Nonlinear and Nonparametric Regression and Instrumental Variables

We consider regression when the predictor is measured with error and an instrumental variable is available. The regression function can be modeled linearly, nonlinearly, or nonparametrically. Our major new result shows that the regression function and all parameters in the measurement error model are identified under relatively weak conditions, much weaker than previously known to imply identifiability. In addition, we develop an apparently new characterization of the instrumental variable estimator: it is in fact a classical ”correction for attenuation” method based on a particular estimate of the variance of the measurement error. This estimate of the measurement error variance allows us to construct functional nonparametric regression estimates, by which we mean that no assumptions are made about the distribution of the unobserved predictor. The general identifiability results also allow us to construct structural methods of estimation under parametric assumptions on the distribution of the unobserved predictor. The functional method uses SIMEX and the structural method uses Bayesian computing machinery. The Bayesian estimator is found to significantly outperform the functional approach.