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NULL RECURRENT UNIT ROOT PROCESSES

Published online by Cambridge University Press:  02 August 2011

Terje Myklebust ,
Hans Arnfinn Karlsen  and
Dag Tjøstheim
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Abstract

The classical nonstationary autoregressive models are both linear and Markov. They include unit root and cointegration models. A possible nonlinear extension is to relax the linearity and at the same time keep general properties such as nonstationarity and the Markov property. A null recurrent Markov chain is nonstationary, and β-null recurrence is of vital importance for statistical inference in nonstationary Markov models, such as, e.g., in nonparametric estimation in nonlinear cointegration within the Markov models. The standard random walk is an example of a null recurrent Markov chain.

In this paper we suggest that the concept of null recurrence is an appropriate nonlinear generalization of the linear unit root concept and as such it may be a starting point for a nonlinear cointegration concept within the Markov framework. In fact, we establish the link between null recurrent processes and autoregressive unit root models. It turns out that null recurrence is closely related to the location of the roots of the characteristic polynomial of the state space matrix and the associated eigenvectors. Roughly speaking the process is β-null recurrent if one root is on the unit circle, null recurrent if two distinct roots are on the unit circle, whereas the others are inside the unit circle. It is transient if there are more than two roots on the unit circle. These results are closely connected to the random walk being null recurrent in one and two dimensions but transient in three dimensions. We also give an example of a process that by appropriate adjustments can be made β-null recurrent for any β ∈ (0, 1) and can also be made null recurrent without being β-null recurrent.

Type
Research Article
Information
Econometric Theory , Volume 28 , Issue 1 , February 2012 , pp. 1 - 41
Copyright
Copyright © Cambridge University Press 2011

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Footnotes

We are very grateful to an anonymous referee for two extremely careful and detailed reviews of earlier versions of the paper. These reports have led to very substantial improvements of the paper. We also thank the chief editor Peter Phillips for a number of comments that have improved our presentation by putting our results in better perspective.

