Semi‐parametric Regression under Model Uncertainty: Economic Applications
| Date | 01 October 2019 |
| DOI | http://doi.org/10.1111/obes.12294 |
| Published date | 01 October 2019 |
| Author | Bettina Grün,Gertraud Malsiner‐Walli,Paul Hofmarcher |
1117
©2019 TheAuthors. OxfordBulletin of Economics and Statistics published by Oxford University and John Wiley & Sons Ltd.
Thisis an open access article under the ter ms of the CreativeCommons Attribution License, which permits use, distribution and reproduction in any medium, provided
the original work is properlycited.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 81, 5 (2019) 0305–9049
doi: 10.1111/obes.12294
Semi-parametric Regression under Model
Uncertainty: Economic Applications*
Gertraud Malsiner-Walli,†Paul Hofmarcher,‡ and
Bettina Gr ¨
un§
†Department of Finance, Accounting and Statistics, Vienna University of Economics and
Business (WU), Welthandelsplatz 1, 1020, Vienna, Austria (e-mail: gertraud.malsiner-
walli@wu.ac.at)
‡Salzburg Centre of European Union Studies (SCEUS), Department of Business, Economics
and Social Theory, Paris Lodron University of Salzburg, M¨onchsberg 2a, 5020, Salzburg,
Austria (e-mail: paul.hofmarcher@sbg.ac.at)
§Department of Applied Statistics, Johannes Kepler University Linz, Altenbergerstraße 69,
4040, Linz, Austria (e-mail: bettina.gruen@jku.at)
Abstract
Economic theory does not always specify the functional relationship between depen-
dent and explanatory variables, or even isolate a particular set of covariates. This means
that model uncertainty is pervasive in empirical economics. In this paper, we indicate
how Bayesian semi-parametric regression methods in combination with stochastic search
variable selection can be used to address two model uncertainties simultaneously: (i) the
uncertainty with respect to the variables which should be included in the model and (ii) the
uncertainty with respect to the functional form of their effects. The presented approach en-
ables the simultaneous identification of robust linear and nonlinear effects. The additional
insights gained are illustrated on applications in empirical economics, namely willingness
to pay for housing, and cross-country growth regression.
I. Introduction
Model uncertainty is pervasive in empirical economics. Economic theory does not always
specify or is inconclusive with respect to (i) the covariates which are part of the model,
and (ii) the functional form between dependent and explanatory variables. Bayesianmodel
averaging (henceforth BMA) has become a popular tool to perform robust inference under
model uncertainty (Hofmarcher et al., 2018). The model uncertainty usually accounted
for in the BMA framework only relates to variable uncertainty while assuming that the
relationship between dependent and explanatory variables is linear. For instance, in the
field of long-term per capita GDP growth empirics a huge strand of literature applies BMA
JEL Classification numbers: C14, O10, O40.
*This research was supported by the Oesterreichische Nationalbank (Anniversary Fund, project number: 14663)
and the Austrian Science Fund (FWF, P28740).
1118 Bulletin
methods to identify robust linear determinants for growth.Variable uncertainty arises in this
context due to the so-called Open-endedness of theory (see Brock and Durlauf, 2001) where
competing growth theories do not rule out each other. BMA takes this variableuncer tainty
into account (see, e.g. Fern´andez, Ley and Steel, 2001; Sala-i-Martin, Doppelhofer and
Miller, 2004; Amini and Parmeter, 2012; Ley and Steel, 2012). BMA techniques are also
employed in Mitchell, Pain and Riley (2011) to identify economic and policy drivers of
international migration to the UK, in Eicher, Henn and Papageorgiou (2012) to assess
effects of preferential trade agreements on trade flows, and in Tobias and Li (2004) to
determine returns to education.
Although the insights from these BMA exercises are valuable, theyrely on the assump-
tion that the functional relationship between the potential covariates and the dependent
variable is known. In standard BMA all covariates included in the model are assumed to
havea linear effect on the outcome. A nonlinear functional relationship is modelled by man-
ually creating and adding new covariates, e.g. by including polynomial terms (Henderson
and Parmeter, 2016) or defining a piecewise linear function (Tobias and Li, 2004). Special
care is needed if these additional terms are to be included in the model in a hierarchical or
grouped way. For instance, if interaction effects are only to be included jointly with their
main effects, the constraint has to be imposed that main effects need to be included first
before interaction effects can be included (Crespo Cuaresma, 2010; Montgomery and Ny-
han, 2010). The inclusion of grouped terms representing a nonlinear effect requires that all
terms needed to model nonlinearity are either jointly added or dropped from the model. Due
to this manual tuning required for the inclusion of any nonlinear effect, BMA applications
in general only include a small subset of potential covariates with nonlinear effects.
However, functional misspecification occurs if covariates are only included with linear
effects while in fact their effects are actually nonlinear. In this case, several issues may
arise which preclude the correct robust identification of effects: (i) irrelevant covariates
may be included into the model to compensate for the missing nonlinear effects and are
overestimated in terms of importance, (ii) important nonlinear covariates maynot be iden-
tified due to the functional misspecification, and (iii) effects of covariates, which vary over
the covariate range, are forced to be constant over the entire covariate range. Inference
based on an improperly specified model thus may lead to incorrect conclusions and policy
prescriptions.
Therefore, efforts have been made in empirical economics to develop and apply mod-
elling techniques which uncover also nonlinear effects without the need to know and a
priori explicitly define these nonlinear effects. Non-parametric as well as semi-parametric
methods have been suggested. Non-parametric methods used in economic regression ap-
plications are mainly based on kernel estimation (for an overview see Henderson and
Parmeter, 2015). For instance, local-linear least square regressions are used in Hender-
son, Papageorgiou and Parmeter (2011) to investigate nonlinearities of differenteconomic
growth determinants and in Delgado, Henderson and Parmeter (2014) to study the effect of
education on economic growth. Semi-parametric methods using model matrix expansions
to obtain more flexible regression functions are discussed in Smith and Kohn (1996) to
model housing values, and in Koop and Tobias (2006) to study returns to education.
Compared to BMA, where dozens of potential determinants are simultaneously in-
cluded in the analysis to identify the relevant ones, the non- and semi-parametric methods
©2019 The Authors. Oxford Bulletin of Economics and Statistics published by Oxford University and JohnWiley & Sons Ltd.
To continue reading
Request your trial