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Lecture Econometrics 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 ) -1 with ( X ' X ) -1 the k - th diagonal element of ( X ' X ) -1. kk kk bk - k Hence and ( X ' X ) -1 kk s2 | X ~ N (0,1) (n - K ) 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 ) -1 kk ~ 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 bk - k s ( X ' X ) -1 kk ( X ' X ) -1 kk and . First, is standard normal. Compare tables! 8 Econometrics The fact that bk - k s ( X ' X ) -1 kk ~ 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 ) -1 kk 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 For n = 40, K = 5, = .95 we find t = 2.030 . Hence 100 % confidence interval for k is - t t t [b k - t s ( X ' X ) -1 , bk + t s ( X ' X ) -1 kk kk ] 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 ) -1 kk If H 0 is true, then t ~ t (n - K ) and in particular E (t ) = 0 . 12 Econometrics If H 0 is false, i.e. k = k1 k 0 , then E (t ) 0 , in particular E (t ) > 0 E (t ) < 0 if k = k1 > k 0 if k = k1 < k 0 pdf of t under H 0 and H1 f (t ) k = k1 > k 0 ( H 1 ) (H k = k 0 (H0 ) (H t 13 Econometrics This suggest the decision rule (for some c) Choose H 0 if - c t c Choose H1 if t > c or t < -c Such a decision rule is a test of the hypothesis H 0 . 14 Econometrics Potential errors if we use this decision rule State of nature H0 H1 OK Type I error Type II error OK Decision H0 H1 Type I error is false rejection and type II error is false acceptance. How do we choose c ? 15 Econometrics A hypothesis test is a choice under uncertainty. Usual procedure: minimize expected loss (maximize expected utility). This requires loss (or utility) function. If we do not have a loss (or utility) function, we use other way to choose c : Set c so that the probability of a type I error (false rejection) is equal to . If H 0 is true, t ~ t (n - K and ) Pr(Type I error) = Pr(| t |> c) = Hence Pr(| t |< c) = Pr(-c < t < c) = 1 - and c is the 1 - th quantile of the t (n - K ) distribution. 2 E.g. if n = 40, K = 5, = .05 we have c = .2.030 16 Econometrics Terminology: is called the size of the test (= probability of a type I error). This test is called the t-test of the hypothesis H 0 : k = k 0 . Special case: k 0 = 0 , i.e. k th regressor has no effect on y . The test statistic is bk t= s ( X ' X ) -1 kk i.e. the ratio of bk and s ( X ' X ) -1 , the standard error of the regression kk coefficient. 17 Econometrics If we want to test H0 : k k0 H1 : k > k 0 then the probability of a type II error is smaller if we use a one-sided test, i.e. we reject H 0 if t > c with Pr(t > c) = , i.e. the (1 - ) th quantile of the t (n - K ) distribution. 18 Econometrics Next we consider statistical hypotheses that involve several regression coefficients simultaneously. We restrict attention to linear hypotheses: H 0 : R = q H1 : R q with R a J K matrix of constants with J K and rank( R ) = J . Why these last two requirements? Examples j =0 R = [0 010 0] q=0 position j 19 Econometrics 2 = ... = K = 0 , i.e. all coefficients except intercept are 0 0 0 R= 0 2 + 3 + 1 0 0 1 0 0 0 1 q=0 production function y = e 0 x2 2 ( K - 1) K + K = 1, e.g. constant returns to scale in Cobb-Douglas x K K e . R = [01 1] q =1 20 Econometrics With linear hypothesis R = q we obviously consider Rb - q = R(b - ) = R( X ' X ) -1 X ' Hence because this is linear in Rb - q ~ N (0, 2 R ( X ' X ) -1 R ' ) Using a result on quadratic function in normal random vector (1) ( Rb - q )' R( X ' X ) R ' ( -1 ) -1 ( Rb - q ) Because 2 is unknown this is not a test statistic. 2 ~ 2 (J ) 21 Econometrics We already noted that (2) (n - K ) s 2 2 = e' e 2 = ' M ~ 2 (n - K ) 2 It can be shown that (1) and (2) are stochastically independent and both have chi-squared distribution. 22 Econometrics F-distribution If v ~ 2 (l ) w ~ 2 ( m) v, w stochastically independent then l ~ F (l , m) w m has an F-distribution with l and m degrees of freedom. The distribution only depends on the parameters l and m (order matters) and can be tabulated. F= v 23 Econometrics 24 Econometrics Conclusion: If we divide (1) by J and (2) by n - K , i.e. the df's of their 2 distributions ( Rb - q )' R( X ' X ) -1 R' J F= s2 We reject H 0 if F > f with ( )-1 ( Rb - q) ~ F (J , n - K ) Pr( F > f ) = i.e. f is the 1 - th quantile of the F ( J , n - K ) distribution. This test is the F-test of J linear restrictions on . 25 Econometrics Alternative approach to linear restrictions: Restricted least squares min S ( ) = ( y - X )' ( y - X ) s.t. R = q Solution is the Restricted Least Squares (RLS) estimator b* = b - ( X ' X ) -1 R ' R ( X ' X ) -1 R' The RLS residuals are ( )-1 ( Rb - q) e* = y - Xb* and * * e e = e' e + ( Rb - q )' R( X ' X ) R' ( -1 ) -1 ( Rb - q ) 26 Econometrics Using this result, we can rewrite the F-statistic as e e - e' e J F= e' e n-K The numerator is the increase in the RSS due to imposing the linear restrictions. Why is the numerator nonnegative? * * 27 Econometrics If we divide numerator and denominator by i =1 ( yi - y ) 2 we obtain n R 2 - R *2 J F= 1- R2 n-K with the numerator the decrease in R 2 due to imposing the linear restrictions. 28 Econometrics Special case of restrictions 2 = ... = K = 0 , i.e. R *2 = 0 R2 F= J 2 1- R n-K 29 Econometrics 30

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