Selecting Functional Form in Production Function Analysis

Ronald C. Griffin, John M. Montgomery, and M. Edward Rister

Functional form selection is a sometimes neglected aspect of applied research in production analysis. To provide an improved and uniform basis for form selection, a number of traditional and popular functional forms are catalogued with respect to intrinsic properties. Guidelines for the conduct of form selection are also discussed.

Key words: flexibility, functional form.

The art and science of applied economics crit- building processes. The use of functional forms ically depends on model building, and the in production function applications is empha- practicing economist makes many decisions in sized in order to limit the discussion. Because the course of constructing an appropriate mod- the concept of flexibility is important to form el. Within this process, the practitioner must selection, we begin with a brief summary of often adopt a functional form as the pattern this topic. for one or more continuous (possibly piece- wise) physical or economic relationships. Common examples of such relationships are Flexibility and Maintained Hypotheses production, profit, and cost functions as well as systems of input and/or output supply and Recent advances in developing new functional demand. The continuity of these relationships forms have been dominated by efforts to con- is rarely, if ever, proven but seems sufficiently ceive "flexible" forms, and different technical compelling to be embraced without question definitions of flexibility have arisen as a result in nearly all cases. The researcher, however, is of these pursuits. Because flexibility is a mul- never in a position to know the true functional tidimensioned concept, a given technical def- form, and, as noted by Hildreth, "the principal inition of flexibility may not be adequate in disadvantage of continuous models lies in the all situations. Local flexibility (sometimes biases which may accrue if an inappropriate Diewert flexibility or, simply, flexibility) im- [functional] form is used" (p. 64). plies that an approximating functional form The model builder's task is complicated by conveys zero error (perfect approximation) for the growing number of available functional an arbitrary function and its first two deriva- forms. A compilation of alternative functional tives at a particular point (Fuss, McFadden, forms and a comparison of these alternatives and Mundlak). The locally flexible form places on the basis of selected criteria are presented no restrictions on the value of the function or in this paper. Available selection criteria per- its first or second derivatives at this point. tain to mathematical, statistical, and economic Therefore, no restrictions are imposed on properties and are useful for formalizing the properties that can be expressed in terms of selection of functional form(s) during model derivatives of second-order or less. The authors are an associate professor, a research assistant, and Second-order Taylor series expansions have an associate professor, respectively, in the Department of Agri- dominated the field of locally flexible forms cultural Economics, Texas A&M University. but are not unique in the ability to offer local This research as conducted by the Texas Agricultural Experi- ment Station (Projects 6447 and 6507) with support from the Texas flexibility (Barnett). Ignoring the complica- Rice Research Foundation (Econo-Rice Project). tions of statistical estimation by momentarily Appreciation is expressed to anonymous reviewers of this Jour- nal and the AJAE for helpful comments during various stages of presuming that we wish functionally to ap- this article's development. proximate a known relationship, locally flex-

Western Journal of Agricultural Economics, 12(2): 216-227 Copyright 1987 Western Agricultural Economics Association Griffin, Montgomery, and Rister Selecting FunctionalForm 217 ible forms can impose large errors in the ap- The concept of maintained hypotheses is proximating function and its derivatives away fundamental to any concept of flexibility. As from the point of perfect approximation (Des- pointed out by Fuss, McFadden, and Mund- potakis). Global flexibility (sometimes Sobo- lak, maintained hypotheses "are not them- lev flexibility) is preferred to local flexibility selves tested as part of the analysis, but are in that second-order restrictions are every- assumed true" (p. 222). The choice of func- where absent (Gallant 1981, 1982). tional form immediately renders some hy- Recent attention to the influence of esti- potheses untestable (maintained) while others mation on model development has severely remain testable. "Sometimes the question of reduced the attractiveness of locally flexible whether a specific hypothesis should be tested forms. Because an estimated Taylor series ex- or maintained is critical" (Ladd, p. 9). As an pansion is a fit to data from the true form, example, King's 1979 address to the AAEA rather than an expansion of the true form, there membership is most certainly rooted in his may be no actual point where the true function disappointment with the maintained hypoth- and its first two gradients are perfectly ap- eses of linear models. proximated. Inquiry in this area has produced Attention to the issue of maintained hy- some discomforting results. The examples pro- potheses aids the researcher in developing a vided by White demonstrate that ordinary least more careful and richer analysis. Popular con- squares estimators of Taylor series expansions cerns pertaining to potential maintained hy- are not reliable indicators of the parameter potheses within economic production models vector for the true expansion of a known func- include homogeneity, homotheticity, elasticity tion. As a consequence of these and other find- of substitution, and concavity. Less restrictive ings, predictive properties of locally flexible functional forms would always be desirable, forms have been found to be satisfactory (for were it not for the greater information needed large samples), but inferences involving single to adequately specify such relationships. parameter estimates and functional combi- Greater flexibility can usually be achieved by nations of these estimates are not reliable. Be- adding arbitrary and nonredundant terms to cause estimation is driven by a global (at least any given function, but to do so reduces de- throughout the data domain) criterion such as grees of freedom and may increase collinearity least sum of squared error, it has been a bit and the expense of parameter estimation. Be- unnatural to tout local flexibility as an advan- cause reductions in maintained hypotheses tage in economic analyses. The implications come at a cost, added flexibility is not always here are severe for typical applications in pro- desirable, and there are likely to be cost-effec- duction function analysis. tive opportunities to achieve particular di- To gauge global flexibility in a manner which mensions of flexibility. also acknowledges derivative values, a mea- sure of error is needed which incorporates errors in derivatives as well as errors in the The Choice Set approximating function. The Sobolev norm (a distance measure) satisfies this requirement and Based on a review of traditional and popular has been employed to assess global flexibility literature, twenty functional forms are iden- (Gallant 1981). In applied work, however, the tified for consideration. The names of each of Sobolev norm is intractable for obtaining pa- these functions and their algebraic form are rameter estimates, so estimation of globally listed in the first two columns of table 1. Note- flexible forms still uses traditional distance worthy reference material for these functions measures like least squares. Evidence provided and some of their properties include Heady by Elbadawi, Gallant, and Souza suggests that and Dillon (quadratic, square root, Mitscher- this produces satisfactory results. This finding lich, Spillman, resistance, modified resis- supports the judgment expressed by Rice that tance); Lau (square rootS); Halter, Carter, and form exceeds norm in importance for approx- Hocking (transcendental); Uzawa (CES); Sil- imation (pp. 1, 20). It is still true, though, that berberg (CES); Diewert (generalized Leontief); the effects of selecting a particular form are jointly determined by the inseparable influ- Lau calls this the "generalized version of the 'Generalized Lin- ence of form restrictiveness and estimation ear Function' " (p. 410). Fuss, McFadden, and Mundlak call it the technique. generalized Leontief function. 218 December 1987 Western Journal of Agricultural Economics

