| One of the aspects of statistical inference that is very important to researchers in economics and finance is the validity of the assumed distribution of the data under investigation. Unfortunately, the hypothesized distribution suggested by theory is often different from the actual distribution. If we use traditional tests like the Kolmogorov-Smirnov test, non rejection of the null hypothesis that the predicted and observed distributions are the same do provide some confidence in using the estimated model; rejection of such a hypothesis, however, do not provide any indication of the true distribution.; Neyman (1937) observed that all goodness-of-fit tests could be converted to only one type of hypothesis test if a probability integral transform is used with the probability density function (PDF) under the null hypothesis. The main advantage the smooth test has over its predecessors like the Pearson chi-squared test is that under “smooth” or local alternatives (alternatives that are very close to the null), it can identify the nature and sources of departure from the null hypothesis. The smooth test can also be used both as an omnibus test as well as a more directional test for more specific alternatives.; One of the drawbacks of the original smooth test is that it was designed for a one-sample problem with fully specified null distribution, which is not always possible to have in practice. I propose both parametric (for density forecast evaluation) and non-parametric (for comparing two unknown densities) techniques in formulating tests based on the probability integral transforms. In case of parametric applications of density forecast evaluation we have to account for the effect of parameter estimation and dependent data in the implementation of the smooth test. In the non-parametric case of comparing two densities I used the orders of the relative sizes of the two samples to get a consistent test. Monte Carlo simulation of these tests shows good power of size characteristics. I applied the proposed smooth tests to evaluate S&P 500 density forecasts and compare age distribution of insured population in New York. |
