Use of Mixed Revealed-Preference
and Stated-Preference Models
with Nonlinear Effects in Forecasting
Elisabetta Cherchi and Juan de Dios Ortúzar
dent. However, use of SP data in forecasting is not recommended
because it is hypothetical and may not account for certain types of real
market constraints (4). Another important problem is that apparently
good models can be obtained with almost any SP survey, but if the
technique is not applied appropriately (for example, use of a noncustomized design in a general context instead of focusing on specific
behavior), serious problems may remain undetected until forecasts are
compared with actual outcomes (5).
Thus the recommended approach involves using both data sources
jointly (1, 2, 4). The mixed RP-SP approach has now been used in
many applications, both in research and in practical work, even with
very complex model structures. However, as is often the case, most
of the attention has been given to estimation, leaving the correct use
of these much-improved models in forecasting in need of better
understanding.
An in-depth review of the literature shows that many papers consider the joint RP-SP estimation problem, but few discuss the use of
these models in forecasting. Making predictions with a mixed RP-SP
model does not imply major problems when models are estimated
with all attributes being generic between RP and SP. However, this is
not always the case. Often, some attributes can be measured only in
one data set or, although available, can be estimated properly only
in one data set. Moreover, because of differences in the nature of
the attributes, different RP and SP parameters can be estimated,
both highly significant. In fact, it is likely that complete preference
homogeneity does not hold for multiple data (6). Finally, differences
between RP and SP parameters might be implicit in the need for SP
data, or even what is looked for when SP data are used, especially when
what is wanted is to test in prediction structural change (i.e., depart
from the current real market) and when utilities are not linear in the
attributes. It is interesting that often nonlinearities are included
only in the SP data, so the problems of nonlinear utility specifications
and mixed RP-SP estimation are strictly connected. Moreover,
although some papers (7–9) agree that often the typical linear utility
assumption does not allow one to correctly reproduce individual
behavior, no evidence exists on the effects of not accounting for the
nonlinear part of utility in forecasting.
This paper investigates the problem of using mixed RP-SP models
in forecasting and, in particular, with the effect of nonlinearities estimated in a mixed RP-SP model but specific only for the SP data. The
focus is on two types of nonlinearity that can be estimated in a mixed
RP-SP model and that often imply partial preference homogeneity.
The first case is the specification of interaction terms that because of
correlation problems in the RP data are normally included only in the
SP data. The second case occurs when different parameters are estimated for the RP and SP environments. Since in prediction only the
Estimation of mixed revealed-preference (RP) and stated-preference (SP)
data has become almost common practice. However, most attention has
gone to estimation, with the correct use of these much-improved models
in forecasting left in need of better understanding. In particular, their
potentiality in accounting for nonlinear effect has not been fully explored,
either in estimation or in prediction. The problem of using an RP-SP
mixed model for accounting for nonlinear effects in the attributes in forecasting, especially when the nonlinearities have been specifically estimated only for the SP data, is addressed. By using a mixed RP-SP data set
gathered especially for a modal choice context in Cagliari, Italy, several
nested logit models with nonlinear systematic utility functions were estimated and used to provide empirical evidence on the effect of incorrect
use of RP-SP specifications in forecasting under different policies. It was
concluded that the partial data enrichment approach is well suited to treat
nonlinearities and allows for consistent improvements in model estimation results. More important, nonlinearities might impose severe limitations in the applicability of these models in forecasting because of more
restrictive microeconomic conditions. Failure to treat these problems correctly can lead to serious forecasting errors, even in the case of relatively
mild policies.
The joint revealed-preference (RP) and stated-preference (SP) estimation method was first proposed in the early 1990s (1); since then it has
become recommended practice because it allows one to exploit the
advantages of each type of data and to overcome their limitations (2).
It is well known that RP data, as based on observations of actual
choices, need a large number of observations, may include only
existing alternatives, and require defining the choice set and calculating detailed and costly level-of-service information for all available options. Moreover, variables such as cost and time are often
correlated, and if data are not measured with a high level of precision, appropriate model structures and functional forms may not be
selected, which may lead to unknown bias in forecasting (3).
Conversely, SP experiments allow eliciting of individual preferences for alternatives that do not exist in reality or that individuals
have never experienced. They also allow researchers to have goodquality information (design under the analyst’s control) at a relatively
small cost, since many observations can be obtained for each responE. Cherchi, Centro Ricerche Modelli di Mobilità, Dipartimento di Ingegneria del
Territorio, University of Cagliari, Piazza d’Armi, 16, Cagliari, CA, 09123, Italy.