Table 1. Properties of Selected Functional Forms

(A) (B) (C) (D) Does x, = 0 Does x, = 0 for any Functional Form for all i one i Asymptotic Function (i, j, k= 1, .. , n) Imply y = O? Imply y = 0? dy/Ox, Convergence Leontief y==min[3,x,, 2x, ... , ./xn] >> 0 yes yes 0 or UBC no Linear y= =a + ixi no no UBC no i

Quadratica y= a + ix, + xix, no no U no i i j

a b Cubic y= a + fix, + 2 xixx, no no U no

+ S 2 Sk 7yx;^xi i j k a Generalized Leontief y= yes no UBN no

a Square root y= =a+ ixx + 2 2 6X no no U no

Logarithmic y= =a + 3,iln x, un defined undefined UBN no

Mitscherlich y= =a (1 - exp(fi,x)) yes yes UBN if/ << 0 i

Spillman y = aI (1 - ,Ti) yes yes UBN if 0 << << 1 i

Cobb-Douglas y= a f xiI yes yes UBN no i

Generalized Cobb-Douglas In y = + 2 2 6jln((xi + x)/2) undefined no U no i j(exp( )) Transcendental y: =a II x~i(exp(bixi)) yes yes U no

1 Resistance y- = a + o/i(6, + x,)-, no no UBN yes

a - Modified resistance y- = a + ~ fix, ' + Z 1 'XF'xij' undefined undefined UBN yes i joi

CES Y = a + ix-" undefined undefined UBN no

a Translog In y = a + iBlnx, + 6,bi(lnx)(ln x) undefined undefined U no

1 - Generalized quadratic y= fijx>6 ) yes no UBN no

Generalized power y = a I x{f(x)exp(g(x)) yes yes U no

a Generalized Box-Cox y(O) = a + /,x;(X) + b1jx(X,)xj(X), no no U no 2 where y(8) = (y - 1)/20 x and x(X) = (x - 1)/X

x Augmented Fourier y = fix, + 6 ij i"xj no no U no

+ 2 'Yhexp(i hx), Ihl*'H i where all x, E [0, 21r], 2 _'h = Y + i3', i = --1

Note: U is unrestricted sign (+, 0, or -); UBC, unrestricted in sign but constant; UBN, unrestricted but nonswitching in sign (e.g., if dy/dxk > 0 for some x, > 0, then dy/Ox, > 0 for all x, > 0); and NG, not in general. a Assuming bj = ,j for all i, j. cb Assuming -yk = 7y, = 'Ykii= kjiyjki = 'yj= for all i, j, k. d Some of the stated conditions are sufficient but not necessary for local concavity (see text). Same as for quadratic form plus y, = 0 for all i, j, k. If any two of the following conditions are satisfied: (1) all 3,= 0, (2) all 6, = 0, (3) all y,, = 0. 'Yes, 2 but x, may equal zero for only one i. h V/2(n+ 3n + 3) assuming 0 and Xare free. See appendix. Linear, quadratic, generalized Leontief, square root, logarithmic, Cobb-Douglas, modified resistance, CES, translog. Griffin, Montgomery,and Rister Selecting FunctionalForm 219

