On Distributions of Ratios

Simon Broda and Raymond Kan
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Raymond Kan: Joseph L. Rotman School of Management, University of Toronto, Canada

No 13-211/III, Tinbergen Institute Discussion Papers from Tinbergen Institute

Abstract: 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 est imator of a structural parameter in a just identified equation.

Keywords: Characteristic Function; Inversion Formula; Saddlepoint Approximation; Simultaneous Equations; Instrumental Variables; Weak Instruments; Bootstrap (search for similar items in EconPapers)
JEL-codes: C15 C26 C46 C63 (search for similar items in EconPapers)
Date: 2014-01-06
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https://papers.tinbergen.nl/13211.pdf (application/pdf)

Related works:
Journal Article: On distributions of ratios (2016)
Working Paper: On Distributions of Ratios (2013)
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Persistent link: https://EconPapers.repec.org/RePEc:tin:wpaper:20130211

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