Robust Methods and Asymptotic Theory in Nonlinear Econometrics

1 Introduction.- 1.1 Specification and misspecification of the econometric model.- 1.2 The purpose and scope of this study.- 2 Preliminary Mathematics.- 2.1 Random variables, independence, Borel measurable functions and mathematical expectation.- 2.1.1 Measure theoretical foundation of probability theory.- 2.1.2 Independence.- 2.1.3 Borel measurable functions.- 2.1.4 Mathematical expectation.- 2.2 Convergence of random variables and distributions.- 2.2.1 Weak and strong convergence of random variables.- 2.2.2 Convergence of mathematical expectations.- 2.2.3 Convergence of distributions.- 2.2.4 Convergence of distributions and mathematical expectations.- 2.3 Uniform convergence of random functions.- 2.3.1 Random functions. Uniform strong and weak convergence.- 2.3.2 Uniform strong and weak laws of large numbers.- 2.4 Characteristic functions, stable distributions and a central limit theorem.- 2.5 Unimodal distributions.- 3 Nonlinear Regression Models.- 3.1 Nonlinear least-squares estimation.- 3.1.1 Model and estimator.- 3.1.2 Strong consistency.- 3.1.3 Asymptotic normality.- 3.1.4 Weak consistency and asymptotic normality under weaker conditions.- 3.1.5 Asymptotic properties if the error distribution has infinite variance. Symmetric stable error distributions.- 3.2 A class of nonlinear robust M-estimators.- 3.2.1 Introduction.- 3.2.2 Strong consistency.- 3.2.3 Asymptotic normality.- 3.2.4 Properties of the function h(?). Asymptotic efficiency and robustness.- 3.2.5 A uniformly consistent estimator of the function h(?).- 3.2.6 A two-stage robust M-estimator.- 3.2.7 Some weaker results.- 3.3 Weighted nonlinear robust M-estimation.- 3.3.1 Introduction.- 3.3.2 Strong consistency and asymptotic normality.- 3.3.3 A two-stage weighted robust M-estimator.- 3.4 Miscellaneous notes on robust M-estimation.- 3.4.1 Uniform consistency.- 3.4.2 The symmetric unimodality assumption.- 3.4.3 The function ?.- 3.4.4 How to decide to apply robust M-estimation.- 4 Nonlinear Structural Equations.- 4.1 Nonlinear two-stage least squares.- 4.1.1 Introduction.- 4.1.2 Strong consistency.- 4.1.3 Asymptotic normality.- 4.1.4 Weak consistency.- 4.2 Minimum information estimators: introduction.- 4.2.1 Lack of instruments.- 4.2.2 Identification without using instrumental variables.- 4.2.3 Consistent estimation without using instrumental variables.- 4.2.4 Asymptotic normality.- 4.2.5 A problem concerning the nonsingularity assumption.- 4.3 Minimum information estimators: instrumental variable and scaling parameter.- 4.3.1 An instrumental variable.- 4.3.2 An example.- 4.3.3 A scaling parameter and its impact on the asymptotic properties.- 4.3.4 Estimation of the asymptotic variance matrix.- 4.3.5 A two-stage estimator.- 4.3.6 Weak consistency.- 4.4 Miscellaneous notes on minimum information estimation.- 4.4.1 Remarks on the function $$S_{ - n}^* (theta |gamma )$$.- 4.4.2 A consistent initial value.- 4.4.3 An upperbound of the variance matrix.- 4.4.4 A note on the symmetry assumption.- 5 Nonlinear Models with Lagged Dependent Variables.- 5.1 Stochastic stability.- 5.1.1 Stochastically stable linear autoregressive processes.- 5.1.2 Multivariate stochastically stable processes.- 5.1.3 Other examples of stochastically stable processes.- 5.2 Limit theorem for stochastically stable processes.- 5.2.1 A uniform weak law of large numbers.- 5.2.2 Martingales.- 5.2.3 Central limit theorem for stochastically stable martingale differences.- 5.3 Dynamic nonlinear regression models and implicit structural equations.- 5.3.1 Dynamic nonlinear regression models.- 5.3.2 Dynamic nonlinear implicit structural equations.- 5.4 Remarks on the stochastic stability concept.- 6 Some Applications.- 6.1 Applications of robust M-estimation.- 6.1.1 Municipal expenditure.- 6.1.2 An autoregressive model of money demand.- 6.2 An application of minimum information estimation.- References.