Estimation of Simultaneous Equation Models with Error Components Structure

Economists can rarely perform controlled experiments to generate data. Existing information in the form of real-life observations simply has to be utilized in the best possible way. Given this, it is advantageous to make use of the increasing availability and accessibility of combinations of time-series and cross-sectional data in the estimation of economic models. But such data call for a new methodology of estimation and hence for the development of new econometric models. This book proposes one such new model which introduces error components in a system of simultaneous equations to take into account the temporal and cross-sectional heterogeneity of panel data. After a substantial survey of panel data models, the newly proposed model is presented in detail and indirect estimations, full information and limited information estimations, and estimations with and without the assumption of normal distribution errors. These estimation methods are then applied using a computer to estimate a model of residential electricity demand using data on American households. The results are analysed both from an economic and from a statistical point of view.

1. Introduction.- 1.1 General.- 1.2 Organization of the Book.- 2. A Survey of Panel Data Models.- 2.1 General.- 2.2 Constant Slope Variable Intercept Models.- 2.2.1 Fixed Effects Models.- 2.2.2 Random Effects Models.- 2.2.2.1 Error Components Models: Classical Estimation Methods.- 2.2.2.2 EC Models: Bayesian Analysis.- 2.2.2.3 EC Models Using Stratified Data.- 2.2.3 Random Effects Models with Non-Zero Correlations between Specific Effects and Exogenous Variables.- 2.2.4 Dynamic Random Effects Models.- 2.3 Variable Coefficient Models.- 2.3.1 Fixed Coefficient Component Models.- 2.3.2 Random Coefficient Models.- 2.3.2.1 Purely Random Coefficient Components Model.- 2.3.2.2 Transmitted Variations Model.- 2.3.2.3 Random Coefficient Models with Stochastically Convergent Parameters.- 2.3.2.4 Random Coefficient First-Order Autoregressive Model.- 2.3.3 The “Mixed” Model.- 2.3.4 Quantitative Effects Models.- 2.3.5 General Stratified Effect Component Models.- 2.3.6 Bayesian Analysis of Some Varying Coefficient Models.- 2.4 Estimation of Variance Components in Panel Data Models.- 2.4.1 Analysis of Variance Methods.- 2.4.2 “Fitting of Constants” Method.- 2.4.3 “Swamy and Arora” Method.- 2.4.4 MINQUE.- 2.4.5 Comparison of Alternate Estimation Methods.- 2.5 Estimation of Models using Incomplete Time-Series Cross-Section Data.- 2.6 Extensions.- 2.6.1 SUR with EC.- 2.6.2 SEM with EC.- 3. Presentation of Simultaneous Equations Models with Error Components Structure and Estimation of the Reduced Form.- 3.1 The Model.- 3.1.1 Notations.- 3.1.2 Stochastic Specifications.- 3.2 Estimation of the Reduced Form.- 3.2.1 Derivation of Stochastic Properties of Reduced Form Errors and Interpretation of the Reduced Form.- 3.2.2 Feasible GLS Estimation of Reduced Form Parameters.- 3.2.3 Maximum Likelihood Estimation of the Reduced Form.- Appendix 3.A Proof of the Consistency of the Feasible GLS Estimator of Reduced Form Coefficients.- 3.A.1 Basic Assumptions and Some Preliminary Results.- 3.A.2 Consistency of $$ {{\hat \Pi }_{m\left( {\operatorname{cov} } \right)}} $$.- 3.A.3 Consistency of AOV Estimators of Eigenvalues (and Variance Components) of ?mm’.- 3.A.4 Consistency of the Feasible GLS Estimator of ?.- 3.A.5 Proof of Lemma L-1.- Appendix 3.B Limiting Distribution of the Feasible GLS Estimator of the Reduced Form.- Appendix 3.C. Limiting Distribution of the Reduced Form Maximum Likelihood Estimators.- 3.C.1 The Information Matrix.