tailieunhanh - Book Econometric Analysis of Cross Section and Panel Data By Wooldridge - Chapter 3

Basic Asymptotic Theory This chapter summarizes some definitions and limit theorems that are important for studying large-sample theory. Most claims are stated without proof, as several require tedious epsilon-delta arguments. We do prove some results that build on fundamental definitions and theorems. A good, general reference for background in asymptotic analysis is White (1984). In Chapter 12 we introduce further asymptotic methods that are required for studying nonlinear models. Convergence of Deterministic Sequences Asymptotic analysis is concerned with the various kinds of convergence of sequences of estimators as the sample size grows. We begin with some definitions regarding nonstochastic sequences of. | Basic Asymptotic Theory This chapter summarizes some definitions and limit theorems that are important for studying large-sample theory. Most claims are stated without proof as several require tedious epsilon-delta arguments. We do prove some results that build on fundamental definitions and theorems. A good general reference for background in asymptotic analysis is White 1984 . In Chapter 12 we introduce further asymptotic methods that are required for studying nonlinear models. Convergence of Deterministic Sequences Asymptotic analysis is concerned with the various kinds of convergence of sequences of estimators as the sample size grows. We begin with some definitions regarding nonstochastic sequences of numbers. When we apply these results in econometrics N is the sample size and it runs through all positive integers. You are assumed to have some familiarity with the notion of a limit of a sequence. definition 1 A sequence of nonrandom numbers faN N 1 2 . converges to a has limit a if for all e 0 there exists Ne such that if N Ne then aN a e. We write aN a as N co. 2 A sequence faN N 1 2 . is bounded if and only if there is some b o such that aN b for all N 1 2 . Otherwise we say that faN is unbounded. These definitions apply to vectors and matrices element by element. Example 1 If aN 2 1 N then aN 2. 2 If aN 1 N then aN does not have a limit but it is bounded. 3 If aN N1 4 aN is not bounded. Because aN increases without bound we write aN co. definition 1 A sequence faN is O N at most of order N if N aN is bounded. When l 0 faN is bounded and we also write aN O 1 big oh one . 2 faN is o N if N aN 0. When l 0 aN converges to zero and we also write aN o 1 little oh one . From the definitions it is clear that if aN o N then aN O N in particular if aN o 1 then aN O 1 . If each element of a sequence of vectors or matrices is O N we say the sequence of vectors or matrices is O N and similarly for o N l . Example 1 If aN log N then aN o N for any l