Hybrid discrete choice models: gained insights versus increasing effort Petr Mariel ∗ Department of Applied Economics III (Econometrics and Statistics), University of the Basque Country Avda. Lehendakari Aguirre, 83 E48015 Bilbao, Spain E-mail: [email protected]Tel: +34.94.601.3848 Fax: +34.94.601.3754 Jürgen Meyerhoff a,b a) Institute for Landscape Architecture and Environmental Planning Technical University of Berlin D-10623 Berlin, Germany E-mail: [email protected]b) The Kiel Institute for the World Economy, Duesternbrooker Weg 120, 24105 Kiel, Germany ∗ Corresponding author 1
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Hybrid discrete choice models: gained insights versus
increasing effort
Petr Mariel∗
Department of Applied Economics III (Econometrics and Statistics), University of the Basque Country
that the latent variables have random coefficients with multivariate distributions with
unknown parameters. The model incorporates factors that influence the random coefficients
and can influence each other through links in the structural equations.
We focus in the following on related but different models called HCMs in which the
latent variables represent the characteristics of individuals, typically constructs like attitudes
(Ben-Akiva et al., 2002). These latent variables are treated as endogenous and related to socio-
demographic characteristics in structural equations, but, at the same time, they are
explanatory in measurement equations relating them to observed indicators. This type of
model has been increasingly used in all fields in which discrete choice models are applied.
Nevertheless, criticism of them has increased at the same pace.
The most frequently used latent variable models thus far have been applied and
supported in transportation by, among others, Abou-Zeid et al. (2010), Walker et al. (2010),
Daly et al. (2012), Prato et al. (2012), Glerum et al. (2014), Kamargianni and Polydoropoulou
(2014), Kim et al. (2014) and Paulssen et al. (2014). In a recent paper, also in transportation,
Hess et al. (2013) find better performance of HCMs in terms of efficiency, represented by
lower standard errors, and argue that this approach presents a theoretical advantage in terms
of endogeneity bias and measurement error, but its practical implications seem limited.
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Chorus and Kroesen (2014) go even further in their criticism. They state that HCMs do not
support the derivation of travel demand policies that aim to change travel behaviour through
changes in a latent variable, because of the non-trivial endogeneity of the latent variable
regarding travel choice and the cross-sectional nature of the latent variable which does not
allow for claims concerning changes in the variable at the individual level. The first argument is
probably highly case specific as the endogeneity of the latent variable can be an empirically
non-relevant issue. The second argument definitely needs future research, as it is not obvious
how strongly the cross-sectional nature of the attitudinal information affects the performance
of the HCMs.
Recently, Dekker et al. (2014) investigated to what extent choices for leisure activities
and related travels are driven by the satisfaction of needs of a particular leisure activity. They
include in their choice model latent variables representing the anticipated level of individual
needs-satisfaction by a particular leisure activity. Using a stated choice-dataset involving
choices between leisure activities, they contrast regret-minimisation based discrete choice
models including and excluding the subjective measurements of need-satisfaction. Their
empirical results show that, not unexpected, a big portion of the unobserved heterogeneity
(around 40%) in the activity specific utility levels can be attributed to anticipated needs
satisfaction.
In environmental valuation, the HCM has been applied, among others, by Hess and
Beharry-Borg (2012), Bartczak et al. (2015), Hoyos et al. (2015), Mariel et al. (2015), and
Lundhede et al. (2015). In general, they all support the finding that HCMs provide greater
insights into attitudes as additional drivers of choices. Both Lundhede et al (2015) and Bartczak
et al (2015) found, for example, a significant influence of age on the latent variable and
subsequently on WTP estimates. In some case also gains in efficiency were achieved.
Nevertheless, Dekker et al. (2013), who additionally asked follow-up questions to record
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respondents’ response certainty, note, in a rather critical way, that this additional information
does not significantly improve the explanation of the observed choices.
Kløjgaard and Hess (2014), applying the HCM approach in order to investigate data
from a health survey, also express scepticism about latent variable models. They found that
only a small share of the overall heterogeneity was linked to the latent variable. According to
their interpretation, an explanation for the weak link could be the fact that preference
heterogeneity is unrelated to attitudes and perceptions, or, more precisely, that the specific
attitudinal statements measured in the survey are not directly linked to preference
heterogeneity.
