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32 results

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Brownian Motion, the Fredholm Determinant, and Time Series Analysis
  • Katsuto Tanaka
  • Published online:
    19 December 2024
    Print publication:
    02 January 2025
  • Brownian motion is an important topic in various applied fields where the analysis of random events is necessary. Introducing Brownian motion from a statistical viewpoint, this detailed text examines the distribution of quadratic plus linear or bilinear functionals of Brownian motion and demonstrates the utility of this approach for time series analysis. It also offers the first comprehensive guide on deriving the Fredholm determinant and the resolvent associated with such statistics. Presuming only a familiarity with standard statistical theory and the basics of stochastic processes, this book brings together a set of important statistical tools in one accessible resource for researchers and graduate students. Readers also benefit from online appendices, which provide probability density graphs and solutions to the chapter problems.

  • COMPUTING LIMITING LOCAL POWERS AND POWER ENVELOPES OF PANEL MA UNIT ROOT TESTS AND STATIONARITY TESTS

  • Katsuto Tanaka
  • Journal:
    Econometric Theory / Volume 35 / Issue 5 / October 2019
    Published online by Cambridge University Press:
    18 September 2018, pp. 978-1011

  • The present article discusses panel unit root tests for two classes of models. One is the moving average (MA) model and the other is the error components model. We conduct score type tests for these models, allowing for various types of regressors, and examine the cross-sectional effect explicitly by presenting an efficient way of computing limiting local powers. It is found that the existence of common regressors does not affect the asymptotic behavior of tests, although heterogeneous regressors do. We also derive the limiting power envelopes for some simple panel models, which shows that the score type panel tests are asymptotically efficient, unlike the time series case.

  • DISTRIBUTIONS OF QUADRATIC FUNCTIONALS OF THE FRACTIONAL BROWNIAN MOTION BASED ON A MARTINGALE APPROXIMATION

  • Katsuto Tanaka
  • Journal:
    Econometric Theory / Volume 30 / Issue 5 / October 2014
    Published online by Cambridge University Press:
    02 April 2014, pp. 1078-1109

  • The present paper deals with the distributions related to the fractional Brownian motion (fBm). In particular, we try to compute the distributions of (ratios of) its quadratic functionals, not by simulations, but by numerically inverting the associated characteristic functions (c.f.s). Among them is the fractional unit root distribution. It turns out that the derivation of the c.f.s based on the standard approaches used for the ordinary Bm is inapplicable. Here the martingale approximation to the fBm suggested in the literature is used to compute an approximation to the distributions of such functionals. The associated c.f. is obtained via the Fredholm determinant. Comparison of the first two moments of the approximate with true distributions is made, and simulations are conducted to examine the performance of the approximation. We also find an interesting moment property of the approximate fractional unit root distribution, and a conjecture is given that the same property will hold for the true fractional unit root distribution.

  • LINEAR NONSTATIONARY MODELS—A REVIEW OF THE WORK OF PROFESSOR P.C.B. PHILLIPS

  • Katsuto Tanaka
  • Journal:
    Econometric Theory / Volume 30 / Issue 4 / August 2014
    Published online by Cambridge University Press:
    10 March 2014, pp. 815-838

  • The work of Professor P.C.B. Phillips, even if it is focused on the area of linear nonstationary models, is enormous. So it is hard for me to explore the whole of his work in this paper. Therefore, I have decided to take up only a few results of his work. The topics chosen here are applications of the martingale approximation and the problem of choosing between stochastic and deterministic trends, which I discuss and, hopefully, extend.

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