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Inequalities for Vectors

Regression with Expectation Errors In this section we examine a generated regressor model which includes expectation errors in the regression. This is an important class of generated regressor models, and is relatively straightforward to characterize. The model is yi = w0 i + u 0 i + vi wi = A0zi xi = wi + ui E (zi”i) = 0 E (ui”i) = 0 E ziu 0 i  = 0: The observables are (yi ; xi ; zi). This model states that wi is the expectation of xi (or more generally, the projection of xi on zi) and ui is its expectation error. The model allows for exogenous regressors as in the standard IV model if they are listed in wi , xi and zi . This model is used, for example, to decompose the e§ect of expectations from expectation errors. In some cases it is desired to include only the expecation error ui , not the expecation wi . This does not change the results described here. The model is estimated as follows. First, A is estimated by multivariate least-squares of xi on zi , Ab = (Z 0Z) 1 (Z 0X), which yields as by-products the Ötted values Wc = ZAb and residuals Ub = Xc Wc. Second, the coe¢ cients are estimated by least-squares of yi on the Ötted values wb i and residuals ubi yi = wb 0 i b + ub 0 i b + vbi : We now examine the asymptotic distributions of these estimates. By the Örst-step regression Z 0Ub = 0, Wc0 Ub = 0 and W0Ub = 0. This means that b and b can be computed separately. Notice that b =  Wc0Wc 1 Wc0 y and y = W c + U +  W Wc  + v: Substituting, using Wc0 Ub = 0 and W Wc = Z (Z 0Z) 1 Z 0U we Önd b =  Wc0Wc 1 Wc0  U +  W Wc  + v  =  Ab 0 Z 0ZAb 1 Ab 0 Z 0 (U U + v) =  Ab 0 Z 0ZAb 1 CHAPTER 12. INSTRUMENTAL VARIABLES 441 where ei = vi + u 0 i ( ) = yi x 0 i : We also Önd b =  Ub 0 Ub 1 Ub 0 y: Since Ub 0W = 0, U Ub = Z (Z 0Z) 1 Z 0U and Ub 0 Z = 0 then b =  Ub 0 Ub 1 Ub 0  W +  U Ub  + v  =  Ub 0 Ub 1 Ub 0 v: Together, we establish the following distributional result. Theorem 12.12 For the model and estimates described in this section, with E y 4 i  < 1, E kzik 4 < 1, E kxik 4 < 1, A0E (ziz 0 i ) A > 0, and E (uiu 0 i ) > 0, as n ! 1 p n  b b  d ! N (0;V ) (12.60) where V =  V V V V  and V = A0E ziz 0 i  A 1 A0E ziz 0 i e 2 i  A  A0E ziz 0 i  A 1 V = E uiu 0 i 1 E uiz 0 i eivi  A  A0E ziz 0 i  A 1 V = E uiu 0 i 1 E uiu 0 i v 2 i  E uiu 0 i 1 : The asymptotic covariance matrix is estimated by Vb =  1 n Wc0Wc 1 1 n Xn i=1 wb iwb 0 i eb 2 i !  1 n Wc0Wc 1 Vb =  1 n Ub 0 Ub 1 1 n Xn i=1 ubiwb 0 i ebivbi !  1 n Wc0Wc 1 Vb =  1 n Ub 0 Ub 1 1 n Xn i=1 ubiub 0 i vb 2 i !  1 n Ub 0 Ub 1 where wb i = Ab 0 zi ubi = xbi wb i ebi = yi CHAPTER 12. INSTRUMENTAL VARIABLES 442 Under conditional homoskedasticity, speciÖcally E  e 2 i eivi eivi v 2 i  jzi  = C then V = 0 and the coe¢ cient estimates b and b are asymptotically independent. The variance components also simplify to V = A0E ziz 0 i  A 1 E e 2 i  V = E uiu 0 i 1 E v 2 i  : In this case we have the covariance matrix estimators Vb 0 =  1 n Wc0Wc 1 1 n Xn i=1 eb 2 i ! Vb 0 =  1 n Ub 0 Ub 1 1 n Xn i=1 vb 2 i ! and Vb