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NEARLY EFFICIENT LIKELIHOOD RATIO TESTS OF A UNIT ROOT IN AN AUTOREGRESSIVE MODEL OF ARBITRARY ORDER

Published online by Cambridge University Press:  20 December 2022

Samuel Brien Open the ORCID record for Samuel Brien [Opens in a new window] ,
Michael Jansson Open the ORCID record for Michael Jansson [Opens in a new window]  and
Morten Ørregaard Nielsen Open the ORCID record for Morten Ørregaard Nielsen [Opens in a new window]
Samuel Brien
Affiliation:
Queen’s University
Michael Jansson*
Affiliation:
UC Berkeley and creates
Morten Ørregaard Nielsen
Affiliation:
Aarhus University
*
Address correspondence to Michael Jansson, Department of Economics, University of California, Berkeley, 530 Evans Hall #3880, Berkeley, CA 94720, USA; e-mail: [email protected].
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Abstract

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We study large sample properties of likelihood ratio tests of the unit-root hypothesis in an autoregressive model of arbitrary order. Earlier research on this testing problem has developed likelihood ratio tests in the autoregressive model of order 1, but resorted to a plug-in approach when dealing with higher-order models. In contrast, we consider the full model and derive the relevant large sample properties of likelihood ratio tests under a local-to-unity asymptotic framework. As in the simpler model, we show that the full likelihood ratio tests are nearly efficient, in the sense that their asymptotic local power functions are virtually indistinguishable from the Gaussian power envelopes. Extensions to sieve-type approximations and different classes of alternatives are also considered.

Type
MISCELLANEA
Information
Econometric Theory , Volume 40 , Issue 5 , October 2024 , pp. 1159 - 1183
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Footnotes

We are grateful to Peter Phillips, two anonymous referees, and seminar participants at Queen’s University for comments and discussion, and to the Danish National Research Foundation for financial support (DNRF Chair Grant No. DNRF154).

References

REFERENCES

Andrews, D.W.K. & Ploberger, W. (1994) Optimal tests when a nuisance parameter is present only under the alternative. Econometrica 62, 1383–1414.CrossRefGoogle Scholar
Andrews, D.W.K. & Ploberger, W. (1995) Admissibility of the likelihood ratio test when a nuisance parameter is present only under the alternative. Annals of Statistics 23, 1609–1629.CrossRefGoogle Scholar
Becheri, I.G., Drost, F.C., & van den Akker, R. (2015) Asymptotically UMP panel unit root tests—The effect of heterogeneity in the alternatives. Econometric Theory 31, 539–559.CrossRefGoogle Scholar
Berk, K.N. (1974) Consistent autoregressive spectral estimates. Annals of Statistics 2, 489–502.CrossRefGoogle Scholar
Boswijk, H.P., Jansson, M., & Nielsen, M.Ø. (2015) Improved likelihood ratio tests for cointegration rank in the VAR model. Journal of Econometrics 184, 97–110.CrossRefGoogle Scholar
Bykhovskaya, A. & Phillips, P.C.B. (2018) Boundary limit theory for functional local to unity regression. Journal of Time Series Analysis 39, 523–562.CrossRefGoogle Scholar
Bykhovskaya, A. & Phillips, P.C.B. (2020) Point optimal testing with roots that are functionally local to unity. Journal of Econometrics 219, 231–259.CrossRefGoogle Scholar
Chan, N.H. & Wei, C.Z. (1987) Asymptotic inference for nearly nonstationary AR(1) processes. Annals of Statistics 15, 1050–1063.CrossRefGoogle Scholar
Chang, Y. & Park, J.Y. (2002) On the asymptotics of ADF tests for unit roots. Econometric Reviews 21, 431–447.CrossRefGoogle Scholar
Dickey, D.A. & Fuller, W.A. (1979) Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74, 427–431.Google Scholar
Dickey, D.A. & Fuller, W.A. (1981) Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49, 1057–1072.CrossRefGoogle Scholar
Dudley, R.M. (2002) Real Analysis and Probability , 2nd Edition. Cambridge University Press.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, 813–836.CrossRefGoogle Scholar
Harvey, D.I., Leybourne, S.J., & Taylor, A.M.R. (2009) Unit root testing in practice: Dealing with uncertainty over the trend and initial condition. Econometric Theory 25, 587–636.CrossRefGoogle Scholar
Jansson, M. & Nielsen, M.Ø. (2011) Nearly efficient likelihood ratio tests for seasonal unit roots. Journal of Time Series Econometrics 3(1), Article no. 5.CrossRefGoogle Scholar
Jansson, M. & Nielsen, M.Ø. (2012) Nearly efficient likelihood ratio tests of the unit root hypothesis. Econometrica 80, 2321–2332.Google Scholar
Kallenberg, O. (2002) Foundations of Modern Probability , 2nd Edition. Springer.CrossRefGoogle Scholar
Lieberman, O. & Phillips, P.C.B. (2014) Norming rates and limit theory for some time-varying coefficient autoregressions. Journal of Time Series Analysis 35, 592–623.CrossRefGoogle Scholar
Lieberman, O. & Phillips, P.C.B. (2017) A multivariate stochastic unit root model with an application to derivative pricing. Journal of Econometrics 196, 99–110.CrossRefGoogle Scholar
Lieberman, O. & Phillips, P.C.B. (2020) Hybrid stochastic local unit roots. Journal of Econometrics 215, 257–285.CrossRefGoogle Scholar
Moon, H.R., Perron, B., & Phillips, P.C.B. (2007) Incidental trends and the power of panel unit root tests. Journal of Econometrics 141, 416–459.CrossRefGoogle Scholar
Moon, H.R., Perron, B., & Phillips, P.C.B. (2014) Point-optimal panel unit root tests with serially correlated errors. The Econometrics Journal 17, 338–372.CrossRefGoogle Scholar
Müller, U.K. & Elliott, G. (2003) Tests for unit roots and the initial condition. Econometrica 71, 1269–1286.CrossRefGoogle Scholar
Ng, S. & Perron, P. (2001) LAG length selection and the construction of unit root tests with good size and power. Econometrica 69, 1519–1554.CrossRefGoogle Scholar
Perron, P. & Qu, Z. (2007) A simple modification to improve the finite sample properties of Ng and Perron’s unit root tests. Economics Letters 94, 12–19.CrossRefGoogle Scholar
Phillips, P.C.B. (1987) Towards a unified asymptotic theory for autoregression. Biometrika 74, 535–547.CrossRefGoogle Scholar
Phillips, P.C.B. & Solo, V. (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.CrossRefGoogle Scholar
Said, S.E. & Dickey, D.A. (1984) Testing for unit roots in autoregressive-moving average models of unknown order. Biometrika 71, 599–607.CrossRefGoogle Scholar