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On the (im-)possibility of representing probability distributions as a difference of i.i.d. noise terms


Ewerhart, Christian; Serena, Marco (2023). On the (im-)possibility of representing probability distributions as a difference of i.i.d. noise terms. Working paper series / Department of Economics 428, University of Zurich.

Abstract

A random variable is difference-form decomposable (DFD) if it may be written as the difference of two i.i.d. random terms. We show that densities of such variables exhibit a remarkable degree of structure. Specifcally, a DFD density can be neither approximately uniform, nor quasiconvex, nor strictly concave. On the other hand, a DFD density need, in general, be neither unimodal nor logconcave. Regarding smoothness, we show that a compactly supported DFD density cannot be analytic and will often exhibit a kink even if its components are smooth. The analysis highlights the risks for model consistency resulting from the strategy widely adopted in the economics literature of imposing assumptions directly on a dfference of noise terms rather than on its components.

Abstract

A random variable is difference-form decomposable (DFD) if it may be written as the difference of two i.i.d. random terms. We show that densities of such variables exhibit a remarkable degree of structure. Specifcally, a DFD density can be neither approximately uniform, nor quasiconvex, nor strictly concave. On the other hand, a DFD density need, in general, be neither unimodal nor logconcave. Regarding smoothness, we show that a compactly supported DFD density cannot be analytic and will often exhibit a kink even if its components are smooth. The analysis highlights the risks for model consistency resulting from the strategy widely adopted in the economics literature of imposing assumptions directly on a dfference of noise terms rather than on its components.

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Additional indexing

Item Type:Working Paper
Communities & Collections:03 Faculty of Economics > Department of Economics
Working Paper Series > Department of Economics
Dewey Decimal Classification:330 Economics
JEL Classification:C46, C6
Uncontrolled Keywords:Differences of random variables, density functions, characteristic function, uniform distribution
Scope:Discipline-based scholarship (basic research)
Language:English
Date:October 2023
Deposited On:21 Feb 2023 09:39
Last Modified:06 Mar 2024 14:39
Series Name:Working paper series / Department of Economics
Number of Pages:29
ISSN:1664-7041
Additional Information:Revised version
OA Status:Green
Free access at:Official URL. An embargo period may apply.
Other Identification Number:merlin-id:23393
  • Content: Updated Version
  • Language: English
  • Description: Revised version October 2023