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NULL RECURRENT UNIT ROOT PROCESSES

  • Terje Myklebust (a1), Hans Arnfinn Karlsen (a2) and Dag Tjøstheim (a2)

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.

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Corresponding author

*Address correspondence to Terje Myklebust, Sogn og Fjordane University College, Ali, Pb 133, NO-6851 Sogndal, Norway; e-mail: terjemy@hisf.no.

References

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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, 545576.
Bec, F., Guay, A., & Guerre, E. (2008) Adaptive consistent unit-root tests based on autoregressive threshold model. Journal of Econometrics 127, 94133.
Bhattacharya, R.N. & Rao, R.R. (1976) Normal Approximation and Asymptotic Expansions. Wiley.
Bingham, N.H., Goldie, C.M., & Teugels, J.L. (1987) Regular Variation. Cambridge University Press.
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, 666678.
Chen, J., Gao, J., & Li, D. (2009) Semiparametric Regression Estimation in Null Recurrent Time Series. Manuscript, Department of Economics, University of Adelaide.
Cline, D.B.H. & Pu, H.H. (1998) Verifying irreducibility and continuity of a nonlinear time series. Statistics and Probability Letters 40, 139148.
Cline, D.B.H. & Pu, H.H. (1999) Stability of nonlinear AR(1) time series with delay. Stochastic Processes and Their Applications 82, 307333.
Davidson, J. (2009) When is a time series I(0)? In Castle, J. and Shephard, N. (eds.), A Festschrift for David Hendry, pp. 322342. Oxford University Press.
Escribano, A. (2004) Nonlinear error correction: The case of money demand in the United Kingdom (1878–2000). Macroeconomic Dynamics 8, 76116.
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, 939956.
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, 114.
Gao, J., King, M., Lu, Z., & Tjøstheim, D. (2009) Specification testing in nonlinear time series with nonstationarity. Annals of Statistics 37, 38933928.
Granger, C.W.J. (1995) Modelling nonlinear relationships between extended-memory variables. Econometrica 63, 265279.
Granger, C.W.J. & Hallman, J. (1991a) Long memory series with attractors. Oxford Bulletin of Economics and Statistics 53, 1126.
Granger, C.W.J. & Hallman, J. (1991b) Nonlinear transformations of integrated time series. Journal of Time Series Analysis 12, 207224.
Granger, C.W.J. & Swanson, N. (1996) Further developments in the study of cointegrated variables. Oxford Bulletin of Economics and Statistics 58, 537553.
Hamilton, J.D. (1994) Time Series Analysis. Princeton University Press.
Horn, R.A. & Johnson, C.R. (1990) Matrix Analysis. Cambridge University Press.
Juhl, T. & Xiao, Z. (2005a) Partially linear models with unit roots. Econometric Theory 21, 877906.
Juhl, T. & Xiao, Z. (2005b) Testing for cointegration using partially linear models. Journal of Econometrics 124, 363394.
Kallianpur, G. & Robbins, H. (1954) The sequence of sums of independent random variables. Duke Mathematical Journal 21, 285307.
Karlsen, H.A., Myklebust, T., & Tjøstheim, D. (2007) Nonparametric estimation in a nonlinear cointegration type model. Annals of Statistics 35, 252299. Earlier version (2000) available as a report under Sonderforschungsbereich 373, Humboldt-Universitat zu Berlin.
Karlsen, H.A., Myklebust, T., & Tjøstheim, D. (2010) Nonparametric regression estimation in a null recurrent time series. Journal of Statistical Planning and Inference 140, 36193626.
Karlsen, H.A. & Tjøstheim, D. (2001) Nonparametric estimation in null recurrent time series. Annals of Statistics 29, 372416. Earlier version (1998) available as a report under Sonderforschungsbereich 373, Humboldt-Universitat zu Berlin.
Meyn, S.P. & Tweedie, R.L. (1993) Markov Chains and Stochastic Stability. Springer-Verlag.
Nummelin, E. (1984) General Irreducible Markov Chains and Non-negative Operators. Cambridge University Press.
Park, J.Y. & Phillips, P.C.B. (1999) Asymptotics for nonlinear transformations of integrated time series. Econometric Theory 15, 269298.
Park, J.Y. & Phillips, P.C.B. (2001) Nonlinear regressions with integrated time series. Econometrica 1, 117161.
Phillips, P.C.B. (1987) Time series regression with a unit root. Econometrica 55, 277301.
Saikkonen, P. (2005) Stability results for nonlinear error correction models. Journal of Econometrics 127, 6981.
Saikkonen, P. (2007) Stability of mixtures of vector autoregressions with autoregressive conditional heteroskedasticity. Statistica Sinica 17, 221239.
Saikkonen, P. & Choi, I. (2004) Cointegrating smooth transition regression. Econometric Theory 20, 301340.
Sancetta, A. (2009) Nearest neigbor conditional estimation for Harris recurrent Markov chains. Journal of Multivariate Analysis 100, 22242236.
Schienle, M. (2007) Reaching for Econometric Generality: Nonparametric Nonstationary regression. Working paper, Department of Economics, University of Mannheim.
Spitzer, F. (1964) Principles of Random Walk. Van Nostrand.
Stewart, G.W. & Sun, J.-G. (1990) Matrix Perturbation Theory. Academic Press.
Tjøstheim, D. (1990) Nonlinear time series and Markov chains. Advances in Applied Probability 22, 587611.
Tweedie, R.L. (1976) Criteria for classifying general Markov chains. Advances in Applied Probability 8, 737771.
Wang, Q.Y. & Phillips, P.C.B. (2009) Asymptotic theory for local time density estimation and nonparametric cointegrating regression. Econometric Theory 25, 710738.

NULL RECURRENT UNIT ROOT PROCESSES

  • Terje Myklebust (a1), Hans Arnfinn Karlsen (a2) and Dag Tjøstheim (a2)

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