J. de D. Ortúzar, Department of Transport Engineering, Pontificia Universidad
Católica de Chile, Casilla 306, Cod. 105, Santiago 22, Chile.
Transportation Research Record: Journal of the Transportation Research Board,
No. 1977, Transportation Research Board of the National Academies, Washington,
D.C., 2006, pp. 27–34.
27
28
Transportation Research Record 1977
RP environment is used, when mixed RP-SP models are estimated
under the assumption of partial preference homogeneity, there is a
problem with how to introduce the specific SP attributes in the RP utility. The effect of not accounting for nonlinearities and for SP specific
attributes in forecasting is also explored, providing further empirical
evidence.
Given two sources of data, say, one coming from an RP survey and
the other from an SP survey, omitting individual q for simplicity, let
the random utility functions associated to alternative j be specified as
follows:
U
= β ′X
RP
j
+ α′Y
RP
j
+
RP
j
SP
SP
SP
U SP
j = β ′X j + γ ′ Z j + j
RP
j
≈ ( o,σ
)
)
2
SP
j ≈ ( o, σ SP
(1)
RP ≠ SP because of the different unknown scale parameters
where involved. As noted by Hensher, using mixed RP-SP data to estimate
choice models does not mean to “simply join the data” (10); the
scale factor in the indirect utility function must be considered. As
the scale factor depends on the standard deviation of the error terms
in the sample [for example, in the multinomial logit (MNL) it is
λ = π / 6 σ], two identical models estimated with different data
are very likely to give different estimated parameters, even if the
individual choice process is the same.
An efficient and correct way to combine two different data sources
(1) is to scale one data set to achieve the same variance in both. It does
not matter what utility is scaled; however, commonly the SP utility is
scaled,
(3)
where to comply with the joint estimation requirement, φ must be such
that φ σSP = σRP; or since in the MNL one has
6 λ SP
and
σ RP =
π
6 λ RP
2
φ SP
j ≈ ( 0 , σ RP )
(5)
and the joint likelihood function is
L=∏
e
V jRP
∑e
V jRP
i
j ∈I RP
Finally, RP and SP are random terms associated to the RP and SP util2
2
ities, respectively, the variances of which, σRP
and σSP
, will be generally different. (The variance is associated to the data used to estimate
the utility.)
First, if two different models using only RP or only SP data were
estimated, one would get the following estimates for the parameters:
α RP = λ RP α ; β RP = λ RPβ
(22)
γ SP = λ SP γ ; β SP = λ SPβ
π
(4)
Therefore, the new (scaled) utility function for the SP data set becomes
RP
, , = vectors of true parameters to be estimated,
XRP, XSP = vectors of attributes common to both data sets, and
YRP, ZSP = vectors of attributes specific to each type of data
[including alternative-specific constants (ASC)].
σ SP =
λ SP
λ RP
2
RP
where
= φU SP
U RP
j
j
φ=
SP
SP
SP
SP
U SP
j = φU j = φβ ′ X j + φ′ Z j + φ j
SP
V jSP
j
JOINT ESTIMATION WITH RP-SP DATA
RP
j
the scale factor should be
∏
SP
e
V jSP
∑e
(6)
V jSP
j ∈I SP
where the two choice sets in the RP (I RP) and SP (I SP) data sets might
be different.
The actual effect of scaling the SP utility by φ is that of forcing the
SP utility to have the same scale parameter of the RP utility. Therefore,
when the RP-SP data are used to estimate a model jointly, the overall
log-likelihood function is scaled only by the unknown (inestimable)
RP factor scale of the Gumbel distribution:
L=∏
RP
e
λ RPV jRP
∑e
j ∈I RP
λ
RP
V jRP
i∏
SP
e
(
λ RP φ V jSP
∑e
λ
RP
)
(φ V )
SP
j
(7)
j ∈I SP
In fact, whatever method is used to estimate the parameters, the scale
parameter φ will also be estimated in the process. In particular, in the
simultaneous estimation (2), Equation 7 is solved as a special nested
logit structure, where φ represents the common scale parameter
associated to the SP nests, yielding the following estimates:
αˆ = λ RP α ; βˆ = λ RPβ; γˆ = λ RP γ
(8)
ˆ , ˆ ) their
where (, , ) are the true population parameters and (ˆ , estimates. Note that since φ always multiplies some parameters, its estimated value is the inverse of the SP scale parameter deflected by the
RP scale parameter (φ̂ = λSP/λRP ). Also, the RP scale parameter of
the joint estimation will be a function of both the RP and the SP
environment, that is, λRP = f(, , ; XRP, XSP, YRP, ZSP), and thus,
it generally will be different from the RP scale parameter associated
with use of only the RP data set; in this latter case one would obtain
λRP = f RP (, ; XRP, YRP ).
