Hostname: page-component-74656f9884-rngvg Total loading time: 0 Render date: 2026-11-06T05:11:25.018Z Has data issue: false hasContentIssue false
  • English
  • Français

UNIT ROOT TESTS WITH WAVELETS

Published online by Cambridge University Press:  17 February 2010

Yanqin Fan  and
Ramazan Gençay
Yanqin Fan
Affiliation:
Vanderbilt University
Ramazan Gençay*
Affiliation:
Simon Fraser University
*
*Address correspondence to Ramazan Gençay, Department of Economics, Simon Fraser University, 8888 University Drive, Burnaby, British Columbia, V5A 1S6, Canada; e-mail: rgencay@sfu.ca.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

This paper develops a wavelet (spectral) approach to testing the presence of a unit root in a stochastic process. The wavelet approach is appealing, since it is based directly on the different behavior of the spectra of a unit root process and that of a short memory stationary process. By decomposing the variance (energy) of the underlying process into the variance of its low frequency components and that of its high frequency components via the discrete wavelet transformation (DWT), we design unit root tests against near unit root alternatives. Since DWT is an energy preserving transformation and able to disbalance energy across high and low frequency components of a series, it is possible to isolate the most persistent component of a series in a small number of scaling coefficients. We demonstrate the size and power properties of our tests through Monte Carlo simulations.

Information

Type
ARTICLES
Information
Econometric Theory , Volume 26 , Issue 5 , October 2010 , pp. 1305 - 1331
Copyright
Copyright © Cambridge University Press 2010

