On cointegration for processes integrated at different frequencies

del Barrio Castro, Tomás and Cubada, Ginaluca and Osborn, Denise R. (2020): On cointegration for processes integrated at different frequencies.

Abstract

This paper explores the possibility of cointegration existing between processes integrated at di¤erent frequencies. Using the demodulator operator, we show that such cointegration can exist and explore its form using both complex- and real-valued representations. A straightforward approach to test for the presence of cointegration between processes integrated at di¤erent frequencies is proposed, with a Monte Carol study and an application showing that the testing approach works well.

• Item Type: MPRA Paper
• Language: English
• Keywords: Periodic Cointegration, Polynomial Cointegration, Demodulator Operator.
• Subjects: C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables > C32 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes ; State Space Models

References

Ahn, S.K. and Reinsel G.C. (1994) Estimation of partially nonstationary vector autoregressive models with seasonal behaviour, Journal of Econometrics, 62, 317-350.

Bauer D. and Wagner M. (2012) A state space canonical form for unit root processes, Econometric Theory, 28, 1313-1349.

Bierens, H.J. (2001). Complex unit roots and business cycles: Are they real?, Econometric Theory, 17, 962-983.

Bierens, H.J. and Martins, L.F. (2010). Time-varying cointegration, Econometric Theory, 26, 1453-1490.

Birchenhall, C.R., Bladen-Hovell, R.C., Chui, A.P.L., Osborn, D.R. and Smith, J.P. (1989). A seasonal model of consumption, Economic Journal, 99, 837-843.

Boswijk, H.P. and Franses, P.H. (1996). Unit roots in periodic autoregressions, Journal of Time Series Analysis, 17, 221-245.

Caporale, G.M., Cuñado, J. and Gil-Alana, L.A. (2013). Modelling long-run trends and cycles in �nancial time series data, Journal of Time Series Analysis, 34, 405-421.

Cubadda G. (1999), Common cycles in seasonal non-stationary time series, Journal of Applied Econometrics, 14, 273-291.

Cubadda, G. (2001). Complex reduced rank models for seasonally cointegrated time series, Oxford Bulletin of Economics and Statistics, 63, 497-511. 13

Cubadda, G., and Omtzigt, P. (2005). Small-sample improvements in the statistical analysis of seasonally cointegrated systems, Computational Statististics & Data Analysis, 49, 333�348

del Barrio Castro, T. (2007). Using the HEGY procedure when not all unit roots are present, Journal of Time Series Analysis, 28, 910-922.

del Barrio Castro, T., A. Bodnar, and A. Sansó (2017). Numerical distribution functions for seasonal unit root tests with OLS and GLS detrending, Computational Statistics, 32, 1533�1568.

del Barrio Castro, T. and Osborn, D.R. (2008). Testing for seasonal unit roots in periodic integrated autoregressive processes, Econometric Theory, 24, 1093-1129.

del Barrio Castro, T., Osborn, D.R. and Taylor, A.M.R. (2012). On augmented HEGY tests for seasonal unit roots, Econometric Theory, 28, 1121-1143.

del Barrio Castro, T., Osborn, D.R. and Taylor, A.M.R. (2016). The performance of lag selection and detrending methods for HEGY seasonal unit root tests, Econometric Reviews, 35, 122-168.

del Barrio Castro, T., P.M.M. Rodrigues & Taylor, A.M.R. (2018) Semi-Parametric Seasonal Unit Root Tests, Econometric Theory, 34, 447-476

del Barrio Castro, T., P.M.M. Rodrigues & Taylor, A.M.R. (2019) Temporal Aggregation of Seasonally Near-Integrated Processes, Journal of Time Series Analysis, 40, 873-886

Engle, R.F., Granger, C.W.J., Hylleberg, S. and Lee, H.S. (1993). The Japanese consumption function, Journal of Econometrics, 55, 275-298.

Franses, P.H. (1994). A multivariate approach to modeling univariate seasonal time series, Journal of Econometrics, 63, 133-151.

Franses, P.H. and Paap, R. (2004). Periodic Time Series Models, Oxford University Press, Oxford, UK.

Gardner, W.A., Napolitano, A. and Paura, L. (2006). Cyclostationarity: Half a century of research, Signal Processing, 86, 639-697.

Ghysels, E. and Osborn, D.R. (2001). The Econometric Analysis of Seasonal Time Series, Cambridge University Press, Cambridge, UK.

Granger, C.W.J. and Lee, T.H. (1989). Investigation of production, sales and inventory relationships using multicointegration and non-symmetric error correction models, Journal of Applied Econometrics, 4, Supplement S1, S145-S159.

Gray, H.L., Zhang, N.-F. and Woodward, W.A. (1989). On generalized fractional processes, Journal of Time Series Analysis, 1989, 233-257.

Gregoir, S. (1999a). Multivariate time series with various hidden unit roots, Part I: Integral operator algebra and representation theorem, Econometric Theory, 15, 435-468.

Gregoir, S. (1999b). Multivariate time series with various hidden unit roots, Part II: Estimation and testing, Econometric Theory, 15, 469-518.

Gregoir, S. (2006). E¢ cient tests for the presence of a pair of complex conjugate unit roots in real time series, Journal of Econometrics, 130, 45-100.

Gregoir, S. (2010). Fully modi�ed estimation of seasonally cointegrated processes, Econometric Theory, 26, 1491-1528.

Hall, S.G., Psaradakis, Z. and Sola, M. (1997). Cointegration and changes in regime: The Japanese consumption function, Journal of Applied Econometrics, 12, 151-168.

Hamilton, J.D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica, 57, 357-384.

Hamilton, J.D. (1994). Time Series Analysis, Princeton University Press, Princeton, USA. 14

Hylleberg, S., Engle, R.F., Granger, C.W.J. and Yoo, B.S. (1990). Seasonal integration and cointegration, Journal of Econometrics, 44, 215-238.

Johansen, S. (1996). Likelihood Based Inference on Cointegration in the Vector Autoregressive Model (2nd ed.). Oxford University Press, Oxford.

Johansen, S. and Schaumburg, E. (1999). Likelihood analysis of seasonal cointegration, Journal of Econo- metrics, 88, 301-339.

Osborn, D.R. (1993). Discussion: Seasonal cointegration, Journal of Econometrics, 55, 299-303.

Pagano, M. (1978). On periodic and multiple autoregressions, Annals of Statistics, 6, 1310-1317.

Paap R. and Franses P. H. (1999) On trends and constants in periodic autoregressions, Econometric Reviews, 18, 271-286.

Park, J.Y. and Hahn, S.B. (1999). Cointegrating regressions with time varying coe¢ cients, Econometric Theory, 15, 664-703.

Smith, R.J., Taylor, A.M.R. and del Barrio Castro, T. (2009). Regression-based seasonal unit root tests, Econometric Theory, 25, 527-560.

Tanaka, K. (2008) Analysis of models with complex roots on the unit circle, Journal of the Japan Statistical Society, 38, 145�155

Tiao, G.C. and Grupe, M.R. (1980). Hidden periodic autoregressive-moving average models in time series data, Biometrika, 67, 365-373.