Functional-Coefficient Instrumental Variables Models

                              ZONGWU CAI
                         Department of Mathematics
                         University of North Carolina
                         Charlotte, NC 28223
                         E-mail: zcai@uncc.edu

    Newey and Powell (1988) showed that a nonparametric model with continuously distributed
    endogenous regressions is ill posed, raising difficulties for consistent estimation.
    There is no distribution theory in that work. To overcome this shortcoming, in this
    article, we propose a new class of flexible instrumental variables  models that are
    linear in the endogenous variables but the coefficients are allowed to change over some
    instrumental variables, which generalize the classical instrumental variables regression
    models (a system of linear simultaneous equations). Although our model has a simpler form
    than that in Newey and Powell (1988), the required conditional mean restriction in our
    model is weaker. It is of interest to estimate the coefficient functions. We develop a
    two-stage approach to estimate the coefficient functions by using local linear techniques.
    The first step comprises the nonparametric estimation of the reduced form and the second
    step is the estimation of the primary equation via a functional-coefficient regression
    technique with the estimated reduced form. We derive consistency and asymptotic normality
    results for our estimators, including the optimal nonparametric convergence rates.
    In particular, the asymptotic variance consists of three terms: the first term is for
    the variation of measurement error, the second term addresses the variability of the
    estimated reduced form at the first step, and the third term accounts correctly for the
    asymptotic covariance between the first and second step. This is different from other
    instrumental variable estimators (see, e.g., Newey, Powell and Vella, 1999) and this
    finding seems to be novel in the literature. This is a joint work with Mitali Das,
    Department of Economics, Columbia University, E-mail: mdas@latte.harvard.edu.