Topics In Advanced Econometrics

This book is intended for second year graduate students and professionals who have an interest in linear and nonlinear simultaneous equations mod els. It basically traces the evolution of econometrics beyond the general linear model (GLM), beginning with the general linear structural econo metric model (GLSEM) and ending with the generalized method of mo ments (GMM). Thus, it covers the identification problem (Chapter 3), maximum likelihood (ML) methods (Chapters 3 and 4), two and three stage least squares (2SLS, 3SLS) (Chapters 1 and 2), the general nonlinear model (GNLM) (Chapter 5), the general nonlinear simultaneous equations model (GNLSEM), the special ca'3e of GNLSEM with additive errors, non linear two and three stage least squares (NL2SLS, NL3SLS), the GMM for GNLSEIVl, and finally ends with a brief overview of causality and re lated issues, (Chapter 6). There is no discussion either of limited dependent variables, or of unit root related topics. It also contains a number of significant innovations. In a departure from the custom of the literature, identification and consistency for nonlinear models is handled through the Kullback information apparatus, as well as the theory of minimum contrast (MC) estimators. In fact, nearly all estimation problems handled in this volume can be approached through the theory of MC estimators. The power of this approach is demonstrated in Chapter 5, where the entire set of identification requirements for the GLSEM, in an ML context, is obtained almost effortlessly, through the apparatus of Kullback information.

1 Extension of Classical Methods I.- 1.1 Introduction.- 1.2 A Brief Historical Review.- 1.3 The Nature of the GLSEM.- 1.4 The GLSEM: Assumptions and Notation.- 1.4.1 Assumptions and Conventions.- 1.4.2 Notation.- 1.5 Inconsistency of OLS Estimators.- 1.6 Two Stage Least Squares (2SLS).- 1.6.1 The Original Derivation.- 1.6.2 An Alternative Formulation.- 1.7 Three Stage Least Squares (3SLS).- 1.8 Restricted 2SLS and 3SLS Estimators.- 1.9 Tests of Prior Restrictions.- 1.9.1 Generalities.- 1.9.2 A Restricted Least Squares Interpretation of 2SLS and 3SLS.- Questions and Problems.- Appendix to Chapter 1.- Preliminaries to Hausman's Test.- Examples.- 2 Extension of Classical Methods II.- 2.1 Limiting Distributions.- 2.1.1 Preliminaries.- 2.1.2 Limiting Distributions for Static GLSEM.- 2.1.3 Limiting Distributions for Dynamic GLSEM.- 2.2 Forecasting from the GLSEM.- 2.2.1 Generalities.- 2.2.2 Forecasting from the URF.- 2.2.3 Forecasting from the RRF.- 2.3 The Vector Autoregressive Model (VAR).- 2.4 Instrumental Variables (IV).- 2.4.1 2SLS and 3SLS as IV Estimators.- 2.4.2 2SLS and 3SLS as Optimal IV Estimators.- 2.5 IV and Insufficient Sample Size.- 2.5.1 The Nature of the Problem.- 2.5.2 Iterated Instrumental Variables (IIV).- 2.6 k-class and Double k-class Estimators.- 2.7 Distribution of LM Derived Estimators.- 2.8 Properties of Specification Tests.- 2.8.1 Single Equation 2SLS.- 2.8.2 Systemwide 2SLS and 3SLS.- 2.8.3 Relation to Hausman's Test.- Questions and Problems.- Appendix to Chapter 2.- Convergence of Second Moment Matrices.- Convergence for Dependent Sequences.- Preliminaries and Miscellaneous.- Convergence of Second Moments of Final Form Errors.- 3 Maximum Likelihood Methods I.- 3.1 Introduction.- 3.2 The Identification Problem.- 3.2.1 Generalities.- 3.2.2 The Simple Supply-Demand Model.- 3.2.3 Identification by Exclusion Restrictions.- 3.2.4 Identification by Linear Restrictions.- 3.2.5 Identification and the Reduced Form.- 3.2.6 Covariance and Cross Equation Restrictions.- 3.2.7 A More General Framework.- 3.2.8 Parametric Nonlinearities and Identification.- 3.3 ML Estimation of the RF.- 3.3.1 General Discussion and ILS.- 3.3.2 Estimation of the Reduced Form.- 3.4 FIML Estimation.- 3.5 Simplified FIML Estimators.- 3.6 Properties of Simplified Estimators.- 3.6.1 Consistency.- 3.7 Limiting Distribution of FIML.- Questions and Problems.- 4 LIML Estimation Methods.- 4.1 The "Concentrated" Likelihood Function.- 4.1.1 A Subset of m* Structural Equations.- 4.2 The Single Equation LIML Estimator.- 4.3 Consistency of the LIML Estimator.- 4.4 An Interesting Interpretation of LIML.- 4.5 Indirect Least Squares (ILS).- 4.6 Relation of LIML to Other Estimators.- 4.7 Limiting Distribution of LIML Estimators.- 4.8 Classic Identifiability Tests.- Questions and Problems.- Appendix to Chapter 4.- Limiting Distribution of (T? - 1).- 5 Nonlinear ML Methods.- 5.1 Motivation.- 5.2 A Mathematical Digression.- 5.3 Aspects of Likelihood Functions.- 5.3.1 An Interesting Inequality.- 5.4 Fisher Information.- 5.4.1 Alternative Representation of the Information Matrix.- 5.5 The Cramer-Rao Bounds.- 5.6 Martingale Properties of Likelihood Functions.- 5.7 Kullback Information.- 5.8 Convergence A.C. of ML Estimators.- 5.8.1 Independent Observations.- 5.8.2 Generalizations.- 5.9 The General Nonlinear Model (GNLM).- 5.9.1 Consistency.- 5.9.2 Identification.- 5.9.3 Asymptotic Normality.- 5.10 The GNLM with Restrictions.- 5.11 Tests of Restrictions.- 5.11.1 Generalities.- 5.11.2 The Conformity Test.- 5.11.3 The Likelihood Ratio Test.- 5.11.4 The Lagrange Multiplier Test.- 5.11.5 Equivalence of the Three Tests.- Questions and Problems.- 6 Topics in NLSE Theory.- 6.1 Nonlinear ML.- 6.1.1 Identification.- 6.1.2 Consistency of the ML Estimator.- 6.1.3 Limiting Distribution of ML Estimators.- 6.1.4 Relation of Structural and Covariance Parameter (ML) Estimators.- 6.1.5 Estimators in Structurally Misspecified Models.- 6.2 Nonlinear 2SLS.- 6.2.1 Identification and Consistency of NL2SLS.- 6.2.2 Asymptotic Normality of NL2SLS.- 6.2.3 Choice of an Optimal NL2SLS Estimator.- 6.3 Nonlinear 3SLS.- 6.3.1 Identification and Consistency of NL3SLS.- 6.3.2 Asymptotic Normality of NL3SLS.- 6.3.3 Optimum NL3SLS and Computational Aspects.- 6.4 GMM.- 6.4.1 Reformulation of GMM as NL2SLS and NL3SLS.- 6.4.2 Identification and Consistency.- 6.4.3 Asymptotic Normality.- 6.4.4 Tests of Restrictions.- 6.5 Causality and Related Issues.- 6.5.1 Introduction.- 6.5.2 Basic Concepts.- Questions and Problems.