Econometrica: May, 1991, Volume 59, Issue 3
Best Nonlinear Three-Stage Least Squares Estimation of Certain Econometric Models
P. M. Robinson
A method is presented for the estimation of nonlinear simultaneous equations and transformation models in the presence of disturbance distribution of unknown form. For a given, finite, set of equations, the instrument achieving the lower variance bound for instrumental variables estimates is a nonparametric regression function. We approximate this without use of smoothed nonparametric estimation: our feasible optimal instruments average over the unsmoothed empirical distribution of preliminary residuals. We justify the method under stationary serial dependence. Somewhat different estimates are proposed in each of three settings: independent disturbances and strongly exogenous but possibly serially dependent explanatory variables; independent disturbances and explanatory variables that include lagged response variables; parametrically autocorrelated disturbances and strongly exogenous but possibly serially dependent explanatory variables. In each case large-sample inference rules are justified. The limiting normal distribution of the estimates is found to be unaffected if the optimal instruments are based on only an arbitrarily small, vanishing fraction of the residuals. In the setting of independent observations we investigate theoretically the effect of such computational savings on the goodness of the normal approximation, obtaining a rate of uniform convergence to normality.