Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T18:30:09.790Z Has data issue: false hasContentIssue false
  • English
  • Français

DETECTION OF NONCONSTANT LONG MEMORY PARAMETER

Published online by Cambridge University Press:  16 October 2013

Frédéric Lavancier ,
Remigijus Leipus ,
Anne Philippe  and
Donatas Surgailis
Frédéric Lavancier
Affiliation:
Université de Nantes
Remigijus Leipus*
Affiliation:
Vilnius University
Anne Philippe
Affiliation:
Université de Nantes
Donatas Surgailis
Affiliation:
Vilnius University
*
*Address correspondence to Remigijus Leipus, Vilnius University, Lithuania; e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

This article deals with detection of a nonconstant long memory parameter in time series. The null hypothesis presumes stationary or nonstationary time series with a constant long memory parameter, typically an I (d) series with d > −.5 . The alternative corresponds to an increase in persistence and includes in particular an abrupt or gradual change from I (d1) to I (d2), −.5 < d1 < d2. We discuss several test statistics based on the ratio of forward and backward sample variances of the partial sums. The consistency of the tests is proved under a very general setting. We also study the behavior of these test statistics for some models with a changing memory parameter. A simulation study shows that our testing procedures have good finite sample properties and turn out to be more powerful than the KPSS-based tests (see Kwiatkowski, Phillips, Schmidt and Shin, 1992) considered in some previous works.

Type
ARTICLES
Information
Econometric Theory , Volume 29 , Issue 5 , October 2013 , pp. 1009 - 1056
Copyright
Copyright © Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The second and fourth authors are supported by a grant (No. MIP-11155) from the Research Council of Lithuania. We are grateful to three anonymous referees for their insightful comments that greatly improved the manuscript.

