Lecture 4

Lecture 4 - Econometrics 1 Lecture 4: The CLR model with...

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Unformatted text preview: Econometrics 1 Lecture 4: The CLR model with normally distributed errors CLR model with normal errors We consider the CLR model + = X y with the additional assumption ) , ( ~ 2 I N i.e. the n-vector of random errors has a multivariate normal distribution with mean 0 and variance matrix I 2 . Econometrics 2 Sampling distributions of b and 2 s . Because ' ) ' ( 1 X X X b + = we have ( ) 1 2 ) ' ( , ~ | X X N X b Econometrics 3 Next, because = M e e 2 ' we have by a result for a quadratic function in a normal random vector ) ( ~ | ' 2 2 K n X e e For the joint distribution of b and 2 s , it can be shown that b and 2 s are independent. Econometrics 4 Conclusion: ( ) 1 2 ) ' ( , ~ | X X N X b ) ( ~ | ) ( 2 2 2 K n X s K n b and 2 s are independent (given X ) In general if ) 1 , ( ~ N z ) ( ~ 2 K v v z , stochastically independent then ) ( ~ K t K v z t = has a (Student) t-distribution with K degrees of freedom. Econometrics 5 Because K is the only parameter the distribution can be tabulated. Econometrics 6 Econometrics 7 Let k b be the k th regression coefficient with sampling variance 1 2 ) ' ( kk X X with 1 ) ' ( kk X X the k th diagonal element of 1 ) ' ( X X . Hence ) 1 , ( ~ | ) ' ( 1 N X X X b kk k k and ) ( ~ ) ( 2 2 2 K n s K n Because these random variables are stochastically independent, we have that given X the ratio ) ( ~ ) ' ( ) ' ( 1 2 2 1 K n t X X s b s X X b kk k k kk k k = Econometrics 8 Because the t-distribution does not depend on X , the result is also true unconditionally....
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Lecture 4 - Econometrics 1 Lecture 4: The CLR model with...

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