Table 1. Extended

(E) (F) (G) (H) (I) (J) (K) (L) Constant Linearly Subsumes Elasticity No. of Separable What Linearly of Sub- Distinct c Linearly if any Other Homogenous Ho]mothetic stitution Concave Parameters Separable x = 0 Functions - yes yes a = 0 yes n no no ifa = 0 yes a = oo affine n + 1 yes yes

ifa = 0, if , = 0 or NG NG 1/(n + l)(n + 2) yes yes Linear all bi =0 all6, = 0

d NG NG (n + 3)(n +2) yes yes Linear, *(n + 1)/6 quadratic

yes yes NG if all 1/2n(n + 1) yes yes 6,> 0

if a = 0, if = 0 NG if / > 0, I/(n + l)(n + 2) yes yes Generalized /=0 all , >-0 Leontief no yes NG if/ >> 0 n + 1 yes undefined

no no NG if a > 0, < 0,, + 1o n Spillman exp(,ix,) < 1

no no NG ifa > 0, n + 1 no no Mitscherlich 0 << f < 1, xi?< 1

if :/i= 1 yes a = 1 if a > 0, n + 1 yes no 0 << << 1, Y i, < 1

2 f if 2; , = 1 yes no NG /2(n + n + 2) yes Cobb-Douglas i j

if fli = 1, if 6 = 0 if 6 = 0 if a > 0, 2n + 1 yes no Cobb-Douglas 5=0 0 << < 1, i, < 1 if a = 0, 6 = if = 0 NG if /> 0, 6 > 0 2n + 1 yes yes

2 if a = 0, if all 6, = 0 NG NG 1/2(n + n + 2) yes undefined all 5, = 0

ifa = , v= 1 yes a = 1/ ifa > 0, i > 0, n +3 NG undefined Linear, (1 + p) 0 < v < 1 Cobb-Douglas

if fi = 1, if all NG if >> 0, all ij > 0 1/2(n + l)(n + 2) yes undefined Cobb-Douglas all bij = 0 y es = 0 i i > 2 if v= 1 yes NG if all f3 0, n + 3 NG NG Generalized 0 - <- 1, Leontief, CES 6 1, v < 1

NG NG NG NG indeterminate NG NG Transcendental

if Y f2i if all ,i = 0 NG NG NG NG = 2aX + 1, or all b = 0,

all Xf = 2 I b, i j and all X/i = 2,ii no no NG NG yes yes Linear, quadratic 220 December 1987 Western Journal of Agricultural Economics

