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1 the Nature of Econometrics and Economic Data

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1 the Nature of Econometrics and Economic Data Cover design: Jordi Uriel INTRODUCTION TO ECONOMETRICS Ezequiel Uriel 2019 University of Valencia I would like to thank the professors Luisa Moltó, Amado Peiró, Paz Rico, Pilar Beneito and Javier Ferri for their suggestions for the errata they have detected in previous versions, and for having provided me with data to formulate exercises. Some students have also collaborated in the detection of errata. In any case, I am solely responsible for the errata that have not been detected. Summary 1 Econometrics and economic data ..................................................... 9 1.1 What is econometrics? ................................................................................ 9 1.2 Steps in developing an econometric model .............................................. 10 1.3 Economic data .......................................................................................... 13 2 The simple regression model: estimation and properties............... 15 2.1 Some definitions in the simple regression model ..................................... 15 2.1.1 Population regression model and population regression function .............................. 15 2.1.2 Sample regression function ........................................................................................ 16 2.2 Obtaining the Ordinary Least Squares (OLS) Estimates .......................... 17 2.2.1 Different criteria of estimation ................................................................................... 17 2.2.2 Application of least square criterion .......................................................................... 19 2.3 Some characteristics of OLS estimators ................................................... 21 2.3.1 Algebraic implications of the estimation ................................................................... 21 2.3.2 Decomposition of the variance of y ........................................................................... 22 2.3.3 Goodness of fit: Coefficient of determination (R2) .................................................... 23 2.3.4 Regression through the origin .................................................................................... 25 2.4 Units of measurement and functional form .............................................. 26 2.4.1 Units of Measurement ................................................................................................ 26 2.4.2 Functional Form ......................................................................................................... 27 2.5 Assumptions and statistical properties of OLS ......................................... 33 2.5.1 Statistical assumptions of the CLM in simple linear regression ................................ 33 2.5.2 Desirable properties of the estimators ........................................................................ 35 2.5.3 Statistical properties of OLS estimators ..................................................................... 37 Exercises ......................................................................................................... 41 Annex 2.1 Case study: Engel curve for demand of dairy products ................ 48 Appendixes ..................................................................................................... 54 ˆ Appendix 2.1: Two alternative forms to express β2 ......................................................... 54 22 Appendix 2.2. Proof: rRxy = .......................................................................................... 55 Appendix 2.3. Proportional change versus change in logarithms ....................................... 55 Appendix 2.4. Proof: OLS estimators are linear and unbiased ............................................ 56 ˆ Appendix 2.5. Calculation of variance of β2 : ................................................................... 57 Appendix 2.6. Proof of Gauss-Markov Theorem for the slope in simple regression .......... 58 ) 2 Appendix 2.7. Proof: σ is an unbiased estimator of the variance of the disturbance ....... 59 Appendix 2.8. Consistency of the OLS estimator ............................................................... 61 Appendix 2.9 Maximum likelihood estimator .................................................................... 62 3 Multiple linear regression: estimation and properties.................... 66 3.1 The multiple linear regression model ....................................................... 66 3.1.1 Population regression model and population regression function .............................. 67 3.1.2 Sample regression function ........................................................................................ 68 3.2 Obtaining the OLS estimates, interpretation of the coefficients, and other characteristics ................................................................................................. 69 3.2.1 Obtaining the OLS estimates ...................................................................................... 69 3.2.2 Interpretation of the coefficients ................................................................................ 71 3.2.3 Algebraic implications of the estimation ................................................................... 75 3.3 Assumptions and statistical properties of the OLS estimators ................. 76 3.3.1 Statistical assumptions of the CLM in multiple linear regression) ............................. 76 3.3.2 Statistical properties of the OLS estimator ................................................................ 78 3.4 More on functional forms ......................................................................... 82 3.4.1 Use of logarithms in the econometric models ............................................................ 82 3.4.2 Polynomial functions ................................................................................................. 83 3.5 Goodness-of-fit and selection of regressors. ............................................ 85 3.5.1 Coefficient of determination ...................................................................................... 85 3.5.2 Adjusted R-Squared ................................................................................................... 86 3.5.3 Akaike information criterion (AIC) and Schwarz criterion (SC) ............................... 87 Exercises ......................................................................................................... 89 Appendixes ..................................................................................................... 97 Appendix 3.1 Proof of the theorem of Gauss-Markov ........................................................ 97 ) 2 Appendix 3.2 Proof: σ is an unbiased estimator of the variance of the disturbance ....... 98 Appendix 3.3 Consistency of the OLS estimator ................................................................ 99 Appendix 3.4 Maximum likelihood estimator .................................................................. 101 4 Hypothesis testing in the multiple regression model ................... 104 4.1 Hypothesis testing: an overview ............................................................. 104 4.1.1 Formulation of the null hypothesis and the alternative hypothesis .......................... 104 4.1.2 Test statistic ............................................................................................................. 105 4.1.3 Decision rule ............................................................................................................ 105 4.2 Testing hypotheses using the t test ......................................................... 108 4.2.1 Test of a single parameter ........................................................................................ 108 4.2.2 Confidence intervals ................................................................................................ 118 4.2.3 Testing hypotheses about a single linear combination of the parameters ................ 119 4.2.4 Economic importance versus statistical significance ............................................... 124 4.3 Testing multiple linear restrictions using the F test. .............................. 124 4.3.1 Exclusion restrictions ............................................................................................... 125 4.3.2 Model significance ................................................................................................... 129 4.3.3 Testing other linear restrictions................................................................................ 131 4.3.4 Relation between F and t statistics ........................................................................... 132 4.4 Testing without normality ...................................................................... 133 4.5 Prediction ................................................................................................ 133 4.5.1 Point prediction ........................................................................................................ 133 4.5.2 Interval prediction .................................................................................................... 134 4.5.3 Predicting y in a ln(y) model .................................................................................... 137 4.5.4 Forecast evaluation and dynamic prediction ............................................................ 138 Exercises ......................................................................................................
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