计量经济学导论电子教案ch05章节幻灯片.pptVIP

Economics 20 - Prof. Anderson * Multiple Regression Analysis y = b0 + b1x1 + b2x2 + . . . bkxk + u 3. Asymptotic Properties Economics 20 - Prof. Anderson * Consistency Under the Gauss-Markov assumptions OLS is BLUE, but in other cases it won’t always be possible to find unbiased estimators In those cases, we may settle for estimators that are consistent, meaning as n ? ∞, the distribution of the estimator collapses to the parameter value Economics 20 - Prof. Anderson * Sampling Distributions as n ? b1 n1 n2 n3 n1 n2 n3 Economics 20 - Prof. Anderson * Consistency of OLS Under the Gauss-Markov assumptions, the OLS estimator is consistent (and unbiased) Consistency can be proved for the simple regression case in a manner similar to the proof of unbiasedness Will need to take probability limit (plim) to establish consistency Economics 20 - Prof. Anderson * Proving Consistency Economics 20 - Prof. Anderson * A Weaker Assumption For unbiasedness, we assumed a zero conditional mean – E(u|x1, x2,…,xk) = 0 For consistency, we can have the weaker assumption of zero mean and zero correlation – E(u) = 0 and Cov(xj,u) = 0, for j = 1, 2, …, k Without this assumption, OLS will be biased and inconsistent! Economics 20 - Prof. Anderson * 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 Economics 20 - Prof. Anderson * 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 Economics 20 - Prof. Anderson * Large Sample Inference Recall that under the CLM assumptions, the sampling distributions are normal, so we could derive t and F distributions for te