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A TEST FOR STATIONARITY VERSUS TRENDS AND UNIT ROOTS FOR A WIDE CLASS OF DEPENDENT ERRORS

Published online by Cambridge University Press:  03 November 2006

Liudas Giraitis ,
Remigijus Leipus  and
Anne Philippe
Liudas Giraitis
Affiliation:
University of York
Remigijus Leipus
Affiliation:
Vilnius University Institute of Mathematics and Informatics
Anne Philippe
Affiliation:
Université de Nantes
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Abstract

We suggest a rescaled variance type of test for the null hypothesis of stationarity against deterministic and stochastic trends (unit roots). The deterministic trend can be represented as a general function in time (e.g., nonparametric, linear, or polynomial regression, abrupt changes in the mean). Under the null, the asymptotic distribution of the test is derived, and critical values are tabulated for a wide class of stationary processes with short, long, or negative dependence structure. A simulation study examines the performance of the test in terms of size and power. The empirical performance of the test is illustrated using the S&P 500 data.The authors thank the editor, the referees, and Karim Abadir for helpful comments and Alfredas Račkauskas for drawing our attention to the criterion of Cremers and Kadelka (1986). The first author's work was supported by the ESRC grants R000238212 and R000239538. The last two authors were supported by a cooperation agreement CNRS/LITHUANIA (4714) and by a bilateral Lithuania-France research project Gilibert.

Type
Research Article
Information
Econometric Theory , Volume 22 , Issue 6 , December 2006 , pp. 989 - 1029
Copyright
© 2006 Cambridge University Press