Christensen, Jorgenson, and Lau (translog - A third category of selection criteria in- transcendental logarithmic); Fuss, McFadden, volves data-specific considerations (goodness- and Mundlak (translog, generalized Cobb- of-fit and general conformity to data). Such Douglas, square root); Denny (generalized concerns are necessarily omitted from this dis- quadratic); de Janvry (generalized power); Ap- cussion because comparisons among function plebaum (generalized Box-Cox); Berndt and forms require the use of a specific dataset and Khaled (generalized Box-Cox); and Gallant findings would not be general. In addition, there (augmented Fourier). is a significant body of literature concerning The augmented Fourier form is globally flex- testing of nested and non-nested models (see ible, asymptotically, in that no second-order Judge et al.). restrictions are imposed anywhere in the do- The fourth grouping of selection criteria per- main. It is also nonparametric; the number of tains to application-specific characteristics. For parameters to be estimated varies with sample example, if the resulting equation is to be used size. Because the augmented Fourier form is in simulation or optimization procedures, there not commonly used in production function may be other desirable properties for func- analyses, a few essential details are presented tional form. These considerations are largely in an appendix. excluded from the present discussion because the number of potential criteria within this category is quite large and highly customized. The Criteria For ease of presentation, the discussion em- phasizes selection of a production function As stated above, determination of the true rather than a demand, profit, or cost function. functional form of a given relationship is im- The rationale for this choice stems partially possible, so the problem is to choose the best from the fact that most recent advances in con- form for a given task. This problem leads to ceiving flexible forms are published as cost or consideration of choice criteria, that is, how demand applications. Much of the discussion one functional form may be judged better or presented here can be readily applied to cost more appropriate than another. These criteria or demand studies. System-related properties, may be grouped in four categories according such as symmetry, are not considered because to whether they relate to maintained hypoth- of the production function emphasis. eses, estimation, data, or application. Some functional forms allow as special cases Concerns regarding maintained hypotheses the assumption of all properties of other func- form one set of objectives whereby appropri- tional forms. The more flexible forms may be ateness can be assessed. If the maintained hy- appropriate when information regarding the potheses implied by a certain function are ac- nature of the relationship does not permit cer- ceptable, or even useful, then the function may tain hypotheses to be maintained. Those func- be deemed appropriate. In the absence of a tional forms subsumed by others are indicated strong theoretical or empirical basis for adopt- in the last column of table 1. Employing a ing a given maintained hypothesis, however, nonlinear transformation of coordinates, the a functional form which is unrestrictive with generalized Box-Cox form subsumes nine oth- respect to this hypothesis may be considered er forms as nested cases. They are: translog appropriate. (0 = X = O); CES (vO = X, all Xf = 26,, bi = 0 Second, functional form has implications for for all i # j); modified resistance (6 = -1/2, X = statistical processes of parameter estimation. -1, all b6 = 0); Cobb-Douglas (0 = X = 0, Data availability, data properties, and the Z /i = 1, all bi = 0); logarithmic (0 = /2, X = availability of computing resources can affect 0); square root and generalized Leontief (0 = the choice of functional form for statistical es- X = 1/2); and quadratic and linear (0 = 1/2, X = timation. Moreover, some forms do not per- 1). mit parameter estimation by linear least squares procedures, and alternative proce- Maintained Hypotheses dures typically offer less information concern- ing estimator properties. Criteria relating to Column A of table 1 concerns the assumed maintained hypotheses and statistical esti- effect of inputs specified in the function but mation are discussed below, and important not applied (i.e., those inputs with a zero level). findings have been condensed in table 1. Assuming that one or more inputs are hy- Griffin, Montgomery, and Rister Selecting FunctionalForm 221 pothesized to be related to output and included Two related properties exhibited by certain as exogenous variables, column A indicates for forms are homogeneity and homotheticity. which functional forms output would be zero Functional forms homogenous of degree 1, said if all input levels were zero. Such a maintained to be linearly homogenous, are indicated in hypothesis may not conform to observed phe- column E. If production is found to be prof- nomena. In agronomic fertility studies, for ex- itable for some input combination in a linearly ample, one often encounters experimental data homogenous production function, profit can where nutrient applications are observed, but be increased without bound by increasing in- soil nutrient levels are not. Zero yields are im- puts proportionately. posed by particular functional forms when ob- Homothetic forms are indicated in column servations pertain to applied nutrients and no F of table 1. The interesting characteristic of nutrients are applied. The hypothesis repre- homothetic production functions is that the sented by column A thereby underscores the marginal rate of technical substitution (MRTS) relationship between variable specification and remains constant as all inputs are increased functional form. proportionately. When the marginal rate of A similar but more relevant property is de- technical substitution is constant, as for homo- scribed in column B, which indicates for which thetic functions, any change in the value of functions inputs are essential; that is, no out- output will affect the optimum input levels put occurs if at least one of the inputs of a proportionately as long as input prices are un- multiple-input model is not applied. Again, changed. the importance of this property primarily de- Related to the concept of the marginal rate pends on how the inputs of a specific model of technical substitution is the concept of the are defined. elasticity of substitution. In general, MRTSj First derivatives of production functions are will increase in magnitude as xi is increased important in nearly all applications. Because (and xj is decreased) along any given isoquant. some are cumbersome to compute, derivatives This rate of change can be measured by the for all twenty forms are given in table 2. Col- elasticity of substitution.2 Those functional umn C of table 1 indicates maintained hy- forms assuming constant elasticity of substi- potheses concerning the sign of marginal prod- tution are indicated in column G. uct. Functions in table 1 are characterized by The generalized constant elasticity of sub- marginal products which are unrestricted in stitution (CES) production function, which is sign (U), unrestricted but nonswitching in sign defined not only to have constant elasticity of (UBN), or unrestricted in sign but constant in substitution but also to be homogenous of de- value (UBC). Functional forms with marginal gree v when a = 0, represents a much-studied products that are unrestricted but nonswitch- class of production functions (Arrow et al.; ing in sign allow successive units of applied McFadden; Uzawa). The concept of elasticity input to increase, decrease, or not change total of substitution and the CES class of production output, but any change in total output must functions have been applied primarily to the be in the same direction as the change pro- problem of the distribution of income among duced by the previous unit of input. Thus, the generally defined or highly aggregated factors nine UBN functional forms (as well as the two of production (Arrow et al.; Behrman; Ner- UBC ones) do not allow model estimation to love). With regard to production function determine at what input level output begins to analysis, however, Silberberg states that decrease (assuming the data supports this test- "knowledge of a at any point would undoubt- able hypothesis) but rather maintains the edly be a useful technological datum for em- hypothesis of everywhere positive (or every- where negative) marginal productivity. 2 For any change along an isoquant, d(xi/xj) is the change in the use of xi as compared with that of Xj, and d(dx/dxj) is the corre- Column D indicates which functional forms sponding change in the marginal rate of technical substitution. maintain the hypothesis of asymptotic con- "The ratio of these differentials, expressed in proportional terms vergence to make them independent of units of measurement, is defined as of output towards a maximum as in- the elasticity of substitution between the factors at the combination put levels are increased. In such cases, output of factors considered" (Allen, p. 341). The elasticity of substitution increases as the level of input increases, re- between xi and xj is given by gardless of the data employed to estimate the (f=/I)d(x,/xj) change in output due to a change in the level "j (x,/ j) d(f/l) of input. where fi = dy/9xi andf = ay/dx,. 222 December 1987 Western Journal of Agricultural Economics

