A popular test for heteroscedasticity is the breusch

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A popular test for heteroscedasticity is the Breusch-Pagan 6 test. Given that E [ U 2 j | X j ] ' g ( ( 0 % ( 1 X j ) for some unknown function g (.). (38) the Breusch-Pagan test tests the null hypothesis H 0 : ( 1 ' 0 ] E [ U 2 j | X j ] ' g ( ( 0 ) ' F 2 , say (39) against the alternative hypothesis H 0 : ( 1 0 ] E [ U 2 j | X j ] ' g ( ( 0 % ( 1 X j ) ' R ( X j ), say . (40) Under the null hypothesis (39) of homoskedasticity the test statistic of the Breusch-Pagan test has a distribution 7 , and the test is conducted right-sided. 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
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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 0 ' 1 & E [ SSR ] E [ TSS ] ' 1 & ( n & 2) σ 2 ' n j ' 1 E [( Y j & ¯ Y ) 2 ] . In the case (41), σ 2 ' 1, μ Y ' E ( Y j ) ' 1 % E ( X j ) ' 2, ' n j ' 1 E [( Y j & ¯ Y ) 2 ] ' E ' n j ' 1 ( Y j & μ Y ) & ( ¯ Y & μ Y ) 2 ' E ' n j ' 1 ( Y j & μ Y ) 2 & n ( ¯ Y & μ Y ) 2 ' ( n & 1)var( Y j ) and var( Y j ) ' E [( X j & 1 % U j ) 2 ] ' E [( X j & 1) 2 ] % E [ U 2 j ] ' E [( X j & 1) 2 ] % 1 ' 3, because is distributed and therefore has the same distribution as and it can be shown X j χ 2 1 U 2 j , that for standard normal random variables Thus, the true R 2 in this case U j , E [( U 2 j & 1) 2 ] ' 2. is R 2 0 ' 1 & n & 2 3( n & 1) ' 2 n & 1 3 n & 3 . 0.7037 for n ' 10 0.6700 for n ' 100 0.6670 for n ' 1000 The estimation results involved are given in Table 2:
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21 Table 2 : Artificial regression estimation results ˆ β ˆ α SER ( ' ˆ σ ) R 2 n estimate : 1.11748 0.55912 0.919045 0.8842 10 ( t & value ): (7.817) (1.675) estimate : 1.03309 0.96028 0.992502 0.8284 100 ( t & value ): (21.753) (8.237) estimate : 1.02360 0.98518 0.983608 0.6899 1000 ( t & value ): (47.124) (26.037) Even for a sample size of n = 10 the OLS estimator is already pretty close to its true value 1, ˆ β and the same applies to , but is too far away from the true value α = 1. However, for n = ˆ σ ˆ α 100 the OLS estimators and deviate only about ±4% from their true values α = β = 1, and ˆ β ˆ α ˆ σ deviates about -1% from its true value 1. In the case n = 1000 these deviations reduce to about ±2%. The R 2 's are too high, and only for n = 1000 is the R 2 reasonably close to its true value.
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