Multiple Regression Analysis: OLS Asymptotics (1) 多元回归分析： OLS的渐近性（1）
y = b0 + b1x1 + b2x2 + . . . + bkxk + u
Intermediate Econometricse 本章提纲
What do we mean by asymptotic normality and large sample inference 渐近正态性和大样本推断的含义是什么
The asymptotic normality of OLS OLS的渐近正态
Intermediate Econometrics,
令 Wn是基于样本 y1, y2,..., yn的关于 的估计量。
如果对于任何 >0，当 n 时Pr(|Wn | ) 0
Wn便是 的一个一致估计量。
当作是Wnp具li有m(一Wn致) 性时. ，我们In也ter称medYiaatn为e SEhcoeWnnonm的et概rics率, 极限，写
Intermediate Econometrics,
Yan Shen
5
What is Consistency
什么是一致性
Let Wn be an estimator of based on a sample y1, y2 ,..., yn. Wn is a consistent estimator of if for every >0, Pr(|Wn | ) 0 as n . When Wn is consistent, we also say that is the probability limit of Wn , written as p lim(Wn ) .
Consistency 一致性
Asymptotic Normality and Large Sample Inference 渐近正态和大样本推断
Asymptotic Efficiency of OLS OLS的渐近有效性
Intermediate Econometrics,
Yan Shen
2
Lecture Outline 本课提纲
6
Consistency v.s. unbiasedness
一致性与无偏性
Is it possible for an estimator to be biased in finite sample but consistent in large sample?
一个估计量是否有可能在有限样本中是有偏的但又具有一 致性？
Suppose true value of z=0, a random variable x =z with probability (n-1)/n, and x=n with probability 1/n.
假设Z的真值为0，一个随机变量X以(n-1)/n的概率取值为Z，而以 1/n的概率取值为n。
These properties hold for any sample size n. 样本容量为任意n时，这些性质都成立。
Intermediate Econometrics,
Yan Shen
4
Why consider consistency?
为什么考虑一致性
Since in many situations the error term is not normally distributed, it is important to know the asymptotic properties (large sample properties), i.e., the properties of OLS estimator and test statistics when the sample size grows without bound. 由于在很多情形下误差项可能呈现非正态分布，了解 OLS 估计量和检验统计量的渐近性，即当样本容量任意 大时的特性就是重要的问题。
What do we mean by saying consistency 一致性的含义是什么
Consistency of OLS estimators OLS估计量的一致性
The Inconsistency of OLS when the zero conditional mean assumption fails 当零条件均值假设不成立时OLS没有一致性。
Unbiasedness of OLS estimators (MLR.1-4) 在MLR. 1-4下 OLS估计量具有无偏性 BLUE of OLS estimators (MLR.1-5) 在MLR.1-5下 OLS估计量是最优线性无偏估计量 MVUE of OLS estimators (MLR.1-6) 在MLR.1-6 下OLS估计量是最小方差无偏估计量 The distribution of t (F) statistic is t (F)distribution t(F)统计量的分布为t(F)分布。
Yan Shen
3
Why considering consistency?
为什么考虑一致性
We have discussed the following finite sample (small sample) properties of the OLS estimators and test statistics: 我们已经讨论了有限样本(小样本）中OLS估计量和检验统计 量具有的如下性质：
E(x)=z* (n-1)/n+n* 1/n=1 X的期望为1
plim(x) is the value of x as n goes to infinity. Therefore plim(x)=z=0.
记plim(x) 为n趋向无穷大时x的取值。因此
plim(x)=z=0.
Intermediate Econometrics, Yan Shen