Table 2. First Derivatives of Selected Functional Forms

ay Function axi Function Ox,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Leontief 0 or f, Linear fi Quadratic fi + 2,ijx

Cubic j joi k Generalized Leontief

2 ? ' Square root (1i + Z aijX )x

Logarithmic efixi iXi Mitscherlich -a1,exp(81x,) II (1 - exp(jx)) joi Spillman -apxiln(i) 1I (1 - jpx) joi Cobb-Douglas aiXi-1 nxif

Generalized Cobb-Douglas y Z 2&,/(x, + xj)

Transcendental a(bi + ilx,) II xaiexp(bjxj)

2 2 Resistance ,iY (6i + xi)- 2 Modified resistance y (xi,-2 + 2ijXi2Xi-1) j=i v+p CES ivxI'--P[a + ' oiXiP] p

Translog y[1i + 2 ijln(xi)]/xi v-6 Generalized quadratic vx;-'y v [ f F.jx.xl-, + (1-) j f i '-,xI] i J Generalized power x Generalized Box-Cox x- 'y'-20[Xi + 2 bij(X - 1)]/X

Augmented Fourier Pi + S 2ijx + i ; hihexp(i z hix,) j Ihl*'H i

pirical work" in that the statistic describes the As indicated in column H of table 1, esti- relative ease with which one input may be sub- mated parameters of the model may need to stituted for another (p. 317). be examined before the researcher can ascer- A final property, concavity, is important in tain whether concavity is present. The impor- at least two respects. It is in one sense pertinent tance of concavity for profit maximization im- to description of the production process, re- plies that such an examination should always flecting a situation in which output increases be conducted. While the conditions listed for at a decreasing rate (or decreases at an increas- concavity in table 1 are necessary and sufficient ing rate) as the level of input is increased. The for global concavity, that is, for concavity property may be of primary importance, how- everywhere in the nonnegative orthant, many ever, in the context of economic optimization; of these conditions are not necessary for local only if the function is concave can input levels concavity. In particular, the indicated condi- that maximize profit be computed from first- tions are merely sufficient for local concavity order equations. in the case of the square root, resistance, mod- Griffin, Montgomery, and Rister Selecting FunctionalForm 223 ified resistance, CES, generalized Leontief, dataset. The total number of parameters to be translog, and generalized quadratic forms. estimated by a given form may be calculated with the formulas presented in column I. Statistical Estimation Ease of analysis and availability of resources may also limit the modeling process to func- Certain properties embodied in functional tional forms containing parameters that are all forms have important implications for the linearly separable (indicated in column J) and mathematical procedures employed in the sta- can therefore be estimated by common and tistical estimation of parameters. Pertinent well-developed linear least squares regression properties included in table 1 are the number techniques. Whereas direct estimation of non- of distinct parameters (column I), and linear linearly separable forms may be possible by separability of the parameters (columns J and such techniques as maximum likelihood esti- K). mation, Fuss, McFadden, and Mundlak note Most functions, in their complete forms, re- that "linear-in-parameter systems have a com- quire a geometrically increasing number of pa- putational cost advantage, and have, in addi- rameters to be estimated as the number of vari- tion, the advantage of a more fully developed ables of main effect is increased. This is statistical theory" (p. 225). Linear least squares primarily because of the large number of in- regression provides information regarding the teractions specified among variables of main small-sample accuracy and precision of esti- effect. For example, if two inputs are hypoth- mates, which may not be available from other esized to affect output, a linear function re- techniques. The value of this information must quires only three parameters to estimate that be weighed against any advantages in the use effect, whereas a complete cubic function re- of nonlinearly separable forms. On the other quires estimation of ten parameters. If ten in- hand, the gain in information for linear least puts are hypothesized to affect output, a linear squares estimators may be artificial, since this function requires that eleven parameters be information results from several important as- estimated, but a cubic function requires esti- sumptions. mation of 286 parameters. 3 In addition, the nature of the dataset must Fuss, McFadden, and Mundlak have ex- be considered if a technique such as ordinary pressed concern that "excess" parameters least squares is to be employed. Column K of would "exacerbate" multicollinearity present table 1 shows which functional forms are lin- in survey data (p. 224). If multicollinearity is early separable in parameters when any input high, the variance of the parameter estimates levels take on zero values. For example, even is increased such that it may be impossible to though column J identifies the generalized determine how much variation in the endog- Cobb-Douglas form as being linearly separa- enous variable is explained by different exog- ble, if the dataset used for estimating the func- enous variables. Higher parameter variances tion contains an observation in which none of also enlarge confidence intervals for applica- one input is applied, ordinary least squares tions (such as the computation of an optimal cannot be conducted. input program) involving subsets of the esti- It is notable that the generalized Box-Cox mated parameters. If, however, multicollin- form is not linear in parameters, and this fact earity is irrelevant because the primary pur- tends to limit applications. If coefficients 0 and pose of the model is for prediction (Maddala, X are chosen a priori, the form becomes linear p. 186), then the issue of having many param- in parameters. When this is done, the Box-Cox eters can be quite unimportant. Of-possible form becomes