Aurobindo Ghosh

Neyman's Smooth Test and Its Applications in Econometrics

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 Neymans 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 Neymans 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.

A Smooth Test for the Equality of Distributions

The two-sample version of the celebrated Pearson goodness-of-fit problem has been a topic of extensive research, and several tests like the Kolmogorov-Smirnov and Cramer-von Mises have been suggested. Although these tests perform fairly well as omnibus tests for comparing two probability density functions (PDFs), they may have poor power against specific departures such as in location, scale, skewness, and kurtosis. We propose a new test for the equality of two PDFs based on a modified version of the Neyman smooth test using empirical distribution functions minimizing size distortion in finite samples. The suggested test can detect the specific directions of departure from the null hypothesis. Specifically, it can identify deviations in the directions of mean, variance, skewness, or tail behavior. In a finite sample, the actual probability of type-I error depends on the relative sizes of the two samples. We propose two different approaches to deal with this problem and show that, under appropriate conditions, the proposed tests are asymptotically distributed as chi-squared. We also study the finite sample size and power properties of our proposed test. As an application of our procedure, we compare the age distributions of employees with small employers in New York and Pennsylvania with group insurance before and after the enactment of the “community rating” legislation in New York. It has been conventional wisdom that if community rating is enforced (where the group health insurance premium does not depend on age or any other physical characteristics of the insured), then the insurance market will collapse, since only older or less healthy patients would prefer group insurance. We find that there are significant changes in the age distribution in the population in New York owing mainly to a shift in location and scale.

A Smooth Test for Density Forecast Evaluation

Recently financial econometricians have shifted their attention from point and interval forecasts to density forecasts mainly to address the issue of the huge loss of information that results from depicting portfolio risk by a measure of dispersion alone. One of the major problems in this area has been the evaluation of the quality of different density forecasts. In this paper I propose an analytical test for density forecast evaluation using the Smooth Test procedure for both independent and serially dependent data. Apart from indicating the acceptance or rejection of the hypothesized model, this approach provides specific sources (such as the mean, variance, skewness and kurtosis or the location, scale and shape of the distribution or types of dependence) of departure, thereby helping in deciding possible modifications of the assumed forecast model. I also address the issue of where to split the sample into in-sample (estimation sample) and out-of-sample (testing sample) observations in order to evaluate the A¢â‚¬A“goodness-of-fitA¢â‚¬? of the forecasting model both analytically, as well as through simulation exercises. Monte Carlo studies revealed that the proposed test has good size and power properties. I also further investigate applications to value weighted S&P 500 returns that initially indicates that introduction of a conditional heteroscedasticity model significantly improve the model over one with constant conditional variance. The simplicity of the proposed A¢â‚¬A“parametricA¢â‚¬? test based on the classical score test should also appeal the practitioners