References

REFERENCES

Abadir, K.M., Distaso, W., & Giraitis, L. (2007) Nonstationarity-extended local Whittle estimation. Journal of Econometrics 141, 1353–1384.Google Scholar
Bardet, J.-M. & Kammoun, I. (2008) Detecting abrupt changes of the long-range dependence or the self-similarity of a Gaussian process. Comptes Rendus de I’Academie des Sciences, Series I 346, 789–794.Google Scholar
Bardet, J.-M., Lang, G., Oppenheim, G., Philippe, A., Stoev, S. & Taqqu, M.S. (2003) Semi-parametric estimation of the long-range dependence parameter: A survey. In Doukhan, P., Oppenheim, G. & Taqqu, M.S. (eds.), Theory and Applications of Long-Range Dependence, pp. 557–577. Birkhäuser.Google Scholar
Beran, J. & Terrin, N. (1996) Testing for a change of the long-memory parameter. Biometrika 83, 627–638.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.Google Scholar
Bružaitė, K. & Vaičiulis, M. (2005) Asymptotic independence of distant partial sums of linear processes. Lietuvos Matematikos Rinkinys 45, 479–500.Google Scholar
Busetti, F. & Taylor, A.M.R. (2004) Tests of stationarity against a change in persistence. Journal of Econometrics 123, 33–66.CrossRefGoogle Scholar
Chan, N.H. & Terrin, N. (1995) Inference for unstable long-memory processes with applications to fractional unit root autoregressions. Annals of Statistics 23, 1662–1683.CrossRefGoogle Scholar
Davidson, J. & de Jong, R.M. (2000) The functional central limit theorem and weak convergence to stochastic integrals II: Fractionally integrated processes. Econometric Theory 16, 643–666.Google Scholar
Davidson, J. & Hashimzade, N. (2009) Type I and type II fractional Brownian motions: A reconsideration. Computational Statistics and Data Analysis 53, 2089–2106.CrossRefGoogle Scholar
Davydov, Y.A. (1970) The invariance principle for stationary processes. Theory of Probability and its Applications 15, 487–498.CrossRefGoogle Scholar
Ferger, D. & Vogel, D., (2010) Weak convergence of the empirical process and the rescaled empirical distribution function in the Skorokhod product space. Theory of Probability and its Applications 54, 609–625.CrossRefGoogle Scholar
Giraitis, L., Kokoszka, P., Leipus, R. & Teyssière, G. (2003) Rescaled variance and related tests for long memory in volatility and levels. Journal of Econometrics 112, 265–294.CrossRefGoogle Scholar
Giraitis, L., Koul, H.L. & Surgailis, D. (2012) Large Sample Inference for Long Memory Processes. Imperial College Press.CrossRefGoogle Scholar
Giraitis, L., Leipus, R. & Philippe, A. (2006). A test for stationarity versus trends and unit roots for a wide class of dependent errors. Econometric Theory 22, 989–1029.Google Scholar
Giraitis, L., Leipus, R. & Surgailis, D. (2009) ARCH(∞) models and long memory properties. In Andersen, T.G., Davis, R.A., Kreiss, J.-P. & Mikosch, T. (eds.), Handbook of Financial Time Series, pp. 71–84. Springer-Verlag.CrossRefGoogle Scholar
Giraitis, L., Robinson, P.M. & Surgailis, D. (2000) A model for long memory conditional heteroscedasticity. Annals of Applied Probability 10, 1002–1024.Google Scholar
Hassler, U. & Meller, B. (2009) Detecting a change in inflation persistence in the presence of long memory: A new approach. Working paper, Goethe University.Google Scholar
Hassler, U. & Nautz, D. (2008) On the persistence of the Eonia spread. Economics Letters 101, 184–187.CrossRefGoogle Scholar
Hassler, U. & Scheithauer, J. (2008) On critical values of tests against a change in persistence. Oxford Bulletin of Economics and Statistics 70, 705–710.CrossRefGoogle Scholar
Hassler, U. & Scheithauer, J. (2011) Testing against a change from short to long memory. Statistical Papers 52, 847–870.Google Scholar
Horváth, L. (2001) Change-point detection in long-memory processes. Journal of Multivariate Analysis 78, 218–234.Google Scholar
Horváth, L. & Shao, Q.-M. (1999) Limit theorems for quadratic forms with applications to Whittle’s estimate. Annals of Applied Probability 9, 146–187.Google Scholar
Johansen, S. & Nielsen, M.Ø. (2010). Likelihood inference for a nonstationary fractional autoregressive model. Journal of Econometrics 158, 51–66.CrossRefGoogle Scholar
Kim, J.-Y. (2000) Detection of change in persistence of a linear time series. Journal of Econometrics 95, 97–116.CrossRefGoogle Scholar
Kim, J.-Y., Belaire-Franch, J. & Badillo Amador, R. (2002) Corrigendum to “Detection of change in persistence of a linear time series.” Journal of Econometrics 109, 389–392.CrossRefGoogle Scholar
