ECONOMETRICS FIELD EXAM Michigan State University May 9, 2008 Instructions: Answer all four (4) questions. Point totals for each question are given in parenthesis; there are 00 points possible. Within a question, each part receives equal weight. You may use a calculator, but only for computations -not for storage or retrieval of information. You must show all your working to get credit for your solutions. Be sure to show your work or provide su cient justi cation for your answers. You may use your notes and books.. (25 points) Consider a system of demand and supply curves where q denotes quantity, p denotes price, y denotes income and is the vector of ones where q = 0 + p + 2 y + " d ; " d N 0; 2 di (demand) q = 0 + p + " s ; " s N 0; 2 si (supply) " d and " s are independent of y; but amongst each other they are contemporaneously correlated, with their covariance constant over time n denotes the sample size. cov [ " dt ; " su ] = ( 0 if t 6= u ds if t = u : (a) Which variables are exogenous and which are endogenous? Also, assuming 6=, verify that the reduced-form equations for p and q are as follows p = 0 0 + 2 y + " d " s ; q = 0 0 + 2 y + " d " s : (b) Show that in general one will not get consistent estimates of the supply equation parameters if one regresses q on p (with an intercept). (c) If one estimates the supply function by instrumental variables using y as an instrument for p, the instrumental variables estimator is obtained as e = Argue that e is consistent if 2 6= 0: X (yi y) (q n i q) X (yi y) (p i p) : n
(d) Show that the indirect least squares estimator of is identical to the instrumental variable estimator of part (c). Show also that one can get another consistent estimator of if one regresses q on the predicted values of p from the reduced form equation (this is 2SLS). Show that this estimator is also the same as part (c). (e) Answer the following two questions. i. Since the error terms in the reduced form equations are contemporaneously correlated, could we get more precise estimates if one estimates the reduced form equations as a seemingly unrelated system, instead of OLS? ii. Moreover, so far we have only discussed estimators of the parameters in the supply function. How would you estimate the demand function? 2. (25 points) Consider the univariate time series y t = + u t ; where E(u t ) = 0. Let b = T P T t= y t and de ne bu t = y t b. Let S b t denote the partial sums of bu t : tx bs t = bu j. j= Consider the Bartlett kernel estimator of 2 times the population spectrum of u t at frequency zero: where Consider the KPSS statistic given by b 2 = b 0 + 2 b j = T MX j= TX t=j+ j b M j ; bu t bu t j : KP SS = XT T 2 t= bs 2 t b 2 : (a) Suppose u t is stationary and satis es the FCLT: T =2 [rt ] X u t ) w(r); t= 2
where w(r) is a standard Wiener process and X 2 = 0 + 2 j ; where j = cov(u t ; u t j ). Assume the bandwidth is chosen so that b 2 is a consistent estimator of 2. Derive the asymptotic distribution of KPSS. (b) Now assume that u t is a unit root process u t = u t + t ; 0 = 0; and that u t itself satis es a FCLT of the form T =2 u [rt ] ) w(r); where w(r) is a standard Wiener process and X 2 = 0 + 2 j ; j= j= where j = cov( t; t j ). It can be shown that for standard bandwidth rules T 2 2 b = op () when u t is a unit root process. Using this result and the FCLT, derive an asymptotic result for KPSS in the unit root case. (c) Discuss how the results you obtain in parts (a) and (b) can be used to motivate the use of the KPSS statistic to test the null hypothesis that u t is stationary against the alternative that u t has a unit root. What is the null distribution of the test? Is the test a right tail, a left tail, or a two tail test? (d) Because the distribution theory in parts (a) and (b) does not depend on the bandwidth, M, we might want to derive results under the xed-b framework. Assume that M = bt where b 2 (0; ] is xed. Redo your analysis for part (a) under this assumption. How is your result di erent from part (a)? (e) Using xed-b asymptotics, redo your analysis for part (b). How is your result di erent from part (b)? (f) How do your results in parts (d) and (e) simplify in the case where M = T? Is the KPSS test useful to test the stationarity of u t when the bandwidth is set equal to the sample size? Why or why not? 3
