Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T18:27:15.946Z Has data issue: false hasContentIssue false
  • English
  • Français

FULLY MODIFIED ESTIMATION OF SEASONALLY COINTEGRATED PROCESSES

Published online by Cambridge University Press:  05 March 2010

Stéphane Gregoir
Stéphane Gregoir*
Affiliation:
EDHEC Business School
*
*Address correspondence to Stéphane Gregoir, EDHEC Business School 12 bis rue de la victoire 75009 Paris, France; e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

We extend the framework of the fully modified, ordinary least squares (OLS) estimator introduced by Phillips and Hansen (1990) to the case of seasonally cointegrated processes at a given frequency. First we extend a weak convergence result of sample covariance matrices (Phillips, 1988) to the case of seasonal unit roots. Using a complex number framework, we then show that we can take into account the constraints that exist in a situation of seasonal cointegration as illustrated in Gregoir (1999a) and derive estimates of the cointegration vectors that allow for asymptotic normal inference. This allows us to propose a test whose null hypothesis is the existence of seasonal cointegration. A Monte Carlo exercise investigates the finite sample properties of this test procedure. The paper closes with the analysis of situations in which there exist more than one frequency at which seasonal cointegration can be observed.

Type
ARTICLES
Information
Econometric Theory , Volume 26 , Issue 5 , October 2010 , pp. 1491 - 1528
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Anderson, T.W. (1971) The Statistical Analysis of Time Series. Wiley.Google Scholar
Bierens, H. (2001) Complex unit roots and business cycles: Are they real? Econometric Theory 17(5), 962–983.CrossRefGoogle Scholar
Brockwell, P.J. & Davies, R.A. (1991) Time series. Theory and methods, 2nd ed. Springer-Verlag.CrossRefGoogle Scholar
Caner, M. (1998) A locally optimal seasonal unit-root test. Journal of Business and Economic Statistics 15, 349–356.Google Scholar
Canova, F. & Hansen, B.E. (1995) Are seasonal patterns constant over time? A test for seasonal stability. Journal of Business and Economic Statistics 13, 237–252.Google Scholar
Chan, N.H. & Wei, C.Z. (1988) Limiting distribution of least squares estimates of unstable auto regressive processes. The Annals of Statistics. 16(1), 367–401.CrossRefGoogle Scholar
Cubadda, G. (2001) Complex reduced rank models for seasonally cointegrated time series. Oxford Bulletin of Economics and Statistics 63, 497–511.CrossRefGoogle Scholar
De Jong, D.N., Nankervis, J.C., Savin, N.E., & Whiteman, C.H. (1992) The power problems of unit root tests in time series with autoregressive errors. Journal of Econometrics 53, 323–343.CrossRefGoogle Scholar
Dickey, D.A., Hasza, D.P., & Fuller, W. (1984) Testing for unit roots in seasonal time series. Journal of the American Statistical Association 79, 355–367.CrossRefGoogle Scholar
Elliott, G. (1998) On the robustness of cointegration methods when regressors almost have unit roots. Econometrica 66(1), 149–158.CrossRefGoogle Scholar
Elliott, G. (1999) Efficient tests for a unit root when the initial observation is drawn from its unconditional distribution. International Economic Review 40, 767–783.CrossRefGoogle Scholar
Elliott, G., Rothenberg, T.J., & Stock, J.H. (1996) Efficient tests for an autoregressive unit root. Econometrica 64(4), 813–836.CrossRefGoogle Scholar
Engle, R.F., Granger, C.W.J., & Hallman, J. (1989) Merging short- and long-run forecasts: An application of seasonal co-integration to monthly electricity sales forecasting. Journal of Econometrics 40, 45–62.CrossRefGoogle Scholar
Engle, R.F., Granger, C.W.J., Hylleberg, S., & Lee, H.S. (1993) Seasonal cointegration: The Japanese consumption function. Journal of Econometrics 55, 275–298.CrossRefGoogle Scholar
Goodman, N.R. (1963) Statistical analysis based on a multivariate complex Gaussian distribution (an introduction). Annals of Statistics 34, 152–177.CrossRefGoogle Scholar
Granger, C.W.J. (1981) Some properties of time series data and their use in econometric model specification. Journal of Econometrics 16, 121–130.CrossRefGoogle Scholar
Gregoir, S. (1999a) Multivariate time series with various hidden unit roots: Part I: Integral operator algebra and representation theory. Econometric Theory 15, 435–468.CrossRefGoogle Scholar
