Asymmetric stable Paretian distribution testing

Abstract

Two new tests for the symmetric stable Paretian distribution with tail index 1 α 2 are proposed. The test statistics and their associated approximate p-values are instantly computed and do not require use of the stable density or distribution or maximum likelihood estimation. They exhibit high power against a variety of alternatives, and much higher power than the existing test based on the empirical characteristic function. The two tests are combined to yield a test that has substantially higher power. A fourth test based on likelihood ratio is also studied. Extensions are proposed to address the asymmetric case and are shown to have reasonable actual size properties and high power against several viable alternatives.

Stable-GARCH Models for Financial Returns: Fast Estimation and Tests for Stability

“…For the former, the point estimates of α do not vary much with respect to the aggregation value s, while those for the latter tend to increase. The τ 0 test was augmented by Paolella [34] in two ways. The first issue to realize is that the test statistic τ 0 (and thus the hypothesis test outcome) is not invariant to permutations of the data (though observe thatα Hint is invariant).…”

“…Paolella [34] shows that the τ 20 test is relatively robust (in terms of actual size under the null) when used with mildly asymmetric stable Paretian data (as is typical for financial asset returns), whereas the following test, ALHADI, is not, and so requires modification to be used with asymmetric data, as discussed in Section 4.4 below.…”

“…When applied to data that are not stable Paretian, these estimates could differ substantially. Paolella [34] formalized this, and developed the α-Hat Discrepancy Test, or, in short, ALHADI, with test statistic…”

“…The method to do this is discussed in Paolella [34], where it is also shown that the resulting test, termed A`τ 20 , often has substantially higher power than either of the two constituent tests. We will see below in the empirical application to financial returns data that this test more often rejects the null of stability than does the τ 20 test.…”

“…This procedure is numerically feasible using the fast MLE, cdf, and inverse cdf routines provided in Nolan's toolbox, and clearly asymptotically valid, though in finite samples, and in light of the relative difficulty of estimating β accurately, can result in size distortions, depending on the nature of the test. Paolella [34] demonstrates that, particularly for the A`τ 20 test, the actual size properties are quite reasonable for mild to moderate asymmetry (as appropriate for financial asset returns), but as |β| Ñ 1, the test becomes conservative, i.e., the actual size is lower than the nominal.…”