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Sequential unit root test for first-order autoregressive processes with initial values

Author

Listed:
  • Jianwei Jin

    (Yokohama National University)

  • Keiji Nagai

    (Yokohama National University)

Abstract

This paper examines the effect of initial values and small-sample properties in sequential unit root tests of the first-order autoregressive (AR(1)) process with a coefficient expressed by a local parameter. Adopting a stopping rule based on observed Fisher information defined by Lai and Siegmund (1983), we use the sequential least squares estimator (LSE) of the local parameter as the test statistic. The sequential LSE is represented as a time-changed Brownian motion with drift. The stopping time is written as the integral of the reciprocal of twice of a Bessel process with drift generated by the time-changed Brownian motion. The time change is applied to the joint density and joint Laplace transform derived from the Bessel bridge of the squared Bessel process by Pitman and Yor (1982), by which we derive the limiting joint density and joint Laplace transform for the sequential LSE and stopping time. The joint Laplace transform is needed to calculate joint moments because the joint density oscillates wildly as the value of the stopping time approaches zero. Moreover, this paper also earns the exact distribution of stopping time by Imhof's formula for both normally distributed and fixed initial values. When the autoregressive coefficient is less than 1, the question arises as to whether the local-to-unity or the strong stationary model should be used. We make the decision by comparing joint moments for respective models with those calculated from the exact distribution or simulations.

Suggested Citation

  • Jianwei Jin & Keiji Nagai, 2022. "Sequential unit root test for first-order autoregressive processes with initial values," KIER Working Papers 1085, Kyoto University, Institute of Economic Research.
  • Handle: RePEc:kyo:wpaper:1085
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    References listed on IDEAS

    as
    1. Perron, Pierre, 1991. "A Continuous Time Approximation to the Unstable First-Order Autoregressive Process: The Case without an Intercept," Econometrica, Econometric Society, vol. 59(1), pages 211-236, January.
    2. Kohtaro Hitomi & Keiji Nagai & Yoshihiko Nishiyama & Junfan Tao, 2021. "Joint Asymptotic Properties of Stopping Times and Sequential Estimators for Stationary First-order Autoregressive Models," KIER Working Papers 1060, Kyoto University, Institute of Economic Research.
    3. Keiji Nagai & Yoshihiko Nishiyama & Kohtaro Hitomi, 2018. "Sequential test for unit root in AR(1) model," KIER Working Papers 1003, Kyoto University, Institute of Economic Research.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Stopping time; observed Fisher information; DDS Brownian motion; local asymptotic normality; Bessel process; initial values; exact distributions;
    All these keywords.

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C46 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Specific Distributions

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