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References

REFERENCES

Abadir, K., W. Distaso, & L. Giraitis (2004) Nonstationarity-extended local Whittle estimation. Preprint, University of York.Google Scholar
Abadir, K., W. Distaso, & L. Giraitis (2005) Two Estimators of the Long-Run Variance. Preprint, York University.Google Scholar
Andrews, D.W.K. (1993) Tests for parameter instability and structural change with unknown change point. Econometrica 61, 821–856.CrossRefGoogle Scholar
Atkinson, A.C., S.J. Koopman, & N. Shephard (1997) Detecting shocks: Outliers and breaks in time series. Journal of Econometrics 80, 387–422.CrossRefGoogle Scholar
Bardet, J.-M., G. Lang, G. Oppenheim, A. Philippe, & M.S. Taqqu (2003) Generators of long-range dependent processes: A survey. In P. Doukhan, G. Oppenheim, & M.S. Taqqu (eds.), Theory and Applications of Long-Range Dependence, pp. 579–623. Birkhäuser.Google Scholar
Bhattacharya, R.N., V.K. Gupta, & E. Waymire (1983) The Hurst effect under trends. Journal of Applied Probability 20, 649–662.CrossRefGoogle Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.Google Scholar
Cremers, H. & D. Kadelka (1986) On weak convergence of integral functionals of stochastic processes with applications to processes taking paths in lpe. Stochastic Processes and Their Applications 21, 305–317.CrossRefGoogle Scholar
Dalla, V., L. Giraitis, & J. Hidalgo (2005) Consistent estimation of the memory parameter for non-linear time series. Journal of Time Series Analysis 27, 211–251.CrossRefGoogle Scholar
Davydov, Y.A. (1970) The invariance principle for stationary processes. Theory of Probability and Its Applications 15, 487–498.CrossRefGoogle Scholar
DeJong, D.N., J.C. Nankervis, N.E. Savin, & C.H. Whiteman (1992) The power problems of unit root tests for time series with autoregressive errors. Journal of Econometrics 53, 323–343.CrossRefGoogle Scholar
Diebold, F.X. & G.D. Rudebusch (1991) On the power of Dickey–Fuller tests against fractional alternatives. Economics Letters 35, 155–160.CrossRefGoogle Scholar
Ding, Z. & C.W.J. Granger (1996) Modeling volatility persistence of speculative returns: A new approach. Journal of Econometrics 73, 185–215.CrossRefGoogle Scholar
Dolado, J.J., J. Gonzalo, & L. Mayoral (2002) A fractional Dickey–Fuller test for unit roots. Econometrica 70, 1963–2006.CrossRefGoogle Scholar
Gikhman, I.I. & A.V. Skorokhod (1980) The Theory of Stochastic Processes, vol. 1. Springer-Verlag.Google Scholar
Giraitis, L., P. Kokoszka, R. Leipus, & G. Teyssière (2003) Rescaled variance and related tests for long memory in volatility and levels. Journal of Econometrics 112, 265–294.CrossRefGoogle Scholar
Giraitis, L., P.M. Robinson, & D. Surgailis (2000) A model for long memory conditional heteroscedasticity. Annals of Applied Probability 10, 1002–1024.CrossRefGoogle Scholar
Horowitz, J.L. & V.G. Spokoiny (2001) An adaptive, rate-optimal test of a parametric mean-regression model against a nonparametric alternative. Econometrica 69, 599–631.CrossRefGoogle Scholar
Hurst, H. (1951) Long term storage capacity of reservoirs. Transactions of the American Society of Civil Engineers 116, 770–799.CrossRefGoogle Scholar
Hurvich, C.M., G. Lang, & P. Soulier (2005) Estimation of long memory in the presence of a smooth nonparametric trend. Journal of the American Statistical Association 100, 853–871.CrossRefGoogle Scholar
Ibragimov, I.A. & Y.V. Linnik (1971) Independent and Stationary Sequences of Random Variables. Wolters–Noordhoff.Google Scholar
Künsch, H. (1987) Statistical aspects of self-similar processes. In Y.A. Prohorov & V.V. Sazonov (eds.), Proceedings of the 1st World Congress of the Bernoulli Society, vol. 1, pp. 67–74. VNU Science Press.CrossRefGoogle Scholar
Kwiatkowski, D., P.C.B. Phillips, P. Schmidt, & Y. Shin (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
Lima, L.R. & Z. Xiao (2004) Robustness of Stationarity Tests under Long-Memory Alternatives. Preprint. Ensaios Econômicos EPGE No. 541.Google Scholar
Lo, A. (1991) Long-term memory in stock market prices. Econometrica 59, 1279–1313.CrossRefGoogle Scholar
Lobato, I.N. & N.E. Savin (1998) Real and spurious long-memory properties of stock-market data (with comments). Journal of Business & Economic Statistics 16, 261–283.Google Scholar
Mikosch, T. & C. Stărică (2004) Nonstationarities in financial time series, the long-range dependence, and the IGARCH effects. Review of Economics and Statistics 86, 378–390.CrossRefGoogle Scholar
Moulines, E. & P. Soulier (1999) Broadband log-periodogram regression of time series with long-range dependence. Annals of Statistics 27, 1415–1439.CrossRefGoogle Scholar
Moulines, E. & P. Soulier (2003) Semiparametric spectral estimation for fractional processes. In P. Doukhan, G. Oppenheim, & M.S. Taqqu (eds.), Theory and Applications of Long-Range Dependence, pp. 251–301. Birkhäuser.Google Scholar
Oliveira, P.E. & C. Suquet (1998) Weak convergence in Lp(0,1) of the uniform empirical process under dependence. Statistics and Probability Letters 39, 363–370.CrossRefGoogle Scholar
Park, J. & P.C.B. Phillips (1999) Asymptotics for nonlinear transformations of integrated time series. Econometric Theory 15, 269–298.CrossRefGoogle Scholar
Phillips, P.C.B. & V. Solo (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.CrossRefGoogle Scholar
Ploberger, W., W. Krämer, & K. Kontrus (1989) A new test for structural stability in the linear regression model. Journal of Econometrics 40, 307–318.CrossRefGoogle Scholar
Pötscher, B.M. (2004) Nonlinear functions and convergence to Brownian motion: Beyond the continuous mapping theorem. Econometric Theory 20, 1–22.CrossRefGoogle Scholar
Robinson, P. (1997) Large-sample inference for nonparametric regression with dependent errors. Annals of Statistics 25, 2054–2083.CrossRefGoogle Scholar
Robinson, P.M. (1995a) Gaussian semiparametric estimation of long range dependence. Annals of Statistics 23, 1630–1661.CrossRefGoogle Scholar
Robinson, P.M. (1995b) Log-periodogram regression of time series with long range dependence. Annals of Statistics 23, 1048–1072.CrossRefGoogle Scholar
Shimotsu, K. & P.C.B. Phillips (2005) Exact local Whittle estimation of fractional integration. Annals of Statistics 33, 1890–1933.CrossRefGoogle Scholar
Watson, G.S. (1961) Goodness-of-fit tests on a circle. Biometrika 48, 109–114.CrossRefGoogle Scholar