Skip to main content Accessibility help


  • Tomás del Barrio Castro (a1), Denise R. Osborn (a2) and A.M. Robert Taylor (a3)


In this paper we extend the large-sample results provided for the augmented Dickey–Fuller test by Said and Dickey (1984, Biometrika 71, 599–607) and Chang and Park (2002, Econometric Reviews 21, 431–447) to the case of the augmented seasonal unit root tests of Hylleberg, Engle, Granger, and Yoo (1990, Journal of Econometrics 44, 215–238), inter alia. Our analysis is performed under the same conditions on the innovations as in Chang and Park (2002), thereby allowing for general linear processes driven by (possibly conditionally heteroskedastic) martingale difference innovations. We show that the limiting null distributions of the t-statistics for unit roots at the zero and Nyquist frequencies and joint F-type statistics are pivotal, whereas those of the t-statistics at the harmonic seasonal frequencies depend on nuisance parameters that derive from the lag parameters characterizing the linear process. Moreover, the rates on the lag truncation required for these results to hold are shown to coincide with the corresponding rates given in Chang and Park (2002); in particular, an o(T1/2) rate is shown to be sufficient.


Corresponding author

*Address correspondence to A.M. Robert Taylor, School of Economics, University of Nottingham NG7 2RD, United Kingdom; e-mail:


Hide All
Beaulieu, J.J. & Miron, J.A. (1993) Seasonal unit roots in aggregate U.S. data. Journal of Econometrics 55, 305328.
Berk, K.N. (1974) Consistent autoregressive spectral estimates. Annals of Statistics 2, 389502.
Boswijk, H.P. & Franses, P.H. (1996) Unit roots in periodic autoregressions. Journal of Time Series Analysis 17, 221245.
Box, G.E.P. & Jenkins, G.M. (1976) Time Series Analysis: Forecasting and Control, rev. ed. Holden-Day.
Burridge, P. & Taylor, A.M.R. (2001) On the properties of regression-based tests for seasonal unit roots in the presence of higher-order serial correlation. Journal of Business & Economic Statistics 3, 374379.
Chang, Y. & Park, J.Y. (2002) On the asymptotics of ADF tests for unit roots. Econometric Reviews 21, 431447.
Davidson, J. (1994) Stochastic Limit Theory. Oxford University Press.
Davis, P.J. (1979) Circulant Matrices. Wiley-Interscience.
del Barrio Castro, T. & Osborn, D.R. (2011) HEGY tests in the presence of moving averages. Oxford Bulletin of Economic and Statistics 73, 691704.
del Barrio Castro, T., Osborn, D.R., & Taylor, A.M.R. (2011) On Augmented HEGY Tests for Seasonal Unit Roots. Working paper 11/02, Granger Centre for Time Series Econometrics.
Fuller, W.A. (1996) Introduction to Statistical Time Series, 2nd ed.Wiley.
Ghysels, E., Lee, H.S., & Noh, J. (1994) Testing for unit roots in seasonal time series: Some theoretical extensions and a Monte Carlo investigation. Journal of Econometrics 62, 415442.
Ghysels, E. & Osborn, D.R. (2001) The Econometric Analysis of Seasonal Time Series. Cambridge University Press.
Gray, R.M. (2006) Toeplitz and Circulant Matrices, A Review. Foundation and Trends® in Communications and Information Theory. Now Publishers.
Hall, A.R. (1994) Testing for a unit root in time series with pretest data-based model selection. Journal of Business & Economic Statistics 12, 461470.
Hylleberg, S., Engle, R.F., Granger, C.W.J., & Yoo, B.S. (1990) Seasonal integration and cointegration. Journal of Econometrics 44, 215238.
Ng, S. & Perron, P. (1995) Unit root tests in ARMA models with data dependent methods for selection of the truncation lag. Journal of the American Statistical Association 90, 268281.
Ng, S. & Perron, P. (2001) Lag length selection and the construction of unit root tests with good size and power. Econometrica 69, 15191554.
Osborn, D.R. & Rodrigues, P.M.M. (2002) Asymptotic distributions of seasonal unit root tests: A unifying approach. Econometric Reviews 21, 221241.
Phillips, P.C.B. & Durlauf, S.N. (1986) Multiple time series regression with integrated processes. Review of Economic Studies 53, 473495.
Rodrigues, P.M.M. & Taylor, A.M.R. (2004a) Alternative estimators and unit root tests for seasonal autoregressive processes. Journal of Econometrics 120, 3573.
Rodrigues, P.M.M. & Taylor, A.M.R. (2004b) Asymptotic distributions for regression-based seasonal unit root test statistics in a near-integrated model. Econometric Theory 20, 645670.
Rodrigues, P.M.M. & Taylor, A.M.R. (2007) Efficient tests of the seasonal unit root hypothesis. Journal of Econometrics 141, 548573.
Said, S.E. & Dickey, D.A. (1984) Testing for unit roots in autoregressive-moving average models of unknown order. Biometrika 71, 599607.
Smith, R.J. & Taylor, A.M.R. (1998) Additional critical values and asymptotic representations for seasonal unit root tests. Journal of Econometrics 85, 269288.
Smith, R.J. & Taylor, A.M.R. (1999) Likelihood ratio tests for seasonal unit roots. Journal of Time Series Analysis 20, 453476.
Smith, R.J., Taylor, A.M.R. & del Barrio Castro, T. (2009) Regression-based seasonal unit root tests. Econometric Theory 25, 527560.
Taylor, A.M.R. (2005) Variance ratio tests of the seasonal unit root hypothesis. Journal of Econometrics 124, 3354.


  • Tomás del Barrio Castro (a1), Denise R. Osborn (a2) and A.M. Robert Taylor (a3)


Full text views

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

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed