Skip to main content Accessibility help


  • Tomás del Barrio Castro (a1) and Denise R. Osborn (a2)


Integration for seasonal time series can take the form of seasonal periodic or nonperiodic integration. When seasonal time series are periodically integrated, we show that any cointegration is either full periodic cointegration or full nonperiodic cointegration, with no possibility of cointegration applying for only some seasons. In contrast, seasonally integrated series can be seasonally, periodically or nonperiodically cointegrated, with the possibility of cointegration applying for a subset of seasons. Cointegration tests are analyzed for periodically integrated series. A residual-based test is examined, and its asymptotic distribution is derived under the null hypothesis of no cointegration. A Monte Carlo analysis shows good performance in terms of size and power. The role of deterministic terms in the cointegrating test regression is also investigated. Further, we show that the asymptotic distribution of the error-correction test for periodic cointegration derived by Boswijk and Franses (1995, Review of Economics and Statistics 77, 436–454) does not apply for periodically integrated processes.The authors gratefully acknowledge the comments of participants at the conference on Unit Root and Cointegration Testing, University of the Algave, September–October 2005, and they particularly thank two anonymous referees and Helmut Lütkepohl (co-editor of this issue of Econometric Theory) for their constructive comments, which have substantially improved the generality of the results in the paper. Tomás del Barrio Castro acknowledges financial support from Ministerio de Educación y Ciencia SEJ2005-07781/ECON.


Corresponding author

Address correspondence to Denise Osborn, Economics, School of Social Sciences, University of Manchester, Manchester M13 9PL, United Kingdom; e-mail:


Hide All


Birchenhall, C.R., R.C. Bladen-Hovell, A.P.L. Chui, D.R. Osborn, & J.P. Smith (1989) A seasonal model of consumption. Economic Journal 99, 837843.
Boswijk, H.P. (1994) Testing for an unstable root in conditional and structural error correction models. Journal of Econometrics 63, 3760.
Boswijk, H.P. & P.H. Franses (1995) Periodic cointegration: Representation and inference. Review of Economics and Statistics 77, 436454.
Boswijk, H.P. & P.H. Franses (1996) Unit roots in periodic autoregression. Journal of Time Series Analysis 17, 221245.
Cubadda, G. (2001) Complex reduced rank models for seasonally cointegrated time series. Oxford Bulletin of Economics and Statistics 63, 497511.
Engle, R.F. & C.W.J. Granger (1987) Cointegration and error correction: Representation, estimation and testing. Econometrica 55, 251276.
Engle, R.F., C.W.J. Granger, S. Hylleberg, & H.S. Lee (1993) Seasonal cointegration: The Japanese consumption function. Journal of Econometrics 55, 275303.
Franses, P.H. (1993) A method to select between periodic cointegration and seasonal cointegration. Economics Letters 41, 710.
Franses, P.H. (1994) A multivariate approach to modelling univariate seasonal time series. Journal of Econometrics 63, 133151.
Franses, P.H. (1995) A vector of quarters representation for bivariate time series. Econometric Reviews 14, 5563.
Franses, P.H. (1996) Periodicity and Stochastic Trends in Economic Time Series. Oxford University Press.
Franses, P.H. & T. Kloek (1995) A periodic cointegration model of quarterly consumption. Applied Stochastic Models and Data Analysis 11, 159166.
Ghysels, E. & D.R. Osborn (2001) The Econometric Analysis of Seasonal Time Series. Cambridge University Press.
Haldrup, N., S. Hylleberg, G. Pons, & A. Sansó (2007) Common periodic correlation features and the interaction of stocks and flows in daily airport data. Journal of Business & Economic Statistics 25, 2132.
Hansen, B. (1992) Efficient estimation and testing of cointegration vectors in the presence of deterministic trends. Journal of Econometrics 53, 87121.
Hylleberg, S., R.F. Engle, C.W.J. Granger, & B.S. Yoo (1990) Seasonal integration and cointegration. Journal of Econometrics 44, 215238.
Johansen, S. & E. Schaumburg (1999) Likelihood analysis of seasonal cointegration. Journal of Econometrics 88, 301339.
Kleibergen, F. & P.H. Franses (1999) Cointegration in a Periodic Autoregression. Econometric Institute Report EI-9906/A, Erasmus University of Rotterdam.
Lee, H.S. (1992) Maximum likelihood inference on cointegration and seasonal cointegration. Journal of Econometrics 54, 149.
Löf, M. & P.H. Franses (2001) On forecasting seasonal cointegrated time series. International Journal of Forecasting 17, 601621.
Osborn, D.R. (1991) The implications of periodically varying coefficients for Seasonal Time Series Processes. Journal of Econometrics 48, 373384.
Osborn, D.R. (2002) Cointegration for Seasonal Time Series Processes. Mimeo, University of Manchester.
Paap, R. & P.H. Franses (1999) On trends and constants in periodic autoregressions. Econometric Reviews 18, 271286.
Phillips, P.C.B. & S.N. Durlauf (1986) Multiple time series regression with integrated processes. Review of Economic Studies 53, 473495.
Phillips, P.C.B. & S. Ouliaris (1990) Asymptotic properties of residual based tests for cointegration. Econometrica 58, 165193.


  • Tomás del Barrio Castro (a1) and Denise R. Osborn (a2)


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