ECONOMETRICS II (ECO 2401S) University of Toronto. Department of Economics. Winter 2016 Instructor: Victor Aguirregabiria

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1 ECOOMETRICS II (ECO 24S) University of Toronto. Department of Economics. Winter 26 Instructor: Victor Aguirregabiria FIAL EAM. Thursday, April 4, 26. From 9:am-2:pm (3 hours) ISTRUCTIOS: - This is a closed-book exam. - o study aids, including calculators, are allowed. - Please, answer all the questions. TOTAL MARKS = PROBLEM (4 points). Let w it be the log-wage of worker i at period t. The researcher has a panel dataset fw it : i = ; 2; :::; t = ; 2; ::::; T g where the number of workers is large, and the number of periods T is small, e.g., = 5; and T =. The researcher postulates the following variance-component model: w it = t + i + u it where, 2,..., T are parameters; E( i ) = E(u it ) = ; E( i u it ) = ; var( i ) = 2 ; u it is not serially correlated, it is homoscedastic across individuals, but its variance may vary over time, var(u it ) = 2 u;t. The main interest of the researcher is the analysis of wage inequality, its persistent and the evolution over time. More speci cally, the researcher is interested in estimation of the variance parameters 2 and 2 u;,..., 2 u;t. (a) [5 points] Propose a root- consistent estimator of the parameters, 2,..., T. ASWER: Given the assumptions of the model, we have that for any period t, E(w it jt) = t. Therefore, we can estimate t using a Method of Moments estimator based on this moment condition. This Method of Moments estimator is: b t = By the LL, this estimator converges in probability to t as goes to in nity. And by the CLT, p ( b t t ) converges in distribution to a (; V ar( i + u it )), with V ar( i + u it ) = u;t. (b) [5 points] Propose a root- consistent estimator of the parameters 2, 2 u;,..., 2 u;t. w it ASWER: De ne it i + u it = w it t. Based on the model assumptions, we have that for any period t, E( it it jt) = u;t E( it it jt) = 2

2 These moment conditions imply, 2 = T T E( it it jt) 2 u;t = E( it it jt) T T E( it it jt) Using these moment conditions and the consistent estimators b t from Question a, we can construct consistent Method of Moments estimators of 2 and 2 u;t. That is, b 2 = 2 u;t = T T # 2 w it t b w it b t w it b t # T T w it b t w it b t # Under the condition that the distributions of i and u it have nite moments of order four, these estimators are root- consistent and asymptotically normal. (c) [5 points] Suppose that u it is serially correlated. Does this correlation a ect the consistency of the estimator proposed in Question b? Explain. ASWER: If u it is serially correlated, then the previous moment conditional become: E( it it jt) = u;t E( it it jt) = 2 + E(u it u it jt) Therefore, the previous estimator of 2 is inconsistent because it not only captures the variance 2 but also the covariance E(u it u it jt). For instance, if the serial correlation is positive such that E(u it u it jt) >, then the previous estimator b 2 over-estimates the true 2, i.e., it over-estimates the time invariant component of wage-inequality. Similarly, this bias in the estimation of b 2 implies also a bias in the estimation of 2 u;t. More speci cally, the previous estimator of 2 u;t is a consistent estimator of 2 u;t E(u it u it jt). Under positive correlation, b 2 u;t under-estimates the true 2 u;t. (d) [5 points] Describe a test of the null hypothesis E(u it u i;t ) =. ASWER: Due to the incidental parameters problem, we cannot obtain root- consistent estimators of the unobservables u it = w it t i. Therefore, our test of the null hypothesis E(u it u i;t ) = cannot be based on residuals for u it. However, we can obtain root- consistent estimates for u it u it u it = w it t + t. The Arellano-Bond test of serial correlation is based on the residuals: cu it = w it t b + b t Under the null hypothesis E(u it u i;t ) =, we have that: E (4u it 4 u it 2 ) = 2

