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ON TESTING FOR SERIAL CORRELATION WITH A WAVELET-BASED SPECTRAL DENSITY ESTIMATOR IN MULTIVARIATE TIME SERIES

Published online by Cambridge University Press:  23 May 2006

Pierre Duchesne
Pierre Duchesne
Affiliation:
Université de Montréal
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Abstract

A new one-sided test for serial correlation in multivariate time series models is proposed. The test is based on a comparison between a multivariate spectral density estimator and the spectral density under the null hypothesis of no serial correlation. Duchesne and Roy (2004, Journal of Multivariate Analysis 89, 148–180) considered a multivariate kernel-based spectral density estimator. However, when the spectral density exhibits irregular features (because of strong autocorrelation or seasonality, among other factors), it is expected that a multivariate wavelet-based spectral density estimator will capture more effectively the local behavior of the spectral density. We consider a test based on a wavelet spectral density estimator, which represents a generalization of a test proposed by Lee and Hong (2001, Econometric Theory 17, 386–423). The asymptotic distribution of the new test is established under the null hypothesis, which is N(0,1). We propose and justify a suitable data-driven method to choose the smoothing parameter of the wavelet estimator (called the finest scale in that context). The new test should be powerful when the spectral density contains peaks or bumps. This is confirmed in a simulation study, where kernel-based and wavelet-based estimators are compared.The author thanks the co-editor Pentti Saikkonen and two referees for their constructive remarks and suggestions. Many thoughtful comments of the referees led to significant improvements of the paper. This work was supported by grants from the National Science and Engineering Research Council of Canada and the Fonds québécois de la recherche sur la nature et les technologies du Québec (Canada).

Type
Research Article
Information
Econometric Theory , Volume 22 , Issue 4 , August 2006 , pp. 633 - 676
Copyright
© 2006 Cambridge University Press

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References

REFERENCES

Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817–858.Google Scholar
Andrews, D.W.K. & W. Ploberger (1996) Testing for serial correlation against an ARMA(1,1) process. Journal of the American Statistical Association 91, 1331–1342.Google Scholar
Box, G.E.P. & D.A. Pierce (1970) Distribution of residual autocorrelations in autoregressive integrated moving average time series models. Journal of the American Statistical Association 65, 1509–1526.Google Scholar
Brockwell, P.J. & R.A. Davis (1991) Time Series: Theory and Methods, 2nd ed. Springer-Verlag.
Brown, B.M. (1971) Martingale central limit theorems. Annals of Mathematical Statistics 42, 59–66.Google Scholar
Chitturi, R.V. (1974) Distribution of residual autocorrelations in multiple autoregressive schemes. Journal of the American Statistical Association 69, 928–934.Google Scholar
Chitturi, R.V. (1976) Distribution of multivariate white noise autocorrelations. Journal of the American Statistical Association 71, 223–226.Google Scholar
Daubechies, I. (1992) Ten Lectures on Wavelets. SIAM.
Duchesne, P. & R. Roy (2004) On consistent testing for serial correlation of unknown form in vector time series models. Journal of Multivariate Analysis 89, 148–180.Google Scholar
Durbin, J. & G.S. Watson (1950) Testing for serial correlation in least squares regression, part I. Biometrika 37, 409–428.Google Scholar
Durbin, J. & G.S. Watson (1951) Testing for serial correlation in least squares regression, part II. Biometrika 38, 159–178.Google Scholar
Fuller, W.A. (1996) Introduction to Statistical Time Series, 2nd ed. Wiley.
Hannan, E. (1970) Multiple Time Series. Wiley.
Hernández, E. & G. Weiss (1996) A First Course on Wavelets. CRC Press.
Hong, Y. (1996) Consistent testing for serial correlation of unknown form. Econometrica 64, 837–864.Google Scholar
Hong, Y. (2001) Wavelet-Based Estimation for Heteroskedasticity and Autocorrelation Consistent Variance-Covariance Matrices. Working paper, Department of Economics and Department of Statistical Science, Cornell University.
Hong, Y. & R.D. Shehadeh (1999) A new test for ARCH effects and its finite-sample performance. Journal of Business & Economic Statistics 17, 91–108.Google Scholar
Hosking, J. (1980) The multivariate portmanteau statistic. Journal of the American Statistical Association 75, 602–608.Google Scholar
Hosking, J. (1981a) Equivalent forms of the multivariate portmanteau statistic. Journal of the Royal Statistical Society, Series B 43, 261–262 (Correction published in 1989, vol. 51, p. 303).Google Scholar
Hosking, J. (1981b) Lagrange-multiplier tests of multivariate time series models. Journal of the Royal Statistical Society, Series B 43, 219–230.Google Scholar
Johnson, M.E. (1987) Multivariate Statistical Simulation. Wiley.
Lee, J. & Y. Hong (2001) Testing for serial correlation of unknown form using wavelet methods. Econometric Theory 17, 386–423.Google Scholar
Li, W.K. & A.I. McLeod (1981) Distribution of the residual autocorrelations in multivariate ARMA time series models. Journal of the Royal Statistical Society, Series B 43, 231–239.Google Scholar
Ljung, G.M. & G.E.P. Box (1978) On a measure of lack of fit in time series models. Biometrika 65, 297–303.Google Scholar
Lütkepohl, H. (1993) Introduction to Multiple Time Series Analysis, 2nd ed. Springer-Verlag.
Mallat, S.G. (1989) A theory for multiresolution signal decomposition: The wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence 11, 674–693.Google Scholar
Meyer, Y. (1986) Ondelettes, fonctions splines et analyse graduées. Lectures given at the University of Torino, Italy.
Meyer, Y. (1992) Wavelets and Operators. Cambridge Studies in Advanced Mathematics 37. Cambridge University Press.
Neumann, M.H. (1996) Spectral density estimation via nonlinear wavelet methods for stationary non-Gaussian time series. Journal of Time Series Analysis 17, 601–633.Google Scholar
Paparoditis, E. (2000a) Spectral density based goodness-of-fit tests for time series analysis. Scandinavian Journal of Statistics 27, 143–176.Google Scholar
Paparoditis, E. (2000b) On some power properties of goodness-of-fit tests for time series models. In C.A. Charalambides, M.V. Koutras, & N. Balakrishnan (eds.), Probability and Statistical Models with Applications, 333–348. Chapman and Hall.
Paparoditis, E. (2005) Testing the fit of a vector autoregressive moving average model. Journal of Time Series Analysis 26, 543–568.Google Scholar
Parzen, E. (1957) On consistent estimates of the spectrum of a stationary time series. Annals of Mathematical Statistics 28, 329–348.Google Scholar
Priestley, M.B. (1981) Spectral Analysis and Time Series, vol. 1: Univariate Series. Academic Press.
Priestley, M.B. (1996) Wavelets and time-dependent spectral analysis. Journal of Time Series Analysis 17, 85–103.Google Scholar
Robinson, P.M. (1991a) Testing strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. Journal of Econometrics 47, 67–84.Google Scholar
Robinson, P.M. (1991b) Automatic frequency domain inference on semiparametric and non-parametric models. Econometrica 59, 1329–1363.Google Scholar
Taniguchi, M. & Y. Kakizawa (2000) Asymptotic Theory of Statistical Inference for Time Series. Springer-Verlag.
Vidakovic, B. (1999) Statistical Modeling by Wavelets. Wiley.
Walter, G. (1994) Wavelets and Other Orthogonal Systems with Applications. CRC Press.
Wei, W.W.-S. (1990) Time Series Analysis, Univariate and Multivariate Methods. Addison-Wesley.