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

Published online by Cambridge University Press:  07 August 2013

Adam Mccloskey  and
Pierre Perron
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Abstract

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 that 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- or short-memory 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 trade-offs 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.

Type
ARTICLES
Information
Econometric Theory , Volume 29 , Issue 6 , December 2013 , pp. 1196 - 1237
Copyright
Copyright © Cambridge University Press 2013 

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Footnotes

The authors are grateful to Shinsuke Ikeda for kindly sharing S&P 500 futures realized volatility data. We also thank Morten Nielsen, the co-editor Robert Taylor, and two referees for useful comments.

References

REFERENCES

Anderson, T.G., Bollerslev, T., Diebold, T., & Labys, P. (2001) The distribution of realized exchange rate volatility. Journal of the American Statistical Association 96, 42–55.Google Scholar
Andrews, D. & Guggenberger, P. (2003) A bias-reduced log-periodogram regression estimator for the long-memory parameter. Econometrica 71, 675–712.CrossRefGoogle Scholar
Bhattacharya, R., Gupta, V., & Waymire, E. (1983) The Hurst effect under trends. Journal of Applied Probability 20, 649–662.Google Scholar
Chen, C. & Tiao, G. (1990) Random level-shift time series models, ARIMA approximations and level-shift detection. Journal of Business & Economic Statistics 8, 83–97.Google Scholar
Dahlhaus, R. (1989) Efficient parameter estimation for self similar processes. Annals of Statistics 17, 1749–1766.Google Scholar
Deo, R.S. & Hurvich, C.M. (2001) On the log periodogram regression estimator of the memory parameter in long memory stochastic volatility models. Econometric Theory 17, 686–710.CrossRefGoogle Scholar
Diebold, F. & Inoue, A. (2001) Long memory and regime switching. Journal of Econometrics 105, 131–159.Google Scholar
Dolado, J., Gonzalo, J. & Mayoral, L. (2005) What Is What? A Simple Test of Long-Memory versus Structural Breaks in the Time Domain. Manuscript, Departments of Economics, Universidad Carlos III de Madrid and Universitat Pompeu Fabra.CrossRefGoogle Scholar
Fox, R. & Taqqu, M. (1986) Large sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Annals of Statistics 14, 517–532.CrossRefGoogle Scholar
Frederiksen, P., Nielsen, F.S., & M.Ø. Nielsen (2012) Local polynomial Whittle estimation of perturbed fractional processes. Journal of Econometrics 167, 426–447.CrossRefGoogle Scholar
Garcia, R. & Perron, P. (1996) An analysis of the real interest rate under regime shifts. Review of Economics and Statistics 78, 111–125.CrossRefGoogle Scholar
Geweke, J. & Porter-Hudak, S. (1983) The estimation and application of long memory time series models. Journal of Time Series Analysis 4, 221–238.CrossRefGoogle Scholar
Granger, C.W.J. & Joyeux, R. (1980) An introduction to long memory time series models and fractional differencing. Journal of Time Series Analysis 1, 15–29.CrossRefGoogle Scholar
Granger, C.W.J. & Hyung, N. (2004) Occasional structural breaks and long memory with an application to the S&P 500 absolute stock returns. Journal of Empirical Finance 11, 399–421.CrossRefGoogle Scholar
Haldrup, N. & M.Ø. Nielsen (2007) Estimation of fractional integration in the presence of data noise. Computational Statistics and Data Analysis 51, 3100–3114.Google Scholar
Hosking, J.R.M. (1981) Fractional differencing. Biometrika 68, 165–176.CrossRefGoogle Scholar
Hurst, H.E. (1951) Long-term storage capacity of reservoirs. Transactions of the American Society of Civil Engineers 116, 770–799.Google Scholar
