299.62 6.10 12.29 Processed fru. 2.42 151.22 6.26 9.13 Processed veg. 1.80 138.43 7.63 7.48 Sweets 2.42 378.61 10.48 6.63 Nonalco. bev. 5.90 433.99 5.86 5.92 Fat & oil 1.80 271.50 8.88 6.81 Misc. foods 6.86 273.08 9.97 8.44 FAFH 34.59 317.79 6.63 9.61 a/ Mean is computed by using seasonal adjusted data, b/ CV 1s coefficients of variation, c/ U, is budget share denoted by 10'2 for 1th commodity, d/ P, 1s price of 1th commodity. 50 w £G K tI £ x pE *9 N 9 TI U R I „ 1 JANQ1 JANOS MONTH FOOO AWAY FnCM HCU£ HASH FOOO AT HOME— -TRIANGLE TOTAL FOOO EXPENDITURES— OIUANO Figure 1. Aggregate Food Consumption Patterns by Month 51 e s 4 2 1 j JANQ 1 MONTH g g r p i. .—WASH PORK fRIANGLE POULTRY— 0 \ AitOHO Figure 2. Consumption Patterns of Meats by Month 5 s R o t r I OZn^Hin ins 0.0 Figure 3. Consumption Patterns of Vegetables by Month by Vegetables of Patterns Consumption 3. Figure H PROCESSED VEGETABLE—-TRIANGLE L G N A I R T - — E L B A T E G E V D E S S E C O R P SH A H S E L B A T E G E V H S E R F JAN82 UQNTH 52 53 w £ £ KL Y E X P £ 0H T! U R S£ s JAMS 5 MONTH FRESH FRUITS— HASH PROCESSED FRUITS——-TRIA G E Figure 4. Consumption Patterns of Fruits by Month CHAPTER V METHODOLOGY The neoclassical consumer theory and important issues in empirical demand analysis, data sources and a description of food conumption structure and pattern in the U.S. have been discussed and presented in the previous Chapters. For the purpose of analyzing food consumption and demand in the United States, a methodology is developed in this chapter for implementing a neoclassical demand model. Even though this study focuses only on the allocation of food expenditure among 19 food categories, we recognize that the consumer's utility maximization problem is typically set up for all commodities simultaneously. Consequently, in order to justify the construction of a complete demand system for food, a separability assumption will be required. In this study, we assume that the food commodities can be weakly separable from other nondurable and durable goods and services over the commodity space. Specifically, the utility function can be expressed as U(x, k ) - U(¥(x), x) where x - an n-dimensional vector of quantities of food commodities demanded; x » a k-dimenslonal vector of quantities of other commodities and services demanded; U - utility function for overal1 commodity space; V - subutility function for food commodity space. 5 4 Previous studies have shown that this separability assumption 1s generally valid. In particular, Chern and Lee (1989b) conducted a nonparametric demand analysis to examine the consumer expenditure allocation structure for 11 expenditure categories, using data base from the BLS's CES interview surveys. They found that all expenditure categories are weakly separable. Therefore, it is proper to construct a separate demand system for the components in each expenditure group. They also found that food at home and food away from home may be separated or grouped together in a complete demand system when it includes a reasonable number of categories for other goods and services. Accordingly, the demand system for food to be developed in this study may be considered as the system for the second stage of a multi-stage consumer budgeting process. The research methodology of this dissertation research consists of two main components: (a) a nonparametric approach for Implementing a neoclassical demand model including testing of data consistency, rank testing for data structure, and clustering analysis for grouping food commodities; (b) a parametric approach to model food demand, including testing of model specification and estimation of parameters 1n various complete demand systems. In this section, we first discuss the nonparametric approach, as a methodology developed for narrowing the gap between the theoretical and empirical analysis. The parametric approach based on econometric methods 1s then presented 1n the next section. 56 5.1 Nonparametric Approach Host of previous empirical demand studies have been carried out under various assumptions. For example, most analysts usually assumed their data to be consistent with utility maximization, and consumer's preference structure to follow a specific type. With these typical assumptions, a parametric econometric model was then specified. Also, the errors in the model specification were typically assumed to follow certain parametric distributions, often a normal distribution. One disadvantage of the parametric econometrics based on these assumptions is that it may not be robust if the data are Inconsistent, even if only slightly, with rational behavior and if the preference structure differs from the particular parametric specification. Indeed, an invalid assumption may lead to erroneous conclusions. In view of these problems, we attempt to use a nonparametric approach to examine the data consistency, data structure, and preference structure. It is Important in this, in fact any, empirical study that we place more emphasis on the information contained in the data themselves rather than making assumption about the structure of data generating process under investigation. If we know more about the structure and characteristics of the data we use, we are less likely to suffer from the specification error, and thus the results of estimation will be more reliable. 57 5.1.1 Testing of data consistency As discussed In Chapter II, one can deal with the aggregation across agents problem through model specification. However, these aggregation conditions are appropriate for evaluating alternative models but not the data. It is important that we examine the consistency of data before undertaking the parametric analysis in estimating the structural parameters. This 1s particularly important for the aggregate data used in this study because of the uncertainties about the possible distortion from aggregation across households in the survey. The objective of conducting tests for data consistency is to examine whether or not the data used in the study satisfies the utility maximization hypothesis. One appropriate approach is to use nonparametric techniques in demand analysis. The spirit of nonparametric tests stem from the revealed preference theory and these tests can be used to examine whether or not the data (observed quantities and prices) are consistent with the utility maximization hypothesis and/or other restrictions on preference structure such as separability. The earlier works on the nonparametric demand theory are due to Afriat (1967, 1972, 1973, 1981), Diewert (1973) and Diewert and Parken (1978). This nonparametric approach has recently been popularized by Varian (1982, 1983, and 1985). Recent applications include the studies by Chalfant and A1ston (1988), Swofford and Whitney (1986, 1988), and Chern and Lee (1989). We will conduct a nonparametric analysis primarily for testing the consistency of data used In this research. Varian (1983) adopted the theorem developed by Afriat (1967) and 58 derived the conditions called Generalized Axioms of Revealed Preference (GARP). Given a finite number of observations on p’-(p{,..., p’) denoting the 1-th observation on the prices of g groups of commodities, and x'-txj,..., xj) being the corresponding observations on the quantities, Varian provides the following two definitions: Definition 1: A ytl.lity function u(x) rationalizes a set of observations (p'^x’J, 1-1,2,...,n, if u(x ) k u(x) for all x such that p x *p x. Definition 2: The data sa tjsfie.s GARP if xf 1s revealed preferred to xj, written as x' R xJ, then p V * pjxJ. Furthermore, Varian shows that the Afriat's Theorem can be stated in the following four equivalent conditions: (1) There exists a nonsatiated utility function that rationalizes the data. (2) The data satisfy "cyclical consistency," that is, prxr > KK' P8x >P8x \ . ..., p V > p V implies p x - p x® , p8x ■ p®xl, ..., pqxq - pqxr. (3) There exist numbers U{, X, > 0, i-1,2,3,.... n,such that Uj i Uj + AjPj (Xj - Xj) (1,j=l,2,3..• .n)• (4) There exists a nonsatiated, continuous, concave, monotonic utility function that rationalizes the data. Varian also shows that cyclical consistency is equivalent to GARP. In practice, the test of GARP is easier to conduct than the Afriat's cyclical consistency test and therefore 1t is used in this study. If our data set satisfies this test, i.e, GARP, there exists a well-behaved utility function which rationalizes the data used in the study. That is to say, the data were generated from the utility maximization. These tests can be applied to the whole group or subgroups of food commodities under study. The necessary conditions for weak separability require both the entire group and subgroups satisfy the GARP. 59 5.1.2 Testing of rank 1n the demand system The rank test of a demand system is a relatively new area in demand analysis. The definition of the rank of a demand system is originally introduced by Gorman (1981), and then extended and applied by Lewbel (1988, 1989). Theoretically speaking, the rank of a matrix is closely related to the dimensionality of a system of the variables involved in that matrix. The rank test of a demand system is a useful tool as a prespecification test in empirical demand analysis. It provides valuable information about the degree of separability, aggregation structure, and the structure of cost function that are consistent with a given data set. However, very little empirical work on rank exists. Lewbel indicated the following reasons: (a) the amount of useful information embodied in the rank has not been recognized; (b) it was not known that the concept of rank could be applied to all demand systems; and (c) no advanced testing method for rank was known, except in a very limited approach of using the nested parameter forms. To our r knowledge, only Hausman et al. (1988) and Lewbel (1988,1989) conducted research on the rank test of demand systems. This section deals with the theory and testing of rank condition in demand analysis. Let Z»C(U,V) be the log of cost function describing any demand system for N goods, where U 1s a utility level, V is a (Nxl) vector of the logged prices of goods, and Z-log(M) with M being total expenditure. Assume that the cost function satisfies the regularity conditions, that is, continuity and differetlability. By Shephard's lemma, we can obtain the budget shares which can be written as 60 w = = d(z, v) <5) let R(V) denote the space spanned by d(Z,V) for a given V over all real Z. Define the rank of the demand system local to V to be r[R(V)], and the global rank K of the demand system by K - sup{ r[R(V)J | V real } (6) By the definition of the rank, budget share, therefore, can be expressed as a linear combination of a basis of R(V). More specifically, W = A(V) *g(z,V) (7) where A(V) is an N by K matrix with r(A) - K for some V, and the K- dimensional vector g(Z,V) 1s a basis for R(V). Note that the space spanned by g includes the constant function because the budget shares sum to one. Under a given price regime ¥*, Eq. (7) can be thought of as a budget share form of Engel curves and be written as W = A*G{Z) (8) So Engel curves are linear in K functions of expenditure, though the precise form of the expenditure functions G typically depends on the prevailing price regime V*. Note that r(A)» r[R(V)]. Therefore, we can estimate the r(A) to obtain the rank of the demand system. Lewbel shows that a demand system has a rank K if and only if K is the smallest integer such that the cost function is of the form C(U,V) * if(C7/6 1 (V) ,0 2 (v) .... ,Qk(v)) (9 ) 61 for some functions 6,, 02, ... , 6k and H. That is, a demand system with rank K has a cost function which can be expressed in terms of K price indices, 01,02,...,0k. Due to the fact that cost functions are linearly homogeneous in prices, each function 0k(V) equals the log of a homogeneous of degree one function of prices, and so can be considered as the expenditure on some composite commodities produced under constant returns. The function H therefore can be interpreted as the 1ogged cost function for the composite commodities. For estimating the rank of a demand system, Eq. (8) with an error term added is expressed as Wt = A*G(zt) +et (10) where et is an error vector with E(et) <= 0, and E(et,Zt) = 0. No assumption of distribution about the error terms will be made except that e'l = 0 due to the adding-up condition, where 1 is an n vector of ones. The specific functional form G(Z) is unknown. However, the structure of Eq. (10) resembles that of a factor analysis, where G is a set of unknown factors that explain W, and the rank K equals the unknown number of factors. Unfortunately, the factor analysis requires restrictions on E(ee') be diagonal. It is unreasonable to impose these restrictions on Eq. (10) because e'l = 0 requires the existence of nonzero covariances. Therefore, a test for the number of factors in factor analysis can not be used to identify K. Two alternative approaches can be employed to estimate the rank of a demand system. One is the parametric approach which entails specifying and estimating G(Zt) using various functional forms. Hausman et al., 62 (1988) employed various polynomial Engel curves to estimate rank and found the evidence of a rank of three for the CES data (BLS's interview survey data, 1982). Another approach is a nonparametric method which was developed and applied by Lewbel and Gill (1988). To nonparametrically estimate K, Lewbel and Gill suggest using the information of factors G(Z) which are all a function of Z. Let Q(Z) be a vector of N or more functions having a finite means. Post multiplying Eq. (10) by Q(Zt)' gives WtQ(Zt)' = A*G(Zt)Q(Zt)'+etQ(Zt)/ (11) Now, let Y=E[WQ(Z)'], which is assumed to exist. Since E[eQ(Z)]=0, we have, Y = A*E[G{Zt)Q{Zt)'] (12) Hence, r(Y)=K, unless by coincidence some components of G is orthogonal to all the elements of Q, or the price regime locally has a rank less than K. In either case, the result would yield r(Y) V [yfTif - Y)] ~ JV(0,E) (13) for some covariance matrix E, where v is a vector. Note that if G was known then Eq.(11) would yield an ordinary instrumental variable estimate of A. G and A are unknown and not estimated, but it is only necessary to estimate r(Y) to get an estimate of r(A). In order to estimate r(Y) given 9, the Gaussian elimination method, which is usually called LDU decomposition, can be used. The key idea of this approach is the rank of any matrix equals the number of nonzero elements of D in LDU decomposition. Following Gill and Lewbel(1988), there exists a unique decomposition of the form PYQ =LDU, where P and Q are premutation matrices, L and U' are low triangular matrices, D is a diagonal matrix, and the elements of D decrease in magnitude along the diagonal. The corresponding decomposition for Y is PYQ=LDU. Let 4 i 0 0 4 o 0 UX1 t?12 £36 = 4i 4 o o 4 , o o tf22 (14) £ 3 1 £ 3 2 l* _ p 0 0 0, 0 0 , where the row partition is by r, p-r, and k-p rows, and the column partition of L and D is by r, p-r, k-p column, and for 0 is r and p-r columns. Let <)2 = Diag(D2), and let A be the (p-r)2 by (p-r) matrix such that vec(D2) = Ac)2. Define the following matrices H - [-L22L21L11 I £21 I 0] (15) (16) 0 u2l 0 (17) 64 Under the null hypothesis, H0 : r(Y) - r, that is, d2 = 0, we can construct the Chi-Square test statistic T<%a - H - *j_r (18) Since d2 = 0 is equivalent to r(Y) s r, this test is consistent against the alternative hypothesis, H,: r(Y) > r. In empirical application the above statistic can be calculated for each possible rank. 5.1.3 Factor and clustering analysis for commodity grouping As discussed above, from a practical point of view, there are limitations in using separability assumptions or the composite commodity theorem to perform commodity grouping in empirical studies. Most of previous studies analyzing demand for a subset of commodities were usually based on an assumption of a utility tree, a concept pioneered by Sono (1945) and Leontief (1947), and named as such by Strotz (1957, 1969) and elaborated by Gorman (1959), Pearce (1961, 1964), Goldman and Uzawa (1964) and Blackorby et al. (1970). A detailed discussion of this utility tree approach is provided by Barten (1975). The basic idea of a utility tree is that one can classify goods into some fami 1iar groups by examining the relationships among goods in terms of substitutability, complementarity and independence. Under the condition of utility maximization, the second partial derivatives of a utility function, which are directly related to the Hessian matrix of the utility function and therefore reflect information about the preference ordering, can be used to determine two goods as complements, if Ufj>0, where Ujj=a2U/axiaxj; as substitutes if Ufj<0, or independent if UfJ = 0. Applying this concept of a utility tree, Theil (1967) developed a theorem which states that under certain conditions the covariance between the variation in the demand for two goods, holding constant prices and income, can be used to measure their Slutsky-Hicks substitution effects up to a negative scale. Barten (1968), O'Brian (1974) and Philip (1972) used the Theil's framework to analyze the consumer preference structure and substitutability among commodities. Therefore, the residual variation around a demand system (or more specifically an Engel curve) can be applied to group goods into familiar commodity groups. The main disadvantage of using the utility tree approach is that a utility function must be specified, and then separability is assumed or implied by investigators. For this reason, we attempt to look for a direct method to group commodities by means of their correlations among expenditures of various commodities under investigation. In order to achieve this goal, we employ a nonparametric procedure to tackle the problem of commodity grouping. The new demand theory is adopted as the theoretical foundation for analyzing the structure of consumer preference14. Two multivariate statistical techniques are then used for grouping the entire commodity set into "similar" commodity categories. The new demand theory refers to the household production function developed by Becker (1965) and the characteristics model developed by Lancaster (1971). These new approaches are potentially more fruitful than the traditional theory for commodity grouping (Michael and Becker, 1973, and Young, 1977). Based on the Lancaster's model, we can restate the consumer choice problem as follows: 66 Max U(Z) Z S. T. zg = Z (19) p'x = M where Z » a g-dimerssional vector of basic characteristics of interest to consumers are interested; Zg= production technology function of consumer units. Under this framework, the relations for commodity groupings can be represented from the production relations in Eq. (19). Following Lancaster (1971) and Becker (1965), the explicit function for production technology is assumed to have a linear form. The consumer model can thus be reformulated as: Max U(Z) z (20) ST. X = AZ p'x = M where A = an n x g dimensional matrix of structural coefficients. Obviously, the vector of Z is unobserved. We adopt the following econometric procedure which is usually used for treating the problem of unobserved variables15. Following Griliches (1974) and Young (1977), we rewrite Eq. (20) as X = AZ + e (21) where e is an n x 1 vector representing a residual error term. We also assume that E(Ze)=0 and E(ee')=¥. Equation (21) is of a form for factor analysis16. Adopting the terminology in factor analysis, Z denotes a set of common factors, e represents specific factors, and A is a matrix of factor loading. The 67 factor analysis is aimed to describing the covariance relationships among many variables in terms of a few underlying, but unobservable, random variables called factors. As such, it is assumed that variables such as x,, x2,...., xH can be grouped by their correlations. That is, all variables within a particular group are highly correlated among themselves but have relatively small correlations with variables in a different group (Johnson and Wichern, 1988). Therefore, it is an appropriate method to explore the relationships between market commodities and their characteristics. In this study, we will use the configuration of factor loading to determine the structure of demand with which commodities can be aggregated. Based on Eq. (21) and the distributional assumptions for e, the covariance matrix of X, denoted by £, can be expressed as: 2=E(XX') =A®A'+W (22) where $ denotes the covariance matrix of common factors. In the sense of Eq.(22), the factor analysis attempts to explain or generate the covariance of X by using the hypothetical common and specific factors according to the factor analysis model in Eq. (21). Under this framework, the model essentially represents n random variables, x.,, x2, ,xn, in terms of g+n hypothetical common and specific factors. Furthermore, if the § = I which means the common factors are q random variables with covariance 0 and variance 1, then the common factors are described as standardized and Eq. (22) reduces to £ = AA'+V (23) 68 Accordingly, the variance of Xf can thus be written as VaziXj) = EiXjX'j) = X^+Xj2+, +X&T1 Xlj+Wl (24) where X2fj is the squared loading which measures the contribution of standardized common factor Zj to the variance of X{. XX2fj- is the sumof the squares of the loading of X 5 on the g common factors, which is usually called communality of X{. Therefore, the variance of X{ is composed of two components, communality and specific variance. For estimating the parameters in Eq. (23), there are two commonly known methods, the principal component method and the maximum likelihood method. In this study, the principal component method is employed. Theoretically, by the spectral decomposition, we know a square matrix can be expressed in terms of its eigenvalues and eigenvectors. To apply the principal component method, Eq.(23) is rewritten as 0 0 0 0 T ! = t ^ i eil^2©2 I • ■ • • (25) a >f^qeg_ where (X^e,) are eigenvalue-eigenvector pairs of £, and X.,>A2> ...>Xq>l are the first q eigenvalues. The first principal factor accounts for the largest percentage of the common variance, with its associated eigenvalue being equal to the sum of the squared loadings. The representation in Eq. (25), when applied to the empirical factor analysis, is known as the principal component solution. 69 Young (1977) pointed out that factor analysis is only appropriate for the case that each commodity can be clearly identified by a uni factor in the structure of common factors in commodity grouping. Difficulties in defining separable groups will arise when the variables are linked to several factors. For this reason, the clustering analysis will be further employed to overcome the problem of multiple characteristics in commodities grouping in this study. The basic objective of a cluster analysis is to discover natural grouping of the items through maximizing within group similarities and between group differences. Generally speaking, alternative clustering techniques can be classified into two approaches, hierarchical clustering and non-hierarchical clustering17. The method proposed in this study is a hierarchical clustering technique which provides a nested sequences of clusters. The procedure attempts to divide a set of variables into nonoverlapping clusters in such a way that each cluster can be interpreted as essentially unidimensional. Using this procedure, a large set of variables can be replaced by the set of cluster components with little loss of information, and the result can provide a uni factor structure to the factor matrix. In fact, this procedure is a type of oblique component analysis related to multiple group factor analysis (Harman, 1976). More specifically, the method of cluster formation used in this study includes the following steps: (1) The analysis begins with all expenditure categories in a single cluster. 70 (2) A cluster is chosen for splitting, which has smallest percentage of variations explained by its cluster component. (3) The chosen cluster is split into two clusters by finding the first two components and assigning each variable to the component with which it has a higher square correlation. (4) Expenditure categories are iteratively reassigned18, which is required to maintain a hierarchical structure, to clusters to maximize the variance accounted for by the cluster components. The steps (2)-(4) are repeated until the cluster satisfies a prespecified criteria such as the percentage of variation accounted for. The structure is thus described in terms of cluster-defined dimensions, the cluster being composites of scores on the definers and in general correlated with each other. In economic sense, categories with similar characteristics, AZ in Eq. (20), are likely to go together to form a particular cluster. Thus categories within a cluster are likely to be either stronger substitutes or complements to each other than to those outside the cluster. Note that the budget constraint in the model of new demand theory can be described in terms of hedonic prices and amount of characteristics, that is, PAZ=M. Therefore, two categories can be thought of as strong substitute goods if the correlation coefficient between the expenditures of two categories appears to be largely negative. As mentioned earlier, one can group goods together in terms of the degree of strength on their substitutability or complementarility. In summary, the factor analysis will help us find the salient dimensions of the commodity space and the cluster analyses is an 71 objective approach to deal with the problem of aggregation. It is concluded that factor and clustering analyses can provide not only a partition of commodity space but also a meaningful interpretation of the groups, reflecting the basic characteristics of the consumer utility function, given the behavioral process postulated in the new demand theory. 5.2 Parametric Approach 5.2.1 Estimating demand system under two-stage budgeting Based on above proposed nonparametric procedures, it is assumed that we can construct a demand system for food with a two-stage budgeting structure. The utility function for food is a separable form and consumers allocate their food budget first to broad food commodity groups and then to food items within each group. For simplicity, let us assume that there are only two food groups. The utility function can then be written as fol1ows: U(xltx2...... ,xn) = .tPiXj)) (26) where U1 and U2 are subutility functions and homogenous of degree one in x,- and Xj,respectively, each of which indicates the consumption level of food items; x{ and Xj are vectors of commodities. With this structure we can interpret Uk(k=l,2) as a quantity index of k-th commodity group. The indirect utility function corresponding to Eq. (26) can be expressed as V = v{v'(E±) ,v 2{2±)) (27) M M where p{ and Pj are vectors of prices and H is total expenditure. Note that V1 is homogenous of degree minus one in M and of degree one in p{. Thus, Eq. (27) can be rewritten as (28) where P1 (1-1,2) can be interpreted as a price index for the i-th commodity group. By applying Roy's identity, we can derive estimable demand equations (Caves and Christensen, 1980b). For instance, the demand function for xp a commodity in the first group, is given by (29) D where (30) The total expenditure for the first commodity group, Mp can be evaluated by (31) where i denotes commodities in the first group. Substituting Eq. (29) 73 Combining Eq. (32) with Eq. (31) to obtain the estimable demand equation for Xj as follows: (33) Note that x{ in Eq. (33) is only a function of the prices of commodities in the first commodity group, and total expenditure in the first group. It does not depend on either total expenditure or prices in the second commodity group. Consequently, under the two-stage budgeting preference structure, the demand equation of x{ can be estimated using only the data on the prices of commodities within the first commodity group,pj, and total group expenditure, Mr This is a very attractive feature in empirical demand analysis, especially for a system involving a large number of commodities. Expenditure allocations within each group can be determined solely by within group relative prices and the total group expenditure. 5.2.2 Testing of model specification For empirical estimation, an explicit functional form needs to be determined a priori. As mentioned before, a flexible function form is preferable for this study. The primary reason for using a flexible functional form is that the underlying preference function has enough free parameters to be able to approximate an arbitrary, continuous, twice-differentiable preference function to the second order, and thus it permits inference without prior constraints on demand elasticities. 74 Many alternative flexible demand systems have been developed and estimated in the literature as reviewed in Appendix A. Gorman (1961) proves that all demand system with linear Engel function must be of the Gorman Polar form: v = f{Mi (35) 1 cF(M) +dG(M) Essentially, the Lewbel's demand system is proportional to the two 75 components demand systems. In order to choose an appropriate complete demand system for analyzing the U.S. food demand structure in the study, we first conduct statistical tests for alternative specifications of Engel curve which are derived from well-known complete demand systems. We firstly specify a generalized Engel function in terms of budget share as fol1ows (Aasness and Asbjorn, 1983): Wd = ai+JbiATx+4>Ciirtf* (36) where is the Box-Cox transformation of M and is defined as M x-1 m k = — — foz k*° (37) ln(Af) for A= 0 Note that M'lMx * 1 under the Box-Cox transformation. For satisfying the adding-up conditions, the following restriction must be held in the demand system: Sa,-=1, Ebj=0, and Ec,=0. This generalized Engel function encompasses the LES, QES, PIGL, and PIGLOG (price independent generalized logarithm) models as special cases (see Figure 5). For instance, when X=1 then the Engel function can be derived from the QES model. Engel function of the PIGL form is also a special case of Eq. (36) with $=0. In addition, if $-0 and X=0 then we obtain an Engel function of the PIGLOG form which is corresponding to the almost ideal demand system (AIDS) and translog demand system (TL). Comparison of alternative specifications of an Engel function involves a comparison of various models of multivariate regression. The likelihood ratio test is employed to gauge which model provides a better 76 approximation to some underlying preference structure. It is, therefore, appropriate that various functional forms are tested against each other. The first step in computing the likelihood ratio statistics is to compute the maximum likelihood estimates of all the parameters in the system, both under the null hypothesis and the alternative. The likelihood ratio test is only applicable when the null hypothesis can be seen as a special case of the alternative with the value of one or more parameters set to a fixed number. Based on the results of the likelihood ratio test, the best demand system can then be selected for analyzing the food demand structure in the U.S.. Since the specific functional form of the demand system will depend on the empirical test results, the one to be used in this study will be selected and presented later in Chapter VII. 77 <}>=0 X=1 Quad!og W f==a|+bfl n (M)+4>cfl n (M) x=o X=1 LES Figure 5. Relationships between Different Systems of Budget Share Equation 78 5.2.3 Model estimation Estimation of a complete demand system is often more complex than that for a single demand equation. Some of the typical problems related to a complete demand system include (a) nonlinearity in parameters and explanatory variables, (b) restrictions on parameters across equations in the system, and (c) joint estimation of equations in a system which is not a typical simultaneous equation system. For empirical estimation, two alternative estimation procedures are appropriate for solving this type of problems (Lee and Chern, 1989). They are the minimum distance (MD) estimation and maximum likelihood (ML) estimation. These procedures are described below: For ease in interpretation, we can write a system of demand equations as: wit = fiiVt'Qi) +5it i =1,2, . . ,N; t=l,2, .. ,T (38) where N = number of commodities; T = number of time periods or observations; Wit= the expenditure shares for the i-th commodity at time t; vt ■ a k-dimensional vector of explanatory variables; 0,- = a p-dimensional vector of unknown parameters to be estimated; Sit= unobserved random disturbance. We assume that the random disturbance terms are identically and independently distributed with mean zero and positive definite covariance matrix £. Note that there are no other assumptions concerning the distribution of the random disturbance term. It does not need to be normally distributed. If lt is written as £t ■ [?1t,52t,...,(;Nt]' then I = E[Ct5t'J, t - 1, 2,...,T; and 79 bu t=s (39) £ t 5 ie5JS] - { * * t*s with 6jj denoting the ij-th element of £. The system consists of N separate univariate nonlinear regression equations. Each equation, therefore, can be expressed as: Wd = fd (%]) +5* i=l,2, . .,N (40) with E($i$j)=&ijl, and U {, f,(0 ,-), and 5,- are all T-dimensional vectors expressed as *11 ■fi For simplicity and convenience, the equations in the system also can be stacked together as a single equation and written as: w = f(0*) +$ (43) with E($?')