one of the subsumed functional concern to the researcher may be the expense forms, and its generality is sacrificed. of estimating a large number of parameters or the loss of degrees of freedom. Either of these problems could actually preclude the use of Formalizing the Selection of certain forms, depending on the number of Functional Form variables of main effect, computing resources available to the researcher, and/or size of the This paper describes why the researcher should be concerned with the functional form of a 3 In some cases, additional maintained hypotheses can be intro- continuous input-output model and provides duced to eliminate certain interactions. For example, any real in- teraction between the quantities of irrigation water and harvest some guidance for assessing whether some labor in crop production may be thought, a priori, to be absent. functional forms may be considered more ap- 224 December 1987 Western Journal of Agricultural Economics propriate than others. As a result of the math- the investigator to choose one model over the ematical properties inherent in each functional other. While it is clear that such data-related form, hypotheses are maintained about the objectives can form an important set of cri- production relationship whenever a specific teria, allowing statistical evidence to dictate function is selected for estimation. 4 Any po- model selection is regarded as a questionable tential hypothesis which is not maintained and and often unnecessary practice. can be expressed as one or more linear rela- The following observation by Hildreth is of tions among function parameters will be test- interest: "It is particularly disconcerting that, able.5 Such relations are indicated in table 1. in many instances in which several alternative Inherent mathematical properties of a given assumptions [as to functional form] have been functional form can have important implica- investigated, alternative fitted equations have tions for statistical estimation and economic resulted which differ little in terms of conven- application. tional statistical criteria such as multiple cor- A functional form may be appropriate be- relation coefficients or F tests of the deviation, cause of the correspondence of maintained hy- but differ much in their economic implication" potheses with generally held theories of the (p. 64). Given the possible differences in eco- true input-output relationship, possibility and nomic implications, it is often advisable to ease of statistical estimation, possibility and explore the sensitivity of calculated economic ease of application, general conformity to data, optima to the choice of functional form. As an or a combination of these criteria. None of the interesting example, in an analysis of a single many criteria guarantee that the true relation- application of nitrogen and potassium on corn, ship will be discovered, nor do any allow a Bay and Schoney have assessed the costs of totally objective choice to be made. Subjective "incorrectly" choosing among four possible judgment is a necessary aspect of choice re- functional forms. Similarly, in an analysis of garding functional form. Thus, formalization multiple nitrogen applications on rice, Griffin of the selection process requires deliberative et al. have examined the sensitivity of optimal choice and frank presentation. fertilizer programs to functional form. Having selected two or more estimable func- tional forms with acceptable theoretical and application properties, the researcher may wish Conclusions to base final selection upon a statistical crite- rion. Such criteria entail data-specific consid- It is often difficult to ascertain why particular erations and may comprise a decision rule as functional forms are chosen for the models simple as choosing the model with the highest presented in published economic research. coefficient of multiple determination. Strictly Possibly, researchers confine their attentions statistical information can be used for the pro- to particular functional forms because they are cess of model selection since several formal/ most comfortable and experienced with these informal opportunities are available for both forms or because these forms are in vogue. In nested and nonnested testing. Judge et al. note fairness, there are probably many examples that tests of nested models measure "how well where the research process is extensive and the models fit the data after some adjustment formal in the selection of functional form, but for parsimony" (p. 862). Tests of nonnested a description of these procedures is omitted models ask the question: Is the performance from published results. These omissions may of model 1 "consistent with the truth" of mod- be due to a propensity of authors, reviewers, el 2 (Pesaran and Deaton, p. 678)? Thus, the or editors to consider such material extra- testing of each model against the evidence pro- neous. Applied economists can enrich their vided by the other will not necessarily allow analysis by selecting functional forms from as broad a choice set as possible and by consid- 4 Also, properties which are not inherent to a given form can ering a number of selection criteria (other than often be incorporated as maintained hypotheses by the appropriate merely data-related criteria). Formal state- choice of linear restrictions. Table 1 indicates the appropriate re- strictions for the considered criteria. Moreover, different functional ment of these procedures in published output forms can be simultaneously developed in order to test different is desirable so that readers might better gauge hypotheses (as in Shumway's paper). research results (particularly sensitivity to 5 However, such tests require the additional (augmenting) hy- pothesis that the true form is of the class under consideration form). (Gallant 1984). The need "to make research processes overt" Griffin, Montgomery, and Rister Selecting FunctionalForm 225