3 Therefore, we can (indirectly) test for no-serial correlation in u it by testing for no second-order serial correlation in 4u it. Let r 2t be the auto-covariance of order 2 at period t of f4u it g: i.e., r 2t E (4u it 4 u it 2 ). And its sample counterpart: br 2t = cu it c uit 2 We can obtain br 2t for any t 2 f4; 5; :::; T g. ote that we need T 4. Let r 2 T r 2t, and let br 2 be its sample counterpart. Arellano & Bond (99) prove that under the null hypothesis br 2 is root- asymptotically normal with mean zero, and they derive the ression for the asymptotic variance V ar(br 2 ). Then, under H : r 2 =. bm 2 br 2 se(br 2 ) a (; ) t=4 PROBLEM 2 (3 points). Consider the Binary choice model, Y i = f i + W i + i g where f:g is the indicator function, i is independent of i but it may be correlated with W i. (a) [2 points] Describe the Rivers-Vuong approach to estimate consistently and (up to scale) and to test for the exogeneity of W i. Make the assumptions of this approach licit. ASWER: Consider the model: () Y = f + W + > g (2) W = Z + u where and u are independent of and Z, but cov(; u) 6=, and therefore and W are not independent. Suppose that (; u) are jointly normal. Then, we have that: = u + where (a) = u = 2 u; (b) is normally distribution as (, 2 2 ) where is the correlation between and u; (c) is independent of u; (d) since is independent of and Z, we have that is independent of, Z, and u, and therefore it is independent of W. Then, we can write the probit model: Y = f + W + u + > g And given that is normally distributed and independent of, W, and u, we have that: + W + u Pr(Y = j; W; u) = We do not know u, but we can obtain a consistent estimate of u as the residual ^u = Y Z ^. 3

4 Rivers and Vuong (988) propose the following procedure: Step. Estimate the regression of W on Z and obtain the residual ^u; Step 2. Run a probit for Y on, W and ^u. Using this procedure we obtain consistent estimates of and only if cov(; u) 6=. Therefore, a t-test of H :,, and. ote that 6= if = is a test of the endogeneity of W. (b) [ points] Discuss how to combine the approach by Rivers-Vuong with a Maximum Score approach to obtain a consistent estimator of and that relaxes the parametric assumption in the distribution of i. ASWER: Suppose that is independent of and Z. Let g(u) be the median of conditional on u, and assume that g(:) is a smooth function that is unknown to the researcher. De ne = g(u). Given these assumptions and de nitions, we have that median(j; W; u) =. Therefore, we have the Binary choice model: Y = f + W + g(u) + > g with median(j; W; u) =. More precisely, let b(u) = (u; u 2 ; :::; u q ) be a polynomial basis in u, and let be a vector of parameters associated to the polynomial terms such that g(u) is approximated using b(u). Then, given residuals bu we have the model: Y = f + W + b(bu) + > g The Maximum Score Estimator is the value of (;,) that maximizes the score function: S(; ; ) = n y i x i + w i + b(bu i ) +( y i ) x i + w i + b(bu i ) < PROBLEM 3 (3 points). Consider the Random Utility Model, Y n = arg max j2f;;:::;jg [ j + Z n j + nj ], where n is the index for individuals/observations, and j is the index for choice alternatives. (a) [5 points] Describe the Logit model and the Maximum Likelihood estimator of the parameters of the model. Comment on the properties of this model. ASWER: In the Logit model jn are i.i.d. over (n; j) Type Extreme Value. For any j, we have that the CDF is F ( j ) = f f j gg. Under this assumption on the distribution of, we have the following form for the Conditional Choice Probabilities (CCPs): P j (; Z n ; ) = f j + Z n j g P J i= f i + Z n i g where = (; ; :::; J ) and is normalized to zero. The log-likelihood function is: J fj l () = fy n = jg ln + # Z n j g P J i= f i + Z n i g n= j= 4