Some of the issues related to the use of latent variables in HCMs might be avoided by
learning from the SEM literature. Cliff (1983), for example, gives some warnings and advice to
structural modellers, reminding them of four principles of elementary scientific inference that
are perfectly applicable to discrete choice models with latent variables. The first principle is
that data do not confirm a model; they only fail to refute it. That is, an estimated model cannot
tell us about what is not in it. Generally, it is thus recommended to estimate multiple
specifications and functional forms of a model in order to better understand the underlying
generating process. The second principle is that post hoc does not imply propter hoc; that is, a
significant coefficient in an estimated model does not always mean causality. That principle
can be related to critique by Chorus and Kroesen (2014) regarding the cross-sectional nature of
the latent variable. Due to this characteristic it is not appropriate for analysis of changes in the
variable at the individual level.
The third principle is crucial in HCM as it states that just giving something a name does
not mean that we understand it. This is directly related to the definition of a latent variable,
which usually is defined through associations with a set of indicators. Cliff (1983, p. 121) states:
“... we can only interpret our results very cautiously unless or until we have included enough
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indicators of a variable in our analysis, and have satisfied not only ourselves but sceptical
colleagues and critics that we have done so”. The meaning of the latent variable will always, to
some extent, be wrong, and our indicators will, to some extent, be unreliable. Moreover, in
HCMs the definition of the latent variables is usually neither based on theoretical foundations
nor proved through empirical work. There are, however, accepted scales to measures, for
example, attitudes with a tested set of questions, like locus of control (Rotter, 1975) or
environmental beliefs (Stern, 2000), which can easily be incorporated in choice models. If the
set of follow-up questions has not been based on theoretical findings, a preliminary
exploratory multivariate analysis should at least be applied to confirm the structure of the
underlying constructs.
The fourth principle is that ex post facto explanations are untrustworthy. If a model
has been adjusted on the basis of its fit or lack of fit to a particular data set, its statistical status
is precarious until it can be tested on a new data set. Regarding that principle, a simple
prediction, such as the one used in this application, can help in model comparison and can
shed light on the real performance of the model and on how close the model is to the true
data-generating process.
3. Model specification
We use two model specifications in this paper to investigate the influence of the
design dimensionality on stated choices. The first is a HCM consisting, apart from
measurement equations for attitudinal indicators, of two types of structural equation, one for
the choice model and one for the latent variable model. The structural equation for the choice
model is based on random utility theory (RUM), which is used to link the deterministic model
with a statistical model of human behaviour. Under this framework, the utility 𝑈𝑖𝑛𝑡 of
alternative 𝑖 for respondent 𝑛 in choice situation 𝑡 (from a total of 𝑇𝑛 choice occasions) is given
by:
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𝑈𝑖𝑛𝑡 = 𝑉𝑖𝑛𝑡 + 𝜀𝑖𝑛𝑡 , (1)
where 𝑉𝑖𝑛𝑡 in a classical logit model depending on observable explanatory variables, which are
usually attributes (𝑥𝑖𝑛𝑡) and vectors of attribute parameters 𝛽. The term 𝜀𝑖𝑛𝑡 is a random
variable following an extreme value distribution with location parameter 0 and scale
parameter 1. In a HCM, 𝑉𝑖𝑛𝑡 also depends on the latent variable 𝐿𝑉𝑛 and a vector of
parameters 𝛼 usually representing the interaction terms of the latent and explanatory
variables. Now let 𝑗𝑛,𝑡 be the alternative chosen by consumer 𝑛 in choice situation 𝑡, such that
𝑃𝑛,𝑡(𝑗𝑛,𝑡) gives the logit probability of the observed choice for consumer 𝑛 in choice situation
𝑡. The logit probability of consumer 𝑛’s observed sequence of choices is 𝑃𝑛 = ∏ 𝑃𝑛,𝑡(𝑗𝑛,𝑡)𝑇𝑛𝑡=1 .
The second structural equation for the latent variable is given by
𝐿𝑉𝑛 = ℎ(𝑍𝑛, 𝛾) + 𝜔𝑛, (2)
where ℎ(𝑍𝑛, 𝛾) represents the determinist part of 𝐿𝑉𝑛 and the specification is ℎ(·), which is in
our case linear, with 𝑍𝑛 being a vector of the socio-demographic variables of respondent 𝑛,
and 𝛾 being a vector of parameters. Additionally, 𝜔𝑛 is a normally distributed random
disturbance with zero mean and standard deviation 𝜎𝜔. In our case, the latent variable should
represent the level of impulsivity of the respondents.
Measurement equations use the values of the attitudinal indicators as dependent
variables, and explain their values with the help of the latent variables. The ℓ𝑡ℎ indicator (of
the total of 𝐿 indicators) for respondent 𝑛 is therefore defined as:
𝐼ℓ𝑛 = 𝑚(𝐿𝑉𝑛, 𝜁) + 𝑣𝑛, (3)
where the indicator 𝐼ℓ𝑛 is a function of the latent variable 𝐿𝑉𝑛 and a vector of parameters 𝜁.
