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Unformatted text preview: Economics 241B Hypothesis Testing: Large Sample Inference Statistical inference in large-sample theory is based on test statistics whose dis- tributions are known under the truth of the null hypothesis. Derivation of these distributions is easier than in &nite-sample theory because we are only concerned with the large-sample approximation to the exact distribution. In what follows we assume that a consistent estimator of S exists, which we term ^ S . Recall that S = E ( g t g t ) , where g t = X t U t . Testing Linear Hypotheses Consider testing a hypothesis regarding the k-th coe cient & k . Proposition 2.1, which established the asymptotic distribution of the OLS estimator, implies that under H : & k = & & k , p n & B k & & & k ! d N (0 ;Avar ( B k )) and \ Avar ( B k ) ! p Avar ( B k ) : Here B k is the k-th element of the OLS estimator B and Avar ( B k ) is the ( k;k ) element of the K K matrix Avar ( B ) . The key issue here is that we have not assumed conditional homoskedasticity, hence \ Avar ( B k ) = S & 1 XX ^ S S & 1 XX ; which is the (heteroskedasticity-consistent) robust asymptotic variance. Under the Slutsky result (Lemma 2.4c), the resultant robust t-ratio t k p n & B k & & & k q \ Avar ( B k ) = & B k & & & k SE ( B k ) ! d N (0 ; 1) ; where the robust standard error is SE = q 1 n \ Avar ( B k ) . Note this robust t- ratio is distinct from the t-ratio introduced under the &nite-sample assumptions in earlier lectures. To test H : & k = & & k , simply follow these steps: Step 1: Calculate the robust t-ratio Step 2: Obtain the critical value from the N (0 ; 1) distribution Step 3: Reject the null hypothesis if j t k j exceeds the critical value There are several di/erences from the &nite-sample test that relies on conditional homoskedasticity. & The standard error is calculated in a di/erent way, to accommodate condi- tional heteroskedasticity. & The normal distribution is used to obtain critical values, rather than the t distribution. & The actual (or empirical) size of the test is not necessarily equal to the nominal size. The di/erence between the actual size and the nominal size is the size distortion. Because the asymptotic distribution of the robust t ratio is standard normal, the size distortion shrinks to zero as the sample size goes to in&nity. To summarize these results, together with the behavior of the Wald statistic let us briey recall the assumptions required for Proposition 2.1: Assumption 2.1 (linearity): Y t = X t & + U t ( t = 1 ;:::;n; ) where X t is a K-dimensional vector of regressors, & is a K-dimensional vector of...
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- Fall '08