Change-in-mean tests in long-memory time series: a review of recent developments

It is well known that standard tests for a mean shift are invalid in long-range dependent time series. Therefore, several long-memory robust extensions of standard testing principles for a change-in-mean have been proposed in the literature. These can be divided into two groups: those that utilize consistent estimates of the long-run variance and self-normalized test statistics. Here, we review this literature and complement it by deriving a new long-memory robust version of the sup-Wald test. Apart from giving a systematic review, we conduct an extensive Monte Carlo study to compare the relative performance of these methods. Special attention is paid to the interaction of the test results with the estimation of the long-memory parameter. Furthermore, we show that the power of self-normalized test statistics can be improved considerably by using an estimator that is robust to mean shifts.

[1]  Domenico Marinucci,et al.  Alternative forms of fractional Brownian motion , 1998 .

[2]  Lihong Wang.  Change-in-mean problem for long memory time series models with applications , 2008 .

[3]  P. Sibbertsen,et al.  A Multivariate Test Against Spurious Long Memory , 2017 .

[4]  C. Granger,et al.  Spurious regressions with stationary series , 1998 .

[5]  Annika Betken.  Testing for Change‐Points in Long‐Range Dependent Time Series by Means of a Self‐Normalized Wilcoxon Test , 2014, 1403.0265.

[6]  Pierre Perron,et al.  Modified local Whittle estimator for long memory processes in the presence of low frequency (and other) contaminations , 2014 .

[7]  Pierre Perron,et al.  MEMORY PARAMETER ESTIMATION IN THE PRESENCE OF LEVEL SHIFTS AND DETERMINISTIC TRENDS , 2012, Econometric Theory.

[8]  L. Horváth,et al.  The effect of long-range dependence on change-point estimators , 1997 .

[9]  P. Robinson.  Gaussian Semiparametric Estimation of Long Range Dependence , 1995 .

[10]  Javier Hidalgo,et al.  Testing for structural change in a long-memory environment☆ , 1996 .

[11]  Yixiao Sun.  Let’s fix it: Fixed-bb asymptotics versus small-bb asymptotics in heteroskedasticity and autocorrelation robust inference , 2014 .

[12]  T. Mikosch,et al.  Nonstationarities in Financial Time Series, the Long-Range Dependence, and the IGARCH Effects , 2004, Review of Economics and Statistics.

[13]  C. Granger,et al.  Occasional structural breaks and long memory with an application to the S&P 500 absolute stock returns , 2004 .

[14]  Nicholas M. Kiefer,et al.  A NEW ASYMPTOTIC THEORY FOR HETEROSKEDASTICITY-AUTOCORRELATION ROBUST TESTS , 2005, Econometric Theory.

[15]  X. Shao,et al.  A self‐normalized approach to confidence interval construction in time series , 2010, 1005.2137.

[16]  Jeffrey R. Russell,et al.  True or Spurious Long Memory? A New Test , 2008 .

[17]  X. Shao,et al.  Testing for Change Points in Time Series , 2010 .

[18]  Zhongjun Qu,et al.  A Test Against Spurious Long Memory , 2009 .

[19]  H. R. Kuensch.  Statistical Aspects of Self-Similar Processes , 1986 .

[21]  J. Geweke,et al.  THE ESTIMATION AND APPLICATION OF LONG MEMORY TIME SERIES MODELS , 1983 .

[22]  W. Newey,et al.  A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelationconsistent Covariance Matrix , 1986 .

[23]  P. Phillips,et al.  Optimal Bandwidth Selection in Heteroskedasticity-Autocorrelation Robust Testing , 2005 .

[24]  Walter Krämer,et al.  Testing for structural change in the presence of long memory , 2000 .

[25]  X. Shao,et al.  A simple test of changes in mean in the possible presence of long‐range dependence , 2011 .

[26]  Yoshihiro Yajima,et al.  On Estimation of a Regression Model with Long-Memory Stationary Errors , 1988 .

[27]  D. Bauer.  Constructing Confidence Sets Using Rank Statistics , 1972 .

[28]  P. Robinson.  Log-Periodogram Regression of Time Series with Long Range Dependence , 1995 .

[29]  Non‐Parametric Change‐Point Tests for Long‐Range Dependent Data , 2013 .

[30]  F. Diebold,et al.  Long Memory and Regime Switching , 2000 .

[31]  W. Newey,et al.  A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelationconsistent Covariance Matrix , 1986 .

[32]  J. Durbin,et al.  Techniques for Testing the Constancy of Regression Relationships Over Time , 1975 .

[33]  P. Robinson.  ROBUST COVARIANCE MATRIX ESTIMATION: HAC ESTIMATES WITH LONG MEMORY/ANTIPERSISTENCE CORRECTION , 2004, Econometric Theory.

[34]  Pierre Perron,et al.  Long-Memory and Level Shifts in the Volatility of Stock Market Return Indices , 2008 .

[35]  L. Giraitis,et al.  Two Estimators of the Long-Run Variance: Beyond Short Memory , 2009 .

[36]  D. Andrews.  Tests for Parameter Instability and Structural Change with Unknown Change Point , 1993 .

[37]  A FIXED‐ b TEST FOR A BREAK IN LEVEL AT AN UNKNOWN TIME UNDER FRACTIONAL INTEGRATION , 2014 .

[38]  D. Andrews.  Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation , 1991 .

[39]  Jonathan H. Wright.  Testing for a Structural Break at Unknown Date with Long‐memory Disturbances , 1998 .