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Unformatted text preview: Economics 241B Hypothesis Testing: Large Sample Inference Statistical inference in largesample 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 &nitesample theory because we are only concerned with the largesample 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 kth 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 kth 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 (heteroskedasticityconsistent) robust asymptotic variance. Under the Slutsky result (Lemma 2.4c), the resultant robust tratio 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 tratio introduced under the &nitesample assumptions in earlier lectures. To test H : & k = & & k , simply follow these steps: Step 1: Calculate the robust tratio 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 &nitesample 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 Kdimensional vector of regressors, & is a Kdimensional vector of...
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 Fall '08
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 Economics

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