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An inverted beta approximation to a MPI unit root test. (English) Zbl 1400.62206
Summary: We extend a result of A. T. A. Wood [Commun. Stat., Simulation Comput. 18, No. 4, 1439–1456 (1989; Zbl 0695.62028)] to a random linear combination of chi-squared variables. The statistic being studied is King’s small-sample, pointwise, most powerful, and invariant (MPI) test [M. L. King, Ann. Stat. 8, 1265–1271 (1980; Zbl 0441.62049)] of spherically symmetric versus elliptically symmetric distributions in a unit root context. References to the latter are J. D. Sargan and A. Bhargava [Econometrica 51, 153–175 (1983; Zbl 0516.62099)], A. Bhargava [Rev. Econ. Stud. 53, 369–384 (1986; Zbl 0602.62074), Biometrika 83, No. 4, 944–949 (1996; Zbl 0883.62096)], and P. A. Shively [J. Appl. Econom. 16, 537–551 (2001)]. With no need either for iterative, numerical inversion of the characteristic function or asymptotic theory, the added value of our paper is to approximate the MPI statistic by a marginal density obtained from a mixture of conditional Inverted Beta (IB) densities with a weight function given by an inverted chi-square density. Critical values and powers are supplied. We apply the IB approximation to test the unit root hypothesis to simulated data, to the Nelson and Plosser data set, to the extended Nelson and Plosser data set and to test the Purchase Power Parity in the real Euro/Dollar exchange rate.
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62F15 Bayesian inference
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