Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T18:18:24.587Z Has data issue: false hasContentIssue false
  • English
  • Français

EFFICIENT LIKELIHOOD INFERENCE IN NONSTATIONARY UNIVARIATE MODELS

Published online by Cambridge University Press:  05 March 2004

Morten Ørregaard Nielsen
Morten Ørregaard Nielsen
Affiliation:
Cornell University
Get access Rights & Permissions [Opens in a new window]

Abstract

Recent literature shows that embedding fractionally integrated time series models with spectral poles at the long-run and/or seasonal frequencies in autoregressive frameworks leads to estimators and test statistics with nonstandard limiting distributions. However, we demonstrate that when embedding such models in a general I(d) framework the resulting estimators and tests regain desirable properties from standard statistical analysis. We show the existence of a local time domain maximum likelihood estimator and its asymptotic normality—and under Gaussianity asymptotic efficiency. The Wald, likelihood ratio, and Lagrange multiplier tests are asymptotically equivalent and chi-squared distributed under local alternatives. With independent and identically distributed Gaussian errors and a scalar parameter, we show that the tests in addition achieve the asymptotic Gaussian power envelope of all invariant unbiased tests; i.e., they are asymptotically uniformly most powerful invariant unbiased against local alternatives. In a Monte Carlo study we document the finite sample superiority of the likelihood ratio test.I am grateful to Bent Jesper Christensen, Niels Haldrup, Pentti Saikkonen (the co-editor), and two anonymous referees for many useful comments and suggestions that significantly improved this paper. This work was done while the author was at the University of Aarhus, Denmark.

Type
Research Article
Information
Econometric Theory , Volume 20 , Issue 1 , February 2004 , pp. 116 - 146
Copyright
© 2004 Cambridge University Press

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

Agiakloglou, C. & P. Newbold (1994) Lagrange multiplier tests for fractional difference. Journal of Time Series Analysis 15, 253–262.CrossRefGoogle Scholar
Bierens, H.J. (2001) Complex unit roots and business cycles: Are they real? Econometric Theory 17, 962–983.Google Scholar
Brown, B.M. (1971) Martingale central limit theorems. Annals of Mathematical Statistics 42, 59–66.CrossRefGoogle Scholar
Chung, C.F. (1996) Estimating a generalized long memory process. Journal of Econometrics 73, 237–259.CrossRefGoogle Scholar
Dickey, D.A. & W.A. Fuller (1979) Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74, 427–431.Google Scholar
Doornik, J.A. (2001) Ox: An Object-Oriented Matrix Language, 4th ed., Timberlake Consultants Press.
Doornik, J.A. & M. Ooms (2001) A Package for Estimating, Forecasting, and Simulating Arfima Models: Arfima Package 1.01 for Ox. Working Paper, Nuffield College, Oxford.
Elliott, G., T.J. Rothenberg, & J.H. Stock (1996) Efficient tests for an autoregressive unit root. Econometrica 64, 813–836.CrossRefGoogle Scholar
Engle, R.F. (1984) Wald, likelihood ratio, and Lagrange multiplier tests in econometrics. In Z. Griliches & M.D. Intriligator (eds.), Handbook of Econometrics, vol. 2, pp. 775–826. North-Holland.
Gil-Alana, L.A. (2001) Testing stochastic cycles in macroeconomic time series. Journal of Time Series Analysis 22, 411–430.CrossRefGoogle Scholar
Gray, H., N. Zhang, & W.A. Woodward (1989) On generalized fractional processes. Journal of Time Series Analysis 10, 233–257.CrossRefGoogle Scholar
Hall, P. & C.C. Heyde (1980) Martingale Limit Theory and Its Application. Academic Press.
Hylleberg, S., R.F. Engle, C.W.J. Granger, & B.S. Yoo (1990) Seasonal integration and cointegration. Journal of Econometrics 44, 215–238.CrossRefGoogle Scholar
Lehmann, E.L. (1986) Testing Statistical Hypotheses, 2nd ed., Springer-Verlag.
Ling, S. & W.K. Li (1997) Fractional ARIMA-GARCH time series models. Journal of the American Statistical Association 92, 1184–1194.CrossRefGoogle Scholar
Ling, S. & W.K. Li (2001) Asymptotic inference for nonstationary fractionally integrated autoregressive moving-average models. Econometric Theory 17, 738–764.CrossRefGoogle Scholar
Nielsen, M.Ø. (2003) Optimal residual based tests for fractional cointegration and exchange rate dynamics. Journal of Business and Economic Statistics, forthcoming.Google Scholar
Phillips, P.C.B. (1987) Time series regression with a unit root. Econometrica 55, 277–301.CrossRefGoogle Scholar
Phillips, P.C.B. & Z. Xiao (1998) A primer on unit root testing. Journal of Economic Surveys 12, 423–469.CrossRefGoogle Scholar
Robinson, P.M. (1991) Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regressions. Journal of Econometrics 47, 67–84.CrossRefGoogle Scholar
Robinson, P.M. (1994) Efficient tests of nonstationary hypotheses. Journal of the American Statistical Association 89, 1420–1437.CrossRefGoogle Scholar
Saikkonen, P. & R. Luukkonen (1993a) Point optimal tests for testing the order of differencing in ARIMA models. Econometric Theory 9, 343–362.Google Scholar
Saikkonen, P. & R. Luukkonen (1993b) Testing for a moving average unit root in autoregressive integrated moving average models. Journal of the American Statistical Association 88, 596–601.Google Scholar
Sargan, J.D. & A. Bhargava (1983) Maximum likelihood estimation of regression models with first order moving average errors when the root lies on the unit circle. Econometrica 51, 799–820.CrossRefGoogle Scholar
Sowell, F.B. (1990) The fractional unit root distribution. Econometrica 58, 495–505.CrossRefGoogle Scholar
Tanaka, K. (1999) The nonstationary fractional unit root. Econometric Theory 15, 549–582.CrossRefGoogle Scholar