Kokoszka, P. & Leipus, R. (2003) Detection and estimation of changes in regime. In Doukhan, P., Oppenheim, G. & Taqqu, M.S. (eds.), Theory and Applications of Long-Range Dependence pp. 325–337. Birkhäuser.Google Scholar
Kruse, R. (2008) Rational bubbles and changing degree of fractional integration. Diskussionspapiere der Wirtschaftswissenschaftlichen Fakultät der Universität Hannover, No. dp-394.Google Scholar
Kumar, M.S. & Okimoto, T. (2007) Dynamics of persistence in international inflation rates. Journal of Money, Credit and Banking 39, 1457–1479.CrossRefGoogle Scholar
Kwiatkowski, D., Phillips, P.C.B., Schmidt, P. & Shin, Y. (1992) Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root? Journal of Econometrics 54, 159–178.CrossRefGoogle Scholar
Lavielle, M. & Ludeña, C. (2000) The multiple change-points problem for the spectral distribution. Bernoulli 6, 845–869.CrossRefGoogle Scholar
Leipus, R. & Surgailis, D. (2013) Asymptotics of partial sums of linear processes with changing memory parameter. Lithuanian Mathematical Journal 53, 196–219.CrossRefGoogle Scholar
Leybourne, S., Taylor, R. & Kim, T.-H. (2007) CUSUM of squares-based tests for a change in persistence. Journal of Time Series Analysis 28, 408–433.CrossRefGoogle Scholar
Liu, M. (1998) Asymptotics of nonstationary fractional integrated series. Econometric Theory 14, 641–662.CrossRefGoogle Scholar
MacNeill, I. (1978) Properties of sequences of partial sums of polynomial regression residuals with applications to tests for change of regression at unknown times. Annals of Statistics, 6, 422–433.CrossRefGoogle Scholar
Marinucci, D. & Robinson, P.M. (1999) Alternative forms of fractional Brownian motion. Journal of Statistical Planning and Inference 80, 111–122.CrossRefGoogle Scholar
Marinucci, D. & Robinson, P.M. (2000) Weak convergence of multivariate fractional processes. Stochastic Processes and their Applications 86:103–120.CrossRefGoogle Scholar
Martins, L.F. & Rodrigues, P.M.M. (2012) Testing for persistence change in fractionally integrated models: An application to world inflation rates. Computational Statistics & Data Analysis. doi:10.1016/j.csda.2012.07.021.Google Scholar
Peligrad, M. & Utev, S. (1997) Central limit theorem for linear processes. Annals of Probability 25, 443–456.Google Scholar
Philippe, A., Surgailis, D. & Viano, M.-C. (2006a) Almost periodically correlated processes with long-memory. In Bertail, P., Doukhan, P. & Soulier, P. (eds.), Dependence in Probability and Statistics, vol. 187 of Lecture Notes in Statistics, pp. 159–194. Springer-Verlag.CrossRefGoogle Scholar
Philippe, A., Surgailis, D. & Viano, M.-C. (2006b) Invariance principle for a class of non stationary processes with long memory. Comptes Rendus de I’Academie des Sciences, Series I 342, 269–274.Google Scholar
Philippe, A., Surgailis, D. & Viano, M.-C. (2008) Time-varying fractionally integrated processes with nonstationary long memory. Theory of Probability and Its Applications 52, 651–673.CrossRefGoogle Scholar
Pollard, D. (1984) Convergence of Stochastic Processes. Springer-Verlag.CrossRefGoogle Scholar
Robinson, P.M. (1991) Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regressions. Journal of Econometrics 47, 67–84.CrossRefGoogle Scholar
Robinson, P.M. (1994) Efficient tests of nonstationary hypotheses. Journal of American Statistical Association 89, 1420–1437.CrossRefGoogle Scholar
Shimotsu, K. (2006) Simple (but effective) tests of long memory versus structural breaks. Working papers no. 1101, Queen’s University.Google Scholar
Sibbertsen, P. & Kruse, R. (2009) Testing for a break in persistence under long-range dependencies. Journal of Time Series Analysis 30, 263–285.CrossRefGoogle Scholar
Surgailis, D. (2003) Non-CLTs: U-statistics, multinomial formula and approximations of multiple Itô-Wiener integrals. In Doukhan, P., Oppenheim, G. & Taqqu, M.S. (eds.), Theory and Applications of Long-Range Dependence, pp. 129–142. Birkhäuser.Google Scholar
Surgailis, D. (2008) Nonhomogeneous fractional integration and multifractional processes. Stochastic Processes and Their Applications 118, 171–198.CrossRefGoogle Scholar
Taqqu, M.S. (1979) Convergence of integrated processes of arbitrary Hermite rank. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 50, 53–83.CrossRefGoogle Scholar
Whitt, W. (1970) Weak convergence of probability measures on the function space C[0,∞). Annals of Mathematical Statistics 41, 939–944.CrossRefGoogle Scholar
Yamaguchi, K. (2011) Estimating a change point in the long memory parameter. Journal of Time Series Analysis 32, 304–314.CrossRefGoogle Scholar