Econometrics 1Lecture 4: The CLR model with normally distributed errors CLR model with normal errors We consider the CLR model εβ+=Xy with the additional assumption ),0(~2INσε i.e. the n-vector of random errors εhas a multivariate normal distribution with mean 0 and variance matrix I2σ.Econometrics 2Sampling distributions of b and 2s. Because εβ')'(1XXXb−+= we have ()12)'(,~|−XXNXbσβEconometrics 3Next, because ⎟⎠⎞⎜⎝⎛′⎟⎠⎞⎜⎝⎛=σεσεσMee2' we have by a result for a quadratic function in a normal random vector )(~|'22KnXee−χσ For the joint distribution of b and 2s, it can be shown that b and 2s are independent.Econometrics 4Conclusion: • ()12)'(,~|−XXNXbσβ • )(~|)(222KnXsKn −−χσ • b and 2s are independent (given X) In general if )1,0(~Nz )(~2Kvχ vz, stochastically independent then )(~ KtKvzt = has a (Student) t-distribution with Kdegrees of freedom.Econometrics 5Because Kis the only parameter the distribution can be tabulated.Econometrics 6Econometrics 7Let kb be the −kth regression coefficient with sampling variance 12)'(−kkXXσ with 1)'(−kkXX the −kth diagonal element of 1)'(−XX . Hence )1,0(~|)'(1NXXXbkkkk−−σβ and )(~)(222KnsKn −−χσ Because these random variables are stochastically independent, we have that given Xthe ratio )(~)'()'(1221KntXXsbsXXbkkkkkkkk−−=−−−βσσβEconometrics 8Because the t-distribution does not depend on X, the result is also true unconditionally. Note similarity of 1)'(−−kkkkXXbσβ and 1)'(−−kkkkXXsbβ. First, is standard normal. Compare tables!Econometrics 9The fact that )(~)'(1KntXXsbkkkk−−−β can be used for 1. Confidence interval for kβ 2. Hypothesis test for kβEconometrics 10Ad 1. Confidence interval From the table of the t-distribution we find αt such that αβαα=≤−≤−−))'(Pr(1tXXsbtkkkk Hence αt is the −⎟⎠⎞⎜⎝⎛−−211αth quantile of the )(Knt − distribution (see graph), i.e. αt satisfies 211))(Pr(αα−−=≤− tKntEconometrics 11area is α αt−αtttt For 95.,5,40 ===αKn we find 030.2=αt . Hence 100α% confidence interval for kβis []11)'(,)'(−−+−kkkkkkXXstbXXstbααEconometrics 12Ad 2. Hypothesis test We consider the hypothesis 0100::kkkkHHββββ≠= Now consider 10)'(−−=kkkkXXsbtβ If 0H is true, then )(~Kntt− and in particular 0)(=tE.Econometrics 13If 0H is false, i.e. 01kkkβββ≠=, then 0)(≠tE, in particular 0)( >tE if 01kkkβββ>= 0)( <tE if 01kkkβββ<= )(00Hkkββ=)(101Hkkkβββ>=)(tftpdf of t under 0H and 1H )(H)(HtHHEconometrics 14This suggest the decision rule (for some )c Choose 0H if ctc≤≤− Choose 1H if ct > or ct−< Such a decision rule is a test of the hypothesis 0H .Econometrics 15Potential errors if we use this decision rule State of nature Decision 0H 1H 0H OK Type II error 1H Type I error OK Type I error is false rejection and type II error is false acceptance. How do we choose c?Econometrics 16A 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 α. If 0H is true, )(~Kntt − and α=>= )|Pr(|error)IPr(Type ct Hence α−=<<−=< 1)Pr()|Pr(| ctcct and c is the ⎟⎠⎞⎜⎝⎛−21αth quantile of the )(Knt− distribution. E.g. if 05.,5,40===αKn we have 030.2.=cEconometrics 17 Terminology: α is called the size of the test (= probability of a type I error). This test is called the t-test of the hypothesis 00:kkHββ=. Special case: 00=kβ, i.e. kth regressor has no effect on y. The test statistic is 1)'(−=kkkXXsbt i.e. the ratio of kb and 1)'(−kkXXs , the standard error of the regression coefficient.Econometrics 18If we want to test 0100::kkkkHHββββ>≤ then the probability of a type II error is smaller if we use a one-sided test, i.e. we reject 0H if ct > with α=> )Pr( ct, i.e. the )1(α−th quantile of the )(Knt− distribution.Econometrics 19Next we consider statistical hypotheses that involve several regression coefficients simultaneously. We restrict attention to linear hypotheses: qRH=β:0 qRH≠β:1 with R a KJ× matrix of constants withKJ≤ and JR=)rank( . Why these last two requirements? Examples • 0=jβ 0=q []jRposition00100↑=Econometrics 20 • 02===Kββ…, i.e. all coefficients except intercept are 0 ⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=10001000010R 0=q KK×− )1( • 132=+++Kβββ , e.g. constant returns to scale in Cobb-Douglas production function εβββexxeyKK202= . []101=R 1=qEconometrics 21With linear hypothesis qR=β we obviously consider εβ')'()(1XXXRbRqRb−=−=− Hence because this is linear in ε )')'(,0(~12RXXRNqRb−−σ Using a result on quadratic function in normal random vector (1) ())(~)(')'()'(2211JqRbRXXRqRbχσ−−−− Because 2σis unknown this is not a test statistic.Econometrics 22We already noted that (2) )(~'')(22222KnMeesKn−==−χσεεσσ It can be shown that (1) and (2) are stochastically independent and both have chi-squared distribution.Econometrics 23F-distribution If )(~2lvχ )(~2mwχ wv, stochastically independent then ),(~ mlFmwlvF = 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.Econometrics 24Econometrics 25Conclusion: If we divide (1) by Jand (2) by Kn − , i.e. the df’s of their 2χdistributions ()),(~)(')'()'(211KnJFsJqRbRXXRqRbF −−−=−− We reject 0H if αfF > with αα=> )Pr( fF i.e. αf is the α−1 th quantile of the ),(KnJF− distribution. This test is the F-test of Jlinear restrictions on β.Econometrics 26Alternative approach to linear restrictions: Restricted least squares )()'()(minβββXyXyS−−= s.t. qR =β Solution is the Restricted Least Squares (RLS) estimator ())(')'(')'(111*qRbRXXRRXXbb −−=−−− The RLS residuals are **Xbye −= and ())(')'()'('11**qRbRXXRqRbeeee −−+=′−−Econometrics 27 Using this result, we can rewrite the F-statistic