References

REFERENCES

Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817–858.CrossRefGoogle Scholar
Berkes, I., Hormann, S., & Horvath, L. (2008) The functional central limit theorem for a family of garch observations with applications. Statistics & Probability Letters 78, 2725–2730.CrossRefGoogle Scholar
Bhargava, A. (1986) On the theory of testing for unit roots in observed time series. Review of Economic Studies 53, 369–384.Google Scholar
Cai, Y. & Shintani, M. (2006) On the alternative long-run variance ratio test for a unit root. Econometric Theory 22, 347–372.CrossRefGoogle Scholar
Chan, N.H. & Wei, C.Z. (1987) Asymptotic inference for nearly nonstationary AR(1) processes. Annals of Statistics 15, 1050–1063.Google Scholar
Choi, I. & Phillips, P.C.B. (1993) Testing for a unit root by frequency domain regression. Journal of Econometrics 59, 263–286.Google Scholar
Coifman, R.R. & Donoho, D.L. (1995) Translation invariant denoising. In Antoniadis, A. & Oppenheim, G. (eds.), Wavelets and Statistics, vol. 103, pp. 125–150, Springer-Verlag.Google Scholar
Daubechies, I. (1992) Ten Lectures on Wavelets, vol. 61 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM.CrossRefGoogle Scholar
Davidson, R., Labys, W.C., & Lesourd, J.-B. (1998) Wavelet analysis of commodity price behavior. Computational Economics 11, 103–128.CrossRefGoogle Scholar
Dickey, D.A. & Fuller, W.A. (1979) Distributions of the estimators for autoregressive time series with a unit root. Journal of American Statistical Association 74, 427–431.Google Scholar
Duchesne, P. (2006a) On testing for serial correlation with a wavelet-based spectral density estimator in multivariate time series. Econometric Theory 22, 633–676.Google Scholar
Duchesne, P. (2006b) Testing for multivariate autoregressive conditional heteroskedasticity using wavelets. Computational Statistics & Data Analysis 51, 2142–2163.CrossRefGoogle Scholar
Dufour, J.M. & King, M. (1991). Optimal invariant tests for the autocorrelation coefficient in linear regressions with stationary and nonstationary errors. Journal of Econometrics 47, 115–143.Google Scholar
Elliott, G., Rothenberg, T.J., & Stock, J.H. (1996) Efficient tests for an autoregressive unit root. Econometrica 64, 813–836.Google Scholar
Fan, Y. (2003) On the approximate decorrelation property of the discrete wavelet transform for fractionally differenced processes. IEEE Transactions on Information Theory 49, 516–521.Google Scholar
Fan, Y. & Whitcher, B. (2003) A wavelet solution to the spurious regression of fractionally differenced processes. Applied Stochastic Models in Business and Industry 19, 171–183.Google Scholar
Gençay, R., Selçuk, F., & Whitcher, B. (2001) An Introduction to Wavelets and Other Filtering Methods in Finance and Economics. Academic Press.Google Scholar
Gençay, R., Selçuk, F., & Whitcher, B. (2003) Systematic risk and time scales. Quantitative Finance 3, 108–116.Google Scholar
Gençay, R., Selçuk, F., & Whitcher, B. (2005) Multiscale systematic risk. Journal of International Money and Finance 24, 55–70.CrossRefGoogle Scholar
Granger, C.W.J. (1966) The typical spectral shape of an economic variable. Econometrica 34, 150–161.CrossRefGoogle Scholar
Hamilton, J.D. (1994) Time Series Analysis. Princeton University Press.CrossRefGoogle Scholar
Hong, Y. (2000) Wavelet-based estimation for heteroskedasticity and autocorrelation consistent variance-covariance matrices. Technical report, Cornell University.Google Scholar
Hong, Y. & Kao, C. (2004) Wavelet-based testing for serial correlation of unknown form in panel models. Econometrica 72, 1519–1563.Google Scholar
Hong, Y. & Lee, J. (2001) One-sided testing for ARCH effects using wavelets. Econometric Theory 17, 1051–1081.CrossRefGoogle Scholar
Lee, J. & Hong, Y. (2001) Testing for serial correlation of unknown form using wavelet methods. Econometric Theory 17, 386–423.Google Scholar
MacKinnon, J.G. (2000) Computing numerical distribution functions in econometrics. In Pollard, A., Mewhort, D., & Weaver, D. (eds.), High Performance Computing Systems and Applications, pp. 455–470. Kluwer.Google Scholar
Mallat, S. (1989) A theory for multiresolution signal decomposition: The wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence 11, 674–693.Google Scholar
Mallat, S. (1998) A Wavelet Tour of Signal Processing. Academic Press.Google Scholar
Nason, G.P. & Silverman, B.W. (1995) The stationary wavelet transform and some statistical applications. In Antoniadis, A. & Oppenheim, G. (eds.), Wavelets and Statistics, Vol. 103, pp. 281–300. Lecture Notes in Statistics. Springer Verlag.Google Scholar
Nelson, C.R. & Plosser, C.I. (1982) Trends and random walks in macroeconomic time series: Some evidence and implications. Journal of Monetary Economics 10, 139–162.Google Scholar
Newey, W.K. & West, K.D. (1987) A simple positive semidefinite heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55, 703–708.CrossRefGoogle Scholar
Ng, S. & Perron, P., (2001) Lag length selection and the construction of unit root tests with good size and power. Econometrica 69, 1519–1554.Google Scholar
Park, H. & Fuller, W. (1995) Alternative estimators and unit root tests for the autoregressive process. Journal of Time Series Analysis 16, 415–429.CrossRefGoogle Scholar
Park, J.Y. & Phillips, P.C.B. (1988) Statistical inference in regressions with integrated processes: Part 1. Econometric Theory 4, 468–497.Google Scholar
Park, J.Y. & Phillips, P.C.B. (1989) Statistical inference in regressions with integrated processes: Part 2. Econometric Theory 5, 95–131.CrossRefGoogle Scholar
Percival, D.B. (1995) On estimation of the wavelet variance. Biometrica 82, 619–631.Google Scholar
Percival, D.B. & Mofjeld, H.O. (1997) Analysis of subtidal coastal sea level fluctuations using wavelets. Journal of the American Statistical Association 92, 868–880.Google Scholar
Percival, D.B. & Walden, A.T. (2000) Wavelet Methods for Time Series Analysis. Cambridge Press.Google Scholar
Phillips, P.C.B. (1986) Understanding spurious regressions in econometrics. Journal of Econometrics 33, 311–340.Google Scholar
Phillips, P.C.B. (1987a) Time series regression with a unit root. Econometrica 55, 277–301.CrossRefGoogle Scholar
Phillips, P.C.B. (1987b) Towards a unified asymptotic theory for autoregression. Biometrica 74, 535–547.Google Scholar
Phillips, P.C.B. & Ouliaris, S. (1990) Asymptotic properties of residual based tests for cointegration. Econometrica 58, 165–193.CrossRefGoogle Scholar
Phillips, P.C.B. & Perron, P., (1988) Testing for a unit root in time series regression. Biometrica 75, 335–346.Google Scholar
Phillips, P.C.B. & Solo, V. (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.Google Scholar
Phillips, P.C.B. & Xiao, Z. (1998) A primer on unit root testing. Journal of Economic Surveys 12, 423–469.Google Scholar
Ramsey, J.B. (1999) The contribution of wavelets to the anlaysis of economic and financial data. Philosophical Transactions of the Royal Society of London A 357, 2593–2606.Google Scholar
Ramsey, J.B. (2002) Wavelets in economics and finance: Past and future. Studies in Nonlinear Dynamics & Econometrics 3, 1–29.Google Scholar
Ramsey, J.B., Zaslavsky, G., & Usikov, D. (1995) An analysis of U.S. stock price behavior using wavelets. Fractals 3, 377–389.CrossRefGoogle Scholar
Ramsey, J.B. & Zhang, Z. (1997) The analysis of foreign exchange data using waveform dictionaries. Journal of Empirical Finance 4, 341–372.Google Scholar
Sargan, J.D. & Bhargava, A. (1983) Testing residuals from least squares regression for being generated by the Gaussian random walk. Econometrica 51, 153–174.Google Scholar
Schmidt, P. & Phillips, P.C.B. (1992) LM tests for a unit root in the presence of deterministic trends. Oxford Bulletin of Economics and Statistics 54, 257–288.Google Scholar
Sims, C.A., Stock, J.H., & Watson, M.W. (1990) Inference in linear time series models with some unit roots. Econometrica 58, 113–144.Google Scholar
Stock, J.H. (1995) Unit roots, structural breaks and trends. In Engle, R.F. & McFadden, D. (eds.), Handbook of Econometrics, pp. 2739–2841. North-Holland.Google Scholar
Stock, J.H. (1999) A class of tests for integration and cointegration. In Engle, R.F. & White, H. (eds.), Cointegration, Causality, and Forecasting Festschrift in Honour of Clive W. J. Granger, chap. 6. Oxford University Press.Google Scholar