Generalized runs tests for heteroscedastic time series

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Abstract
The problem of testing for nonhomogeneous white noise (i.e., independently but possibly nonidentically distributed observations, with a common, specified or unspecified, median) against alternatives of serial dependence is considered. This problem includes as a particular case the important problem of testing for heteroscedastic white noise. When the value of the common median is specified, invariance arguments suggest basing this test on a generalized version of classical runs: the generalized runs statistics. These statistics yield a run-based correlogram concept with exact (under the hypothesis of nonhomogeneous white noise) p-values. A run-based portmanteau test is also provided. The local powers and asymptotic relative efficiencies (AREs) of run-based correlograms and the corresponding run-based tests with respect to their traditional parametric counterparts (based on classical correlograms) are investigated and explicitly computed. In practice, however, the value of the exact median of the observations is seldom specified. For such situations, we propose two different solutions. The first solution is based on the classical idea of replacing the unknown median by its empirical counterpart, yielding aligned runs statistics. The asymptotic equivalence between exact