Advanced Econometrics Takeshi Amemiya

Large Sample Theory

Large sample theory plays a major role in the theory of econometrics because econometricians must frequently deal with more complicated models than the classical linear regression model of Chapter 1. Few finite sample results are known for these models, and, therefore, statistical inference must be based on the large sample properties of the estimators. In this chapter we shall present a brief review of random variables and the distribution function, discuss various convergence theorems including laws of large numbers and central limit theorems, and then use these theorems to prove the consistency and the asymptotic normality of the least squares estimator. Additional examples of the application of the convergence theorems will be given.

3.1 A Review of Random Variables and the Distribution Function

This section is not meant to be a complete discussion of the subject; the reader is assumed to know the fundamentals of the theory of probability, random variables, and distribution functions at the level of an intermediate textbook in mathematical statistics.1 Here we shall introduce a few concepts that are not usually dealt with in intermediate textbooks but are required in the subse­quent analysis, in particular, the rigorous definition of a random variable and the definition of the Stieltjes integral.2