HEC Lausanne - Advanced Econometrics

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1 HEC Lausanne - Advanced Econometrics Christohe HURLI Correction Final Exam. January 4. C. Hurlin Exercise : MLE, arametric tests and the trilogy (3 oints) Part I: Maximum Likelihood Estimation (MLE) Question ( oints): since the random variales X ; ::; X are i:i:d: (.5 oint), the loglikelihood of the samle x ; ::; x is de ned as to e: with (.5 oint) ln f X (x i ; ) ` (; x) ln () X ln f X (x i ; ) () ln (c) + ln (x i ) () So, we have ( oint): ` (; x) ln () ln (c) + X ln (x i ) (3) Question ( oints): the ML estimator of is de ned as to e (.5 oint): The log-likelihood equation (FOC) is the following: g ; x arg max ` (; x) (4) R ln ` (; (5) with (.5 ln ` (; By solving this system, we get: + ln (c) X ln (x i ) (6) ln (c) X ln (x i ) (7)

2 Mid-term 3. Advanced Econometrics. HEC Lausanne. C. Hurlin The SOC is ased on the Hessian numer: H ; ln ` (; 3 ln (c) X ln (x i ) (8) Given the FOC, the Hessian numer is equal to (.5 oint): H ; x 3 ln (c) X ln (x i ) (9) 3 < () This numer is negative, so we have a maximum. The maximum likelihood estimators of the arameter is de ned as to e (.5 oint): ln (c) X ln (X i ) () Question 3 ( oints): The sequence of i:i:d: (.5 oint) random variales ln (X ) ; ::; ln (X ) satisfy E (ln (X i )) ln (c) : Given the WLL, we have (.5 oint): X ln (X i ) E (ln (X i )) ln (c) () By using the CMP theorem for a function g (z) ln (c) g z (.5 oint), we have: X ln (X i ) g (ln (c) ) (3) or equivalently ln (c) As a consequence (.5 oint): X ln (X i ) ln (c) ln (c) + (4) The estimator is (weakly) consistent. Question 4 ( oints): Since the rolem is regular (.5 oint), we have: d ; I ( ) (5) where denotes the true value of the arameter and I( ) the (average) Fisher information matrix for one oservation. The Fisher information matrix associated to the samle

3 Mid-term 3. Advanced Econometrics. HEC Lausanne. C. Hurlin 3 is equal to: I ( ) E ( H ( ; X)) (6) E + X 3 ln (c) ln (X i ) (7) ln (c) X E (ln (X i )) (8) (ln (c) ln (c) + ) (9) () () Since the samle is i:i:d:, we have I ( ) I () () As a consequence (.5 oint): d ; (3) or equivalently asy ; (4) Question 5 ( oints). The Fisher information matrix for one oservation is equal to: I ( ) (5) A rst natural consistent estimator is given y (.5 oint): I (6) A second ossile estimator is the BHHH estimator ased on the cross-roduct of the gradients: I X g i x i ; (7) with g i x i ; + ln (c) ln (x i) (8) So, a second consistent estimator is given y (.5 oint): I X + ln (c) ln (x i) (9)

4 Mid-term 3. Advanced Econometrics. HEC Lausanne. C. Hurlin 4 The asymtotic variance of is equal to (.5 oint): V asy I ( ) So, two alternative estimator of the asymtotic variance of are given y (.5 oint for each estimator): V asy (3) X V asy + ln (c) ln (x i) (3) Part II: Parametric tests Question 6 ( oints). Given the eyman Pearson lemma the rejection region of the UMP test of size is given y (.5 oint): (3) L ( ; x) L ( ; x) < K (33) where K is a constant determined y the size or equivalently It gives (.5 oint): ln ( ) ` ( ; x) ` ( ; x) < ln (K) (34) X ln (c)+ ln (x i )+ ln ( )+ ln (c) X ln (x i ) < ln (K) X ln (x i ) < K (35) where K ln (K) + (ln ( ) ln ( )) + ln (c) is a constant term. Since < ; we have ( oint): X ln (x i ) < K (36) with K 3 K ( X ln (x i ) > K 3 (37) ) : This inequality can e re-exressed as: with K 4 ln (c) form ( oint): ln (c) X ln (x i ) < K 4 (38) K 3 : The rejection region of the UMP test of size has the general W n x : o (x) < A where A is a constant determined y the size : Remark: the exressions of the constant terms K ; K 3 and K 4 are useless (no oint for these exressions). (39)