- 3.C.2 Limit of the Information Matrix.- 3.C.3 The Limiting Distribution.- 4 Estimation of the Structural Form — Part 1.- 4.1 Generalised Two Stage Least Squares — A Single Equation Method.- 4.1.1 Estimation in the Case of Known Variance Components.- 4.1.2 Estimation in the Case of Unknown Variance Components.- 4.2 Generalised Three Stage Least Squares — A System Method.- Appendix 4.A Proof of the Consistency of the 2SLS Covariance Estimators $$ {{\hat a}_{m\left( {\operatorname{cov} } \right)}} $$ and $$ {{\hat a}_{m\left( {\operatorname{cov} } \right)}} $$.- 4.A.1 Consistency of $${{\hat a}_{m,C2SLS}}$$.- 4.A.2 Consistency of $$ {{\hat \alpha }_{m,C2SLS}} $$.- Appendix 4.B Proof of the Consistency of AOV Estimators of Eigenvalues and Variance Components of ?mm.- 4.B.1 Method 1.- 4.B.2 Method 2.- Appendix 4.C Proof of the Consistency of the Feasible (and pure) G2SLS Estimator.- Appendix 4.D Limiting Distribution of the Feasible G2SLS Estimator.- Appendix 4.E Limiting Distribution of the Feasible G3SLS Estimator.- 5 Estimation of the Structural Form — Part 2.- 5.1 Full Information Maximum Likelihood (FIML) Estimation of the Structural Form.- 5.2 Limited Information Maximum Likelihood (LIML) Estimation of the Structural Form.- Appendix 5.A Limiting Distribution of the FIML Estimators.- 5.A.1 The Information Matrix.- 5.A.2 Limit of the Information Matrix.- 5.A.3 The Limiting Distribution.- 6 The Just-Identified Case and Indirect Estimation of Structural Parameters.- 6.1 The Identification Problem.- 6.2 Derivation of the Indirect Estimators of Structural Coefficients and their Limiting Distributions.- 6.2.1 The Case of a Single Just-Identified Structural Equation.- 6.2.1.1 The Indirect Estimation Method.- 6.2.1.2 The Indirect Covariance Estimator.- 6.2.1.3 The Indirect Feasible GLS Estimator.- 6.2.2 The Case of a Just-Identified System.- 6.3 Comparison of the IfGLS Estimator with the fG2SLS and fG3SLS Estimators.- 6.3.1 The Case of a Single Just-Identified Equation.- 6.3.2 The Case of a Just-Identified System.- 6.3.2.1 Comparison between fG2SLS and fG3SLS Estimators.- 6.3.2.2 Comparison between fG2SLS, fG3SLS and IfGLS Estimators.- Appendix 6.A Limiting Distribution of the Indirect Feasible GLS Estimator.- 7 Bias of the Feasible Estimators of Reduced Form and Structural Variance Components and Coefficients.- 7.1 The Unbiasedness of the Feasible AOV Estimators of Reduced Form Variance Components.- 7.2 The Unbiasedness of the Feasible GLS Estimator of the Reduced Form Coefficients.- 7.3 Bias of Structural Variance Components Estimators.- 7.3.1 A General Note.- 7.3.2 Bias of $$ {{\hat \sigma }_{\varepsilon 11}} $$.- 7.3.3 Bias of $$ {{\hat \sigma }_{\mu 11}} $$.- 7.4 Bias of Structural Coefficients Estimators.- 7.4.1 A General Note.- 7.4.2 Bias of the Covariance 2SLS Estimator.- 7.4.3 Bias of the Feasible G2SLS Estimator.- Appendix 7.A Preliminary Computations of Orders.- 7.A.1 Some Basic Order Calculations.- 7.A.2 Further Results on Orders.- Appendix 7.B Derivations of Expectations.- 7.B.1 Expectation of $$ u{'_1}\bar FU'{N_4}{u_1} $$.- 7.B.2 Expectation of $$ u{'_1}\bar FU'\bar FU'{N_4}{u_1} $$.- 7.B.3 Expectation of $$ u{'_1}\bar HU'\bar RU'{N_4}{u_1} $$.- 7.B.4 Expectation of $$ u{'_1}CU\bar RU'{N_4}{u_1} $$.- 7.B.5 Expectation of $$ u{'_1}\bar FU'{N_4}U\bar F'{u_1} $$.- Appendix 7.C Order Calculations Involved in the Determination of the Bias of the Feasible G2SLS Estimator.- Appendix 7.D Expectation of ?i11X’Njul for i=1,4 and j=1,4.- 8 Application to a Model of Residential Electricity Demand.- 8.1 The Model.- 8.2 The Data.- 8.3 Estimation Methods.- 8.4 Results.- Appendix 8.A Computer Programs of Estimation Methods.- 9 Conclusions.- References.