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Distribution Theory for the Studentized Mean for Long, Short, and Negative Memory Time Series

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  • McElroy, Tucker S.
  • Politis, Dimitris N.

Abstract

We consider the problem of estimating the variance of the partial sums of a stationary time series that has either long memory, short memory, negative/intermediate memory, or is the first-difference of such a process. The rate of growth of this variance depends crucially on the type of memory, and we present results on the behavior of tapered sums of sample autocovariances in this context when the bandwidth vanishes asymptotically. We also present asymptotic results for the case that the bandwidth is a fixed proportion of sample size, extending known results to the case of flat-top tapers. We adopt the fixed proportion bandwidth perspective in our empirical section, presenting two methods for estimating the limiting critical values - both the subsampling method and a plug-in approach. Extensive simulation studies compare the size and power of both approaches as applied to hypothesis testing for the mean. Both methods perform well - although the subsampling method appears to be better sized - and provide a viable framework for conducting inference for the mean. In summary, we supply a unified asymptotic theory that covers all different types of memory under a single umbrella.

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Bibliographic Info

Paper provided by Department of Economics, UC San Diego in its series University of California at San Diego, Economics Working Paper Series with number qt35c7r55c.

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Date of creation: 01 May 2012
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Handle: RePEc:cdl:ucsdec:qt35c7r55c

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Keywords: Social and Behavioral Sciences; Kernel; Lag-windows; Overdifferencing; Spectral estimation; Subsampling; Tapers; Unit-root problem;

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References

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  1. Politis, D N & McElroy, Tucker S, 2009. "Fixed-b asymptotics for the studentized mean from time series with short, long or negative memory," University of California at San Diego, Economics Working Paper Series qt70c4x0sq, Department of Economics, UC San Diego.
  2. Sun, Yixiao, 2003. "A Convergent t-statistic in Spurious Regressions," University of California at San Diego, Economics Working Paper Series qt150457tv, Department of Economics, UC San Diego.
  3. Francis X. Diebold & Glenn D. Rudebusch, 1990. "On the power of Dickey-Fuller tests against fractional alternatives," Finance and Economics Discussion Series 119, Board of Governors of the Federal Reserve System (U.S.).
  4. Kiefer, Nicholas M. & Bunzel, Helle & Vogelsang, Timothy & Vogelsang, Timothy & Bunzel, Helle, 2000. "Simple Robust Testing of Regression Hypotheses," Staff General Research Papers 1832, Iowa State University, Department of Economics.
  5. Doukhan, Paul & Louhichi, Sana, 1999. "A new weak dependence condition and applications to moment inequalities," Stochastic Processes and their Applications, Elsevier, vol. 84(2), pages 313-342, December.
  6. Pham, Tuan D. & Tran, Lanh T., 1985. "Some mixing properties of time series models," Stochastic Processes and their Applications, Elsevier, vol. 19(2), pages 297-303, April.
  7. Donald W.K. Andrews, 1988. "Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation," Cowles Foundation Discussion Papers 877R, Cowles Foundation for Research in Economics, Yale University, revised Jul 1989.
  8. Robinson, P.M., 2005. "Robust Covariance Matrix Estimation: Hac Estimates With Long Memory/Antipersistence Correction," Econometric Theory, Cambridge University Press, vol. 21(01), pages 171-180, February.
  9. Tanaka, Katsuto, 1990. "Testing for a Moving Average Unit Root," Econometric Theory, Cambridge University Press, vol. 6(04), pages 433-444, December.
  10. Kiefer, Nicholas M., 2001. "Heteroskedasticity-Autocorrelation Robust Standard Errors Using the Bartlett Kernel without Truncation," Working Papers 01-13, Cornell University, Center for Analytic Economics.
  11. McElroy, Tucker S & Politis, D N, 2011. "Distribution Theory for the Studentized Mean for Long, Short, and Negative Memory Time Series," University of California at San Diego, Economics Working Paper Series qt0dr145dt, Department of Economics, UC San Diego.
  12. Kiefer, Nicholas M. & Vogelsang, Timothy J., 2005. "A New Asymptotic Theory for Heteroskedasticity-Autocorrelation Robust Tests," Working Papers 05-08, Cornell University, Center for Analytic Economics.
  13. Yixiao Sun & Peter C. B. Phillips & Sainan Jin, 2006. "Optimal Bandwidth Selection in Heteroskedasticity-Autocorrelation Robust Testing," Cowles Foundation Discussion Papers 1545, Cowles Foundation for Research in Economics, Yale University.
  14. Dickey, David A & Fuller, Wayne A, 1981. "Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root," Econometrica, Econometric Society, vol. 49(4), pages 1057-72, June.
  15. Lopez, J. Humberto, 1997. "The power of the ADF test," Economics Letters, Elsevier, vol. 57(1), pages 5-10, November.
  16. Abadir, Karim M. & Distaso, Walter & Giraitis, Liudas, 2009. "Two estimators of the long-run variance: Beyond short memory," Journal of Econometrics, Elsevier, vol. 150(1), pages 56-70, May.
  17. Hurvich, Clifford M., 2002. "Multistep forecasting of long memory series using fractional exponential models," International Journal of Forecasting, Elsevier, vol. 18(2), pages 167-179.
  18. Marinucci, D. & Robinson, P. M., 2000. "Weak convergence of multivariate fractional processes," Stochastic Processes and their Applications, Elsevier, vol. 86(1), pages 103-120, March.
  19. Parker, Cameron & Paparoditis, Efstathios & Politis, Dimitris N., 2006. "Unit root testing via the stationary bootstrap," Journal of Econometrics, Elsevier, vol. 133(2), pages 601-638, August.
  20. Agnieszka Jach & Tucker McElroy & Dimitris N. Politis, 2012. "Subsampling inference for the mean of heavy‐tailed long‐memory time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 33(1), pages 96-111, 01.
  21. Findley, David F, et al, 1998. "New Capabilities and Methods of the X-12-ARIMA Seasonal-Adjustment Program," Journal of Business & Economic Statistics, American Statistical Association, vol. 16(2), pages 127-52, April.
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Cited by:
  1. McElroy, Tucker S. & Politis, Dimitris N., 2012. "Distribution Theory for the Studentized Mean for Long, Short, and Negative Memory Time Series," University of California at San Diego, Economics Working Paper Series qt35c7r55c, Department of Economics, UC San Diego.

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