Information Equivalence among Transformations of Semi‐parametric Nonlinear Panel Data Models*
| Published date | 01 December 2023 |
| Author | Nicholas Brown |
| Date | 01 December 2023 |
| DOI | http://doi.org/10.1111/obes.12569 |
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 85, 6 (2023) 0305-9049
doi: 10.1111/obes.12569
Information Equivalence among Transformations of
Semi-parametric Nonlinear Panel Data Models*
NICHOLAS BROWN
Department of Economics, Queen’s University, 94 University Ave, 209 Dunning Hall Kingston,
K7L 3N6, Ontario, Canada(e-mail: n.brown@queensu.ca)
Abstract
This paper considers transformations of nonlinear semi-parametric mean functions that
yield moment conditions for estimation. Such transformations are said to be information
equivalent if they yield the same asymptotic efficiency bound. I derive a unified theory of
algebraic equivalence for moment conditions created by a given linear transformation. The
main equivalence result states that under standard regularity conditions, transformations
that create conditional moment restrictions in a given empirical setting need only to
have an equal rank to reach the same efficiency bound. I compare feasible and infeasible
transformations of both nonlinear models with multiplicative heterogeneity and linear
models with unobserved factor structures.
I. Introduction
In the standard linear panel data model with additive unobserved heterogeneity, it is
well known that numerous transformations can be used to eliminate the heterogeneity
prior to estimation. The most common methods are the within and first-differencing
transformations. Similarly, when the heterogeneity appears as a multiplicative term in
the conditional mean like in certain Generalized Linear Model settings, modified within
and differencing transformations can control for the heterogeneity and provide moment
conditions for estimation. There exist other transformations that control for heterogeneity
but are clearly absurd. For example, multiplying all the data by zero eliminates the
heterogeneity along with all information for estimation. For a less trivial example, suppose
the population model is linear with a single additive effect and homoskedastic errors.
Then second-differencing is still consistent but less efficient than first-differencing. These
examples raise the question of how to evaluate methods for eliminating heterogeneity
while preserving information for estimation.
This paper considers conditional mean models with unobserved heterogeneity. The
general framework encompasses a large class of both linear and strictly nonlinear models.
JEL Classification numbers: C14, C33, C36.
*I would like to thank Jeffrey Wooldridge and Peter Schmidt for their guidance and advice. I would also like to
thank the participants of the AEASP seminar series, 2020 Red Cedar Conference, MSU Econometrics Reading
Group, and 2021 MEA conference participants for their insightful questions and comments. All errors are my
own.
1341
©2023 The Authors. Oxford Bulletin of Economics and Statistics published by Oxford University and John Wiley & Sons Ltd.
This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and
distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.
1342 Bulletin
The models are referred to as ‘‘semiparametric’’ in the sense that nothing is assumed
about the relationship between the heterogeneity and observables other than regularity
conditions needed for asymptotic analysis. In place of assumptions on the conditional
distribution of the heterogeneity, these models often require a transformation to control
for the unobservables.
I provide a unified framework for comparing such transformations in terms of the
information they preserve. I show that transformations yielding conditional moment
restrictions, given certain regularity assumptions, will provide the same √N-asymptotic
efficiency bound if they have equal rank. This result is useful because once the researcher
has multiple transformations that satisfy the conditions in my theory, they can choose
one based solely on concerns of feasible inference, computational demand, and finite-
sample bias reduction. I also demonstrate that for the examples in this paper, infeasible
transformations that are functions of unobservables are often available to eliminate
heterogeneity. An additional implication of my results is that once a researcher has a
feasible transformation that is the same rank as the infeasible one, they do not need
to search for additional transformations in order to reach the efficiency bound of the
infeasible moment restrictions.
As mentioned above, the within and first-differencing transformations are commonly
used to eliminate additive heterogeneity. When the covariates are strictly exogenous,
these transformations provide conditional moment restrictions that can be exploited for
estimation of the population parameters. For a given conditional variance matrix, Arellano
and Bover (1995) suggest that Generalized Least Squares (GLS) on the demeaned
equations is equivalent to the efficient Three-Stage Least Squares (3SLS) estimator. This
claim was later proven in Im et al. (1999), along with a proof that the GLS estimators on
the demeaned and first-differenced data are equivalent.
