Modelling Category Inflation with Multiple Inflation Processes: Estimation, Specification and Testing1
| Published date | 01 December 2021 |
| Author | Sarah Brown,Christopher Spencer,Mark N. Harris |
| DOI | http://doi.org/10.1111/obes.12366 |
| Date | 01 December 2021 |
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©2020 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 82, 6 (2020) 0305–9049
doi: 10.1111/obes.12366
Modelling Category Inflation with Multiple Inflation
Processes: Estimation, Specification and Testing*
Sarah Brown,†Mark N. Harris‡ and Christopher Spencer§
†Department of Economics, University of Sheffield, 9 Mappin Street, Sheffield, S1 4DT,
UK. (e-mail: sarah.brown@sheffield.ac.uk)
‡School of Economics, Finance and Property, Faculty of Business and Law,
Curtin University, Perth, WA 6845, Australia. (e-mail: mark.harris@curtin.edu.au)
§School of Business and Economics, Loughborough University, Ashby Road, Loughborough,
Leicestershire, LE11 3TU, UK. (e-mail: c.g.spencer@lboro.ac.uk)
Abstract
Zero-inflated ordered probit (ZIOP) and middle-inflated ordered probit (MIOP) models
are finding increasing favour in the discrete choice literature. We propose generalizations
to these models – which collapse to their ZIOP/MIOP counterparts under a set of simple
parameter restrictions – with respect to the inflation process. These generalizations form
the basis of a new specification test of the inflation process in ZIOP and MIOP models.
Support for our generalization framework is principally demonstrated by revisiting a key
ZIOP application from the economics literature, and reinforced by the reassessment of
an important MIOP application from political science. Our specification test supports the
generalized models over the original ZIOP/MIOP ones, suggesting an important role for
it in modelling zero- and middle-inflation processes.
I. Introduction
Recent advances in discrete choice modelling have witnessed the development of inflated
ordered probit models. These models draw inspiration from the suite of hurdle and double-
hurdle models for continuous and count outcome variables developed to address an ex-
cess of zero observations (Cragg, 1971; Mullahey, 1986; Lambert, 1992; Heilbron, 1994;
Mullahey, 1997). Their use, which hinges on an assumption that the data are generated
by two distinct data generation processes, is typically motivated by the fact that in some
ordered choice situations, a large proportion of empirical observations fall into a single
choice category which appears ‘inflated’ relative to the others. Underpinning the impor-
JEL Classification numbers: C12, C35.
*The authors thank Benjamin Bagozzi and Bumba Mukherjee for making their data available on the Harvard
Dataverse, which can be found at https://dataverse.harvard.edu/dataverse/pan (accessed 25 October 2016). Brown
and Harris thank the Australian Research Council for financial support. Wealso thank the editor and three referees
for their comments, which have greatly improvedthe paper.
Modelling category inflation 1343
tance of accounting for the presence of suspected category inflation is the fact that failing
to do so can lead to model mis-specification, biased estimates and incorrect inference.
Inflated ordered probit models have been applied in fields such as economics, political
science and medical statistics, and can be divided into two main variants. The first is the
zero-inflated ordered probit (ZIOP ) model, in which an excess of observations is observed
at one end of the choice spectrum (Harris and Zhao, 2007; Meyerhoefer and Zuvekas,
2010; Downward, Lera-Lopez and Rasciute, 2011; Gurmu and Dagne, 2012; Habib et al.,
2012; Jiang et al., 2013; Peng, Meyerhoefer and Zuvekas, 2013; Akcura, 2015; Bagozzi
et al., 2015; and Falk and Katz-Gerro, 2016). The second is the more recently developed
middle-inflated orderedprobit (MIOP) model, which is characterized by a middle outcome
being inflated (Bagozzi and Mukherjee, 2012; Brooks, Harris and Spencer, 2012; Bagozzi
et al., 2014; Miwa, 2015; and Zirogiannis et al., 2015).
This paper adds to this growing literature in several important ways. We propose gen-
eralizations of inflated ordered probit models that preserve the ordering of outcomes while
still explicitly accounting for the maintained inflation process. Instead of having a single
‘splitting equation’ in a setting with Jcategorical outcomes (see Harris and Zhao, 2007),
our generalizations require J−1 of these latent equations to be estimated. These equations
capture the propensity to be pushed away from the model’s non-inflated outcomes towards
the inflated one. We refer to these models as the generalized zero-inflated ordered pro-
bit (GZIOP) and the generalized middle-inflated ordered probit (GMIOP ). These models
collapse to their associated ZIOP and MIOP counterparts when the parameter vectors of
the J−1 splitting equations are restricted to be equal. As these generalized models nest
their ZIOP/MIOP counterparts, it is possible to use standard testing paradigms to test if
the nested model specifications are too restrictive.
This aspect of our contribution is significant, as insufficient attention is devoted to this
issue in the literature.1We derive the appropriate Lagrange multiplier (LM ) tests, which
can be used without having to estimate the more general models (c.f., the likelihood ratio
(LR) test, for example). To explore the performance of our proposed generalization and
testing framework, we consider the original data and model specifications from Harris
and Zhao (2007), who model tobacco consumption at the individual level. LM tests based
on the data in their ZIOP application appear correctly sized in Monte Carlo experiments
and have good power properties, and typically exhibit good quasi-power in identifying
mis-specified models. This suggests that our LM tests are good general specification tests.
When the generalized model is estimated using the original data and specification in Harris
and Zhao (2007), our specification test favours the generalized model.
To complement the above analysis, we also explored the performance of our general-
ization framework in a middle-inflation setting. The same methodological approach was
applied to a dataset from Bagozzi and Mukherjee (2012), who use a MIOP framework to
model ‘face-saving’middle-category responses in a commonly studied Eurobarometer sur-
vey question (European Commission, 2002a,b). The associated LM tests were also found
to have desirable statistical properties, and favoured the generalized model when it was
estimated on the original data and specification in their paper. Taken together, our findings
1Our testing framework focuses on instances whereone inflated model nests another. In relation to the problem of
zero-inflation in the Poisson counts literature, Wilson(2015) argues that the widespread practice of using the Vuong
test as a test of zero-inflation in a non-nested setting is erroneous.
©2020 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd
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