Lecture 5

1. Lecture 5 Topics in Applied Econometrics A

2. Asymptotics • Until now, we covered finite sample or small sample properties of the OLS estimators • For example, we derived the unbiasedness of the OLS estimator under Assumptions 1 – 4. • It’s a finite sample property b/c it holds for any n (n > k+1) • We also added an additional assumption that u is normally distributed and independent of the explanatory variables. • This allowed us to show that the OLS estimators have normal sampling distributions, which lead to the t and F distributions for t and F statistics

3. Asymptotics • In addition to finite sample properties, it is important to also know the asymptotic or large sample properties of estimators and test statistics. • These are properties that are defined as the sample size grows without bound • One important finding is that even without the normality assumption, t and F statistics have approximately t and F distributions

4. Detour on asymptotics • Though not all useful estimators are unbiased, virtually all economists agree that a minimal requirement for an estimator is consistency • meaning as n  ∞, the distribution of the estimator collapses to the parameter value • We generally have a fixed sample so consistency involves a thought experiment about what would happen as the sample size gets large • If obtaining more data does not get us closer to the parameter value of interest, then we are using a poor estimation procedure.

5. Sampling Distributions as n n3 n1 < n2 < n3 n2 n1 b1

6. 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

7. A few words on plims • Consistency: Let Wn be an estimator of theta based on a sample Y1, Y2, …, Yn of size n. Then, Wn is a consistent estimator of θ if for every ε > 0 • P(|Wn – θ| > ε )0 as n  ∞ • When Wn is consistent, we also say that θ is the probability limit of Wn, written as plim(Wn) = θ • In other words, the distribution of Wn becomes more and more concentrated about theta as n grows. • If an estimator is not consistent, then it does not help us to learn about θ, even with an unlimited amount of data.

8. Example: plim • Unbiased estimators are not necessarily consistent, but those whose variances shrink to zero as the sample size grows are consistent. • Formally: If Wn is an unbiased estimator of θ and Var(Wn)  0 as n ∞ then plim(Wn) = θ. • We know the sample average is unbiased for μ. We also know that the variance of ybar is equal to σ2/n. • As n  ∞, σ2/n  0 so ybar is a consistent estimator of μ (as well as being unbiased)

9. Proving Consistency of the OLS Estimator

10. Here we applied the Law of Large Numbers (numerator and denominator converge to their population parameters)

11. 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!

12. 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

13. 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 we add data

14. 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

15. 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, because a normal distribution is symmetric • Normality assumption not needed to conclude OLS is BLUE, only for inference (t-stats, confidence intervals, etc)

16. 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)F(z) as n , or P(Z<z)  F(z) • The central limit theorem states that the standardized average of any population with mean m and variance s2 is asymptotically ~N(0,1), or

17. Asymptotic Normality

18. 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 with a large sample, we do still need homoskedasticity

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 nR2 stat • Definition: auxiliary regression means a regression used to compute a test statistic (for example) where we don’t really care about the estimated coefficients.

21. LM Statistic (cont) • Suppose we have a standard model, y = b0 + b1x1 + b2x2 + . . . bkxk + u and our null hypothesis is • H0: bk-q+1 = 0, ... , bk = 0 • First, we just run the restricted model

22. 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 LM Statistic (cont) • If our null hypothesis is true, the R2 from a regression of ũ on the x’s should be “close” to zero. • 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