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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 ε β + = X y with the additional assumption ) , ( ~ 2 I N σ ε i.e. the nvector of random errors ε has a multivariate normal distribution with mean 0 and variance matrix I 2 σ . Econometrics 2 Sampling distributions of b and 2 s . Because ε β ' ) ' ( 1 X X X b − + = we have ( ) 1 2 ) ' ( , ~  − X X N X b σ β Econometrics 3 Next, because ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ′ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = σ ε σ ε σ 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 − χ σ 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 σ β • ) ( ~  ) ( 2 2 2 K n X s K n − − χ σ • b and 2 s are independent (given X ) In general if ) 1 , ( ~ N z ) ( ~ 2 K v χ v z , stochastically independent then ) ( ~ K t K v z t = has a (Student) tdistribution 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 σ 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 − − σ β and ) ( ~ ) ( 2 2 2 K n s K n − − χ σ 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 − − = − − − β σ σ β Econometrics 8 Because the tdistribution does not depend on X , the result is also true unconditionally....
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This note was uploaded on 07/22/2008 for the course ECON 513 taught by Professor Rashidian during the Fall '07 term at USC.
 Fall '07
 Rashidian
 Econometrics

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