(Tweeten, p. 549) pertains as much to the choice Can Be Estimated Consistently without A Priori of functional form as to other aspects of com- Knowledge of Functional Form." Econometrica monly reported research methodology. That 51(1983):1731-51. the true form cannot be known and that it is Fuss, M., D. McFadden, and Y. Mundlak. "A Survey of impossible to measure how well any chosen Functional Forms in the Economic Analysis of Pro- duction." ProductionEconomics: A DualApproach to form approximates the true form lends im- Theory and Applications, ed. M. Fuss and D. Mc- portance to the selection process. Formaliza- Fadden, pp. 219-68. Amsterdam: North-Holland tion of this process requires the development Publishing Co., 1978. of an explicit and, hopefully, exhaustive choice Gallant, A. R. "On the Bias in Flexible Functional Forms set and some acceptable criteria for conducting and an Essentially Unbiased Form." J. Econometrics the selection. The objective of this paper has 15(1981):211-45. been to develop such a choice set and offer an . "The Fourier Flexible Form." Amer. J. Agr. Econ. initial listing of plausible criteria. Extension of 66(1984):204-15. this compilation and discussion to settings . "Unbiased Determination of Production Tech- other than production function applications nologies." J. Econometrics 20(1982):285-323. remains as a needed endeavor. Griffin, R. C., M. E. Rister, J. M. Montgomery, and F. T. Turner. "Scheduling Inputs with Production Func- [Received December 1985; final revision tions: Optimal Nitrogen Programs for Rice." S. J. Agr. received August 1987.] Econ. 17(1985):159-68. Halter, A. N., H. O. Carter, and J. G. Hocking. "A Note on the Transcendental Production Function." J. Farm Econ. 39(1957):966-74. Heady, E. 0., and J. L. Dillon, eds. Agricultural Produc- References tion Functions. Ames: Iowa State University Press, 1961. Allen, R. G. D. Mathematical Analysis for Economists. Hildreth, C. G. "Discrete Models with Qualitative De- New York: St. Martin's Press, 1938. scriptions." Methodological Procedures in the Eco- Applebaum, E. "On the Choice of Functional Forms." nomic Analysis ofFertilizerUse Data, ed. E. L. Baum, Int. Econ. Rev. 20(1979):449-58. E. O. Heady, and J. Blackmore, pp. 62-75. Ames: Arrow, K. J., H. B. Chenery, B. S. Minkas, and R. M. Iowa State College Press, 1955. Solow. "Capital Labor Substitution and Economic Judge, G. G., W. E. Griffiths, R. C. Hill, H. Liitkepohl, Efficiency." Rev. Econ. and Statist. 63(1961):225-50. and T. Lee. The Theory andPracticeofEconometrics, Barnett, W. A. "New Indices of Money Supply and the 2nd ed. New York: John Wiley & Sons, 1985. Flexible Laurent Demand System." J. Bus. and Econ. King, R. A. "Choices and Consequences." Amer. J. Agr. Statist. 1(1983):7-23. Econ. 61(1979):839-48. Bay, T. F., and R. A. Schoney. "Data Analysis with Com- Ladd, G. W. "Artistic Tools for Scientific Minds." Amer. puter Graphics: Production Functions." Amer. J. Agr. J. Agr. Econ. 61(1979): 1-11. Econ. 64(1982):289-97. Lau, L. J. "Testing and Imposing Monotonicity, Con- Behrman, J. R. "Selectoral Elasticities of Substitution vexity and Quasi-Convexity Constraints." Production Between Capital and Labor in a Developing Econo- Economics: A Dual Approach to Theory and Appli- my: Time Series Analysis in the Case of Postwar cations, ed. M. Fuss and D. McFadden, pp. 409-53. Chile." Econometrics 40(1972):311-26. Amsterdam: North-Holland Publishing Co., 1978. Berndt, E. R., and M. S. Khaled. "Parametric Productiv- McFadden, D. "Constant Elasticity of Substitution Pro- ity Measurement and Choice Among Flexible Func- duction Functions." Rev. Econ. Stud. 30(1963):73- tional Forms." J. Polit. Econ. 87(1979):1220-45. 83. Christensen, L. R., D. W. Jorgenson, and L. J. Lau. Maddala, G. S. Econometrics. New York: McGraw-Hill "Transcendental Logarithmic Production Frontiers." Book Co., 1977. Rev. Econ. and Statist. 55(1973):28-45. Nerlove, M. "Recent Empirical Studies of the CES and de Janvry, A. "The Generalized Power Function." Amer. Related Production Functions." The Theory andEm- J. Agr. Econ. 54(1972):234-37. piricalAnalysis of Production,ed. M. Brown, pp. 55- Denny, M. "The Relationship Between Forms of the Pro- 122. New York: Columbia University Press, 1967. duction Function." Can. J. Econ. 7(1974):21-31. Pesaran, M. H., and A. S. Deaton. "Testing Non-Nested Despotakis, K. A. "Economic Performance of Flexible Nonlinear Regression Models." Econometrica Functional Forms." Eur. Econ. Rev. 30(1986): 1107- 46(1978):677-94. 43. Rees, C. S., S. M. Shah, and C. V. Stanojevic. Theory Diewert, W. E. "An Application of the Shephard Duality and Applications of Fourier Analysis. New York: M. Theorem: A Generalized Leontief Production Func- Dekker, 1981. tion." J. Polit. Econ. 79(1971):481-507. Rice, J. R. The Approximation ofFunctions, vol. 1. Read- Elbadawi, I., A. R. Gallant, and G. Souza. "An Elasticity ing MA: Addison-Wesley Publishing Co., 1964. 226 December 1987 Western Journal of AgriculturalEconomics