=£®I. where W, f(0*) and 5 are all TNxl vectors and can be, respectively, written more concisely as: *x A (01)' *2 m * ■“ MJ* s2 A (01) n (43) W = • , f(Q*) = • ■ *N A , <’©*). Based on the above assumptions of the disturbance term, a minimum distance estimation procedure can be used to estimate a vector of parameters in Eq. (40). An estimate of parameters, say QT(£), can be 80 obtained from the solutions to the problem of minimizing the following quadratic function, given a positive definite covariance matrix £: q t (B,E) = [&r-£<0*)] (44) Gallant (1987) suggested that the following procedure could be used to obtain an estimator of £, say £. First is to obtain the estimates of parameters 0i for each equation by computing 6{ from minimizing SSEiiBj) = (0p ] (0p ] for i=l,2, .. ,N (45) Then, using the estimates 6,-, the elements of £ can be estimated by &U = j,lwi-fi(Bl)]'[wj-fj (B'j)] i, j=i,2,.. ,N (46) Equivalently, we can define the residual vector as: %i = wi-fi(B\) i = i , 2 ,.. ,N (47) or for all N equations as | = [ I,,..., |N ]' then £ can be estimated by £ = T'1 . Therefore, we can substitute £ in Eq. (46) or Eq. (47) and then into Eq. (44) to obtain the MD estimate of 0. The adding-up constraint requires that the sum of the expenditure shares from each group in the system must be equal to one. That is £^<=0, and thus, the covariance matrix is singular. Therefore, one of equations in Eg. (38) is completely redundant because it can be derived from any N-l equations by an appropriate linear combination ( Parks, 1971 ). It is noted that the parameter estimates obtained by minimum distance estimation are not invariant to the equation deleted (Berndt 81 and Savin, 1975). However, the property of invariance can be obtained by using an estimated matrix of £ which is maintained throughout the iterative estimation process and is defined as \ ] [V-/<0;> 1' (48) where 0y* is the value of 0* obtained from minimizing QT(0,£) at the y-th iteration. Alternatively, if the distribution of the random disturbances for the sample of observations is known, then a maximum likelihood estimation procedure can be employed. If the random disturbances are assumed to follow a multivariate normal distribution. Following Parks (1971), under this distributional assumption, the joint density function can be written as: T T f(5|E) = (2tt)” A transformation from the ?'s to W's gives (for simplicity, let 0 = 0* as defined in Eq. (44)) f (w|E,6) = Jc|E|"T exp [-A [w-f (0'] '(E-^J) [w-f (6) ] dw\j\T (50> where Jacobian |J|=1 and k is a constant of proportionality. The logarithm of the likelihood function is L = £'+ (-£) log IE"1 |-|[&r-f(0)]' (E-1® J) [w-f(0)l (51) A 82 The maximum likelihood estimators for 6 and E are those values that maximize L. Thus, the maximization of the logarithm of likelihood function, Eq. (51), is equivalent to the minimization of Eq. (44). Malinvaud (1980) shows that if we iterate the estimates QT(£) by replacing £ at each iteration with the inverse of the moment matrix of residuals from the previous iteration, we will eventually arrive at the maximum likelihood estimator. Barten (1969) also shows that when disturbances are serially independent, then the estimates can be derived from Eq. (51) and are invariant to the equation deleted. However, if there exists autocorrelation in the estimation of system of nonlinear equations, such as E(£it£js)*0 for t#s. For simplicity, consider the first-order autoregressive scheme, AR(1). Thus the error structure follows: Sit = PiSit-i+ e it for t= 2 ,3 T, 1=1,2 / ■ ■ / N. (52) where eit's are independently, identically distributed normal random vectors with mean zero and contemporaneous covariance £; and q , is the coefficient of autocorrelation in the i-th equation and |e,| s 1. With the formulation of AR(1), Parks (1967) shows that o 12B1B2 • • • * aNNBNBK where B is the block matrix [BJ and its diagonal submatrix can be expressed as: Using the Koyck transformation, i.e., by lagging Eq. (38) and multiplying it through by q { and subtracting the resulting equation from Eq. (38), we can obtain wit = (Vc, - p ^ (Vt_^»0j) +c t=2, . ,T; i =1,2, . ,N Eq. (55) can be verified by reference to AR(1) scheme. Without autocorrelation across equations, Berndt and Savin (1975) have shown that the adding-up restriction implies all diagonal elements to be identical, that is, e^e for all i. Note that the system of equations in Eqs. (44) and (51) to be solved are nonlinear in parameters and an explicit formula for solving parameters does not exist. Consequently, numerical methods are required for solutions. 5.2.4 Test statistic on parameters restriction To test a restriction on parameters in a nonlinear equation system, an asymptotically valid chi-square test can be constructed by using the statistics of minimizing a quadratic function in Eq. (57) under the assumption that the consistently estimated covariance, 2, to be the same for the unrestricted full model and the restricted reduced 84 model (Gallant and Jorgenson,1979). Suppose the statistics of minimizing the quadratic function of full model is QT(8f,2) while that of the reduced model is QT(8r,2), then the test statistic is x = (?r It can be shown that % is distributed asymptotically as a chi- square with degree of freedom being equal to the difference in the number of free parameters in the two models. In addition, if the density function of disturbance term is known and the maximum likelihood estimation approach is used, then the likelihood ratio test also can be employed to test the parameter restrictions. The likelihood ratio is the ratio of the maximum value of the 1ikelihood function for the restricted model to the maximum value of 1ikelihood function for the unrestricted model. For normally distributed disturbances, the 1ikelihood ratio is equal to the ratio of the determinant of the restricted estimator of the variance-covariance matrix of residuals to the determinant of the unrestricted estimator, each raised to the power -(T/2), where T is number of observations. More specifically, the test statistic of 1ikelihood ratio can be expressed as -2 ln($) = r(-^£) (57) “U where Qu is log of determinant of vari ance-covari ance matrix of residuals for the unrestricted estimator, and Qr is the restricted estimator. Under the null hypothesis this test statistic is 85 asymptotically distributed as chi-square with the number of degrees of freedom equal to the number of restrictions to be tested. 5.2.5 Computation of elasticities The basic characteristics of consumer preferences can be summarized by a set of price (own and cross) and income elasticities. In general, the formulas of elasticities for each commodity in a complete demand system can be calculated analytically as follows: The income elasticity, qf, of commodity i is ^ = '-5 ^ 1 + 8« (59> where j is Kronecker delta. From the Slutsky equation, the relation between the price and income elasticities can be expressed as q^WjOy-Wjq,, where ofj are the Allen elasticities of substitution. Note that Allen elasticities of substitution are symmetric, that is o,j= ojM; and the product term, WjOy-q*!], is the compensated price elasticity. Therefore, the compensated price elasticities, q*?j, can be computed from the income elasticity and the uncompensated price elasticities. They are expressed as 86 nij = i. (60) Consequently, after obtaining the estimates of parameter in Eq. (38), the elasticities for each food commodity in a demand system can be computed numerically. However, in case of a demand system within a two-stage budgeting structure we can not evaluate full elasticities by means of the above formulas directly from the estimated parameters in the second stage demand equations. For instance, estimation of Eq. (33) yields only the estimates of partial elasticities which are conditional on the expenditure level of the first commodity group (M,). This is analogous to tiy in Eq. (59) which is conditional on total expenditure (M). Following Caves and Christensen (1980), and Kohler (1983), we can show that the full price elasticity within a subgroup under a two-stage budgeting structure can be disaggregated and expressed as fol1ows: d M P d P P j (61) d P M d p j P where P = price index for the separable subgroup, iwm = expenditure elasticity of i-th good in the subgroup, il ,-j = the partial price elasticity holding total expenditure on the separable subgroup constant. Eq. (61) can be rewritten as (62) where n,,=the price elasticity for the aggregate commodity group I. Under homothetic condition, qim*=l, and setting (3P/3Pj)(pj/P)=Wj, Eq. (62) reduces to the following expression: 87 (63) Consider next the computation of expenditure elasticities with respect to total expenditure under a two-stage budgeting process. The following formula was suggested by Bieri and de Janvry (1972) and by Manser (1976): tli = n i * (64) where t|„ : the first-stage expenditure elasticity of I-th commodity group with respect to total expenditure; f)] : the second-stage expenditure elasticity of i-th commodity given expenditure of group I being constant. Furthermore, to compute cross-price elasticities between commodities in different groups under a two-stage budgeting, we can use the Slutsky substitution terms, Sfj-, to derive the interdependent relationships between commodities. Following Phlips (1974), and Deaton and Muellbauer (1980), a utility function is weakly separable if and only if the Slutsky terms are of the form (65) where (66) and all i e I, j e J, I # J. is the compensated derivative of expenditure on the group I with respect to a proportional change of all price i in group J. Note that under given conditions, XtJ can be considered as the intergroup substitution terms (see Deaton and Muellbauer, 1980, Page 129). For the purpose of deriving a formula to 88 interdependent relationships in terms of elasticities, Eq. (65) can be first written, by substituting &,j in Eq. (66), as follows: 3<3Ti _ , i d Multiplying both sides of Eq. (67) by [Pj/qf], and rearranging the terms, we can obtain the following equation: *1 ij = " t^tli <6 8 > where W, = budget share of aggregate group I at the first stage; w,- = budget share of i-th commodity ; t){ = expenditure elasticity with respect to total expenditure for i-th commodity. Consequently, Eq. (67) has a direct linkage between the first- stage and second-stage for computing the cross-price elasticities between commodities in different groups. Furthermore, in order to obtain the standard deviations for the associated elasticity estimates. The approaches of approximating the standard errors of elasticities based on the asymptotic theory are used in this study. Let an elasticity ij be expressed as a function of the parameter vector,®. By the asymptotic theory, the estimates of parameters, § , have the following distribution: 8 - 2V(0O, Var(0)) (69) where § » E(8); Var(0) - a variance-covariance matrix of 8 . Taking exact first-order Taylor series expansion of q(8) around ti(80), 89 we have 11(6) =t|(6o)+ii|i|,.(8-0o) (70) 0* lies between 0O and 0, and 3(0)/30'|e* is the gradient vector at 0*. Eq. (70) is a linearized transformation of a normal random variable. Hence, the distribution of q(0) can be expressed as follows: il(0) ~ N(r\ (0O), -^rle* Var(B) -^|Me*) (71) Theresult of Eq. (71), sometimes called the delta method, is a well-known approach for deriving standard deviations and confidence intervals for elasticities. Due to the complexibility of the system selected for empirical estimation, this method is only implemented for computing the standard error of elasticities for the LA/AIDS model under one-stage procedure in this study. CHAPTER VI RESULTS OF NONPARMETRIC ANALYSES This chapter presents the results from nonparametric analyses. Data consistency testing based on the GARP are reported first. Second, the rank condition of 19 food categories is examined by using four common specifications of an Engel function. Third, the results of commodity grouping through factor and cluster analyses are provided. Fourth, according to the clustering hierarchical structure, a basic set of six commodity groups is constructed. Finally, some concluding remarks are presented. 6.1 Data Consistency Test Results As discussed in Chapter V, the nonparametric test is employed to examine whether or not our data series (quantities and prices of food commodities) are consistent with the utility maximization hypothesis. The consistency test, i.e., GARP, is performed for two alternative groups. The first group includes all 19 expenditure categories of food at home and food away from home. The second group includes only the 18 expenditure categories for food at home. The tests are conducted for alternative sample periods. Note that the computer software used in this study is a PC version developed and provided by Varian, which can not accommodate more than 75 observations. Since our sample of 84 monthly observations exceeds slightly this limit, we construct various 90 91 data sets for alternative sample periods. In fact, these data sets constructed from different time periods allow us to examine whether or not our data series is stable during the entire sample period. The results of the consistency test are summarized in terms of the number of violations of GARP. Table 9 presents these testing results. It is noted that the maximum number of violations exceeds k N(N-l), where N is the number of observations under investigation. Since N=75, \ N(N-l) is equal to 2,775. Table 9 provides the following important findings. First, the number of violations is fairly small for both groups under alternative sample periods, especially for the 19 commodities group. Note that the power of nonparametric test in empirical demand analysis has been an issue gaining considerable attention among researchers. Landsburg (1980), Varian (1982), and Chalfant and A1ston (1985), point out that when the nonparametric method is applied to aggregate consumption series, one may not easily find a violation of the revealed preference axioms because the budget lines drawn for annual observations rarely cross. Aggregate consumption of every good is rising over time in such applications that each current bundle of goods would be revealed preferred to all previous ones; that is, even if the underlying preference structure is not rational, the data might not reject the GARP. However, Chalfant and Alston (1985) further indicate that it seems reasonable to expect the power of nonparametric tests to be higher for disaggregated goods such as those in this study than for more aggregated goods. This is because quantities of various food items consumed do not all rise or fall uniformly with time, and thus, price variation relative to variation in real expenditure, is likely to be 92 Table 9. Results of Nonparametric GARP Test Number of Violations Sample Periods 19 Commodities 18 Commodities® 1980 - 1985 6 16 1981 - 1986 2 15 1980 - 1986b 4 0 1980 - 1986c 0 15 a/ Excluding food away from home; b/ Deletes 9 observations including January, and February of 1980, and December of all years; c/ Deletes 9 observations including January, February, April, May and August of 1980, May of 1981, June of 1982, June of 1984 and April of 1985. 93 greater, than that for the aggregated bundles of goods. In fact, we were surprised to find such a small number of violations for the data used in this study because our data is relatively disaggregated and is constructed from a monthly basis rather than a quarterly or annual basis. Secondly, considering two different sample periods, i.e., 1980- 1985 and 1981-1986, the numbers of violations are relatively small for both groups. We also noticed that the violations occur with almost the same observations. This evidence implies that our data structure is relatively stable during the entire sample period of 1980-1986. Third, we investigate further the causes of violations occurred in the 18 commodities group and discover that the main cause is the inclusion of the December observations. In order to verify this finding, we exclude all December observations together with January and February of 1980. The number of violations reduces to zero. This finding has a very important implication for modeling a demand system just for food at home. The result indicates that the appropriate approach to model a demand system for categories of food at home using monthly data is to either exclude the December observations or to include both food at home and food away from home categories (as in the 19 commodities group). This result, in fact, would confirm our intuitive belief that the behavior of purchasing is somewhat different between the regular months and December with the holiday seasons. In summary, there are two important findings from the nonparametric testing, (a) The data set for the 19 expenditure categories data appear to be consistent with the hypothesis that they 94 are generated by the maximization of a utility function by a representative consumer. This group of food at home and away from home may constitute a weakly separable group, (b) The 18 expenditure categories for food at home can satisfy the GARP only under the condition of excluding December observations. This implies that the consumption of food at home in December may be somewhat different from other months in the year. 6.2 Results of Rank Test As discussed earlier, the rank of a demand system can be determined by estimating the Engel curves. Lewbel indicates that the rank test is an valuable prespecification examination in empirical demand analysis. This is because rank has several useful implications for characterizing a demand system and preference structure. Specifically, a demand system has rank K=1 if and only if the system is homothetic. Under this situation, budget shares are independent of the level of income. A demand system has a rank K=2 if and only if the system is the generalized linear system (GL, Muellbauer, 1975). Also, an indirect subutility function in Gorman Polar Form (Gorman, 1959) would yield demand systems having a rank K=2. These systems include the translog and AIDS demand models. Furthermore, the maximum possible rank of the exactly aggregable class of demand system is K=3 (Gorman, 1981) while the maximum rank of the def1ated income class of systems is K=4 (Lewbel, 1989). Finally the rank of a demand system is a lower bound on of degree of separability (Lewbel, 1989). The preceding results indicate that the lower the rank, the more 95 stringent the assumption on separability and the related utility structure possessed by aggregation across agents and commodities. It is therefore important to know the rank of demands in order to properly specify the empirical model which can appropriately reflect the preference structure consistent with data used in the study. To estimate the rank, we employ four common specifications of Engel function, representing special cases of Eq.(36). (1) W± = ai+Jbilog(Af)+ci[log(W)]2 (72) (2) w± = a ^ b ^ + C j M (73) (3) Wj « aj+Jbj log(M) (74) (4) wd = (75) Eq. (72) is a quadratic generalization of the Leser-Working Engel curve specification in which the budget share is assumed to be a function of the logged total expenditure (or income) and its square term (Leser, 1963; Working, 1943). Eq. (73) is a quadratic expenditure system in which budget share is a function of both the inverse of total expenditure and its square (Poliak and Wales, 1979,1980). Eq. (74) belongs to a PI6L0G class of demand and Eq. (75) corresponds to the LES. These later two equations are special cases of Eqs. (72) and (73), respectively, as the square terms are vanished. Note that Eqs. (72) and (73) satisfy the rank condition of K=3 (with 3 parameters in the equation) while the Eqs. (74) and (75) have rank K=2 with two parameters in the equation. 96 The results of these specification test for the entire set of 19 commodities are summarized in Table 10. In general, Table 10 reveals that Eqs. (72) and (73) which have a rank K-3 have almost identical estimated results in terms of the log of likelihood function as well as the log of the determinant of covariance matrix of the residuals. The models with rank K=2, Eqs. (74) and (75), also show similar results. Comparing the values of 1ogged likelihood function, the equation with a rank K=3 have siightly higher values than those with a rank K=2. For further examination of model specification, a joint test of the null hypothesis that the square terms are zero in the equations with a rank K=3 is conducted. The Wald statistics are computed and presented in Table 10. This statistics has a Chi-square distribution with the degrees of freedom equal to the number of restricted parameters. The results show that the Wald statistics is 38.18 for Model (1) and 39.84 for Model (2). Both test results reject the null hypothesis, implying Eqs. (72) and (73) more appropriate than Eqs. (74) and (75). In conducting these tests, we treat all the budget share equations to be symmetric in all goods. In order words, the specifications of functional form for all goods are the same. Johansen (1981) suggested that while estimating a demand system it may be useful to assume a functional form of the utility function treating different goods asymmetrically. In this case budget share equations may not be symmetrical. Indeed, we notice that only two coefficients of the square terms,i.e., for eggs and non-alcoholic beverages, among the 18 equations estimated in the system are significant at c=l%. We therefore estimate a system excluding eggs and nonalcoholic beverage, and then recalculate 97 Table 10. Results of Specification of Engel Functions for 19 commodities MODEL* CASEb In Lc fid R2 WS® (1) A 6218.88 -216.92 0.86 38.18 B 5382.26 -188.93 0.84 26.32 (2) A 6218.84 -216.92 0.86 39.84 B 5382.21 -188.93 0.84 26.91 (3) A 6203.45 -216.51 0.79 B 5370.98 -188.63 0.78 (4) A 6202.87 -216.49 0.79 B 5370.71 -188.63 0.78 a/ Model (1): W, - a.+b.logtMJ+cJloglM) j2; Model (2): W { = a*+bs (l/H)+cf (M) 5 Model (3): U { «= a.+bflog(M)$ Model (4): U { - a,+b-(l/M). b/ Case A: including 19 food categories in the system; 6: including 17 food categories in the system without eggs and nonalcoholic beverage, c/ Values of log-1ikelihood function; d/ Log of determinant of covariance matrix of the residuals, e/ R =l-[Det(E'E)/Det(W'W)] where E'E is the cross product matrix of residuals; W'W is the cross product matrix of dependent variable; Det denotes the determinant of a matrix, f/ WS: Wald Statistics (Chi-square statistic) is defined as X - -N(log(l-R )); N denotes sample size. 98 the Wald statistics. The values of Wald statistic are substantially reduced to 26.32 for Model (1) and 26.91 for Model (2). These revised test results can not reject the null hypothesis that the quadratic terms in the system without eggs and non-alcoholic beverage are zero. One important implication of this result is that the functional form of Engel curves may be different among different commodities in the system. It is important to verify the Gorman theorem that the rank of the matrix of coefficients for the polynomial terms in total expenditure is at most three under the condition of the exactly aggregable function in the Engel analysis. To accomplish this task, we specify a more general model which extends Eq. (72) to include a third degree term in log of total expenditure. Therefore, the expenditure share can be expressed as: W1=ai*bi log (Af) +cilog (Af) 2+di log (Af)3 (76) The rank restriction takes the form that the ratio of coefficients of the cubic terms of the quadratic terms will be constant across equations (Gorman,1986; Hausman, et al., 1988). The estimated "Gorman statistic" (df/cf) are presented in Table 11. The results in the Table 123 clearly reveal the estimated ratios to be remarkably close to a constant, 0.051, even though there are considerable variations in the estimates of the c/s and d/s. This finding will support the Gorman theorem. Therefore, we can conclude that the rank of the demand system based on the data used in the study is at most three while the actual rank of demand system is equal to two if eggs and nonalcoholic beverages are excluded. 99 Table 11. Estimated Gorman Statistics8 Commodities Ratios (10'1) SE(10‘3)b Cereal -0.51 0.15 Bakery Products -0.50 0.13 Beef -0.50 0.12 Pork -0.56 0.90 Other Meats -0.44 4.34 Poultry -0.42 7.92 Seafood -0.56 1.01 Eggs -0.56 0.86 Milk Products -0.34 23.87 Other Diary -0.51 0.24 Fresh Fruits -0.52 0.15 Fresh Vegetables -0.49 0.49 Processed Fruits -0.50 0.21 Processed Veg. -0.52 0.13 Sweets -0.51 0.23 Nonalc. Bev. -0.51 0.23 Fat & oil -0.52 0.13 Misc. Food -0.51 0.12 FAFH -0.51 0.12 In Lc 6398.04 fld -203.42 a/ The statistics are the ratios of d,/c, for equation W, = af+bjlog(M)+cf[log(M)] +dj[log(M)] ; b/ Standard Errors; c/ The values of log of likelihood function; d/ The values of log of determinant of covariance matrix of the residuals. 100 6.3 Results of Commodity Grouping Traditionally, the grouping of food Items Is based on their physical similarity and usage in meals and diets. However, from a prospective of consumer demand theory, forming such composites by the traditional way may not fully reflect the consumer preference structure. Consumers pursue utility from good characteristics which Include many technical factors such as nutrition, food preparation and serving and other factors such as taste. To group the 19 categories of food expenditure into some composites with a property of "wlthin-group similarity, and between group difference", two multivariate statistical techniques discussed in Chapter V are employed. The first analysis uses the procedure of principal component factor analysis. The results of this principal component factor analysis are presented in Table 12. The four largest eigenvalues are 6.695, 2.858, 2.284, and 1.105, which together account for 68% of the standardized variance. Therefore, the first four principal components explain about two thirds of the variations of expenditure data for the 19 food items. Note that we retain factor four components on the basis of the eigenvalues-greater-than-one rule since the fifth eigenvalue is only 0.932. After the factors were estimated, it is necessary to interpret them because the factor solutions can help us discover salient general dimension that reproduce the interrelations among the variables. These factors may suggest what characteristics of consumer interest might be, and what are likely definers of the clusters. Interpretation usually means assigning to each common factor a name that reflects the 101 Table 12. Results of Principal Component Factor Analysis Factor Eigenvalue Difference Proportion Cumulative 1 6.695 3.836 0.352 0.352 2 2.858 0.574 0.150 0.503 3 2.284 1.179 0.120 0.623 4 1.105 0.172 0.058 0.681 5 0.932 0.159 0.049 0.730 6 0.774 0.086 0.041 0.771 7 0.687 0.085 0.036 0.807 8 0.602 0.090 0.032 0.839 9 0.512 0.043 0.027 0.866 10 0.468 0.032 0.025 0.890 11 0.437 0.091 0.023 0.913 12 0.345 0.014 0.018 0.931 13 0.332 0.093 0.018 0.949 14 0.239 0.030 0.013 0.962 15 0.209 0.030 0.011 0.973 16 0.179 0.044 0.010 0.982 17 0.136 0.004 0.007 0.989 18 0.132 0.057 0.007 0.996 19 0.074 - 0.004 1.000 102 importance of the factor in predicting each of the observed variables. In general, the interpretation involves identifying the relatively high loadings, usually in absolute value, on that factor. The highly positive loadings can then help to define one end of the underlying dimension, while the highly negative loading (if any) define the opposite end. It is also noted that factor interpretation is a subjective process. Table 13 indicates that the first component have all positive loading except for beef and eggs while the 1argest positive loadings are obtained for cereal, bakery products, other dairy, fresh vegetables, processed fruits, nonalcoholic beverages, fats and oils, miscellaneous foods, and food away from home. Specifically, the correlations with cereal (0.784), bakery products (0.896), dairy products (0.789), processed fruits (0.807), and miscellaneous foods (0.889) are all relatively high. Therefore, this factor may be interpreted as related to necessity food commodities. The second component represents a contrast of beef (0.392), pork (0.57), poultry (0.437), eggs (0.737), milk products (0.42), processed vegetables (0.655) against fresh fruits (-0.599), fresh vegetables (-0.223), nonalcoholic beverages (-0.36), and food away from home (-0.466) with a relatively small loading on cereal (-0.036), bakery products (-0.020), seafood (-0.03), and miscellaneous foods (-0.079). Considering the sign and magnitude of loading, this factor may be interpreted as related to characteristics of high fat and cholesterol because of positive loadings for red meats and eggs and negative signs for fresh fruits and fresh vegetables. The third and fourth components also show to have a contrasting structure with different magnitudes and signs for various goods. Table 13. Results of Factor Pattern Categories Factorl Factor2 Factors Factor4 FC" Cereal 0.784 -0.036 -0.191 -0.290 0.737 Bakery 0.896 0.020 -0.071 -0.109 0.820 Beef -0.239 0.392 0.749 0.013 0.772 Pork 0.223 0.576 0.397 0.477 0.766 Other Meats 0.304 0.042 0.701 0.031 0.586 Poultry 0.474 0.437 0.059 0.085 0.427 Seafood 0.422 -0.030 -0.171 0.449 0.410 , Eggs -0.031 0.737 0.099 -0.071 0.559 ? Milk 0.490 0.429 0.302 -0.349 0.638 Dairy Pro. 0.798 -0.130 0.122 0.301 0.759 Fresh Fru. 0.442 -0.599 0.541 0.063 0.850 Fresh Veg. 0.620 -0.223 0.469 -0.095 0.663 Pro. Fru. 0.807 0.172 -0.157 -0.017 0.706 Pro. Veg 0.434 0.655 -0.312 -0.239 0.772 Sweet 0.434 0.381 -0.370 0.455 0.678 Nona. Bev. 0.755 -0.361 0.129 -0.087 0.724 Fat & Oil 0.607 0.260 0.022 -0.251 0.499 Mis. Food 0.889 -0.079 -0.294 0.036 0.885 FAFH. 0.678 -0.466 -0.094 0.072 0.691 VEb 6.695 2.858 2.284 1.105 a/ FC denotes final communality; b/ VE denotes variance explained by each factor. The final communality estimates show that all the goods are relatively well accounted for by the first four components, as they range from 0.885 for miscellenous foods to 0.410 for seafood. The results of the principal component factor analysis reveal valuable information regarding the preference structure expressed by salient factors. However, the loadings of a principal component show each good has multifactor characteristics. That is, the variance of each variable could not result from a single factor loading. For example, the loadings of pork in four principal components are fairly close. Therefore, a separable partition for commodity grouping from the results of principal component factor analysis is not Immediately forthcoming. As we mentioned earlier, the factor analysis is only appropriate for the case that each commodity can be clearly identified by a uni factor in the structure of common factors in commodity grouping. In order to find a more clear-cut commodity grouping for goods with multiple characteristics, the cluster procedure is therefore used. In this study, the centroid component approach is used to perform the cluster analysis. Table 14 presents the cluster listing which includes the information about the variables in each cluster, the squared correlation of the variable with its own cluster component, and the next highest squared correlation of the variable with a cluster component, and 1-R2 ratio-. Note that variables refer to 19 food categories. The 1-R2 ratio is defined as the ratio of 1-R2 with own cluster to 1-R2 with next highest. Therefore, if the clusters are well-separated then the next highest squared correlation should have a low value. Also, the 1arger the squared correlation of own cluster component is, the better 105 Table 14. Results of Cluster Listing R-Sauare with Cluster Variable Own Cluster Next Closest 1-R2 Ratiob 1 Cereal 0.744 0.396 0.423 Bakery 0.850 0.533 0.322 Pro. Fruits 0.702 0.368 0.472 Misc. Food 0.860 0.575 0.329 2 Pro. Veg. 1.000 0.250 0.000 3 Pork 1.000 0.233 0.000 4 Sweets 1.000 0.193 0.000 5 Other Dairy 0.797 0.466 0.380 Nona. Bev. 0.797 0.473 0.385 6 Other Meats 1.000 0.180 0.000 7 Milk Products 1.000 0.179 0.000 8 Seafood 1.000 0.119 0.000 9 Poultry 1.000 0.190 0.000 10 Fresh Fruits 0.829 0.404 0.287 Fresh Veg. 0.829 0.357 0.266 11 Fat & Oil 1.000 0.329 0.000 12 FAFH 1.000 0.405 0.000 13 Eggs 1.000 0.205 0.000 14 Beef 1.000 0.232 0.000 a/ The next highest Rz of the variable with a cluster component; b/ 1-R2 ratio is one minus the value in the own cluster column to one minus the value in the next closest. 