LINREG3

Γ ? β γ δ x j u j j 12 n 37 clearly only and can

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γ . η ) % ( β % γ . δ ). X j % U j , j ' 1,2,. ...., n , (37) Clearly, only and can be estimated by OLS, but not separately without α % γ . ηβ % γ . δα , β and γ knowing η and δ . This case is ruled out by Assumption 1. 6. Heteroscedasticity Recall that the errors of a regression model are heteroskedastic if the conditional U j variance of given the explanatory variables is not constant, but a function of the explanatory U j variables. In particular, the error terms in model (7) are heteroskedastic if there exists a non- constant function ψ such that E [ U 2 j | X 1, j , X 2, j ,..., X k & 1, j ] ' ψ ( X 1, j , X 2, j X k & 1, j (38) Heteroscedasticity often occurs in practice. It is actually the rule rather than the exception. One of the problems of heteroscedasticity is that the standard errors and t-values of the OLS parameter estimators are no longer valid. However, this problem can easily be cured by replacing the standard errors (24) with the heteroscedasticity consistent (H.C.) standard errors: ˜ σ i ' ' n j ' 1 w 2 i , j ˆ U 2 j ( ' H . C . standard error of ˆ β i ). (39) and the t-values with the heteroscedasticity consistent (H.C.) t-values ˜ t i ' ˆ β i ˜ σ i ( ' H . C . t & value of ˆ β i ).. (40) The F and Wald tests in Proposition 4 are also no longer valid under heteroscedasticity, but the cure for this is difficult to explain at the undergraduate level. To test joint hypotheses under heteroscedasticity with EasyReg you have to increase the econometrics level to “Intermediate”. Then after running your regression you will get the option to conduct Wald tests of linear parameter restrictions. This option gives you two versions of the Wald test, one for the homoscedastic case and one for the heteroskedastic case. See the guided tour on OLS estimation.
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4 Breusch, T. and A. Pagan (1979), "A Simple Test for Heteroscedasticity and Random Coefficient Variation", Econometrica 47, 1287-1294. 14 To decide whether the errors of model (7) are homoscedastic or heteroskedastic, use the Breusch-Pagan 4 test. Given that E [ U 2 j | X 1, j , X 2, j ,.... , X k & 1, j ] ' g ' k & 1 i ' 1 γ i X i , j % γ k for some unknown function g (.). (41) the Breusch-Pagan test tests the null hypothesis H 0 : γ 1 ' .... ' γ k & 1 ' 0 ] E [ U 2 j | X 1, j , X 2, j ,.... , X k & 1, j ] ' g ( γ k ) ' σ 2 , (42) against the alternative hypothesis H 0 : E [ U 2 j | X 1, j , X 2, j ,.... , X k & 1, j ] ' g ' k & 1 i ' 1 γ i X i , j % γ k g ( γ k ) (43) Under the null hypothesis (42) of homoskedasticity the test statistic of the Breusch-Pagan test has a distribution, and the test is conducted right-sided. χ 2 k & 1 APPENDIX Proof of (14): Substituting (2) in (3) yields ˆ β ' ' n j ' 1 ( X j & ¯ X )( α % β . X j % γ . Z j % U j ) ' n j ' 1 ( X j & ¯ X ) 2 ' α ' n j ' 1 ( X j & ¯ X ) ' n j ' 1 ( X j & ¯ X ) 2 % β ' n j ' 1 ( X j & ¯ X ) X j ' n j ' 1 ( X j & ¯ X ) 2 % γ ' n j ' 1 ( X j & ¯ X ) Z j ' n j ' 1 ( X j & ¯ X ) 2 % ' n j ' 1 ( X j & ¯ X ) U j ' n j ' 1 ( X j & ¯ X ) 2 ' β % γ ' n j ' 1 ( X j & ¯ X )( Z j & ¯ Z ) ' n j ' 1 ( X j & ¯ X ) 2 % ' n j ' 1 ( X j & ¯ X ) U j ' n j ' 1 ( X j & ¯ X ) 2 (44) where is the sample mean of the Z j ’s. Note that the last equality in (4) follows from the fact ¯ Z that ' n j ' 1 ( X j & ¯ X ) ' 0.
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γ β γ δ X j U j j 12 n 37 Clearly only and can be...

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