Skip to main content
    • Aa
    • Aa


  • Yanqin Fan (a1) and Ramazan Gençay (a2)

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.

Corresponding author
*Address correspondence to Ramazan Gençay, Department of Economics, Simon Fraser University, 8888 University Drive, Burnaby, British Columbia, V5A 1S6, Canada; e-mail:
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

D.W.K. Andrews (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817858.

I. Berkes , S. Hormann , & L. Horvath (2008) The functional central limit theorem for a family of garch observations with applications. Statistics & Probability Letters 78, 27252730.

A. Bhargava (1986) On the theory of testing for unit roots in observed time series. Review of Economic Studies 53, 369384.

N.H. Chan & C.Z. Wei (1987) Asymptotic inference for nearly nonstationary AR(1) processes. Annals of Statistics 15, 10501063.

I. Choi & P.C.B. Phillips (1993) Testing for a unit root by frequency domain regression. Journal of Econometrics 59, 263286.

R.R. Coifman & D.L. Donoho (1995) Translation invariant denoising. In A. Antoniadis & G. Oppenheim (eds.), Wavelets and Statistics, vol. 103, pp. 125150, Springer-Verlag.

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

R. Davidson , W.C. Labys , & J.-B. Lesourd (1998) Wavelet analysis of commodity price behavior. Computational Economics 11, 103128.

D.A. Dickey & W.A. Fuller (1979) Distributions of the estimators for autoregressive time series with a unit root. Journal of American Statistical Association 74, 427431.

P. Duchesne (2006b) Testing for multivariate autoregressive conditional heteroskedasticity using wavelets. Computational Statistics & Data Analysis 51, 21422163.

J.M. Dufour & M. King (1991). Optimal invariant tests for the autocorrelation coefficient in linear regressions with stationary and nonstationary errors. Journal of Econometrics 47, 115143.

G. Elliott , T.J. Rothenberg , & J.H. Stock (1996) Efficient tests for an autoregressive unit root. Econometrica 64, 813836.

Y. Fan & B. Whitcher (2003) A wavelet solution to the spurious regression of fractionally differenced processes. Applied Stochastic Models in Business and Industry 19, 171183.

R. Gençay , F. Selçuk , & B. Whitcher (2003) Systematic risk and time scales. Quantitative Finance 3, 108116.

R. Gençay , F. Selçuk , & B. Whitcher (2005) Multiscale systematic risk. Journal of International Money and Finance 24, 5570.

C.W.J. Granger (1966) The typical spectral shape of an economic variable. Econometrica 34, 150161.

Y. Hong & C. Kao (2004) Wavelet-based testing for serial correlation of unknown form in panel models. Econometrica 72, 15191563.

J. Lee & Y. Hong (2001) Testing for serial correlation of unknown form using wavelet methods. Econometric Theory 17, 386423.

S. Mallat (1989) A theory for multiresolution signal decomposition: The wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence 11, 674693.

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

G.P. Nason & B.W. Silverman (1995) The stationary wavelet transform and some statistical applications. In A. Antoniadis & G. Oppenheim (eds.), Wavelets and Statistics, Vol. 103, pp. 281300. Lecture Notes in Statistics. Springer Verlag.

C.R. Nelson & C.I. Plosser (1982) Trends and random walks in macroeconomic time series: Some evidence and implications. Journal of Monetary Economics 10, 139162.

W.K. Newey & K.D. West (1987) A simple positive semidefinite heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55, 703708.

S. Ng & P. Perron , (2001) Lag length selection and the construction of unit root tests with good size and power. Econometrica 69, 15191554.

H. Park & W. Fuller (1995) Alternative estimators and unit root tests for the autoregressive process. Journal of Time Series Analysis 16, 415429.

D.B. Percival (1995) On estimation of the wavelet variance. Biometrica 82, 619631.

D.B. Percival & H.O. Mofjeld (1997) Analysis of subtidal coastal sea level fluctuations using wavelets. Journal of the American Statistical Association 92, 868880.

P.C.B. Phillips (1986) Understanding spurious regressions in econometrics. Journal of Econometrics 33, 311340.

P.C.B. Phillips (1987a) Time series regression with a unit root. Econometrica 55, 277301.

P.C.B. Phillips (1987b) Towards a unified asymptotic theory for autoregression. Biometrica 74, 535547.

P.C.B. Phillips & S. Ouliaris (1990) Asymptotic properties of residual based tests for cointegration. Econometrica 58, 165193.

P.C.B. Phillips & P. Perron , (1988) Testing for a unit root in time series regression. Biometrica 75, 335346.

P.C.B. Phillips & V. Solo (1992) Asymptotics for linear processes. Annals of Statistics 20, 9711001.

P.C.B. Phillips & Z. Xiao (1998) A primer on unit root testing. Journal of Economic Surveys 12, 423469.

J.B. Ramsey (1999) The contribution of wavelets to the anlaysis of economic and financial data. Philosophical Transactions of the Royal Society of London A 357, 25932606.

J.B. Ramsey , G. Zaslavsky , & D. Usikov (1995) An analysis of U.S. stock price behavior using wavelets. Fractals 3, 377389.

J.B. Ramsey & Z. Zhang (1997) The analysis of foreign exchange data using waveform dictionaries. Journal of Empirical Finance 4, 341372.

J.D. Sargan & A. Bhargava (1983) Testing residuals from least squares regression for being generated by the Gaussian random walk. Econometrica 51, 153174.

P. Schmidt & P.C.B. Phillips (1992) LM tests for a unit root in the presence of deterministic trends. Oxford Bulletin of Economics and Statistics 54, 257288.

C.A. Sims , J.H. Stock , & M.W. Watson (1990) Inference in linear time series models with some unit roots. Econometrica 58, 113144.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
  • URL: /core/journals/econometric-theory
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 21 *
Loading metrics...

Abstract views

Total abstract views: 131 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 28th March 2017. This data will be updated every 24 hours.