Concept‐Based Bayesian Model Averaging and Growth Empirics
| Published date | 01 December 2014 |
| Author | Wendun Wang,Jan R. Magnus |
| Date | 01 December 2014 |
| DOI | http://doi.org/10.1111/obes.12068 |
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©2014 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 76, 6 (2014) 0305–9049
doi: 10.1111/obes.12068
Concept-Based Bayesian Model Averaging and
Growth Empirics*
Jan R. Magnus† and Wendun Wang‡
†Department of Econometrics and Operations Research, VU University Amsterdam,
The Netherlands (e-mail: jan@janmagnus.nl)
‡Econometric Institute, Erasmus University Rotterdam, The Netherlands (e-mail:
wangwendun@gmail.com)
Abstract
In specifying a regression equation, we need to specify which regressors to include, but also
how these regressors are measured. This gives rise to two levels of uncertainty: concepts
(level 1) and measurements within each concept (level2). In this paper we propose a hierar-
chical weighted least squares (HWALS) method to address these uncertainties. We examine
the effects of different growth determinants taking explicit account of the measurement
problem in the growth regressions. We find that estimates produced by HWALS provide
intuitive and robust explanations. We also consider approximation techniques which are
useful when the number of variables is large or when computing time is limited.
I. Introduction
In applied econometrics, when estimating a regression equation, one has to decide which
concepts (say inflation) to include in the regression: the ‘specification’problem. In addition,
one has to decide which measurementsof these concepts to use (e.g. CPI-based or PPI-based
inflation): the ‘measurement’ problem. The measurement problem is common in practice
because most economic variables can be measured in various ways. Climate, for example,
as a potential determinant of growth, can be measured by the fraction of a country lying in
the tropics, the area of a country lying in the tropics, or absolute latitude.Another example is
the concept of market concentration, typically thought of as a factor that affectsthe financial
stability of individual firms, which can be measured by the Herfindahl-Hirschman index
but also by the market share of, say, the four largest firms.
The measurement problem raises at least three issues. First, different choices of mea-
surements produce different estimates for the same concept, leading to ambiguity in
explanation and policy implications. Second,multiple measurements typically cause multi-
collinearity if they are included in one regression model, so that the estimates for individual
*We are grateful to participants at seminars at Tilburg University, Erasmus University Rotterdam, Groningen
University, and the 23rd (EC)2conference in Maastricht; to Gerda Claeskens, John Einmahl, and Arthur van Soest;
and to the editor and two referees for constructive and helpful comments.
JEL Classification numbers: C51, C52, C13, C11.
Concept-based Bayesian model averaging 875
measurements lack precision, and statistical inference on a concept based on these esti-
mates can, therefore, be misleading. Third, including multiple measurements in one model
can also cause a problem of dimensionality when the number of explanatory variables is
close to or even exceeds the number of observations.
The current paper addresses the measurement problem by introducing hierarchical (two-
level) model averaging, where we perform model averaging over concepts and measure-
ments. From here on we shall denote concepts as groups, and measurements as variables.
We propose a method called hierarchical weighted least squares (HWALS), a generaliza-
tion of weighted-average least squares (WALS) developed in Magnus, Powell and Pr¨ufer
(2010). In hierarchical model averaging we introduce prior probabilities for the variables
in each group, and treat the regression parameters as hierarchical random variables. We are
uncertain about the error term, about which groups to select, and about which variables
to select. All three levels of uncertainty are explicitly taken into account in hierarchical
WALS estimation.
The HWALS procedure has several advantages. It provides an estimate and standard
deviation for each group, which facilitates statistical inference and enables us to analyze
the effect of each group; it combines model selection and estimation and thus avoids the
problems associated with pretesting (see Danilov and Magnus (2004) for a discussion and
review of these problems); it allowsresearchers to assign various types of priors depending
on the strength of their information and beliefs; it limits the extent of multicollinearity and
dimensionality problems because it only considers models with one variablein each g roup;
and its computational burden is very light, especially compared to standard Bayesianmodel
averaging (BMA) and Bayesian averaging of classical estimates (BACE).
In the empirical growth literature the three types of uncertainty are especially important,
because there is little consensus in this literature on which regressors to include, and, even
if there is an agreement on a regressor (group), there is still a disagreement on which
measurement (variable) of that regressor to use. In addition, the number of variables in
growth empirics is large and may even exceed the number of observations. For example,
Durlauf, Johnson and Temple (2005) listed 145 candidate variables, while the number of
countries is typically less in cross-country growth studies. Our paper employs HWALS to
re-investigate the effects of various growth determinants.
We mainly compare our estimates with those of Sala-i-Martin, Doppelhofer and Miller
(2004) and with the WALS estimates of Magnus et al. (2010). Our hierarchical model
averaging estimates produce more intuitive signs and they are more robust. This is the
benefit we gain from not ignoring the measurement problem, so that correlated variables
within one group are not at all included in the regression. Our empirical results also
provide several new insights. For example, we find – in contrast to the current literature
– that education and relative government size (government’s economic activities) are not
robust, because some of the variables in these groups have poor explanatory power in the
growth regressions.
The paper is organized as follows.A literature review is provided in section II. In section
III we present the hierarchical estimation strategy. Section IV describes the data, grouping
and scaling. We apply our estimation strategyto the data in section V and discuss the results.
Next we address the potential problem that the number of variables is too large to apply
the HWALS technique directly. In that case, approximations are required and these are
©2014 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd
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