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ASYMPTOTIC DISTRIBUTIONS FOR REGRESSION-BASED SEASONAL UNIT ROOT TEST STATISTICS IN A NEAR-INTEGRATED MODEL

Published online by Cambridge University Press:  01 August 2004

Paulo M.M. Rodrigues  and
A.M. Robert Taylor
Paulo M.M. Rodrigues
Affiliation:
University of Algarve
A.M. Robert Taylor
Affiliation:
University of Birmingham
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Abstract

In this paper we derive representations for the limiting distributions of the regression-based seasonal unit root test statistics of Hylleberg, Engle, Granger, and Yoo (1990, Journal of Econometrics 44, 215–238) and Beaulieu and Miron (1993, Journal of Econometrics 55, 305–328), inter alia, when the underlying process displays near seasonal integration. Our results generalize those presented in previous studies by allowing for an arbitrary seasonal periodicity (including the nonseasonal case), a wide range of possible assumptions on the initial conditions, a range of (seasonal) deterministic mean effects, and finite autoregressive behavior in the driving shocks. We use these representations to simulate the asymptotic local power functions of the seasonal unit root tests, demonstrating a significant dependence on serial correlation nuisance parameters in the case of the pairs of t-statistics, but not the associated F-statistic, for unit roots at the seasonal harmonic frequencies. Monte Carlo simulation results are presented that suggest that the local limiting distribution theory provides a good approximation to the finite-sample behavior of the statistics. Our results lend further weight to the advice of previous authors that inference on the unit root hypothesis at the seasonal harmonic frequencies should be based on the F-statistic, rather than on the associated pairs of t-ratios.We are grateful to Bruce Hansen and two anonymous referees for their helpful comments and suggestions on earlier versions of this paper.

Type
Research Article
Information
Econometric Theory , Volume 20 , Issue 4 , August 2004 , pp. 645 - 670
Copyright
© 2004 Cambridge University Press

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References

REFERENCES

Beaulieu, J.J. & J.A. Miron (1993) Seasonal unit roots in aggregate U.S. data. Journal of Econometrics 55, 305–328.Google Scholar
Burridge, P. & A.M.R. Taylor (2001) On the properties of regression-based tests for seasonal unit roots in the presence of higher-order serial correlation. Journal of Business and Economic Statistics 19, 374–379.Google Scholar
Burridge, P. & A.M.R. Taylor (2004) Bootstrapping the quarterly HEGY seasonal unit root tests. Journal of Econometrics, forthcoming.Google Scholar
Canjels, E. & M.W. Watson (1997) Estimating deterministic trends in the presence of serially correlated errors. Review of Economics and Statistics 79, 184–200.Google Scholar
Chan, N.H. (1989) On the nearly nonstationary seasonal time series. Canadian Journal of Statistics 17, 279–284.Google Scholar
Chan, N.H. & C.Z. Wei (1987) Asymptotic inference for nearly non-stationary AR(1) processes. Annals of Statistics 15, 1050–1063.Google Scholar
Dickey, D.A., D.P. Hasza, & W.A. Fuller (1984) Testing for unit roots in seasonal time series. Journal of the American Statistical Association 79, 355–367.Google Scholar
Elliott, G, T.J. Rothenberg, & J.H. Stock (1996) Efficient tests for an autoregressive unit root. Econometrica 64, 813–836.Google Scholar
Fuller, W.A. (1996) Introduction to Statistical Time Series, 2nd ed. Wiley.
Ghysels, E., H.S. Lee, & J. Noh (1994) Testing for unit roots in seasonal time series: Some theoretical extensions and a Monte Carlo investigation. Journal of Econometrics 62, 415–442.Google Scholar
Gregoir, S. (2001) Efficient Tests for the Presence of a Pair of Complex Conjugate Unit Roots in Real Time Series. Working paper, CREST-INSEE.
Hylleberg, S., R.F. Engle, C.W.J. Granger, & B.S. Yoo (1990) Seasonal integration and cointegration. Journal of Econometrics 44, 215–238.Google Scholar
Jeganathan, P. (1991) On the asymptotic behaviour of least-squares estimators in AR time series with roots near the unit circle. Econometric Theory 7, 269–306.Google Scholar
Nabeya, S. (2000) Asymptotic distributions for unit root test statistics in nearly integrated seasonal autoregressive models. Econometric Theory 16, 200–230.Google Scholar
Nabeya, S. (2001a) Approximation to the limiting distribution of t- and F-statistics in testing for seasonal unit roots. Econometric Theory 17, 711–737.Google Scholar
Nabeya, S. (2001b) Unit root seasonal autoregressive models with a polynomial trend of higher degree. Econometric Theory 17, 357–385.Google Scholar
Nabeya, S. & K. Tanaka (1990) Limiting powers of unit-root tests in time-series regression. Journal of Econometrics 46, 247–271.Google Scholar
Pantula, S.G., G. Gonzalez-Farias, & W.A. Fuller (1994) A comparison of unit root criteria. Journal of Business and Economic Statistics 13, 449–459.Google Scholar
Perron, P. (1989) The calculation of the limiting distribution of the least-squares estimator in a near-integrated model. Econometric Theory 5, 241–255.Google Scholar
Perron, P. (1992) The limiting distribution of the least-squares estimator in nearly integrated seasonal models. Canadian Journal of Statistics 20, 121–134.Google Scholar
Phillips, P.C.B. (1987) Towards a unified asymptotic theory for autoregression. Biometrika 74, 535–547.Google Scholar
Phillips, P.C.B. (1988) Regression theory for near-integrated time series. Econometrica 56, 1021–1043.Google Scholar
Phillips, P.C.B. & C.C. Lee (1996) Efficiency gains from quasi-differencing under nonstationarity. In P.M. Robinson & M. Rosenblatt (eds.), Athens Conference on Applied Probability and Time Series: Essays in Memory of E.J. Hannan pp. 300–314. Springer-Verlag.
Rodrigues, P.M.M. (2001) Near seasonal integration. Econometric Theory 17, 70–86.Google Scholar
Rodrigues, P.M.M. & A.M.R. Taylor (2003) Asymptotic Distributions for Regression-Based Seasonal Unit Root Test Statistics in a Near-Integrated Model. Discussion Papers in Economics, 03/03, University of Birmingham.
Smith, R.J. & A.M.R. Taylor (1998) Additional critical values and asymptotic representations for seasonal unit root tests. Journal of Econometrics 85, 269–288.Google Scholar
Smith, R.J. & A.M.R. Taylor (1999a) Regression-Based Seasonal Unit Root Tests. Department of Economics Discussion paper 99–15, University of Birmingham.
Smith, R.J. & A.M.R. Taylor (1999b) Likelihood ratio tests for seasonal unit roots. Journal of Time Series Analysis 20, 453–476.Google Scholar
Tanaka, K. (1996) Time Series Analysis: Nonstationary and Noninvertible Distribution Theory. Wiley.
Taylor, A.M.R. (1998) Testing for unit roots in monthly time series. Journal of Time Series Analysis 19, 349–368.Google Scholar