Permutation testing for dependence in time series

Given observations from a stationary time series, permutation tests allow one to construct exactly level $\alpha$ tests under the null hypothesis of an i.i.d. (or, more generally, exchangeable) distribution. On the other hand, when the null hypothesis of interest is that the underlying process is an uncorrelated sequence, permutation tests are not necessarily level $\alpha$, nor are they approximately level $\alpha$ in large samples. In addition, permutation tests may have large Type 3, or directional, errors, in which a two-sided test rejects the null hypothesis and concludes that the first-order autocorrelation is larger than 0, when in fact it is less than 0. In this paper, under weak assumptions on the mixing coefficients and moments of the sequence, we provide a test procedure for which the asymptotic validity of the permutation test holds, while retaining the exact rejection probability $\alpha$ in finite samples when the observations are independent and identically distributed. A Monte Carlo simulation study, comparing the permutation test to other tests of autocorrelation, is also performed, along with an empirical example of application to financial data.

[1]  S. Holm.  A Simple Sequentially Rejective Multiple Test Procedure , 1979 .

[2]  Y. Rinott.  On normal approximation rates for certain sums of dependent random variables , 1994 .

[3]  C. Stein.  A bound for the error in the normal approximation to the distribution of a sum of dependent random variables , 1972 .

[4]  Joseph P. Romano,et al.  EXACT AND ASYMPTOTICALLY ROBUST PERMUTATION TESTS , 2013, 1304.5939.

[5]  T. Breusch.  TESTING FOR AUTOCORRELATION IN DYNAMIC LINEAR MODELS , 1978 .

[6]  J. Wolfowitz,et al.  An Exact Test for Randomness in the Non-Parametric Case Based on Serial Correlation , 1943 .

[7]  E. Fama.  EFFICIENT CAPITAL MARKETS: A REVIEW OF THEORY AND EMPIRICAL WORK* , 1970 .

[8]  G. E. Noether,et al.  Asymptotic Properties of the Wald-Wolfowitz Test of Randomness , 1950 .

[9]  A. Mokkadem.  Mixing properties of ARMA processes , 1988 .

[10]  Joseph P. Romano,et al.  Robust Permutation Tests For Correlation And Regression Coefficients , 2017 .

[11]  I. Ibragimov,et al.  Some Limit Theorems for Stationary Processes , 1962 .

[12]  Thomas E. Nichols,et al.  Nonparametric permutation tests for functional neuroimaging: A primer with examples , 2002, Human brain mapping.

[13]  G. Box,et al.  On a measure of lack of fit in time series models , 1978 .

[14]  R. C. Bradley.  Basic properties of strong mixing conditions. A survey and some open questions , 2005, math/0511078.

[15]  G. Neuhaus,et al.  Conditional Rank Tests for the Two-Sample Problem Under Random Censorship , 1993 .

[16]  Joseph P. Romano,et al.  Uncertainty in the Hot Hand Fallacy: Detecting Streaky Alternatives to Random Bernoulli Sequences , 2019, The Review of Economic Studies.

[17]  L. Godfrey.  TESTING AGAINST GENERAL AUTOREGRESSIVE AND MOVING AVERAGE ERROR MODELS WHEN THE REGRESSORS INCLUDE LAGGED DEPENDENT VARIABLES , 1978 .

[18]  Markus Pauly,et al.  Testing equality of spectral densities using randomization techniques , 2015, 1506.00827.

[19]  A. Janssen,et al.  How do bootstrap and permutation tests work , 2003 .

[20]  A. Janssen,et al.  Studentized permutation tests for non-i.i.d. hypotheses and the generalized Behrens-Fisher problem , 1997 .

[21]  Marek Omelka,et al.  Testing equality of correlation coefficients in two populations via permutation methods , 2012 .

[22]  G. Box,et al.  Distribution of Residual Autocorrelations in Autoregressive-Integrated Moving Average Time Series Models , 1970 .

[23]  B. Malkiel.  The Ef cient Market Hypothesis and Its Critics , 2003 .

[24]  Andrey A. Ptitsyn,et al.  Permutation test for periodicity in short time series data , 2006, BMC Bioinformatics.