多元回归分析OLS的渐进性.ppt

Economics 20 - Prof. Anderson 多元回归分析：渐进性 y = b0 + b1x1 + b2x2 + . . . bkxk + u 渐进性的含义 如果误差并非正态分布，对任何的样本容量而言，t统计量、F统计量并非恰好服从t分布、F分布。 幸运的是，即使没有正态性假定，t统计量和F统计量仍然渐进的服从t分布、F分布，，至少在大样本情况下使如此。 一致性 在高斯-马尔科夫假定下，OLS估计是最优线性无偏的，但我们并非总能得到无偏的估计量。 一致性是对一个估计量最起码的要求。在无法满足无偏性的情况下，我们可以搜集尽可能多的样本，即使n→ ∞,参数估计值的分布将逼近真实参数值。 一致性的正式定义 一致性的直观理解 当样本容量增加时的样本分布 OLS的一致性 在高斯-马尔科夫假定下，OLS估计值是一致且无偏的。 类似的，我们可以像无偏性一样证明一致性，为此需要引入概率极限。 简单回归中证明一致性 一个较弱的假定 为了得到无偏性，我们需要零条件均值假设E(u|x1, x2,…,xk) = 0 为了得到一致性，我们仅需要较弱的假定：零均值和零相关:E(u) = 0 ,Cov(xj,u) = 0, for j = 1, 2, …, k. 不满足上述条件，OLS是有偏和不一致的。 Deriving the Inconsistency Just as we could derive the omitted variable bias earlier, now we want to think about the inconsistency, or asymptotic bias, in this case Asymptotic Bias (cont) So, thinking about the direction of the asymptotic bias is just like thinking about the direction of bias for an omitted variable Main difference is that asymptotic bias uses the population variance and covariance, while bias uses the sample counterparts Remember, inconsistency is a large sample problem – it doesn’t go away as add data Summary of Direction of Asymptotic bias Large Sample Inference Recall that under the CLM assumptions, the sampling distributions are normal, so we could derive t and F distributions for testing This exact normality was due to assuming the population error distribution was normal This assumption of normal errors implied that the distribution of y, given the x’s, was normal as well Large Sample Inference (cont) Easy to come up with examples for which this exact normality assumption will fail Any clearly skewed variable, like wages, arrests, savings, etc. can’t be normal, since a normal distribution is symmetric Normality assumption not needed to conclude OLS is BLUE, only for inference Theorem 5.2 Asymptotic Normality Law of large numbers Asymptotic Normality (cont) Because the t distribution approaches the normal distribution for large df, we can also say that Lagrange Multiplier statistic Once we are us