IDEAS home Printed from
   My bibliography  Save this paper

Nonstationary Density Estimation and Kernel Autoregression




An asymptotic theory is developed for the kernel density estimate of a random walk and the kernel regression estimator of a nonstationary first order autoregression. The kernel density estimator provides a consistent estimate of the local time spent by the random walk in the spatial vicinity of a point that is determined in part by the argument of the density and in part by initial conditions. The kernel regression estimator is shown to be consistent and to have a mixed normal limit theory. The limit distribution has a mixing variate that is given by the reciprocal of the local time of a standard Brownian motion. The permissible range for the bandwidth parameter h_{n} includes rates which may increase as well as decrease with the sample size n, in contrast to the case of a stationary autoregression. However, the convergence rate of the kernel regression estimator is at most n^{1/4}, and this is slower than that of a stationary kernel autoregression, in contrast to the parametric case. In spite of these differences in the limit theory and the rates of convergence between the stationary and nonstationary cases, it is shown that the usual formulae for confidence intervals for the regression function still apply when h_{n} -> 0.

Suggested Citation

  • Peter C.B. Phillips & Joon Y. Park, 1998. "Nonstationary Density Estimation and Kernel Autoregression," Cowles Foundation Discussion Papers 1181, Cowles Foundation for Research in Economics, Yale University.
  • Handle: RePEc:cwl:cwldpp:1181