These results shows that two commonly used methods of estimation preserve the
same information in the linear case. However, they limit their investigation to a small
number of estimators and only allow for a single additive effect. My approach nests
the results of Im et al. (1999), but also applies to more general interactive fixed effects
models. Because some of the estimators for these models rely on nonlinear first-step
estimation, it is beneficial to show that two transformations have the same information
bound, so the empirical researcher can choose the one that is easier to compute and has
better finite-sample properties.
One approach to estimation of nonlinear models with a multiplicative heterogeneity
term is the fixed effects Poisson (FEP) estimator. Hausman, Hall, and Griliches (1984)
derive the FEP as the conditional maximum likelihood estimator of a multinomial
distribution1. Wooldridge (1999) shows that the FEP is in fact consistent under a much
weaker strict exogeneity assumption due the likelihood’s implicit transformation of the
data. Another approach is the generalized next-differencing transformation first studied
by Chamberlain (1992) and Wooldridge (1997), which subtracts from a time period the
next period outcome, weighted by the quotient of the mean functions. While generalized
next-differencing was originally proposed for a sequential exogeneity setting, I study it
1Similar to the linear fixed effects estimator, the FEP estimator is a true fixed effects procedure as it can be derived
by estimating via pooled Poisson regression and treating the multiplicative terms as parameters to estimate.
©2023 The Authors. Oxford Bulletin of Economics and Statistics published by Oxford University and John Wiley & Sons Ltd.
Get this document and AI-powered insights with a free trial of vLex and Vincent AI
Get Started for FreeStart Your Free Trial of vLex and Vincent AI, Your Precision-Engineered Legal Assistant
-
Access comprehensive legal content with no limitations across vLex's unparalleled global legal database
-
Build stronger arguments with verified citations and CERT citator that tracks case history and precedential strength
-
Transform your legal research from hours to minutes with Vincent AI's intelligent search and analysis capabilities
-
Elevate your practice by focusing your expertise where it matters most while Vincent handles the heavy lifting
Start Your Free Trial of vLex and Vincent AI, Your Precision-Engineered Legal Assistant
-
Access comprehensive legal content with no limitations across vLex's unparalleled global legal database
-
Build stronger arguments with verified citations and CERT citator that tracks case history and precedential strength
-
Transform your legal research from hours to minutes with Vincent AI's intelligent search and analysis capabilities
-
Elevate your practice by focusing your expertise where it matters most while Vincent handles the heavy lifting
Start Your Free Trial of vLex and Vincent AI, Your Precision-Engineered Legal Assistant
-
Access comprehensive legal content with no limitations across vLex's unparalleled global legal database
-
Build stronger arguments with verified citations and CERT citator that tracks case history and precedential strength
-
Transform your legal research from hours to minutes with Vincent AI's intelligent search and analysis capabilities
-
Elevate your practice by focusing your expertise where it matters most while Vincent handles the heavy lifting
Start Your Free Trial of vLex and Vincent AI, Your Precision-Engineered Legal Assistant
-
Access comprehensive legal content with no limitations across vLex's unparalleled global legal database
-
Build stronger arguments with verified citations and CERT citator that tracks case history and precedential strength
-
Transform your legal research from hours to minutes with Vincent AI's intelligent search and analysis capabilities
-
Elevate your practice by focusing your expertise where it matters most while Vincent handles the heavy lifting
Start Your Free Trial of vLex and Vincent AI, Your Precision-Engineered Legal Assistant
-
Access comprehensive legal content with no limitations across vLex's unparalleled global legal database
-
Build stronger arguments with verified citations and CERT citator that tracks case history and precedential strength
-
Transform your legal research from hours to minutes with Vincent AI's intelligent search and analysis capabilities
-
Elevate your practice by focusing your expertise where it matters most while Vincent handles the heavy lifting