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Bimodal t-Ratios



This paper studies the sampling distribution of the conventional t-ratio when the sample comprises independent draws from a standard Cauchy (0,1) population. It is shown that this distribution displays a striking bimodality for all sample sizes and that the bimodality persists asymptotically. An asymptotic theory is developed in terms of bivariate stable variates and the bimodality is explained by the statistical dependence between the numerator and denominator statistics of the t-ratio. This dependence also persists asymptotically. These results are in contrast to the classical t statistic constructed from a normal population, for which the numerator and denominator statistics are independent and the denominator, when suitably scaled, is a constant asymptotically. Our results are also in contrast to those that are known to apply for multivariate spherical populations. In particular, data from an n dimensional Cauchy population are well known to lead to a t-ratio statistic whose distribution is classical t with n-1 degrees of freedom. In this case the univariate marginals of the population are all standard Cauchy (0,1) but the sample data involves a special form of dependence associated with the multivariate spherical assumption. Our results therefore serve to highlight the effects of the dependence in component variates that is induced by a multivariate spherical population. Some extensions to symmetric stable populations with exponent parameter alpha does not equal 1 are also indicated. Simulation results suggest that the sampling distributions are well approximated by the asymptotic theory even for samples as small as n = 20.

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  • Peter C.B. Phillips & Vassilis A. Hajivassiliou, 1987. "Bimodal t-Ratios," Cowles Foundation Discussion Papers 842, Cowles Foundation for Research in Economics, Yale University.
  • Handle: RePEc:cwl:cwldpp:842
    Note: CFP 1385.

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    References listed on IDEAS

    1. Phillips, P.C.B., 1983. "Exact small sample theory in the simultaneous equations model," Handbook of Econometrics,in: Z. Griliches† & M. D. Intriligator (ed.), Handbook of Econometrics, edition 1, volume 1, chapter 8, pages 449-516 Elsevier.
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    Cited by:

    1. Phillips, Peter C. B., 1988. "Conditional and unconditional statistical independence," Journal of Econometrics, Elsevier, vol. 38(3), pages 341-348, July.
    2. Peter C.B. Phillips & Mico Loretan, 1990. "Testing Covariance Stationarity Under Moment Condition Failure with an Application to Common Stock Returns," Cowles Foundation Discussion Papers 947, Cowles Foundation for Research in Economics, Yale University.
    3. Phillips, Peter C. B. & Loretan, Mico, 1991. "The Durbin-Watson ratio under infinite-variance errors," Journal of Econometrics, Elsevier, vol. 47(1), pages 85-114, January.
    4. Phillips, P.C.B., 1990. "Time Series Regression With a Unit Root and Infinite-Variance Errors," Econometric Theory, Cambridge University Press, vol. 6(01), pages 44-62, March.

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