Rank-based tests for randomness against first-order serial dependence

Optimal rank-based procedures were derived in Hallin, Ingenbleek, and Puri (1985, 1987) and Hallin and Puri (1988) for some fundamental testing problems arising in time series analysis. The optimality properties of these procedures are of an asymptotic nature, however, whereas much of the attractiveness of rank-based methods lies in their small-sample applicability and robustness features. Accordingly, the objective of this article is twofold: (a) a study of the finite-sample behavior of the asymptotically optimal tests for randomness against first-order autoregressive moving average dependence proposed in Hallin et al. (1985), both under the null hypothesis (tables of critical values) and under alternatives of serial dependence (evaluation of the power function), and (b) an (heuristic) investigation of the robustness properties of the proposed procedures (with emphasis on the identification problem in the presence of “outliers”). We begin (Sec. 2) with a brief description of the rank-based measures of serial dependence to be considered throughout: (a) Van der Waerden, (b) Wilcoxon, (c) Laplace, and (d) Spearman—Wald—Wolfowitz autocorrelations. The article is mainly concerned with first-order (lag 1) coefficients of these types. Tables of the critical values required for performing tests of randomness are provided (Sec. 3), and the finite-sample power of the resulting tests is compared with that of their parametric competitors (Sec. 4). Although the exact level of classical parametric procedures is only approximately correct (whereas the distribution-free rank tests are of the correct size), the proposed rank-based tests compare quite favorably with the classical ones, and appear to perform at least as well as (often strictly better than) their classical counterparts. The examples of Section 5 emphasize the robustness properties of rank-based tests with respect to departures from modeling assumptions, outliers, and gross errors (Secs. 5.1 and 5.3), as well as their insensitivity to spurious end effects (Sec. 5.2). Discrepancies between rank-based and the usual parametric tests also may provide an indication that an intervention analysis should be considered (Sec. 5.4), and rank-based correlograms may detect serial dependence in series where standard methods fail to do so (Sec. 5.5). These examples show how classical Gaussian methods that take normal white noise for granted can yield misleading diagnostic information—spurious autocorrelation or failure to detect significant serial dependencies—when the data have outliers, atypical start-up behavior, and so on. Rank-based tests exhibit much better resistance to aberrations of this type, and the conclusions drawn from the methods proposed here are thus likely to be more reliable in the model-identification process than those resulting from an inspection of traditional correlograms. © 1976 Taylor & Francis Group, LLC.