106 1s the clustering. Meanwhile, low ratios of 1-R2 would Indicate well- separated clusters. Based on the prespecified criterion, each cluster has only a single eigenvalue greater than one. Using this criterion, the maximum of 14 clusters are determined as shown In Table 16. Under this grouping of 14 clusters, many characteristics of 19 food categories are preserved. For Instance, various kinds of meats, seafood, eggs, milk, and oils appear to have a relatively high possibility to be independent. On the other hand, we also observe several categories to be strongly correlated, and they form three commodity groups. The first group includes cereal, bakery products, processed fruits and miscellaneous foods which form a single cluster. This group may be appropriately interpreted as a staple food group. The next group consists of other dairy products and non-alcoholic beverages. This is a group which may be defined as taste oriented food group. Finally, fresh fruits and fresh vegetables form a group which may be is nutrition related as a healthy diet group. Table 15 and Table 16 present and illustrate the cluster structure. Table 15 shows the correlation coeffiicients between each variable and each cluster component. Considering Cluster 1, cereal (0.863), bakery products (0.922), processed fruits (0.838) and miscellaneous goods (0.927) have the highest positive association with cluster 1, while beef (-0.330) and eggs (-0.046) show a negative correlation. Note that the figures for cluster 1 in the Table 15 and figures for factor 1 in Table 13 show a very similar pattern. The correlations between the cluster components, i.e., inter-cluster 107 Table IS. Cluster Structure Cluster Variable 1 2 3 4 5 6 7 Cereal 0.863 0.409 -0.025 0.293 0.629 0.107 0.347 Bakery 0.922 0.413 0.174 0.360 0.730 0.218 0.432 Beef -0.330 -0.054 0.483 -0.206 -0.215 0.366 0.165 Pork 0.121 0.229 1.000 0.340 0.159 0.276 0.285 Other Meats 0.160 -0.046 0.276 -0.077 0.303 1.000 0.365 Poultry 0.435 0.430 0.388 0.277 0.311 0.111 0.362 Seafood 0.344 0.213 0.096 0.236 0.320 0.074 0.073 Eggs -0.046 0.452 0.319 0.168 -0.181 0.063 0.306 Milk 0.407 0.423 0.285 0.168 0.297 0.365 1.000 Dairy Pro. 0.683 0.131 0.226 0.379 0.893 0.362 0.277 Fresh Fru. 0.293 -0.375 -0.000 -0.186 0.635 0.371 0.103 Fresh Veg. 0.451 0.086 0.090 -0.019 0.597 0.401 0.352 Pro. Fru. 0.838 0.528 0.175 0.464 0.606 0.162 0.417 Pro. Veg 0.500 1.000 0.229 0.339 0.101 -0.046 0.423 Sweets 0.439 0.339 0.340 1.000 0.283 -0.077 0.168 Nona. Ber. 0.688 0.050 0.058 0.126 0.893 0.180 0.254 Fat & Oil 0.574 0.356 0.217 0.302 0.431 0.215 0.394 Mis. Food 0.927 0.426 0.103 0.441 0.758 0.081 0.249 FAFH. 0.632 0.067 -0.116 0.163 0.637 0.145 0.114 Table 15 (Continued) Cluster Variable 8 9 10 11 12 13 14 Cereal 0226 0336 0 340 0 492 0 511 -0 046 -0 272 Bakery 0300 0424 0 407 0 587 0 609 -0 085 -0 188 Beef -0 234 0 114 0 077 -0 035 -0 382 0 304 1 000 Pork 0 096 0 388 0 049 0 217 -0 116 0 319 0 483 Other Meats 0 074 0 111 0 424 0 215 0 145 0 063 0 366 Poultry 0 128 1 000 0 111 0 244 0 087 0154 0114 Seafood 1 000 0 128 0 192 0 182 0 323 -0 014 -0 234 Eggs -0 013 0 154 -0 201 0 127 -0 290 1 000 0 304 Milk 0 073 0 362 0 250 0 394 0 114 0 306 0 165 Dairy Pro. 0 352 0 343 0 570 0 351 0 579 -0 027 -0 218 Fresh Fru. 0 113 0 013 0 911 0 073 0 491 -0 325 0 063 Fresh Veg. 0 237 0 189 0 911 0 334 0 499 -0 042 0 077 Pro. Fru. 0 309 0 404 0 347 0 472 0 445 0 078 -0 272 Pro. Veg 0 213 0 430 -0 159 0 356 0 067 0 452 -0 054 Sweets 0 236 0 277 -0 112 0302 0 163 0 168 -0 206 Nona. Ber. 0 220 0 212 0638 0418 0557 -0 297 -0 166 Fat & Oil 0 182 0 244 0 224 1 000 0 155 0 127 -0 035 Mis. Food 0 388 0 382 0 357 0 486 0 677 -0 111 -0 439 FAFH. 0 323 0 087 0 543 0 155 1 000 -0 290 -0 382 108 Table 16. Inter-Cluster Structure Cluster 1 2 3 4 5 6 7 1 1 000 0 500 0 121 0 439 0 767 0 160 0.407 2 0 500 1 000 0 229 0 339 0 101 -0 046 0 423 3 0 121 0 229 1 000 0 340 0 159 0 276 0 285 4 0439 0339 0340 1000 0 283 -0 077 0 168 5 0 767 0 101 0 159 0 283 1 000 0 303 0 297 6 0 160 -0 046 0 276 -0 077 0 303 1 000 0 365 7 0 407 0 423 0 285 0 168 0 297 0 365 1 000 8 0 344 0 213 0 096 0 236 0320 0074 0073 9 0 435 0 430 0 388 0 277 0 311 0 111 0 362 10 0 409 -0 159 0 049 -0 112 0 677 0 424 0 250 11 0 574 0 356 0 217 0 302 0 431 0 215 0 394 12 0 632 0 067 -0 116 0 163 0 637 0 145 0 114 13 -0 046 0 452 0 319 0 168 -0 181 0 063 0 306 14 -0 330 -0 054 0 483 -0 206 -0 215 0 366 0 165 Table 16 (Continued) Cluster 8 9 10 11 12 13 14 1 0.344 0 435 0 409 0 574 0 632 -0 046 -0.330 2 0.213 0 430 -0 159 0 356 0 067 0 452 -0.054 3 0.096 0388 0049 0217 -0116 0319 0.483 4 0.236 0 277 -0 112 0 302 0 162 0 168 -0.206 5 0.320 0311 0 677 0 431 0 637 -0 181 -0.215 6 0.074 0 111 0 424 0 215 0 145 0 063 0.366 7 0.073 0362 0250 0 394 0 114 0 306 0.165 8 1.000 0128 0 192 0 182 0 323 -0 014 -0.234 9 0.128 1 000 0 111 0 244 0 087 0 154 0.114 10 0.192 0111 1 000 0 224 0 54<4 -0 201 0.077 11 0.182 0 244 0 224 1000 0 155 0 127 -0.035 12 0.323 0 087 0 544 0 155 1 000 -0 290 -0.382 13 -0.014 0 154 -0 201 0 127 -0 290 1 000 0.304 14 -0.234 0 114 0 077 -0 035 -0 382 0 304 1.000 109 correlations, are reported in Table 16. Note that a high negative correlation coefficient between cluster components would suggests a large degree of substitutability in consumption between these groups. A summary of the results of this oblique centroid component cluster analysis is reported in Table 17. Table 18 indicates that total variation explained by 14 clusters is 17.042, accounting for 91.6%. Figure 6 provides a hierarchical structure of the clustering analysis for 19 expenditure categories. The hierarchical structure gives us a fundamental framework of preference structure to identify the goods into similar commodity groups. In order to establish the validity of these groupings, we further examine whether or not well-behaved subutilities of alternative commodity groups related to this hierarchical framework exist, using nonparametric tests. Table 18 presents the results of consistency test in terms of the number of violations with the GARP for selected commodity groupings. The test results show that a commodity group involving more than three kinds of meat products and eggs is more likely to have a high number of violations. This would imply that the traditional grouping of classifying all meats products into a single group may not be justified as it does not satisfy the consumer preference structure. In addition, we also observed that various sample periods tend to have relatively close numbers of violations of GARP except for the commodity group including only beef and pork, (i.e. Group G), which has no violation for the 1980-1985 period but 23 violations for the 1981-1986 period. no Table 17. Summary of Oblique Centroid Component Cluster Analysis No® CTVb PCTVc MPTV* MR2® M(l-R2)f 1 5.527 0.291 0.291 0.0001 • 2 8.289 0.436 0.328 0.1921 0.903 3 9.645 0.508 0.446 0.2139 0.816 4 10.479 0.552 0.515 0.2139 0.829 5 11.628 0.612 0.583 0.2692 0.808 6 12.361 0.651 0.596 0.2692 0.808 7 13.025 0.686 0.597 0.2692 0.808 8 13.979 0.736 0.639 0.5219 0.641 9 14.702 0.774 0.646 0.5219 0.641 10 15.281 0.804 0.707 0.5219 0.566 11 15.897 0.837 0.718 0.7015 0.538 12 16.337 0.860 0.726 0.7015 0.472 13 16.884 0.889 0.741 0.7015 0.472 14 17.042 0.916 0.787 0.7015 0.472 a/ Number of cluster; b/ Cumulative total variation explained by clusters; c/ Proportion of total variation explained by cumulative clusters; d/ Minimum proportion of total variation explained by a cluster; e/ minimum R for a variable; f/ Maximum 1-R2 ratio for a variable. 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XXXXXXX XXXXXXX XXXXXXXXXXXXX XXXXXXXXXXXXX XXXXXXXXXXXXX XXXXXXXXXXXXX MMMMXXXMXXXXX N A M E O F V A R I A B L E O R C L U S T E R XXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXX xxxxxxxxxxxxxxxxxxxxxxm XXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX M H H XXXXXXX XXXXXXX MMMM M XXXXXXX XXXXXXX XXXXXXX XXXXXXX XXXXXXX XXXXXXXXXXXXX XXXXXXXXXXXXX XXXXXXXXXXXXX XXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXX XXXXXXX XXXXXXX XXXXXXX XXXXXXX XXXXXXX XXXXXXX XXXXXXX XXXXXXX XXXXXXX 111 112 Table 18. Results of Consistency (GARP) Test for Selected Commodity Groupings Alternative Commodity Groupings Items A B1 B2 Cl C2 D1 02 E F G 1. Cereal ABE 2. Bakery Products ABE 3. Beef ABC G 4. Pork A BC G 5. Other Meats A - B C 6. Poultry ABC D 7. Sea Food AB E 8. Eggs A BCD 9. Milk Products A BCD 10 Other Diary A B F 11 Fresh Fruits AB F 12 Fresh Veg. AB F 13 Processed Fru. AB E 14 Processed Veg. ABCD 15 Sweets A B C D 16 Non-alc. Bev. A B F 17 Oils A BE 18 Misc. Foods A B E 19 Food Away from Home AB F Sample period Number of Violations 1980 - 1985 6 18 0 4 10 2 11 4 4 0 1981 - 1986 2 22 2 8 10 7 15 7 4 23 1980 - 1986® 0 20 0 8 8 9 13 15 4 22 a/ Deletes 9 observations including January* February, April, May and August of 1980, May of 1981, June of 1982, June of 1984 and April of 1985. 113 6.4 Construction of The Basic Set of Six Commodity Groups According to the clustering hierarchical structure presented in Figure 6, the 19 categories of food expenditure can be separated into at most 14 commodity groups. However, the Indication from the previous rank test results for 19 categories shows that the rank of our data is likely to be three. This implies the lower bound of the number of commodity groups for data used in this study to be three. Taking consideration of computational burden in parametric analysis and rank condition, we will construct a data set with six commodity groups based on the clustering results. In addition, we will compare this grouping with another classification of six food groups conventionally used by BLS. Note taht based on our clustering hierarchical structure, six commodity groups are slightly modified. Specifically, the final six groups of food categories are defined as: (a) cereal, bakery products, processed fruits, oils and miscellaneous foods; (b) beef, pork, other meats, eggs and processed vegetables; (c) poultry, milk products and sweets; (d) seafood and other dairy products; (e) fresh fruits, fresh vegetables and non-alcoholic beverages; and (f) food away from home. On the other hand, according to the BLS's classification, their six commodity groups are (a) cereal and bakery products; (b) beef, pork, other meats, poultry, seafood, and eggs; (c) milk products and other dairy products; (d) fresh fruits, fresh vegetables, processed fruits, and processed vegetables; (e) sweets, non-alcoholic beverages, oils and miscellaneous foods; and (f) food away from home. As discussed earlier, one way of justifying the construction of a two-stage budgeting is based on separability conditions. More specifically, according to the clustering results, the 19 food commodities in this study can be partitioned into six broad commodity groups, if the utility function can be expressed as U(xlfx. t ^ 1 9 ) = C^( (Xj i Xj t ^j.3 f ^ 1 7 t X^j) i v2 (Xj t Xj i Xj i X q t x ^ ,)t (77) ^ 3 (Xs ’ "^9 ’ ^3.5) r ^ 4 (Xj 1 Xh q ) , V5 ( X j j § • X^ j ) / V g ( X y ) ) where U(.) and V,(.) satisfy the conditions of a well-behaved utility function. With this structure, we can interpret V, as a quantity index of i-th commodity group. The indirect utility function corresponding to Eg. (77) can be written as 78) Af Note that the function V, is homogenous of degree minus one in H and of degree one in p {. Thus Eq. (78) can be rewritten as P ± ( P i • Pz • • • tPlt) M M M where P1 (i=l,2,..,6) can be thought of as a price index for 1-th commodity group. In order to test whether the functional structure of Eq. (77) is consistent with our data with six food groups, we can perform a two- stage GARP testing. At first stage, the subutility functions of Vf(.) are examined, on the basis of nonparametric test, to see whether or not these functions are consistent with utility maximization hypothesis. Then, the same consistency test can be performed for checking whether these six aggregate groups can be each rationalized by a well-behaved utility function. 115 The results of the first-stage test for existence of subutility function using both the clustering groups and the BLS classifications are presented as Table 19. Several interesting findings are revealed (a) The total numbers of violations are higher for the set of grouping used by BLS than that based on cluster analysis, (b) The number of violations are higher for the 1980-1985 period than for the 1981-1986 period. In fact, the clustering groups results in only four violations when the I sample period of 1981-1986 is used, (c) The meat group (Group B) classified by the BLS has many violations with the GARP especially when the sample period extends to 1986. This is an indication that all these six meats categories may not belong to the same group with a well- behaved subutility function. An aggregate index for aggregating quantities and prices over commodities, must be formulated for conducting the second-stage GARP testing. The most commonly used index formulas are Paasche, Laspeyres, Fisher, and Divisia indexes. Fisher and Divisia are superlative indexes15. Diewert (1978) showes that the superlative Indexes are numerically consistent in aggregation to the second order under certain conditions. Moreover, the Divisia index has been shown to track any aggregation following a theoretical exact aggregator function with no error at all in continuous time series. This remarkable result holds, no matter what the form of the unknown exact aggregation function may be, so long as the assumptions necessary for the existence of an exact aggregator function hold (Barnett and Serietis,1990). For an overview of index number theory, see Diewert (1976,1980). For this reason, Divisia index is adopted to compute aggregate indexes for six food groups. 116 Table 19. Comparison of GARP Test Results Between Cluster and BLS Groupings Expenditure GrouDina bv Items Cluster Analysis BLS Classification 1. Cereal AA 2. Bakery A A Products 3. Beef B B 4. Pork B B 5. Other Meats B B 6 . Poultry C B 7. Sea Food D B 8 . Eggs B B 9. Milk Products C C 10 Other Diary D C 11 Fresh Fruits E D 12 Fresh Veg. E D 13 Processed Fru. A D - 14 Processed Veg. BD 15 Sweets C E 16 Nonalcoholic E E Beverage 17 Fat and Oil A E 18 Misc. Foods AE 19 Food Away FF from Home Sample period Number of Violations 1980 - 1985 2 2 12 9 2 0 0 2 17 6 14 0 1981 - 1986 0 4 0 0 0 0 0 16 4 6 6 0 1980 - 1986° 0 0 0 0 0 0 0 24 8 4 6 0 a/ deletes 9 observations including Jan. Feb. Apr. May and Aug. of 1980, May of 1981, June of 1982, June of 1984 and April of 1985. 117 The Divisia price index in discrete time is based on Tornqvist approximation and calculated using the following formula: log(Pt) -log (iVi) - ■§■£ [log (pit) -log (pit.x) ] (80> where Sit=(pitxit)/E(pu x?t), which is the budget share of commodity 1 at time t. It is noted that the growth rates of the aggregate is the weighted average of the growth rates, using budget shares as weights. After Divisia indexes are calculated, the GARP test is again employed to test consistency for six aggregate food groups. Table 20 presents the test results. The results show little differences between the two groupings at the aggregate level. The BLS grouping has slightly more violations for the sample of 1980-85. It is somewhat surprising that the total number of violation for the data base constructed from the clustering structure Is identical to the 19 category data base in Table 9 for the 1980-1986 period with 9 observations excluded. That implies that data structure may not be affected by using Divisia index for aggregation over commodities, which is consistent with the aggregation property of Divisia index. This evidence strongly indicates that this data set is fairly consistent with commodity aggregation Barnhart and Whiteney (1988) demonstrate that a nonparametric analysis can be used to improve parametric estimation. They found evidence that the performance of parametric estimation of a demand system in terms of the number of violations to monotonicity and/or convexity was improved by dropping outliers identified by a nonparametric analysis. Taking consideration of the consistency test 118 Table 20. Results of Consistency (GARP) Test for Six Food Groups Number of Violations Sample Periods Data T Data IIb 1980 - 1985 6 9 1981 - 1986 2 2 1980 - 1986c 4 2 a/ Data I is six food groups classified by cluster analysis; b/ Data 11 is six food groups classified by BLS. c/ Deletes 9 observations including Jan. and Feb. of 1980, and December of all years. 119 results reported here, the six commodity groups from the clustering result are therefore employed as the basic groups for further parametric estimation. Following Barnhast and Whiteney, we will also delete those observations (early months of 1980) which exhibit violations in our nonparametric analysis. Consequently, we will only use monthly observations from September 1980 to December 1986 in the sample. As mentioned earlier, the rank condition of a demand system can provide valuable information for model specification. To examine the rank condition of the basic six food groups, a nonparametric rank testing procedure is further employed. Lewbel (1990) indicated that the error in the rank estimation procedure is minimized if Q(Z) in Eq. (11) contains the actual G(Z) functions. Therefore, we choose several commonly used Engel specifications including 1, M, log(M), 1/M, M2, and log(M)2 for our testing (M is total expenditure). Each element of Q(Z) function was divided by its sample mean to ensure that Y (Y*E(WQ(Z)') in Eq. (12) was not ill conditioned as a result of the enormous range of magnitudes in the functions comprising Q(Z) (Lewbel, 1990). The first four largest pivots, which are diagonal elements of D in the LDU decomposition of Y, are 0.355, 0.003, 3.885*10*6, and 2.102*10*7, respectively. Table 21 presents the results of applying the LDU rank test to the estimated pivots. The result clearly shows a fairly small probability (1.545*10'9) that all but one pivots are zero, but a relatively high possibility (about 100%) that all but two pivots are zero. Note that the X2 statistic drops dramatically from K=1 to K=2. The rank tests are sequential, and the test for each rank is only consistent against greater values of the rank. Therefore, these results indicate, 120 Table 21. Results of Nonparametric Rank Test for Six Food Groups Test H0: Rank <« i 1 Chi-square ~ P-valued 1 93.45 1.545*10*9 2 0.00023 1.00000 3 3.79*10’5 1.00000 4 2.45*10-'5 1.00000 121 with high precision, a rank of two for the demand system of the basic six food groups. 6.5 Concluding Remarks The main merit of using a nonparametric procedure in demand analysis is that it is concerned with the information contained in the data themselves and makes no assumption of the specific form of the I underlying utility function. Therefore, the misspecification error can be alleviated. Several manifest and important findings from the nonparametric analyses on data consistency, data structure and commodity groupings are summarized as follows: (1) The data series for the 19 food categories appear to be consistent with the utility maximization hypothesis. Meanwhile, this group of food at home and food away from home may consist of weakly separable subgroups. However, the 18 expenditure categories for food at home can satisfy the consistency testing (GARP) only under a special condition of excluding December observations. (2) The rank test shows that the rank of the demand system in this study is at most three while the actual rank is equal to two if eggs and nonalcoholic beverage are excluded. (3) Based on clustering analysis, we can partition 19 food categories into six broad commodity groups. The GARP testing also shows that these six basic commodity groups can each be rationalized by a well-behaved utility function. Furthermore, the nonparametric rank test results indicate a rank of two for the demand system of the basic six food groups. These nonparametric results provide useful information for formulating the parametric analysis. However, two questions concerning the use of nonparametric procedures in this study must be mentioned as they may lead to more works in future studies. One is that the GARP testing procedure used in this study is a deterministic and heuristic test. In other words, it is an "all or nothing" type of test, which is not a statistical hypothesis testing procedure. Data can pass the GARP test only if all data points are completely consistent with the hypothesis. In case of data with measurement errors, or other stochastic factors, the GARP test will erroneously reject the hypothesis because it is not a true violation. Therefore, the GARP test may fail to detect the actual structure changes in preference. Being a deterministic test type, we, therefore, can not evaluate the power of test and confidence intervals. In fact, Varian (1985) developed another GARP testing procedure, which is a stochastic test, taking into consideration measurement error and other stochastic influences. However, from practical points of view, there is a limitation in appling this procedure in empirical demand study because it requires a priori the known error variance structure which is usually unknown. Accordingly, further work is necessary to develop a statistical procedure to deal with test power, measurement error, and other kinds of stochastic factors. These development would be a challenging area of research in nonparametric demand analysis. Another question is on the use of multivariate statistical techniques in commodity grouping. In this study, we attempt to apply the concepts of household production theory and employ a factor analysis and 123 a clustering procedure to partition 19 food categories into several "similar" commodity groups. We further use the consistency testing procedure to verify whether or not the groupings generated by multivariate statistical analysis satisfy the utility maximization hypothesis. The estimation methods used in this study are principal component factor analysis as well as the hierarchical clustering procedure. Indeed, several other alternative estimation methods can also be used. Different estimation methods may yield different commodity grouping results. Therefore, the commodity grouping may be sensitive to the chooice of estimation method. Unfortunately, there is no single theoretical criterion to judge which method is the best. How to choose an appropriate procedure to conduct commodity grouping is still unsettled in empirical demand analysis. Due to the fact that separability plays an important role in empirical demand analysis, especially for a large demand system, further evaluation of methods for commodity grouping is an interesting and critical area of research in future demand study. CHAPTER VII RESULTS FROM PARAMETRIC ESTIMATION The primary objective of this chapter is to present the empirical results obtained from parametric analyses. First, a model specification test is conducted. Second, based on model specification test results, the most appropriate demand system of the PIGL06 type, is thus selected. To explore and estimate all flexible functional forms for 19 food categories is not possible because of insurmountable problem in nonlinear estimation. We therefore take a more practical approach to estimate the LA/AIDS for 19 food categories under one-stage budgeting procedure. Seven alternative versions of the LA/AIDS model are estimated. Under this procedure, of course, no separability assumptions are required. Third, three PIGLOG formulations including the Lewbel's full model, indirect translog, and LA/AIDS are employed to estimate structure parameters under a two-stage preference structure. Fourth, the comparisons of elasticities are presented. Finally, the predictive performance of seven LA/AIDS models for 19 food categories is evaluated on the basis of five statistical criteria. 7.1 Model Specification Test To explicitly specify a functional form for parametric analysis, a general Engel form like Eq. (36) in Chapter V is thus employed. Note that this general Engel specification is nonlinear in exogenous 124 variables. However, for a given value of X, the budget shares are linear in unknown parameters (a,,b{,c() and each parameter appears in just one equation. For implementation we first set X=0 and X=1 which we are interested in and employ seemingly unrelated regression (SUR) to estimate the parameters in the system. The SUR estimates of two special cases give us fairly good initial values in the iterative algorithm to estimate the parameters in the general case. It is well known that, if each equation in the system has the same exogenous variables, then the estimates obtained from ordinary least squares (OLS) identical to those from the maximum likelihood method (ML) or SUR in spite of the error terms being correlated (Theil, 1971,P.309; Johnston, 1984, P.338). The iterative algorithm we used was the Davidon-FIetcher-Powel1 algorithm in SHAZAM, which is a modified Newton-Raphson technique. The estimates are obtained by minimizing the logarithm of the determinant of the variance-covariance matrix of the residuals. In addition, we conduct a likelihood ratio test for the nested hypothesis that all c{ are zero in the system. Table 22 presents the estimation results and chi-square test statistic. The estimates of A's for the general model and PIGL are both insignificant at a conventional level a=0.05. These estimates suggest the specification of log form is superior. Moreover, the chi- square statistic reveals that the null hypothesis, i.e., all c{ being zero, can not be rejected for all three nested hypotheses. Based on these empirical results, we can conclude that the most appropriate model for this study is the PIGLOG demand system. 126 Table 22. Results of Specification of Engel Function for Six Food Groups Estimated or Restricted Systems X c, In L fla R2 X2b 1. General 3.09*10’6 NA 1261.12 -49.221 0.525 4.26 Model (4.38)c 2. Quadlog 0 NA 1261.12 -49.221 0.525 4.26 3. QES 1 NA 1261.10 -49.220 0.525 5.72 4. PIGL -2.626 0 1259.84 -49.171 0.501 (1.83) 5. PIGLOG 0 0 1259.05 -49.163 0.497 6. LES 1 0 1258.35 -49.143 0.487 a/ The values of log of determinant of covariance matrix of the residuals. b/ X = -N(log(l-R )), N denotes sample size. c/ Figures in the parenthesis are asymptotical standard error. NA * Not applicable. 127 7.2 Specification of PIGLOG Demand System The preceding model specification test results suggest that the PIGLOG demand system is likely to be most appropriate for this study. Therefore, a PIGLOG consumer model in budget share form is adopted for our parametric estimation of demand parameters. Following Lewbel (1989), the PIGLOG demand model can be expressed as : w _ ai+ci^+J^i (cf+aV+0. 5VfcV) - (cil+b1 (l+v'cl)) Zi 1 " l+v'cl where a and b are vectors, d is a scale, c is a symmetric matrix, 1 is a vector of ones; V is a N-vector with component of vf and v,«1og(p{); and Z=log(M). The adding-up, homogeneity and symmetric conditions require that a'l*=0, b'l=0, and F cl= 0. Lewbel (1989) shows that these budget shares are derived from the following indirect utility function in log form: log (U(V,Z)) = b'v+logid+a'v+O.SV'cV-ia'l+V'cDZ) <82) It is noted that two widely known PIGLOG models, translog demand system (TL) and almost ideal demand system (AIDS), are nested within Eq. (94) as special cases. Specifically, the restriction b f»0 for all i reduces the system to the exactly aggregatable translog model, whereas the restriction c/1-0 for all i leads to the AIDS model. These restrictions can be tested to assess the adequacy and relative explanatory power of the AIDS and TL for a particular data base. In empirical studies, the AIDS has often been estimated using a simple linear approximation to avoid nonlinearity of the system (see, e.g. Deaton and Muellbauer 1980). This approximation essentially amounts to replacing the term d+a'V+0.5V'cV in Eq. (81) with some mechanical price Index such as the Stone index defined as logPVEwjtlog(pJt). Deaton and Muellbauer (1980) note that In most cases the approximation Is fairly close, particularly if wide variations In prices do not occur In the sample period. Anderson and Blundell (1983) also provide the evidence that the use of a Stone index has little effect on the value of the log likelihood function. Other studies have noted similar results especially in the area of food consumption (B1and fort1 and Green, 1983). Initial attempt was to estimate the original version of the Lewbel's general form and AIDS. But convergency failed to satisfy after 300 Iterations. Consequently, in this study we follow others in using the Stone index to provide a linear approximation to the term, d+a'V+0.5V'cV» in the original specification. In order to avoid simultaneity problem, the Stone's index is computed by using lagged budget shares as suggested by Eales and Unnevehr (1988). That is, v'-iogP'-Ewjit.,Pj,t. Note that the data used in this study is aggregate data (over households). Muellbauer (1975) indicates that aggregation over different expenditure levels, M,, in a PIGLOG model results in aggregate budget shares that depend on E(Hilog(Mi))/E(Hi), which 1s usually called a "representative" total expenditure level, Instead of M,. Furthermore, Lewbel (1990) shows that If PIGLOG demand functions W*»a(V)+b(V)log(M), satisfy the equality constraints implied by utility maximization then W«a(V)+b(V)(k+log(M)) also satisfies these constraints for any constant 129 k. Therefore, following Lewbel, let us set Mt« Eh(^). and Wt«=Eh (WhtMht )/Mt» w ^ere Eh denotes the averaging operator across households. Mt is average (per household) total expenditures and Wt is the vector of aggregate budget shares in the econonjy. Define M*t by log(M*t)«{Eh(Hhtlog(Mht)/Mt)-k for some constant k. Summing up the PIGLOG demand functions and being weighted by Mht/Mt over households (h) gives Wt=a(Vt)+b(Vt)(k+log(M*t). Under the assumption of all agents maximizing utility, if the distribution of M is such that, over time, Mt « M*t for any constant k then macroeconomic demands will resemble those of a utility maximizing representative consumer. Since we will use aggregate data obtained from households having different total expenditure levels, M,, we first examine the average aggregation error of using M, weighted average total expenditures, rather than EtMjlogtMjJJ/EfM,-). We construct the Mt and Eh(Mhtlog(Mht)) for each month t as the sample mean over all households (h) of Mht and Mhtlog(Mht), respectively. For the sample period, we have 84 observations, using this information to run the following regression: . J?+(Sin(Me) +ec (83) The resultant R2 is 0.9575 implying the average aggregation error of using Mt in pi ace of M*t is less than 5%. This result indicates that the distribution of total consumption expenditure for food commodities during the sample period is relatively stable. For the remaining estimation of structure parameters in the demand system, the weighted monthly average expenditure level, Mt, will be used. i 130 In order to incorporate the Impacts of demographic variables into the system, we follow a demographic translating procedure developed by Poliak and Wales (1978). Accordingly, the a, parameters are specified as a linear function of household characteristic variables, Dh: a i s aio+j£anPh (84) Two demographic variables, household size and age of household head, are considered in this research. The other demographic variables can not be averaged across households on a monthly basis. Furthermore, many researchers argue that demand models should incorporate or test for dynamic behavior of consumer (see, e.g. Anderson and Blundell (1982, 1983), Blanciforti, et al. (1986)). In fact, several previous demand studies have shown that a dynamic form of demand system provides a better approximation to consumer behavior than a static formulation. These studies include Manser (1976), Pope (1980), Lamm (1982), Anderson and B1undel1 (1983), and B1ancifort, et al. (1986). The dynamics is often assumed to reflect persistence in consumption pattern and to capture the changes in taste over time. Several alternative approaches have been used to involve the dynamic factors of consumer behavior into the model specification (see, Johnson, et a l., 1984). In this study, a trend variable (T) and a 1agged quantity variable (Qt_1) are included to capture the impacts of persistence in consumption pattern and/or changes in taste over time. Consequently, using the Stone's index and incorporating the demographic variables as well as dynamic factors, the share equations, Eq. (81), can be extended and expressed as: 131 wi=taio+aii£>£-i+ai2Di+ai5I?2+ai4r+Jbiv,*+cir- (85) (c^i+Jbjd+t^cl)) Z] [l+V'cl] ~1 where V* is Stone's index. The adding-up restriction requires that E a i0«=l; £ a {1«0; X a i2«0; E a S3=0; and E a i4“0. Eq. (85) is called Lewbel's full model or simply the full model. Preliminary estimation of Eq. (85) for 19 food commodities in one stage reveals that the model is too large to handle. It would be an unmanageable task to estimate the full model due to nonlinearity in estimation. The only possibility to estimate a system for 19 commodities together in one stage is to restrict c'l=0 to avoid the troublesome of nonlinear estimation. This imposition of c'l=0 in the Eq. (85) reduces the model to a linear approximation AIDS (LA/AIDS). This restricted version of Eq. (85) can be rewritten as: (8 6 ) = aiO+ailQt-l+al2Dl+alp2+auT+jL CiJVi+t>i Accordingly, the LA/AIDS model will be utilized first to estimate the structural parameters under one stage preference structure while the full model and two nested models, LA/AIDS and TL, are employed to estimate the structural parameters under a two-stage budgeting procedure. For empirical estimation, a disturbance term (|t) is added to Eqs. (85) and (86) to obtained a stochastic form for an aggregate PIGLOG demand system. To allow for serially correlated errors, we further specify the error structure following the first-order autoregressive scheme, that is,Et « c5t-i+et» t-2,3,..,T. 132 7.3 Estimation of The LVAIDS for 19 Food Categories in One Stage This section presents the estimation results of LA/AIDS demand system for 19 food categories under the one stage structure. In order to evaluate the sensitivity and robustness of parametric estimates, various specifications of the LA/AIDS model are compared. Note that a dummy variable is included in the model specification to take into account of observations in violation with GARP under the previous nonparametric ? testing. Specifically, we define D is a dummy variable having a value of one for months of Hay of 1981, June of 1982, June of 1984 and April of 1985, and zero otherwise. This variable is added to Eq. (85) with a parameter a{5. The seven alternative specifications of the LA/AIDS are classified into two groups as follows: Static models: I - including the first order autocorrelation, AR(1), family size (FS), age of household head (AGE),and dummy variable (D); II - including FS, AGE, and D; III - including AR(1), and D; Dynamic models: I - including AR(1), FS, AGE, D, 1agged quantity variable (Qt.,), and time trend (T); II - including AR(1), Qt ., FS, AGE,and D; III - including AR(1), T, FS, AGE, and D; IV - including AR(1), Qt.1} and D. Since the LA/AIDS model plays a prominent role in the empirical analysis, an interpretation about this model is appropriate and helpful. To begin with the budget share in Eq. (86) is expressed in terms of intercept, dynamic factors, demographic factors, one dummy variable, prices and real expenditures. The intercept term, ai0, represents an average budget share when all logarithmic prices and real expenditure, 133 dynamic factors, demographic variables and a dummy variable are equal to zero. The effects of habit and changes In taste on budget allocation are reflected In the coefficients of a{1 and ai4, respectively. If both coefficients of habit and taste variables are statistically insignificant then the dynamic model is the same as static model. The effects of demographic variables on food budget allocation are reflected in the coefficients of a{2 and a{3. The b{ portrays the change in the 1th budget share with respect to a change in real expenditure holding other independent variables constant. Note that the coefficients of b, sum to zero, and a negative b{ implies that the commodity is a necessity, i.e., q,='l-i*(bf/w i) < 1 , while a positive one indicates a 1uxury commodity. Thus the expenditure share, w(, will increase with an increase In total expenditure for b{>0 while the opposite will be true for b,<0. The Cfj, price coefficients, delineates the change in the ith budget share for a given proportional change in price j with other independent variables being held constant. Note that the number of parameters in the LA/AIDS demand system for 19 commodities is quite large. For instance, the number of parameter to be estimated for Dynamic Model II with homogeneity, symmetry and adding-up restrictions imposed is 280. 