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Unit root tests considering initial values and a concise method for computing powers

Author

Listed:
  • Kohtaro Hitomi

    (Kyoto Institute of Technology)

  • Jianwei Jin

    (Yokohama National University)

  • Keiji Nagai

    (Yokohama National University)

  • Yoshihiko Nishiyama

    (Institute of Economic Research, Kyoto University)

  • Junfan Tao

    (Institute of Economic Research, Kyoto University)

Abstract

The Dickey-Fuller (DF) unit root tests are widely used in empirical studies on economics. In the local-to-unity asymptotic theory, the effects of initial values vanish as the sample size grows. However, for a small sample size, the initial value will affect the distribution of the test statistics. When ignoring the effect of the initial value, the left-sided unit root test sets the critical value smaller than it should be. Therefore, the size and power of the test become smaller. This paper investigates the effect of the initial value for the DF test (including the t test). Limiting approximations of the DF test statistics are the ratios of two integrals which are represented via a one-dimensional squared Bessel process. We derive the joint density of the squared Bessel process and its integral, enabling us to compute this ratio's distribution. For independent normal errors, the exact distribution of the Dickey-Fuller coefficient test statistic is obtained using the Imhof (1961) method for non-central chi-squared distribution. Numerical results show that when the sample size is small, the limiting distributions of the DF test statistics with initial values fit well with the exact or simulated distributions. We transform the DF test with respect to a local parameter into the test for a shift in the location parameter of normal distributions. As a result, a concise method for computing the powers of DF tests is derived.

Suggested Citation

  • Kohtaro Hitomi & Jianwei Jin & Keiji Nagai & Yoshihiko Nishiyama & Junfan Tao, 2022. "Unit root tests considering initial values and a concise method for computing powers," KIER Working Papers 1084, Kyoto University, Institute of Economic Research.
  • Handle: RePEc:kyo:wpaper:1084
    as

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    References listed on IDEAS

    as
    1. Peter C.B. Phillips, 1987. "Multiple Regression with Integrated Time Series," Cowles Foundation Discussion Papers 852, Cowles Foundation for Research in Economics, Yale University.
    2. Abadir, Karim M., 1995. "The Limiting Distribution of the t Ratio Under a Unit Root," Econometric Theory, Cambridge University Press, vol. 11(4), pages 775-793, August.
    3. 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.
    4. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    5. 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.
    6. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    7. 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

    Dickey-Fuller tests; Squared Bessel process; joint density; powers approximated by normal distribution; exact distribution;
    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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