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Nonstationary Density Estimation and Kernel Autoregression

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Author Info
Peter C.B. Phillips () (Cowles Foundation, Yale University)
Joon Y. Park (School of Economics, Seoul National University)

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Abstract

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.

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File URL: http://cowles.econ.yale.edu/P/cd/d11b/d1181.pdf
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Publisher Info
Paper provided by Cowles Foundation, Yale University in its series Cowles Foundation Discussion Papers with number 1181.

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Length: 27 pages
Date of creation: Jun 1998
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Handle: RePEc:cwl:cwldpp:1181

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Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

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Related research
Keywords: Brownian sheet; kernel regression; local time; martingale embedding; mixture normal; nonstationary density; occupation time; quadratic variation; unit root autoregression;

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References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
  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(05), pages 615-645, October. [Downloadable!]
  2. Hardle, W. & Vieu, P., 1990. "Kernel regression smoothing of time series," CORE Discussion Papers 1990031, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  3. Ait-Sahalia, Yacine, 1996. "Nonparametric Pricing of Interest Rate Derivative Securities," Econometrica, Econometric Society, vol. 64(3), pages 527-60, May. [Downloadable!] (restricted)
    Other versions:
  4. repec:cup:etheor:v:13:y:1997:i:5:p:615-45 is not listed on IDEAS
  5. Wolfgang Hardle & Oliver Linton, 1994. "Applied Nonparametric Methods," Cowles Foundation Discussion Papers 1069, Cowles Foundation, Yale University. [Downloadable!]
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  6. 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. [Downloadable!] (restricted)
  7. 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. [Downloadable!] (restricted)
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  1. Oliver Linton & Enno Mammen, 2006. "Nonparametric Transformation to White Noise," STICERD - Econometrics Paper Series /2006/503, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE. [Downloadable!]
    Other versions:
  2. Emmanuel Guerre & Hyungsik Roger Moon, 2005. "A Study of a Semiparametric Binary Choice Model with Integrated Covariates," IEPR Working Papers 05.37, Institute of Economic Policy Research (IEPR). [Downloadable!]
    Other versions:
  3. H. Karlsen & T. Myklebust & D. Tjostheim, . "Nonparametric Estimation in a Nonlinear Cointegration Type Model," Sonderforschungsbereich 373 2000-33, Humboldt Universitaet Berlin.
  4. Qiying Wang & Peter C. B. Phillips, 2009. "Asymptotic Theory for Zero Energy Density Estimation with Nonparametric Regression Applications," Cowles Foundation Discussion Papers 1687, Cowles Foundation, Yale University. [Downloadable!]
  5. Qiying Wang & Peter C.B. Phillips, 2006. "Asymptotic Theory for Local Time Density Estimation and Nonparametric Cointegrating Regression," Cowles Foundation Discussion Papers 1594, Cowles Foundation, Yale University. [Downloadable!]
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  6. Peter C. B. Phillips, 2001. "Descriptive econometrics for non-stationary time series with empirical illustrations," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 16(3), pages 389-413. [Downloadable!]
    Other versions:
  7. Peter C.B. Phillips, 2008. "Local Limit Theory and Spurious Nonparametric Regression," Cowles Foundation Discussion Papers 1654, Cowles Foundation, Yale University. [Downloadable!]
  8. Guerre, 2004. "Design-Adaptive Pointwise Nonparametric Regression Estimation For Recurrent Markov Time Series," Econometrics 0411007, EconWPA. [Downloadable!]
    Other versions:
  9. Peter C.B. Phillips, 1998. "Econometric Analysis of Fisher's Equation," Cowles Foundation Discussion Papers 1180, Cowles Foundation, Yale University. [Downloadable!]
  10. Joon Y. Park & Yoon-Jae Whang, 1999. "Random Walk or Chaos: A Formal Test on the Lyapunov Exponent," Working Paper Series no9, Institute of Economic Research, Seoul National University. [Downloadable!]
  11. Peter C.B. Phillips & Sainan Jin & Ling Hu, 2005. "Nonstationary Discrete Choice: A Corrigendum and Addendum," Cowles Foundation Discussion Papers 1516, Cowles Foundation, Yale University. [Downloadable!]
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  12. Federico Bandi & Peter C. B. Phillips, 2000. "Accelerated Asymptotics for Diffusion Model Estimation," Econometric Society World Congress 2000 Contributed Papers 1656, Econometric Society. [Downloadable!]
  13. Seung Hyun Hong & Peter C. B. Phillips, 2005. "Testing Linearity in Cointegrating Relations with an Application to Purchasing Power Parity," Cowles Foundation Discussion Papers 1541, Cowles Foundation, Yale University. [Downloadable!]
  14. Andrew Jeffrey & Linton, Oliver Linton & Thong Nguyen & Peter C.B. Phillips, 2001. "Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach," Cowles Foundation Discussion Papers 1311, Cowles Foundation, Yale University. [Downloadable!]
  15. Ted Juhl & Zhijie Xiao, 2000. "N-Consistent Semiparametric Regression: Partially Linear Models with Unit Roots," Econometric Society World Congress 2000 Contributed Papers 1532, Econometric Society. [Downloadable!]
  16. Mototsugu Shintani & Oliver Linton, 2000. "Is There Chaos in the World Economy? A Nonparametric Test Using Consistent Standard Errors," Working Papers 0111, Department of Economics, Vanderbilt University, revised Jun 2001. [Downloadable!]
    Other versions:
  17. Joon Y. Park & Peter C.B. Phillips, 1998. "Nonlinear Regressions with Integrated Time Series," Cowles Foundation Discussion Papers 1190, Cowles Foundation, Yale University. [Downloadable!]
    Other versions:
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