Under the null hypothesis and using the variance of

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) under the null hypothesis, and using the variance of this difference to determine whether or not a given-sized difference is significant. If the difference is significant at some chosen level of test size, we reject the null hypothesis. If there is a significant difference between the two sets of estimates, one can regard this as evidence supporting the hypothesis that there is correlation between regressors and the disturbance. In such a case, OLS is inconsistent, and GIVE should be used. 6.2 THE WU-HAUSMAN TEST The Hausman test can not be implemented in all econometrics packages. A similar procedure which can be done in almost any package makes use of the Wu-Hausman test. This is sometimes described as an “exogeneity” test, but is more properly thought of as a test of regressor “orthogonality”; that is, a test of the assumption that there is zero asymptotic correlation between the regressors and the disturbance term. The null and alternative hypotheses are again H 0 : plim 1 T X u = 0 H a : plim 1 T X u 0 The test is implemented in a series of steps. For simplicity, in the presentation which follows we suppose that only one of the regressors might be asymptotically correlated with the disturbance. Suppose that this is the variable X 4t in the regression model: Y X X X u t t t t t = + + + + β β β β 1 2 2 3 3 4 4 We also assume that the researcher has available two instruments, W 1 and W 2 , for the potentially correlated regressor X 4 . STEP 1 : Estimate by OLS the “ instrumenting regression ”, and save the residuals. This is the regression of the potentially correlated regressor on all instruments. Note carefully that by “instruments”, we mean here all explanatory variables in the original regression model that are not being treated as potentially correlated, together with all additional variables being used as instruments for the potentially correlated variable. So in our example, the “instrumenting regression” to be estimated by OLS is X X X W W t t t t t t 4 1 2 2 3 3 4 1 5 2 = + + + + + α α α α α ε Denote the saved residuals as ε t , for t=1,...,T. STEP 2 : Estimate by OLS the original regression model: Y X X X u t t t t t = + + + + β β β β 1 2 2 3 3 4 4 STEP 3 : Use a variable addition test to test for the significance of the residuals when added as a regressor to the original regression model. That is we wish to test H 0 : θ = 0 against H A : θ 0 in the OLS regression
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11 Y X X X u t t t t t t = + + + + + β β β β θε 1 2 2 3 3 4 4 If we reject H 0 , we regard this as evidence in favour of the alternative hypothesis that X 4t is correlated (asymptotically) with the disturbance term, and so should use GIVE rather than OLS to obtain our parameter estimates. If we cannot reject the null, the OLS estimator is preferred.
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