5 Mid-term 3. Advanced Econometrics. HEC Lausanne. C. Hurlin 5 Question 7 ( oints). Given the de nition of the error of tye I (.5 oint): Pr (Wj H ) (4) So, here we have (.5 oint): Pr asy < A ; (4) Then, we have: Pr < A asy A (; ) (4) where (:) denotes the cdf of the standard normal distriution. From this exression, we can deduce the critical value of the UMP test of size ( oint): A + () (43) Question 8 ( oints). Consider the test H : H : with < : The rejection region of the UMP test of size is ( oint): W x : (x) < + () (44) W does not deend on (.5 oint). It is also the rejection region of the UMP one-sided test (.5 oint): H : H : < Question 9 ( oints). Consider the one-sided tests: Test A: H : against H : < Test B: H : against H : > The non rejection regions of the UMP tests of size are (.5 oint): W A x : (x) > + (45) W B x : (x) < + So, non rejection region of the two-sided test of size is ( oint): W x : + < (x) < + (46) (47)

6 Mid-term 3. Advanced Econometrics. HEC Lausanne. C. Hurlin 6 Since () ( ) ; this region can e rewritten as: W x : (x) < (48) The region rejection of the two-sided test of size is (.5 oint): W x : (x) > (49) Question ( oints). By de nition of the ower function, we have (.5 oint): Under the alternative: P () Pr (Wj H ) 8 6 (5) asy H ; with 6 : The ower function can e exressed as: P () Pr W H Pr + Pr < + + The ower function is (.5 oint): < < + + Pr < (5) (5) P () (53) When tends to in nity, two cases have to e considered. If >, then we have (.5 oint): lim P () ( ) + ( ) (54) If <, then we have (.5 oint): lim P () (+) + (+) + (55) Whatever the value of ; the ower function tends to one: The test is consistent. Part III: The trilogy lim P () (56) Question (3 oints). The LR test statistic is de ned as to e (.5 oint): LR H ` ; x H ` ; x (57)

7 Mid-term 3. Advanced Econometrics. HEC Lausanne. C. Hurlin 7 where H and H resectively denote the ML estimator of the arameter under the null and the alternative. We know that under the alternative H (x) ln (c) As a consequence, we have (.5 oint): X ln (x i ) ln (c) X ln (x i ) (58) H (x) (4 ) (59) The log-likelihood of the samle under the alternative is equal to (.5 oint): H ` ; x ln H ln (c) + H X ln (x H i ) H ln () :3 (6) Under the null, the arameter is known ( :5) and the log-likelihood of the samle is equal to (.5 oint) H ` ; x ln ( ) ln (c) + X ln (x i ) :5 ln (:5) :5 4 + :5 373:87 (6) The LR test statistic (realisation) is equal to (.5 oint): LR (x) H ` ; x H ` ; x ( 373: :3) 9:3 (6) Under some regularity conditions and under the null, we have: LR d () (63) since there is only one restriction imosed. The critical region for a 5% signi cance level is: W x : LR (x) > :95 () 3:845 (64) Conclusion: for a 5% signi cance level, we reject the null H : :5 (.5 oint). Question (3 oints). In this case, the Wald test statistic (realisation) is equal to (.5 oint): H :5 H :5 ( :5) Wald (x) V H asy H 6:5 (65) Under some regularity conditions and under the null, we have (.5 oint): Wald d () (66)

8 Mid-term 3. Advanced Econometrics. HEC Lausanne. C. Hurlin 8 since there is only one restriction imosed. The critical region for a 5% signi cance level is (.5 oint): W x : Wald (x) > :95 () 3:845 (67) Conclusion: for a 5% signi cance level, we reject the null H : :5 (.5 oint). Question 3 (4 oints). The LM test statistic is de ned as to e (.5 oint): LM s ( c ; x i) I ( c ) (68) where s ( c ; x) is the score of the unconstrained model evaluated at c : In our case, the score is de ned y (.5 oint): s (; ln ` (; + ln (c) The realisation of the score evaluated at c is equal to ( oint) X ln (X i ) (69) s ( c ; x) + X c ln (c) c ln (x i ) c :5 + :5 4 :5 : (7) The estimate of the Fisher information numer associated to the samle I ( c ) is equal to (.5 oint): I ( c ) 44:4444 (7) c :5 So, the realisation of the LM test statistic is (.5 oint): LM (x) s ( c; x) I ( c ) : 44:4444 : (7) Under some regularity conditions and under the null, we have (.5 oint): LM d () (73) since there is only one restriction imosed. The critical region for a 5% signi cance level is: W x : LM (x) > :95 () 3:845 (74) Conclusion: for a 5% signi cance level, we reject the null H : :5 (.5 oint).

