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# Nonparametric Instrumental Variables Regression

Finite Sample Theory In Chapter 5 we reviewed the rich exact distribution available for the linear regression model under the assumption of normal innovations. There was a similarly rich literature in econometrics which developed a distribution theory for IV, 2SLS and LIML estimators. An excellent review of the theory, mostly developed in the 1970s and early 1980s, is reviewed by Peter Phillips (1983). This theory was developed under the assumption that the structural error vector e and reduced form error u2 are multivariate normally distributed. Even though the errors are normal, IV-type estimators are are non-linear functions of these errors and are thus the estimators non-normally distributed. Formulae for the exact distributions have been derived, but are unfortunately functions of model parameters and hence are not directly useful for Önite sample inference. One important implication of this literature is that it is quite clear that even in this optimal context of exact normal innovations, the Önite sample distributions of the IV estimators are nonnormal and the Önite sample distributions of test statistics are not chi-squared. The normal and chisquared approximations hold asymptotically, but there is no reason to expect these approximations to be accurate in Önite samples. A second important result is that under the assumption of normal errors, most of the estimators do not have Önite moments in any Önite sample. A clean statement concerning the existence of moments for the 2SLS estimator was obtained by Kinal (1980) for the case of joint normality. Let b 2sls;2 be the 2SLS estimators of the coe¢ cients on the endogeneous regressors. Theorem 12.7 If (yi ; xi ; zi) are jointly normal, then for any r, E b 2sls;2 r < 1 if and only if r < `2 k2 + 1: This result states that in the just-identiÖed case the IV estimator does not have any Önite order integer moments. In the over-identiÖed case the number of Önite moments corresponds to the number of overidentifying restrictions (`2 k2). Thus if there is one over-identifying restriction the 2SLS estimator has a Önite mean, and if there are two over-identifying restrictions then the 2SLS estimator has a Önite variance. The LIML estimator has a more severe moment problem, as it has no Önite integer moments (Mariano, 1982) regardless of the number of over-identifying restrictions. Due to this lack of moments, Fuller (1977) proposed the following modiÖcation of LIML. Instead of (12.39), Fullerís estimator is b Fuller = X0 (In KMZ) X 1 X0 (In KMZ) y  K = b CHAPTER 12. INSTRUMENTAL VARIABLES 432 for some C  1. Fuller showed that his estimator has all moments Önite under suitable conditions. Hausman, Newey, Woutersen, Chao and Swanson (2012) propose an estimator they call HFUL which combines the ideas of JIVE and Fuller which has excellent Önite sample properties