Neyman's Smooth Test and Its Applications in Econometrics

Abstract

The smooth test originally proposed by Neyman (1937) deserves a renewed attention in the context of the current applications in Econometrics. Our paper attempts to put Neyman's smooth test into perspective with the existing literature on goodness-of fit tests and other procedures based on probability integral transforms suggested by early pioneers such as R. A. Fisher (1930, 1932) and Karl Pearson (1933, 1934). 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 briefly reviewed recent advances in different locally most powerful unbiased tests, their differential geometric interpretations as the curvature of the power hypersurface and their relationship with Neyman's smooth test. Finally, we venture into some applications in econometrics by evaluating density forecast estimation and calibrations discussed by Diebold, Gunther and Tay (1998) and others. We reviewed the use of smooth tests in survival analysis by Pena (1998), Gray and Pierce (1985). We also proposed the use of smooth type tests on the 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.