k ver 10232012 p kolm 11 large sample inference

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Unformatted text preview: that is1 E (u ) = 0 Cov(x j , u ) = 0, for j = 1, 2,..., k VER. 10/23/2012. © P. KOLM 11 Large Sample Inference (1/2) Recall: • Under the classical linear model (CLM) assumptions, the sampling distributions are normal, so we could derive t - and F -distributions for hypothesis testing • The normality was due to assuming the population error was normal • The assumption of normal errors implied that the distribution of the dependent variable, y , given the independent variables, x 1,..., x k , was normal as well VER. 10/23/2012. © P. KOLM 12 Large Sample Inference (2/2) • There are many situations in which this exact normality assumption will fail o For example: Any skewed variable (like hedge fund returns, option returns, wages, savings, etc.) cannot be normal as a normal distribution is symmetric • Note that the normality assumption is not needed to conclude OLS is BLUE o We only used it for inference (hypothesis testing) VER. 10/23/2012. © P. KOLM 13 Central Limit Theorem • Recall: The central limit theorem states that the standardized average of any population with mean μ and variance σ 2 is asymptotically ∼ N (0,1), that is Y −μ a Z= ∼ N (0,1) σ n • Asymptotic normality means that P(Z < z ) → Φ(z ) as n → ∞ (where Φ(z ) denotes a normal CDF) • Based on the central limit theorem, we can show that OLS estimators are asymptotically normal. In particular, we have the important result on the next page VER. 10/23/2012. © P. KOLM 14 Asymptotic Normality of OLS (1/2) Facts: Under the Gauss-Markov assumptions, we have that 1. For each j = 1,..., k (excluding the intercept, j=0) ˆ (β a j s ˆ where se (β j ) = 2 j SSTj (1 − R ) ˆ − β j ) se (β j ) ∼ N (0,1) (the usual OLS standard error) n 1 ˆ 2. s = ∑ ui2 is a consistent estimator of σ2 n − k − 1 i =1 2 3. ( a ˆ n (β j − β j ) ∼ N 0, Avar where Avar ˆ ( n (β − β ))) j j ˆ ( n (β − β )) is the asymptotic variance of j j ˆ n (β j − β j ) (for the ˆ slope coefficients).2 We say that β j is asymptotically normally distributed VER. 10/23/2012. © P. KOLM 15 Remarks: • Because the t -distribution approaches the normal distribution for large df , we can write a ˆ ˆ (βj − βj ) se (βj ) ∼ tn −k −1 ∼ N (0,1) • Note that while we no longer need to assume normality with a large sample, we do still need homoscedasticity (For you: Why?) • Hypothesis testing: Asymptotic normality implies that . . . o . . . the t -statistic is approximately normal if the sample size is large enough → we can conduct t-tests of single regression coefficients using the normal distribution o . . . the F -statistic is approximately F distributed if the sample is large enough3 → thus for testing exclusion restrictions or other multiple hypothesis, nothing changes from what we have done before VER. 10/23/2012. © P. KOLM 16 OLS is Asymptotically Efficient • Under the Gauss-Markov assumptions, the OLS estimators will have the smallest asymptotic variances • We...
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This document was uploaded on 02/17/2014 for the course COURANT G63.2751.0 at NYU.

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