On Distributions of Ratios

Tinbergen Institute Discussion Paper 13-211/III

Broda, Simon A. and Kan, Raymond (2016). On distributions of ratios, Biometrika, Volume 103, Issue 1, pp. 205-218. https://doi.org/10.1093/biomet/asv052

44 Pages Posted: 8 Jan 2014 Last revised: 11 Jun 2020

See all articles by Simon A. Broda

Simon A. Broda

University of Zurich - Department of Banking and Finance

Raymond Kan

University of Toronto - Rotman School of Management

Date Written: January 7, 2014

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 estimator of a structural parameter in a just identified equation.

Keywords: Characteristic Function, Inversion Formula, Saddlepoint Approximation, Simultaneous Equations, Instrumental Variables, Weak Instruments, Bootstrap

JEL Classification: C15, C26, C46, C63

Suggested Citation

Broda, Simon A. and Kan, Raymond, On Distributions of Ratios (January 7, 2014). Tinbergen Institute Discussion Paper 13-211/III, Broda, Simon A. and Kan, Raymond (2016). On distributions of ratios, Biometrika, Volume 103, Issue 1, pp. 205-218. https://doi.org/10.1093/biomet/asv052, Available at SSRN: https://ssrn.com/abstract=2375638 or http://dx.doi.org/10.2139/ssrn.2375638

Simon A. Broda (Contact Author)

University of Zurich - Department of Banking and Finance ( email )

Plattenstr 32
Zurich, 8032
Switzerland

Raymond Kan

University of Toronto - Rotman School of Management ( email )

105 St. George Street
Toronto, Ontario M5S3E6
Canada
416-978-4291 (Phone)
416-971-3048 (Fax)

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