Suggested Solution for PS #5

Transcription

1 Cornell University Department of Economics Econ 62 Spring 28 TA: Jae Ho Yun Suggested Solution for S #5. (Measurement Error, IV) (a) This is a measurement error problem. y i x i + t i + " i t i t i + i The problem is that " i and i might be correlated. If this is the case, we cannot obtain consistent estimators by OLS, which is called endogeneity problem. (b) Get bt i by regressing t i on z. Then, get b from estimating the model, y i x i + bt i + " i: (c) We can use the Wu-Hasman test. Under the null hypothesis that there is no endogeneity, p N( b OLS b 2SLS ) follows normal distribution asymptotically. This is equivalent to testing a signi cance of e in the following eqatation. y i x i + t i + e t i + " i The test-statistics is t-test for e : 2. (robit Model) (a) We can obtain NLS estimator by solving the following problem: Arg min (d i (x i )) 2 : i The corresponding First Order Condition (or Normal equations) is as follows: 2 (d i (x i ))(x i )x i ; i where (x i ) is a normal pdf. The resulting NLS estimator is biased due to its nonlinear feature, but consistent as discussed in the class. It is not the most e cient estimator, since MLE can achieve Cramer-Rao bound.

2 (b) Conditional variance of d i is (x i ) [ (x i )] ; which indicates conditional heteroskedasticity. (c) Yes, we can improve the e ciency, since we take into account its conditional variance strucutre. Think of GLS First, estimate e as shown in part (a). Then, solve the following problem: Arg min i (d i (x i )) 2 (x i e ) (x i e ) : (d) For MLE, construct the loglikelihood function as follows: 3. (2SLS) l() [d i log (x i ) + ( d i ) log( (x i )] : i By FOC (score function), we can get the ML estimator: i d i (x i ) (x i ) (a) First, get b Y 2 as follows: (x i ) ( d i ) x i : (x i ) by 2 X(X X) X Y 2 X Y 2 : Then, we can obtain 2SLS estimator by b (Z X Z) Z X y : (b) See the following equations showing that b is same as b : b (Z X Z) Z X by (Z X Z) Z X X y (Z X Z) Z X y b : 4. (IV Estimation) y X + Y + where X is k N and Y is G N and X is k N: 2

3 (a) In order to identify ; k should be at least, k + G: (b) OLS estimator: b (Z Z) Z y (Z Z) Z (Z + ) ( b ) (Z Z) Z Applying the asymptotic theory, we will have the following. p N( b d Z ) N ; 2 Z p lim N Or you can just say that the (approximated) asymptotic variance of b is 2 (Z Z) : (c) IV estimator: b IV (Z MZ) Z My where M X(X X) X Asymptotic Variance of IV estimator: b IV (Z MZ) Z My (Z MZ) Z M(Z + ) b IV (Z MZ) Z M Using the similar technique, we have: p N( b IV ) d Z N ; 2 MZ p lim N Or you can just say that the (approximated) asymptotic variance of b IV is 2 (Z MZ) : (d) Asymptotic Variance of b b IV : b b IV f + (Z Z) Z g f + (Z MZ) Z Mg Note that E( b b IV ) : (Z Z) Z (Z MZ) Z M 3

4 V ar( b b IV ) E[( b b IV )( b b IV ) ] E[f(Z Z) Z (Z MZ) Z Mgf(Z Z) Z (Z MZ) Z Mg ] E[(Z Z) Z Z(Z Z) (Z MZ) Z M Z(Z Z) (Z Z) Z MZ(Z MZ) + (Z MZ) Z MMZ(Z MZ) 2 [(Z Z) (Z Z) (Z Z) + (Z MZ) ] 2 [(Z MZ) (Z Z) ] (e) The rank of the covariance matrix: The covariance matrix is 2 [(Z MZ) (Z Z) ]: The rank of this matrix is same as that of (Z Z) (Z MZ); since pre- or postmultiplication of some matrix (in our case, [(Z MZ) (Z Z) ]) by a nonsingular matrix does not change its rank. remultiply the covariance matrix by (Z MZ) and postmultiply it by (Z Z):Then, we are end up with (Z Z) (Z MZ): (Z Z) (Z MZ) Z (I M)Z X Y I M X Y X I M X X I M Y Y I M X Y I M Y Note that I M X Y I M Y Since Y I M Y has a full rank, the rank of the covariance matrix is G. (f) The asymptotic covariance between b and b b IV : Cov( b ; b b IV ) E[( b )( b b IV ) ] E[f(Z Z) Z gf(z Z) Z (Z MZ) Z Mg ] E[(Z Z) Z Z(Z Z) (Z Z) Z MZ(Z MZ) ] 2 [(Z Z) (Z Z) ] : 5. (Order Condition in SEM) Note that for identi cation of equation, we should have K K +G : (a) f(x) + x: Here, we have that K 2;(Strictly, the rank of X is two) since f(x) is a linear combination of explanatory variables 4

5 in equation. However, K + G 3: Therefore, this is not identi ed. (b) f(x) + x + x 2 : Now, K 3: This is just-identi ed case. (c) f(x) + exp(x): Now, K 3: This is also just-identi ed case. 6. (K-variable Regression) (a) Matrix X, will look like the following. X ; C A 2.. Using the usual way of getting the variances of LS estimator, we have; V ar( b OLS ) (X X) X nodd X where n n odd + : Note that, if n is even, n odd and if n is odd, n odd +: Therefore, V ar( b OLS ) n n 2 (b) We are trying to solve the following equations. (n odd ) X odd(y i x i ) ( ) X even(y i x i ) Then, because of the strange(?) feature of X matrix, we have; (n odd ) X odd(y i ) ( ) X even(y i 2 ) Therefore, 5

6 2 odd y i n odd even y i Next, let s gure out the covariance matrix of the above estimator. odd yi n odd even yi odd ( +"i) n odd even ( 2 +"i) 2 + odd "i n odd even "i Hence, V ar( ) E " odd "i n odd even "i n odd odd "i n odd : even "i # The variance of the OLS estimator is (X X). This is same as the above. 6