1 Instrumental Variables Estimation with Flexible Distributions* Christian Hansen Graduate School of Business University of Chicago James B. McDonald Brigham Young University Department of Economics Whitney K. Newey Department of Economics M.I.T. October 2, 2006 Abstract Instrumental variables are often associated with low estimator precision. This paper explores efficiency gains which might be achievable using moment conditions which are nonlinear in the disturbances and are based on flexible parametric families for error distributions. We show that these estimators can achieve the semiparametric efficiency bound when the true error distribution is a member of the parametric family. Monte Carlo simulations demonstrate low efficiency loss in the case of normal error distributions and potentially significant efficiency improvements in the case of thick-tailed and/or skewed error distributions. Research assistance provided by Brigham Frandsen, Samuel Dastrup, and Randall Lewis is gratefully appreciated.21. Introduction Instrumental variables (IV) estimation is important in economics. A common finding is that the precision of IV estimators is low. This paper explores potential efficiency gains that might result from using moment conditions that are nonlinear in the disturbances. It is known that this approach can produce large efficiency gains in regression models. The hope is that such efficiency gains might also be present when models are estimated by IV. These gains could help in overcoming the low efficiency of IV estimators. A simple approach to improving efficiency in IV estimation based on nonlinear functions of the residuals is to use flexible parametric families of disturbance distributions. This approach has proven useful in a variety of settings. For example, McDonald and Newey (1988) present a generalized t distribution which can be used to obtain partially adaptive estimators of regression parameters. McDonald and White (1993) use the generalized t and an exponential generalized beta distribution to show substantial efficiency gains can be obtained from partially adaptive estimators in applications characterized by skewed and/or thick tailed error distributions. Hansen, McDonald, and Theodossiou (2005) consider some additional partially adaptive regression estimators and find similar efficiency gains. Here we follow an iterative approach to estimation with flexible distributions. We use residuals from a preliminary IV estimator to estimate the location, shape, and scale parameters for a density. We do this by quasi maximum likelihood on the residual distribution although other ways to estimate the parameters could be used. The product of the instrumental variables and the location score for the density, evaluated at the estimated distributional parameters, is then used to form moment conditions for nonlinear IV estimation. We give consistency and asymptotic normality results for the estimator of the structural parameters. We also show that the3asymptotic variance of the structural slope estimator does not depend on the estimator of the distributional parameters. To help motivate the form of our estimator we derive the semiparametric efficiency bound for the structural slope estimators when the disturbance is independent of the instruments and the reduced form is unrestricted. This bound depends on the marginal distribution of the error and on the conditional expectation of the endogenous variable. When the reduced form for the endogenous right-hand side variables happens to be linear and additively separable in an independent disturbance, our nonlinear IV estimator achieves the semiparametric bound when the true distribution is included in the parametric family. Thus, the estimator has a "local" efficiency property, attaining the semiparametric bound is some cases. To evaluate efficiency gains in practice we consider several empirical examples and carry out some Monte Carlo work. The empirical applications are taken from Card (1995) and Angrist and Krueger (1991). We find that there can be substantial efficiency gains in estimation from using more flexible distributions. We also find evidence of potentially large efficiency gains in the Monte Carlo work. Previous work on IV estimation with nonlinear functions of the residuals includes Newey(1990a, 1990b), Chernozhukov and Hansen (2005), and Honore and Hu (2004). Newey (1990a, 1990b) considers efficiency in nonlinear simultaneous equations with disturbances independent of instruments, which specializes to the case considered here. Chernozhukov and Hansen (2005) consider IV estimation where the residual function corresponds to regression quantiles. Honore and Hu (2004) also consider estimation based on residual ranks. Section 2 of the paper introduces the model and estimators. The flexible distributions we consider are described in Section 3. Section 4 gives the asymptotic theory, including thesemiparametric variance bound. Section 5 reports results from the empirical applications with the results from the Monte Carlo simulations included in Section 6. Section 7 concludes. 2. The Model and Estimators The model we consider is a regression model with a disturbance that is independent of instruments. This model takes the form 0[ ] 0, ( ), and independent.ii i i i ii iyX E ZZz zβεε ε′=+ = = (1) where yi is a left-hand side endogenous variable, Xi is a p×1 vector of right-hand side variables, β is a p×1 parameter vector, iε is a scalar disturbance, and Zi is an m×1 vector of instrumental variables that is a function of variables zi that are independent of the disturbance. We will assume throughout that the first element of Xi and of Zi is 1, so that the mean zero restriction is just a normalization. The nonlinear instrumental variables estimators (NLIV) we consider are based on a parametric family of pdf's. Let ),(γεf denote a member of this family with parameter vector γ. This parameter vector may include location, scale, and shape parameters. In keeping with the normalization adopted above we will restrict the parameters so that the f(ɛ,γ) has mean zero. Also, let ()(),ln,/f .ρεγ εγ ε=∂∂ If Xi were exogenous we could form an estimator of the parameters by maximizing ),~,(ln1∑=′−niiiXyfγβ where γ~is a preliminary estimator of γ. This estimator has a first-order condition (10niiiiXyX),ρβγ=′=−∑. We generalize this