Neyman's Smooth Test and its Applications in Econometrics

Abstract

The following essay is a reappraisal of the role of the smooth test proposed by Neyman (1937) in the context of current applications in econometrics. We revisit the derivation of the smooth test and put it into the perspective of the existing literature on tests based on probability integral transforms suggested by early pioneers such as R.A.Fisher (1930, 1932) and Karl Pearson (1933, 1934) and the other tests for goodness-of-fit. Our discussion touches data-driven and other methods of testing and inference on the order of the smooth test and the motivation and choice of orthogonal polynomials used by Neyman and others. We review other locally most powerful unbiased tests and look at their differential geometric interpretations in terms of Gaussian curvature of the power hypersurface and review some recent advances. Finally, we venture into some applications in econometrics by evaluating density forecast calibrations discussed by Diebold, Gunther and Tay (1998) and others. We discuss the use of smooth tests in survival analysis as done by Pena (1998), Gray and Pierce (1985) and in tests based on p-values and other probability integral transforms suggested in Meng (1994). Uses in diagnostic analysis of stochastic volatility models are also mentioned. Along with our narrative of the smooth test and its various applications, we also provide some historical anecdotes and sidelights that we think interesting and instructive.

Keywords: smooth test, goodness-of-fit tests, probability integral transform, unbiased test, score test, density forecast evaluation, calibration, orthogonal polynomials, predictive density, data-driven methods

JEL Classification: C12, C52, C53