Parametric Identification of Multiplicative Exponential Heteroscedasticity
| DOI | http://doi.org/10.1111/obes.12280 |
| Published date | 01 June 2019 |
| Date | 01 June 2019 |
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©2018 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 81, 3 (2019) 0305–9049
doi: 10.1111/obes.12280
Parametric Identification of Multiplicative
Exponential Heteroscedasticity*
Alyssa Carlson†
†Department of Economics, Michigan State University, East Lansing, MI 48824-1038, USA
(e-mail: carls405@msu.edu)
Abstract
Harvey (1976) first proposed multiplicative exponential heteroscedasticity in the context
of linear regression. These days it is more commonly seen in latent variable models such
as Probit or Logit where correctly modelling the heteroscedasticity is imperativefor consis-
tent parameter estimates (Yatchew and Griliches, 1985). However, it appears the literature
lacks a formal proof of point identification for the parametric model. This paper presents
several examples that show the conditions presumed throughout the literature are not
sufficient for identification. As a contribution, this paper discusses when identification
can and cannot be easily obtained and provides proofs of point identification in common
specifications.
I. Introduction
Multiplicative exponential heteroscedasticity was first proposed by Harvey (1976) in the
context of a linear conditional mean model. Estimation is undertaken in two stages, re-
quiring first an argument that the conditional variance parameters are identified and then
showing that a weightedleast squares estimator identifies the conditional mean parameters.
More recently, multiplicative exponential functions are used to model heteroscedasticity
in the latent errors of binary response models. However, with the cases of heteroscedastic
Logit and Probit, the parameters in the conditional variance function are estimated con-
currently with the coefficients of interest requiring joint identification of the parameters.
Standard textbooks such as Greene (2011) and Wooldridge (2010) state that these models
are estimable under fairly standard conditions and are more flexible in the specification
of the conditional mean function compared to standard Probit and Logit models leading
to widespread use in empirical work. However, the literature has yet to provide proofs of
parametric identification. To fill the gap in the literature, this paper explores the issues of
identifications in models with exponential heteroscedasticity.
JEL Classification numbers: C10, C25, C51.
*This is a revised section of myPh.D. dissertation. I thank Professor Kyoo Il Kim and Professor JeffreyWooldridge
for their thoughtful comments.
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