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13 Hypothesis Testing

# 13 Hypothesis Testing - Economics 241B Hypothesis Testing...

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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 0 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 0 : ° 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 ² 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 0 : ° 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

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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. ³ The standard error is calculated in a di/erent way, to accommodate condi- tional heteroskedasticity. ³ The normal distribution is used to obtain critical values, rather than the t distribution. ³ 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 brie±y recall the assumptions required for Proposition 2.1: Assumption 2.1 (linearity): Y t = X 0 t ° + U t ( t = 1 ; : : : ; n; ) where X t is a K -dimensional vector of regressors, ° is a K -dimensional vector of coe¢ cients and U t is the latent error.
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13 Hypothesis Testing - Economics 241B Hypothesis Testing...

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