134 7.3.1 Regression results A careful comparison of all regression results from seven alternative specifications indicates that Dynamic Model I provides the most piausible results and it is used as the basic model for further discussion and comparison. Table 23 presents the estimated regression results of this model. The regression results of other alternative LA/AIDS specifications are not presented in this dissertation except the results of Dynamic Model II are reported in Appendix C. The coefficients of the habit effects show that only three out of eighteen coefficients are statistically significant at e=5%. The coefficient for milk is positive while fresh fruits and sweets have a negative sign. The positive coefficient ref1ects persistence in budget share allocation while a negative coefficient implies that past purchases tend to lower current budget allocation. Since the choice of one month lag is subjective, we also try a yearly lag structure, i.e., using the quantity 1agged for twelve months as an independent variable to capture the habit effects. The results show no improvement over the original results. The coefficients of a{2 ,age of head of household, measure the effect of the stage of family life cycle on budget allocation. This coefficients is significant and negative for beef and is significant but positive for poultry and eggs. These results reveal that elderly family is likely to allocate more budget in purchasing poultry and eggs but less in beef than young family. The effects of family size on budget allocation are measured by ai3. The estimates of this coefficient show that only one of eighteen commodities, miscellaneous foods, is highly significant. The coefficients of ai4 reflects the time trend effects Table 23. Regression Results of The LA/AIDS Model ______w i t h AR ( 1 ) , 0 r AGE, FS, T. and Da ______Commodities ______Parameters” 1 2____ |____ 4____ §____ §____ 7____ 8 ____ 9 af0(10 ) 04 5765 7758 F795 7712 04 Ol 06 T J f (0.05)c(2.84) (1.43) (0.71) (1.54) (0.30) (0.68) (3.02) (1.55) a{1 0.26 -0.04 -0.43 -0.13 0.41 1.96 -0.67 -0.52 0.58 (1.10)(-0.59)(-1.04)(-0.33) (0.98) (3.13)(-0.85) (1.74) (2.48) ai2(10*4) 0.46 2.04 -18.55 2.22 -1.14 4.17 3.36 1.48 -1.24 (0.33) (0.77)(-2.89) (0.71)(-0.52) (2.30) (1.37) (2.09)(-0.62) ai3(10 ) 3.03 0.04 14.80 8.36 2.03 -0.48 2.25 -0.24 3.94 (1.41) (0.01) (1.55) (1.77) (0.62)(-0.18) (0.63)(-0.24) (1.31) a,,(10*3) 1.77 -0.00 -3.47 -1.71 0.50 -0.44 -0.01 -0.53 1.85 (1.21)(-0.12)(-1.85)(-2.09) (0.26)(-0.56)(-0.06)(-1.78) (0.97) ai5(10*3) 2.78 5.00 -7.52 -1.50 2.04 -0.73 1.02 -0.29 1.42 (2.00) (1.85)(-1.17)(-0.48) (0.94)(-0.41) (0.43) (0.73) (1.90) b,(10 } -0.12 0.23 -2.13 0.09 -0.51 0.45 -1.10 0.06 -0.96 (-0.31) (0.31)(-1.21) (0.11K-0.86) (0.86) (-1.57) (0.29)(-1.76) c „ d -2.01 (-0.53) c >2 0.12 0.63 (0.78) (2.30) C13 0.04 -0.22 4.26 (0.03)(-0.37) (0.97) cIA -1.12 -0.42 -1.43 0.82 (-1.20)(-1.50)(-0.93) (0.90) Cje 6.46 -0.47 0.69 1.48 -3.87 (2.41)(-2.02) (0.38) (1.24)(-0.94) ci6 0.22 -0.05 0.27 0.82 -0.59 1.53 (0.24)(-0.28) (0.22) (1.39)(-0.52) (1.77) c,, 3.56 -0.54 -0.04 -0.59 0.33 0.97 -2.41 (2.69)(-2.22)(-0.03)(-0.85) (0.26) (1.36)(-2.02) C:a 0.25 0.05 -0.64 0.35 -0.16 0.13 -0.16 0.87 (0.81) (0.66)(-1.42) (1.58)(-0.40) (0.51)(-0.64) (7.42) C19 -4.99 -0.10 -1.51 -1.33 3.66 0.67 1.38 0.31 1.96 (-1.66)(-0.46)(-0.88)(-1.22) (1.16) (0.55) (1.03) (0.71) (0.41) c il0 2.24 -0.40 -4.03 1.25 -2.57 -1.60 1.11 0.22 5.89 (0.54)(-1.68)(-2.10) (0.94)(-0.64)(-1.18) (0.68) (0.47) (1.22) cni -0.23 0.26 -0.79 1.10 -0.43 0.57 0.55 0.19 -0.00 (-0.35) (1.17)(-0.70) (2.11)(-0.50) (1.22) (0.96) (1.06)(-0.01) ci12 0.16 -0.45 0.49 -0.19 0.39 -0.59 -0.64 -0.14 0.35 (0.48)(-2.76) (0.58)(-0.551 (0.80)(-1.83)(-1.69)(-1.23) (0.78) c i13 4.17 -0.04 1.07 -0.58 -0.80 -0.29 -1.52 -0.24 1.49 (3.08)(-0.25) (0.96)(-0.97)(-0.50)(-0.43)(-2.00)(-1.01) (0.95) cfU 0.26 0.27 0.46 0.02 0.93 0.27 -0.72 0.15 3.79 (0.12) (1.72) (0.38) (0.03) (0.42) (0.35)(-0.78) (0.55) (1.56) Cj.c -0.98 0.46 4.20 -1.12 -0.19 -0.15 -1.79 -0.35 0.05 (-0.71) (1.64) (2.23)(-1.24)(-0.11)(-0.18)(-1.78)(-1.13) (0.03) C{*£ 3.32 0.13 -3.26 -1.18 1.55 0.25 -0.91 -0.31 2.57 (1.88) (0.37)(-1.44)(-1.08) (0.67) (0.21)(-0.63)(-0.70) (1.13) ci17 -3.89 0.11 -0.55 0.11 0.79 1.05 -0.40 -0.02 -1.38 (-0.42) (0.90)(-0.61) (0.21) (0.70) (1.79)(-0.68)(-0.08)(-1.24) c i18 4.34 0.35 -0.74 1.81 6.48 -4.37 -0.83 0.65 6.68 (0.97) (0.92)(-0.25) (1.03) (1.28)(-2.26)(-0.36) (0.97) (1.31) rho -0.21 (-4.91) R2 0.56 0.19 0.76 0.48 0.49 0.43 0.08 0.87 0.76 Table 23 (Continued) Commodities Parameters M. 11 12 J l. 11 15. 16 17 18. a j0(10'2) -5 18 5 86 3 32 425 572 1.41 2 64 1 12 8 51 (-2 71) (2 32) (1 78) (1 58) (0 46) (0.65) (0 61) (1 13) (1 13) *11 -0 .44 -1 .31 ■■0 .30 0 .20 0 .10 ■1.69 1 .26 0 .40 0 .40 (-1 89)(-2 52)(-0 71) (0 84) (0 39)(-2 32) (2 04) (0 70) (1 0 1 ) a l2(10'4) 1 77 1 14 -1 37 2 49 0 56 -1 44 3 38 -0 27 1 07 (0 8 6 ) (0 50)(-0 80) (1 72) (0 39)(-0 53) (1 00)(-0 24) (0 31) a„(10-3) 0 07 -0 02 4 99 2 38 3 45 3 92 0 31 1 19 13 50 (0 02)(-0 0 1 ) (1 91) (1 1 1 ) (1 62) (0 96) (0 06) (0 71) (2 62) a«(10'4) -5 30 -0 04 0 19 0 16 2 19 2 87 0 96 -0 00 8 97 (-2 18)(-0 60) (0 45) (0 17) (1 6 8 ) (2 56) (0 56)(-0 03) (2 92) a,5(10-5) 3 85 -4 62 -0 16 -0 26 -0 89 3 37 3 00 1 29 6 11 (1 90)(-1 97)(-0 09)(-0 19)(-0 64) 0 27) (0 91) (1 18) (1 84) b,{10'2) -1 19 -0 13 -0 00 0 15 -0 04 0 60 0 25 -0 30 0 59 (-2 15)(-0 2 0 ) (0 0 0 ) (0 40)(-0 1 1 ) (0 81) (0 28)(-0 99) (0 64) C !1 CI2 Ci3 Ci4 Cf5 CI6 Ci7 Ci8 Ci9 'f 10 -17 39 (-2 06) C f 11 2 41 0 27 (2 25) (0 44) C i 12 0 11 -0 24 1 00 21)(-0 79) 33) (0 (3o -0 76 0 21 1 28 -2 40 '113 O i 40) (0 43)(-0 94)(-2 01) -l 31 -0 44 0 55 0 93 2 71 '114 o (-0 1 74) (i 81) (0 83) (1 17) 'f 15 0 38 -2 01 -0 46 -0 19 1 15 0 54 (0 18)(-2 92)(-1 01)(-0 20) (0 90) (0 32) 'i 16 -4 67 -0 95 0 01 -2 71 0 78 7 45 -1 61 (-1 76)(-1 00) (0 01K-1 84) (0 52) (4 36)(-0 47) 0 46 -0 04 -0 02 0 90 -1 17 0 11 -0 11 1.97 to o '117 Pa“4 (0 34)(-0 10)(-0 10) (1 31)(-1 42) (0 1 11) (2!.92) -7 63 -0 93 1 80 0 47 4 76 1 75 6 86 4L 42 18.78 Cf 18 to o (-1 1 65) (2 29) (0 18) (1 32) (0 62) (1 77) (2!.24) (1.60) R 0 30 0 09 0 38 0 34 0 30 0 22 0 30 01.59 0.82 a/ AR(1) is first order autocorrelation, Qt_1 is lagged quantity variable, FS is family size, AGE is age of household head, T 1s time trend, and D is dummy variable, b/ Estimation method is ITSUR; c/ Figures in the parenthes are asymptotic t-ratio*, d/ All values of c{J (i,j«l,2,.,18) are presented by multiplying 10 with the symmetric condition (Cy«Cj,) only half of parameters need to be estimated, e/ The same auto correlation coefficient is assumed for every equation. 137 which can be thought of as taste changes. Among 19 food categories estimated, four time trend coefficients are statistically significant at a»5%. The signs of ai4 for pork and dairy products are negative while sweets and miscellaneous foods have a positive sign. The dummy variable is used to take care the observations with viol ation in nonparameteric testing. This variable has a significant coefficient in only two out of 18 equations. The estimated coefficient of expenditure effect, b{, for dairy products is highly significant with a negative sign, indicating a necessity for this food category. Other food commodities in the system with an insignificant real expenditure effect tend to have unitary expenditure elasticity. It is noted that if all food categories in the system have unitary expenditure elasticity then the corresponding utility function will be homothetic. If a preference is homethetic then doubling quantities would double utility, or in other words, the composition of the budget is independent of total expenditure or of utility. Consider next the price effects. Twelve of eighteen own price coefficients have the expected positive sign but only four of them are statistically significant at a=5%. For cross price effects only about 10% (16 out of 153) of cross price coefficients are statistically significant. Generally speaking, for a large demand system, this is a common result that the estimates of parameters are not always •statistically significant. However, the estimated elasticities are mostly statistically significant based on the computed asymptotic standard errors of elasticities as to be discussed later. 138 The coefficient of autocorelation of residual terms, rho, is statistically significant with a f-value of -3.56. Moreover, other five LA/AIDS models with AR(1) specification also show that the hypothesis of no autocorelation In the disturbance term 1s rejected. These evidences strongly support that we should incorporate the serial correlation scheme. Otherwise the model will be subject to misspecification errors. Finally, Table 23 also reports the goodness of fit criterion (R2). i R2 offers information complementary to the theoretical and statistical evaluation of parameter estimates. The R2 values range from 0.08 for seafood to 0.87 for eggs. The low R2 for some commodities may be due to a short run disequilibrium and/or supply constraints for those food items in a short time period. Both possibilities can occur, especially when monthly data are used. Another reason for low R2 in this study may be measurement errors or extreme values in data series. We carefully examined the original data series and found some observations to be outliers. For example, observations of expenditures for beef in February of 1982, for other meats in October of 1982, August of 1983, and for fresh fruits in May of 1986 (see the comparison of actual and estimated expenditure series for commodifies in the Appendix H) are either unusually high or low. However, the exact causes for a low R2 in several categories remain to be determined. To determine whether or not dynamic factors, demographic effects and autocorrelation should be included in the model, a chi-square test is used. Note that this test statistic based on Eg. (56) is computed only for those models being nested. Table 24 presents the test results. Three important findings are obtained from these test results. First, the dynamic factors are important. The test statistics of Model D1 versus three static models (SI, S2, and S3) all reject a null hypothesis of no dynamic factor in the system. Second, demographic variables are shown to have significant Impacts on food budget allocation. The test statistics of Model D2 versus D4 and SI versus S3 are 70 and 40, respectively. Both values are greater than the corresponding critical values at a*5%, thus indicating a rejection of the null hypothesis of no effects of demographic variables. Third, autocorrelation appears to be present. This result is confirmed by the test statistics, X2«40, of Model SI versus Model S2. Indeed, this result is consistent with the fact that the autocorrelation parameter is significantly different from zero in regression. Let us consider only those of dynamic models. The test results show that Model D1 is statistically different from 02 while Models 01 and D3 are not statistically different. Consequently, this test suggests that the dynamic model with first order autocorrelation and demographic variables is the most preferred specification. 140 Table 24. Test Statistics of Alternative LA/AIDS Model Specifications Hypothesis test X2 df* Model8 D1 vs. SI 78.98* 36 Model 01 vs. S2 119.89* 37 Model 01 vs. S3 154.89* 72 Model D1 vs. D2 53.89* 18 Model D1 vs. D3 25.56 18 Model D1 vs. D4 123.89* 54 Model D2 vs. SI 26.00 18 Model D3 vs. SI 54.33* 18 Model D 2 vs. D4 70.00* 36 Model SI vs. S3 75.00* 36 Model SI vs. S2 40.00* 1 a/ Model D1 Including first-order autocorrelation, AR(1), 1agged quantity variable (Qt .), time trend (T), family size (FS), age of head of household (AGE), Dummy variable (D); Model SI including AR(1), FS, AGE, D; Model S2 including FS, AGE, D; Model S3 including AR(1), D; Model D2 including AR(1), Qt_., FS, AGE, D; Model D3 including AR(1), T, FS, AGE, D; Model D4 including AR(1), Q*..,, D; b/ Degrees of freedom are equal to the difference in the numbers of parameters between the base model and the model in question. */ Significant at a * 5%. 141 7.3.2 Comparisons of Key Elasticities for 19 Food Categories One of the important objectives of a parametric demand analysis is to summarize the observed data by important demand elasticities. It is therefore important that own-price and cross-price elasticities, expenditure elasticity, and the elasticities for family size and age of household head are computed and evaluated. All elasticities presented in this section are computed using the estimated structural parameters and sample means of the associated variables. Table 25 compares the estimated uncompensated own-price elasticities obtained from various LA/AIDS specifications. The main findings from these results are summarized below: (1) As one can see, among seven LA/AIDS specifications, no one has all expected signs for uncompensated own-price elasticities. Every model has at least two elasticities with an unexpected sign. Fat and oil consistently have a wrong sign for its own-price elasticities even though they are not statistically significant (estimated asymptotic standard errors are discussed 1ater). Two possible explanations for these results. Either there are data problems or these food categories have an asymmetric functional form in the system. (2) Several food commodities including cereal, bakery, pork, other meats, poultry, seafood, fresh vegetables and processed fruits, have fairly stable uncompensated own-price elasticities among all LA/AIDS specifications. For other commodities, their uncompensated own-price elasticities are sensitive to model specification. This evidence indicates that price and expenditure variables can largely explain the consumption behavior of those commodities with stable uncompensated own- 142 Table 25. Comparison of Uncompensated Own-Price Elasticities Estimated for Different LA/AIDS Specifications® Static Model Dynamic Model Commodities I II III I II III IV Cereal -2.23 -2.04 -2.19 -1.71 -2.19 -2.11 -2.12 Bakery -0. 8 8 -0 . 8 8 -0 . 8 8 -0.90 -0.89 -0 . 8 8 -0.89 Beef 0.15 0.09 0.24 -0.39 0.24 -0.20 0.31 Pork -0. 6 8 -0.63 -0.70 -0.80 -0.64 -0.79 -0.65 Other meats -2.55 -2.40 -2.29 -2.37 -2.74 -2.07 -2.11 Poultry -0.56 -0.44 -0.59 -0.46 -0.42 -0.60 -0.49 Seafood -2.18 -1.97 -2.34 -2.15 -2.14 -2.12 -2.21 Eggs -0.10 -0.08 -0.06 -0.23 -0.17 -0.17 -0.11 Milk -0.58 -0.75 -0.17 -0.53 -1.55 0.19 -0.99 Other dairy -0.25 -1.30 0.07 -5.04 -0.60 -3.63 -0.08 Fresh fruit -0.89 -0.88 -0.82 -0.92 -1.05 -0.78 -1.01 Fresh veg. -0.70 -0.70 -0.70 -0.68 -0.71 -0.66 -0.71 Proc. fruit -1.93 -2.01 -2.04 -1.99 -1.99 -1.95 -2.06 Proc. veg. 0.29 0.29 0.27 0.51 -0.75 1.31 -0.80 Sweets -1.45 -1.09 -1.51 -0.78 -1.43 -0.76 -1.36 Nonal. bev. -0.96 -0.82 -0.90 -1.28 -1.18 -1.02 -1.07 Fat & oil 0.10 0.03 0.08 0.11 0.10 0.15 0.12 Misc. foods -1.17 -1.30 -1.70 1.70 -0.80 1.44 -0.88 FAFH -0.42 -0.39 -0.64 1.77 -0.39 2.35 -0.54 a/ Static Model I : including the first autocorrelation, AR(1), Dummy variable (D), family size (FS), and age of household head (AGE); Static Model II : including D, FS, AGE; Static Model III: including AR(1), D; Dynamic Model I : including AR(1), D, FS, AGE, 1agged quantity variable (Qt.«)»and time trend (T); Dynamic Model II: including AR(1), D, Qt.,, FS, AGE; Dynamic Model III: including AR(1), D, T, FS, AGE; Dynamic Model IV : including AR(1), D, Qt.r b/ FAFH= food away from home. 143 price elasticities for alternative model specifications. Unfortunately, these results also Indicate that for other food commodities, the demand structure may not be stable. (3) Comparisons with four dynamic models, reveal that alternative ways of incorporating dynamic factors, with a 1 agged quantity variable and/or a time trend, yield notable differences in the estimated uncompensated own-price elasticities for beef, milk, other dairy products, processed vegetables, sweets, miscellaneous foods and food away from home. (4) The incorporation of demographic variables affects the estimated own-price elasticities for milk and other dairy products (see Dynamic Models II and IV). (5) For beef, the uncompensated own-price elasticity has the expected negative sign only under the specification including the time trend variable. Beef appears to be one of the most troublesome category for which to estimate demand structure in this study. It is surprising to note that the seasonally unadjusted expenditures of beef have been declining during the sample period. (6 ) Use of the time trend as a dynamic factor results in unexpected positive own-price elasticities for milk, processed vegetables, miscellaneous foods, and food away from home. One possible explanation of this result is that the time trend variable may compete with other independent variables (especially the price and demographic variables) in the system and thus the effects of prices may be distorted. Table 26 presents the estimated expenditure elasticities from 144 Table 26. Comparison of Expenditure Elasticities Estimated for Different LVAIDS Specifications8 Static Model Dynamic Model Commodities I II III I II III IV Cereal 0.95 0 . 8 8 0.99 0.96 0.97 0.95 1.02 Bakery 1.00 0.90 0.98 1.02 1.00 1.03 0.98 Beef 0.78 0 . 8 8 0.98 0.79 0.81 0.81 0.96 Pork 1.06 1.09 1.14 1.02 1.05 1.02 1.15 Other meats 0.77 0.82 0.82 0.76 0.77 0.75 0.83 Poultry 0.94 1.08 0.95 1.05 1.10 0.90 1.06 ? Seafood 0.75 0.96 0.77 0.65 0.66 0 . 6 8 0.67 Eggs 1.02 1.01 1.01 1.05 1.01 1.03 1.00 Milk 0.73 0.76 0.78 0.76 0.75 0.71 0.80 Other dairy 0.74 0.68 0.73 0.71 0.73 0.73 0.73 Fresh fruit 0.93 0.86 0.91 0.96 0.95 0.96 0.92 Fresh veg. 0.99 1.00 1.06 1.01 1.01 0.99 1.08 Pro. fruit 1.05 0.97 1.09 1.10 1.07 1.10 1.09 Proc. veg. 1.01 0.94 1.07 1.01 1.00 1.03 1.08 Sweets 1.08 1.03 1.11 1.15 1.14 1.10 1.20 Nonal. bev. 1.01 0.92 1.03 1.01 1.01 1.02 1.02 Fat & oil 0.83 0.73 0.85 0.89 0.86 0.88 0.88 Misc. foods 1.06 1.05 1.15 1.10 1.09 1.08 1.19 FAFHb 1.13 1.15 1.04 1.11 1.11 1.13 1.02 a/ Static Model I : Including the first autocorrelation, AR(1), dummy variable (D), family size (FS), and age of household head (AGE); Static Model II : including D, FS, AGE; Static Model III : including AR(1), D; Dynamic Model I : including AR(1), D, FS, AGE, 1agged quantity variable (Qt .),and time trend (T); Dynamic Model II : Including AR(1), D, Qt ., FS, AGE; Dynamic Model III: including AR(1), D, T, FS, AGE; Dynamic Model IV : including AR(1), D, Qt_.,. b/ FAFH=food away from home. 145 various LA/AIDS specifications. The results show that the estimated expenditure elasticities appear to be quite similar among alternative specifications. It is noted that the range of expenditure elasticities is relatively smal1. In particular, cereal, bakery, eggs, fresh vegetable, fresh fruits, processed vegetables and nonalcoholic beverage have approximately unitary expenditure elasticities. For Dynamic Model II, the range of expenditure elasticities varies from 0.66 for seafood to 1.14 for sweets. Tables 27 and 28 present the estimated elasticities for age of household head and family size, respectively. Several interesting findings can be summarized as follows: (1) Generally speaking, the estimated elasticities for the age of household head and family size are fairly stable regardless of model specifications. (2) Estimated elasticities for family size are positive for all food categories expect food away from home. Among the 19 food commodities, beef, pork, processed vegetables and miscellaneous foods appear to have higher family size elasticities than other food commodities. (3) Food away from home is the only category with a strong negative elasticity of family size. This can be logically explained by household production theory. The 1arger the family size, the more likely household would consume food at home, perhaps, due to the economies of scale in food preparation at home. (4) Six commodities, i.e., beef, other meats, milk, fresh vegetables, sweets, and fat and oil, have negative estimated 146 Table 27. Comparison of Estimated Elasticities for AGE with Different LA/AIDS Specifications" Static Model Dynamic Model Commodities I 11 1 11 III Cereal 0.12 0.10 0.08 0.08 0.12 Bakery 0.12 0.04 0.16 0.18 0.13 Beef -1.43 -1.37 -1.18 -1.49 -1.18 Pork 0.13 0.19 0.25 0.10 0.29 Other meats -0.37 -0.31 -0.19 -0.40 -0.21 Poultry 0.70 0.74 0.69 0.67 0.70 Seafood 0.70 0.50 0.74 0.72 0.73 Eggs 0.38 0.31 0.60 0.45 0.53 Milk -0.21 -0.21 -0.13 -0.19 -0.15 Other dairy 0.09 0.05 0.19 0.12 0.18 Fresh fruit 0.21 0.21 0.16 0.24 0.16 Fresh veg. -0.13 -0.11 -0.20 -0.17 -0.17 Proc. fruit 0.49 0.50 0.47 0.48 0.45 Proc. veg. 0.16 0.16 0.14 0.15 0.13 Sweets -0.03 0.02 -0.27 -0.15 -0.12 Nonal. bev. 0.28 0.29 0.26 0.33 0.22 Fat & oil -0.09 -0.17 -0.07 -0.08 -0.06 Misc. foods 0.16 0.11 0.07 0.06 0.09 FAFH 0.06 0.08 -0.00 0.09 -0.00 a/ Static Model I : including the first autocorrelation, AR(1), dummy variable (D), family size (FS), and age of household head (AGE); Static Model II : including D, FS, AGE; Static Model III : including AR(1), D; Dynamic Model I : including AR(1), D, FS, AGE, 1agged quantity variable (Q,.),and time trend (T); Dynamic Model II ; including AR(1), D, Qt ., FS, AGE; Dynamic Model III: including AR(1), D, T, FS, AGE; Dynamic Model IV : including AR(1), D, Qt.r b/ FAFH=food away from home. 147 Table 28. Comparison of Estimated Elasticities for Family Size with Different LA/AIDS Specifications* Static Model Dvnamic Model Commodities I II I II III Cereal 0.30 0.33 0.28 0.25 0.28 Bakery 0.04 0.14 0.00 0.01 0.01 Beef 0.60 0.59 0.53 0.63 0.50 Pork 0.50 0.39 0.53 0.50 0.49 Other meats 0.24 0.26 0.19 0.22 0.20 Poultry 0.07 -0.00 -0.04 -0.03 0.08 Seafood 0.27 0.11 0.28 0.26 0.26 Eggs 0.01 0.04 -0.06 -0.03 -0.02 Milk 0.27 0.28 0.24 0.20 0.30 Other dairy -0.03 0.02 0.00 -0.02 -0.04 Fresh fruit -0.05 0.03 -0.00 0 . 0 0 -0.07 Fresh veg. 0.31 0.19 0.41 0.39 0.34 Proc. fruit 0.31 0.37 0.25 0.25 0.29 Proc. veg. 0.48 0.45 0.49 0.47 0.50 Sweets 0.29 0.35 0.42 0.41 0.27 Nonal. bev. 0.13 0.23 0.01 0.00 0.13 Fat & oil 0.21 0.21 0.17 0.20 0.18 Misc. foods 0.54 0.48 0.50 0.47 0.54 FAFH -0.51 -0.52 -0.47 -0.47 -0.48 a/ Static Model I : including the first autocorrelation, AR(l), dummy variable (D), family size (FS), and age of household head (AGE); Static Model II : including D, FS, AGE; Static Model III : including AR(1), D; Dynamic Model I : including AR(1), D, FS, AGE, 1agged quantity variable (Qt_«),and time trend (T); Dynamic Model II : including AR(1), D, Qt_., FS, AGE; Dynamic Model III: including AR(1), D, T, FS, AGE; Dynamic Model IV : including AR(1), D, Qt.r 148 elasticities for age of household head. This result from Dynamic Model II indicates that a 1% increase in age of household head would decrease the quantities of beef, other meats, milk demanded by 1.49%, 0.40%, and 0.19%, respectively. Among the 19 food commodities, beef has the 1argest negative elasticity with respect to the age of household head. (5) Poultry and seafood appear to have strong positive age elasticities. This may be due to the fact that elderly families tend to have more health concerns than younger families. Overal1 the estimated elasticities from Dynamic Model II are more plausible than other specifications. Note that this conclusion is not consistent with the previous finding from the model specification test which suggests Dynamic Models I and III to be more preferable. Since we are more concerned with the accuracy of the estimated elasticities, Dynamic Model II incorporating first order autocorrelation, 1agged quantity variable, family size and age of household head, and one dummy variable will be used for further analysis. 149 7.3.3 A complete demand elasticity matrix for 19 food commodities Many previous demand system studies concentrated only on the estimation of own-price and expenditure elasticities and paid little attention on Interdependent relationships among commodities in the system. One possible reason for this lack of alternation is that it is very difficult to evaluate the plausibility of the estimated cross-price elasticities. Theoretically, the cross-price elasticities can be positive or negative. There are no expected signs. However, one main merit of a system approach is to permit us systematically analyze the substitutability and complementarity among commodities in the system. Formulating a complete elasticities matrix will be useful to analyze food consumption pattern and consumer behavior. Table 29 gives a complete matrix of uncompensated price and expenditure elasticities and their asymptotic standard errors for 19 food commodities, obtained from Dynamic Model II. Other uncompensated price and expenditure elasticities matrices from alternative LA/AIDS model specifications are provided in Appendix F for reference. The signs of uncompensated cross-price elasticities indicate the pattern of substitutabi1ity/complementarity according to the Houthakker definition. As an estimated uncompensated cross-price elasticity is negative (positive), nfj<0 , commodities i and j are said to be specific complements (substitutes). As mentioned earlier, this definition Is based on the sign of a second-partial derivative of the utility function, or in other words, the element of the inverse of the Hessian matrix of the utility function. These partial derivatives are related to the nature of the interactions among goods in the utility function. The 150 Table 29. Uncompensated Price and Expenditure Elasticities Matrix for 19 Food Commodities Using LA/AIDS Model with AR(1), Qt.1t FS, AGE and Da Other Sea Dairy Cereal Bakery Beef Pork meats Poultry food Eggs Milk product (1) (2) (3) (4) (5) (6) (?) (8) (9) (10) 1 -2.19 t 0.04 0.14 -0.40 1.92 0.35 1.23 0.08 -2.99 1.44 (0.97) (0 .0 0 )(0 .11) (0.06) (0.52) (0.04) 0 .1 2 )(0 .0 1 ) 0.50) (1.08) 2 0.02 -0.89 -0.02 -0.07 -0.07 -0.01 0.09 0.01 0.01 -0.03 (0 .0 0 )(0 .0 0 )(0 .0 0 )(0 .0 0 )(0 .0 0 )(0 .0 0 ) 0 .0 0 )(0 .0 0 ) 0 .0 0 ) (0 .0 0 ) 3 0.06 0.00 0.24 0.12 0.33 0.05 0.08 -0.01 0.06 -0.51 (0 .0 2 )(0 .0 0 )(0 .1 0 )(0 .0 2 ) (0.03) (0 .0 1 ) 0 .0 2 ) (0 .0 0 ) 0.03) (0.-04) 4 -0.28 -0.10 0.19 -0.64 0.57 0.26 0.09 0.11 0.38 0.69 (0.03) (0.00)(0.05) (0.03) (0.05) (0 .0 1 ) 0 .0 2 ) (0 .0 0 ) 0.03) (0.05) 5 1.92 -0.15 0.83 0.84 -2.74 -0.20 0.15 0.04 1.66 -1.10 (0.52) (0.00}(0.19) (0.10) (1 .10) (0.06) 0 .10)(0 .0 1 ) 0.59) (0.99) 6 0.35 -0.03 0.10 0.37 -0.21 -0.42 0.34 0.01 0.41 -1.16 (0.04) (0.00)(0.08) (0 .0 2 ) (0.06) (0.04) 0.03) (0 .0 0 ) 0.06) (0.09) 7 1.68 -0.24 -0.26 -0.15 0.22 0.47 2.14 -0.07 0.98 0.43 (0 .2 2 ) (0.01)(0.22) (0.06) (0.18) (0.06) 0.18) (0 .0 1 ) 0 .2 0 ) (0.29) 8 0.19 0.06 -0.10 0.39 0.08 0.02 0.14 -0.17 0.45 0.85 (0.04) (0.00)(0.09) (0 .0 2 ) (0.05) (0 .0 2 ) 0.03) (0 .0 1 ) 0.06) (0.07) 9 -1.81 -0.01 -0.01 -0.29 0.70 0.33 0.44 0.10 0.99 1.56 (0 .2 2 ) (0.00)(0.08) (0.03) (0.25) (0.03) 0.05) (0 .0 0 ) 0.46) (0.72) 10 0.95 -0.03 -0.87 0.66 -0.72 -0.74 0.20 0.23 2.06 -0.60 (0.46) (0 .0 0 )(0 .10) (0.05) (0.42) (0.04) 0.07) (0 .0 1 ) 0.72) (1.93) 11 0.10 0.08 -0.38 0.30 0.02 0.09 0.15 0.03 0.25 0.30 (0 .0 1 ) (0.00)(0.04) (0 .0 1 ) (0 .0 1 ) (0 .0 1 ) 0 .0 1 ) (0 .0 0 ) 0 .0 2 ) (0 .0 2 ) 12 -0.03 -0.15 0.24 -0.08 0.08 -0.16 0.18 -0.04 0.04 0.10 (0.01) (0 .0 0 )(0 .0 2 ) (0 .0 1 ) (0 .0 1 ) (0 .0 1 ) 0 .0 1 )(0 .0 0 ) 0 .0 1 ) (0 .0 1 ) 13 1.59 -0.02 0.40 0.29 1.71 -0.18 0.61 -0.05 0.91 0.11 (0.19) (0.00)(0.15) (0.16) (0.29) (0.04) 0.07) (0 .0 1 ) 0.19) (0.29) 14 -0.93 0.14 0.39 -0.02 -0.15 0.60 0.32 0.08 0.10 0.71 (0.70) (0.00)(0.29) (0 .11) (0.79) (0.09) 0.15) (0 .0 2 ) 0.71) (1.35) 15 -0.95 0.14 1.71 -0.57 -0.87 0.37 0.65 -0.22 1.43 0.19 (0.19) (0.01)(0.30) (0.08) (0.25) (0.07) 0 .1 0 )(0 .0 1 ) 0.23) (0.36) 16 0.42 0.01 -0.79 -0.20 0.17 0.00 0.14 -0.02 0.37 -0.30 (0.04) (0.00)(0.07) (0 .0 2 ) (0.05) (0 .0 2 ) 0.03) (0 .0 0 ) 0.05) (0.07) 17 -0.17 0.07 -0.36 0.07 0.58 0.56 0.29 -0.61 1.37 -0.16 (0.16) (0.00)(0.17) (0.06) (0.21) (0.06) 0.07) (0.82) 3.71) (0.29) 18 0.30 0.00 0.05 0.30 -0.21 -0.25 0.16 0.03 0.13 -2.21 (0.23) (0.00)(0.10) (0.04) (0.25) (0.03) 0.05) (0 .0 0 ) 0.24) (0.44) 19 -0.18 0.00 -0.23 -0.16 -0.10 -0.09 0.07 -0.08 0.18 0.15 (0.04) (0 .0 0 )(0 .0 2 )(0 .0 1 ) (0.04) (0 .0 1 ) 0 .0 1 )(0 .0 0 ) 0.04) (0.08) 151 Table 29 (Continued) FreshFreshPro.Pro.Nonal. FatMisc.Expen. fruit veg. fruit veg. Sweets beve. oil food FAFH elasti (11) (12) (13) (14) (15) (16) (17) (18) (19) 1 0 10 0 08 -0 37 0 30 0.02 0.09 0.15 0.03 0 25 0 98 (0 0 1 ) (0 0 0 ) (0 04) (0 0 1 )(0 .0 1 )(0 .0 1 ) (0 .0 1 ) (0 .00)(0 0 2 ) (0 0 1 ) 2 0 04 -0 08 -0 01 0 04 0.06 0.01 0.02 0.01 0 04 1 03 (0 0 0 ) (0 0 0 ) (0 0 0 ) (0 0 0 )(0 .0 0 )(0 .0 0 ) (0 .0 0 ) (0 .00)(0 0 1 ) (0 0 1 ) 3 -0 16 0 12 0 14 0 10 0.58 -0.63 -0.09 0.08 -1 01 0 72 (0 0 1 ) (0 0 0 ) (0 0 2 ) (0 0 2 ) (0.03) (0.05) (0 .0 1 ) (0.09)(0 50) (0 .02) 4 0 23 -0 07 -0 09 -0 01 -0.33 -0.29 0.02 0.52 -1 37 1 05 (0 0 1 ) (0 0 0 ) (0 0 2 ) (0 0 2 ) (0.03) (0.04) (0 .0 1 ) (0.10)(0 50) (0 0 2 ) 5 0 03 0 09 -0 47 -0 10 -0.73 0.36 0.37 -0.51 -1 14 0 86 (0 0 2 ) (0 0 1 ) (0 16) (0 32) (0.19) (0 .2 2 ) (0.08) (1 .50)(6 17) (0 0 2 ) 6 0 10 -0 18 -0 16 0 39 0.32 -0.01 0.35 -0.62 -1 12 1 17 (0 0 1 ) (0 01)(0 03) (0 04) (0.05) (0.08) (0 .0 2 ) (0.18)(0 8 8 ) (0 05) 7 0 25 -0 25 -0 69 -0 26 -0.74 -0.35 -0.24 -0.49 1 33 0 51 (0 03) (0 0 2 ) (0 09) (0 1 1 ) (0.13) (0.26) (0.05) (0.60)(2 73) (0 05) 8 0 07 -0 12 -0 11 0 13 -0.45 -0.10 0.01 0.21 -2 34 1 04 (0 0 2 ) (0 01)(0 03) (0 04) (0.05) (0.09) (0 .0 2 ) (0.17)(0 82) (0 0 2 ) 9 0 19 0 04 0 52 -0 04 -0.79 0.52 -0.19 0.23 -1 33 0 79 (0 0 1 ) (0 0 0 ) (0 06) (0 1 2 ) (0.07) (0 .1 0 ) (0.03) (0.63)(2 87) (0 0 1 ) 10 0 23 0 08 0 07 0 30 0.12 -0.40 -0.06 -3.56 1 30 0 77 (0 0 1 ) (0 01) (0 09) (0 23) (0 .1 1 ) (0.13) (0.05) (1.14)(5 57) (0 0 1 ) 11 -1 05 -0 04 0 11 0 09 -0.33 -0.18 -0.01 0.56 -1 04 0 95 (0 0 2 ) (0 0 0 ) (0 0 1 ) (0 0 1 ) (0 .0 2 ) (0.03) (0 .0 1 ) (0.05)(0 30) (0 0 2 ) 12 -0 04 -0 71 -0 11 0 07 -0.25 -0.02 -0.00 0.22 0 02 1 00 (0 0 0 ) (0 0 0 ) (0 0 1 ) (0 01) (0 .0 1 ) (0 .0 2 ) (0 .0 0 ) (0.03)(0 16) (0 0 1 ) 13 0 14 -0 14 -1 99 0 28 -0.32 -1.07 0.31 -1.28 1 53 1 08 (0 0 2 ) (0 0 1 ) (0 16) (0 14) (0 .1 1 ) (0.19) (0.06) (0.65)(2 53) (0 0 2 ) 14 0 15 0 12 0 37 -0 75 -0.37 -0.10 -0.59 0.48 -0 73 1 00 (0 03) (0 0 2 ) (0 26) (0 89) (0.30) (0.39) (0.14) (2.24)(8 49) (0 03) 15 -0 45 -0 34 -0 33 -0 28 -1.43 2.55 0.02 -1.01 2 28 1 28 (0 04) (0 0 2 ) (0 1 1 ) (0 17) (0.29) (0.27) (0.06) (0.75)(3 29) (0 04) 16 -0 10 -0 01 -0 44 -0 03 1.05 -1.18 -0.02 0.55 -0 38 1 04 (0 0 1 ) (0 01)(0 03) (0 04) (0.05) (0.15) (0 .0 2 ) (0.17)(0 83) (0 0 1 ) 17 -0 01 0 00 0 43 -0 59 0.04 -0.05 0.10 2.35 -2 95 0 83 (0 0 2 ) (0 01)(0 1 1 ) (0 14) (0 .1 1 ) (0 .2 0 ) (0 .1 1) (0 .66)(2 58) (0 0 2 ) 18 0 25 0 09 -0 45 0 12 -0.35 0.46 0.60 -0.80 0 94 1 14 (0 01) (0 0 1 ) (0 08) (0 15) (0.09) (0 .1 2) (0.04) (1.38)(3 25) (0 0 1 ) 19 -0 10 -0 00 0 11 -0 04 0.16 -0.07 -0.15 0.19 -0 39 1 08 (0 0 0 ) (0 0 0 ) (0 0 1 ) (0 0 2 ) (0 .0 2 )(0 .0 2 ) (0 .0 1 ) (0.13)(0 46) (0 0 0 ) a/ AR(1) is the first order autocorrelation, Q,.,, is the lagged quantity variable, FS is family size, AGE is age of household head, D is a dummy variable, b/ Figures in the parenthes are asymptotic standard errors. 152 results show that most of the estimated elasticities are statistically significant. As we observe the signs of the cross-price elasticities, approximately half of the elasticities are positive, Indicating these food categories are specific substitutes. Table 30 presents the compensated price elasticities for 19 food commodities estimated from Dynamic Model II. The compensated price elasticities are computed from uncompensated price elasticities, expenditure elasticities and budget shares following Eq. (60). The definition of substitutability and complementarity in Hick's sense is based on the sign of the total compensated substitution effect including specific and general cross-price effects for a given level of utility. If a compensated cross-price elasticity, q^j, is positive, then a rise in the absolute price of j-th commodity would cause consumption of i-th commodity to increase, holding real income and other prices constant. Accordingly, goods i and j are said to be Hicksian substitutes (complements) if q *u > 0 (<0 ). The main findings from the estimated compensated price elasticities among 19 commodities are summarized as follows: (1) Cereal has strong substitute relationships with other meats (1.94), seafood (1.25), and dairy products (1.49) and has a complement relationship with milk (-2.95). (2) Bakery products appear to have fairly weak relationships with other commodities in the system. The compensated cross-price elasticities range from 0.4 with respect to food away from home to -0.07 with seafood. (3) Among three red meats, the estimates of compensated cross- 153 Table 30. Compensated Price Elasticities Matrix for 19 Food Commodities Using LA/AIDS Model with AR(1), Qt.„ FS, AGE, and Da Other Sea Dairy Cereal Bakery Beef Pork meats Poultry food Eggs Milk product (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 1 -2.16 0.10 0.21 -0.36 1.94 0.38 1.25 0.09 -2.95 1.49 2 0.05 -0.83 0.05 -0.02 -0.05 0.02 -0.07 0.02 0.03 0.01 3 0.08 0.05 0.30 0.15 0.35 0.07 -0.06 -0.00 -0.03 -0.48 4 -0.25 -0.04 0.27 -0.60 0.60 0.29 -0.07 0.12 -0.33 0.73 5 1.95 -0.10 0.89 0.87 -2.72 -0.17 0.17 0.05 1.69 -1.07 6 0.38 0.14 0.19 0.42 -0.18 -0.39 0.36 0.02 0.46 -1.11 7 1.70 -0.20 -0.22 -0.13 0.23 0.48 -2.13 -0.06 1.00 0.46 8 0.22 0.12 -0.03 0.44 0.11 0.05 -0.12 -0.15 0.50 0.89 9 -1.93 0.04 -0.06 -0.32 1.11 0.30 0.48 0.13 -1.51 2.09 10 0.97 0.02 -0.81 0.69 -0.70 -0.72 0.22 0.23 2.09 -0.56 11 0.13 0.13 -0.30 0.34 0.05 0.12 0.17 0.04 0.29 0.34 12 0.00 -0.09 0.32 -0.04 0.11 -0.13 -0.15 -0.03 0.09 0.14 13 1.62 0.05 0.48 0.33 1.74 -0.15 -0.58 -0.04 0.96 0.16 14 -0.90 0.20 0.46 0.02 -0.13 0.63 -0.29 0.10 -0.06 0.76 15 -0.91 0.22 1.80 -0.52 -0.83 0.40 -0.63 -0.20 -1.38 0.24 16 0.44 0.08 -0.71 -0.16 0.19 0.03 -0.11 -0.01 0.42 -0.26 17 -0.15 0.12 -0.30 0.10 0.60 0.58 -0.27 -0.60 1.40 -0.12 18 0.33 0.07 0.13 0.35 -0.18 -0.22 -0.14 0.05 0.18 -2.16 19 -0.15 0.07 -0.16 -0.12 -0.07 -0.06 0.09 -0.06 -0.13 0.19 154 Table 30 (Continued) Fresh Fresh Pro. Pro. Nonal. Fat Hi sc. fruit veg. fruit veg. Sweets beve. oil food FAFH (11) (12) (13) (14) (15) (16) (17) (18) (19) 1 0 13 0 13 -0.30 0 34 0 05 0 12 0 17 0 04 0 29 2 0 07 -0 05 0.02 0 06 0 09 0 07 0 04 0 08 0 40 3 -0 14 0 14 0 .16 0 12 0 60 -0 58 -0 07 0 13 -0 76 4 0 27 -0 03 -0 07 0 01 -0 31 -0 23 0 04 0 59 -1 00 5 0 05 0 12 -0 45 -0 08 -0 71 0 41 0 38 -0 45 -0 84 6 0 14 -0 14 -0 13 0 41 0 35 0 06 0 37 -0 54 -0 71 7 0 27 -0 23 -0 68 -0 25 -0 73 -0 32 -0 23 -0 46 1 51 8 0 11 -0 08 -0 08 0 15 -0 43 -0 04 0 03 0 28 -1 97 9 0 22 0 06 0 54 -0 03 -0 77 0 57 -0 17 0 28 -1 05 10 0 25 0 10 0 09 0 32 0 13 -0 35 -0 05 -3 50 1 57 11 -1 02 -0 01 0 13 0 10 -0 31 -0 12 0 01 0 62 -0 71 12 -0 01 -0 68 -0 08 0 09 -0 23 0 03 0 02 0 29 0 37 13 0 17 -0 11 -1 96 0 30 -0 29 -1 00 0 33 -1 20 1 91 14 0 19 0 16 0 40 -0 73 -0 34 -0 04 -0 58 0 55 -0 38 15 -0 41 -0 30 -0 30 -0 25 -1 40 2 63 0 04 -0 92 -2 73 16 -0 07 0 02 -0 41 -0 01 1 08 -1 12 -0 00 0 62 -0 02 17 0 02 0 03 0 45 -0 58 0 06 -0 00 0 11 2 41 -2 66 18 0 29 0 13 -0 42 0 14 -0 32 0 53 0 62 -0 72 -1 34 19 -0 07 0 03 0 13 -0 02 0 19 -0.00 -0 14 0 27 -0 01 a/ AR(1) is the first order autocorrelation, QM is the lagged quantity variable, FS is family size, AGE is age of household head, D is a dummy variable. 