Hostname: page-component-cb9f654ff-65tv2 Total loading time: 0 Render date: 2025-08-11T19:33:01.681Z Has data issue: false hasContentIssue false
  • English
  • Français

NULL RECURRENT UNIT ROOTPROCESSES

Published online by Cambridge University Press:  02 August 2011

Terje Myklebust ,
Hans Arnfinn Karlsen  and
Dag Tjøstheim
Get access Rights & Permissions [Opens in a new window]

Abstract

The classical nonstationary autoregressive models areboth linear and Markov. They include unit root andcointegration models. A possible nonlinear extensionis to relax the linearity and at the same time keepgeneral properties such as nonstationarity and theMarkov property. A null recurrent Markov chain isnonstationary, and β-nullrecurrence is of vital importance for statisticalinference in nonstationary Markov models, such as,e.g., in nonparametric estimation in nonlinearcointegration within the Markov models. The standardrandom walk is an example of a null recurrent Markovchain.

In this paper we suggest that the concept of nullrecurrence is an appropriate nonlineargeneralization of the linear unit root concept andas such it may be a starting point for a nonlinearcointegration concept within the Markov framework.In fact, we establish the link between nullrecurrent processes and autoregressive unit rootmodels. It turns out that null recurrence is closelyrelated to the location of the roots of thecharacteristic polynomial of the state space matrixand the associated eigenvectors. Roughly speakingthe process is β-null recurrent ifone root is on the unit circle, null recurrent iftwo distinct roots are on the unit circle, whereasthe others are inside the unit circle. It istransient if there are more than two roots on theunit circle. These results are closely connected tothe random walk being null recurrent in one and twodimensions but transient in three dimensions. Wealso give an example of a process that byappropriate adjustments can be madeβ-null recurrent for anyβ ∈ (0, 1) and can also be madenull recurrent without being β-nullrecurrent.

Information

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

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.)

Article purchase

Temporarily unavailable

Footnotes

We are very grateful to an anonymous referee fortwo extremely careful and detailed reviews ofearlier versions of the paper. These reports haveled to very substantial improvements of the paper.We also thank the chief editor Peter Phillips fora number of comments that have improved ourpresentation by putting our results in betterperspective.

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