    Download full text from publisher

    File URL:
    Download Restriction: no

    References listed on IDEAS

    1. Jiang, George J. & Knight, John L., 1997. "A Nonparametric Approach to the Estimation of Diffusion Processes, With an Application to a Short-Term Interest Rate Model," Econometric Theory, Cambridge University Press, vol. 13(5), pages 615-645, October.
    2. Hardle, Wolfgang & Linton, Oliver, 1986. "Applied nonparametric methods," Handbook of Econometrics, in: R. F. Engle & D. McFadden (ed.), Handbook of Econometrics, edition 1, volume 4, chapter 38, pages 2295-2339, Elsevier.
    3. Collomb, Gérard & Härdle, Wolfgang, 1986. "Strong uniform convergence rates in robust nonparametric time series analysis and prediction: Kernel regression estimation from dependent observations," Stochastic Processes and their Applications, Elsevier, vol. 23(1), pages 77-89, October.
    4. P. M. Robinson, 1983. "Nonparametric Estimators For Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 4(3), pages 185-207, May.
    5. Wolfgang Härdle & Philippe Vieu, 1992. "Kernel Regression Smoothing Of Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 13(3), pages 209-232, May.
    6. repec:cup:etheor:v:13:y:1997:i:5:p:615-45 is not listed on IDEAS
    7. Ait-Sahalia, Yacine, 1996. "Nonparametric Pricing of Interest Rate Derivative Securities," Econometrica, Econometric Society, vol. 64(3), pages 527-560, May.
    8. Yacine Ait-Sahalia, 1998. "Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed-Form Approach," NBER Technical Working Papers 0222, National Bureau of Economic Research, Inc.
    9. Phillips, Peter C B & Ploberger, Werner, 1996. "An Asymptotic Theory of Bayesian Inference for Time Series," Econometrica, Econometric Society, vol. 64(2), pages 381-412, March.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Broadie, Mark & Detemple, Jerome & Ghysels, Eric & Torres, Olivier, 2000. "Nonparametric estimation of American options' exercise boundaries and call prices," Journal of Economic Dynamics and Control, Elsevier, vol. 24(11-12), pages 1829-1857, October.
    2. Bonsoo Koo & Oliver Linton, 2010. "Semiparametric Estimation of Locally Stationary Diffusion Models," STICERD - Econometrics Paper Series 551, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    3. GHYSELS, Eric & PATILEA, Valentin & RENAULT, Eric & TORRES, Olivier, 1997. "Nonparametric methods and option pricing," LIDAM Discussion Papers CORE 1997075, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Ghysels, E. & Ng, S., 1996. "A Semi-Parametric Factor Model for Interest Rates," Cahiers de recherche 9612, Universite de Montreal, Departement de sciences economiques.
    5. Broadie, Mark & Detemple, Jerome & Ghysels, Eric & Torres, Olivier, 2000. "American options with stochastic dividends and volatility: A nonparametric investigation," Journal of Econometrics, Elsevier, vol. 94(1-2), pages 53-92.
    6. Franke, Jürgen & Kreiss, Jens-Peter & Mammen, Enno, 1997. "Bootstrap of kernel smoothing in nonlinear time series," SFB 373 Discussion Papers 1997,20, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    7. Kristensen, Dennis, 2004. "Estimation in two classes of semiparametric diffusion models," LSE Research Online Documents on Economics 24739, London School of Economics and Political Science, LSE Library.
    8. Matthew Pritsker, 1997. "Nonparametric density estimation and tests of continuous time interest rate models," Finance and Economics Discussion Series 1997-26, Board of Governors of the Federal Reserve System (U.S.).
    9. Wolfgang Hardle & Torsten Kleinow & Alexander Korostelev & Camille Logeay & Eckhard Platen, 2008. "Semiparametric diffusion estimation and application to a stock market index," Quantitative Finance, Taylor & Francis Journals, vol. 8(1), pages 81-92.
    10. Qi Li & Jeffrey Scott Racine, 2006. "Nonparametric Econometrics: Theory and Practice," Economics Books, Princeton University Press, edition 1, number 8355.
    11. Ichimura, Hidehiko & Todd, Petra E., 2007. "Implementing Nonparametric and Semiparametric Estimators," Handbook of Econometrics, in: J.J. Heckman & E.E. Leamer (ed.), Handbook of Econometrics, edition 1, volume 6, chapter 74, Elsevier.
    12. Zhijie Xiao & Oliver Linton & Raymond J. Carroll & E. Mammen, 2002. "More Efficient Kernel Estimation in Nonparametric Regression with Autocorrelated Errors," Cowles Foundation Discussion Papers 1375, Cowles Foundation for Research in Economics, Yale University.
    13. Dabo-Niang, Sophie & Francq, Christian & Zakoïan, Jean-Michel, 2010. "Combining Nonparametric and Optimal Linear Time Series Predictions," Journal of the American Statistical Association, American Statistical Association, vol. 105(492), pages 1554-1565.
    14. Chen, Bin & Song, Zhaogang, 2013. "Testing whether the underlying continuous-time process follows a diffusion: An infinitesimal operator-based approach," Journal of Econometrics, Elsevier, vol. 173(1), pages 83-107.
    15. Cai, Zongwu, 2003. "Nonparametric estimation equations for time series data," Statistics & Probability Letters, Elsevier, vol. 62(4), pages 379-390, May.
    16. Aït-Sahalia, Yacine & Park, Joon Y., 2016. "Bandwidth selection and asymptotic properties of local nonparametric estimators in possibly nonstationary continuous-time models," Journal of Econometrics, Elsevier, vol. 192(1), pages 119-138.
    17. Oliver Linton & Douglas Steigerwald, 2000. "Adaptive testing in arch models," Econometric Reviews, Taylor & Francis Journals, vol. 19(2), pages 145-174.
    18. Renò, Roberto, 2008. "Nonparametric Estimation Of The Diffusion Coefficient Of Stochastic Volatility Models," Econometric Theory, Cambridge University Press, vol. 24(5), pages 1174-1206, October.
    19. Gao, Jiti & King, Maxwell, 2003. "Estimation and model specification testing in nonparametric and semiparametric econometric models," MPRA Paper 11989, University Library of Munich, Germany, revised Feb 2006.
    20. Kristensen, Dennis, 2008. "Estimation of partial differential equations with applications in finance," Journal of Econometrics, Elsevier, vol. 144(2), pages 392-408, June.


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:cwl:cwldpp:1181. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: . General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Brittany Ladd (email available below). General contact details of provider: .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.