Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T18:12:51.185Z Has data issue: false hasContentIssue false
  • English
  • Français

PARTIALLY LINEAR MODELS WITH UNIT ROOTS

Published online by Cambridge University Press:  22 August 2005

Ted Juhl  and
Zhijie Xiao
Ted Juhl
Affiliation:
University of Kansas
Zhijie Xiao
Affiliation:
Boston College
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper studies the asymptotic properties of a nonstationary partially linear regression model. In particular, we allow for covariates to enter the unit root (or near unit root) model in a nonparametric fashion, so that our model is an extension of the semiparametric model analyzed in Robinson (1988, Econometrica 56, 931–954). It is proved that the autoregressive parameter can be estimated at rate N even though part of the model is estimated nonparametrically. Unit root tests based on the semiparametric estimate of the autoregressive parameter have a limiting distribution that is a mixture of a standard normal and the Dickey–Fuller distribution. A Monte Carlo experiment is conducted to evaluate the performance of the tests for various linear and nonlinear specifications.We thank Bruce Hansen, Roger Koenker, Helmut Lütkepohl, Peter Phillips, three referees, and participants of the 8th World Congress of the Econometric Society and the 10th Midwest Econometrics Group Meeting for helpful comments on an earlier version of this paper. This investigation was supported by the University of Kansas General Research Fund allocation 2301789-003.

Type
Research Article
Information
Econometric Theory , Volume 21 , Issue 5 , October 2005 , pp. 877 - 906
Copyright
© 2005 Cambridge University Press

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. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817–858.Google Scholar
Beelders, O. (1999) Adaptive Estimation and Unit Root Tests. Working paper, Emory University.
Beveridge, S. & C.R. Nelson (1981) A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the “business cycle.” Journal of Monetary Economics 7, 151–174.Google Scholar
Chan, N.H. & C.Z. Wei (1987) Asymptotic inference for nearly nonstationary AR(1) processes. Annals of Statistics 15, 1050–1063.Google Scholar
Chen, H. (1988) Convergence rates for parametric components. Annals of Statistics 16, 135–146.Google Scholar
Doornik, J.A. (1998): Ox—An Object Oriented Matrix Programming Language. Timberlake Consultants Press.
Elliot, G., T.J. Rothenberg, & J.H. Stock (1996) Efficient tests for an autoregressive unit root. Econometrica 64, 813–836.Google Scholar
Engle, R.F., C.W.J. Granger, J. Rice, & A. Weiss (1986) Semiparametric estimates for the relation between weather and electricity sales. Journal of the American Statistical Association 81, 310–320.Google Scholar
Fan, Y. & Q. Li (1999) Root-N-consistent estimation of partially linear time series models. Journal of Nonparametric Statistics 11, 251–269.Google Scholar
Fan, Y., Q. Li, & T. Stengos (1995) Root-N-consistent semiparametric regression with conditional heteroskedastic disturbances. Journal of Quantitative Economics 11, 229–240.Google Scholar
Granger, C.W.J. (1995) Modeling nonlinear relationships between extended-memory variables. Econometrica 63, 265–279.Google Scholar
Granger, C.W.J. & T. Terasvirta (1993) Modelling Nonlinear Economic Relationships. Oxford University Press.
Hamilton, J.D. (1994) Time Series Analysis. Princeton University Press.
Hansen, B.E. (1995) Rethinking the univariate approach to unit root testing. Econometric Theory 11, 1148–1171.Google Scholar
Hasan, M.N. & R.W. Koenker (1997) Robust rank tests of the unit root hypothesis. Econometrica 65, 133–161.Google Scholar
Heckman, N.E. (1986) Spline smoothing in a partly linear model. Journal of the Royal Statistical Society, Series B 48, 244–248.Google Scholar
Juhl, T. & Z. Xiao (2000) Partially Linear Models with Unit Roots. Working paper, University of Illinois.
Linton, O.B. (1995) Second order approximation in the partially linear regression model. Econometrica 63, 1079–1112.Google Scholar
Lucas, A. (1995) Unit root tests based on M estimators. Econometric Theory 11, 331–346.Google Scholar
Ng, S. & P. Perron (2001) Lag length selection and the construction of unit root tests with good size and power. Econometrica 69, 1519–1554.Google Scholar
Pham, D. & L. Tran (1985) Some mixing properties of time series models. Stochastic Process and Applications 19, 297–303.Google Scholar
Phillips, P.C.B. (1987) Time series regression with a unit root. Econometrica 55, 277–301.Google Scholar
Phillips, P.C.B. & J.Y. Park (1998) Nonstationary Density Estimation and Kernel Autoregression. Working paper, Yale University.
Powell, J.L., J.H. Stock, & T.M. Stoker (1989) Semiparametric estimation of index coefficients. Econometrica 57, 1403–1430.Google Scholar
Rice, J. (1986) Convergence rates for partially linear spline models. Statistics and Probability Letters 4, 203–208.Google Scholar
Robinson, P.M. (1988) Root-N-consistent semiparametric regression. Econometrica 56, 931–954.Google Scholar
Schick, A. (1986) On asymptotically efficient estimation in semiparametric models. Annals of Statistics 14, 1139–1151.Google Scholar
Seo, B. (1999) Distribution theory for unit root tests with conditional heteroskedasticity. Journal of Econometrics 91, 113–144.Google Scholar
Shiller, R.J. (1984) Smoothness priors and nonlinear regression. Journal of the American Statistical Association 72, 420–423.Google Scholar
Shin, D.W. & B.S. So (1999) Unit root tests based on adaptive maximum likelihood estimation. Econometric Theory 15, 1–23.Google Scholar
Speckman, P. (1988) Kernel smoothing in partial linear models. Journal of the Royal Statistical Society, Series B 50, 413–436.Google Scholar
Tong, H. (1990) Nonlinear Time Series: A Dynamical System Approach. Oxford University Press.
Wahba, G. (1986) Partial and interaction spline models for the semiparametric estimation of functions of several variables. In T.J. Boardman (ed.), Proceedings of Computer Science and Statistics: 18th Symposium on the Interface. American Statistical Association.
Yoshihara, K. (1976) Limiting behavior of U-statistics for stationary, absolutely regular processes. Zeitschrift für Wahrscheinlichskeitstheorie und Verwandte Gebiete 35, 237–252.Google Scholar