Rescaling is based on the data enrichment paradigm; this has the
implicit assumptions that SP data are pooled only to enrich RP
information that may be deficient and that the true parameters are
invariant whatever method is used to elicit the individual preferences. Notwithstanding, there are various reasons why, after controlling for scale differences, the common parameters may still show
some differences.
Differences in the common parameters for both data sets do not
prevent use of the mixed RP-SP approach. As long as there is at least
one parameter common to both data sets, a partial data enrichment
approach can be applied. This is an important point, as one of the
major powers of the joint RP-SP estimation is that it allows use of
each source of data to capture those aspects of the choice process
for which it is superior. In particular, this is crucial to account for
Cherchi and Ortúzar
29
nonlinearities because measurement problems, high correlations,
and limited trade-offs often make RP data unsuitable for testing
nonlinear behavior.
FORECASTING WITH MIXED RP-SP MODEL
WITH NONLINEARITIES
To use a model estimated with mixed RP-SP data for prediction purposes, only the RP environment should be considered, since it represents real behavior. Thus, even if a joint RP-SP model is built to get
better estimates, when the joint model is used in forecasting, all information must refer to the RP environment. When all attributes are
generic for both sets, making predictions with a mixed RP-SP
model does not imply additional problems, because the parameters
are constrained to be the same in the RP and SP environments and
are estimated by using both data sets.
However, in the partial preference homogeneity approach, when
parameters are estimated specific for SP data, one must address the
problems of whether to use in forecasting the RP or the SP parameters and whether the SP parameters should be scaled. The common
rule that all parameters moved from the SP to the RP environment
should be scaled refers to the case of the ASC and has been explicitly discussed in the literature (11–13). The same consideration cannot be extended to parameters associated to real variables, as these
are of a different nature to the ASC. In fact, the ASC are not associated to data; rather, they tend to reproduce the market shares in the
sample (12).
Scaling an SP parameter by φ corresponds to measuring the individual preferences in the SP environment. Since any parameter estimated in a mixed RP-SP model is deflected by the unknown RP scale
parameter (see Equation 8), when specific SP parameters are included
in the RP utility function they should not be scaled. To see this, consider the simple case of an MNL and two utility functions specified
as in Equation 1. When RP and SP data are estimated jointly, the SP
utility is scaled as in Equation 7 through the maximization of the log
likelihood function:
However, as stated, the key point is that scaling is required because
the different data show different market equilibriums. In fact, what
is scaled in Equation 5 and following is the whole random utility, that
is, both the systematic utility (parameters and attributes together) and
the error term. At the same time, beyond a problem of scale, differences between RP and SP parameters associated to the same attribute
are often generated by differences in the attribute trade-offs implied
in each data set.
Therefore, the problem of using parameters estimated specific only
for SP in the RP probability (10) should be regarded as a problem of
having ˆ ZPj consistent with the new RP environment and with the
trade-offs implied in the SP data. In particular, since ˆ = λRP , this
guarantees consistency with the scale of the logit model used for prediction. The consistency with the trade-offs, instead, affects the measure of the Z attributes used in the RP environment. In fact, the
attribute values should belong to the range of variation of the SP
attribute used to estimate the parameter in both cases, that is, even if
Z can be observed in the RP environment or if only an engineering
estimate is available. In this way, the attribute trade-offs implied in
Equation 10 will be consistent with the ratio between the parameters
associated to them.
When nonlinear utilities are specified, a third set of conditions must
be verified: the microeconomic conditions on the marginal utilities. In
nonlinear modal utilities, the effect of a unit change in an attribute is
not simply equivalent to its associated parameter, as the marginal utility is also a function of the other parameters involved in the nonlinearity, including the individual characteristics. This limits the range
of variation of Z in forecasting. It is also important to note that the
microeconomic conditions need to be verified for both the estimated
model and the model to be used in forecasting; in the case of partial
preference homogeneity, these two models are not the same.
