Importance of Non-parametric Density Estimation in Econometrics with Illustrations

Econometrics deals with joint and conditional probability density functions and the mean value of the latter, the regression equation. When these densities take nonparametric form we get rich results. In this paper we retrace the work we did in mid seventies and recast them in today's perspectives. We also raise a few very interesting research issues that could be of some interest for newcomers to this field. Even before Efron wrote his first influential paper on bootstraps the senior author used simulated data generated from a Monte Carlo exercise and estimated the entire sampling distribution of estimators whose small sample distributions were not known and used them for statistical inference. One important result the authors showed then was that such an empirically estimated sampling distribution could approximate reasonably well the exact sampling distribution and its asymptotic approximation in known case derived by Anderson and Sawa in a two endogenous equations case. This raised hopes that the method should be useful in other cases when we do not know the exact small sample distributions of econometric estimators, setting the stage for bootstraps use in econometrics. One question that is often raised against the nonparametric inference is how one would formulate null hypotheses and test them. This issue is addressed in this paper and a suggestion is offered. Another issue that is addressed in this paper is the rate of convergence of different nonparametric density estimators to their true distribution.