Hurvich, C.M. & Beltrao, K.I. (1994) Automatic semiparametric estimation of the memory parameter of a long-memory time series. Journal of Time Series Analysis 15, 285–302.Google Scholar
Hurvich, C.M., Deo, R. & Brodsky, J. (1998) The mean squared error of Geweke and Porter-Hudak’s estimator of the memory parameter of a long-memory time series. Journal of Time Series Analysis 19, 19–46.Google Scholar
Hurvich, C.M., Lang, G. & Soulier, P. (2005) Estimation of long memory in the presence of a smooth nonparametric trend. Journal of the American Statistical Association 100, 853–871.CrossRefGoogle Scholar
Iacone, F. (2010) Local Whittle estimation of the memory parameter in presence of deterministic components. Journal of Time Series Analysis 31, 37–49.CrossRefGoogle Scholar
Ikeda, S. (2013) Two Scale Realized Kernels: A Univariate Case. Manuscript, Faculty of Economics, National Graduate Institute of Policy Studies.Google Scholar
Künsch, H.R. (1986) Discriminating between monotonic trends and long-range dependence. Journal of Applied Probability 23, 1025–1030.CrossRefGoogle Scholar
Künsch, H.R. (1987) Statistical aspects of self-similar processes. In Prohorov, Y. & Sazarov, V. (eds.), Proceedings of the First World Congress of the Bernoulli Society, vol. 1, pp. 67–74. VNU Science Press.Google Scholar
Lu, Y.K. & Perron, P. (2010) Modeling and forecasting stock return volatility using a random level shift model. Journal of Empirical Finance 17, 138–156.CrossRefGoogle Scholar
Mikosch, T. & Stărică, C.; (2004) Nonstationarities in financial time series, the long-range dependence, and the IGARCH effects. Review of Economics and Statistics 86, 378–390.Google Scholar
Ohanissian, A., Russell, J.R., & Tsay, R.S. (2004) True or Spurious Long Memory in Volatility: Does It Matter for Pricing Options?, Manuscript, Booth School of Business, University of Chicago.Google Scholar
Ohanissian, A., Russell, J.R., & Tsay, R.S. (2008) True or spurious long memory? A new test. Journal of Business & Economic Statistics 26, 161–175.CrossRefGoogle Scholar
Perron, P. (1989) The great crash, the oil price shock, and the unit root hypothesis. Econometrica 57, 1361–1401.Google Scholar
Perron, P. & Qu, Z. (2010) Long-memory and level shifts in the volatility of stock market return indices. Journal of Business & Economic Statistics 28, 275–290.CrossRefGoogle Scholar
Qu, Z. (2011) A test against spurious long memory. Journal of Business & Economic Statistics 29, 423–438.CrossRefGoogle Scholar
Robinson, P.M. (1995) Log-periodogram regression of time series with long range dependence. Annals of Statistics 23, 1048–1072.CrossRefGoogle Scholar
Robinson, P.M. (1997) Large-sample inference for nonparametric regression with dependent errors. Annals of Statistics 25, 2054–2083.CrossRefGoogle Scholar
Shimotsu, K. (2006) Simple (but effective) Tests of Long Memory versus Structural Breaks, Working paper 1101, Department of Economics, Queen’s University.Google Scholar
Skovgaard, I.M. (1986) On multivariate Edgeworth expansions. International Statistical Review 54, 169–186.Google Scholar
Smith, A. (2005) Level shifts and the illusion of long memory in economic time series. Journal of Business & Economic Statistics 23, 321–335.CrossRefGoogle Scholar
Sun, Y. & Phillips, P.C.B. (2003) Nonlinear log-periodogram regression for perturbed fractional processes. Journal of Econometrics 115, 355–389.CrossRefGoogle Scholar
Taylor, S.J. (2000) Consequences for Option Pricing of a Long Memory in Volatility. Manuscript, Department of Accounting and Finance, Lancaster University.Google Scholar
Velasco, C. (1999) Non-stationary log-periodogram regression. Journal of Econometrics 91, 325–371.CrossRefGoogle Scholar
Velasco, C. (2000) Non-Gaussian log periodogram regression. Econometric Theory 16, 44–79.Google Scholar