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Cholesky Decomposition

Generated Regressors The ìtwo-stageî form of the 2SLS estimator is an example of what is called ìestimation with generated regressorsî. We say a regressor is a generated if it is an estimate of an idealized regressor, or if it is a function of estimated parameters. Typically, a generated regressor wb i is an estimate of an unobserved ideal regressor wi . As an estimate, wb i is a function of the sample, not just observation i. Hence it is not ìi.i.d.îas it is dependent across observations, which invalidates the conventional regression assumptions. Consequently, the sampling distribution of regression estimates is a§ected. Unless this is incorporated into our inference methods, covariance matrix estimates and standard errors will be incorrect. The econometric theory of generated regressors was developed by Pagan (1984) for linear models, and extended to non-linear models and more general two-step estimators by Pagan (1986). Independently, similar results were obtained by Murphy and Topel (1985). Here we focus on the linear model: yi = w0 i + vi (12.50) wi = A0zi E (zivi) = 0: The observables are (yi ; zi). We also have an estimate Ab of A. Given Ab we construct the estimate wb i = Ab 0 zi of wi , replace wi in (12.50) with wb i , and then estimate by least-squares, resulting in the estimator b = Xn i=1 wb iwb 0 i !1 Xn i=1 wb iyi ! : (12.51) The regressors wb i are called generated regressors. The properties of b are di§erent than leastsquares with i.i.d. observations, since the generated regressors are themselves estimates. This framework includes the 2SLS estimator as well as other common estimators. The 2SLS model can be written as (12.50) by looking at the reduced form equation (12.14), with wi = 0zi , A = , and Ab = b is (12.19). The examples which motivated Pagan (1984) and Murphy and Topel (1985) emerged from the macroeconomics literature, in particular the work of Barro (1977) which examined the impact of ináation expectations and expectation errors on economic output. For example, let i denote realized ináation and zi be the information available to economic agents. A model of ináation expectations sets wi = E (i jzi) = 0zi and a model of expectation error sets wi = i E (i jzi) = i 0zi . Since expectations and errors are not observed they are replaced in applications with the Ötted values wbi = b 0 zi or residuals wbi = i b 0 zi where b is a coe¢ cient estimate from a regression of i on zi . The generated regressor framework includes all of these examples. The goal is to obtain a distributional approximation for b in order to construct standard errors, conÖdence intervals and conduct tests. Start by substituting equation (12.50) into (12.51). We obtain b = Xn i=1 wb iwb 0 i !1 Xn i=1 wb i w0 i + vi  ! : Next, substitute w0 i = wb 0 i + (wi wb i) 0 . We obtain b = Xn i=1 wb iwb 0 i !1 Xn i=1 wb i (wi wb i) 0 + vi  CHAPTER 12. INSTRUMENTAL VARIABLES 437 E§ectively, this shows that the distribution of b has two random components, one due to the conventional regression component wb ivi , and the second due to the generated regressor (wi wb i) 0 . Conventional variance estimators do not address this second component and thus will be biased. Interestingly, the distribution in (12.52) dramatically simpliÖes in the special case that the ìgenerated regressor termî (wi wb i) 0 disappears. This occurs when the slope coe¢ cients on the generated regressors are zero. To be speciÖc, partition wi = (w1i ; w2i), wb i = (w1i ; wb 2i); and = ( 1 ; 2 ) so that w1i are the conventional observed regressors and wb 2i are the generated regressors. Then (wi wb i) 0 = (w2i wb 2i) 0 2 . Thus if 2 = 0 this term disappears. In this case (12.52) equals b b = Xn i=1 wb iwb 0 i !1 Xn i=1 wb ivi ! : This is a dramatic simpliÖcation. Furthermore, since wb i = Ab 0 zi we can write the estimator as a function of sample moments: p n  b  = Ab 0 1 n Xn i=1 ziz 0 i ! Ab !1 Ab 0 1 p n Xn i=1 zivi ! : If Ab p ! A we Önd from standard manipulations that p n  b  d ! N (0;V ) where V = A0E ziz 0 i  A 1 A0E ziz 0 i v 2 i  A  A0E ziz 0 i  A 1 : (12.53) The conventional asymptotic covariance matrix estimator for b takes the form Vb = 1 n Xn i=1 wb iwb 0 i !1 1 n Xn i=1 wb iwb 0 i vb 2 i ! 1 n Xn i=1 wb iwb 0 i !1 (12.54) where vbi = yi wb 0 i b. Under the given assumptions, Vb p ! V . Thus inference using Vb is asymptotically valid. This is useful when we are interested in tests of 2 = 0 . Often this is of major interest in applications. To test H0 : 2 = 0 we partition b =  b 1 ; b 2  and construct a conventional Wald statistic W = n b 0 2 hVb