• Bernoulli
  • Volume 20, Number 1 (2014), 207-230.

Uniform convergence rates for a class of martingales with application in non-linear cointegrating regression

Qiying Wang and Nigel Chan

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For a class of martingales, this paper provides a framework on the uniform consistency with broad applicability. The main condition imposed is only related to the conditional variance of the martingale, which holds true for stationary mixing time series, stationary iterated random function, Harris recurrent Markov chains and $I(1)$ processes with innovations being a linear process. Using the established results, this paper investigates the uniform convergence of the Nadaraya–Watson estimator in a non-linear cointegrating regression model. Our results not only provide sharp convergence rate, but also the optimal range for the uniform convergence to be held. This paper also considers the uniform upper and lower bound estimates for a functional of Harris recurrent Markov chain, which are of independent interests.

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Bernoulli, Volume 20, Number 1 (2014), 207-230.

First available in Project Euclid: 22 January 2014

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Harris recurrent Markov chain martingale non-linearity non-parametric regression non-stationarity uniform convergence


Wang, Qiying; Chan, Nigel. Uniform convergence rates for a class of martingales with application in non-linear cointegrating regression. Bernoulli 20 (2014), no. 1, 207--230. doi:10.3150/12-BEJ482.

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  • [1] Andrews, D.W.K. (1995). Nonparametric kernel estimation for semiparametric models. Econometric Theory 11 560–596.
  • [2] Bosq, D. (1998). Nonparametric Statistics for Stochastic Processes: Estimation and Prediction, 2nd ed. Lecture Notes in Statistics 110. New York: Springer.
  • [3] Cai, Z., Li, Q. and Park, J.Y. (2009). Functional-coefficient models for nonstationary time series data. J. Econometrics 148 101–113.
  • [4] Chen, J., Li, D. and Zhang, L. (2009). Robust estimation in nonlinear cointegrating model. J. Multivariate Anal. 101 707–717.
  • [5] Chen, X. (2000). On the limit laws of the second order for additive functionals of Harris recurrent Markov chains. Probab. Theory Related Fields 116 89–123.
  • [6] de la Peña, V.H. (1999). A general class of exponential inequalities for martingales and ratios. Ann. Probab. 27 537–564.
  • [7] Diaconis, P. and Freedman, D. (1999). Iterated random functions. SIAM Rev. 41 45–76.
  • [8] Fan, J. and Yao, Q. (2003). Nonlinear Time Series: Nonparametric and Parametric Methods. Springer Series in Statistics. New York: Springer.
  • [9] Gao, J., King, M., Lu, Z. and Tjøstheim, D. (2009). Nonparametric specification testing for nonlinear time series with nonstationarity. Econometric Theory 25 1869–1892.
  • [10] Gao, J., King, M., Lu, Z. and Tjøstheim, D. (2009). Specification testing in nonlinear and nonstationary time series autoregression. Ann. Statist. 37 3893–3928.
  • [11] Gao, J., Li, D. and Tjøstheim, D. (2011). Uniform consistency for nonparametric estimates in null recurrent time series. Working Paper 0085, School of Economics, Univ. Adelaide.
  • [12] Hansen, B.E. (2008). Uniform convergence rates for kernel estimation with dependent data. Econometric Theory 24 726–748.
  • [13] Karlsen, H.A., Myklebust, T. and Tjøstheim, D. (2007). Nonparametric estimation in a nonlinear cointegration model. Ann. Statist. 35 252–299.
  • [14] Karlsen, H.A. and Tjøstheim, D. (2001). Nonparametric estimation in null recurrent time series. Ann. Statist. 29 372–416.
  • [15] Kasparis, I. and Phillips, P.C.B. (2009). Dynamic misspecification in nonparametric cointegrating regression. Discussion Paper 1700, Cowles Foundation.
  • [16] Kristensen, D. (2009). Uniform convergence rates of kernel estimators with heterogeneous dependent data. Econometric Theory 25 1433–1445.
  • [17] Liero, H. (1989). Strong uniform consistency of nonparametric regression function estimates. Probab. Theory Related Fields 82 587–614.
  • [18] Marmer, V. (2008). Nonlinearity, nonstationarity, and spurious forecasts. J. Econometrics 142 1–27.
  • [19] Masry, E. (1996). Multivariate local polynomial regression for time series: Uniform strong consistency and rates. J. Time Series Anal. 17 571–599.
  • [20] Nummelin, E. (1984). General Irreducible Markov Chains and Nonnegative Operators. Cambridge Tracts in Mathematics 83. Cambridge: Cambridge Univ. Press.
  • [21] Nze, P.A. and Doukhan, P. (2004). Weak dependence: Models and applications to econometrics. Econometric Theory 20 995–1045.
  • [22] Park, J.Y. and Phillips, P.C.B. (1999). Asymptotics for nonlinear transformations of integrated time series. Econometric Theory 15 269–298.
  • [23] Park, J.Y. and Phillips, P.C.B. (2001). Nonlinear regressions with integrated time series. Econometrica 69 117–161.
  • [24] Peligrad, M. (1992). Properties of uniform consistency of the kernel estimators of density and of regression functions under dependence assumptions. Stochastics Stochastics Rep. 40 147–168.
  • [25] Tong, H. (1990). Nonlinear Time Series: A Dynamical System Approach. Oxford Statistical Science Series 6. New York: Oxford Univ. Press.
  • [26] Wang, Q. (2011). Martingale limit theorems revisited and non-linear cointegrating regression. Working paper.
  • [27] Wang, Q. and Phillips, P.C.B. (2009). Asymptotic theory for local time density estimation and nonparametric cointegrating regression. Econometric Theory 25 710–738.
  • [28] Wang, Q. and Phillips, P.C.B. (2009). Structural nonparametric cointegrating regression. Econometrica 77 1901–1948.
  • [29] Wang, Q. and Phillips, P.C.B. (2011). Asymptotic theory for zero energy functionals with nonparametric regression applications. Econometric Theory 27 235–259.
  • [30] Wang, Q. and Phillips, P.C.B. (2012). A specification test for nonlinear nonstationary models. Ann. Statist. 40 727–758.
  • [31] Wang, Q. and Wang, R. (2013). Non-parametric cointegrating regression with NNH errors. Econometric Theory 29 1–27.
  • [32] Wu, W.B., Huang, Y. and Huang, Y. (2010). Kernel estimation for time series: An asymptotic theory. Stochastic Process. Appl. 120 2412–2431.
  • [33] Wu, W.B. and Shao, X. (2004). Limit theorems for iterated random functions. J. Appl. Probab. 41 425–436.