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Nonstationary Density Estimation and Kernel Autoregression

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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.

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
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    File URL: http://cowles.yale.edu/sites/default/files/files/pub/d11/d1181.pdf
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    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.
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    3. 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).
    4. 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.
    5. repec:cup:etheor:v:13:y:1997:i:5:p:615-45 is not listed on IDEAS
    6. Ait-Sahalia, Yacine, 1996. "Nonparametric Pricing of Interest Rate Derivative Securities," Econometrica, Econometric Society, vol. 64(3), pages 527-560, May.
    7. 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.
    8. 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.
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