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LINREG3

Now suppose you want to test the joint hypothesis

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Now suppose you want to test the joint hypothesis that, for example, for i = 1,. ., m , β i ' 0
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3 Again, this means that for any constant K > 0, = 1 and lim n 64 P ( ˆ F > K ) = 1. lim n 64 P ( ˆ W > K ) 9 where m # k ! 1, against the alternative hypothesis that the null hypothesis is false: for at β i 0 least one index i # m . One possible way of testing this hypothesis is to conduct m separate two- sided t tests for i = 1,. ., m . However, the problem is that the left-hand side random variables in (23) are in general not independent, hence under the null hypothesis the test β 1 ' .... ' β m ' 0 statistics are in general not independent. In particular, it is impossible to select a critical ˆ t 1 ,..., ˆ t m value t * such that for a given significance level α ×100%, P [| ˆ t 1 |> t ( ,| ˆ t 2 t ( ,...,| ˆ t m t ( ] ' α , because we do not know the joint distribution of ˆ t 1 ,..., ˆ t m . The solution of this problem is the following. Consider the restricted regression model Y j ' β m % 1 X m % 1, j % β m % 2 X m % 2, j % ... % β k & 1 X k & 1, j % β k % U j , j ' 1,2,. .., n . (27) Then it can be shown that: Proposition 4 : Under the null hypothesis and the conditions of Proposition 2, β 1 ' .... ' β m ' 0 ˆ F ' ( SSR 0 & SSR )/ m SSR /( n & k ) - F m , n & k , (28) and under the conditions of Proposition 2, ˆ W ' m . ˆ F ' SSR 0 & SSR SSR /( n & k ) - χ 2 m , (29) where SSR is the sum of squared residuals of the unrestricted model (7) and SSR 0 is the sum of squared residuals of the restricted model (27). Moreover, under the alternative hypothesis that for at least one index i # m , the test statistics converge in probability to 4 as β i 0, ˆ F and ˆ W n 6 4 3 . The test based on is called, for obvious reasons, the F test, and the test based on is called ˆ F ˆ W the Wald test, named after the statistician with that name who proposed this test. The tests involved are conducted right-sided. In particular in the case of the Wald test the null hypothesis involved is rejected at say the 5% significance level if > c , where the critical value c is chosen ˆ W
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10 such that for a distributed random variable W , P [ W > c ] = 0.05. χ 2 m If m = k ! 1 then the restricted model (27) takes the form Y j ' β k % U j , j ' 1,2,. .., n . (30) The sum of squares residuals of this model, SSR 0 , is then equal to the total sum of squares of model (7), TSS ' ' n j ' 1 ( Y j & ¯ Y ) 2 , (31) where The F test involved then has test statistic ¯ Y ' (1/ n ) ' n j ' 1 Y j . ˜ F ' ( TSS & SSR )/( k & 1) SSR /( n & k ) , (32) which has an distribution under the null hypothesis that and the F k & 1, n & k β 1 ' .... ' β k & 1 ' 0 conditions of Proposition 2. This test is called the overall F test. Its null hypothesis amounts to the hypothesis that none of the explanatory variables have an effect on the X i , j , i ' 1,. .., k & 1, dependent variable Y j . 4. The adjusted R 2 The R 2 in the multiple regression case is defined the same as in the two-variable regression case: R 2 ' 1 & SSR TSS . (33) The problem with the R 2 is that it can be inflated towards 1 by including more explanatory variables in the model, because min ˆ β 1 ,..., ˆ β k ' n j ' 1 ( Y j & ' k i ' 1 ˆ β i X i , j ) 2 > min ˆ β 1 ,..., ˆ β k , ˆ β k % 1 ' n j ' 1 ( Y j & ' k % 1 i ' 1 ˆ β i X i , j ) 2 .
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Now suppose you want to test the joint hypothesis that for...

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