However without assumption iv we need an additional

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However, without Assumption IV * we need an additional condition in Proposition 6 in order to use the central limit theorem, namely: Proposition 10 . If the sample size n is large then under the assumptions I * - III * and the additional condition the approximate normality results in Proposition 7 carry over. E [ X 2 j ] < 4 Moreover, without Assumption IV * the Propositions 6 and 8 are no longer true. As to Proposition 6, this not a big deal, as in large samples we can still use Proposition 7, but without Assumption IV * we can no longer derive confidence intervals for the forecasts, as these confidence intervals are based on Proposition 8. It is therefore important to test the normality assumption. 10. Testing the normality assumption For a normal random variable U with zero expectation and variance σ 2 it can be shown that
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3 Jarque, C.M.and A.K. Bera, (1980), "Efficient Tests for Normality, Homoscedasticity and Serial Independence of Regression Residuals". Economics Letters 6, 255 BB 259. 4 Kiefer, N. and M. Salmon (1983), "Testing Normality in Econometric Models", Economic Letters 11, 123-127. 5 Also spelled as "Heteroskedasticity." 18 Kurtosis ' def . E [ U 4 ]/ F 4 & 3 ' 0, Skewness ' def . E [ U 3 ] ' 0 (34) Therefore, the normality condition can be tested by testing whether the kurtosis and the skewness of the model errors are zero, using the residuals. This is the idea behind the Jarque-Bera 3 and Kiefer-Salmon 4 tests. Under the null hypothesis (34) the test statistic involved has a χ 2 2 distribution 11. Heteroscedasticity 5 We say that the errors U j of regression model (2) are heteroskedastic if assumption III * does not hold: E [ U 2 j | X j ] ' R ( X j ) for some function R (.). (35) Heteroscedasticity often occurs in practice. It is actually the rule rather than the exception. The main consequence of heteroscedasticity is that the conditional variance formulas in Propositions 2 and 3 do no longer hold, although the unbiasedness result in Proposition 1 is not affected by heteroscedasticity. Therefore, the Propositions 4-8 are no longer valid as well. In particular, the conditional variance of [see (60)] under heteroscedasticity takes the form ˆ β var( $ $ | X 1 ,..., X n ) ' E [( $ $ & $ ) 2 | X 1 ,..., X n ] ' ' n j ' 1 ( X j & ¯ X ) 2 R ( X j ) ' n i ' 1 ( X i & ¯ X ) 2 2 . (36) A cure for the heteroscedasticity problem is to replace the standard error of by ˆ β
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6 Breusch, T. and A. Pagan (1979), "A Simple Test for Heteroscedasticity and Random Coefficient Variation", Econometrica 47, 1287-1294. 7 In the multiple regression case the degrees of freedom is equal to the number of parameters minus 1 for the intercept. 19 # F $ $ ' n n & 2 ' n j ' 1 ( X j & ¯ X ) 2 $ U 2 j ' n i ' 1 ( X i & ¯ X ) 2 2 . (37) This is known as the Heteroscedasticity Consistent (H.C.) standard error. The H.C. t-value then becomes Under the null hypothesis β = 0 this t-value is no longer t distributed, but ˜ t ˆ β ' ˆ β σ ˆ β . the standard normal approximation remains valid if the sample size n is large.
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