The specification of 𝑣𝑛 determines the behaviour of the measurement model and depends on
the nature of the indicator. Responses to impulsivity statements in our case study are collected
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using a Likert type response scale, so that the measurement equations are given by typical
ordinal logit (Mariel et al., 2015) in which, apart from the parameters 𝜁, the corresponding
thresholds 𝜏 need to be estimated.
The model is finally estimated by maximum simulated likelihood. The estimation
involves maximizing the joint likelihood of the observed sequence of choices (𝑃𝑛) and the
observed answers to the attitudinal questions (𝐿𝐼ℓ𝑛). The two components are conditional on
the given realization of the latent variable 𝐿𝑉𝑛. Accordingly, the log-likelihood function of the
model is given by integration over 𝜔𝑛:
𝐿𝐿(𝛽, 𝛾, 𝜁, 𝜏) = ∑ 𝑙𝑛𝑁𝑛=1 ∫ (𝑃𝑛 ∏ 𝐿𝐼ℓ𝑛)𝐿
ℓ=1𝜔 𝑔(𝜔)𝑑𝜔. (4)
Thus, the joint likelihood function (4) depends on the parameters of the utility functions
included in (1), the parameters for the socio-demographic interactions in the latent variable
specification defined in (2), and the parameters for the measurement equations defined in (3).
Daly et al. (2012) describe different identification procedures. We follow the Bolduc
normalization by setting σω equal to 1. All model components are estimated simultaneously
and are contrasted using PythonBiogeme (Bierlaire, 2003, 2008) and Ox (Doornik, 2001).
The benchmark model for the hybrid setting described above is a typical RPL model in
which we assume that 𝛽𝑛 is a vector of the true, but unobserved, taste coefficients for
consumer 𝑛. We assume that 𝛽𝑛 is distributed over consumers with density 𝑔(𝛽,Ω). In this
case, if 𝑃𝑛,𝑡𝑅 (𝑗𝑛,𝑡|𝛽) gives the logit probability of the observed choice for consumer 𝑛 in choice
situation 𝑡, the logit probability of consumer 𝑛’s observed sequence of choices is:
𝑃𝑛𝑅(Ω) = ∫ ∏ 𝑃𝑛𝑅�𝑗𝑛,𝑡|𝛽�𝑇𝑛𝑡=1 𝑔(𝛽|Ω)𝑑𝛽 𝛽 . (5)
The log-likelihood function for the observed choices is then:
𝐿𝐿(Ω) = ∑ 𝑙𝑛𝑁𝑛=1 (𝑃𝑛𝑅(Ω)) . (6)
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4. Case study
The survey aimed at measuring preferences for land use changes in Germany.1 Thus,
the selected choice attributes comprise share of forest, land consumption, biodiversity
conservation and a price attribute (Table 4.1). All attributes except those concerning
biodiversity conservation were presented in all designs, while the biodiversity attributes were
used to adjust the number of attributes according to the design plan proposed by Hensher
(2004). Following this approach, 16 separate efficient designs were created using C-efficiency
allowing for minimizing the variance of WTP (Scarpa and Rose 2008). The designs were
optimized for a MNL model.
Table 4.1: Attributes used in the Choice Experiment
Attribute Description FOREST Percentage changes in the share of forest (positive and negative) LAND Percentage changes in land conversion for housing development and
traffic (positive and negative) BIO Biodiversity in the whole landscape including all landscape types BIO_AGRAR Agricultural landscape biodiversity BIO_FOREST Forest landscape biodiversity BIO_URBAN Urban area biodiversity BIO_OTHER1 Biodiversity in other landscape types: forests, urban areas,
mountains, water BIO_OTHER2 Biodiversity in other landscape types: urban areas, mountains, water BIO_OTHER3 Biodiversity in other landscape types: mountains, water COST Contribution to a landscape fund in € per year
Table 4.2 provides an overview of the 16 designs and of how the dimensions of the
choice sets vary across designs. All choice tasks included an SQ alternative, i.e., a zero price
option with no environmental changes, plus two or more alternatives depending on the
design-of-designs plan. Choices in the choice experiment regarding landscape changes had to
be made by considering the landscape within a distance of about 15 kilometres from the
respondent’s place of residence. Respondents for the nationwide online survey were recruited
1 See Meyerhoff et al. (2015) for more details of the design of the choice experiment and the survey.
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from a panel of a survey company. Each respondent was randomly allocated to one of the 16
designs.
