Lecture 16 Introuction to Asymptotics讲座16介绍的渐近性.pptVIP

Lecture 16:Introduction to Asymptotics (Chapter 12.1–12.3) Agenda The Asymptotic Perspective (Chapter 12.1) Asymptotic Unbiasedness (Chapter 12.2) Consistency (Chapter 12.2) Probability Limits (Chapter 12.2) Consistency of OLS (Chapter 12.3) The Asymptotic Perspective (Chapter 12.1) This lecture marks a major shift in the mathematical underpinnings of the course. So far, we have focused on finding unbiased and efficient estimators. That is, we want estimators that will give you the right answer on average over all possible samples, and that have the lowest possible variance. The Asymptotic Perspective (cont.) So far, we have focused on finding unbiased and efficient estimators. This lecture introduces a set of new criteria for judging estimators, based on large-sample (asymptotic) properties. Instead of looking at the properties of the estimator averaging over all possible samples, we look at properties as the sample size gets very, very large. The Asymptotic Perspective (cont.) Instead of looking at the properties of the estimator averaging over all possible samples, we will start looking at properties as the sample size gets very, very large. Why the shift? The math is going to be much more convenient. Cross-sample properties become intractable as we relax the last (and most crucial) Gauss–Markov assumption, that the X ’s are fixed across samples. The Asymptotic Perspective (cont.) Large-sample (asymptotic) properties are mathematically tractable. We just have to hope that estimators that we prove work well with a near-infinite number of observations will also work well with the finite datasets we actually observe. The Asymptotic Perspective (cont.) One use of Monte Carlo techniques is to study computationally the small sample properties of estimators that have been derived asymptotically. Estimators designed for large samples don’t tend to work well in small samples, but are appropriate for “reasonably large” samples. The Asymptotic Perspective (cont.) What