Estimation of Simultaneous Equation Models with Error Components Structure

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.