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Econometrics Lecture 4 The CLR model with normally distributed errors CLR model with normal errors We consider the CLR model y X with the additional assumption N 0 2 I i e the n vector of random errors has a multivariate normal distribution with mean 0 and variance matrix 2 I 1 Econometrics Sampling distributions of b and s 2 Because b X X 1 X we have b X N 2 X X 1 2 Econometrics Next because e e 2 M we have by a result for a quadratic function in a normal random vector e e 2 X 2 n K For the joint distribution of b and s 2 it can be shown that b and s 2 are independent 3 Econometrics Conclusion b X N 2 X X 1 s2 n K 2 X 2 n K b and s 2 are independent given X In general if z N 0 1 v 2 K z v stochastically independent then z t K v K has a Student t distribution with K degrees of freedom t 4 Econometrics Because K is the only parameter the distribution can be tabulated 5 Econometrics 6 Econometrics Let bk be the k th regression coefficient with sampling variance 2 X X kk1 with X X kk1 the k th diagonal element of X X 1 Hence bk k X X kk1 X N 0 1 and n K s2 2 2 n K Because these random variables are stochastically independent we have that given X the ratio bk k 1 X X kk s 2 bk k s X X kk1 t n K 2 7 Econometrics Because the t distribution does not depend on X the result is also true unconditionally Note similarity of bk k s X X kk1 bk k X X kk1 and First is standard normal Compare tables 8 Econometrics The fact that bk k s X X kk1 t n K can be used for 1 Confidence interval for k 2 Hypothesis test for k 9 Econometrics Ad 1 Confidence interval From the table of the t distribution we find t such that Pr t bk k s X X kk1 t 1 Hence t is the 1 th quantile of the t n K distribution see 2 graph i e t satisfies 1 Pr t n K t 1 2 10 Econometrics area is t t t For n 40 K 5 95 we find t 2 030 Hence 100 confidence interval for k is b k t s X X kk1 bk t s X X kk1 11 Econometrics Ad 2 Hypothesis test We consider the hypothesis H0 k k0 H1 k k 0 Now consider t bk k 0 s X X kk1 If H 0 is