5 This log-likelihood function if globally concave in. Furthermore, the gradient and Hessian of this function have simple closed form ressions. Therefore, the numerical computation of the MLE can be implemented in a simple way using ewton s method. In a Logit j = P j [ P j ]. Taking this into account, we can show that the likelihood equations for this model are: ()=@ = : j [fy n = jg P j (; Z n ; )] A = n= j= And for ()=@ j = with j = ; 2; :::; J: Z n [fy n = jg P j (; Z n ; )] = n= Independence of Irrelevant Alternatives. The logit model imposes the restriction that the ratio between the probabilities of two alternatives, say j and i, depends OLY on the utilities of these alternatives, and not on utilities of other alternatives: P jn = f j + Z n j g P in fj + Z n i g Therefore, if we change the choice set, by adding or/and removing alternatives, the ratios between probabilities should not change. This property can generate unrealistic predictions. (b) [5 points] Describe a ested Logit model and a two-step consistent estimator of the parameters of the model. Propose a simple approach to obtain an asymptotically e cient estimator using this two-step estimator. ASWER: The ested Logit was proposed to relax the IIA property of the logit model but keeping its computational convenience. Suppose that the set J = f; ; :::; Jg of choice alternatives is partitioned into G mutually exclusive groups of alternatives, that we index by g. Let J g be the set of alternatives in group g such that: J = [ G J g. The idea is that alternatives within a group g= share some common unobserved features that make them closer substitutes that alternatives in di erent groups. The key assumption is that the vector of unobservables = ( ; ; :::; J ) has a Generalized Extreme Value (GEV) distribution: 8 >< F () = >: G g= j j2j g g 9 g >= >; where,, 2,..., G are positive parameters, with. Consider the RUM Y = arg max j2j fj + Zn j + jn g where n = ( n ; n ; :::; Jn ) has a GEV distribution. The CCPs of this model have the following form: P j (; Z n ; ) = P g () (; Z n ; ) P (2) jjg (; Z n; ) 5

6 with P (2) jjg (; Z n; ) = P () g (; Z n ; ) = and I g;n are the group inclusive values: I g;n = j2j g The likelihood function of the model, l() = two likelihoods: l () () + l (2) () j + Z n j g i + Zn i i2j g g n g o I g;n P n G g = g o I g ;n j + Zn j A g n= ln Pr(Y nj; Z n ; ) can be written as the sum of l() = + G n= g= n= j2j y () n fy () n = gg ln P () g (; Z n ; ) fy (2) n = jg ln P (2) jjy n () (; Z n ; ) where y n () 2 f; 2; :::; Gg represents the observed group-choice of individual n, and y n (2) 2 J () y n represents the observed within group choice of individual n. ote that l() = l () ()+l (2) () where: l () () is the between-group likelihood function for the choice variable y n () ; and l (2) () is the within-group likelihood function for the choice variable y n (2). We can estimate a combination of the parameters in by maximizing l () (), and other combination of parameters by maximizing l (2) (). This two-step procedure is not statistically e cient but it is computationally very convenient because each step consists of a standard ML estimation (i.e., globally concave likelihood function). Step : Maximization of within-group likelihood function l (2) () with probabilities: P jjg;n = f j g + Z n j;g g P i2j g f i g + Z n i;g g where the estimated parameters are: g and j;g j. g g Step 2: Construct the estimated inclusive values: bi g;n = f j g b + Z n b j;g ga j2j g 6

7 And maximization of between-group likelihood function l () () with probabilities: n g I P g;n = b o g;n P n G g = g o The estimated parameters are g, with one of these parameters normalized to zero within each group. Given this consistent two-step estimator, b 2 step, we can construct an e cient estimator, and a valid variance-covariance matrix by doing one ewton or BHHH iteration in the estimation of the full likelihood function: b eff = 2 l( b # 2 step b 2 step ) bi g ;n 7