3. (25 points) Consider a panel data model with a single explanatory variable, with a unitspeci c slope: y it = a i + b i x it + u it ; t = ; :::; T; where you have access to a large random sample of size N from the cross section. In what follows assume that E(u it jx i ; :::; x it ; a i ; b i ) = 0; t = ; :::; T; and that fx it : t = ; :::; T g varies over time. Let = E(b i ) be the population average of the slope. (a) If we act as if b i = and just use the usual xed e ects estimator that removes a i, is the FE estimator always consistent for (with xed T )? When would it be? (b) If the FE estimator is consistent for, explain why it would be necessary to use a variance estimator that is robust to heteroskedasticity and serial correlation. (c) Now suppose E(b i jx i ; :::; x IT ) = 0 + x i, where x i is the time average. Propose a way to consistently estimate 0 and : (d) After having done the estimation in part (c), how would you estimate? Sketch how you would obtain the asymptotic variance of your estimator. (e) Suppose that you estimate 0 ;, 0 and in the unobserved e ects equation y it = 0 + x i + 0 x it + x i x it + f i + r it ; t = ; :::; T by standard random e ects. When will this estimator be consistent for 0 and? Will it always be asymptotically e cient? 4. (25 points) Provide an answer for each of the following six questions. You must support any agree/disagree answer with a careful explanation. (a) (5 points) Consider the model y = X + " where y is an n matrix of data, X is a n (k + ) matrix of data and is a (k + ) vector of parameters. The rst column of X contains the intercept regressor. Assume that the data represents a random sample from the population, and assume that V ar("jx) is 4
nite. We also have a matrix W of instrumental variables which satis es the following three conditions p lim n W 0 " = 0; p lim n W 0 W = Q exists, is nonrandom and positive de nite and p lim n W 0 X = D exists, is nonrandom and has full column rank. i. Show that the instrumental variables estimator e given as e = X 0 W (W 0 W ) W 0 X X 0 W (W 0 W ) W 0 y is consistent (Hint: Write e = B n n W 0 "; and show that B n has a well de ned p lim). ii. Add now the stronger assumption that n =2 W 0 "! d N 0; 2 Q : Show that (b) (5 points) Assume p n e! d N y i = + " i 0; 2 D 0 Q D : where is nonrandom, E (" i ) = 0; var (" i ) = 2 ; and cov [" i ; " j ] = 2 for i 6= j and therefore the variance covariance matrix is 2 3 ::: E [" " 0 ] = = 2 ::: 6 4 ::: ::: ::: ::: 7 5 : ::: If 0; then these errors terms could have been obtained as follows: " = z + u where z N (0; 2 I) and u N (0;! 2 ) independent of z: I is the identity matrix and is the vector of ones. Answer the following questions. i. Show that the covariance matrix of " is E [" " 0 ] = 2 I +! 2 0 : ii. What are the values of 2 and! 2 in terms of and 2 so that " has covariance structure of (i)?. 5
(c) (3 points) Suppose an econometric model has a well de ned likelihood function that is twice continuously di erentiable. Discuss how the model can be placed into the GMM framework and estimates obtained. What choice of weighting matrix will make the GMM estimates as asymptotically e cient as the MLE estimates? Formal proofs are not required but please sketch the details. (d) (4 points) Consider the simple two variable regression model y i = 0 + x i + u i where the usual Gauss-Markov assumptions hold except that cov(x; u) = xu 6= 0. Let z i be an instrument such that cov(z; x) = zx. Suppose z i is not a valid instrument because cov(z; u) = zu 6= 0. Derive a condition that depends on population variances and correlations that determines when the asymptotic bias of OLS is less than the asymptotic bias of IV. Relate your answer to case where z is a weak instrument. (e) (5 points) Consider a standard cross section linear equation with a single exogenous explanatory variable (x i ) and a single binary endogenous explanatory variable (w i ), written for a random draw from the population: y i = 0 + x i + 2 w i + d i x i + u i ; where d i is unobserved heterogeneity that interacts with the exogenous variable. Let z i be an omitted exogenous variable such that w i = [ 0 + x i + 2 z i + v i > 0]: Consider the following two-step estimation method: (i) Run probit of w i on (; x i ; z i ) and save the tted probabilities, ^ i. (ii) Estimate the equation y i = 0 + x i + 2 w i + e i by IV, using instruments (; x i ; ^ i ). Without worrying about regularity conditions, provide a set of su cient conditions for this procedure to consistently estimate the j coe cients. (f) (3 points) Agree or Disagree with the following statement, and provide justi cation: Most of the time with large microeconomic data sets, one should use least absolute deviations for estimating linear models to guard against outliers. 6