1 Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved. CH5 Multiple Regression Analysis: OLS Asymptotic 

2 5.1 Consistency  y =  0 +  1 x 1 +  2 x  k x k + u (5.1) Under the Gauss-Markov assumptions (MLR1~5) OLS is BLUE, but in other cases it won’t always be possible to find unbiased estimators.

3 5.1 Consistency The OLS estimators have normal sampling distributions, which led directly to the t and F distributions for t and F statistic. If the error is not normality distribution, the distribution of a t and F statistic is not exactly t and F statistic for any sample size.

4 5.1 Consistency In addition to finite sample properties, it is important to know the asymptotic properties or large sample properties of estimators and test statistics. These properties are defined the sample size grows without bound.

5 Therom 5.1 Consistency of OLS Even without the normality assumption (Assumption MLR.6.), t and F statistic have approximately t and F distribution at least in large sample size.

6 Consistency of OLS Under the Gauss-Markov assumptions, the OLS estimator is consistent (and unbiased) Consistency is a minimal requirement for an estimator.

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8 Therom 5.1 Consistency of OLS Under Assumption MLR1~MLR4, the OLS estimator 0, 1,…, k.

9 Proving Consistency for Simple Regression (5.2) (5.3)

10 Assumption MLR4 ’ Assumption MLR.4’ requires only that each x j is uncorrelated with u. For consistency, we can have the weaker assumption of zero mean and zero correlation – E(u) = 0 and Cov(x j,u) = 0, for j = 1, 2, …, k E(y|x 1, x 2,…,x k )= β 0 +β 1 x β k x k, Assumpt MLR.4’, β 0 +β 1 x β k x k, need not represent the population regression function. Without this assumption, in the finite sample, OLS will be biased and inconsistent!

11 Deriving the Inconsistency in OLS Just as we could derive the omitted variable bias earlier, now we want to think about the inconsistency, or asymptotic bias, in this case

12 Deriving the Inconsistency in OLS We can views the inconsistency as being the same as the bias. An import point about inconsistency in OLS estimators. Remember: inconsistency is a large sample problem it doesn’t go away as add data, the problem gets worse with more data.

Asymptotic Normality and Large Sample Inference Recall that under the CLM assumptions (MLR.1~6), 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 (MLR6) implied that the distribution of y, given the x’s, was normal.

14 Large Sample Inference Easy to come up with examples for which this exact normality assumption will fail The wages, arrests, savings, etc. can’t be normal, since a normal distribution is symmetric. Normality assumption not needed to conclude OLS is BLUE, but exact inference based on t and F statistics requires MLR6.

15 Central Limit Theorem Based on the central limit theorem, we can show that OLS estimators are asymptotically normal Asymptotic Normality implies that P(Z<z)  (z) as n , or P(Z<z)   (z) The central limit theorem states that the standardized average of any population with mean  and variance  2 is asymptotically ~N(0,1), or

16 Theorem 5.2 Asymptotic Normality of OLS

17 Asymptotic Normality (cont) Because the t distribution approaches the normal distribution for large df., we can also say that Note that while we no longer need to assume normality (MLR6.) with a large sample, we do still need assume zero conditional mean (MLR.4.) and homoskedasticity of u (MLR.5.)

18 Asymptotic Normality (cont) when u is not normally distribution, if the sample size is not very large, then the t distribution can be a poor approximation to the distribution of the t statistics. The central limit theorem delivers a useful approximation. It is very import require the homoskedasticity assumption. Var(y|X) is not constant, the usual t statistic and confidence interval are invalid no matter how large the sample size is.

19 Asymptotic Standard Errors If u is not normally distributed, we sometimes will refer to the standard error as an asymptotic standard error, since So, we can expect standard errors to shrink at a rate proportional to the inverse of √n

20 Lagrange Multiplier statistic Once we are using large samples and relying on asymptotic normality for inference, we can use more that t and F stats The Lagrange Multiplier or LM statistic is an alternative for testing multiple exclusion restrictions Because the LM statistic uses an auxiliary regression it’s sometimes called an nR u 2 stat.

21 LM Statistic (step by step) Suppose we have a standard model, y = β 0 + β 1 x 1 + β 2 x β k x k + u (5.11) (unrestricted) and our null hypothesis is H 0 : β k-q+1 = 0,..., β k = 0 (5.12) (1) we just run the restricted model

22 LM Statistic (cont) With a large sample, the result from an F test and from an LM test should be similar Unlike the F test and t test for one exclusion, the LM test and F test will not be identical

23 Example 5.3 We use the LM statistic to test the null hypothesis that avgsen and tottime have no effect on narr86 once the other factors have been controlled for.

24 Example 5.3

Asymptotic Efficiency of OLS Under the Gauss-Markov assumptions, the OLS estimators will have the smallest asymptotic variances. The OLS estimators are BLUE and is also asymptotically efficient.

Asymptotic Efficiency of OLS In the simple regression case. In the model

Asymptotic Efficiency of OLS Can apply the law of large number to the numerator and denominator, which converge in probability to Cov(z, u) and Cov(z, x)

Asymptotic Efficiency of OLS Under the Gauss-Markov assumptions