3.5 Extensions -- 3.5.1 4D and Beyond -- 3.5.2 Mixed FE-RE Models -- 3.6 Testing -- 3.7 Conclusion -- Appendix 1 -- Example for normalizing with 1: Model (3.14), → ∞ -- Example for normalizing with √ 1 2/ : Model (3.2), 1, 2 → ∞ -- Appendix 2: Proof of formula (3.19) -- Appendix 3: Inverse of (3.34), and the estimation of the variance components -- References -- Chapter 4 Estimation of Sparse Variance-Covariance Matrix -- 4.1 Introduction -- 4.2 Basic Methods -- 4.3 Multi-dimensional Panel Data and Sparse Variance-Covariance Matrix -- 4.4 Estimating Variance-Covariance Matrix -- 4.4.1 Estimating Variance-Covariance Matrices in Higher Dimensions -- 4.4.1.1 Case 1: -- 4.4.1.2 Case 2: -- 4.4.1.3 Case 3: -- 4.4.1.4 Case 4: -- 4.4.2 Shrinkage Approximation -- 4.4.3 Regularistion -- 4.5 Testing for Misspecification in the Unobserved Heterogeneity -- 4.5.1 Finite Sample Performance of the Misspecification Test -- 4.6 Further Considerations -- 4.6.1 Unbalanced Panel -- 4.6.2 Higher Dimension -- 4.6.3 Computation -- Appendix -- References -- Chapter 5 Models with Endogenous Regressors -- 5.1 Introduction -- 5.2 The Hausman-Taylor-like Instrument Variable Estimator -- 5.2.1 A Simple Approach -- 5.2.2 Sources of Endogeneity -- 5.2.3 The Hausman-Taylor Estimator -- 5.2.3.1 Extending the Hausman-Taylor Two-Stage Least Squares Estimator -- 5.2.3.2 The More Efficient Hausman-Taylor Estimator -- 5.2.4 Time Varying Individual Specific Effects -- 5.2.5 Properties -- 5.2.6 Common Correlated Effects Pooled with Hausman-Taylor -- 5.2.7 Using External Instruments -- 5.3 The Non-linear Generalized Method of Moments Estimator -- 5.4 Mixed Effects Models -- 5.5 Exogeneity Tests -- 5.5.1 Testing for Endogeneity -- 5.5.2 Testing for Instrument Validity -- 5.5.3 Testing in the Case of Fixed Effects -- 5.5.3.1 Improper Model Specifications -- 5.5.3.2 Conventional Endogeneity. |