Unit Root Tests with Wavelets

Gencay, Ramazan and Fan, Yanqin (2007): Unit Root Tests with Wavelets.

Abstract

This paper develops a wavelet (spectral) approach to test 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.

• Item Type: MPRA Paper
• Language: English
• Keywords: Unit root tests, discrete wavelet transformation, maximum overlap wavelet transformation, energy decomposition.
• Subjects: C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General; C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics

References

Andrews, D. W. K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica, 59, 817–858.

Bhargava, A. (1986). On the theory of testing for unit roots in observed time series. Review of Economic Studies, 53, 369–384.

Cai, Y. and Shintani, M. (2006). On the alternative long-run variance ratio test for a unit root. Econometric Theory, 22, 347–372.

Chan, N. H. and Wei, C. Z. (1987). Asymptotic inference for nearly nonstationary AR(1) processes. Annals of Statistics, 15, 1050–1063.

Coifman, R. R. and Donoho, D. L. (1995). Translation invariant denoising. Wavelets and Statistics, ed. A. Antoniadis and G. Oppenheim, Vol. 103, New York, Springer-Verlag, pages 125–150.

Daubechies, I. (1992). Ten Lectures on Wavelets, volume 61 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia.

Davidson, R., Labys, W. C., and Lesourd, J.-B. (1998). Walvelet analysis of commodity price behavior. Computational Economics, 11, 103–128.

Dickey, D. A. and Fuller, W. A. (1979). Distributions of the estimators for autoregressive time series with a unit root. Journal of American Statistical Association, 74, 427–431.

Duchesne, P. (2006a). On testing for serial correlation with a wavelet-based spectral density estimator in multivariate time series. Econometric Theory, 22, 633–676.

Duchesne, P. (2006b). Testing for multivariate autoregressive conditional heteroskedasticity using wavelets. Computational Statistics & Data Analysis, 51, 2142–2163.

Dufour, J. M. and King, M. (1991). Optimal invariant tests for the autocorrelation coefficient in linear regressions with stationary and nonstationary errors. Journal of Econometrics, 47, 115–143.

Elliott, G., Rothenberg, T. J., and Stock, J. H. (1996). Efficient tests for an autoregressive unit root. Econometrica, 64, 813–836.

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.

Fan, Y. and Whitcher, B. (2003). A wavelet solution to the spurious regression of fractionally differenced processes. Applied Stochastic Models in Business and Industry, 19, 171–183.

Gen¸cay, R., Sel¸cuk, F., and Whitcher, B. (2001). An Introduction to Wavelets and Other Filtering Methods in Finance and Economics. Academic Press, San Diego.

Gen¸cay, R., Sel¸cuk, F., and Whitcher, B. (2003). Systematic risk and time scales. Quantitative Finance, 3, 108–116.

Gen¸cay, R., Sel¸cuk, F., and Whitcher, B. (2005). Multiscale systematic risk. Journal of International Money and Finance, 24, 55–70.

Granger, C. W. J. (1966). The typical spectral shape of an economic variable. Econometrica, 34, 150–161.

Hamilton, J. D. (1994). Time Series Analysis. Princeton University Press, Princeton, New Jersey.

Hong, Y. (2000). Wavelet-based estimation for heteroskedasticity and autocorrelation consistent variance-covariance matrices. Ph.D. thesis, Working Paper, Department of Economics and Department of Statistical Science, Cornell University.

Hong, Y. and Kao, C. (2004). Wavelet-based testing for serial correlation of unknown form in panel models. Econometrica, 72, 1519–1563.

Hong, Y. and Lee, J. (2001). One-sided testing for ARCH effects using wavelets. Econometric Theory, 17, 1051–1081.

Lee, J. and Hong, Y. (2001). Testing for serial correlation of unknown form using wavelet methods. Econometric Theory, 17, 386–423.

MacKinnon, J. G. (2000). Computing numerical distribution functions in econometrics. High Performance Computing Systems and Applications, ed. A. Pollard, D. Mewhort, and D. Weaver, Amsterdam, Kluwer, pages 455–470.

Mallat, S. (1989). A theory for multiresolution signal decomposition: The wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11, 674–693.

Mallat, S. (1998). A Wavelet Tour of Signal Processing. Academic Press, San Diego.

Nason, G. P. and Silverman, B. W. (1995). The stationary wavelet transform and some statistical applications. Wavelets and Statistics, Volume 103 of Lecture Notes in Statistics, ed. A. Antoniadis and G. Oppenheim, Springer Verlag, New York, pages 281–300.

Nelson, C. R. and Plosser, C. I. (1982). Trends and random walks in macroeconomic time series: Some evidence and implications. Journal of Monetary Economics, 10, 139–162.

Newey, W. K. and West, K. D. (1987). A simple positive semidefinite heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica, 55, 703–708.

Ng, S. and Perron, P. (2001). Lag length selection and the construction of unit root tests with good size and power. Econometrica, 69, 1519–1554.

Park, H. and Fuller, W. (1995). Alternative estimators and unit root tests for the autoregressive process. Journal of Time Series Analysis, 16, 415–429.

Park, J. Y. and Phillips, P. C. B. (1988). Statistical inference in regressions with integrated processes: Part 1. Econometric Theory, 4, 468–497.

Park, J. Y. and Phillips, P. C. B. (1989). Statistical inference in regressions with integrated processes: Part 2. Econometric Theory, 5, 95–131.

Percival, D. B. (1995). On estimation of the wavelet variance. Biometrica, 82, 619–631.

Percival, D. B. and Mofjeld, H. O. (1997). Analysis of subtidal coastal sea level fluctuations using wavelets. Journal of the American Statistical Association, 92, 868–880.

Percival, D. B. and Walden, A. T. (2000). Wavelet Methods for Time Series Analysis. Cambridge Press, Cambridge.

Phillips, P. C. B. (1986). Understanding spurious regressions in econometrics. Journal of Econometrics, 33, 311–340.

Phillips, P. C. B. (1987). Time series regression with a unit root. Econometrica, 55, 277–301.

Phillips, P. C. B. and Ouliaris, S. (1990). Asymptotic properties of residual based tests for cointegration. Econometrica, 58, 165–193.

Phillips, P. C. B. and Perron, P. (1988). Testing for a unit root in time series regression. Biometrica, 75, 335–346.

Phillips, P. C. B. and Solo, V. (1992). Asymptotics for linear processes. Annals of Statistics, 20, 971–1001.

Phillips, P. C. B. and Xiao, Z. (1998). A primer on unit root testing. Journal of Economic Surveys, 12, 423–469.

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.

Sargan, J. D. and Bhargava, A. (1983). Testing residuals from least squares regression for being generated by the Gaussian random walk. Econometrica, 51, 153–174.

Schmidt, P. and 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.

Sims, C. A., Stock, J. H., and Watson, M. W. (1990). Inference in linear time series models with some unit roots. Econometrica, 58, 113–144.

Stock, J. H. (1995). Unit roots, structural breaks and trends. Handbook of Econometrics, ed. R. F. Engle and D. McFadden, Amsterdam, North-Holland, pages 2739–2841.

Stock, J. H. (1999). A class of tests for integration and cointegration. Cointegration, Causality, and Forecasting Festschrift in Honour of Clive W. J. Granger, ed. R. F. Engle and H. White, Oxford, Oxford University Press, Chapter 6.