Hostname: page-component-745bb68f8f-b6zl4 Total loading time: 0 Render date: 2025-01-15T00:09:47.320Z Has data issue: false hasContentIssue false
  • English
  • Français

UNIFORM CONSISTENCY FOR NONPARAMETRIC ESTIMATORS IN NULL RECURRENT TIME SERIES

Published online by Cambridge University Press:  03 November 2014

Jiti Gao ,
Shin Kanaya ,
Degui Li  and
Dag Tjøstheim
Jiti Gao*
Affiliation:
Monash University
Shin Kanaya
Affiliation:
University of Aarhus
Degui Li
Affiliation:
University of York
Dag Tjøstheim
Affiliation:
University of Bergen
*
*Address correspondence to Jiti Gao, Department of Econometrics and Business Statistics, Monash University, Caulfield East, VIC 3145, Australia; e-mail: jiti.gao@monash.edu.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper establishes uniform consistency results for nonparametric kernel density and regression estimators when time series regressors concerned are nonstationary null recurrent Markov chains. Under suitable regularity conditions, we derive uniform convergence rates of the estimators. Our results can be viewed as a nonstationary extension of some well-known uniform consistency results for stationary time series.

Type
ARTICLES
Information
Econometric Theory , Volume 31 , Issue 5 , October 2015 , pp. 911 - 952
Copyright
Copyright © Cambridge University Press 2014 

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.)

References

REFERENCES

Andrews, D.W.K. (1995) Nonparametric kernel estimation for semiparametric models. Econometric Theory 11, 560–596.CrossRefGoogle Scholar
Bandi, F. & Moloche, G. (2008) On the Functional Estimation of Multivariate Diffusion Processes. Working paper at the University of Chicago.Google Scholar
Bosq, D. (1998) Nonparametric Statistics for Stochastic Processes: Estimation and Prediction, 2nd ed. Lecture Notes in Statistics 110. Springer-Verlag.CrossRefGoogle Scholar
Cai, Z., Li, Q., & Park, J. (2009) Functional-coefficient models for nonstationary time series data. Journal of Econometrics 148, 101–113.CrossRefGoogle Scholar
Chen, J., Gao, J., & Li, D. (2012) Estimation in semiparametric regression with nonstationary regressors. Bernoulli 18, 678–702.CrossRefGoogle Scholar
Chen, J., Li, D., & Zhang, L. (2010) Robust estimation in a nonlinear cointegration model. Journal of Multivariate Analysis 101, 706–717.CrossRefGoogle Scholar
Chung, K.L. (1967) Markov Chains with Stationary Transition Probabilities, 2nd ed. Springer-Verlag.Google Scholar
Doukhan, P. (1994) Mixing, Properties and Examples. Lecture Notes in Statistics 85. Springer-Verlag.Google Scholar
Doukhan, P. & Ghindés, M. (1980) Estimations dans le processus: “X n+1 = f(X n)+ε n”. Proceedings of the French Academy of Sciences Series A–B 291, A61–A64.Google Scholar
Fan, J. & Gijbels, I. (1996) Local Polynomial Modelling and Its Applications. Chapman & Hall.Google Scholar
Fan, J. & Yao, Q. (2003) Nonlinear Time Series: Nonparametric and Parametric Methods. Springer.CrossRefGoogle Scholar
Gao, J. (2007) Nonlinear Time Series: Semiparametric and Nonparametric Methods. Chapman & Hall/CRC.CrossRefGoogle Scholar
Gao, J., King, M.L., Lu, Z., & Tjøstheim, D. (2009a) Specification testing in nonstationary time series autoregression. Annals of Statistics 7, 3893–3928.Google Scholar
Gao, J., King, M.L., Lu, Z., & Tjøstheim, D. (2009b) Nonparametric specification testing for nonlinear time series with nonstationarity. Econometric Theory 25, 1869–1892.CrossRefGoogle Scholar
Gao, J., Tjøstheim, D., & Yin, J. (2013) Estimation in threshold autoregressive models with nonstationarity. Journal of Econometrics 172, 1–12.CrossRefGoogle Scholar
Hansen, B.E. (2008) Uniform convergence rates for kernel estimation with dependent data. Econometric Theory 24, 726–748.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.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. (1998) Nonparametric Estimation in Null Recurrent Time Series. Discussion paper available at Sonderforschungsbereich 373 50, Humboldt University.Google Scholar
Karlsen, H.A. & Tjøstheim, D. (2001) Nonparametric estimation in null recurrent time series. Annals of Statistics 29, 372–416.CrossRefGoogle Scholar
Kristensen, D. (2009) Uniform convergence rates of kernel estimators with heterogenous dependent data. Econometric Theory 25, 1433–1445.CrossRefGoogle Scholar
Liebscher, E. (1996) Strong convergence of sums of α-mixing random variables with applications to density estimation. Stochastic Processes and Their Applications 65, 69–80.CrossRefGoogle Scholar
Liero, H. (1989) Strong uniform consistency of nonparametric regression function estimates. Probability Theory and Related Fields 82, 587–614.CrossRefGoogle Scholar
Lin, G. (1998) On the Mittag-Leffler distributions. Journal of Statistical Planning and Inference 74, 1–9.CrossRefGoogle Scholar
Masry, E. (1996) Multivariate local polynomial regression for time series: Uniform strong consistency and rates. Journal of Time Series Analysis 17, 571–599.CrossRefGoogle Scholar
Meyn, S.P. & Tweedie, R.L. (2009) Markov Chains and Stochastic Stability, 2nd ed. Cambridge University Press.CrossRefGoogle Scholar
Nummelin, E. (1984) General Irreducible Markov Chains and Non-negative Operators. Cambridge University Press.CrossRefGoogle Scholar
Ould-Saïd, E. & Cai, Z. (2005) Strong uniform consistency of nonparametric estimation of the censored conditional mode function. Journal of Nonparametric Statistics 17, 797–806.CrossRefGoogle Scholar
Phillips, P.C.B. & Park, J. (1998) Nonstationary Density Estimation and Kernel Autoregression. Cowles Foundation Discussion paper 1181.Google Scholar
Roussas, G.G. (1990) Nonparametric regression estimation under mixing conditions. Stochastic Processes and Their Applications 36, 107–116.CrossRefGoogle Scholar
Schienle, M. (2008) Nonparametric Nonstationary Regression. Ph.D. thesis, University of Mannheim, Germany.Google Scholar
Schienle, M. (2011) Nonparametric Nonstationary Regression with Many Covariates. SFB 649 Discussion paper available athttp://sfb649.wiwi.hu-berlin.de/papers/pdf/SFB649DP2011-076.pdf.Google Scholar
Stone, C.J. (1980) Optimal rates of convergence for nonparametric estimators. Annals of Statistics 8, 1348–1360.CrossRefGoogle Scholar
Tjøstheim, D. (1990) Nonlinear time series and Markov chains. Advances in Applied Probability 22, 587–611.CrossRefGoogle Scholar
van der Vaart, A.W. & Wellner, J. (1996) Weak Convergence and Empirical Processes with Applications to Statistics. Springer.CrossRefGoogle Scholar
Wang, Q.Y. & Phillips, P.C.B. (2009a) Asymptotic theory for local time density estimation and nonparametric cointegrating regression. Econometric Theory 25, 710–738.CrossRefGoogle Scholar
Wang, Q.Y. & Phillips, P.C.B. (2009b) Structural nonparametric cointegrating regression. Econometrica 77, 1901–1948.Google Scholar