Table 4.2: Design overview
Design Sets Alternatives Attributes Levels Range Interviews completed
Note: The number of interviews does not include those respondents who always chose the SQ option.
The questionnaire also included scales to capture different attitudes or personality
traits of the respondents. One of these was a scale developed for measuring impulsivity. The
scale is meant to provide a measurement instrument that allows the psychological trait of
impulsivity to be recorded in an economic way, i.e., in a way that consumes only a small
amount of interview time. The scale follows the UPPS (Urgency Premeditation Perseverance
and Sensation Seeking Impulsive Behavior Scale) approach. Kovaleva et al. (2012) point out
that there is still no standard definition of impulsiveness but that it is assumed that the
construct is multidimensional and thus comprises various aspects of impulsive behaviour.
These include, among others, i) the tendency to act without thinking and without sufficient
information for a decision, ii) the tendency to prefer a smaller immediate reward, and iii) the
tendency to choose riskier alternatives or the inability to assess the risks associated with
decisions correctly. Therefore, the UPPS approach comprises the four subscales urgency,
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intention, endurance and willingness to take risks. Each subscale is addressed using two items.
Table 4.3 reports the wording of the attitudinal statements and the direction of the association
with the latent construct impulsivity. Kovaleva et al. (2012) show that their scale performs well
and allows a reliable and valid measurement of impulsivity.
Table 4.3: Attitudinal questions
impul1 urgency + Sometimes I do things impulsively that I shouldn't do impul2 + I sometimes do things to cheer myself up that I later regret impul3 intention - I usually think carefully before I act impul4 - I usually consider things carefully and logically before I make up
my mind impul5 endurance - I always bring to an end what I have started impul6 - I plan my schedule so that I get everything done on time impul7 willingness to
take risks + I am willing to take risks
impul8 + I am happy to take chances
The scale was added to the survey in order to shed light on the link between
respondents’ psychological traits and their stated choices in the survey. We expect that
respondents who tend to be more impulsive are more likely to choose alternatives with a
positive price, i.e., not the SQ option, and that this intensifies when the choice sets become
more complex with a higher dimensionality. The reason for this is that people who are said to
be more impulsive are, among other things, expected to be more likely to act without
reflecting on the consequences and to be more likely to take risks (Kovaleva et al., 2012). To
some extent, however, the scale, which was provided by a leading social science research
centre in Germany (GESIS - Leibnitz Institute for the Social Sciences), was added in an
experimental manner as we expected it to be a reliable measurement instrument enabling us
to estimate HCMs. The literature applying latent variable models indicates that not using
reliable measurement instruments reduces the possibility of estimating an HCM.
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5. Results
Table 5.1 describes the variables used in the econometric models, along with their
descriptive statistics. Non-responses to items mean that the useable sample comprises 23,118
responses from 1,661 individuals. Briefly, the mean age is 42.3 years, the share of female
respondents is 53% and the mean disposable income of the respondents’ households is 17,500
Euros. As the survey was conducted as an online survey, we did not expect the sample to be
representative for the population in Germany. Not all people have access to the Internet and,
above all, not all use it regularly. Obvious deviations exist for the variables education and
income. Compared to the German population, the share of respondents with higher education
is too large and thus the disposable incomes are also too high. However, as we did not plan to
aggregate, for example, welfare measures based on the model results, we assume for the
following that the model comparison is not affected by the sample composition.
Table 5.1: Summary statistics
Variable (Attribute) Description Mean Std.Dev. Min Max
AGE Age 42.31 13.53 19 84 MAN Gender: Male 0.47 0.50 0 1 HIGHEDUC Level of education > secondary 0.39 0.49 0 1 INCOME Income 17,500.00 34,959.09 450 100,000 POSITION Position of the choice set 9.18 6.22 1 24 ALTERNATIVES Number of alternatives 3.85 0.81 3 5 ATTRIBUTES Number of attributes 5.32 1.08 4 7 WIDE Wide level range 0.29 0.45 0 1 NARROW Narrow level range 0.33 0.47 0 1 LEVEL3 Three level range 0.28 0.45 0 1 LEVEL4 Four level range 0.33 0.47 0 1
In addition to the socio-economic information, the respondents were asked a series of
attitudinal questions regarding impulsivity, as presented in Table 4.3. Table 5.2 shows the
response distributions on a 5-point Likert scale. For each statement, values closer to five would
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equate to stronger agreement while values closer to one would equate to stronger
disagreement.
Table 5.2: Responses to the impulsivity attitudinal questions