155 price effects show that they are substitutes. For Instance, quantity demanded for beef could increase by 0.15% due to a 1% Increase 1n pork price, and by 0.35% resulting from a 1% Increase In other meats. Pork quantity demanded would increase by 0.27% and 0.60%, respectively, due to a 1% Increase In the prices of beef and other meats. In addition, a 1% increase in the price of beef and pork could increase the quantity demanded for other meats by 0.89% and 0.87%, respectively. (4) Poultry is found to be substitutes with beef, pork, and seafood but as a complement with other meats. The results also show that poultry and dairy products have a relatively strong complementary relationship and similarity with food away from home as well. (5) Seafood has a negative compensated cross-price elasticity with beef and pork indicating a complementarity with these two red meat products but has a substitute relationship with other meats. We also observe that the relationships between seafood and cereal, as well as milk, dairy products, and food away from home are substitutes. As a 1% increase in the prices of food away from home and cereal would result in 1.51% and 1.71% increases, respectively, of quantity demanded for seafood. (6 ) Egg and milk, as well as dairy products are found to be substitutes, but egg and food away from home are complements. Cross price estimates between egg and beef, other meats, poultry, seafood do not show significant interdependent relationshlps. (7) The cross-price elasticities of milk and dairy products indicate that both commodities are substitutes. The relationship between milk and cereal, as well as milk and food away from home Indicates they 156 are strong complements. The results also Indicate that the dairy products category has significant Interdependent relationships with cereal, red meats, poultry, miscellaneous foods, and food away from home. (8 ) In general, fresh fruits and fresh vegetables do not exhibit strong cross-price relationships with other commodities except miscellaneous foods and food away from home. (9) The magnitudes of cross-price elasticities indicate strong substitution between processed fruits and cereal, as well as other meats, milk, and food away from home. Complementary relationships between processed fruits and nonalcoholic beverage as well as miscellaneous foods are also observed. (10) The cross-price elasticities among processed vegetables, sweets, nonalcoholic beverages, fat and oil, and miscellaneous foods indicate the following pairs of commodities having strong substitution relationships: processed vegetables and dairy products; processed vegetables and miscellaneous foods; sweets and beef; sweets and nonalcoholic beverage; oil and milk; oil and miscellaneous foods. However, the following pairs are found to be complements: sweets and milk; sweets and food away from home; nonalcoholic beverages and sweets; oil and food away from home; miscellaneous foods and dairy products; miscellaneous foods and food away from home. (11) The estimated cross-price elasticities for food away from home do not show significant interdependent relationships with other commodities in the system. 157 7.4 Estimated Results from The Two-Stage Procedure The primary objective of this section is to present the estimation results obtained from a two-stage procedure. From practical points of view, the main merit of assuming a two-stage budgeting process is to permit us to focus on subsystems of food demand. More important, it allows us to use a more flexible functional form for estimating structural parameters. For a small system, the convergency can be more easily attained and the problem of model misspecification can b'e alleviated. Undoubtedly, these econometric gains can not be obtained totally free. Two-stage budgeting and weak separability place severe restrictions on the degree of substitutability between commodities in different groups. For instance, the whole groups will be substitutes or complements for one another and all pairs of goods, one from each group, must also be either substitutes or complements as the case may be (Deaton and Muellbauer, 1980). Three alternative PIGLOG models including the full model in Eq. (85), and two nested models, TL and LA/AIDS, are used to estimate structural parameters in the first-stage aggregate group and the subgroups in the second stage. To save the space, we present only the test statistics and a complete matrix of uncompensated price and expenditure elasticities. The detailed regression results and compensated price elasticities matrix are given in Appendices D and G. To test whether the nested models are significantly different from the full model, a chi-square test is used. We employ two alternative approaches to compute chi-square test statistics. One is based on minimum objective function in Eq. (56). The other is called the 158 likelihood ratio approach using Eq. (57). Table 31 presents test results obtained from using the minimum objective function. The results clearly show that there is no significant difference between the full model and nested models for the first-stage aggregate groups and all six groups in the second-stage. On the other hand, Table 32 presents test results from the likelihood ratio approach. The chi-square statistics Indicate that there exist significant differences between the full model and nested models for the aggregate group in the first stage and subgroups I and III in the second stage. Indeed, the regression results reveal that several parameters restricted to be zero in the nested models are statistically significant from zero in the full model. These results imply rejecting the restricted model for the aggregate group in the first stage and subgroups I, III, and V in the second stage. Moreover, the estimated elasticities presented in Tables 33-38 reveal similar notable differences between the full model and nested models. Comparing the two test approaches, we found that the likelihood ratio approach appears to have a higher test power than the minimum objective function approach. Table 33 presents a complete elasticities matrix obtained from three PIGLOG models for the aggregate group in the first stage. Overal1, the results show that the TL and LA/AIDS produce very similar estimates of elasticities. However, significant differences between the full model and nested models are also observed. The estimated food expenditure elasticities show that Groups IV and VI are income-elastic under all three model specifications. The results indicate that these two groups are luxuries. These findings are reasonable because Group IV includes 159 Table 31. Test Statistics based on Objective Function for Three PIGLOG Model Specifications Objective Function8 X2 Statistic1* FULLTLLA/AIDS FULL vs. TL FULL vs. LA/AIDS First-stage 318.94 324.12 324.12 5.18 5.18 Second-stage I 256.78 261.22 261.22 4.44 4.44 II 256.78 261.22 261.22 4.44 4.44 » III 130.24 131.72 131.72 1.48 1.48 IV 65.12 65.86 65.86 0.74 0.74 V 130.98 131.72 131.72 0.74 0.74 a/ The value of minimum objective function in Eq.(57) using same estimated v|riance-covariance matrix for three models. b/ Critical value X,,5 0 O5«11.01; Critical value X?, ' * =9.49; Critical value X2,2'0[05=5.99. 16G Table 32. Test Statistics Based on Likelihood Ratio for Three PIGLOG Model Specifications In 0“X2 Statistic FULL TL l a /a i d s FULL vs. TL FULL vs. LA/AIDS First-stage -53.7017 -53.5157 -53.5198 13.764b 13.4606b Second-stage I -40.9478 -40.6101 -40.6269 24.9898c 23.7466° II -38.7281 -38.7650 -38.7486 2.7306 1.4760 III -16.8868 -16.7737 -16.7845 8.3694d 7.5702d IV -7.6580 -7.6691 -7.6732 0.8184 1.1233 V -17.7215 -17.6595 -17.6639 4.5880 4.2624 a/ Log of determinant of covariance matrix,of residuals; b/ Significant at & ■ 5% (critical value X2,5=11.01); c/ Significant at a = 5% (critical value X2U =9.49); d/ Significant at s - 5% (critical value X ,2“5.99). 161 Table 33. Uncompensated Price and Expenditure Elasticities Matrix For Aggregate Group in First Stage ______G r o u p " ______Expend1- Group Model I II III IV V VI ture I Full -0 81 -0 12 0 02 -0 04 0 30 0 12 0 90 TL -0 86 -0 02 -0 06 -0 23 -0 02 0 38 0 81 LA/AIDS -0 86 -0 01 -0 06 -0 22 -0 02 0 37 0 81 II Full 0 06 -0 01 -0 04 0 15 -0 22 -0 81 1 07 TL -0 03 -0 02 -0 14 0 12 -0 17 -0 53 0 77 LA/AIDS -0 02 -0 03 -0 14 0 12 -0 17 -0 53 0 77 III Full -0 13 -0 12 -0 53 0 13 -0 30 0 08 0 94 TL -0 20 -0 21 -0 69 0 22 -0 19 0 26 0 81 LA/AIDS -0 20 -0 21 -0 69 0 22 -0 19 0 26 0 82 IV Full -0 11 0 30 -0 04 -0 25 -0 44 -1 19 1 20 TL -0 30 0 04 0 07 0 36 0 00 -1 17 1 00 LA/AIDS -0 30 0 05 0 07 0 36 0 01 -1 18 1 00 V Full -0 08 -0 29 -0 19 0 17 -0 41 -0 04 0 83 TL -0 03 -0 14 -0 10 0 04 -0 50 -0 08 0 80 LA/AIDS -0 03 -0 14 -0 10 0 04 -0 50 -0 08 0 80 VI Full -0 04 -0 23 0 00 -0 44 -0 05 -0 27 1 01 TL 0 12 -0 19 0 02 -0 62 -0 09 -0 52 1 27 LA/AIDS 0 12 -0 19 0 02 -0 62 -0 09 -0 51 1 28 a/ Group I : cereal, bakery, processed fruit, fat & oil, and misc. foods; Group II : beef, pork, other meats, eggs, and processed vegetables; Group III: poultry, milk, and sweets; Group IV : seafood, and dairy products; Group V : fresh fruits, fresh vegetables, and nonalcoholic beverage; Group VI : food away from home. \ 162 seafood and dairy products while Group VI Is food away from home. Table 35 also shows that the demand for food In Group II appears to be Income- elastic 1n the full model. The estimated direct price elasticities all have an expected signs from the full model but one unexpected sign appears in Group IV from the nested models (Table 34). Cross-price elasticities are generally small with the exception of Groups II and IV related to food away from home. Furthermore, the cross-price elasticities indicate that most groups are classified as gross complements except for the gross substitutes exhibited between Group I and Group VI (food away from home), Group III and Group VI, Groups II and III, as well as Groups III and IV. In the second stage, the demand systems for five groups are estimated separately. Group VI contains only one single item and therefore it does not need to be estimated separately. The estimated elasticities by group are discussed here. Note that all estimated elasticities presented here are only those within each group in the second stage rather than the net of interaction in the entire food commodity space. Table 34 presents the results for Group I. This group contains cereal, bakery products, processed fruits, fat and oil, and miscellaneous foods. It is clear that miscellaneous foods has a expenditure elasticity greater than one while others are less than one. Two out of five estimated own-price elasticities, cereal, and oil, have an incorrect sign regardless of model specification. Under the full model, the estimated own-price elasticities for bakery, processed fruits, and miscellaneous foods are -0.92, -0.79 and -0.48, respectively. In addition, the estimated cross-price elasticities for 163 Table 34. Uncompensated Price and Expenditure Elasticities Matrix . in Second Stage,Group I Commodities® Expendi- Commodities Model 1 2 13 17 18 ture 1 Full 0.78 0.02 -0.48 -0.29 -1.08 0.91 TL 0.65 -0.01 -0.56 -0.10 -0.81 0.83 LA/AIDS 0.55 -0.02 -0.53 -0.09 -0.72 0.81 2 Full 0.01 -0.92 -0.01 0.01 -0.05 0J83 TL 0.02 -0.91 -0.00 -0.00 0.26 0.63 LA/AIDS 0.02 -0.91 -0.00 0.00 0.27 0.62 13 Full -0.57 0.01 -0.79 0.04 0.07 0.90 TL -0.65 -0.07 -1.17 0.30 0.76 0.83 LA/AIDS -0.63 -0.08 -1.24 0.32 0.75 0.88 17 Full -0.46 0.07 0.00 0.04 -0.67 0.72 TL -0.12 0.01 0.44 0.25 -1.16 0.58 l a /a i d s -0.12 0.01 0.47 0.24 -1.21 0.61 18 Full -0.40 -0.06 0.16 -0.15 -0.48 1.29 TL -0.43 -0.08 0.17 -0.39 -0.83 1.55 LA/AIDS -0.40 -0.05 0.18 -0.40 -0.89 1.54 a/ Commodities: 1- cereal; 2- bakery; 13- processed fruits; 17- fat & oil; 18- misc. foods. 164 cereal show complementary relationships with processed fruits, fat and oil, and miscellaneous foods. Processed fruits and miscellaneous foods are shown to have substitutability relation. Table 35 presents the estimates of elasticity for Group II. Beef, pork, other meats, eggs and processed vegetables are included in this group. The elasticity of expenditure for beef demand is 1.24 indicating a 1% increase of group expenditure would Increase the quantity of beef demanded by 1.24%. The remaining expenditure elasticities for pork, other meats, eggs and processed vegetables are 0.99, 0.87, 0.71 and 0.46, respectively. Own-price elasticities, generally speaking, show an inelastic demand for commodities in this group except for other meats. Other meats has the highest (in absolute value) own-price elasticity of -1.29, while the estimate for eggs is the lowest, -0.33 in the full model. The cross-price elasticity estimates show that food commodities in this group do not exhibit significant interdependent relationships except between processed vegetables and beef. This significant cross price effect reinforces the validity of the nonparametric analysis and supports linking beef and processed vegetables in the same group. It is also noted that the estimated elasticities in this group are quite similar among three PIGLOG models which are consistent with the results of likelihood ratio test for this group. The estimated elasticities for Groups III and IV are presented in Tables 36 and 37, respectively. Among these five food commodities, sweets and seafood are expenditure-elastic goods as might be expected. The estimated own-price elasticities of sweet have an unexpected sign for all three models. The estimated own-price elasticities in the full Table 35. Uncompensated Price and Expenditure Elasticities Matrix in Second Stage, Group II Commodities* Expendi Commodities Model 3 4 5 8 14 ture 3 Full -0.84 -0.18 0.08 -0.00 -0.24 1.24 TL -0.84 -0.18 0.03 -0.00 -0.26 1.25 l a /a i d s -0.83 -0.18 0.04 -0.01 -0.27 1.25 4 Full -0.24 -0.74 0.03 -0.05 0.01 0.99 TL -0.19 -0.75 0.04 -0.06 -0.01 0.97 LA/AIDS -0.20 -0.75 0.05 -0.06 -0.00 0.96 5 Full 0.13 0.05 -1.29 -0.18 0.32 0.87 TL 0.25 0.09 -1.42 -0.17 0.41 0.84 LA/AIDS 0.27 0.10 -1.45 -0.17 0.41 0.84 8 Full 0.35 -0.13 -0.52 -0.33 -0.08 0.71 TL 0.18 -0.17 -0.41 -0.33 -0.03 0.76 LA/AIDS 0.15 -0.17 -0.41 -0.33 -0.02 0.77 14 Full -0.56 0.14 0.40 -0.00 -0.51 0.46 TL -0.71 0.10 0.69 -0.00 -0.56 0.48 LA/AIDS -0.75 0.11 0.71 0.01 -0.56 0.48 a/ Commodities: 3- beef; 4- pork; 5- other meats; 8 - eggs; 14- processed vegetables. 166 Table 36. Uncompensated Price and Expenditure Elasticities Matrix In Second Stage, Group III Commodities a Expendi Commodities Model 6 9 15 ture 6 Full -0.57 -0.87 -0.37 0.89 TL -0.58 -0.26 -0.10 0.94 LA/AIDS -0.68 -0.13 0.01 0.80 9 Full -0.09 -0.38 -0.56 0.69 TL -0.09 -0.10 -0.49 0.69 LA/AIDS -0.06 -0.01 -0.66 0.73 1 o 15 Full -0.33 O • M 0.42 1.68 TL -0.32 -1.34 0.03 1.63 LA/AIDS -0.26 -1.62 0.17 1.71 a/ Commodities: 6 - poultry; 9- milk; 15- sweets; 167 Table 37. Uncompensated Price and Expenditure Elasticities Matrix in Second Stage, Group IV Commodities® Expendi Commodities Model I II ture 7 Full -0.94 -0.56 1.51 TL -0.99 -0.52 1.51 LA/AIDS -0.94 -0.58 1.52 10 Full -0.03 -0.73 0.75 TL -0.04 -0.80 0.75 LA/AIDS -0.03 -0.72 0.75 a/ Commodities: 7- seafood; 10- dairy products. 168 model reveal that an increase in own price by 1% would depress its demand by 0.57% for poultry, by 0.38% for milk, by 0.94% for seafood, and for dairy products by 0.73%. All cross-price elasticities are negative for commodities in both groups, which are somewhat surprising. However, they are not theoretically incorrect. Finally, the estimated elasticities for Group V (fresh fruits, fresh vegetables and nonalcoholic beverages) are contained in Table 38. Both fresh fruits and nonalcoholic beverages have expenditure elasticities close to unity. The own-price elasticities for three commodities in the group are very similar, i.e., fresh fruits (-0.79), fresh vegetables (-0.82), nonalcoholic beverages (-0.80). The estimated cross-price elasticities from the full model show that fresh fruits is a complement to fresh vegetables and nonalcoholic beverages. A complementary relationship is also found between fresh vegetables and nonalcoholic beverages. However, the cross-price elasticities of nonalcoholic beverages with respect to the price of fresh fruits and fresh vegetables are close to zero, indicating the quantity demanded for nonalcoholic beverages is not affected by changes in the prices of fresh fruits and fresh vegetables. Note that above discussed estimates of elasticities are calculated within each group holding group expenditure constant, and thus they are partial elasticities. To obtain full elasticities with respect to the entire food system, a linkage between the first-stage and second-stage must be constructed. As mentioned earlier, Eq. (62) can be used to compute the full own-price and cross-price elasticities within each group, while Eq. (6 8 ) can be utilized to generate the cross-price 169 Table 38. Uncompensated Price and Expenditure Elasticities Matrix in second Stage, Group V Commodities8 Expend1- Commodities Model 11 12 16 ture 11 Full -0.79 -0.18 -0.25 1.04 TL -0.96 -0.23 0.11 1.01 l a /a i d s -0.90 -0.23 0.10 1.03 12 Full -0.22 -0.82 -0.12 0.94 TL -0.21 -0.86 0.15 0.92 LA/AIDS -0.21 -0.85 0.15 0.90 16 Full 0.00 0.01 -0.80 1.01 TL 0.06 0.05 -1.14 1.04 LA/AIDS 0.05 0.05 -1.14 1.03 a/ Commodities: 11- fresh fruit; 12- fresh vegetables; 16- nonalcoholic beverage. 170 elasticities cross different groups. Furthermore, full expenditure elasticities can be computed with respect to total expenditure. For this purpose, Eq. (64) can be used. Three complete matrices of estimated full system elasticities obtained from the full, TL, and LA/AIDS models under a two-stage budgeting procedure are given in Tables 39, 40, and 41, respectively. Let us compare the estimated elasticities from the three models. The results exhibit several unexpected positive signs for the estimated uncompensated own-price elasticities. For example, the three models all yield a wrong sign for cereal, sweet, and fat and oil. The TL and LA/AIDS models also provide an unreasonal magnitude of own-price elasticity for milk. In addition, all three models indicate demand for other meats to be price-elastic as the estimated price elasticity is - 1.15 from the full model, -1.28 from the TL model, and -1.32 from the LA/AIDS model. Generally speaking, the estimated cross-price elasticities are similar between TL and LA/AIDS models whereas there are significant differences between the full model and two nested models. With all three models, the estimated expenditure elasticities for seafood, sweets, miscellaneous foods, and food away from home are greater than one. The same, however, is not true for beef, while the full model gives an estimated expenditure elasticity of 1.33, and TL and LA/AIDS models suggest beef being expenditure-inelastic. In sum, the comparison of elasticities across the three models reveal that no model conforms exactly with the a priori theoretical expectation. However, the estimated expenditure elasticities from the two-stage budgeting structure seem to provide more plausible results in 171 Table 39. Uncompensated Price and Expenditure Elasticities Matrix for 19 Food Commodities Using Full Model under Two-Stage Budgeting Procedure Other Sea Dairy Cereal Bakery Beef Pork meats Poultry food Eggs Milk product (1) (2) (3) (4) (S) (6 ) (7) (8 ) (9) (10) 1 0.80 0.07 -0.19 -0.09 -0.06 0.11 -0.01 -0.02 0.12 -0.03 2 0.03 -0.87 -0.17 -0.08 -0.05 0.10 -0.01 -0.02 0.11 -0.02 3 0.39 0.75 -0.32 -0.11 0.28 0.12 0.73 0.08 0.13 0.74 4 0.31 0.60 0.17 -0.50 0.19 0.09 0.58 0.01 0.10 0.58 5 0.27 0.53 0.49 0.26 -1.15 0.08 0.51 -0.12 0.09 0.51 6 -0.14 -0.27 -0.47 -0.22 -0.13 -0.45 0.78 -0.05 -0.68 0.79 7 0.10 0.18 3.29 1.46 0.87 0.06 -0.57 0.28 0.05 0.20 8 0.22 0.43 0.65 0.04 -0.40 0.07 0.42 -0.28 0.07 0.41 9 -0.11 -0.21 -0.37 -0.17 -0.10 0.01 0.60 -0.04 -0.23 0.61 10 0.05 0.09 1.64 0.73 0.43 0.03 0.15 0.14 0.03 -0.35 11 0.06 0.12 -1.73 0.78 -0.47 -0.53 0.48 -0.16 -0.64 0.47 12 0.06 0.11 -1.57 -0.71 -0.43 -0.48 0.43 -0.14 -0.58 0.43 13 -0.05 0.06 -0.18 -0.09 -0.06 0.11 -0.01 -0.02 0.12 -0.03 14 0.14 0.28 -0.37 -0.25 0.48 0.04 0.27 0.03 0.05 0.27 15 -0.26 -0.51 -0.89 -0.41 -0.25 -0.10 1.47 -0.09 0.26 1.48 16 0.06 0.12 -1.69 -0.77 -0.46 -0.52 0.47 -0.15 -0.63 0.46 17 0.44 0.11 -0.15 -0.07 -0.05 0.09 -0.01 -0.02 0.10 -0.02 18 -0.37 0.01 -0.26 -0.12 -0.08 0.15 -0.01 -0.03 0.17 -0.04 19 0.03 0.06 -0.42 -0.20 -0.12 0.06 -0.17 -0.04 0.06 -0.20 172 Table 39 (Continued) Fresh Fresh Pro. Pro. Nonal. Fat Misc. Expen. fruit veg. fruit veg. Sweets beve. Oil food FAFH elasti. (11) (12) (13) (14) (15) (16) (17) (18) (19) 1 0 49 0 12 -0 46 -0 .78 -0 47 -0 .53 -0 .27 -1 02 -0 .64 0 .82 2 0 45 0 39 0 01 -0 .02 0 18 0 .81 0 .02 0 01 1 .51 0 .75 3 -0 35 -0 31 0 33 -0 10 0 22 -0 64 0 19 1 35 -3 65 1 33 4 -0 28 -0 25 0 27 0 11 0 18 -0 50 0 15 1 08 -2 91 1 06 5 -0 25 -0 22 -0 23 0 41 0 15 -0 44 0 14 0 95 -2 56 0 93 6 -0 77 -0 68 -0 12 -0 05 -0 26 -1 38 -0 07 -0 44 5 75 0 84 7 -0 77 -0 69 0 08 0 27 0 14 -1 39 0 04 0 37 -7 38 1 81 8 -0 20 -0 18 0 19 -0 01 0 13 -0 36 0 11 0 77 -2 09 0 76 9 -0 60 -0 53 -0 09 -0 04 -0 48 -1 07 -0 06 -0 34 4 45 0 65 10 -0 38 -0 34 0 04 0 14 0 07 -0 69 0 02 0 19 -3 67 0 90 11 -0 63 -0 02 0 05 -0 17 -0 84 0 04 0 03 0 24 1 84 0 86 12 -0 07 -0 68 0 05 -0 15 -0 76 0 15 0 03 0 22 1 67 0 78 13 0 49 0 43 -0 77 -0 03 0 19 0 87 0 06 0 13 1 63 0 81 14 -0 13 -0 12 0 12 -0 46 0 08 -0 23 0 07 0 50 -1 35 0 49 15 -1 45 -1 28 -0 22 -0 10 0 62 -2 60 -0 14 -0 83 10 81 1 58 16 0 16 0 16 0 05 -0 16 -0 82 -0 51 0 03 0 23 1 79 0 84 17 0 39 0 34 0 .02 -0 02 0 16 0 70 0 05 -0 62 1 31 0 65 18 0 68 0 59 0 .19 -0 04 0 27 1 21 -0 13 -0 39 2 28 1 13 19 0 03 0 02 0 .03 -0 .05 0 .11 0 .05 0 .01 0 14 -0 27 1 01 173 Table 40. Uncompensated Price and Expenditure Elasticities Matrix for 19 Food Commodities Using Translog Model under Two-Stage Budgeting Procedure Other Sea Dairy Cereal Bakery Beef Pork meats Poultry food Eggs Milk product (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 1 0 .67 0 .02 0 18 0 07 0.04 -0 04 -0 08 0 01 -0 05 -0 10 2 0 .03 -0 88 0 14 0 06 0.03 -0 03 -0 06 0 01 -0 04 -0 08 3 0 16 0 25 -0 32 0 11 0 23 -0 52 0 56 0 08 -0 60 0 55 4 0 13 0 19 0 21 -0 51 0 20 -0 41 0 43 0 00 -0 48 0 43 5 0 11 0 17 0 60 0 29 -1 28 -0 35 0 38 -0 12 -0 41 0 31 6 -0 09 -0 16 -2 08 -0 92 -0 55 -0 49 1 03 -0 20 -0 13 1 04 7 -0 15 -0 26 1 04 0 44 0 26 0 46 -0 32 0 09 0 51 0 86 8 0 10 0 15 0 49 0 01 -0 29 -0 32 0 34 -0 28 -0 37 0 34 9 -0 07 -0 12 -1 53 -0 68 -0 40 -0 03 0 76 -0 15 -0 00 0 77 10 -0 07 -0 13 0 51 0 22 0 13 0 23 0 29 0 05 0 25 -0 11 11 0 10 0 16 -0 53 -0 24 -0 14 -0 21 0 25 -0 05 -0 25 0 24 12 0 10 0 14 -0 48 -0 22 -0 13 -0 19 0 23 -0 05 -0 23 0 22 13 -0 63 -0 04 0 18 0 07 0 04 -0 04 -0 08 0 01 -0 05 -0 10 14 0 06 0 10 -0 51 0 21 0 77 -0 20 0 21 0 03 -0 23 0 21 15 -0 16 -0 28 -3 60 -1 60 -0 95 -0 17 1 78 -0 35 -1 11 1 81 16 0 11 0 16 -0 54 -0 25 -0 15 -0 21 0 25 -0 06 -0 25 0 24 17 -0 11 0 03 0 13 0 05 0 03 -0 03 -0 06 0 01 -0 03 -0 07 18 -0 40 -0 02 0 34 0 14 0 08 -0 07 -0 16 0 03 -0 09 -0 19 19 0 09 0 12 -0 28 -0 13 -0 08 0 09 -0.24 -0 03 0.09 -0 27 174 Table 40 (Continued) Fresh Fresh Pro. Pro. Nonal. Fat Misc. Expen. fruit veg. fruit veg. Sweets beve. Oil food FAFH elastl. (11) (12) (13) (14) (15) (16) (17) (18) (19) o 1 0 .07 0 .16 -0.55 -0 24 -0 .14 -0 .21 -0 09 1 77 -0 .25 0 .67 2 0 05 0 04 0 01 0 01 -0 03 0 10 0 01 0 29 1 .88 0 51 o 3 -0 30 -0 27 0 14 -0 13 -0 76 1 57 0 07 0 80 -3 54 0 96 o 4 -0 24 -0 21 0 11 0 09 -0 59 1 44 0 05 0 62 -2 76 0 95 5 -0 20 -0 18 0 09 0 50 -0 52 -0 38 0 05 0 54 -2 39 0 65 6 -0 41 -0 36 -0 08 -0 20 -0 03 -0 76 -0 05 -0 39 8 24 0 76 7 0 26 0 23 -0 13 0 08 0 72 0 50 -0 07 -0 58 -7 72 1 51 i 8 -0 19 -0 17 0 09 0 05 -0 47 O 35 0 04 0 49 -2 17 0 59 9 -0 30 -0 27 -0 06 -0 15 -0 44 -0 56 -0 03 -0 28 6 07 0 56 10 0 13 0 11 -0 06 0 04 0 36 0 25 -0 04 -0 29 -3 84 0 75 11 -0 83 -0 10 0 09 -0 06 -0 30 0 35 0 04 0 53 1 39 0 81 12 -0 09 -0 74 0 08 -0 05 -0 27 0 37 0 04 0 48 1 27 0 74 13 0 07 0 06 -1 16 0 01 -0 04 0 13 0 31 0 80 2 47 0 67 14 -0 12 -0 10 0 05 -0 51 -0 29 -0 22 0 03 0 31 -1 36 0 37 15 -0 70 -0 63 -0 14 -0 35 0 16 -1 33 -0 08 -0 67 14 31 1 32 1 16 0 20 0 18 0 09 -0 06 -0 30 o 89 0 04 0 54 1 42 0 83 17 0 05 0 04 0 45 0 01 -0 03 0 09 0 26 -1 13 1 73 0 47 18 0 13 0 11 0 20 0 02 -0 08 0 24 -0 37 -0 75 4 64 1 26 19 0 01 0 01 0 08 -0 04 0 16 0 03 0 03 0 48 -0 52 1 27 175 Table 41. Uncompensated Price and Expenditure Elasticities Matrix for 19 Food Cononodities Using LA/AIDS Model under Two-Stage Budgeting Procedure Other Sea Dairy Cereal Bakery Beef Pork meats Poultry food Eggs Milk product (1 ) (2) (3) (4) (5) (6) (7) (8) (9) (1 0 ) 1 0.57 0.01 0.18 0.07 0.04 -0.04 -0.08 0.01 -0.05 -0.10 2 0.03 -0.88 0.14 0.06 0.03 -0.03 -0.06 0.01 -0.04 -0.08 3 0.16 0.25 -0.32 0.11 0.24 -0.51 0.56 0.07 -0.60 0.55 t 4 0.13 0.19 0.19 -0.53 0.20 -0.40 0.43 0.00 -0.47 0.43 5 0.11 0.17 0.61 0.30 -1.32 -0.35 0.38 -0.11 -0.40 0.37 6 -0.09 -0.16 -1.91 -0.85 -0.50 -0.61 1.01 -0.19 -0.02 1.03 7 -0.15 -0.26 1.04 0.44 0.26 0.46 -0.27 0.09 0.51 0.81 8 0.10 0.15 0.46 0.01 -0.29 -0.32 0.34 -0.28 -0.37 0.34 9 -0.07 -0.12 -1.41 -0.62 -0.37 0.01 0.75 -0.14 0.09 0.76 10 -0.07 -0.13 0.51 0.22 0.13 0.23 0.30 0.05 0.25 -0.03 11 0.10 0.16 -0.53 -0.24 -0.15 -0.21 0.26 -0.05 -0.24 0.25 12 0.10 0.14 -0.48 -0.22 -0.13 -0.19 0.24 -0.05 -0.22 0.23 13 -0.61 -0.04 0.18 0.07 0.04 -0.04 -0.08 0.01 -0.05 -0.10 14 0.06 0.10 -0.55 0.22 0.79 -0.20 0.21 0.04 -0.23 0.21 15 -0.16 -0.28 -3.31 -1.47 -0.87 -0.10 1.76 -0.32 -1.38 1.78 16 0.11 0.16 -0.54 -0.25 -0.15 -0.21 0.27 -0.06 -0.25 0.26 17 -0.11 0.04 0.13 0.05 0.03 -0.03 -0.06 0.01 -0.03 -0.07 18 -0.37 0.01 0.34 0.14 0.08 -0.07 -0.16 0.03 -0.09 -0.19 19 0.09 0.12 -0.28 -0.13 -0.08 0.09 -0.24 -0.03 0.09 -0.27 176 Table 41 (Continued) FreshFreshPro.Pro.Nonal. FatMisc.Expen. fruit veg. fruit veg. Sweets beve. Oil food FAFH elasti. (11) (12) (13) (14) (15) (16) (17) (18) (19) 1 0 .07 0 16 -0 52 -0 .24 -0 .14 -0 .21 -0 08 -0 68 -0 24 0 74 2 0 05 0 04 0 01 0 01 -0 03 0 10 0 01 0 30 1 83 0 64 3 -0 30 -0 27 0 14 -0 14 -0 75 -0 57 0 07 0 80 -3 51 0 92 4 -0 24 -0 21 0 11 0 10 -0 59 -0 44 0 05 0 62 -2 74 0 73 5 -0 20 -0 18 0 09 0 50 -0 51 -0 38 0 05 0 54 -2 38 0 65 6 -0 40 -0 36 -0 08 -0 19 0 08 -0 75 -0 05 -0 38 7 92 0 62 7 0 26 0 23 -0 13 0 08 0 71 0 50 -0 07 -0 58 -7 75 1 40 8 -0 19 -0 17 0 09 0 06 -0 46 -0 35 0 04 0 49 -2 16 0 55 9 -0 30 -0 26 -0 06 -0 14 -0 60 -0 56 -0 03 -0 28 5 83 0 56 10 0 13 0 11 -0 06 0 04 0 35 0 25 -0 04 -0 29 -3 85 0 69 11 -0 76 -0 10 0 09 -0 06 -0 29 0 35 0 04 0 53 1 37 0 86 12 -0 09 -0 73 0 08 -0 05 -0 27 0 37 0 04 0 48 1 28 0 73 13 0 07 0 06 -1 23 0 01 -0 04 0 13 0 33 0 79 2 40 0 86 14 -0 12 -0 10 0 05 -0 51 -0 29 -0 22 0 03 0 31 -1 35 0 43 15 -0 70 -0 62 -0 14 -0 33 0 30 -1 31 -0 08 -0 66 13 75 1 34 16 0 19 0 18 0 09 -0 06 -0 30 -0 89 0 04 0 54 1 41 0 81 17 0 05 0 .04 0 48 0 01 -0 03 0 09 0 25 -1 18 1 69 0 67 18 0 13 0 .11 0 20 0 02 -0 08 0 24 -0 38 -0 81 4 52 1 03 19 0 01 0 .01 0 07 -0 04 0 15 0 03 0 03 0 47 -0 51 1 32 177 terms of estimated magnitude and our Intuitive expectations than those from one-stage procedure. 7.5 Further comparison of elasticities In empirical demand studies, the estimates of elasticity can vary substantially due to different model specifications and the restrictions on the underlying utility structure. In this section, several comparisons of elasticities are conducted. We will first compare the various sets of estimated elasticities obtained in this study, and then compare our estimates with those obtained from other studies. Note that an entirely appropriate and sound comparison is not an easy task and the comparisons are often inconclusive, especially when making comparison with other studies. One can easily face with many difficulties such as differences in details of model formulation, in estimation methods, in definitions of commodities, sample periods, and data used. Nonetheless, such comparisons may still provide useful insights. Table 42 provides the estimated uncompensated own-price elasticities obtained from both one-stage and two-stage budgeting procedures. The results clearly reveal that the estimates of price elasticity are fairly different between the two different preference structures except the estimated elasticities for bakery products, pork, poultry, eggs, fresh vegetables and miscellaneous foods have a striking similarity. Indeed, the similarities of the own-price elasticities of bakery products and fresh vegetables are also observed among different LA/AIDS model specifications under the one-stage procedure discussed earlier. One plausible explanation of this finding is that the changes Table 42. Comparison of Uncompensated Own-Prlce Elasticities for 19 Food Commodities from One-Stage and Two-Stage Procedures Commodities r IIb IIIC IV* Cereal -2.19 0.80 0.67 0.57 Bakery -0.89 -0.87 -0.88 -0.88 Beef 0.24 -0.32 -0.32 -0.32 Pork -0.64 -0.50 -0.51 -0.53 Other meats -2.74 -1.15 -1.28 -1.32 Poultry -0.42 -0.45 -0.49 -0.61 Seafood -2.14 -0.57 -0.32 -0.27 Eggs -0.17 -0.28 -0.29 -0.28 Milk -1.55 -0.23 -0.00 .0.09 Other dairy -0.60 -0.35 -0.11 -0.03 Fresh fruit -1.05 -0.63 -0.83 -0.76 Fresh veg. -0.71 -0.68 -0.74 -0.73 Proc. fruit -1.99 -0.77 -1.16 -1.23 Proc. veg. -0.75 -0.46 -0.51 -0.51 Sweets -1.43 0.62 0.16 0.30 Nonal. bev. -0.07 -0.51 -0.89 -0.89 Fat & oil 0.10 0.05 0.26 0.25 Misc. foods -0.80 -0.39 -0.75 -0.81 FAFH -0.39 -0.27 -0.52 -0.51 a7 I is LA/AIDS model with AR(1), Qt.1t FS, AGE, and D under one-stage procedure. b/ II is the full model with AR(1), Qt.1} FS, AGE and D under two-stage procedure. c/ III 1s the translog model with AR(1), Qt.1t FS, AGE and D under two- stage procedure. d/ IV is LA/AIDS model with AR(1), Qt.lf FS, AGE and D under two-stage procedure. 179 in consumption for both commodities have been largely due to changes in their own-price rather than other factors. The results further show that the three alternative PIGLOG models under a two-stage procedure providerelatively similar estimates of the own-price elasticity for all except milk, dairy products and processed fruits. For example, the estimates of price elasticity for beef is -0.32 from all three PIGLOG models. The similarity is especially notable between the TL and LA/AIDS models. The comparison of the estimated expenditure elasticities between one-stage and two-stage structure is also presented in Table 43. In general, there are differences between the two preference structures. The results show that the income elasticities for sweets are greater than one in all cases, indicating it being a 1uxury good. Similar income effects are also observed for seafood in all the three PIGLOG models and for beef from the full model, and for food away from home from the TL and LA/AIDS. Note that the income elasticities for 19 food commodities are computed by using the income elasticity of food from our previous study based on the LA/AIDS model using the BLS's CES Interview Survey data for 1980-1986. Specifically, the estimated income elasticity of food is 0.84. Based on these results presented in Tables 42 and 43, one can conclude that the estimated demand elasticities are very sensitive to the assumed underlying preference structure and model specification. Large variations of estimated elasticities may be found between different preference structures and models even though the data used are the same. Table 43. Comparison of Expenditure and Income Elasticities from One-Stage and Two-Stage Procedures for 19 Food Commodities Expenditure Income! Commodities I6 II IIId IV6 I II III IV Cereal 0 98 0 82 0 67 0 74 0 82 0 69 0 56 0 62 Bakery 1 03 0 75 0 51 0 64 0 87 0 67 0 43 0 54 Beef 0 72 1 33 0 96 0 92 0 60 1 12 0 81 0 77 Pork 1 05 1 06 0 75 0 73 0 88 0 89 0 63 0 61 Other meats 0 86 0 93 0 65 0 65 0 72 0 78 0 55 0 55 Poultry 1 10 0 84 0 76 0 62 0 92 0 71 0 64 0 52 Seafood 0 51 1 81 1 51 1 40 0 43 1 52 1 27 1 18 Eggs 1 04 0 76 0 59 0 55 0 87 0 64 0 50 0 46 Milk 0 79 0 65 0 56 0 56 0 66 0 55 0 47 0 47 Other dairy 0 77 0 90 0 75 0 69 0 65 0 76 0 63 0 58 Fresh fruit 0 95 0 86 0 81 0 86 0 80 0 69 0 68 0 72 Fresh veg. 1 00 0 78 0 74 0 73 0 84 0 66 0 46 0 61 Proc. fruit 1 08 0 81 0 67 0 86 0 91 0 68 0 56 0 72 Proc. veg. 1 00 0 49 0 37 0 43 0 84 0 41 0 31 0 36 Sweets 1 28 1 58 1 32 1 34 1 08 1 33 1 11 1 13 Nonal. bev. 1 08 0 84 0 83 0 81 0 91 0 71 0 70 0 68 Fat & oil 0 83 0 65 0 47 0 67 0 70 0 55 0 39 0 56 Misc. foods 1 14 1 13 1 26 1 03 0 96 0 95 1 06 0 87 FAFH 1 08 1 01 1 27 1 32 0 91 0 85 1 07 1 11 a/ Income elasticities are computed by using income elasticity for food, 0.84, obtained from LA/AIDS model using 80-86 BLS interview survey data, b/ I is LA/AIDS model with AR(1), Qt_1, FS, AGE, and 0 under one-stage procedure, c/ II is full model with AR(1), FS, AGE and D under two-stage procedure, d/ III is translog model with AR(1), Qt„,,, FS, AGE and D under two-stage procedure, e/ IV is LA/AIDS model with AR(1), Qt_.,, FS, AGE and D under two-stage procedure. 