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A TEST FOR WEAK STATIONARITY IN THE SPECTRAL DOMAIN

Published online by Cambridge University Press:  20 July 2018

Javier Hidalgo  and
Pedro C. L. Souza
Javier Hidalgo*
Affiliation:
London School of Economics
Pedro C. L. Souza
Affiliation:
University of Warwick
*
*Address correspondence to Javier Hidalgo, Economics Department, London School of Economics, London WC2A 2AE, UK; e-mail: [email protected]
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Abstract

We examine a test for weak stationarity against alternatives that covers both local-stationarity and break point models. A key feature of the test is that its asymptotic distribution is a functional of the standard Brownian bridge sheet in [0,1]2, so that it does not depend on any unknown quantity. The test has nontrivial power against local alternatives converging to the null hypothesis at a T−1/2 rate, where T is the sample size. We also examine an easy-to-implement bootstrap analogue and present the finite sample performance in a Monte Carlo experiment. Finally, we implement the methodology to assess the stability of inflation dynamics in the United States and on a set of neuroscience tremor data.

Type
ARTICLES
Information
Econometric Theory , Volume 35 , Issue 3 , June 2019 , pp. 547 - 600
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

We thank the Associate Editor and two referees for very helpful comments. Any remaining errors are our sole responsibility.

References

REFERENCES

Anderson, T.W. & Walker, A.M. (1964) On the asymptotic distribution of the autocorrelations of a sample from a linear stochastic process. Annals of Mathematical Statistics 35, 1296–1303.CrossRefGoogle Scholar
Aue, A. & Horvarth, L. (2013) Structural breaks in time series. Journal of Time Series Analysis 34, 1–16.CrossRefGoogle Scholar
Bickel, P.J. & Wichura, M.J. (1971) Convergence criteria for multiparameter stochastic processes and its applications. Annals of Mathematical Statistics 42, 1656–1670.CrossRefGoogle Scholar
Bandyopadhy, A.Y., Jentsch, C., & Subba Rao, S. (2017) Spectral domain test for stationarity of spatio-temporal data. Journal of Time Series Analysis 38, 326–351.CrossRefGoogle Scholar
Brillinger, D.R. (1981) Time Series, Data Analysis and Theory. Holden-Day.Google Scholar
Brockwell, P.J. & Davis, R.A. (1991) Time Series: Theory and Methods. Springer-Verlag.CrossRefGoogle Scholar
Dahlhaus, R. (1996) On the Kulback-Leibler information divergence of locally stationary processes. Stochastic Processes and its Applications 62, 139–168.CrossRefGoogle Scholar
Dahlhaus, R. (1997) Fitting time series models to nonstationary processes. Annals of Statistics 25, 1–37.Google Scholar
Dahlhaus, R. (2009) Local inference for locally stationary time series based on the empirical spectral measure. Journal of Econometrics 151, 101–112.CrossRefGoogle Scholar
Dahlhaus, R. & Polonik, W. (2009) Empirical spectral processes for locally stationary time series. Bernoulli 15, 1–39.CrossRefGoogle Scholar
Dalla, V., Giraitis, L., & Hidalgo, J. (2005) Consistent estimation of the memory parameter for nonlinear time series. Journal of Time Series Analysis 27, 211–255.CrossRefGoogle Scholar
Davis, R.A., Huang, D., & Yao, Y. (1995) Testing for a change in the parameter values and order of an autoregressive model. Annals of Statistics 23, 282–304.CrossRefGoogle Scholar
Delgado, M.A., Hidalgo, J., & Velasco, C. (2005) Distribution free goodness-of-fit tests for linear processes. Annals of Statistics 33, 2568–2609.CrossRefGoogle Scholar
Dette, H., Preuss, P., & Vetter, M. (2011) A measure of stationarity in locally stationary processes with applications to testing. Journal of the American Statistical Association 106(495), 1113–1124.CrossRefGoogle Scholar
Dwivedi, Y. & Subba Rao, S. (2011) A test for second-order stationarity of a time series based on the discrete Fourier transform. Journal of Time Series Analysis 32, 68–91.CrossRefGoogle Scholar
Fragkeskou, M. & Paparoditis, E. (2016) Inference for the fourth-order innovation cumulant in linear time series. Journal of Time Series Analysis 37, 240–266.CrossRefGoogle Scholar
Giacomini, R., Politis, D.N., & White, H. (2013) A warp-speed method for conducting Monte Carlo experiments involving bootstrap estimators. Econometric Theory 29, 567–589.CrossRefGoogle Scholar
Grenander, U. & Rosenblatt, M. (1957) Statistical Analysis of Stationary Time Series. Wiley.CrossRefGoogle Scholar
Hannan, E.J. (1970) Multiple Time Series. Wiley.CrossRefGoogle Scholar
Härdle, W. & Mammen, E. (1993) Comparing nonparametric versus parametric regression fits. Annals of Statistics 21, 1926–1947.CrossRefGoogle Scholar
Hidalgo, J. (2007) Specification testing for regression models with dependent data. Journal of Econometrics 143, 143–163.CrossRefGoogle Scholar
Hidalgo, J. & Seo, M. (2015) Specification with lattice processes. Econometric Theory 31, 294–336.CrossRefGoogle Scholar
Ibragimov, I.A. & Rozanov, Y.A. (1978) Gaussian Random Processes. Springer-Verlag.CrossRefGoogle Scholar
Jentsch, C. & Subba Rao, S. (2015) A test for second order stationarity of a multivariate time series. Journal of Econometrics 185, 124–161.CrossRefGoogle Scholar
Lucas, R.E. (1976) Econometric policy evaluation: A critique. In Brunner, K. & Meltzer, A. (eds.), Carnegie-Rochester Conference Series on Public Policy, Vol. 1, pp. 19–46. North Holland Publishing Company.Google Scholar
Paparoditis, E. (2009) Testing temporal constancy of the spectral structure of a time series. Bernoulli 15, 1190–1221.CrossRefGoogle Scholar
Perron, P. (2006) Dealing with structural breaks. In Hassani, H., Mills, T. C., & Patterson, K. (eds.), Pelgrave Handbook of Econometrics, Vol. 1, Econometric Theory, pp. 278–352. Palgrave Macmillan UK.Google Scholar
Picard, D. (1985) Testing and estimating change points in time series. Advances in Time Series Analysis 17, 841–867.Google Scholar
Preuss, P., Vetter, M., & Dette, H. (2013) A test for stationarity based on empirical processes. Bernoulli 19, 2715–2749.CrossRefGoogle Scholar
Priestley, M.B. (1965) Evolutionary spectra and non-stationary processes. Journal of the Royal Statistical Society, Series B 62, 204–237.Google Scholar
von Sachs, R. & Neumann, M.H. (2000) A wavelet-based test for stationarity. Journal of Time Series Analysis 21(5), 597–613.CrossRefGoogle Scholar
Welch, P.D. (1967) The use of fast Fourier Transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms. IEEE Transaction of Audio and Electroacoustic 15, 70–73.CrossRefGoogle Scholar
Whittle, P. (1963) Prediction and Regulation. Van Nostrand.Google Scholar