Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-14T22:50:07.376Z Has data issue: false hasContentIssue false
  • English
  • Français

ASYMPTOTIC THEORY FOR LOCAL TIME DENSITY ESTIMATION AND NONPARAMETRIC COINTEGRATING REGRESSION

Published online by Cambridge University Press:  01 June 2009

Qiying Wang  and
Peter C.B. Phillips
Get access Rights & Permissions [Opens in a new window]

Abstract

Asymptotic theory is developed for local time density estimation for a general class of functionals of integrated and fractionally integrated time series. The main result provides a convenient basis for developing a limit theory for nonparametric cointegrating regression and nonstationary autoregression. The treatment directly involves local time estimation and the density function of the processes under consideration, providing an alternative approach to the Markov chain and Fourier integral methods that have been used in other recent work on these problems.

Type
Research Article
Information
Econometric Theory , Volume 25 , Issue 3 , June 2009 , pp. 710 - 738
Copyright
Copyright © Cambridge University Press 2009

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.)

Footnotes

The authors thank the co-editor and two referees for helpful comments on the original version. Wang acknowledges partial research support from the Australian Research Council. Phillips acknowledges partial research support from a Kelly Fellowship and the NSF under grant, SES 04-142254 and SES 06-47086. Wang can be contacted at qiying@maths.usyd.edu.au.

References

REFERENCES

Akonom, J. (1993) Comportement asymptotique du temps d’occupation du processus des sommes partielles. Ann. Inst. H. Poincaré Probab. Statist. 29, 57–81.Google Scholar
Bandi, F. (2004) On Persistence and Nonparametric Estimation (with an Application to Stock Return Predictability). Manuscript, Graduate School of Business, Chicago.Google Scholar
Berkes, I. & Horváth, L. (2006) Convergence of integral functionals of stochastic processes. Econometric Theory 22, 304–322.CrossRefGoogle Scholar
Borodin, A.N. & Ibragimov, I.A. (1995) Limit Theorems for Functionals of Random Walks. Proc. Steklov Inst. Math. 2.Google Scholar
de Jong, R. (2004) Addendum to: “Asymptotics for nonlinear. transformations of integrated time series.” Econometric Theory 20, 627–635.CrossRefGoogle Scholar
de Jong, R. & Wang, C.-H. (2005) Further results on the asymptotics for nonlinear transformations of integrated time series. Econometric Theory 21, 413–430.CrossRefGoogle Scholar
Feller, W. (1971) An Introduction to Probability Theory and Its Applications, vol. II, 2nd ed. Wiley.Google Scholar
Geman, D., & Horowitz, J. (1980) Occupation densities. Annals of Probability 8, 1–67.CrossRefGoogle Scholar
Gorodetskiĭ, V.V. (1977) Convergence to semistable Gaussian processes. Teor. Verojatnost. i Primenen. 22, 513–522.Google Scholar
Guerre, E. (2004) Design-Adaptive Pointwise Nonparametric Regression Estimation for Recurrent Markov Time Series. Manuscript, Queen Mary College, London.10.2139/ssrn.619303Google Scholar
Hall, P. & Heyde, C.C. (1980) Martingale Limit Theory and Its Application. Academic.Google Scholar
Hannan, E.J. (1979) The central limit theorem for time series regression. Stochastic Process. Appl. 9, 281–289.CrossRefGoogle Scholar
Hu, L. & Phillips, P.C.B. (2004) Dynamics of the federal funds target rate: a nonstationary discrete choice approach. Journal of Applied Econometrics 19, 851–867.10.1002/jae.747CrossRefGoogle Scholar
Jacod, J. & Shiryaev, A.N. (2003) Limit Theorems for Stochastic Processes, 2nd ed. Springer- Verlag.CrossRefGoogle Scholar
Jeganathan, P. (2004) Convergence of functionals of sums of r.v.s to local times of fractional stable motions. Annals of Probability 32, 1771–1795.CrossRefGoogle Scholar
Karlsen, H.A. & Tjøstheim, D. (2001) Nonparametric estimation in null recurrent time series. Annals of Statistics 29, 372–416.CrossRefGoogle Scholar
Karlsen, H.A., Myklebust, T. & Tjøstheim, D. (2007) Nonparametric estimation in a nonlinear cointegration model. Annals of Statistics 35, 252–299.10.1214/009053606000001181CrossRefGoogle Scholar
Lukács, E. (1970) Characteristic Functions. Hafner.Google Scholar
Park, J.Y. & Phillips, P.C.B. (1999) Asymptotics for nonlinear transformation of integrated time series. Econometric Theory, 15, 269–298.CrossRefGoogle Scholar
Park, J.Y. & Phillips, P.C.B. (2001) Nonlinear regressions with integrated time series. Econometrica 69, 117–161.CrossRefGoogle Scholar
Phillips, P.C.B. (2001) Descriptive econometrics for non-stationary time series with empirical applications. Journal of Applied Econometrics 16, 389–413.CrossRefGoogle Scholar
Phillips, P.C.B. (2005) Econometric analysis of Fisher’s equation. American Journal of Economics and Sociology 64, 125–168.CrossRefGoogle Scholar
Phillips, P.C.B. & Park, J.Y. (1998) Nonstationary Density Estimation and Kernel Autoregression. Cowles Foundation Discussion paper 1181.Google Scholar
Pötscher, B.M. (2004) Nonlinear functions and convergence to Brownian motion: Beyond the continuous mapping theorem. Econometric Theory 20, 1–22.CrossRefGoogle Scholar
Revuz, D. & Yor, M. (1999) Continuous Martingales and Brownian Motion. Fundamental Principles of Mathematical Sciences 293. Springer-Verlag.CrossRefGoogle Scholar
Shorack, G.R. & Wellner, J.A. (1986) Empirical Processes with Applications to Statistics. Wiley.Google Scholar
Taqqu, M.S. (1975) Weak convergence to fractional Brownian motion and to the Rosenblatt process. Zeitschrift für Wahrscheinlichskeitstheorie and Verwandte Gebiete 31, 287–302.CrossRefGoogle Scholar
Wang, Q., Lin, Y.-X., & Gulati, C.M. (2003a) Strong approximation for long memory processes with applications. Journal of Theoretical Probability 16, 377–389.CrossRefGoogle Scholar
Wang, Q., Lin, Y.-X., & Gulati, C.M. (2003b) Asymptotics for general fractionally integrated processes with applications to unit root tests. Econometric Theory 19, 143–164.CrossRefGoogle Scholar