MEMORY PARAMETER ESTIMATION IN THE PRESENCE OF LEVEL SHIFTS AND DETERMINISTIC TRENDS

We propose estimators of the memory parameter of a time series that are robust to a wide variety of random level shift processes, deterministic level shifts and deterministic time trends. The estimators are simple trimmed versions of the popular log-periodogram regression estimator that employ certain sample size-dependent and, in some cases, data-dependent trimmings which discard low-frequency components. We also show that a previously developed trimmed local Whittle estimator is robust to the same forms of data contamination. Regardless of whether the underlying long/shortmemory process is contaminated by level shifts or deterministic trends, the estimators are consistent and asymptotically normal with the same limiting variance as their standard untrimmed counterparts. Simulations show that the trimmed estimators perform their intended purpose quite well, substantially decreasing both finite sample bias and root mean-squared error in the presence of these contaminating components. Furthermore, we assess the tradeoffs involved with their use when such components are not present but the underlying process exhibits strong short-memory dynamics or is contaminated by noise. To balance the potential finite sample biases involved in estimating the memory parameter, we recommend a particular adaptive version of the trimmed log-periodogram estimator that performs well in a wide variety of circumstances. We apply the estimators to stock market volatility data to find that various time series typically thought to be long-memory processes actually appear to be short or very weak long-memory processes contaminated by level shifts or deterministic trends.

[1]  C. Velasco,et al.  NON-GAUSSIAN LOG-PERIODOGRAM REGRESSION , 2000, Econometric Theory.

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

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

[4]  Aaron Smith,et al.  Level Shifts and the Illusion of Long Memory in Economic Time Series , 2004 .

[5]  J. Dolado,et al.  What is What?: A Simple Time-Domain Test of Long-Memory vs. Structural Breaks , 2005 .

[6]  Stephen J. Taylor.  Consequences for Option Pricing of a Long Memory in Volatility , 2000 .

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

[8]  C. Granger,et al.  AN INTRODUCTION TO LONG‐MEMORY TIME SERIES MODELS AND FRACTIONAL DIFFERENCING , 1980 .

[9]  P. Perron,et al.  The Great Crash, The Oil Price Shock And The Unit Root Hypothesis , 1989 .

[10]  É. Moulines,et al.  Log-Periodogram Regression Of Time Series With Long Range Dependence , 1999 .

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

[12]  C. Velasco,et al.  Non-stationary log-periodogram regression , 1999 .

[13]  P. Robinson,et al.  Large-Sample Inference for Nonparametric Regression with Dependent Errors - (Now published in 'Annals of Statistics', 28 (1997), pp.2054-2083.) , 1997 .

[14]  G. C. Tiao,et al.  Random Level-Shift Time Series Models, ARIMA Approximations, and Level-Shift Detection , 1990 .

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

[16]  Rohit S. Deo,et al.  The mean squared error of Geweke and Porter‐Hudak's estimator of the memory parameter of a long‐memory time series , 1998 .

[17]  Philippe Soulier,et al.  Estimation of Long Memory in the Presence of a Smooth Nonparametric Trend , 2002 .

[18]  M. Taqqu,et al.  Large-Sample Properties of Parameter Estimates for Strongly Dependent Stationary Gaussian Time Series , 1986 .

[19]  Fabrizio Iacone,et al.  Local Whittle estimation of the memory parameter in presence of deterministic components , 2010 .

[20]  H. Künsch.  Discrimination between monotonic trends and long-range dependence , 1986 .

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

[22]  René Garcia,et al.  Série Scientifique Scientific Series an Analysis of the Real Interest Rate under Regime Shifts , 2022 .

[23]  F. Diebold,et al.  The Distribution of Realized Exchange Rate Volatility , 2000 .

[24]  H. E. Hurst,et al.  Long-Term Storage Capacity of Reservoirs , 1951 .

[25]  P. Perron,et al.  Modeling and forecasting stock return volatility using a random level shift model , 2010 .

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

[27]  Katsumi Shimotsu,et al.  Simple (but effective) tests of long memory versus structural breaks , 2006 .

[28]  Donald W. K. Andrews,et al.  A BIAS-REDUCED LOG-PERIODOGRAM REGRESSION ESTIMATOR FOR THE LONG-MEMORY PARAMETER , 2003 .

[29]  R. Dahlhaus.  Efficient parameter estimation for self-similar processes , 1989, math/0607078.

[30]  Iliyan Georgiev,et al.  Functional Weak Limit Theory for Rare Outlying Events , 2002 .

[31]  Jeffrey R. Russell,et al.  True or Spurious Long Memory in Volatility : Does it Matter for Pricing Options ? , 2004 .

[32]  Niels Haldrup,et al.  Estimation of Fractional Integration in the Presence of Data Noise , 2003, Comput. Stat. Data Anal..

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

[34]  Ib M. Skovgaard,et al.  On Multivariate Edgeworth Expansions , 1986 .

[35]  C. Hurvich,et al.  ON THE LOG PERIODOGRAM REGRESSION ESTIMATOR OF THE MEMORY PARAMETER IN LONG MEMORY STOCHASTIC VOLATILITY MODELS , 2001, Econometric Theory.

[36]  Peter C. B. Phillips,et al.  Nonlinear Log-Periodogram Regression for Perturbed Fractional Processes , 2002 .

[37]  Clifford M. Hurvich,et al.  AUTOMATIC SEMIPARAMETRIC ESTIMATION OF THE MEMORY PARAMETER OF A LONG‐MEMORY TIME SERIES , 1994 .

[38]  F. Nielsen,et al.  Local Polynomial Whittle Estimation of Perturbed Fractional Processes , 2008 .

[39]  R. Bhattacharya,et al.  THE HURST EFFECT UNDER TRENDS , 1983 .