ECONOMETRICS II (ECO 2401S) University of Toronto. Department of Economics. Winter 2014 Instructor: Victor Aguirregabiria

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1 ECONOMETRICS II (ECO 2401S) University of Toronto. Department of Economics. Winter 2014 Instructor: Victor guirregabiria SOLUTION TO FINL EXM Monday, pril 14, From 9:00am-12:00pm (3 hours) INSTRUCTIONS: This is a closed-book exam. No study aids, including calculators, are allowed. The exam consists of ve sets of questions. Try to answer all the questions. TOTL MRKS = 100 PROBLEM 1 (25 points). Consider the Censored Linear Regression Model (CLRM), Y = maxfx + " ; 0g where " is independent of X and it has a pdf f(:) and a cdf F (:) that is strictly monotonically increasing over the real line. (a) [5 points] Obtain the expression of the selection bias function s(x ) E(maxfX+ "; 0g j X) X. NSWER: For notational simplicity, let z X. Then, s(z) = E(maxf"; zg) = Pr(" > z) E(" j " > z) + Pr(" < z) ( z), such that: s(z) = [1 F ( z)] = z F ( z) + Z +1 z Z +1 z f(") " d" + F ( z) ( z) 1 F ( z) " f(")d" (b) [5 points] Show that the selection bias function s(x ) is a strictly decreasing. NSWER: Given the conditions on the CDF F, the sample selection function s(z) is continuously di erentiable. The derivative function of s(z) is: s 0 (z) = F ( z) + z f( z) z f( z) = F ( z) < 0 (c) [5 points] Suppose that this CLRM is such that X = X 1, where X 2 2 f0; 1g is a binary variable. Obtain the expression of the parameter E(Y jx 1 = 1) E(Y jx 1 = 0) in terms of 1 and s(:). NSWER: By de nition of the selection function, we have that E(Y jx) = X + s(x). Then, in this speci c model we have that: = E(Y jx 1 = 1) E(Y jx 1 = 0) = [ s( )] [ 0 + s( 0 )] = 1 + s( ) s( 0 ) 1

2 (d) [5 points] In the model of question (c), show that the OLS estimator of 1 in the linear regression Y = X 1 + e (that ignores the selection problem) is a consistent estimator of. NSWER: The OLS estimator of 1 is P n (x 1i x 1 )y i = P n (x 1i x 1 ) 2, that by the LLN converges in probability to E((X 1 E(X 1 ))Y ) = V (X 1 ). Given that X 1 is binary, we have that: (1) E(X 1 ) = p where p Pr(X 1 = 1); (2) V (X 1 ) = p(1 p); and (3) E((X 1 E(X 1 ))Y ) = p (1 p) E(Y jx 1 = 1) (1 p) p E(Y jx 1 = 0). Therefore, E((X 1 E(X 1 ))Y ) = V (X 1 ) = E(Y jx 1 = 1) E(Y jx 1 = 0) =. (e) [5 points] Using the expressions that you have derived in (c) and (d), obtain the sign of the bias of the OLS estimator of 1 when 1 > 0, and when 1 < 0. NSWER: Given that s(:) is strictly decreasing, we have that: (1) if 1 > 0, then s( ) s( 0 ) < 0 and < 1, i.e., the OLS estimator provides a downward bias estimate of the true 1 ; and (2) if 1 < 0, then s( ) s( 0 ) > 0 and > 1, i.e., the OLS estimator provides an upward bias estimate of the true 1. In both cases we have that jj < j 1 j, i.e., the OLS estimator implies an attenuation bias. PROBLEM 2 (20 points). Let be a vector S 1 parameters, and let be a vector of R 1 parameters, where R > S. Suppose that an econometric model implies the following relationship between these two vector of parameters =, where is a R S matrix of constants that is full column rank. Let b be a root-n consistent and asymptotically normal estimator of, such that p n( b )! d N(0; V ), and let V b be a root-n consistent estimator of V. (a) [5 points] De ne the Minimum Distance Estimator (MDE) of when the restrictions = are equally weighted. NSWER: b EMD = arg min h b i 0 h b i This a quadratic criterion function that is globally convex in. The estimator is characterized by rst order conditions of optimality 2 0 h b b EMD i = 0, and solving for b EMD we get: b EMD = 0 1 0b (b) [5 points] De ne the MDE of when the restrictions are optimally weighted. NSWER: h i 0 h b EMD = arg min b bv 1 b i 2