181 Consider next a comparison of our results to those found in other studies. Undoubtedly, this comparison with other studies would be more difficult than the preceding comparison within this study. Numerous empirical studies of the food demand In the U.S. were conducted as discussed in the chapter of literature review. Table 44 provides an extensive, but not exhaustive, survey from previous estimates of food demand elasticities. As one can see the estimated elasticities are fairly different among these studies as may be expected. For instance, the estimated own-price elasticities for beef vary widely as they are -1.58 from Capps and Havlick (1984), -1.39 from Menkhaus et al. (1985), -1.09 from Christensen and Manser (1977), -1.05 from Moschini and Meike (1989), -0.96 and -0.95 from Heien (1982, 1983), -0.66 from Dahlgram (1987), -0.62 from Huang (1985), -0.57 from Eales and Unnevehr (1988), and -0.27 from B1ackorby et al. (1978). The range of the estimated expenditure elasticities for beef varies from the highest value of 1.74 (Menkhaus et al., 1985) to the 1owest 0.34 ( Eales and Unnevehr, 1988 ). The similar findings can also be reached for other food commodities shown in Table 44. In addition, we also note that the level of aggregation for commodity definition and number of commodities included in the system are arbitrary and quite differently among studies. They were usually dependent on researchers' interest and data availability. However, as Klevmarben (1979) indicated, the estimated elasticities are in general not independent of the level of aggregation. In general, a large demand system of highly disaggregate commodities 1s more likely to have elasticity estimates with an unexpected sign. For 182 Table 44. Survey of Estimated Price and Expenditure Elasticities of Food Demand in U.S. Study Commodity Base period or basis Elasticities for computation own-price expenditure Manser meats 1959 -0.53“ 1.11 (1976) fruit & veg. -0.35 0.29 cereal & bakery -0.65 0.18 others -0.94 1.65 Christensen fish 1971 -0.17b 0.37 & Manser beef -1.09 1.45 (1977) poultry -0.71 0.93 seafood -0.38 0.40 B1ackorby, fish 1968 -0.64° 0.87 Boyce & beef -0.27 1.04 Russell poultry -0.63 1.01 (1978) pork -0.69 1.13 Eastwood food at home 1978 -0.23d 0.36 & Craven food away from -0.42 0.63 (1981) home Heien beef constant -0.96 1.27 (1982) pork -0.51 0.33 chicken -0.80 0.33 eggs -0.26 -0.39 fresh fruit -3.02 1.99 fresh vegetable -0.35 0.26 Lanm food at home 1980 -0.16 0.11 (1982) food away from -0.11 0.12 home B1anciforti meats sample mean -0.57* 0.78 & Green fruits & veg. (1948-1978) -0.60 0.67 (1983) cereal & bakery -0.55 0.36 others -1.01 1.62 Heien beef constant -0.95 0.94 (1983) pork -0.95 0.32 broilers -0.47 0.65 milk -0.33 0.24 eggs -0.15 0.52 183 Table 44 (continued) Study Commodity Based period or basis Elasticities for computation own-price expenditure Huang & meat constant -0.53 0.36 Haidacher poultry -0.68 0.14 (1983) fish 0.06 -0.06 eggs -0.14 -0.07 dairy -0.30 0.18 fat -0.15 0.57 fruit -0.37 0.63 vegetable -0.21 0.21 Pro. fru. & veg. -0.39 0.43 cereal -0.35 -0.29 sugar -0.11 0.44 nonalc. bev. -0.33 0.14 Uohlgenant food sample mean -0.36 0.46 (1984) (1946-1968) Capps & ground beef sample mean -1.58 1.38 Havlicek roasts (1972-1974) -1.83 1.66 (1984) steaks -1.69 1.51 pork -1.30 1.11 other meat -1.46 1.28 poultry -1.25 1.10 seafood -2.24 1.96 Huang beef constant -0.62 0.45 (1985) pork -0.73 0.44 other meats -1.37 0.06 chicken -0.53 0.36 turkey -0.68 0.32 fresh and frozen fish 0.01 0.12 canned and cured fish 0.04 0.00 eggs -0.15 -0.03 chess -0.33 0.59 fluid milk -0.26 -0.22 evaporated and dry milk -0.83 -0.27 other fats and oil -0.22 0.37 sugar -0.05 -0.18 other 27 food categories (not repeated here) Menkhaus, beef 1980 -1.39 1.74 Clair and pork -0.69 0.23 Hallingbye chicken -0.68 -0.69 (1985) 184 Table 44 (continued) Study Commodity Based period or basis Elasticities for computation own-price expenditure Kokoski cereal log mean -0.77 0.88 (1986) meats (1972-73 & -0.79 1.23 dairy 1980-1981) -0.77 0.75 fruits & veg -0.72 1.08 others -0.77 ' 0.96 Huang & cereal & bakery sample mean -0.71 0.85 Raunikar dairy (1977-1978) -0.81 0.93 (1987) red meats -0.95 1.21 poultry -0.83 0.85 fish & shellfish -0.68 0.85 eggs -0.69 0.80 fruits & veg. -0.87 1.00 others -0.92 1.01 Craren & food at home sample mean -0.46c 0.31 Haidacher food away from (1955-1978) -0.49 0.73 (1987) home Dahlgram beef 1985 -0.66 0.44 (1987) pork -0.58 -0.05 chicken -0.60 0.20 Chern & food at home sample mean -0.98f 0.67 Lee food away from (1980-1986) -1.43 1.51 (1989) home Eales & chicken sample mean -0.28° 0.53 Unnevehr beef (1965-1985) -0.57 0.34 (1988) pork -0.76 0.28 non-meat food -0.64 0.48 Heien & steak sample mean -0.73 1.14 Pompel1 i roast (1977) -1.11 1.37 (1988) ground -0.85 0.69 Moschini beef sample mean -1.059 1.39 & Meike pork (1967I-1987IV) -0.84 0.85 (1989) chicken -0.10 0.21 fish -0.43 0.31 a/ Indirect translog with habit; e/ AIDS with autocorrelation model; b/ Indirect translog with explicit f/ Quadratic expenditure system; additivity; g/ For period after structure change, c/ Compensate price elasticity; d/ Extended linear expenditure system with habit formation; 185 example, the study conducted by Huang (1985) has two positive own-price elasticities and twelve negative expenditure elasticities which may not be plausible. Unfortunately, we can not completely eliminate such a problem because of the large size of the system we attempt to estimate. 7.6 Predictive Performance Evaluation The preceding discussions of empirical results focus on the assessment of the signs, magnitude, and precision of estimated elasticities obtained from alternative model specifications. This section will evaluate the predictive performance of alternative LA/AIDS model specifications under one-stage procedure. Similar evaluations for three PIGLOG models under a two-stage budgeting can also be conducted, but they are not presented In this dissertation. The comparison would suggest which model formulation has the best goodness of fit and can be best used to project U.S. food demand in the future. Five different criteria are used to evaluate the predictive performance of alternative specifications. Note that the evaluation could be made by examining budget shares, expenditure levels, quantities demanded, or relative changes in quantities. For this study, the comparison of expenditure levels are more appropriate and useful than other variables. One may be more interested in quantities than expenditures. Unfortunately, the actual quantities can not be derived in this study. The quantity index is not as useful as the expenditure levels. In principle, the evaluation is based on an ex post simulation or historical simulation, and computation of the differences between the actual and predict values of expenditure levels over the sample period 186 for each food commodity in the system. The first criterion is the measure of average information inaccuracy. More specifically, the measure or index (I) as developed by Strobel (1982) can be expressed as follows: Jj-sE <•§£>] (87) x t-l where Ait * predicted expenditure levels for commodity i at time t; Mj, » actual expenditure levels; T « the number of sample period. Note that I{ is always positive as exception only occurs in a perfect prediction, that Is A,t =M,t. As the formula suggests, smaller the value of I{ is, less will be the inaccuracy. Therefore, the smaller the value of I{ 1s, the better the predictive performance. We can also construct a similar measure for assessing the relative performance for the system as a whole. Based on the measure of If, the average information inaccuracy for the entire system can be computed by summing up all individual I{ in the system, that is I - El,. Thus the share of information inaccuracy for commodity 1 to the entire system can also be eval uated by I,/I. The computation results of average information inaccuracy are presented in Table 45. A comparison of the average information inaccuracy measures by commodities reveal that beef, seafood, sweets and food away from home have relatively poor fit of the data. These four commodities appear to be the most difficult ones to predict with the estimated models in this study. They account for about 50% of the model inaccuracy of the system. In general, other commodities in the system Table 45. Comparison of Average Information Inaccuracies for Different LA/AIDS Specifications* Commo Model s'> Accuracv' Rankina dities SI S2 S3 D1 02 D3 D4 1 2 3 4 5 6 7 Cereal 7 .45 2 .45 2 .52 2 .23 2 .25 2 .23 2 .36 D1 03 02 04 S2 S3 SI Bakery 18 .44 3 .96 4 .26 4 .53 4 .59 4 .42 4 .55 S2 S3 03 01 04 02 SI Beef 43 .90 20 .31 24 .51 18 .05 21 .22 17 .81 25 .26 D3 01 S2 02 S3 04 SI Pork 14 .35 9 .76 9 .97 8 .71 9 .56 8 .49 10 .12 D3 01 02 S2 S3 04 SI 0 . meats 9 .40 5 .67 6 .34 5 .72 6 .90 5 .54 6 .91 D3 S2 01 S3 iD2 04 SI Poultry 3 .92 3 .48 3 .96 4 .01 3 .90 3 .73 4 .19 S2 03 02 SI S3 01 04 Seafood 28.41 10 .40 9 .31 9 .30 9 .12 9 .12 9 .41 D2 03 01 S3 04 S2 SI Eggs 6 .18 1 .66 1 .67 1 .47 1 .52 1 .51 1 .61 D1 03 02 04 S2 S3 SI Milk 13 .67 2 .86 3 .06 2 .93 3 .28 2 .69 3 .22 D3 S2 01 S3 04 02 SI 0 . dairy 2 .81 2 .82 2 .81 3 .04 3 .03 2 .84 3 .09 SI S3 S2 03 02 01 D4 F. fruit 5 .39 5 .35 5 .33 5 .97 6 .30 5 .31 6 .24 03 S3 S2 SI 01 04 02 F. veg. 3 .30 3 .40 3 .41 3 .33 3 .31 3 .31 3 .49 SI 02 03 01 S2 S3 D4 P. fruit 2 .95 3 .04 3 .31 2 .84 2 .98 2 .80 3 .32 03 01 SI 02 S2 S3 04 P. veg. 3 .68 3 .65 3 .93 3 .63 3 .71 3 .54 3 .92 03 01 S2 SI 02 04 S3 Sweets 11 .87 11 .74 11 .97 10 .08 11 .06 10 .92 11 .35 01 03 02 04 S2 SI S3 N. bev. 6 .32 6 .53 6 .28 6 .47 6 .73 5 .99 6 .47 03 S3 SI 01 04 S2 D2 Fat & oil 2 .38 2 .41 2 .44 2 .24 2 .28 2 .27 2 .34 01 03 02 04 SI S2 S3 M. foods 5 .99 6 .18 7 .16 5 .13 5 .95 5 .22 6 .79 01 03 02 SI S2 04 S3 FAFH 24 .62 25 .02 29 .79 23 .55 25 .98 23 .53 29 .37 03 01 SI 04 S2 02 S3 Entire System0 215 131 142 123 134 121 144 03 01 S2 02 S3 04 SI a/ These presented figures were obtained by multiplying the original numbers by 10 . b/ Model SI : including the first autocorrelation, AR(1), a dummy variable (D), family size (FS), and age of household head (AGE); 52 : including FS, AGE, D; 53 : including AR(1), D; D1 : including AR(1), D, FS, AGE, 1agged quantity variable (Qt.J»and time trend (T); D2 : including AR(1), D, Qt ., FS, AGE; D3 : including AR(1), D, T, FS, AGE; D4 : including AR(1), D, Qt .. c/ I« 221,, I, is average information inaccuracy of commodity i. 188 show a fairly close fit to the data. Overall Models D3 and D1 yield the best fit to the data, whereas Model SI has the worst fit. However, the accuracy ranking among these models is not exactly the same for all commodities. For example, SI (not 03 or 01) model ranks first for other dairy products and fresh vegetables. The second criterion is the % root-mean-square error (RMS-E). The RMS-E for commodity 1 is defined as The RMS-E is thus a measure of the deviation of the predicted value from its actual value. Note that magnitude of RMS-E can be evaluated only by comparising it with the average value of the variable in question. To obtain an overal1 RMS-E measure to the entire system, the weighted RMS-E is computed by using budget shares as weighting factor. Table 46 provides the results of MRS-E measure for alternative model specifications. The two 1argest RMS-E appear in food away from home and beef while eggs and oil have the two smallest RMS-E in the system. Other commodities in the system also show a fairly good predictive performance. In general, the measures of RMS-E exhibit a similar pattern of relative predictive performance among models as those of average information inaccuracy. A minor difference is the overal1 performance of the system as whole. Model S3 has the worst goodness of fit. These results based on MRS-E also suggest that dynamic models are preferred over static models. Among the dynamic models, Model D1 is the best. 189 Table 46. Comparison of Root-Mean-Square Errors for Different IA /A I0S Specifications Modelsa Accuracy Rankina Commodities SI S2 S3 01 D2 D3 04 1 2 3 4 5 6 7 Cereal 0 22 0 12 0 12 0 11 0 11 0 11 0.12 D 1 02 03 S2 S3 04 SI Bakery 0 41 0 22 0 22 0 23 0 23 0 23 0 23 S2 S3 01 02 03 04 SI Beef 0 72 0 57 0 62 0 54 0 57 0 54 0 63 D1 03 S2 02 S3 04 SI Pork 0 33 0 28 0 28 0 27 0 28 0 26 0 29 D3 01 S2 S3 02 04 SI Other meats 0 22 0 18 0 19 0 18 0 20 0 18 0 20 S2 01 03 S3 02 04 SI Poultry 0 15 0 14 0 15 0 15 0 15 0 14 0 15 S2 03 SI S3 01 02 04 Seafood 0 46 0 21 0 20 0 20 0 20 0 20 0 20 S3 01 02 03 04 S2 S3 Eggs 0 13 0 06 0 06 0 06 0 06 0 06 0 06 S2 S3 01 02 03 04 SI Milk 0 40 0 16 0 16 0 16 0 17 0 15 0 17 D3 S2 S3 01 02 04 SI Other dairy 0 16 0 16 0 16 0 16 0 16 0 16 0 16 SI S2 S3 01 02 03 04 Fresh fruit 0 19 0 18 0 18 0 20 0 20 0 18 0 20 S2 S3 03 SI 01 02 04 Fresh veg. 0 14 0 15 0 15 0 15 0 14 0 14 0 15 SI 02 03 S2 S3 01 04 Proc. fruit 0 12 0 12 0 13 0 12 0 12 0 12 0 13 SI S2 01 02 03 S3 D4 Proc. veg. 0 12 0 11 0 12 0 11 0 12 0 11 0 12 S2 01 03 SI S3 02 04 Sweets 0 24 0 24 0 24 0 22 0 23 0 23 0 23 D1 02 03 04 SI S2 S3 Nonal. bev. 0 27 0 28 0 27 0 27 0 28 0 26 0 28 03 SI S3 01 S2 02 04 Fat & oil 0 09 0 09 0 09 0 09 0 09 0 09 0 09 SI S2 S3 01 02 03 04 Misc. foods 0 29 0 30 0 32 0 27 0 29 0 27 0 31 01 03 SI D2 S2 04 S3 FAFH 1 31 1 32 1 44 1.28 1 35 1 28 1 43 D1 03 SI S2 02 04 S3 Entire System 0.66 0.62 0.67 0.60 0.63 0.60 0.66 01 03 S2 02 SI 04 S3 a/ Model SI : including the first autocorrelation, AR(1), a dummy variable (D), family size (FS), and age of household head (AGE); 52 : including D, FS, AGE; 53 : including AR(1), D; D1 : including AR(1), D, FS, AGE, 1agged quantity variable (Qta1),and time trend (T); D2 : including AR(1), D, Qt.,, FS, AGE; D3 : including AR(1), D, T, FS, AGE; D4 : including AR(1), D, Qt.r b/ RMS-E is computed by weighted average RMS-E of each commodity. 190 The RMS-E measure is based on absolute term, and thus the measure 1s likely to be affected by the magnitude of expenditure levels. Therefore, a higher expenditure levels may result in a higher RMS-E. To overcome this shortcoming, another useful criterion is based on the root-mean-square percentage error (RMS-PE). The measure of RMS-PE Is defined as RMS-PEj-. i * ( „ 100 (89) This is also a measure of deviation of the predicted value from its actual value but expressed in percentage term. Table 47 presents the results of RMS-PE calculation. Again, the results reveal that sweets, seafood and beef have a relatively high RMS- PE in comparison with other commodities in the system. On the contrary, food away from home has a relatively low RMS-PE although its RMS-E is high. This result confirms our concern about the shortcoming of RMS-E expressed in absolute term. In general, the three dynamic models Dl, D2, and D3 show that the percentage errors for 19 food commodities are all less than 10%. Actually, the percentage errors for several commodities are only about 4% which imply a very good performance especially for monthly data. Considering the entire system, the weighted average RMS- PE 's show that Models D3 and Dl are ranked first and second, respectively. These values differ only by four one-hundredth of a point. Basically, the above discussed three measures of predictive performance, I, RMS-E and RMS-PE, are all stemmed from statistical deviation concept. These three criteria have been commonly used for Table 47. Comparison of Root-Mean-Square Percentage Errors for Different LA/AIDS Specifications Models® Accuracy Rankina Commodities SI S2 S3 Dl D2 D3 D4 1 2 3 4 5 6 7 Cereal 8 .18 4 .22 4 .28 4 .02 4. 04 4,.02 4 .15 Dl D3 D2 D4 S2 S3 SI Bakery 6 .81 3 .69 3 .82 3 .95 3. 98 3 .90 3 .96 S2 S3 D3 Dl D4 D2 SI Beef 9.82 7 .17 7 .87 6 .80 7.51 6 .65 8 .02 D3 Dl S2 D2 S3 D4 SI Pork 8 .18 7 .04 7 .18 6 .66 6 .99 6 .58 7 .25 D3 Dl D2 S2 S3 D4 SI Other meats 8 .13 6 .64 7 .06 6 .59 7. 30 6 .51 7.31 D3 Dl S2 S3 D2 D4 SI Poultry 5.35 5 .06 5 .45 5 .47 5. 37 5 .25 5 .62 S2 D3 SI D2 S3 Dl D4 Seafood 2 0 .89 10 .42 9 .65 9 .65 9. 55 9 .54 9 .75 D3 D2 Dl S3 D4 S2 SI Eggs 15. 04 5 .48 5 .53 5 .24 5. 29 5 .27 5 .47 Dl D3 D2 D4 S2 S3 SI Milk 9. 41 3 .67 3 .82 3 .70 3. 96 3 .55 3 .91 D3 S2 Dl S3 D4 D2 SI Other dairy 3. 62 3 .62 3 .62 3 .76 3. 77 3 .62 3 .81 SI S2 S3 D3 Dl D2 D4 Fresh fruit 6 .00 5 .99 5 .98 6 .26 6 .43 5 .94 6 .40 D3 S3 S2 SI Dl D4 D2 Fresh veg. 4. 51 4 .58 4 .58 4 .54 4. 52 4 .52 4 .64 SI D2 D3 Dl S2 S3 D4 Proc. fruit 5. 18 5 .26 5 .50 5 .07 5. 18 5 .05 5 .49 D3 Dl SI D2 S2 D4 S3 Proc. veg. 6 .55 6 .55 6 .76 6 .57 6 .58 6 .50 6 .75 D3 SI S2 Dl D2 04 S3 Sweets 1 0 .36 10 .33 10 .46 9 .47 9.92 9 .99 10 .09 Dl D2 D3 D4 S2 SI S3 Nonal. bev. 4. 70 4 .76 4 .68 4 .74 4.82 4 .60 4 .72 D3 S3 SI D4 S2 Dl D2 Fat & oil 5. 29 5 .33 5 .35 5 .10 5. 16 5 .14 5 .22 Dl D3 D2 D4 SI S2 S3 Misc. foods 4. 12 4 .18 4 .59 3 .83 4. 11 3 .86 4 .44 Dl D3 D2 SI S2 D4 S3 FAFH 3. 72 3 .76 4 .09 3 .64 3. 81 3 .65 4 .06 Dl D3 SI S2 D2 D4 S3 Entire System 6.00 4.87 5 .10 4.75 4.95 4.71 5 .13 D3 Dl S2 D2 S3 D4 SI a/ Model SI : including the first autocorrelation, AR(1), a dummy variable (D), family size (FS), and age of household head (AGE); 52 : including D, FS, AGE; 53 : including AR(1), D; Dl : including AR(1), D, FS, AGE, 1agged quantity variable (Qt-i)»an if _,>0 (91) otherwise Combining Eqs. (90) and (91), we therefore can define another index variable which can be used to measure the correct prediction on the changes of direction as follow: 1 if Dit=DPit (92) 0 otherwise Consequently, the number of correct predictions in direction change for commodity 1 is T TP, - E TPic (93) t-2 Note that we can also write Eq. (93) in terms of the ratio of correct prediction in direction change as TPR,«TP,/T-1. The computation results of TP measure are presented In Table 48. The results reveal that the ratios of correction in predicting turning points are higher than sixty percent, and the number of correct predictions In direction change varies from 45 to 60 for different commodities under alternative model specifications. It 1s also noted that the turning points on beef consumption can be predicted very well by various model specifications. This result 1s a sharp contrast to the findings based on the previous three statistical deviation criteria. For this reason, we carefully examine the data series for beef. We observe there are three unusual, in fact abnormal observations of beef consumption during the sample period. They may be considered as outliers in statistical sense because they may result from measurement error or other unaccounted factors. Generally speaking, Models Dl and D2 are preferred, according to this criterion for most commodities in the system. Another measure is used to evaluate whether the model has the ability to correctly predict the direction of expenditure level movement. To perform this test, we adopt Merton's framework (Merton, 1981) which is usually called the Merton market timing test. We specify first the following regression equation for commodity 1 based on the direction variables defined previously: (94) where the coefficient of slope, b{, is the probability of a correct prediction on direction change of expenditure level minus one. As a result, if b, is significantly greater than zero, then the model has the 194 Table 48. Comparison of Accuracy Number on Turning Point for Different LA/AIDS Specifications Model sa______Accuracy Ranking Commodities SI S2 S3 Dl D2 D3 D4 1 2 3 4 5 6 7 Cereal 44 .41 43 50 47 49 43 DlD3D2SIS3 D4 D2 (0.57) (0.54)(0.57)(0.66)(0.62)(0.64)(0.57) Bakery 45 46 52 57 56 55 54 DlD2 D3 D4 S3 S2 SI (0.58) 0.61)(0.68)(0.75)(0.74)(0.72)(0.71) Beef 49 46 51 54 49 53 52 Dl D3 D4 S3 SI D2 S2 (0.63) 0.61)(0.67)(0.71)(0.64)(0.70)(0.68) Pork 44 44 49 53 50 54 46 D3Dl D2 S3 D4 SI S2 (0.57) 0.58)(0.64)(0.70)(0.66)(0.71)(0.61) Other meats 45 43 45 43 46 44 39 D2 SI S3 D3 S2 Dl D4 (0.58) 0.57)(0.59)(0.57)(0.61)(0.58)(0.51) Poultry 47 51 43 48 49 52 40 03 S2 D2 Dl SI S3 D4 (0.61) 0.67)(0.57)(0.63)(0.64)(0.68)(0.53) Seafood 44 41 42 46 50 44 47 D 2 D4 Dl SI D3 S3 SI (0.57) 0.54)(0.55)(0.61)(0.66)(0.58)(0.62) Eggs 52 50 55 58 57 57 55 DlD2 D3 S3 D4 SI S2 (0 .6 8 ) 0.66)(0.72)(0.76)(0.75)(0.75)(0.72) Milk 46 43 43 43 43 45 42 SID3 S2 S3 Dl D2 D4 (0.60) 0.57)(0.57)(0.57)(0.57)(0.59)(0.55) Other dairy 46 44 45 46 49 43 49 D2 D4 SI Dl S3 S2 D3 (0.60) 0.58)(0.59)(0.61)(0.64)(0.57)(0.64) Fresh fruit 44 45 44 46 48 43 46 D2Dl D4 S2 SI S3 D3 (0.57) 0.59)(0.58)(0.61)(0.63)(0.57)(0.61) Fresh veg. 45 39 49 48 46 45 48 S3Dl D4 D2 SI D3 S2 (0.58) 0.51)(0.64)(0.63)(0.61)(0.59)(0.63) Proc. fruit 54 50 50 53 53 52 49 SI Dl D2 D3 S2 S3 D4 (0.70) 0.66)(0.66)(0.70)(0.70)(0.68)(0.64) Proc. veg. 46 41 47 48 47 45 44 DlS3D2SID3D4 S2 (0.60) 0.54)(0.62)(0.63)(0.62)(0.59)(0.58) Sweets 41 31 40 50 50 50 46 DlD2D4 D3 SI S3 S2 (0.53) 0.41)(0.53)(0.66)(0.66)(0.61)(0.63) Nonal. bev. 47 43 45 44 42 43 42 SI S3 S2 Dl D3 D2 D4 (0.61) 0.57)(0.59)(0.58)(0.55)(0.57)(0.55) Fat & oil 54 46 52 54 55 54 55 D2 D4 SI Dl D3 S3 S2 (0.70) 0.61)(0.68)(0.71)(0.72)(0.71)(0.72) Misc. foods 49 49 46 50 50 54 45 D3 Dl D2 SI S2 S3 D4 (0.64) 0.64)(0.61)(0.66)(0.66)(0.71)(0.59) FAFH 55 52 49 56 51 58 47 D3 Dl SI S2 D2 S3 D4 (0.71) 0.68)(0.64)(0.74)(0.67)(0.76)(0.62) Entire 897 845 890 947 938 936 891 Dl D2 D3 SI S3 D4 S2 System0 (0.61) 0.59)(0.62)(0.66)(0.65)(0.65)(0.62) a/ Model SI : including AR(1), D, FS, and AGE; 52 : including D, FS, AGE; 53 : including AR(1), D; Dl : including AR(1), D, FS, AGE, CL.,and T; D2 : including AR(1), D, Qt.„ FS, AGE; D3 ; including AR(1), D, T, FS, AGE; D4 : including AR(1), D, Qt.r b/ Figures in parentheses are accuracy ratios, c/ TP is sum of accuracy number on turning point of each commodity. 195 ability to correctly predict the direction of expenditure level movement (Breen, et al., 1989). The estimates of slope coefficient along with their t-rat1on are provided In Table 49. As one can see, the numbers of significant coefficients of bf are different among alternative model specifications. For example, 15 out of 19 coefficients are significant for Model D2 whereas only 5 out of 19 are significant for Model S2 at a conventional level of c=5%. It 1s noted that the estimates of slope coefficients are insignificant for other meats, milk, and fresh fruits regardless of model specification. In fact, the results indicate that all seven specifications of LA/AIDS model do not have ability to correctly predictthe direction of expenditure level movement for those commodities. According to the Merton market timing test, Model D2 Is the most accurate specification in predicting direction change. Finally, Appendix I presents the graphical comparisons of actual and predicted values of expenditure levels for 19 food categories during the entire sample period, using Model D2. These graphs generally confirm the assessments of predictive performance based on the five statistical criteria discussed above. In conclusion, based on above five criteria, no single model specification can outperform other specifications for every commodities in the system. Usually each specification may be appropriate only for some particular commodities. For instance, we found Model S2 has the best predictive power for bakery products and poultry whereas Model SI is good for other dairy products and fresh vegetables. In fact, the results from these predictive performance evaluations are consistent 196 with previous results of elasticity comparisons. To put together a general ranking of these seven specifications of the LA/AIDS model with respect to the entire system, we would need subjective judgements. The ranking would depend on the criteria selected. One general conclusion is that the dynamic models tend to outperform the static models regardless of criteria used. If the three statistical deviation measures are only the criteria used, then Models Dl and D3 would be selected as superior. Models S2 and D2 would be both classified as the next best while Models SI, S3, and D4 would be considered as inferior. However, if we also take into consideration of the ability to correctly predict the turning points, then Model D2 would be the best model specification. 197 Table 49. Comparison of Slope coefficients in the Merton Market Timing Test for Different LA/AIDS Specifications* 'Models*' Commodities SI S2 S3 Dl D2 D3 D4 Cereal 0.17 0.10 0.16 0.32 0.26 0.32 0.13 (1.46)c 0 .8 8 ) (1.36) (2.90) (2.35) (2 .8 8 ) (1.12) Bakery 0.17 0.21 0.39 0.51 0.48 0.46 0.44 (1.49) 1.80) (3.56) (4.99) (4.61) (4.43) (4.16) Beef 0.29 0.22 0.30 0.42 0.30 0.39 0.34 (2.56) 1 .8 8 ) (2.81) (3.97) (2 .6 8 ) (3.63) (3.14) Pork 0.13 0.15 0.29 0.39 0.32 0.42 0.21 (1 .2 0 ) 1.36) (2.59) (3.74) (2.87) (4.06) (1.84) Other meats 0.17 0.13 0.18 0.13 0.21 0.15 0.02 (1.46) 1 .1 2 ) (1.60) (1.12) (1.85) (1.36) (0 .2 0 ) Poultry 0.21 0.34 0.13 0.26 0.29 0.36 0.04 (1.89) 3.11) (1.09) (2.31) (2.57) (3.38) (0.39) Seafood 0.14 0.08 0.10 0.21 0.32 0.16 0.24 (1.25) 0.69) (0.91) (1.85) (2 .8 6 ) (1.37) (2.10) Eggs 0.34 0.31 0.43 0.52 0.49 0.49 0.44 (3.15) 2.82) (4.24) (5.27) (4.90) (4.91) (4.27) Milk 0.19 0.12 0.12 0.12 0.12 0.18 0.10 (1.69) 1.04) (1.04) (1.04) (1.08) (1.60) (0.82) Other dairy 0.19 0.16 0.18 0.21 0.29 0.13 0.29 (1.72) 1.38) (1.60) (1.85) (2.60) (1.14) (2.62) Fresh fruit 0.14 0.18 0.16 0.21 0.27 0.13 0.21 (1.25) 1.60) (1.36) (1.88) (2.46) (1.16) (1 .8 8 ) Fresh veg. 0.17 0.03 0.29 0.26 0.21 0.19 0.27 (1.51) 0.24) (2.59) (2.35) (1 .8 6 ) (1.63) (2.42) Proc. fruit 0.40 0.32 0.32 0.39 0.39 0.37 0.29 (3.84) 2 .8 8 ) (2.87) (3.71) (3.71) (3.42) (2.61) Proc. veg. 0.19 0.08 0.24 0.26 0.24 0.18 0.16 (1.72) 0.69) (2.10) (2.35) (2.10) (1.63) (1.38) Sweets 0.06 0.18 0.05 0.32 0.32 0.21 0.26 (0.55)( 1.63) (0.45) (2 .8 8 )(2 .8 8 )(1 .8 8 ) (2.36) Nonal. bev. 0.23 0.12 0.20 0.15 0.10 0.14 0.11 (2 .0 2 ) 1.08) (1.76) (1.34) (0.87) (1.18) (0.92) Fat & oil 0.39 0.20 0.36 0.41 0.44 0.41 0.44 (3.76) 1.80) (3.38) (3.98) (4.29) (3.98) (4.27) Misc. foods 0.27 0.29 0.21 0.32 0.32 0.42 0.18 (2.45) 2.61) (1 .8 6 ) (2 .8 8 ) (2.87) (4.00) (1.63) FAFH 0.43 0.36 0.29 0.47 0.34 0.53 0.24 (4.10) 3.38) (2.59) (4.60) (3.15) (5.31) (2.11) a/ Regression model: P..«=Oj+pfAit+e,t, Pit is predict direction variable for commodity i at time t; A,t is actual direction variable for commodity i at time t. b/ Model SI ncluding AR(1), D, FS, and AGE; 52 ncluding D, FS, AGE; 53 ncluding AR(1), D; Dl ncluding AR(1), D, FS, AGE, Qt„«,and T; D2 ncluding AR(l), D, Qt ., FS, AGE; D3 ncluding AR(1), D, T, FS, AGE; D4 . i ncludingiiv. i uvi i n y nn^iAR(l), / , is, D, i Qt| , . ^ • c/ Figures in parentheses are t-statistics. CHAPTER VIII SUMMARY AND CONCLUSIONS This chapter provides a brief summary of the objectives, empirical results, and conclusions; the limitations of the research; and their implications for future research. First, the objectives of this research are briefly summarized. Second, the important empirical findings are reviewed. Finally, the limitations of this research and their implications for further research are discussed. 8.1 Objectives of Research: The primary objective of this dissertation research is to link the nonparametric and parametric approaches in analyzing food consumption behavior. This objective is accomplished by selecting the most appropriate complete demand systems for 19 food categories in the United States. This research uses one of the most complete and consistent data bases available to date and it employs more recently developed and advanced econometric procedures than previously employed. This approach is used to narrow the gap between theoretical and empirical demand analysis. Data used in this study were obtained from the continuing Consumer Expenditure Surveys (CES) conducted by the Bureau of Labor Statistic (BLS) from 1980 to 1986. One of the most unique features of data used in this study is that the expenditure and price series for each of the food 198 categories specified for the study are very consistent because the consumer price index series of CPI-U was constructed using the expenditure weights obtained from the CES. The study also employ several advanced econometric procedures to bridge the gap between theoretical and empirical analysis of food demand. These advance econometric procedures include nonparametric testing on data consistency, multivariate statistical analysis, nonlinear system estimation,> and translating methods. With a consistent data base and valid econometric procedures, the empirical results of this research are consistent with the underlying microeconomic theory. The estimates of demand parameters in this study should accurately reflect the shaping of the U.S. food consumption trends in the sample period of 1980-1986. 8.2 Empirical Findings Total food expenditures (in current dollars) including food at home and food away from home increased during the sample period. However, the share of food at home and away from home changed significantly. This trend can be partly attributed to the changes in demographic variables: more participation of women in the labor force, changing age composition of the population, and new and varied life styles. The results reveal decreases in expenditures on beef, pork, other meats, and eggs. Such changes are likely to reflect consumer's concerns about the effects of cholesterol, saturated fat and the related health risk associated with consumption of red meats. By contrast food items with increases in expenditures appear to be considered as healthful food. Included among these are fish and vegetables with 200 nutritional benefits, as well as convenience goods such as frozen prepared meals and other prepared food items included in the miscellaneous foods category. Other important and interesting findings from nonparametric and parametric analyses are summarized as follows. 8.2.1 Nonparametric results One of the most Important contributions of this study Is that it uses a nonparametric approach to examine the data consistency, data structure, and preference structure. The study places more emphasis on the information contained in the data themselves rather than making assumptions about the structure of the data generating process under investigation. It is important to an empirical Investigation because the more that is known about the structure and characteristics of the data, the less likely the empirical model is to suffer from the specification error and thus the results of estimation would be more reliable. Several interesting findings from nonparametric analyses as described below, in fact, provide useful and important insights for formulating the parametric models for empirical estimation. There are three main findings from the nonparametric GARP testing on data consistency. (1) The GARP testing reveals that the expenditure data from the CES's diary surveys and their corresponding price data from CPI-U are consistent with the utility maximization process during the sample period except the earlier months in 1980 and five other monthly observations. (2) The 19 food expenditure categories seem to constitute a coherent group of commodities satisfying the assumptions of the maximization of a utility function by a representative consumer and 201 weak separability among its subgroups. (3) The 18 expenditure categories of only food at home did not satisfy the consistency test unless all December observations were excluded. This finding implies that the consumption of food at home in December must be different from other months in the year and probably related to food consumption away from home. The rank test using a parametric approach shows that the rank of the demand system for 19 food commodities based on our data base is three while the rank may be equal to two if eggs and nonalcoholic beverage are excluded. This result suggests that the minimum number of forming aggregate groups from the 19 food categories is three. The result further shows that the Engel functional form could be different among food commodities In this food demand system. A factor analysis based on principal component estimation method reveals that the four common factors can explain about 70 percent of total variations of expenditures of the 19 food categories. Meanwhile, the factor loadings indicate that each food category has multi factor characteristics. Furthermore, according to the hierarchical structure of clustering analysis, our 19 food categories can be appropriately classified into six aggregate food groups. They are: (1) cereal, bakery products, processed fruits, fat and oil, miscellaneous foods; (2) beef, pork, other meats, eggs, and processed vegetables; (3) poultry, milk products, and sweets; (4) seafood and other dairy products; (5) fresh fruits, fresh vegetables, and non-alcoholic beverage; and (6) food away from home. The GARP test shows that these six aggregate groups are all consistent with utility maximization. On this basis, this grouping is 202 more valid than the traditional grouping used by the BLS. There are many violations to GARP if the traditional way of grouping meats is followed (i.e., beef, pork, other meats), poultry, seafood, and eggs together. Moreover, the rank test based on a nonparametric approach indicates that the rank condition of these six aggregate groups is two. 8.2.2 Parametric estimation results In order to search for and specify appropriate functional forms for parametric estimation, a model specification test is conducted. The test result suggests that the PIGLOG demand system is the most appropriate model for this study. Seven alternative specifications of the LA/AIDS model are formulated to estimate demand parameters of the 19 food categories under a one-stage budgeting procedure. Among these seven specifications, the test statistics show that dynamic factors, demographic factors, and first-order autocorrelation should be incorporated into model specification. The statistical results related to elasticity estimates obtained from alternative LA/AIDS specifications can be summarized below: (1) The estimated demand elasticities for cereal, bakery products, pork, other meats, poultry, seafood, fresh vegetables and processed fruits are fairly robust among alternative model specifications. (2) The estimated own-price elasticities for beef, milk, other dairy products, processed vegetables, sweets, miscellaneous foods, and food away from home are sensitive to the inclusion or deletion of dynamic factors, 1agged quantity variable, and time trend, in model specification. The estimates for beef have an expected sign only under 203 the specification incorporating the time trend as a dynamic variable. Meanwhile, demographic variables are found to have substantial impacts on the estimates of price elasticity for milk and other dairy products. (3) Food away from home Is the only one category among 19 food commodities, having a significant negative elasticity with respect to family size. This result may be due to the economies of scale in the production of food at home. (4) Six food commodities, i.e., beef, other meats, milk, fresh vegetables, sweets, and fat and oil, have negative elasticities with respect to age of household head. On the other hand, poultry and seafood have positive elasticity estimates for this demographic variable. (5) The estimated cross-price elasticities indicate that interdependent relationships among 19 food commodities reveal some interesting food consumption patterns. For example, among three red meats, the cross-price estimates show that they are substitutes. Poultry is found to be a substitute for beef, pork, seafood but a complement with other meats. Meanwhile, seafood and beef, as well as pork are complements while with other meats and poultry, are substitutes. In order to compare alternative flexible functional forms, three PIGLOG models, the full model based on Lewbel (1989), translog and LA/AIDS, are employed to estimate structural parameters under a two- stage budgeting procedure. The test statistics show that the full model is significantly different from translog and LA/AIDS for the aggregate group in the first stage and for subgroups I and III in the second stage. However, elasticity estimates reveal a great deal of similarity between the translog and LA/AIDS models. With all three models, the 204 estimated expenditure elasticities for seafood, sweets, miscellaneous foods, and food away from home are greater than one, and thus these food commodities are expenditure-elastic. In comparing the estimated elasticities obtained from one-stage and two-stage procedures, the results clearly indicate that the differences are significant. A logical conclusion, therefore, is that the elasticity estimates are sensitive to the assumptions of underlying preference structure and model specifications. Large differences of estimated elasticities could be produced from assuming a different preference structure and model specification even though the data used in the study are the same. Finally, words of caution must be made regarding some implausible and unusual estimates of demand elasticities obtained in this study. (1) Some estimates are unreasonable. For instance, own-price elasticities of fat and oil have unexpective sign. Cereal and sweets also have unexpected positive own-price elasticities under a two-stage budgeting procedure. (2) The estimated own-price elasticities for beef are very sensitive to choice of dynamic factors. The estimates range from -0.39 to 0.31 under one-stage budgeting procedure. However, the estimates from a two-stage budgeting procedure yield -0.32 for all three models using the time trend as a dynamic factor. This evidence seems to indicate that the preference structure of beef consumption may have undergone structural change over time. Therefore, the use of these unstable estimates of price elasticity of beef for policy analysis must be done with caution. (3) The estimated own-price elasticities for milk and dairy products are also sensitive to the choice of dynamic factors, demographic variables, as well as the assumption of preference structure in model specification. The estimated own-price elasticities for milk and other dairy products vary from inelastic to elastic depending upon model specifications. In the case of milk and other dairy products, the magnitude of the own-price elasticity has important policy implications. t \ For instance, if the own-price elasticity of milk and milk products is elastic then producers can gain benefits from reducing its price. In this case, government price subsidy may not be needed to maintain income for milk producers. Therefore, one must be caution in the use of these estimates of price elasticity for milk and other dairy products for policy analysis. Based on this study, an evaluation of government milk price support program will be subject to some degree of uncertainty because the estimated price elasticities for milk are not very robust. 