On some models and tests for heteroscedasticity

Abstract

In the present paper a family of bivariate distributions characterized by standardized symmetric conditional distributions and by a simple scedastic curve (i.e. conditional Variance) is considered. A particular case arises from the first order autoregressive Conditionally Heteroscedastic (ARCH) model of time series analysis; therefore, for this family, inference based on data from simple random sampling and from ARCH models is compared and unified to some extent. Despite the fact that the marginal density is not known is closed form even for typical cases (as conditional Normality), its moments (computed for the general case) show that the marginal distribution has higher kurtosis than the conditional distributions; hence heavy tails of rest data distribution may be disturbing factors in testing heteroscedasticity, especially if one uses a gaussian-score based test. The score based test for independence vs. scedastic regression is shown to be of square correlation type under conditional normal distribution. Moreover, the score based test under Student's t conditional normal distribution is also introduced and proposed as an alternative to the above Normal case test. Although the Student's t test converges to the Normal case for increasing degrees of freedom, it is shown that, using the Normal case test when in fact the conditional is leptokurtic Student's t (e.g. with degrees of freedom between 9 and 12) entails a relevant loss in asymptotic efficiency.