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

Kohtaro Hitomi (hitomi@kit.ac.jp), Jianwei Jin (jin-jianwei-cd@ynu.jp), Keiji Nagai (nagai-keiji-hs@ynu.ac.jp), Yoshihiko Nishiyama (nishiyama@kier.kyoto-u.ac.jp) and Junfan Tao (tao.junfan.7j@kyoto-u.ac.jp)
Additional contact information
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

No 1084, KIER Working Papers from Kyoto University, Institute of Economic Research

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.

Keywords: Dickey-Fuller tests; Squared Bessel process; joint density; powers approximated by normal distribution; exact distribution (search for similar items in EconPapers)
JEL-codes: C12 C22 C46 (search for similar items in EconPapers)
Pages: 20 pages
Date: 2022-11
New Economics Papers: this item is included in nep-ecm and nep-ets
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