References

REFERENCES

Aparicio, F.M., Escribano, A., & Sipols, A.E. (2006) Range unit-root (RUR) tests: Robust against nonlinearities, error distributions, structural breaks and outliers. Journal of Time Series Analysis 27, 545–576.CrossRefGoogle Scholar
Bec, F., Guay, A., & Guerre, E. (2008) Adaptive consistent unit-root tests based on autoregressive threshold model. Journal of Econometrics 127, 94–133.CrossRefGoogle Scholar
Bhattacharya, R.N. & Rao, R.R. (1976) Normal Approximation and Asymptotic Expansions. Wiley.Google Scholar
Bingham, N.H., Goldie, C.M., & Teugels, J.L. (1987) Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
Chan, K.S. & Tong, H. (1985) On the use of the deterministic Lyapunov function for the ergodicity of stochastic difference equations. Advances in Applied Probability 17, 666–678.CrossRefGoogle Scholar
Chen, J., Gao, J., & Li, D. (2009) Semiparametric Regression Estimation in Null Recurrent Time Series. Manuscript, Department of Economics, University of Adelaide.Google Scholar
Cline, D.B.H. & Pu, H.H. (1998) Verifying irreducibility and continuity of a nonlinear time series. Statistics and Probability Letters 40, 139–148.CrossRefGoogle Scholar
Cline, D.B.H. & Pu, H.H. (1999) Stability of nonlinear AR(1) time series with delay. Stochastic Processes and Their Applications 82, 307–333.CrossRefGoogle Scholar
Davidson, J. (2009) When is a time series I(0)? In Castle, J. and Shephard, N. (eds.), A Festschrift for David Hendry, pp. 322–342. Oxford University Press.Google Scholar
Escribano, A. (2004) Nonlinear error correction: The case of money demand in the United Kingdom (1878–2000). Macroeconomic Dynamics 8, 76–116.Google Scholar
Escribano, A., Sipols, A.E., & Aparicio, F.M. (2006) Nonlinear cointegration and nonlinear error correction: Record counting cointegration tests. Communications in Statistics—Simulation and Computation 35, 939–956.CrossRefGoogle Scholar
Feigin, P.D. & Tweedie, R.L. (1985) Random coefficient autoregressive processes: A Markov chain analysis of stationarity and finiteness of moments. Journal of Time Series Analysis 6, 1–14.CrossRefGoogle Scholar
Gao, J., King, M., Lu, Z., & Tjøstheim, D. (2009) Specification testing in nonlinear time series with nonstationarity. Annals of Statistics 37, 3893–3928.CrossRefGoogle Scholar
Granger, C.W.J. (1995) Modelling nonlinear relationships between extended-memory variables. Econometrica 63, 265–279.CrossRefGoogle Scholar
Granger, C.W.J. & Hallman, J. (1991a) Long memory series with attractors. Oxford Bulletin of Economics and Statistics 53, 11–26.CrossRefGoogle Scholar
Granger, C.W.J. & Hallman, J. (1991b) Nonlinear transformations of integrated time series. Journal of Time Series Analysis 12, 207–224.CrossRefGoogle Scholar
Granger, C.W.J. & Swanson, N. (1996) Further developments in the study of cointegrated variables. Oxford Bulletin of Economics and Statistics 58, 537–553.CrossRefGoogle Scholar
Hamilton, J.D. (1994) Time Series Analysis. Princeton University Press.CrossRefGoogle Scholar
Horn, R.A. & Johnson, C.R. (1990) Matrix Analysis. Cambridge University Press.Google Scholar
Juhl, T. & Xiao, Z. (2005a) Partially linear models with unit roots. Econometric Theory 21, 877–906.CrossRefGoogle Scholar
Juhl, T. & Xiao, Z. (2005b) Testing for cointegration using partially linear models. Journal of Econometrics 124, 363–394.CrossRefGoogle Scholar
Kallianpur, G. & Robbins, H. (1954) The sequence of sums of independent random variables. Duke Mathematical Journal 21, 285–307.CrossRefGoogle Scholar
Karlsen, H.A., Myklebust, T., & Tjøstheim, D. (2007) Nonparametric estimation in a nonlinear cointegration type model. Annals of Statistics 35, 252–299. Earlier version (2000) available as a report under Sonderforschungsbereich 373, Humboldt-Universitat zu Berlin.CrossRefGoogle Scholar
Karlsen, H.A., Myklebust, T., & Tjøstheim, D. (2010) Nonparametric regression estimation in a null recurrent time series. Journal of Statistical Planning and Inference 140, 3619–3626.CrossRefGoogle Scholar
Karlsen, H.A. & Tjøstheim, D. (2001) Nonparametric estimation in null recurrent time series. Annals of Statistics 29, 372–416. Earlier version (1998) available as a report under Sonderforschungsbereich 373, Humboldt-Universitat zu Berlin.CrossRefGoogle Scholar
Meyn, S.P. & Tweedie, R.L. (1993) Markov Chains and Stochastic Stability. Springer-Verlag.CrossRefGoogle Scholar
Nummelin, E. (1984) General Irreducible Markov Chains and Non-negative Operators. Cambridge University Press.CrossRefGoogle Scholar
Park, J.Y. & Phillips, P.C.B. (1999) Asymptotics for nonlinear transformations of integrated time series. Econometric Theory 15, 269–298.CrossRefGoogle Scholar
Park, J.Y. & Phillips, P.C.B. (2001) Nonlinear regressions with integrated time series. Econometrica 1, 117–161.CrossRefGoogle Scholar
Phillips, P.C.B. (1987) Time series regression with a unit root. Econometrica 55, 277–301.CrossRefGoogle Scholar
Saikkonen, P. (2005) Stability results for nonlinear error correction models. Journal of Econometrics 127, 69–81.CrossRefGoogle Scholar
Saikkonen, P. (2007) Stability of mixtures of vector autoregressions with autoregressive conditional heteroskedasticity. Statistica Sinica 17, 221–239.Google Scholar
Saikkonen, P. & Choi, I. (2004) Cointegrating smooth transition regression. Econometric Theory 20, 301–340.CrossRefGoogle Scholar
Sancetta, A. (2009) Nearest neigbor conditional estimation for Harris recurrent Markov chains. Journal of Multivariate Analysis 100, 2224–2236.CrossRefGoogle Scholar
Schienle, M. (2007) Reaching for Econometric Generality: Nonparametric Nonstationary regression. Working paper, Department of Economics, University of Mannheim.Google Scholar
Spitzer, F. (1964) Principles of Random Walk. Van Nostrand.CrossRefGoogle Scholar
Stewart, G.W. & Sun, J.-G. (1990) Matrix Perturbation Theory. Academic Press.Google Scholar
Tjøstheim, D. (1990) Nonlinear time series and Markov chains. Advances in Applied Probability 22, 587–611.CrossRefGoogle Scholar
Tweedie, R.L. (1976) Criteria for classifying general Markov chains. Advances in Applied Probability 8, 737–771.CrossRefGoogle Scholar
Wang, Q.Y. & Phillips, P.C.B. (2009) Asymptotic theory for local time density estimation and nonparametric cointegrating regression. Econometric Theory 25, 710–738.CrossRefGoogle Scholar