8.2.3 Evaluation of predictive performance Five different criteria are used to evaluate the predictive performance of seven LA/AIDS specifications over the sample period. These five measures are average information inaccuracy, root-mean-square error, root-mean-square percentage error, number of accurate predictions of turning points, and Merton market timing test. Based on these five criteria, it is found that no single model specification can outperform others, and each specification may be appropriate for some particular food categories. In fact, the results obtained from the evaluation of predictive performance are consistent with those of elasticity 206 estimates. Finally, we provide a general ranking tof the seven alternative specifications for the entire system of 19 equations. The dynamic models are always preferred to static models no matter what criterion is used. If three statistical deviation measures are the only criteria, the dynamic models incorporating either a 1agged quantity variable and a time trend or only a time trend would be superior. However, if we also take into consideration of the ability to correctly predict turning points and the plausibility of estimated elasticities, then the dynamic model with only a 1agged quantity variable would be the most appropriate model specification for this study. 8.3 Limitations and Further Study The important contribution of this study is twofolds. One is the development of a methodology for implementing the neoclassic demand model for food commodities. The other is the development of econometric models for analyzing food demand and consumption using a unique monthly data base. However, this research does not provide an answer to every demand-related question. Much work need to be continued in order to gain a better understanding of food demand and consumption in the United States. Some limitations of this study are provided in the following discussion which, in turn, would suggest several interesting areas for future research. First, the demographic variables used in this study are household size and age of household head. These two variables have been shown to have significant impacts on food consumption. However, as mentioned in earlier chapters, many other factors such as race, women of labor force participation, and geographic regions, may also be important determinants to food consumption. Indeed, these socio-economic and geographic information are also available in the BLS's CES public-use tapes. However, these variables can not be simply averaged on a monthly basis. One possible extension of this research would be to construct a monthly weighted average data bases for various groups of households by demographic variables, socio-economic or geographic regions, and to estimate food demand systems for various household groups. The comparison of econometric results among household groups would provide valuable information to decision makers in the food marketing industry and government agencies. Furthermore, one can also integrate CES diary survey data with interview survey data to analyze consumption patterns and structure with respect to entire consumption budget. This integrated model will provide further insights on consumption behavior of food and other nonfood commodities and services in the United States. Second, the expenditure data used in this study are aggregate monthly data which are generated by using weighted average methods. Basically, the use of aggregate data can avoid the problem of zero expenditures in demand estimation. However, this aggregation may distort the data structure because the occurrences of reported zero expenditure may be due to reasons such as infrequency of purchase, the durability of the commodity, or just no consumption. In fact, one can conduct an econometric analysis to directly use the household level data and deal with econometric problems related to limited dependent variables in estimation. Several studies in this area are available in the 208 literature, such as Wales and Woodland (1983), Lee and Pitt (1986), Deaton and Irish (1987), Pudney (1989). However, the problems of zero expenditure and/or corner solution in estimating a system of equations is still not completely solved. In order to use diary survey data effectively for analyzing consumer behavior, further work in the area of zero expenditures and/or corner solutions must be conducted in future research. Third, in estimating the structural parameters of demand systems, we treated prices and total expenditure of food categories as exogenous variables. However, they may be endogenous in the sense that they are correlated with error terms. The reasons for their endogeneity are (1) prices of food categories may be determined by demand and supply simultaneously, especially when monthly data are used as in this study; (2 ) the budget share is defined as the ratio of expenditure of each food category to the total expenditure, and these shares are dependent variables in the study. Therefore, the total expenditure may be considered as endogenous in the system. Since total expenditure is one of the right hand side independent variables, one most rightfully treat the demand model as a simultaneous equations system and has to deal with such problems as specification, identification, and estimation. For further research, one may adopt the approach developed by Jorgensen et al. (1982) to solve the endogeneity of price variables in order to avoid involving supply side into demand system. Furthermore, one may follow the approach developed by Attfield (1985) to deal with the endogeneity in total expenditure variable. Fourth, two multivariate statistical techniques are employed to perform commodity grouping in this study. Based on clustering analysis, 19 food categories are classified into six food groups. Essentially, we expect the categories within each food group to be stronger substitutes or complements to each other than to those food categories outside the group. However, the estimated cross-price elasticities obtained from a two-stage budgeting procedure are not entirely consistent with this expectation. This finding may be due to a possibility that the clustering results do not really reflect the preference structure in the data. For further research, one can try to use different estimation methods in multivariate statistics for conducting commodity grouping. This is a very interesting and important area in empirical demand analysis especially for a large system with many commodities. Finally, in this research we only focus on the analysis of food demand structure and consumption pattern. Forecasting or projecting food demand is another important objective in conducting a demand analysis. For the purpose of forecasting food demand in the United States under various scenarios, one can use the estimated demand parameters obtained in this research and adopt the synthetic approach used by Brandow (1961) to develop a more required matrix of price and expenditure elasticities for the 19 food categories under investigation. For further research, simulation approaches such as the bootstrap method can also be used to forecast U.S. food demand in the future. FOOTNOTES 1. For the detailed proof, see Varian (1984, pp 113). 2. See Bieri and de Janvry (1972), Henderson and Quant (1980). 3. See Deaton and Muellbauer (1980) and Varian (1984). 4. Advantages of using duality in empirical demand studies are: (a) it allows us to derive a system of demand functions simply by differenting a function (indirect utility function or expenditure function), (b) a system of demand functions can be expressed in terms of only exogenous variables; (c) the indirect utility function can be applied in welfare calculations or in the construction of true cost-of-livingindex number; (d) the duality approach permits us to study the structure of utility functions; (e) The indirect utility function facilitates the incorporation of additional nonprice variables in the analysis of consumer demand. ( see Diewert (1974), Lau (1974)). 5. For references about various forms of separability, see Bieri and de Janvry (1972), Brown and Deaton (1972), Pudney (1981), and Johnson, Hassan and Green (1984). 6 . For a more detailed discussion, see Howe (1977), Derrick and Wolken (1982). 7. Equivalent scales was originally developed by Engel (1895). A well-known example was given by Paris and Houthakker (1955) using commodity specific household equivalent scales and a general scale for deflating total expenditure in this classical study. Equivalent scales were also formulated and discussed by many researchers such as Barten (1964),Gorman (1976), Muellbauer (1974), Poliak and Wales (1980,1981), Chern and Soberon-Ferrer(1986) and Jorgensen and Slesnick (1987). 8 . A consumer unit refers to one of the following: (a) the collection of all members of a household who are related by blood, marriage, adoption, or other legal arrangement; (b) a person living alone or sharing a household with others or living as a roomer in a private home or lodging house or in a permanent living quarters in a hotel or motel, but who is financially independent; or (c) two or more persons who live together and pool their income to make joint expenditure decisions. 2 1 0 211 9. More detailed information about sampling design and Important survey concepts are provided by the BLS in their Diary Survey publlc-use tapes documentations. 10. Weighting-- each consumer unit in the CES's sample represents a given proportion of families in the U.S. population, which is considered as the universe. The weights were derived from a weighting procedure based on the urban control totals of the Current Population Surveys (CPS). Several factors are involved in determining the weights: the probability of selection of the housing unit; a weight control factor to adjust for subsampling in the field; a noninterview adjustment which is a function of region, tenure, consumer unit size and race; a national rat1o-estimate adjustment for 12 age, 2 sex and 2 race levels; and a final adjustment based on CU composition. The Weighting procedure used in CES is similar to that used in other large household surveys such as CPS. However, in order to improve weighting method, BLS has been using the new weighting method, called the Generalized Least Squares (GLS) procedure, to determine the weights for CES' data since 1985. More detailed information about the GLS procedure, see Garner and Zieschang (1986), " Weighting the Consumer Expenditure Survey Data: The GLS Study,”. 11. For more information about weighting factors, consumer unit weight 21, see Diary Survey public-use tape documentation. 12. See Diary Survey public-use tape documentation for detailed UCC items. 13. For more information about X-ll procedure, see the SAS/ETS manual and BLS (1977) technical paper. 14. New demand theory (household production functional model) postulates that consumer is interested in a set of entities defined variously as 'basic commodity' (Becker, 1965; Michael and Backer, 1973) and 'characteristics' (Lancaster, 1971) from which utility is directly obtained rather than market goods. 15. See Goldberger (1974) and Griliches (1974). 16. A more complete discussion of factor analysis 1s provided in Morrison (1976), and Johnson and Wichern (1988). 17. For more detailed discussion in clustering methods, see Anderberg, (1973), Harman (1976), as well as Johnson and Wichern (1988, pp. 543 - 570). 18. The iterative reassignment of variables to clusters proceeds in two phases: (a) a nearest component sorting phase; and (b) a search algorithmn in which each variable in turn is tested to see if assigning it to a different cluster increases the amount of variance explained.(see SAS/ETS manual) 19. An Index number functional form is said to be "superlative” if it is exact (i.e., consistent with) for a flexible aggregator functional form. More detailed information about exact and superlative index, is available in Diewert (1976). 9 APPENDIX A REVIEW ON COMPLETE DEMAND SYSTEMS 213 214 Review on Complete Demand Systems There are three alternative approaches which can be employed to derive an explicit system of demand functions for an empirical analysis. In this Appendix we will only focus on demand systems derived from specifying a direct utility function as well as by the duality approach. Two relatively comprehensive review articles on consumer economics have been published in the economics literatures. One article, written by Brown and Deaton (1972), mainly deals with the micro aspects of consumer economics. The other article by Ferber (1973) focuses on the macro aspects of consumer economics. Both articles provide basic concepts for empirical demand analysis. For gaining an overview of complete demand systems, a good introduction to this subject is provided by Barten (1977). A more formal introduction is available in texts by Powell (1974), Philips (1974), and Theil (1975, 1976). Unfortunately, many new developments in complete demand systems using the duality approach are not covered in these review articles. In this Appendix, we will attempt to synthesize recent developments in this subject area. We will first deal with the approach based on a direct utility function, then turn to the duality theory. 215 I. Utility Specification Approach The first well-known demand system was the linear Expenditure System (LES) developed by Stone (1954), which was derived from a direct utility function suggested by Klein and Rubin (1947-1948). Several proposed demand systems in this category can be characterized by a Johasen utility function which is expressed as follows: where a{, b{, r, are parameters, and a, < 1 , b{ > 0 , rf < q,. Several widely known demand systems are special cases of this general utility function (see Figure 7). For instance, if r,**0 then the resulting demand system is a direct addilog function which was first introduced by Houthakker (1960). If a^O the relevant term in above equation is replaced by its limit for af - 0 then the resulting demand system is a LES. Other demand systems with additive utility function also can be derived from this specific form of utility function. Moreover, if a quadratic term is added into the equation, then a direct translog demand system can be generated (Christensen et al.,1975). Many previous demand studies used the family of the LES model such as Stone (1954), Houthakker(1957), Barten(1968), Parks(1971), Powell(1974), « Pol1ak and Wales(1969), Gamaletson(1973), Goldberger and Gamaletson (1970), Wales (1971), Lluch (1973), Lluch and William (1975), Lluch and Powel1 (1975), B1ancifori, et al. (1986), Chern and Lee (1989a,b). The most attractive feature of this kind of demand systems is parameters parsimony because they assume the goods to be strongly separable. This assumption results in only the direct specific substitution effect to exist empirically; the cross specific substitution effects are zero while the general substitution effects together with the income effect is negligible. Consequently, when we use this type of demand system for empirical demand analysis, we must pay for a price for economizing the parameters. Moreover, the proportionality of the direct price elasticity and the income elasticity imposed into a demand system may not be valid for many goods and services such as food. r,.=0 Direct Addilog System a,-?0 Linear 1-Branch Cobb-Douglas Constant Expenditure Demand Demand Elasticities System System System System Figure 7. Relationships between Additive Demand Systems 217 II. Duality Approach In recent years, a wealth of various flexible function forms for modeling demand systems have been developed by applied econometricans1. These developments are mainly based on the duality theory in consumption analysis. Two relatively complete surveys in this topic were done by Diewert (1974) and Lau (1977). In order to formulate an integral demand system, either an indirect utility function or a expenditure function is postulated and then differentiated to obtain the associated system of demand functions. The main reason for the popularity and wide adoption of these flexible demand systems for empirical demand analysis is that they do not impose a priori constrains on price and income elasticities over parameters and data space, and thus the statistical inference of demand structure would be more reliable. The first demand system derived from the duality approach was the indirect addilog model (Houthakker,1960). Note that the indirect addilog demand system is not a flexible function form. The best known flexible demand systems include the following. First, the generalized Leontief (GL) demand system was developed by Diewert (1969, 1974). The important applications of this model include Diewert (1974), and Darrough (1977). Another popular flexible form is the indirect translog demand system (ITL), introduced by Christensen, Jorgenson and Lau (1975). The ITL model has been widely applied in empirical studies including Christen at el.(1975), Jorgenson and Lau(1975), Lau and Mitchell (1970), 1 An algebraic functional form for a complete demand system is said to be flexible if the parameters of a complete demand system can be chosen so that the consumer demand functions and their own and cross prices and income elasticities are capable of assuming arbitrary values at given set of prices of commodities and income subject to the requirements of theoretical consistency (Lau, 1988). 218 Jorgenson et al. (1982), Ewis and Fisher (1984). Next the almost ideal demand system (AIDS) is developed and estimated by Deaton and Muellerbau (1980). The AIDS is derived from a cost function with a PIGLOG form of preference. Many applications in empirical demand analysis have been conducted by using a linear approximation of the AIDS such as Blanciforti, Green and King(1980), Chern, et a l. (1988), Eales and Unnevehr(1988), Moschini and Meilke (1989). These three flexible demand systems, GL, ITL and AIDS, can be classified as a second-order Taylor's series approximations2. Two other demand systems being closely related to these models are minflex generalized Leontief and minflex translog demand systems. The minflex GL and the minflex TL models stemmed from the Laurent expansion series are originated and analyzed by Barnett and his colleagues. Barnett and his colleagues further devoted their efforts to examine the flexibility of above mentioned flexible demand systems. They showed (1) all of above mentioned these systems have a very small region satisfying the theoretical regularity conditions and often perform poorly in Monte Cairo experiments; (2) the minflex Laurent systems appears to violate the theoretically approximate curvature conditions less often than the ITL and GL, and (3) the minflex laurent system still exhibits regions where the curvature conditions are violated. These results are fully documented in Barnett (1981,1982, 1983a, 1983b,1985), Barnett and Lee 2 Two definitions for second-order approximation are (a) the definition in mathematics: V* is a second order local approximation to V at the point Vo,if (V*(v) -V(v))/||v-v0| 2 - 0 as v - v0. (Barnett, 1983); (b) the definition in economics: V* is a seco d order approximation to v at v 1fV*(v )=V(v0), av*/av =3V/3v| and d*V /avav.l^-afySvav' l ^ , (Diewert(1971),Christensen,et al. (1973)). 219 (1985, 1987), Barnett et al. (1985,1987). Most recently, two normalized quadratic systems, proposed and estimated by Diewert and Wales(1988), are derived from a normalized quadratic reciprocal indirect utility function and a normalized quadratic expenditure function, respectively. Diewert and Wales showed that under the certain conditions both systems not only have good properties in flexibility also satisfy the appropriate curvature conditions globally. Another recent significant development is due to Lewbel (1987) who developed a class of demand systems, called fractional demand systems, which not only possess good property of global flexibility but also can characterize several commonly known demand systems such ITL and AIDS as special cases. Generalized Cobb-Douglas demand system is also a flexible demand system, proposed by Diewert (1973), which is based on neither Taylor's series nor Laurent series. It has been empirically implemented by Diewert (1973) and Berndt, Darrough and Diewert(1977). In addition, another well-known demand system derived from an indirect utility function but without the property of flexibility is quadratic expenditure demand systems (QES), introduced and estimated by Howe, Pol1ak and Wales (1979), and Poliak and Wales (1978, 1980) which are theoretically plausible demand functions. The QES is a generalization of LES in that the demand functions are quadratic in total expenditure. The empirical applications of the QES model include Kokosk1(l986), Chern and Lee(1989a, 1989b), Lee and Chern (1989). All above mentioned demand systems can be considered as parametric demand systems since the functional forms of demand system have been specified. Alternatively, two other demand systems appeared in the economics literature can be treated as semiparametric models. One is the Fourier demand system which is originally introduced and estimated by Gallant (1981, 1982, 1984). The fourier form has the property of Sobolev flexibility, which means the specification error bias can be made arbitrarily small by including additional terms in the Fourier series as the sample size increase. Note that Sobolev flexible form can estimate elasticities consistently (Elbadawiet et al. 1983). Empirical applications of the Fourier model is due to Gallant (1981), Ewis and Fisher (1985). Chalfant(1987) incorporates the Fourier series into the AIDS model and developed a globally flexible version of the AIDS model. Another semi parametric model is the asymptotically ideal model (AIM), developed and estimated by Barnett and Yue (1988), which is derived from the Muntz-Szatz series expansion. The Muntz-Szatz series expansion possesses good properties in both global flexibility in the same sense as Fourier model and ease in testing or imposition of theoretical regularity. Some theoretical characteristics of above mentioned systems of demand function are summarized in Table 51. 221 Table 50. Theoretical Characteristics of Selected Complete Demand Systems Demand Systems Mathematical Flexibility Minimality Global Expansion Propertity Propertity0 Regularity® 1 . Generalized Taylor's Local Yes Leontief Leontief 2 . Generalized None c c Cobb-Douglas Cobb-Douglas 3. Indirect Taylor's Local Yes Cobb-Douglas Translog 4. Quadratic None Local Yes Not Relevant Expenditure 5. Normalized Taylor's Local Yes May be imposed Quadratic Expenditure 6 . Almost Ideal Taylor's Local Yes May be imposed Demand System 7. Minflex Laurent Taylor's Local Yes d Systems 8 . Fourier Demand Fourier Global Not applicable® May be imposed Systems 9. Almost Ideal Muntz-Szatz Global Not applicable May be imposed Models 10 . Fractional Demand Systems Various Global Various May be imposed a/ Minimality propertity : A function possesses just enough parametric freedom to attain local flexibility, b/ Global regularity indicates the special case of the functional form which is globally regular, c/ Only in the class of homogeneous indirect utility function, d/ Minflex Laurent functional form is a special case of a second-order Laurent series expansion. The full Laurent expansion has the global regularity, e/ The number of estimated parameters is determined by a sample size rule to assure consistent estimation. 222 III. Concluding Remarks Some theoretical considerations on alternative flexible systems of demand functions in empirical demand study can be summarized as fol1ows: (1) The local approximation interpretation has important limitations for statistical properties (White, 1980). Therefore, one can not arbitrarily choose a point in the data space such as sample mean and assert that the elasticity matrix will be estimated consistently at that point regardless of what the true state of nature actually is. (2) There is no theoretical basis for restricting approximation on the second-order series expansion. In fact, as more data is available, it may be desirable to increase the order of the series approximation. For instance, a third-order translog utility function may be used to empirically test the propositions related to a partial strong separability, whereas second-order expansions yield ambiguous test results (Hayes,1986). Undoubtedly, the addition of higher-order terms would dramatically increase the number of parameters to be estimated. Thus the theoretical desirability may result in empirical difficulties. Therefore, one must determine which approximation methods provides the most flexibility at the least cost. (3) From practical point of view, there remain some unresolved problems such as difficulty in interpreting initial parameter estimates and the problems of convergence in estimation and achieving global regularity conditions. (4) There exists no one best parametric functional form for all purposes. A functional form may well-suited for a specific application, but poorly-suited for other situations. In reality, an empirical demand analysis is carried out with a specific purpose in mind. Each situation has specific conditions such as the number of observations and objectives such as forecasting which must be met in the specification of an econometric model. Therefore, selecting an appropriate functional form for empirical estimation has become not only a science but an art. Some criterion are usually used as the basis for selection: (a) theoretical consistency; (b) domain of applicability; (c) flexibility; (d) computational facility; and (e) factual conformity (Lau, 1986)3. Nevertheless, the main determinant for selecting a functional form is the objective of research. In choosing an appropriate functional form for analyzing and projecting the structure of food demand, King (1979) pointed out two areas of concern: (a) changing elasticities over time, and (d) relationship between income and price elasticities. APPENDIX B FOOD EXPENDITURE CATEGORIES 2 2 4 225 Table 51. Food Expenditure Categories CATEGORIES ITEMS 1. Cereals ready to eat and cooked cereals, pasta, flour, prepared flour mixes and other cereal products 2. Bakery Products bread, crackers and cookies, biscuits and rolls, cakes, cupcakes,bread and cracker products, pies,tarts, sweet rolls, coffee cakes, doughnuts, other specified frozen and refrigerated bakery products 3. Beef ground beef,roasts,steaks,veal and other cuts of beef excluding canned beef 4. Pork bacon,pork chops,ham (including canned), roasts, sausage and other cuts of pork 5. Other Meats frankfurters, lunch meats (cold cuts) such as bologna, liverwurst; also lamb, mutton, goat and game meat 6 . Poultry fresh and frozen chickens and other specified fresh and frozen poultry 7. Fish and Seafood canned fish and seafood, fresh or frozen fish and shellfish 8 . Eggs fresh eggs, powdered eggs, and egg substitutes 9. Fresh Milk and Cream fresh whole milk,other fresh milk fresh sour cream,fresh sour cream dressings 10. Other Dairy Products butter, cheese, ice cream and ice cream product, yogurt, powdered milk, condensed and evaporated milk, liquid and powered diet beverages, malted milk, milk shakes, chocolate milk and other specified dairy products 11. Fresh Fruits all fresh fruits 12. Fresh Vegetables all fresh vegetables 13. Processed Fruits all frozen fruits, fruits juices, canned and dried fruits, and canned or bottled fruits juices 14. Processed Vegetables all frozen vegetables, canned and dried vegetables, and vegetable juices 226 Table 51 (Continued) 15. Sugar and Other sugar, candy and chewing gun,artificial Sweets sweeteners,jams,jellies,preserves, fruits butters, syrup, fudge mixes, icings and other specified sweets 16. Nonalcoholic diet and non-diet carbonated drinks, coffee, Beverages tea, other beverages,nonalcoholic beverages, and other specified nonalcoholic beverages 17. Fats and Oils margarine, shortening and sal ad dressings, nondairy cream substitutes and imitation milk and peanut butter 18. Miscellaneous Foods frozen prepared meals and other foods, canned and packaged soups, potato chips, nuts and other snacks, and seasonings condiments, and seasonings, baking needs and other specified condiments, other canned and packaged prepared foods 19. Food Away from Home lunch, dinner, breakfast and brunch, snacks and non-alcoholic beverages at restaurants and carry outs plus meals at school in home city, board, meals for someone away at school and catered affairs APPENDIX C REGRESSION RESULTS OF LA/AIDS DYNAMIC MODEL II 2 2 7 Table 52. Regression Results of The LA/AIDS Model with AR(1), Qt.,, AGE, FS, and Da Estimation Method: ITSUR ______Commodities______Parameters 1 2 3 4 5 6 7 8 9 aiO(10’2) -0.0 t 0 .06 0 .11 0 .04 0 .06 0 01 0 02 0 .03 0.03 (-0.99) (2.96) (2 .14) (1.45) (2.09) (0 6 6 ) (1 .03) (4.75) (0.90) 0.37 -0 05 -0 03 0 30 0 67 2 11 -0 58 -0 40 0.65 aii■ i -0 / (1.56)( -0 65) (-0 08) (0 71) (1 67) (3 60) ( 74) (-1 42) (2.77) *12(10 > 0.47 2 31 -23 45 0 89 -2 48 4 05 3 26 1 12 -1.79 0 88 • 9 (0.33) (0 90) (-3 60) (0 28) (-1 09) (2 31) (1 37) (1 60) (- . ) *13(10 > 2.77 0 22 17 60 7 93 2 42 -0 33 2 13 -0 15 3.33 (1.31) (0 05) (1 78) (1 63) (0 70) (-0 12) (0 60) (-0 15) (1.08) 3.19 5 09 -4 90 -0 92 3 07 -0 50 0 96 -0 12 2.14 *«(I f ' 1 0 3) “P (2.33) (1 90) (-0 73) (-0 28) (1 34) (-0 30) (0 41)( -0 18) (1.07) b i dO'3) -0.43 1 97 -20 20 2 08 -3 92 4 84 -10 09 0 49 -8.88 (-0 .11) (0.27)( -1 ID (0.24)( -0 62) (0.95)( -1 47) (0.26)( -1.58) ciiC -3.35 (-0.92) Ci2 0.11 0 64 (0.75) (2 39) Ci3 0.38 -0 10 8 81 (0.31)( -0 23) (2 90) Ci4 -1.13 -0 39 0 80 1 47 (-1.26)( -1 44) (0 64) (1 68) Ci5 5.39 -0 44 2 31 2 33 -4 90 (2.16)( -2 36) (1 58) (2 17) (-1 44) Ci6 0.99 -0 06 0 33 1 06 -0 57 1 62 (1.27)( -0 32) (0 31) (1 91) (-0 70) (2 15) Ci7 3.47 -0 55 -0 61 -0 35 0 42 0 94 -2 38 (2.69)( -2 36) -0 47) (-0 52) (0 39) (1 40) (-2 03) Ci8 0.22 0 07 -0 11 0 45 0 10 0 03 -0 15 0 94 (0.75) (1 05) -0 27) (2 09) (0 32) (0 13) (-0 61) (8 38) Ci9 -8.40 -0 06 -0 55 -1 53 4 64 1 15 1 99 0 51 -2.38 (■3.17)( -0 32) -0 34) (-1 57) (1 74) (1 19) (1 59) (1 39) (-0.62) C i10 4.05 -0 18 -3 79 2 80 -3 11 -3 20 0 86 0 96 8.81 (1.05)( -0 83) -2 09) (2 30) (-0 8 8) (-2 8 6 ) (0 58) (2 36) (1.56) c i 11 0.33 0 23 -1 21 0 95 0 06 0 29 0 49 0 08 0.80 (0.72) (1 05) -1 39) (1 99) (0 12) (0 72) (0 97) (0 51) (1.38) Ci12 -0.08 -0 46 0 77 -0 27 0 24 -0 49 -0 55 -0 13 0.14 (-0.26)( -2 92) (1 16) (-0 75) (0 69) (-1 62) (-1 50) (-1 14) (0.36) Cf 13 3.86 -0 03 0 98 -0 37 -1 33 -0 43 -1 46 -0 12 2.22 (2.90)( -0 24 (0 92) (-0 63) (-1 Ql)( -0 72) (-1 97) (-0 53) (1.61) C114 -1.67 0 24 0 70 -0 04 -0 28 1 08 -0 57 0 15 -0.19 (-0 .8 8 ) (1 6 8 ) (0 61) (-0 05) (-0 15) (1 57) (-0 65) (0 56) (-0 .10) Ci 15 -2.26 0 39 4 15 -1 36 -2 06 0 90 -1 56 -0 51 -3.42 (-1.70) (1 35) (2 43) (-1 53) (-1 47) (1 08) (-1 56) (-1 65) (-2.25) Ci16 2.44 0 09 -4 61 -1 16 0 98 0 01 -0 79 -0 11 2.20 (1.58) (0 28) (-2 25) (- 1 13) (0 63) (0 oi)( -0 58) (-0 28) (1.18) Ci17 -0.31 0 11 -0 66 0 11 1 02 0 99 -0 52 0 01 -0.81 (-0.34) (1 00) (-0 76) (0 22) (1 12) (1 93) (-0 92) (0 08) (-0.81) ^i18 2.17 0 08 0 40 2 13 -1 45 -1 70 -1 19 0 24 0.92 J (0.50) (0 .22) (0 14) (1 25) (-0 35) (-1 03) (-0 51) (0.38) (0 .20) rho -0.22 (5.15) R2 0.56 0 .18 0 .72 0 .43 0 .39 0 .44 0 .10 0 .87 0.73 Table 52 (Continued) Commodities Parameters 10 11 12 13 14 15 16 17 18 a i0 0.12 0.06 0 03 0 05 0 02 -0 03 0 02 0.01 -0.07 (3.12) (3.26) (2 06) (2 80 (0 93) (-1 32) (0 75) (0.79)( -1.29) -0 0 0 -1 88 1 a ni 1 -0.25 -1.42 19 26 27 43 0.42 0.82 / (-1.04)( -2.85)( -0 46) (1 10 (1 13) (-2 48) (2 33) (0.77) (2.08) *12(10 j 1.13 1.66 -1 20 2 54 0 57 -0 80 4 25 -0.33 0.87 (0.56) (0 . 