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Unformatted text preview: [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 6 Breusch, T. and A. Pagan (1979), "A Simple Test for Heteroscedasticity and Random Coefficient Variation", Econometrica 47, 12871294. 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. tvalue then becomes Under the null hypothesis β = 0 this tvalue 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. A popular test for heteroscedasticity is the BreuschPagan 6 test. Given that E [ U 2 j  X j ] ' g ( ( % ( 1 X j ) for some unknown function g (.). (38) the BreuschPagan test tests the null hypothesis H : ( 1 ' ] E [ U 2 j  X j ] ' g ( ( ) ' F 2 , say (39) against the alternative hypothesis H : ( 1 … ] E [ U 2 j  X j ] ' g ( ( % ( 1 X j ) ' R ( X j ), say . (40) Under the null hypothesis (39) of homoskedasticity the test statistic of the BreuschPagan test has a distribution 7 , and the test is conducted rightsided. χ 2 1 12. How close are OLS estimators ? The ice cream data in Table 1 is not based on any actual observations on sales and temperature; I have picked the numbers for and quite arbitrarily. Therefore, there is no way X j Y j to find out how close the OLS estimates are to the unknown parameters α ˆ α ' & 0.25, ˆ β ' 1.5 and β . Actually, we do not know either whether the linear regression model (2) and its assumptions are applicable to this artificial data. In order to show how well OLS estimators approximate the corresponding parameters I 8 Via the EasyReg International menus File 6 Choose an input file 6 Create artificial data. Rather than generating one random sample of size n = 1000 and then using subsamples of sizes n = 10 and n = 100, these samples have been generates separately for n = 10, n = 100 and n = 1000. 20 have generated random samples 8 for three sample sizes: n = 10, n = 100 and ( Y 1 , X 1 ),...,( Y n , X n ) n = 1000, as follows. The explanatory variables have been drawn independently from the X j χ 2 1 distribution, the regression errors have been drawn independently from the N(0,1) U j distribution, and the ‘s have been generated by Y j Y j ' 1 % X j % U j , j ' 1,2,..., n . (41) Thus, in this case the parameters α and β in model (2) are α = 1 and β = 1, and the standard error of is σ = 1. Moreover, note that the Assumptions I *IV * hold for model (41). U j The true R 2 can be defined by R 2 ' 1 & E [ SSR ] E [ TSS ] ' 1 & ( n & 2) σ 2 ' n j ' 1 E [( Y j & ¯ Y ) 2 ] ....
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 Fall '10
 H.Bierens
 Economics, Econometrics, Normal Distribution, Regression Analysis, Null hypothesis, Statistical hypothesis testing, j Uj, j'1

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