Bernoulli

  • Bernoulli
  • Volume 18, Number 2 (2012), 678-702.

Estimation in semi-parametric regression with non-stationary regressors

Jia Chen, Jiti Gao, and Degui Li

Full-text: Open access

Abstract

In this paper, we consider a partially linear model of the form Yt = Xtτθ0 + g(Vt) + ϵt, t = 1, …, n, where {Vt} is a β null recurrent Markov chain, {Xt} is a sequence of either strictly stationary or non-stationary regressors and {ϵt} is a stationary sequence. We propose to estimate both θ0 and g(⋅) by a semi-parametric least-squares (SLS) estimation method. Under certain conditions, we then show that the proposed SLS estimator of θ0 is still asymptotically normal with the same rate as for the case of stationary time series. In addition, we also establish an asymptotic distribution for the nonparametric estimator of the function g(⋅). Some numerical examples are provided to show that our theory and estimation method work well in practice.

Article information

Source
Bernoulli, Volume 18, Number 2 (2012), 678-702.

Dates
First available in Project Euclid: 16 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.bj/1334580729

Digital Object Identifier
doi:10.3150/10-BEJ344

Mathematical Reviews number (MathSciNet)
MR2922466

Zentralblatt MATH identifier
1238.62044

Keywords
asymptotic theory nonparametric estimation null recurrent time series semi-parametric regression

Citation

Chen, Jia; Gao, Jiti; Li, Degui. Estimation in semi-parametric regression with non-stationary regressors. Bernoulli 18 (2012), no. 2, 678--702. doi:10.3150/10-BEJ344. https://projecteuclid.org/euclid.bj/1334580729


Export citation

References

  • [1] Bandi, F.M. and Phillips, P.C.B. (2003). Fully nonparametric estimation of scalar diffusion models. Econometrica 71 241–283.
  • [2] Bhattacharya, P.K. and Zhao, P.L. (1997). Semiparametric inference in a partial linear model. Ann. Statist. 25 244–262.
  • [3] Bickel, P.J., Klaassen, C.A.J., Ritov, Y. and Wellner, J.A. (1993). Efficient and Adaptive Estimation for Semiparametric Models. Baltimore, MD: Johns Hopkins Univ. Press.
  • [4] Chen, H. (1988). Convergence rates for parametric components in a partly linear model. Ann. Statist. 16 136–146.
  • [5] Chen, J., Gao, J. and Li, D. (2010). Supplement to “Estimation in semi-parametric regression with non-stationary regressors.”DOI:10.3150/10-BEJ344SUPP.
  • [6] Chen, X. (2007). Large sample sieve estimation of semi–nonparametric models. In Handbook of Econometrics VI (J.J. Heckman and E.E. Leamer, eds.) 5549–5632. Amsterdam: Elsevier.
  • [7] Engle, R., Granger, C., Rice, J. and Weiss, A. (1986). Semiparametric estimates of the relation between weather and electricity sales. J. Amer. Statist. Assoc. 81 310–320.
  • [8] Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Monographs on Statistics and Applied Probability 66. London: Chapman and Hall.
  • [9] Fan, J. and Yao, Q. (2003). Nonlinear Time Series. Nonparametric and Parametric Methods. New York: Springer.
  • [10] Gao, J. (2007). Nonlinear Time Series. Semiparametric and Nonparametric Methods. Monographs on Statistics and Applied Probability 108. Boca Raton, FL: Chapman and Hall/CRC.
  • [11] Gao, J., King, M., Lu, Z. and Tjøstheim, D. (2009). Nonparametric specification testing for nonlinear time series with nonstationarity. Econometric Theory 25 1869–1892.
  • [12] Gao, J., King, M.L., Lu, Z. and Tjøstheim, D. (2009). Specification testing in nonstationary time series autoregression. Ann. Statist. 37 3893–3928.
  • [13] Green, P.J. and Silverman, B.W. (1994). Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach. Monographs on Statistics and Applied Probability 58. London: Chapman & Hall.
  • [14] Hall, P. and Heyde, C.C. (1980). Martingale Limit Theory and Its Application. New York: Academic Press.
  • [15] Härdle, W., Liang, H. and Gao, J. (2000). Partially Linear Models. Heidelberg: Physica-Verlag.
  • [16] Härdle, W., Lütkepohl, H. and Chen, R. (1997). A review of nonparametric time series analysis. Internat. Statist. Rev. 65 49–72.
  • [17] Hastie, T.J. and Tibshirani, R.J. (1990). Generalized Additive Models. Monographs on Statistics and Applied Probability 43. London: Chapman and Hall.
  • [18] Juhl, T. and Xiao, Z. (2005). Partially linear models with unit roots. Econometric Theory 21 877–906.
  • [19] Karlsen, H.A., Myklebust, T. and Tjøstheim, D. (2007). Nonparametric estimation in a nonlinear cointegration type model. Ann. Statist. 35 252–299.
  • [20] Karlsen, H.A. and Tjøstheim, D. (2001). Nonparametric estimation in null recurrent time series. Ann. Statist. 29 372–416.
  • [21] Li, Q. and Racine, J.S. (2007). Nonparametric Econometrics. Theory and Practice. Princeton, NJ: Princeton Univ. Press.
  • [22] Lin, Z. and Lu, C. (1996). Limit Theory for Mixing Dependent Random Variables. Mathematics and Its Applications 378. Dordrecht: Kluwer Academic.
  • [23] Linton, O. (1995). Second order approximation in the partially linear regression model. Econometrica 63 1079–1112.
  • [24] Nummelin, E. (1984). General Irreducible Markov Chains and Nonnegative Operators. Cambridge Tracts in Mathematics 83. Cambridge: Cambridge Univ. Press.
  • [25] Phillips, P.C.B. and Park, J. (1998). Nonstationary density estimation and kernel autoregression. Cowles Foundation Discussion Paper No. 1181, Yale Univ.
  • [26] Robinson, P.M. (1983). Nonparametric estimators for time series. J. Time Ser. Anal. 4 185–207.
  • [27] Robinson, P.M. (1988). Root-N-consistent semiparametric regression. Econometrica 56 931–954.
  • [28] Robinson, P.M. (1989). Hypothesis testing in semiparametric and nonparametric models for econometric time series. Rev. Econom. Stud. 56 511–534.
  • [29] Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition. Proc. Natl. Acad. Sci. U.S.A. 42 43–47.
  • [30] Schienle, M. (2008). Nonparametric nonstationary regression. Working paper, Department of Economics, Univ. Mannheim, Germany.
  • [31] Silverman, B.W. (1986). Density Estimation for Statistics and Data Analysis. London: Chapman and Hall.
  • [32] Wang, Q. and Phillips, P.C.B. (2009). Asymptotic theory for local time density estimation and nonparametric cointegrating regression. Econometric Theory 25 710–738.
  • [33] Wang, Q. and Phillips, P.C.B. (2009). Structural nonparametric cointegrating regression. Econometrica 77 1901–1948.
  • [34] Withers, C.S. (1981). Conditions for linear processes to be strong-mixing. Z. Wahrsch. Verw. Gebiete 57 477–480.

Supplemental materials

  • Supplementary material: Proofs of the theorems. We provide this supplemental document in case the reader may want to have a look at the detailed proofs of Theorems 3.1 and 3.2 and Lemma 6.1. The details are available from Chen, Gao and Li [5].