Gregoir, S. (1999b) Multivariate time series with various hidden unit roots: Part II: Estimation and testing. Econometric Theory 15, 469–518.CrossRefGoogle Scholar
Gregoir, S. (2006) Efficient tests for the presence of a pair of complex conjugate unit roots in real time series. Journal of Econometrics 130(1), 45–100.CrossRefGoogle Scholar
Gregoir, S. & Laroque, G. (1993). Multivariate time series: A polynomial error correction representation theorem. Econometric Theory 9, 329–342.CrossRefGoogle Scholar
Gregoir, S. & Laroque, G. (1994) Polynomial cointegration estimation and test. Journal of Econometrics 63 (1), 183–214.CrossRefGoogle Scholar
Hall, P. & Heyde, C.C. (1980) Martinale limit theory and its application. Academic Press.Google Scholar
Hannan, E.J. (1970) Multiple Time Series. Wiley-Interscience.CrossRefGoogle Scholar
Hansen, B.E. (1992) Convergence to stochastic integrals for dependent heterogeneous processes. Econometric Theory 8(4), 489–500.CrossRefGoogle Scholar
Harris, D. (1997) Principal components analysis of cointegrated time series. Econometric Theory 13, 529–557.CrossRefGoogle Scholar
Hasza, D.P. & Fuller, W. (1982) Testing for nonstationarity parameter specifications: Seasonal time series models. The Annals of Statistics 10, 1209–1216.CrossRefGoogle Scholar
Hylleberg, S., Engle, R.F., Granger, C.W.J., & Yoo, B.S. (1991) Seasonal integration and cointegration. Journal of Econometrics 44, 215–238.CrossRefGoogle Scholar
Johansen, S. (1988) Statistical analysis of cointegration vectors. Journal of Economics Dynamics and Control 12, 231–254.CrossRefGoogle Scholar
Johansen, S. (1991) Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica 59, 1551–1580.CrossRefGoogle Scholar
Johansen, S. & Schaumburg, E. (1998) Likelihood analysis of seasonal cointegration. Journal of Econometrics 88(2), 301–339.CrossRefGoogle Scholar
Kwiatkowski, D., Phillips, P.C.B., Schmidt, P., & Shin, Y. (1992) Testing the null hypothesis of stationarity against the alternative of a unit root. Journal of Econometrics 54, 159–178.CrossRefGoogle Scholar
Lee, J.H. (1992) Maximum likelihood inference on cointegration and seasonal cointegration. Journal of Econometrics 54, 1–48.CrossRefGoogle Scholar
Maravall, A. (1993) Stochastic linear trends. Journal of Econometrics, 56, 5–37.CrossRefGoogle Scholar
Müller, U.K. & Elliott, G. (2003) Tests for unit roots and the initial condition. Econometrica 71, 1269–1286.CrossRefGoogle Scholar
Newey, W.K. & West, K.D. (1994) Automatic lag selection in covariance matrix estimation. Review of Economic Studies 61, 631–653.CrossRefGoogle Scholar
Park, J.Y. (1992) Canonical cointegration regression. Econometrica 60, 119–143.CrossRefGoogle Scholar
Phillips, P.C.B. (1988) Weak convergence of sample covariance matrices to stochastic integrals via martingale approximation. Econometric Theory 4, 528–533.CrossRefGoogle Scholar
Phillips, P.C.B. (1991) Optimal inference in cointegrated systems. Econometrica 59(2), 283–306.CrossRefGoogle Scholar
Phillips, P.C.B. & Hansen, B.E. (1990) Statistical inference in instrumental variables regression with I(1) processes. Reviews of Economic Studies 57, 99–125.CrossRefGoogle Scholar
Phillips, P.C.B. & Perron, P. (1988) Testing for a unit root in time series regression. Biometrika 75(2), 971–1001.CrossRefGoogle Scholar
Phillips, P.C.B. & Solo, V. (1992) Asymptotics for linear processes. Annals of Statistics 20(2); 971–1001.CrossRefGoogle Scholar
Saikkonen, P. (1991) Asymptotically efficient estimation of cointegration regression. Econometric Theory 7, 1–21.CrossRefGoogle Scholar
Schwert, G.W. (1989) Test for unit roots: A Monte Carlo investigation. Journal of Business and Economic Statistics 7, 147–159.Google Scholar
Shin, Y. (1994) A residual-based test of the null of cointegration against the alternative of no cointegration. Econometric Theory 10, 91–115.CrossRefGoogle Scholar
Smith, R.J. & Taylor, A.M.R. (1998) Additional critical values and asymptotic representations for seasonal unit root tests. Journal of Econometrics 85, 269–288.CrossRefGoogle Scholar
Stock, J.H. & Watson, M.W. (1993) A simple estimator of cointegrating vectors in higher order integrated systems. Econometrica 61, 783–820.CrossRefGoogle Scholar
Taylor, A.M.R. (2003) On the asymptotic properties of some seasonal unit root tests. Econometric Theory 19, 311–321.CrossRefGoogle Scholar
Tsay, R.S. & Tiao, G.C. (1990) Asymptotic properties of multivariate nonstationary processes with applications to auto regressions. Annals of Statistics 18, 220–250.CrossRefGoogle Scholar