3 This a quadratic criterion function that is globally h convex ini. The estimator is characterized by rst order conditions of optimality 2 0 V b 1 b b OMD = 0, and solving for b OMD we get: h i 1 b OMD = 0 V b 1 0 V b 1 b (c) [10 points] Derive the asymptotic variance of the estimator of, both for the Equally Weighted and for the Optimally Weighted MDE. NSWER: For the EMD p n( b EMD ) = p n 0 1 0b p n = p n 0 1 0b p n = p n( b ) Therefore, such that p n( b EMD )! d N(0; V 0 1 ) var( b EMD ) = V 0 1 For the OMD p n( b OMD ) = p i 1 n h 0 V b 1 0 V b 1 b p i 1 n h 0 V b 1 0 V b 1 = p h i 1 n 0 V b 1 0 V b 1 b p i 1 n h 0 V b 1 0 V b 1 h i 1 = 0 V b 1 0 V b 1p n( b ) Therefore, by the consistency of b V 1 (Slutsky s theorem): p i 1 h i 1) n( b OMD )! d N(0; h 0 V 1 0 V 0 V 1 h i 1) = N(0; 0 V 1 such that var( b OMD ) = h 0 V 1 i 1 PROBLEM 3 (30 points). Consider a random coe cients multinomial choice model with J +1 choice alternatives f0; 1; :::; Jg. The "utility" of alternative j = 0 is normalized to zero. The utility of alternative choice j > 0 for individual i is U ij = + i Z j + " ij, where: is a parameter; Z j is an observable attribute of alternative j; i is a random coe cient with that is i.i.d. over individuals with a normal distribution N( ; 2 ); and " ij is an unobservable that is i.i.d. over i and over j with an extreme value type 1 distribution. The researcher observes product attributes fz 1, Z 2,..., Z J g and a random 3

4 sample of n individuals with their optimal choices fy i : i = 1; 2; :::; ng with y i 2 f0; 1; :::; Jg. We are interested in using this sample to estimate the parameters of the = f; ; g. (a) [5 points] Write the expression of the choice probability P j () Pr(Y = jj) in terms of the primitives of the model. NSWER: We can write i = + v i, where v i is iid standard normal with pdf (:). Then: Z P j () = exp + + v i Zj 1 + P J k=1 exp + (v i ) dv i + v i Zk (b) [5 points] Write the log-likelihood function of this model and data. NSWER: l() = = nx 1fy i = jg ln P j () n j ln P j () where n j P n 1fy i = jg. (c) [5 points] Show that if we do not impose the model restrictions on the choice probabilities, the MLE of P j is the frequency estimator. NSWER: Without restrictions on the choice probabilities, we have a multinomial model with J free parameters/probabilities P = fp j : j = 1; 2; :::; Jg. The log likelihood function is: l(p) = = nx 1fy i = jg ln P j + 1fy i = 0g ln(1 P 1 ::: P J ) j=1 n j ln P j + n 0 ln(1 P 1 ::: P J ) j=1 The likelihood equations are: l(p) P j = n j bp j n 0 bp 0 = 0 Or n j b P0 = n 0 b Pj. Summing this expression over j > 0, we have that (n n 0 ) b P 0 = n 0 (1 b P0 ), and solving for b P 0, we get b P 0 = n 0 =n. Plugging this expression into the likelihood equations we get that b P j = n j =n, and this is the frequency estimator. (d) [5 points] Propose a simulator of the choice probability P j (). De ne the simulated log-likelihood function of this model, and the simulated MLE. 4