Rudin, W. Principles of MathematicalAnalysis, 2nd ed. that y(x) in equation (A. 1) is real-valued, it is suf- New York: McGraw-Hill Book Co., 1964. ficient to impose Shumway, C. R. "Supply, Demand, and Technology in a Multiproduct Industry: Texas Field Crops." Amer. (A.2) , =0,= Yh= Y'-h, and yr = -'y . J. Agr. Econ. 65(1983):748-60. Equation (A. 1), conditions (A.2), and the identity Silberberg, E. The Structure of Economics:A Mathemat- ical Analysis. New York: McGraw-Hill Book Co., exp(i hix,) = cos(2 hix,) + i sin(2 hx,i) 1978. Tolstov, G. P. Fourier Series. Englewood Cliffs NJ: Pren- can be used to obtain tice-Hall, 1962. (A.3) y(x) = 1 ['ycos(2 hx,) Tweeten, L. "Hypothesis Testing in Economic Science." Ihl-

Because complex-valued trigonometric polyno- (A.4) y= fx, + Oxdxj mials of order H, i i j + Z [-yYcos( hx) - y, sin(2 hxI)]. Ihl*-H (A. 1) y(x)= 'Yhexp(i ihix, ) , Ihl\-H i can be used to approximate any known periodic The constant term, a, is omitted to avoid redun- function, and because the approximation is perfect dancy since it is contained in the Fourier expression when H = oo, trigonometric polynomials have a (h = 0). The number of parameters in the augmented number of fruitful applications in physics and the Fourier form depends on n and H, but we have engineering sciences. The following additional in- been unable to obtain a completely general formula formation is needed to interpret equation (A.1): h for the number of parameters in the Fourier portion is an n-dimensional vector of integers (..., -1, 0, of (A.4). The following formula applies for n = 1, 2, 3, or 4. 1, ... ) with "length" I hl* = hi ; H is exoge- n(n + 3) m = ml + m2= + m2, nously chosen by the researcher and serves to limit 2 the number of terms in (A. 1). Each coefficient Yh is 2 complex-valued, Yh ="h + i y , where i = -1 and where m is the total number of coefficients in the 'hand Yh are the real and complex components of augmented Fourier form, ml is the number of terms Y,, respectively. Thus, each 7Y really represents two in the quadratic portion (excluding the constant separate parameters. term), and m2 is the number of terms (including the If y(x) is known and the terms Yh are chosen in constant term) in the Fourier portion; m2 is given a certain manner (see Tolstov, pp. 13-14, 174; or by an nth degree polynomial in H: Rees, Shah, and Stanojevic, p. 225), then these terms m2 = ao + alH1 + a H2 + ... + anHn are Fourier coefficients, and equation (A.1) identi- 2 fies a Fourier series approximating y(x). For all pos- with coefficients identified in the following table for sible trigonometric polynomials of limited order ap- n up to 4. proximating y(x), the Fourier series provides the best approximation as measured by least sum-of- Table A.1. Polynomial Coefficients for m2 square differences across the domain (Rudin, pp. l 172-73; Tolstov, p. 174). ao a a2 a3 a4 In production analysis we are not interested in periodic or complex-valued functions. Because pro- 1 1 2 0 0 0 duction functions are not expected to be periodic, 2 1 2 2 0 0 3 1 %/3 2 4/3 0 the exogenous factors, the xi, must be scaled so that 4 4 3/111 536/63 -2/2/3 3 164//99 2/3 they lie in an interval of length less than 2ir. This is typically accomplished by scaling or normalizing xi into [0, 27r] (Gallant 1981). In order to guarantee The number of terms in the Fourier portion can Griffin, Montgomery, and Rister Selecting FunctionalForm 227 be quite large if either n or H is large. To illustrate cavity of the augmented Fourier form can be de- the effect of increasing n when H is small: if H = 2 rived by following the procedure of Gallant (1981). and n = 2, ml = 5 and m2 = 13; ifH = 2 and n = To do so requires substantial additional notation 4, ml = 14 and m2 = 41. and will not be pursued here. Sufficient (but not necessary) conditions for con-