The Annals of Statistics

Nonparametric estimation in a nonlinear cointegration type model

Hans Arnfinn Karlsen, Terje Myklebust, and Dag Tjøstheim

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We derive an asymptotic theory of nonparametric estimation for a time series regression model Zt=f(Xt)+Wt, where {Xt} and {Zt} are observed nonstationary processes and {Wt} is an unobserved stationary process. In econometrics, this can be interpreted as a nonlinear cointegration type relationship, but we believe that our results are of wider interest. The class of nonstationary processes allowed for {Xt} is a subclass of the class of null recurrent Markov chains. This subclass contains random walk, unit root processes and nonlinear processes. We derive the asymptotics of a nonparametric estimate of f(x) under the assumption that {Wt} is a Markov chain satisfying some mixing conditions. The finite-sample properties of (x) are studied by means of simulation experiments.

Article information

Ann. Statist., Volume 35, Number 1 (2007), 252-299.

First available in Project Euclid: 6 June 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62G08: Nonparametric regression 91B84: Economic time series analysis [See also 62M10]
Secondary: 60J05: Discrete-time Markov processes on general state spaces

Cointegration nonstationary time series models null recurrent Markov chain nonparametric kernel estimators transfer function model


Karlsen, Hans Arnfinn; Myklebust, Terje; Tjøstheim, Dag. Nonparametric estimation in a nonlinear cointegration type model. Ann. Statist. 35 (2007), no. 1, 252--299. doi:10.1214/009053606000001181.

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  • Aparicio, F. M. and Escribano, A. (1998). Information-theoretic analysis of serial dependence and cointegration. Stud. Nonlinear Dynam. Econom. 3 119–140.
  • Bandi, F. M. (2003). Persistence and nonparametric estimation: Some observations. Technical report, Grad. School Business, Univ. Chicago.
  • Bec, F. and Rahbek, A. (2002). Vector equilibrium correction models with nonlinear discontinuous adjustments. Technical report, Dept. Statistics and Operations Research, Univ. Copenhagen.
  • Bolthausen, E. (1980). The Berry–Esseén theorem for functionals of discrete Markov chains. Z. Wahrsch. Verw. Gebiete 54 59–73.
  • Bolthausen, E. (1982). The Berry–Esseén theorem for strongly mixing Harris recurrent Markov chains. Z. Wahrsch. Verw. Gebiete 60 283–289.
  • Engle, R. F. and Granger, C. W. J. (1987). Co-integration and error correction: Representation, estimation and testing. Econometrica 55 251–276.
  • Escribano, A. (2004). Nonlinear error-correction: The case of money demand in the United Kingdom (1878–2000). Macroeconomic Dynamics 8 76–116.
  • Escribano, A. and Mira, S. (2002). Nonlinear error correction models. J. Time Ser. Anal. 23 509–522.
  • Feigin, P. D. and Tweedie, R. L. (1985). Random coefficient autoregressive processes: A Markov chain analysis of stationarity and finiteness of moments. J. Time Ser. Anal. 6 1–14.
  • Granger, C. W. J. (1983). Co-integrated variables and error-correcting models. UCSD discussion paper, Univ. California, San Diego.
  • Granger, C. W. J. (1995). Modelling nonlinear relationships between extended-memory variables. Econometrica 63 265–279.
  • Granger, C. W. J. and Hallman, J. (1991). Long memory series with attractors. Oxford Bulletin Economics and Statistics 53 11–26.
  • Granger, C. W. J. and Lee, T. H. (1989). Investigation of production, sales and inventory relationships using multicointegration and non-symmetric error correction models. J. Appl. Econometrics 4 145–159.
  • Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.
  • Hamilton, J. D. (1994). Time Series Analysis. Princeton Univ. Press, Princeton, NJ.
  • Hansen, B. E. and Seo, B. (2002). Testing for two-regime threshold cointegration in vector error-correction models. J. Econometrics 110 293–318.
  • Hendry, D. and Ericsson, N. (1991). An econometric analysis of U.K. money demand in Monetary Trends in the United States and the United Kingdom by Milton Friedman and Anna J. Schwartz. Amer. Economic Review 81 8–38.
  • Johansen, S. (1991). Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica 59 1551–1580.
  • Johansen, S. (1995). Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. Oxford Univ. Press.
  • Karlsen, H. A., Myklebust, T. and Tjøstheim, D. (2006). Nonparametric estimation in a nonstationary framework. Technical report, Dept. Mathematics, Univ. Bergen. In preparation.
  • Karlsen, H. A. and Tjøstheim, D. (1998). Nonparametric estimation in null recurrent time series. Sonderforschungsbereich 373, Humboldt-Universität zu Berlin.
  • Karlsen, H. A. and Tjøstheim, D. (2001). Nonparametric estimation in null recurrent time series. Ann. Statist. 29 372–416.
  • Myklebust, T., Karlsen, H. A. and Tjøstheim, D. (2003). A Markov chain characterization of unit root processes and the problem of nonlinear cointegration. Under revision.
  • Nummelin, E. (1984). General Irreducible Markov Chains and Nonnegative Operators. Cambridge Univ. Press.
  • Park, J. Y. and Phillips, P. C. B. (2001). Nonlinear regressions with integrated time series. Econometrica 69 117–161.
  • Phillips, P. C. B. and Park, J. Y. (1998). Nonstationary density estimation and kernel autoregression. Cowles Foundation Discussion Paper 1181, Yale Univ.
  • Xia, Y. (1998). On some nonparametric and semiparametric approaches to time series modelling. Ph.D. dissertation, Univ. Hong Kong.