5 NSWER: Let fv r : r = 1; 2; :::Rg be R independent random draws from a standard normal distribution. Then, the simulator of P j () is: ep R j () = 1 R RX exp + + v r Z j 1 + P J k=1 exp + + v r Z k The simulated log-likelihood function is: e l R () = n j ln e P R j () The simulated MLE is the value of that maximizes e l R (). (e) [5 points] Obtain the simulated likelihood equations with respect to ; ; and. NSWER: The likelihood equations are P J n j e P R j () ep R j 1 () = 0, where e P R j () = e P R j (), P e j R(), P e j R()! 0 and with P e j R() = 1 P R R j(v r ; ) [1 j (v r ; )] P e j R() = 1 P R R j(v r P ; )[Z J j k=1 Z k k (v r ; )] P e j R() = 1 P R R vr j (v r P ; )[Z J j k=1 Z k k (v r ; )] j (v r exp + + v r Z j ; ) 1 + P J k=1 exp + + v r Z k (f) [5 points] Describe a simulated-based version of the BHHH algorithm to compute the simulated MLE. NSWER: We start with an initial vector b 0. Then, we generate the recursively the sequence f b K : K 1g where at iteration K 1, we obtain update b K using the formula: " nx b K = b K 1 + e li R(b K 1 ) e # 1 " li R nx (b K 1 ) e # l R 0 i (b K 1 ) where e l R i () is the simulated score based with e l R i () = 1fy i = jg ln P e j R () 5

6 such that e l R i () = 1fy i = jg e P R j () 1 ep R j () PROBLEM 4 (5 points). Consider the single-equation econometric model Y = g(x; ";) where g is a known real valued function, X is a vector of observable explanatory variables, " is a vector of unobservables, and is a vector of unknown parameters. Function g is continuous in all its arguments and monotonic in ", though may not be strictly monotonic. The unobservables " are independent of X and have a continuous and strictly increasing CDF over the Euclidean space. The distribution of " is unknown to the researcher, i.e., the model is semiparametric. Let fy i ; x i : i = 1; 2; :::; ng be a random sample of Y and X. (a) [5 points] Propose a consistent estimator of. NSWER: Under the assumption that the unobservables have medians independent of X, the Least bsolute Deviations (LD) estimator provides a consistent estimator of Y = g(x; ";) in a model with non-additively-separable unobservables such as Y = g(x; ";). This estimator is de ned as: nx b LD = arg min jy i g(x i ; 0;)j PROBLEM 5 (15 points). Consider the static linear panel data model Y it = i + i X it +u it, where X it is a explanatory variable that is strictly exogenous with respect to u it. (a) [5 points] Suppose that i and X it are independently distributed, but i and X it are not independent. The researcher does not want to make any assumption about the joint distribution of i and X it. Propose an estimator of E( i ) that is consistent as N! 1 and T is xed. NSWER: De ne the random variable v i i, that by construction has zero mean, and by assumption is independent of X it. We can write the model as Y it = i + X it +e it, where e it u it + v i X it. Note that for any two periods t and s we have that: E(X it e is ) = E (X it [u is + v i X is ]) = E (X it u is ) + E (v i X it X is ) = = 0 Therefore, X it is strictly exogenous with respect to e it in the model Y it = i + X it +e it. Under this condition, we know that the OLS estimator in the rst di erences transformed equation, or the OLS estimator in the within-groups transformed equation are consistent as N! 1 and T is xed. 6

7 (b) [5 points] Prove that the estimator proposed in (a) is consistent. NSWER: For instance, in the rst di erences transformed equation, Y it Y it 1 = ( ) + (e it e it 1 ) and E([ ] [e it e it 1 ]) = 0. Under this condition, and given that there are not incidental parameters, the OLS estimator is consistent. (c) [5 points] Suppose that both i and i are NOT independently distributed of X it. The researcher does not want to make any assumption on the joint distribution of ( i ; i ) and X it. Propose an estimator of E( i ) that is consistent as N! 1 and T is xed, and prove its consistency. NSWER: OLS in rst di erence or OLS in within-groups transformed model are inconsistent because now the error term e it u it + v i X it is correlated with X it. However, we can consider the following transformation of the model. First, we take rst di erences: Y it Y it 1 = i ( ) + (u it u it 1 ) nd provided that ( ) 6= 0 (this has zero probability mass if X it is a continuous random variable) we can divide right-hand-side and left-hand-side by ( ) to get: or Y it Y it 1 = i + u it u it 1 Y it Y it 1 = + it where it v i + u it u it 1. It is clear that E( it ) = 0. Therefore, the sample mean of Y it Y it 1 is a consistent estimator of as N! 1 and T is xed, and prove its consistency. For instance, suppose that T = 2: b = 1 N NX Yi2 It is clear that by LLN b converges in probability to E Yi2 Y i1 X i2 X i1 = E( + i2 ) =. X i2 Y i1 X i1 7