9 Mid-term 3. Advanced Econometrics. HEC Lausanne. C. Hurlin 9 Exercise : Logit model and MLE (5 oints) Question ( oints). the conditional distriution of the deendent variale Y i is a Bernoulli distriution with a (conditional) success roaility equal to i Pr (y i j x i ) : (.5 oint) y i j x i Bernoulli ( i ) Since the variales fy i ; x i g are i:i:d: (.5 oint), the conditional log-likelihood of the samle fy i ; x i g corresonds to the sum of the (log) roaility mass functions associated to the conditional distriutions of y i given x i for i ; ::; (.5 oint): with ` (; yj x) X ln f yjx (y i j x i ; ) (75) f yjx (y i j x i ; ) yi i ( i ) yi (76) As a consequence, the (conditional) log-likelihood of the samle fy i ; x i g is equal to (.5 oint): ` (; yj x) X y i ln (x i ) + X ( y i ) ln ( (x i )) (77) or equivalently ` (; yj x) X ex (x i y i ln ) + ex (x i ) + X ( y i ) ln ex (x i ) + ex (x i ) (78) Question ( oints). By de nition of the score vector, we have ( oint): s (; yj (; yj X (x i y i (x i i X y i (x i ) (x i )x i X (x i ( y i ) (x i i X (x i ( y i ) ) (x i )x i (79) Since (x) (x) ( (x)) ; this exression can e simli ed as follows (.5 oint): s (; yj x) X (x i ) ( y (x i )) i (x i ) X y i ( (x i )) x i x i X X ( y i ) (x i ) x i X (y i y i (x i ) (x i ) + y i (x i )) x i ( y i ) (x i ) ( (x i )) (x i ) x i X (y i (x i )) x i (8)

10 Mid-term 3. Advanced Econometrics. HEC Lausanne. C. Hurlin So, the score vector of the logit model is simly de ned y (.5 oint): s (; yj x) X x i (y i (x i )) (8) Question 3 ( oints). The Hessian matrix is a K K matrix given y (.5 oint): So, we have (.5 oint): H (; yj (; (; yj (8) H (; yj x) X x i (x i (83) The Hessian matrix is de ned as to e (oint) : H (; yj x) X x i (x i ) x i (84) or equivalently (since (x i ) is a scalar) H (; yj x) X (x i ) x ix i (85) Question 4 (4 oints). The rst ste consists in writing a function Matla to de ne the oosite of the log-likelihood. The following syntax ( oints) is roosed, cf. Figure : Figure : Log-Likelihood function The second ste consists in using the numerical otimisation algorithm of Matla y using the following syntax, cf. Figure ( oints).

11 Mid-term 3. Advanced Econometrics. HEC Lausanne. C. Hurlin Figure : Main rogram Question 5 ( oints). Since the rolem is regular, the asymtotic variance covariance matrix of the MLE estimator is (.5 oint): V asy I I A consistent estimator for average Fisher information matrix, ased on the Hessian matrix, is (.5 oint): I X H i ; yi j x i H ; yj x (87) So, an estimator of the asymtotic variance covariance matrix of the MLE estimator is given y (.5 oint): In this case, we have (.5 oint): V asy I H ; yj x (86) (88) V asy X x i x i x i (89) Question 6 (3 oints). The following syntax is roosed, cf. Figure 3:

12 Mid-term 3. Advanced Econometrics. HEC Lausanne. C. Hurlin Figure 3: Asymtotic standard errors Exercise 3: OLS and heteroscedasticity (5 oints) The aim of this code is to comute the OLS estimate of the arameter of a linear model, denoted eta in the code (.5 oint) and the corresonding White consistent standard errors ( oint). The matrix M denotes the estimator de ned y (.5 oint): M X " i x i x i (9) where " i denotes the OLS residual for the i th unit and x i the corresonding vector of exlicative variales. The code uses a nite samle correction (.5 oint) for M (Davidson and MacKinnon, 993). The matrix V corresonds to the White consistent estimate of the asymtotic variance covariance matrix of the OLS estimator ( oint). The vector std corresonds to the roust asymtotic standard errors (.5 oint).

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