71) (-0 71) (1 77 (0 40) (-0 28) (1 27) (-0.30) (0.24) a j3(10‘3) -0.33 0.00 4 78 2 34 3 26 3 86 0 00 1.37 12.68 • m (-0 .11) (0 .00) (1 8 6 ) (1 08 (1 54) (0 91) (0 00) (0.83) (2.33) aj4(101 *T ) 2.87 -5.06 -0 10 -0 25 -0 44 3 76 2 54 1.35 8.03 (1. 47)( -2 .12)( -0 06) (-0 18 (-0 32) (1 37) (0 77) (1.24) (2.29) b,(10 ) -9.93 -1.67 -0 03 1 93 0 00 6 65 2 44 -3.04 9.51 (-1.83)( -0.26)( -0 0 1) (0 49 (0 00) (0 87) (0 27) (-1 .0 0 ) (0.98) cn Ci2 Ci3 Ci4 Ci5 Ci6 Ci7 Ci8 Ci9 'i 10 1.69 (0.22) C i 11 0.94 -0 16 (1 -49)( -0 28) Ci12 0.31 -0 13 0.91 (0.72)( -0 47) (3.09) Ci13 0.28 0 33 -0.34 -2 38 (0.17) (0 87) (-1.23)( -2 11) Ci14 1.28 0 28 0.22 0 67 0.46 (0.48) (0 67) (0.81) (0 62) (0.23) Ci15 0.47 -1 07 -0.79 -0 77 -0.66 -1 02 (0.26)( -1 75) (-1.76)( -0 81) (-0.57)( -0 61) Ci16 -1.77 -0 58 -0.08 -2 57 -0.17 6 18 -1.05 (-0 .8 8 )( -0 75) (-0.13)( -2 04) (-0.13) (3 75) (-0.35) Ci17 -0.30 -0 02 -0.01 0 76 -1.06 0 06 -0.10 1.94 (-0.24)( -0 08) (-0.04) (1 14) (-1.36) (0 08) (-0 .11) (3.00) C:118 -15.34 1 78 0.69 3 08 0.86 -2 39 3.25 4.16 1.47 (-2.49) (1 67) (0.95) (1 22) (0.26) (0 8 6) (0.99) (2.26) (0.13) 0.30 0 03 0.39 0 31 0.28 0 14 0.27 0.59 0.79 a/ AR(1) is first order autocorrelation, Qt., is lagged quantity variable, FS is family size, AGE is age of household head, and D is dummy variable, b/ Figures in the parenthes are asymptotic t-ratio; c/ All values of c,j (i,j=l,2,...,18) are presented by multiplying 10'3 with the symmetric condition (cjj=cJ-i) only half of parameters need to be estimated. d/ The same auto correlation coefficient is assumed for every equation. APPENDIX D REGRESSION RESULTS OF FULL MODEL UNDER TWO-STAGE PROCEDURE 230 231 Table 53. Estimates of First-Stage Structural Parameters, Full Model Estimation Method :ITSUR Commodity Groups- Parameters I II III IV V VIb aiO 0.46 0.13 0.08 -0.30 0.18 0.45 (3.12)c (1.62) (1.36) (-2.31) (1.81) ,(10 a 1 1 ’ ) 1.36 -2.62 2.15 -0.89 -1.05 1.05 / (0 .6 8 ) (-1.16) (0.83) (-0.45) (-0.41) ai2(10'4) 7.41 3.56 4.33 15.88 -1.79 -29.39 (1 .02) (0.81) (1.26) (-1.69) (0.33) ai3 0.03 0.01 -0.00 0.02 0.01 -0.07 (2.60) (1.52) (-0.65) (1.45) (1.52) ai4 0.02 0.01 0.01 -0.01 0.01 -0.04 (3.12) (1.06) (2.97) (-0 .8 6 ) (1.53) bi -0.73 -0.19 -0.05 0.90 0.02 0.05 (-2.81) (-1.44) (-0.40) (7.19) (0.13) C i1 -2.09 (-1.29) Ci2 -0.52 -0.02 (-1 .0 2) (-0.08) ci3 -0.11 -0.03 0.03 (-0.29) (-0.34) (0.57) Table 54:Estimates of Second-Stage Structural Parameters for Group I, Full Model Estimation Method: ITSUR Commodities" Parameters 1 2 13 17 18b 0.21 0.34 0.14 0.09 0.22 a i1 0V (2.84) (3.88) (2 .02) (1.76) a ll(10 ) 1.18 -0.37 0.60 3.68 -5.09 (1.28) (-1.45) (0.62) (1.87) a,2(10-‘> -0.91 -3.32 7.10 -4.49 1.62 (-0.18) (-0.45) (1.30) (-1.25) a i3(10-3) 7.55 -13.71 -0.81 0.78 6.19 (0.95) (-1.17) (-0.09) (-0.14) au 1.99 5.14 -4.54 -0.28 -5.46 (0.40) (0.69) (-0.83) (-0.08) bf -0.07 -0.15 -0.15 -0.10 0.47 *■» (-1.94) (-3.72) (-4.47) (-3.55) cn l l O ) 0.21 (2.90) c i2(10'3) 0.02 0.05 ■» (4.20) (7.35) 3 10 -0.05 0.01 0.01 ci1 ( -3) *y (-1.98) (0.99) (0 .21) C { 10 ) -0.03 0.01 -0.01 0.07 1 "t (-1.36) (1.89) (-0.05) (3.88) 9 -0.08 0.07 0.18 0.06 c is1 «/ (10 ) (-1.04) (1.59) (3.62) (1.52) rho -0.14 (-1.76) R2 0.31 0.39 0.31 0.71 a/ Commodities include:!- cereal; II- bakery; III- processed fruit; IV- fat and oil; V- misc. food, b/ The coefficients are derived from restrictions, c/ Figures in the parenthese are asymptotic t-ratios. d/ All budget share equations have the same autocorrelation coefficients of rho. 233 Table 55. Estimates of Second-Stage Structural Parameters for Group II,Full Model Estimation Method: ITSUR commodities" Parameters 3 4 5 8 14b ai0 0.87 0.06 0.17 0.01 -0.11 (3.56)c (0.42) (1.49) (0 .11) 0.86 &!*!i i -0.17 -1.41 1.45 0.99 - / (-0 .22) (-1.36) (0.85) (0.55) *12(10 ) -5.35 3.38 0.30 0.59 1.08 1 (-2.96) (2.36) (0.33) (1.35) *13(10 > -0.02 0.01 -0.00 -0.01 0.02 •9 (-0 .66) (0 .68) (-0.33) (-1.06) *14(10 > -0.02 0.00 0.02 0.00 0.00 (-1.14) (0.15) (1.85) (0.34) 0. 0.27 -0.01 -0.18 0.00 -0.08 (1.18) (-0.04) (-1.19) (0 .0 0 ) cil(10 ) -0.39 (-0.89) c,2(1 0 3) -0.12 0.06 (-0.85) (0.93) c,3(1 0 3) 0.18 0.05 -0.06 •9 (0.87) (0.87) (-0.54) c l4(10-3) -0.00 -0.01 -0.02 0.04 (-0 .02) (-0.45) (-1.04) (5.27) ci5<10 ) 0.01 0.03 0.04 0.00 (0.03) (0.73) (0.63) (0.31) rho 0.08 •y (1.08) R2 0.59 0.29 0.29 0.60 a/ Commodities include: 3- beef; 4- pork; 5- other meats; 8 - eggs; 14- processed vegetables, b/ The coefficients are derived from restrictions, c/ Figures in the parenthese are asymptotic t-ratios. d/ All budget share equations have the same autocorrelation coefficients of rho. 234 Table 56. Estimates of Second-Stage Structural Parameters for Group III, Full Model Estimation Method: ITSUR Commodities0 Parameters 6 9 15° ai0 1.70 1.21 -1.91 (1.50)° (2.64) a« -0.61 0.37 0.24 (-0.08) (0.08) a,-2 0.01 -0.01 0.00 (2.17) (-1.78) aj3 -0.05 0.05 0.00 (-0.99) (1.08) a,-4 -0.04 0.02 0.02 (-1.37) (0.63) 0.55 0.32 -0.87 «T (2.90) (2.92) e„(10-3) -1.14 (-1.22) ci2(icr3) -1.17 0.07 (-2.23) (0.15) CjstlO ) 1.30 0.76 (1.49) (1.27) rho 0.16 •% (1.51) R2 0.25 0.56 a/ Commodities include : 6- poultry; 9- milk; 15- sweets; b/ The coefficients are derived from restrictions, c/ Figures in the parenthese are asymptotic t-ratios. d/ All budget share equations have the same autocorrelation coefficients of rho. Table 57. Estimates of Second-Stage Structural Parameters for Group IV, Full Model Estimation Method: ITSUR Commoodities" Parameters 7 1(F ai0 0.91 0.09 (3.08) ai1 -8.36 8.36 (-0.67) <*i2 0.00 0.00 (0.63) ai3 -0.01 0.01 (-0.26) ai4 -0.00 0.00 (-0.11) bi -0.19 0.19 (-0.55) C i1 0.08 (0.54) -0.06 C i21 mm J (-0.22) rho 0.32 (1.69) R2*% 0.21 a/Commodities include : 7- seafood, 10- other dairy products; b/ The coefficients are derived from restrictions, c/ Figures in the parenthese are asymptotic t-ratios. d/ All budget share equations have the same autocorrelation coefficients of rho. 236 Table 58. Estimates of Second-Stage Structural Parameters for Group V, Full Model Estimation Method: ITSUR Commodities Parameters 11 12 16b a i0 0.45 0.40 0.15 (3.09)c (2.75) a iii i 2.96 -0.42 -2.54 (0.82) (-0.14) ai2(10'3) 1.27 -0.21 -1.06 (1.22) (-0.21) -0.02 0.01 0.01 a i3 (-1.37) (0.37) -0.01 -0.01 0.02 a i4 (-1.26) (-0.52) *>i -0.25 -0.30 -0.55 (-2.21) (-3.41) 0.07 (1.82) C|21 c. -0.03 0.03 (-1.30) (0.56) ^i3i 0.17 0.27 (1-19) (2.41) rho -0.04 (-0.35) R2 0.19 0.35 a/ Commodities including: 11- frest fruits; 12- fresh vegetables; 16- nonalcoholic beverage, b/ The coefficients are derived from restrictions, c/ Figures in the parenthese are asymptotic t-ratios. d/ All budget share equations have the same autocorrelation coefficients of rho. APPENDIX E: ELASTICITIES FORMULAS FOR THE FULL MODEL 237 238 The full model used in this study is expressed as „ _ a10+ailQt_1+Ai2D1+ai3D2+au T+civ+biV'-(cil+bi(l+v,cl'))Z 1=------U F Z i ------where V==log(P,); Z =log(M); 1^= a vector of ones; V = Stone's index; a,b = a vector of parameters to be estimated; c - a symmatric matrix of parameters to be estimated. The expenditure, price, and demographic elasticities for the full model can be evaluated by using following formulas: The expenditure elasticities are tli = l - - | - [ - Cif +bj] i+v'cl 1 The price elasticities are ^ ' Til1 (l*v'cl)^ U l +V'cl) clla10+ailOt-l+ai2Dl+a13D2+a14T+cfiV b^*- (cil+bi(l+V,cl)) z] ] -6^ where a _ / 1 i=j °ij ~ { o i+j The demographic elasticities are — — r 1 i APPENDIX F UNCOMPENSATED PRICE AND EXPENDITURE ELASTICITIES MATRIX FOR 19 FOOD CATEGORIES 239 Table 59. Uncompensated Price and Expenditure Elasticities Matrix for 19 Food Commodities Using LA/AIDS Model with AR(1), FS, AGE and D° Other Sea Dairy Cereal Bakery Beef Pork meats Poultry food Egg Milk product (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 1 -2.23 0.04 0.18 -0 30 1 86 0 20 1 40 0 11 -2 37 0.82 2 0.02 -0.88 -0.01 -0 07 -0 06 -0 01 -0 09 0 01 -0 02 -0.02 3 0.08 0.00 0.15 0 19 0 28 0 04 -0 08 -0 02 0 07 -0.58 4 -0.22 -0.10 0.32 -0 68 0 51 0 23 -0 07 0 11 -0 29 0.39 5 1.86 -0.13 0.71 0 75 -2 55 -0 11 0 12 0 04 0 90 0.41 6 0.20 -0.02 0.09 0 34 -0 11 -0 56 0 26 -0 02 0 34 -0.89 7 1.90 -0.23 -0.27 -0 12 0 17 0 36 -2 18 -0 04 0 60 0.34 8 0.28 0.04-0.13 038 009 -0 04 -0 08 -0 10 0 41 0.67 9 -1.55 -0.01 0.11 -0 26 0 59 0 22 0 29 0 11 -0 58 1.01 10 0.54 -0.02 -0.98 039 0 27 -0 57 0 16 0 18 1 01 -0.25 11 0.16 0.07 -0.48 0 28 0 01 0 10 0 20 0 04 0 27 0.24 12 -0.04 -0.14 0.24 -0 10 0 05 -0 12 -0 18 -0 05 0 03 0.12 13 1.59 -0.01 0.37 036 1 68 -0 10 -0 75 -0 07 0 74 0.04 14 -1.21 0.12 0.49 0 01 -0 38 0 64 -0 08 0 03 -0 67 1.05 15 -1.06 0.08 1.68 -0 56 -0 75 0 53 -0 32 -0 19 -1 45 -0.02 16 0.41 0.02 -0.66 -0 17 -0 00 -0 12 -0 11 -0 02 0 40 -0.28 17 0.02 0.08 -0.32 0 15 0 35 0 55 -0 33 -0 14 0 67 0.01 18 0.06 0.02 -0.13 0 28 -0 14 -0 21 -0 04 0 05 0 05 -1.56 19 -0.13 0.00 -0.22 -0.15 -0 14 -0 08 0 03 -0 .08 -0 12 0.06 a/ AR(1) is the first order autocorrelation, (L.., is the lagged quantity variable, FS is family size, AGE is age of noushold head, and D is a dummy variable. 241 Table 59 (Continued) Fresh Fresh Pro. Pro. Nonal. Fat Misc. Expen. fruit veg. fruit veg. Sweet beve. oil food FAFH elasti (11) (12) (13) (14) (15) (16) (17) (18) (19) 1 0.16 0.07 -0 48 0. 28 0. 01 0.10 0.20 0.04 0.27 0.95 2 0.03 -0.08 -0 00 0 04 0 04 0.02 0.02 0.02 0 05 1.00 3 -0.21 0.11 0 13 0 13 0 57 -0.52 -0.08 -0.10 -0 93 0.78 4 0.21 -0.08 -0 13 0 00 -0 33 -0.25 0.06 0.48 -1 24 1.06 5 0.01 0.07 -0 26 -0 24 -0 64 0.00 0.22 -0.32 -1 68 0.85 6 0.11 -0.14 -0 09 0 42 0 46 -0.26 0.35 -0.51 -0 92 0.95 7 0.32 -0.27 -0 86 -0 06 -0 35 -0.28 -0.28 -0.10 0 68 0.69 8 0.12 -0.13 -0 15 0 05 -0 40 -0.08 0.03 0.32 -2 31 1.03 9 0.20 0.03 0 42 -0 27 -0 80 0.56 -0.08 0.11 -0 85 0.74 10 0.18 0.09 0 03 0 44 -0 00 -0.37 0.01 -2.50 0 64 0.75 11 -0.89 -0.06 0 10 0 06 -0 25 -0.14 0.00 0.70 -1 31 0.89 12 -0.06 -0.70 -0 08 0 08 -0 29 0.11 -0.01 0.21 -0 04 0.99 13 0.13 -0.10 -1 93 0 13 -0 40 -0.90 0.29 -0.67 1 14 1.04 14 0.11 0.13 0 18 0 29 0 01 -0.27 -0.76 0.24 -0 96 1.01 15 -0.34 -0.38 -0 40 0 01 -1 45 2.42 -0.04 -0.47 1 55 1.17 16 -0.09 0.06 -0 37 -0 08 1 00 -0.96 -0.04 -0.13 0 12 1.02 17 0.01 -0.02 0.39 -0 76 -0 05 -0.12 0.10 1.93 -2 68 0.82 18 0.32 0.09 -0 .24 0 06 -0 16 -0.11 0.49 -1.17 1 25 1.08 19 -0.13 -0.01 0 .08 -0 05 0 11 0.01 -0.14 0.25 -0 42 1.13 242 Table 60. Uncompensated Price and Expenditure Elasticities Matrix for 19 Food Commodities Using LA/AIDS Model with FS, AGE, and Da Other Sea Dairy Cereal Bakery Beef Pork meats Poultry food Egg Milk product (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 1 -2.04 0 05 0 16 -0 32 1 83 0 24 1 36 0 08 -1 85 0 11 2 0.02 -0 88 0 00 -0 06 -0 05 -0 00 -0 07 0 01 -0 02 -0 02 3 0.06 0 01 0 09 0 16 0 31 0 05 -0 11 -0 00 0 06 -0 48 4 -0.23 -0 10 0 27 -0 63 0 43 0 18 0 04 0 11 -0 16 0 08 5 1.83 -0 10 0 80 0 64 -2 40 -0 04 -0 03 0 03 0 42 1 46 6 0.23 -0 01 0 10 0 27 -0 05 -0 44 0 08 -0 04 0 20 -0 66 7 1.84 -0 21 -0 37 0 08 -0 04 0 12 -1 97 0 02 0 31 0 42 8 0.20 0 03 -0 04 0 40 0 06 -0 09 0 04 -0 08 0 43 0 41 9 -1.21 -0 02 0 11 -0 14 0 28 0 14 0 15 0 12 -0 75 1 20 10 0.08 -0 01 -0 79 0 09 0 96 -0 41 0 21 0 11 1 21 -1 30 11 0.13 0 05 -0 38 0 28 -0 02 0 10 027 005 030 013 12 -0.02 -0 13 0 21 -0 09 0 03 -0 15 -0 15 -0 04 0 02 0 16 13 1.61 -0 00 0 27 0 41 1 57 -0 11 -0 74 -0 07 0 26 0 52 14 -1.17 0 13 0 55 0 10 -0 70 0 57 0 02 0 04 -0 41 1 08 15 -0.97 0 07 1 57 -0 38 -1 06 0 47 -0 15 -0 16 -1 17 -0 40 16 0.40 0 01 -0 62 -0 25 0 10 0 00 -0 23 -0 04 0 42 -0 18 17 -0.16 0 08 -0 34 0 22 0 24 0 46 -0 08 0 48 -0 01 -0 34 18 0.05 0 02 -0 08 0 44 -0 47 -0 27 0 12 0 08 0 19 -1 64 19 -0.11 -0 01 -0 24 -0 17 -0 11 -0 08 -0 00 -0 08 -0 17 0 15 a/ FS is family size, AGE is age of household head, and D is dummy variable. 243 Table 60 (Continued) Fresh Fresh Pro. Pro. Nonal. Fat Misc. Expen. fruit veg. fruit veg. Sweet beve. oil food FAFH elasti. (11) (12) (13) (14) (15) (16) (17) (18) (19) 1 0.13 0.05 -0.38 0 28 -0 02 0 10 0.27 005 030 0.90 2 0.02 -0.07 0.00 0 04 0 03 0 01 0.02 0 04 0 05 0.90 3 -0.17 0.10 0.10 0 14 0 53 -0 50 -0.09 -0 07 -1 04 0.84 4 0.21 -0.07 -0.13 0 04 -0 22 -0 37 0.09 074-1 41 1.11 5 0.02 0.04 -0.14 -0 45 -0 90 0 21 0.15 -1 15 -1 25 0.87 6 0.11 -0.17 -0.10 0 37 0 36 0 00 0.29 -0 68 -0 95 1.08 7 0.41 -0.23 -0.86 0 02 -0 17 -0 64 -0.07 0 42 0 02 0.90 8 0.13 -0.13 -0.15 0 06 -o 35 -0 20 0.08 0 49 -2 30 1.02 9 0.23 0.02 0.15 -0 17 -0 65 0 58 0.07 0 33 -1 22 0.77 10 0.10 0.13 0.30 0 46 -0 21 -0 23 -0.14 -2 63 1 39 0.68 11 -0.88 -0.07 0.07 0 06 -0 25 -0 15 0.01 0 67 -1 21 0.83 12 -0.08 -0.70 -0.07 0 07 -0 29 0 07 0.00 0 28 -0 12 1.00 13 0.09 -0.09 -2.01 0 12 -0 35 -0 90 0.37 -0 16 0 58 0.96 14 0.11 0.12 0.17 0 29 0 06 -0 31 -0.76 -0 09 -0 74 0.94 15 -0.34 -0.38 -0.35 0 04 -1 09 2 10 0.08 0 08 1 00 1.08 16 -0.08 0.04 -0.37 -0 09 0 87 -0 82 -0.10 -0 26 0 28 0.93 17 0.03 0.02 0.51 -0 76 0 12 -0 33 0.03 1 98 -2 62 0.74 18 0.30 0.12 -0.06 -0 03 0 03 -0 22 0.50 -1 30 1 15 1.06 19 -0.12 -0.02 0.04 -0.04 0.07 0 03 -0.14 0.22 -0.39 1.15 244 Table 61. Uncompensated Price and Expenditure Elasticities Matrix for 19 Food Commodities Using LA/AIDS Model with AR(1), and Da Other Sea Dairy Cereal Bakery Beef Pork meats Poultry food Egg Milk product (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 1 -2.19 0.05 0 14 -0 38 2. 09 0 19 1 34 0 12 -2 67 1.34 2 0.02 -0.88 -0 03 -0 06 -0. 07 -0 00 -0 08 0 01 -0 02 -0.02 3 0.06 -0.02 0 24 0 19 0 29 0 06 -0 02 -0 02 0 13 -0.63 4 -0.27 -0.10 0 32 -0 70 0 44 0 20 -0 09 0 11 -0 24 0.32 5 2.09 -0.14 075 0 65 -2 29 -0 14 -0 07 0 02 0 46 0.85 6 0.19 -0.01 0 15 0 30 -0 14 -0 59 0 16 -0 01 0 22 -0.79 7 1.82 -0.21 -0 07 -0 16 -0 09 0 22 -2 34 -0 07 0 76 -0.10 8 0.29 0.03 -0 13 0 41 0 04 -0 04 -0 14 -0 06 0 39 0.59 9 -1.74 -0.02 0 22 -0 22 0 30 0 14 0 37 0 11 -0 17 0.53 10 0.88 -0.02 -1 05 0 32 0 56 -0 51 -0 05 0 16 0 53 0.07 11 0.16 0.06 -0 45 0 27 0 03 0 11 0 18 0 07 0 26 0.23 12 -0.03 -0.14 0 13 -0 08 0 06 -0 09 -0 13 -0 05 0 04 0.12 13 1.51 0.00 0 42 0 28 1 80 -0 13 -0 81 -0 06 0 77 -0.11 14 -1.15 0.13 0 37 -0 06 -0 10 0 63 -0 11 0 05 -0 74 1.06 15 -1.26 0.09 1 80 -0 61 -0 78 0 40 -0 27 -0 23 -1 29 0.03 16 0.40 0.03 -0 70 -0 16 -0 03 -0 10 -0 11 -0 00 0 41 -0.37 17 0.03 0.08 -0 40 0 19 0 24 0 56 -0 30 0 07 1 29 -0.63 18 -0.16 0.04 -0 10 0 30 -0 26 -0 26 0 01 0 07 0 05 -1.28 19 -0.09 -0.00 -0 24 -0 12 -0 13 -0.05 0 06 -0 08 -0 09 0.01 a/ AR(1) is first order autocorrelation, and D is a dummy variable. 245 Table 61 (Continued) Fresh Fresh Pro. Pro. Nonal. Fat Misc. Expen. fruit veg. fruit veg. Sweet beve. oil food FAFH elasti. (11) (12) (13) (14) (15) (16) (17) (18) (19) 1 0.16 0 06 -0 45 0 27 0 03 0 11 0 18 007 026 1.00 2 0.03 -0 07 0 00 0 04 0 04 0 03 0 02 0 06 0 01 0.98 3 -0.20 0 06 0 15 0 09 0 61 -0 56 -0 10 -0 08 -1 15 0.94 4 0.21 -0 07 -0 14 0 03 -0 36 -0 24 0 08 0 51 -1 09 1.15 5 0.03 0 07 -0 35 -0 06 -0 66 -0 05 0 15 -0 63 -1 59 0.89 6 0.13 -0 10 -0 11 0 41 0 35 -0 21 0 36 -0 63 -0 63 0.95 7 0.29 -0 18 -0 93 -0 09 -0 30 -0 28 -0 25 0 07 1 20 0.73 8 0.21 -0 13 -0 13 0 08 -0 49 -0 01 0 01 0 47 -2 40 1.01 9 0.19 0 04 0 44 -0 30 -0 71 0 58 -0 04 0 11 -0 62 0.79 10 0.18 0 10 -0 05 0 45 0 03 -0 49 -0 01 -2 04 0 21 0.73 11 -0.82 -0 09 0 10 0 06 -0 32 -0 13 -0 01 0 64 -1 21 0.87 12 -0.10 -0 70 -0 07 0 09 -0 26 0 12 -0 00 0 24 -0 21 1.06 13 0.12 -0 09 -2 04 0 19 -0 48 -0 83 0 31 -0 72 1 50 1.08 14 0.11 0 15 0 26 0 27 -0 14 -0 23 -0 79 -0 12 0 68 1.08 15 -0.44 -0 35 -0 48 -0 11 -1 51 2 32 0 06 -0 92 2 33 1.19 16 -0.08 0 07 -0 34 -0 07 0 96 -0 90 -0 06 -0 07 0 09 1.03 17 -0.02 -0 00 0 43 -0 79 0 09 -0 18 0 08 2 21 -2 97 0.85 18 0.29 0 11 -0 25 -0 03 -0 32 -0 06 0 56 -1 70 1 81 1.18 19 -0.12 -0 02 0 10 -0 03 0 16 0 01 -0 15 0 37 -0 64 1.04 246 Table 62. Uncompensated Price and Expenditure Elasticities Matrix for 19 Food Commodities Uing LA/AIDS Model with AR(1), Q t..,, T, FS, AGE, and Da Other Sea Dairy Cereal Bakery Beef Pork meats Poultry food Egg Milk product (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 1 -1.71 0 05 0 02 -0 40 2. 30 008 127 009-178 0.80 2 0.02 -0 90 -0 04 -0 07 -0 08 -0 01 -0 09 0 01 -0 02 -0.07 3 0.01 -0 01 -0 39 -0 19 0 10 0 05 0 00 -0 09 -0 20 -0.55 4 -0.28 -0 11 -0 35 -0 80 0 36 0 20 -0 15 0 09 -0 33 0.31 5 2.31 -0 16 0 26 0 53 -2 37 -0 20 0 12 -0 05 1 31 -0.91 6 0.07 -0 03 0 08 0 29 -0 22 -0 46 0 35 0 04 0 23 -0.58 7 1.73 -0 23 0 02 -0 26 0 18 0 48 -2 15 -0 07 0 69 0.56 8 0.22 0 04 -0 57 0 31 -0 14 0 11 -0 15 -0 23 0 27 0.19 9 -1.16 -0 01 -0 33 -0 30 0 86 0 16 0 33 0 07 -0 53 1.38 10 0.53 -0 08 -0 92 0 30 -0 59 -0 37 0 27 0 05 1 38 -5.04 11 -0.07 0 08 -0 24 0 34 -0 13 0 18 0 17 0 06 -0 00 0.75 12 0.05 -0 14 0 16 -0 06 0 12 -0 19 -0 20 -0 05 0 11 0.03 13 1.72 -0 02 0 44 0 19 1 73 -0 12 -0 63 -0 10 0 61 -0.32 14 0.15 0 15 0 26 0 01 0 52 0 15 -0 40 0 08 2 11 -0.73 15 -0.42 0 18 1 73 -0 48 -0 08 -0 07 -0 75 -0 15 0 01 0.15 16 0.57 0 02 -0 56 -0 20 0 26 0 04 -0 16 -0 05 0 44 -0.80 17 -0.21 0 07 -0 30 0 07 0 45 0 59 -0 22 -0 46 1 49 0.27 18 0.62 0 05 -0 11 0 26 0 93 -0 63 -0 12 0 09 0 96 -1.10 19 0.44 0 00 0 04 0 00 -0 39 0 02 0 07 -0 03 -0 56 0.75 a/ AR(1) is the first order autocorrelation, Q,.., is the lagged quantity variable, T is time trend variable, FS is family size, AGE is age of houshold head, D is a dummy variable. 247 Table 62 (Continued) Fresh Fresh Pro. Pro. Nonal. Fat Misc. Expen. fruit veg. fruit veg. Sweet beve. oil food FAFH elasti. (11) (12) (13) (14) (15) (16) (17) (18) (19) 1 -0.07 0 08 -0 24 0 34 -0 13 0. 18 0 17 0 06 -0 00 0.96 2 0.04 -0 08 -0 01 0 04 0 08 0 02 0 02 0 06 0 04 1.04 3 -0.10 0 08 0 16 0 07 0 59 -0 44 -0 07 -0 08 0 35 0.70 4 0.27 -0 09 -0 14 0 00 -0 28 -0 29 0 03 0 44 0 04 1.02 5 -0.15 0 14 -0 28 0 33 -006 056 028 2 32 -4 81 0.82 6 0.20 -0 22 -0 11 0 09 -0 06 0 08 0 37 -1 58 0 27 1.16 7 0.28 -029 -0 72 -0 34 -0 85 -0 41 -0 18 -0 36 1 46 0.47 8 0.17 -013 -0 22 0 13 -0 31 -0 28 -0 02 0 57 -1 02 1.05 9 0.01 0 09 0 35 0 89 0 02 0 61 -0 32 1 57 -4 45 0.78 10 0.57 0 03 -0 17 0 30 0 10 -1 07 0 11 -1 76 6 22 0.72 11 -0.92 -0 07 0 07 -0 14 -0 62 -0 29 -0 01 -0 29 0 17 0.96 12 -0.08 -0 68 -0 09 0 18 -0 14 0 00 -0 01 0 57 -0 58 1.00 13 0.09 -0 12 -1 99 0 38 -0 08 -1 12 0 37 0 19 0 22 1.06 14 -0.25 0 31 0 52 0 51 0 64 0 44 -0 65 2 65 -7 44 0.98 15 -0.84 -0 20 -0 09 0 47 -0 78 3 08 0 04 0 71 -3 77 1.25 16 -0.16 -0 00 -0 46 0 13 1 27 -1 28 -0 02 1 17 -1 24 1.04 17 -0.02 -0 01 0 51 -0 66 0 07 -0 05 0 11 2 50 -3 22 0.83 18 -0.14 0 26 0 07 0 68 0 25 0 98 0 63 1 70 -6 46 1.08 19 0.01 -0 .06 0 .01 -0.38 -0 26 -0 21 -0 17 -1 28 -1 77 1.12 248 Table 63. Uncompensated Price and Expenditure Elasticities Matrix for 19 Food Commodities Using LA/AIDS Model with AR(1), T, FS, AGE, and Da Other Sea Dairy Cereal Bakery Beef Pork meats Poultry food Egg Milk product (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 1 -2.11 0 05 0 14 -0 41 2 31 0 02 1 45 0 10 -1 63 -0.07 2 0.02 -0 88 -0 03 -0 08 -0 07 -0 01 -0 09 0 01 -0 03 -0.06 3 0.06 -0 00 -0 20 -0 19 0 10 0 00 0 01 -0 08 -0 09 -Q.58 4 -0.28 -0 11 -0 35 -0 79 0 32 0 22 -0 12 0 07 -0 37 -0.01 5 2.31 -0 13 0 25 0 48 -2 07 -0 28 0 13 -0 03 0 95 0.43 6 0.02 -0 02 -0 00 0 32 -0 29 -0 60 0 25 0 03 0 27 -0.21 7 1.96 -0 23 0 04 -0 22 0 18 0 34 -2 12 -0 04 0 47 0.49 8 0.24 0 03 -0 54 0 24 -0 08 0 07 -0 08 -0 17 0 15 0.03 9 -1.06 -0 02 -0 15 -0 34 0 63 0 18 0 23 0 04 0 19 -0.35 10 -0.04 -0 07 -0 98 0 00 0 28 -0 13 0 24 0 01 -0 35 -3.63 11 0.02 0 08 -0 31 0 32 -0 22 0 20 0 19 0 07 0 09 0.70 12 0.03 -0 14 0 14 -0 07 0 15 -0 14 -0 20 -0 05 0 10 -0.01 13 1.70 -0 02 0 45 0 22 1 41 -0 08 -0 71 -0 12 0 47 -0.61 14 -0.54 0 14 0 31 -0 07 0 63 0 21 -0 14 0 03 1 24 -0.40 15 -0.64 0 12 1 41 -0 48 0 12 0 04 -0 44 -0 13 -0 05 -0.07 16 0.52 0 03 -0 44 -0 19 0 21 -0 10 -0 11 -0 04 0 33 -0.81 17 0.13 0 08 -0 19 0 14 0 34 0 52 -0 24 -0 09 0 41 0.47 18 0.65 0 06 -0 17 0 32 1 14 -0 77 -0 02 0 14 1 29 0.16 19 -0.34 -0 01 -0 01 0 04 -0 54 007 002 -004 -043 0.60 a/ AR(1) is the first order autocorrelation, T is time trend variable, FS is family size, AGE is age of houshold head, D is a dummy variable. 1 249 Table 63 (Continued) Fresh Fresh PrcT Pro. Nonal. Fat Misc. Expen. fruit veg. fruit veg. Sweet beve. Oil food FAFH elasti. (11) (12) (13) (14) (15) (16) (17) (18) (19) 1 0.02 0 08 -0. 31 0. 32 -0. 22 0. 20 0 19 0 07 0. 09 0 94 2 0.04 -0 08 -0.01 0. 04 0 05 0 03 0 02 0 08 0. 01 1 03 3 -0.13 0 07 0 16 0 08 0 48 -0 34 -0 05 -0 14 0 10 0 74 4 0.25 -0 06 -0 19 -0 03 -0 28 -0 28 0 06 0 54 0 39 1 04 5 -0.24 0 17 -0 07 0 41 0 12 0 45 0 22 2 83 -6 70 0 80 6 0.23 -0 16 -0 07 0 13 0 04 -0 20 0 33 -1 93 0 92 0 92 7 0.30 -0 30 -0 82 -0 11 -0 50 -0 28 -0 20 -0 05 0 47 0 62 8 0.19 -0 14 -0 25 0 05 -0 27 -0 21 -0 00 0 86 -1 14 1 03 9 0.07 0 08 0 27 0 52 -0 01 0 47— ■* o 18 2 11 -3 39 0 70 10 0.53 0 00 -0 34 -0 16 -0 03 -1 09 0 20 -0 24 5 06 0 72 11 -0.78 -0 10 0 08 -0 16 -0 56 -0 22 -0 04 -0 49 0 22 0 93 12 -0.11 -0 66 -0 06 0 19 -0 15 0 13 -0 00 0 67 -0 79 0 99 13 0.10 -0 09 -1 95 0 29 -0 13 -1 00 0 36 0 80 -0 10 1 04 14 -0.30 0 33 0 39 1 31 0 96 0 36 -0 69 3 62 -8 37 0 98 15 -0.75 -0 21 -0 13 0 71 -0 76 3 02 0 03 1 39 -4 37 1 18 16 -0.13 0 07 -0 41 0 11 1 24 -1 02 -0 01 0 98 -1 27 1 04 17 -0.07 -0 00 0 50 -0 69 0 05 -0 02 0 15 2 28 -3 85 0 85 18 -0.23 0 30 0 28 0 93 0 49 0 82 0 58 1 44 -8 .12 1 05 19 0.01 -0 08 -0 01 -0 .43 -0 30 -0 22 -0 20 -1 62 2.35 1 15 250 Table 64. Uncompensated Price and Expenditure Elasticities Matrix for 19 Food Commodities Using LA/AIDS Model with AR(1), Qt.1, and D° ______Dairy Cereal Bakery Beef Pork meats Poultry food Egg Milk product (1) (2) (3) (4) (6) (7) (7) (8) (9) (10) 1 -2.12 0. 05 0. 11 -0 46 2 07 0 35 1 11 0 08 -3 33 2.08 2 0.02 -0 89 -0 04 -0 06 -0 08 -0 01 -0 09 0 01 -0 01 -0.03 3 0.05 -0 03 0 31 0 13 0 29 0 05 0 01 -0 01 0 03 -0.59 4 -0.32 -0 10 0 22 -0 65 0 44 0 23 -0 10 0 11 -0 31 0.62 5 2.07 -0 16 0 75 0 65 -2 11 -0 13 -0 13 -0 02 0 99 -0.68 6 0.36 -0 03 0 13 0 34 -0 14 -0 49 0 19 0 01 0 25 -1.06 7 1.51 -0 22 0 05 -0 17 -0 17 0 28 -2 21 -0 11 1 24 -0.07 8 0.20 0 04 -0 06 0 42 -0 06 0 04 -0 21 -0 11 0 45 0.81 9 -2.17 -0 01 0 05 -0 28 0 65 0 17 0 59 0 12 -0 99 1.60 10 1.37 -0 02 -0 98 0 61 -0 44 -0 68 -0 04 0 22 1 60 -0.08 11 0.09 0 07 -0 35 0 30 0 01 0 13 0 14 0 06 0 26 0.31 12 -0.03 -0 14 0 13 -0 08 0 10 -0 12 -0 11 -0 04 0 05 0.11 13 1.47 -0 01 -0 47 0 19 1 82 -0 16 -0 67 -0 05 0 97 -0.01 14 -1.12 0 14 0 25 -0 15 0 29 0 60 -0 30 0 10 -0 21 0.59 15 -1.13 0 17 1 81 -0 63 -0 99 0 25 -0 61 -0 28 -1 20 0.13 16 0.38 0 01 -0 83 -0 20 0 22 0 02 -0 15 0 01 0 30 -0.38 17 -0.09 0 08 -0 38 0 10 0 44 0 55 -0 23 -0 74 2 33 -0.25 18 0.23 0.02 005 029 -0 25 -0 25 -0 20 0.02 0 08 -2.09 19 -0.16 0.00 -0 25 -0 13 -0 .10 -0 07 0 .12 -0.07 -0 .13 0.11 a/ AR(1) is the first order autocorrelation, Qt.n is the lagged quantity variable, and D is a dummy variable. 251 Table 64 (Continued) Fresh Fresh Pro. Pro. Nonal. Fat Misc. Expen. fruit veg. fruit veg. Sweet beve. oil food FAFH elasti. (11) (12) (13) (14) (15) (16) (17) (18) (19) 1 0.09 0.07 -0. 35 0 30 0. 01 0. 13 0 14 0 06 0. 26 1.03 2 0.04 -0.07 -0. 00 004 008 0.01 002 0 03 0 02 1.02 3 -0.15 0.06 0 16 0 07 0 61 -0 67 -0 10 0 07 -1 20 0.90 4 0.23 -0.06 -0 11 -0 07 -0 37 -0 29 0 04 0 50 -1 16 1.15 5 0.01 0.11 -0 57 0 19 -0 84 0 47 0 28 -0 61 -1 18 0.91 6 0.14 -0.13 -0 14 0 39 0 22 0 03 0 35 -0 63 -0 91 1.11 7 0.23 -0.15 -0 77 -0 25 -0 69 -0 39 -0 19 -0 60 2 15 0.54 8 0.18 -0.11 -0 10 0 16 -0 59 0 03 -0 02 0 15 -2 21 1.00 9 0.20 0.05 0 56 -0 08 -0 66 0 42 -0 15 0 16 -1 02 0.81 10 0.24 0.09 0 00 0 25 0 09 -0 50 -0 10 -3 35 0 97 0.76 11 -1.01 -0.08 0 10 0 08 -0 43 -0 12 -0 04 0 41 -0 87 0.95 12 -0.09 -0.71 -0 11 0 08 -0 23 -0 02 0 00 0 25 -0 14 1.09 13 0.13 -0.14 -2 06 0 36 -0 45 -0 92 0 30 -1 50 1 99 1.12 14 0.13 0.14 0 49 -0 80 -0 57 0 03 -0 60 -0 34 -0 46 1.08 15 -0.59 -0.30 -0 46 -0 43 -1 36 2 41 0 14 -1 21 2 94 1.35 16 -0.07 -0.01 -0 38 0 01 1 00 -1 07 -0 06 0 66 -0 51 1.03 17 -0.06 0.01 0 41 -0 60 0 20 -0 18 0 12 2 57 -3 19 0.87 18 0.18 0.11 -0 52 0 09 -0 42 0 55 0 65 -0 88 1 11 1.24 19 -0.08 -0.01 0 14 -0 02 0 21 -0 08 -0 16 0 24 -0 .54 0.99 APPENDIX G COMPENSATED PRICE ELASTICITIES MATRIX FOR 19 FOOD CATEGORIES BASED ON DYNAMIC MODEL I 252 253 Table 65. Compensated Price Elasticities Matrix for 19 Food Commodities Using LA/AIDS Model with AR(1), Qt.1f T, FS, AGE, and D° ______Dairy Cereal Bakery Beef Pork meats Poultry food Egg Milk product (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 1 -1.69 0 10 009 -036 233 0 11 1 29 0 10 -1 73 0.84 2 0.05 -0 83 0 03 -0 03 -0 05 0 02 -0 07 0 02 0 03 -0.02 3 0.03 0 03 -0 34 -0 16 0 12 0 06 0 02 -0 08 -0 17 -0.52 4 -0.25 -0 04 -0 28 -0 76 0 39 0 23 -0 12 0 10 -0 28 0.35 5 2.33 -0 11 0 32 0 57 -2 35 -0 18 0 14 -0 04 1 35 -0.87 6 0.11 0 04 0 17 0 34 -0 18 -0 42 0 37 0 06 0 28 -0.53 7 1.74 -0 20 0 05 -0 24 0 19 0 49 -2 14 -0 07 0 71 0.58 8 0.25 0 10 -0 50 0 35 -0 11 0 14 -0 13 -0 22 0 31 0.24 9 -1.13 0 04 -0 28 -0 27 0 88 0 18 0 34 0 08 -0 50 1.41 10 0.55 -0 03 -0 87 0 33 -0 57 -0 35 0 28 0 06 1 41 -5.01 11 -0.04 0 14 -0 17 0 38 -0 10 0 21 0 19 0 07 0 04 0.79 12 0.08 -0 08 0 23 -0 02 0 15 -0 16 -0 18 -0 04 0 15 0.08 13 1.75 0 04 0 52 0 23 1 76 -0 09 -0 61 -0 09 0 66 -0.27 14 0.17 0 21 0 33 0 05 0 55 0 18 -0 38 0 09 2 15 -0.69 15 -0.38 0 25 1 82 -0 42 -0 05 -0 04 -0 72 -0 13 0 06 0.20 16 0.59 0 08 -0 48 -0 16 0 29 0 07 -0 13 -0 04 0 48 -0.75 17 -0.19 0 12 -0 24 0 10 0 47 0 62 -0 20 -0 45 1 53 0.30 18 0.65 0 11 -0 03 0 30 0 96 -0 60 -0 10 0 10 1 .00 -1.05 19 -0.41 0 07 0 12 0 05 -0 36 0 05 0 10 -0 02 -0 .51 0.79 a/ AR(1) is the fir st order autoc orrelation, Qt_., is the lagged quantity variable, T is ti me trend, FS is family size, AGE is age of houshold head, D is a dummy variable. 254 Table 65 (Continued) Fresh Fresh Pro. Pro. Nonal. Fat Misc. fruit veg. fruit veg. Sweet beve. oil food FAFH (11) (12) (13) (14) (15) (16) (17) (18) (19) 1 -0.04 0.14 -0 17 0 38 -0 10 0. 21 0 19 0.07 0.04 2 0.08 -0.04 0 02 0 06 0 10 0. 08 0 04 0.13 0.40 3 -0.08 0.10 0 17 0 08 0 61 -0 39 -0 06 -0.03 0.59 4 0.30 -0.01 -0 12 0 02 -0 25 -0 23 0 04 0.51 0.40 5 -0.12 0.17 -0 26 0 35 -0 04 0 61 0 30 2.38 -4.53 6 0.24 -0.18 -0 08 0 11 -0 03 0 15 0 39 -1.50 -0.68 7 0.30 -0.28 -0 71 -0 33 -0 84 -0 38 -0 17 -0.33 1.62 8 0.20 -0.10 -0 19 0 15 -0 29 -0 22 0 00 0.65 -0.65 9 0.03 0.11 0 37 090 004 066 -0 30 1.63 -4.18 10 0.59 0.06 -0 15 -0 29 0 11 -1 03 0 13 -1.71 6.47 11 -0.88 -0.04 0 09 -0 12 -0 60 -0 24 0 01 -0.22 0.50 12 -0.04 -0.65 -0 07 0 19 -0 12 0 06 0 01 0.64 -0.23 13 0.12 -0.08 -1 97 0 40 -0 05 -1 06 0 39 0.26 0.59 14 -0.22 0.34 054 0 53 0 66 050 -064 2.72 -7.10 15 -0.80 -0.16 -0 05 0 50 -0 75 3 16 0 06 0.80 -3.33 16 -0.13 0.03 -0 44 0 15 1 29 -1 22 -0 00 1.24 -0.88 17 0.01 0.02 0 53 -0 64 0 09 -0 00 0 13 2.55 -2.93 18 -0.10 0.29 0 09 0 70 0 28 1 05 0 65 1.77 -6.08 19 0.05 -0.02 0 04 -0 36 -0 23 -0 15 -0 15 -1.21 2.16 APPENDIX H COMPENSATED PRICE ELASTICITIES MATRIX FOR AGGREGATE COMMODITY GROUPS 255 256 Table 66. Compensated Price Elasticities Matrix for Aggregate Group in The First Stage Groups8 'OUp Model I II III IV V VI I Full -0.63 -0.03 0.07 0.11 0.41 0.43 TL -0.70 0.06 -0.01 -0.09 0.08 0.66 LA/AIDS -0.70 0.06 -0.01 -0.09 0.08 0.65 II Full 0.27 0.09 0.03 0.33 -0.09 -0.44 TL 0.13 0.05 -0.09 0.25 -0.08 -0.26 LA/AIDS 0.13 0.05 -0.09 0.25 -0.08 -0.26 III Full -0.03 -0.47 0.29 -0.18 0.41 TL -0.14 -0.64 0.36 -0.09 0.55 t t ( 1 o o o • o • ^ • LA/AIDS o o o -0.13 -0.64 0.36 -0.09 0.54 IV Full 0.13 0.42 0.03 -0.05 -0.29 -0.77 TL -0.10 0.14 0.13 0.53 0.13 -0.82 LA/AIDS -0.10 0.14 0.13 0.53 0.13 -0.83 V Full 0.08 -0.21 -0.14 0.31 -0.31 0.25 TL 0.13 -0.06 -0.05 0.17 -0.40 0.20 LA/AIDS 0.13 -0.06 -0.05 0.18 -0.40 0.20 VI Full 0.17 -0.13 0.07 -0.28 0.08 0.08 TL 0.37 -0.07 0.10 -0.40 0.07 -0.07 LA/AIDS 0.37 -0.07 0.10 -0.40 0.07 -0.06 a/ Group I : cereal, bakery, processed fruit, fat & oil,and misc. food; Group II : beef, pork, other meats, eggs, and processed vegetables; Group III: poultry, milk, and sweet; Group IV : seafood, and dairy products; Group V : fresh fruits, fresh vegetables, and nonalcoholic beverage; Group VI : food away from home. Table 67. Compensated Price Elasticities Matrix in The Second Stage, Group I Commodities0 Model 1 2 13 17 18 1 Full 0.91 0.30 -0.37 -0.20 -0.76 TL 0.77 0.24 -0.46 -0.03 -0.52 LA/AIDS 0.66 0.23 -0.43 -0.02 -0.44 2 Full 0.13 -0.67 0.09 0.09 0.24 TL 0.11 -0.72 0.07 0.06 0.48 LA/AIDS 0.11 -0.72 0.07 0.06 0.48 13 Full -0.43 0.28 -0.68 0.12 0.39 TL -0.54 0.18 -1.01 0.38 1.05 LA/AIDS -0.50 0.18 -1.13 0.40 1.06 17 Full -0.36 0.28 0.09 0.10 -0.42 TL -0.04 0.18 0.51 0.31 -0.96 LA/AIDS -0.03 0.19 0.54 0.30 -1.00 18 Full -0.22 0.32 0.32 -0.04 -0.03 TL-0.22 0.39 0.36 -0.25 -0.28 LA/AIDS -0.18 0.42 0.37 -0.26 -0.35 a/ Commodities: 1- cereal; 2- bakery; 13- processed fruits; 17- fat & oil; 18- misc. foods. Table 68. Compensated Price Elasticities Matrix in The Second Stage, Group II Commodities8 Model 3 4 5 8 14 3 Full -0.32 0.12 0.29 0.08 -0.11 TL -0.31 0.12 0.24 0.08 -0.13 LA/AIDS -0.30 0.12 0.24 0.08 -0.14 4 Full 0.18 -0.50 0.19 0.01 0.12 TL 0.22 -0.52 0.20 0.00 0.09 LA/AIDS 0.21 -0.52 0.21 0.01 0.10 5 Full 0.49 0.25 -1.15 -0.12 0.41 TL 0.60 0.29 -1.28 -0.11 0.49 LA/AIDS 0.61 0.30 -1.31 -0.11 0.50 8 Full 0.65 0.04 -0.40 -0.29 -0.00 TL 0.50 0.01 -0.28 -0.28 0.05 LA/AIDS 0.48 0.02 -0.28 -0.28 0.06 14 Full -0.37 0.25 0.48 0.03 -0.46 TL -0.51 0.21 0.77 0.03 -0.51 LA/AIDS -0.54 0.22 0.79 0.04 -0.51 a/ Commodities: 3- beef; 4- pork; 5- other meats; 8 - eggs; 14- processed vegetables. Table 69. Compensated Price Elasticities Matrix in The Second Stage, Group III Commoditiesa Model 6 9 15 6 Full -0.31 -0.47 -0.14 TL -0.31 0.17 0.14 LA/AIDS -0.45 0.24 0.21 9 Full 0.11 -0.07 -0.38 TL 0.11 0.21 -0.32 LA/AIDS 0.15 0.32 -0.47 15 Full 0.16 0.66 0.84 TL 0.16 -0.60 0.44 LA/AIDS 0.24 -0.85 0.60 a/ Commodities: 6 - poultry; 9- milk; 15- sweets; Table 70. Compensated Price Elasticities Matrix in The Second Stage, Group IV Commodities® Model 7 10 7 Full -0.45 0.46 TL -0.50 0.50 LA/AIDS -0.45 0.45 10 Full 0.21 -0.22 TL 0.29 -0.29 LA/AIDS 0.22 -0.22 a/ Commodities: 7- sea foods; 10- dairy products. Table 71. Compensated Price Elasticities Matrix in The Second Stage, Group V Commodities8 Model 11 12 16 11 Full -0.52 0.09 0.25 TL -0.64 0.03 0.59 LA/AIDS -0.64 0.03 0.59 12 Full 0.02 -0.58 0.33 TL 0.03 -0.62 0.59 LA/AIDS 0.02 -0.62 0.62 16 Full 0.26 0.27 -0.57 TL 0.33 0.32 -0.64 LA/AIDS 0.31 0.31 -0.65 a/ Commodities: 11- fresh fruit; 12- fresh vegetables 16- nonalcoholic beverage. APPENDIX I GRAPHS FOR COMPARING ACTUAL AND PREDICTED EXPENDITURE BASED ON DYNAMIC MODEL II 261 OlUlUJ llUVIt lVlllO* fiunvr CWkvc ^biivr OLhVI' 292 263 COMPARISON OK ACTUAI. AND PREDICT EXPENDITURE 13V MONTH n»*crnr rrwoucrs: *vi«*cc mckiv ttrprtmnunrs rtw mrcdrxn Sau»-IC rrnino; err. »nno - ore. s. 7 V ■\ \ J. JAtmj JAM*34 Mntltlt ACHIAt MA'M TRfOICt IRIMHir C OMPARISON OP ACTUAI. AND PREDICT EXPENDITURE T3Y MONTU orrr: a v c r a t c w r , m * r.xr ruor'i'nrn rrr imrriKiio sam ii rm w - *rr iorr - oie. nor. m #-\ A m u r —— h a -.h r n r m c r tniA»tftir 264 COMPARISON OP ACTUAI. AND PREDICT EXPENDITURE DY MONTH 2.9 J n 7 t 7 ft 7 ft 2.4 S 2 2 \ i 7.7 V 7 I 7 n I 9 > ft I 7 I o j«uno uohiii — mash r n t u i c t — COMPARISON OP ACTUAI. AND PREDICT EXPENDITURE DY MONTH OIIHR U(AIS AVIRACf nilklY f*riNf)| IIIRIS r |R MOUSCHUIO SA'^ t i r n u o o : s c r 19 rb - otc. im JA'mn jAuni jaiir? jAiin) jambs jamas jauai UVIIM A C iU A i i u s m r r « ( 0 i c t ( r i m k u 265 COMPARISON OF ACTUAL AND PREDICT EXPENDITURE I3Y MONTH rom tnt: *vin»nt »tC*iY i»ro/OMu \ UTII1II m *5h rmoici COMPARISON o f a c t u a l a n d p r e d ic t EXPENDITURE DY MONTH » n u nt 0 is n UIIIH n*r.M r n u » u : » iriaik-.h 266 COMI’AUISON ()!•• ACTUAI. AND I’UKDICT K.VI’ENDITIIIIIS DY MONTH 0 P* n a> o B? e pi 0 no e 19 n /•i a m * 0 7J I 0 /> r a n l o ;n ' e f.i i n J!* an c«r>7 ii n D A M I < M ^ 0 6? R 0 61 c o m 5 • 59 t p m V 57 o r-6 0 57 e m 0 50 A c m u h a s h r»rnin iRMimir COMI’AlilSON o r ACTUAI, AND I’UKDICT KXPKNDITUUK HY MONTH uiik- Av»nAn KiiKir rurriNHmnis rm uaisfHoio SW«*U M R 100: SCP. 19*0 - dlC. 1986 A 2 \ 7 n * * ’ ’ ' ’ ’ ’ ...... I...... — 1------1- ■- —'- "'T JWIRO JAIIOI JAM82 JAHOJ JAMB* JAMBS JAIIB6 JAMB? |A«lllt a c m i »i h a s h r m o i r i ir i a i a h c 267 COMPARISON OK ACTUAI. AMD PKKDIC.T KXPK.NDITIJUK. BY MONTH n*ii)T Avrn»r,t t>cr*iv c»pr»*>Mmrs Pin nonstnoio SAI4-1C Piniott: 51P. I9B0 - l>EC. 1906 aciipai hash r n i o i c i iriahgic COMPARISON OK ACTUAI. AND PKKMC'T KXPKNOITURI:: HY MONTH *nfsm rni»M<&: AViiiAr.r. wifK iY E*rr.noi l i n e s p e r HrmsriioiO SAu ii remoo- sip. 1900 - dec isos 1 / It Ad (I A r f AmiAi. . — rniutct i h ia ik h i 268 COMI'AlilSON o r ACTUAI. AND I’KKDICT KXl’KNDITUIUi 13V MONTH m t s n vcr.itanus avihact w r m v Evrcmiiinrs t i« tousnoio Sf-iV'tC Sin. 1909 - (i(C. 1900 IA U/ Ariir»i h a s h rntoici • COMPARISON OK ACTUAI. AMI) I’RKOICT KXPKNOITURK 13V MONTH rnrx-f.Mft riimis- AvinAr.r wffKtv r*rinoi i»mrs rrn imisooio rrnioo sir toon - Die. 1900 JAIIflJ 4AH0J 4AHA4 u v m t acih a i — h a s h r m m c i imAtK.ic 269 COMPAKISON OK ACTUAI, AND PHKDICT liXPKNDITUKH DY MONTH mnri&sio vrr,r.Mnics: Avrn*cc urciaY rvrcHmiims pro iious imo io m u h c rcnioo: sen, »«® - otc. i»bs i o? t ** " i o? 5 I OA £ I 05 , i o« Y I 01 I 0.' «r i ono i p o 11 \ I 0 04 II A 91 n * or, \ f i « ««•, A n ® 94 R • 93 v • 9? i 0 31 ' o go i e m 0 04 n n; o m 0 65 e 6« 0 41 e 9; 0 61 n «n 0 /!> 0 7A UJtJIH • mash rnrnici . COMI’AlilSON 01 * ACTUAI, AND PREDICT RXPKNDITURL: 13V MONTH s u m s *vrn*or. orrtdv c m u n m « > ( s rtR »«msriioio sAirir Pii'ino- sip »?no - m e . ifloo JAIIA? j Anas Afiii«i -— h a s h mriiirt — iRiAir.ir 270 COMPARISON OK ACTUAI, AND PREDICT EXPENDITURE NY MONTH I II I 10 I 09 I 09 I tlf • i nr. i I 0'i V t 04 \ I 99 I 0? ?I I ntop (iti1) ui •e <17 u-»* 0 Oft | M l 5 o 01 $ 0 07 I 0 91 o on (* • 99 \ 0 ftft \ ft 97 a 0 pr. o rr. n "i o pi JANft? JAMftJ JAIM4 M PI III ACHIAI -- HASH POtOICT 1RIAHGIC COMPARISON ()| - ACTUAI. AND PHKDICT EX PI' ^ ; I 11Y MONTH 'I i \f V ACIHAl • — 11*511 I R I D I C ! iniAII-.lt 271 COMI'AKISON OK ACTUAI. AND I’RKDICT EXPF’.NDITUHG DY MONTH uisc rows- *vf««cf m m ciriw iiuntsn n housihoid s» ru rt*iu>- sin. 190* > Die. i«ae * h. A 0) JAUM AMIIH • »iash rncoici ------1 COMI’AHISON OK ACTUAI. 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