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On Distributions of Ratios

  • Simon A. Broda
  • Raymond Kan

    (University of Toronto)

A large number of exact inferential procedures in statistics and econometrics involve the sampling distribution of ratios of random variables. If the denominator variable is positive, then tail probabilities of the ratio can be expressed as those of a suitably defined difference of random variables. If in addition, the joint characteristic function of numerator and denominator is known, then standard Fourier inversion techniques can be used to reconstruct the distribution function from it. Most research in this field has been based on this correspondence, but which breaks down when both numerator and denominator are supported on the entire real line. The present manuscript derives inversion formulae and saddlepoint approximations that remain valid in this case, and reduce to known results when the denominator is almost surely positive. Applications include the IV estimator of a structural parameter in a just identified equation.

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Paper provided by Universiteit van Amsterdam, Dept. of Econometrics in its series UvA-Econometrics Working Papers with number 13-10.

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Date of creation: 22 Dec 2013
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Handle: RePEc:ame:wpaper:1310
Contact details of provider: Postal: Dept. of Econometrics, Universiteit van Amsterdam, Valckenierstraat 65, NL - 1018 XE Amsterdam, The Netherlands
Web page: http://www.ase.uva.nl/uva-econometrics
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  1. Soujin Wang, 1990. "Saddlepoint approximations in resampling analysis," Annals of the Institute of Statistical Mathematics, Springer, vol. 42(1), pages 115-131, March.
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  10. Holly, A & Phillips, P C B, 1979. "A Saddlepoint Approximation to the Distribution of the k-Class Estimator of a Coefficient in a Simultaneous System," Econometrica, Econometric Society, vol. 47(6), pages 1527-47, November.
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  14. Simon A. Broda, 2013. "Tail probabilities and partial moments for quadratic forms in multivariate generalized hyperbolic random vectors," UvA-Econometrics Working Papers 13-04, Universiteit van Amsterdam, Dept. of Econometrics.
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