Suppose, for clarity, that the variable Z is indeed the product of X
and Y. The conditions for the estimated model use the utility functions
in Equation 9 and imply two different tests: for the RP data only (Conditions A) and for the SP data only (Conditions B), because the two
data sets are disjointed:
ˆ
L=∏
ˆ
RP
exp ( λ RP) X RP
+
λ
) Y RP
(
j
j
[
∑%
RP
A:
]
∂V jqRP
∂X RP
jq
j
ˆ
i
∏
ˆ
RP
SP
RP
exp φ ( λ ) X j + φ ( λ ) Z SPj
[
∑%
SP
B:
]
(9)
Multiplying the whole SP utility by a scale factor (as in Equation 5)
makes the estimated parameters, in both environments, scaled by the
same RP scale parameter. Therefore, when a specific SP parameter
is used in the RP environment, the model probability is
⎛
⎞
⎜⎝ λ RP ⎟⎠
RP
PjPR =
exp
[ (βˆ ) X
RP
j
]
∂V SP
jq
ˆ + ˆ X SPjq
= ˆ + ˆ Y SP
=
;
jq
∂Y SP
jq
(11)
Microeconomic conditions for the model to be used in forecasting
apply instead to the specification used in Equation 10 and require that
the following conditions be satisfied only by the RP data:
∂V jqRP
∂X RP
jq
⎛
⎞
⎜⎝ λ RP ⎟⎠
+ ( αˆ ) Y Rj P + ( ˆ ) Z Pj
∑%
∂V SP
jq
∂X SPjq
j
(λ)
∂V jqRP
ˆ
= ˆ ;
=
∂Y RP
jq
∂V jqRP
ˆ + ˆ X RP
= ˆ + ˆ Y RP
=
;
jq
jq
∂Y RP
jq
(12)
(10)
j
where ZP denotes an engineering set of attributes that define the policy
to be forecasted.
DATA SET
In the wake of changes all over Europe, the local rail company in
Sardinia, Italy, decided to transform the rail system for an urban
corridor in Cagliari (with 200,000 inhabitants and approximately
30
19,000 trips per day) into a metropolitan-like service by introducing
major changes in speed, frequency, comfort, ticketing, number of
stations, and so forth. The corridor was served by a major highway
and two other transport modes: a railway line and an urban bus, with
quite different levels of service, to the extent that the private car was
by far the dominant mode, having more than 80% of trips versus
fewer than 15% by bus and barely 3% by train. In 1998 a survey was
set up to build a data bank for a modal choice context. The method used
for building the data bank incorporated three kinds of surveys (7):
a qualitative survey using focus groups to gain a good understanding
of the phenomenon, an RP survey for describing current travel, and
an SP survey for evaluating the introduction of a revamped train
commuter option.
To collect data on current trips, a 24-h travel diary survey filled in
personally by each respondent was administered to 900 individuals,
and several socioeconomic characteristics relative to each individual
and family were collected as well. A maximum of 10 trips described
in considerable detail was recorded. Moreover, specific questions
were included to clarify whether the person who made the trip was
the person who paid for it and chose the mode of transport. Also, a
list of nonavailable alternatives was required, stating the reasons why
these were not available. This information allowed distinguishing
between objective and subjective availability, and only the first kind
was considered.
The SP survey, conducted on a selected subsample (300 individuals) of people who answered the RP questionnaire, had the objective of expanding the RP data set and checking commuter responses
to the major changes in the train alternative (i.e., the current train
service with far superior characteristics). A binary choice experiment
between the proposed new train service and the current transport
mode was used. The SP experiment did not consider the current train
users because the new train service proposed in the SP design was
superior to the current system in all characteristics (including the fare).
No other significant categories of user were present in the study corridor. The design included four variables at three levels (trip time,
cost, frequency, and comfort) for bus users and three variables at three
levels (trip time, cost, and frequency) and one variable (comfort) at
two levels for car users. Several other variables were analyzed and
tested in pilot surveys. For example, walking time was not included
in the final design because the railway line was fixed, and thus walking time could not freely be modified without departing from the real
context. However, because of its importance, the walking time for
each individual in the new situation was calculated and included in the
front page of the SP form as general information. Walking time was
computed directly from the distance walked, on the basis of known
origin and destination points. Also, depending on each user’s final
destination, parking availability was checked and average parking
time calculated. The experimental design allowed estimation of twoterm interactions to account for nonadditive effects of cost, frequency,
and travel time.
After a careful check of the data, a final sample (i.e., mixed RP/SP
data set) of 1,396 observations, composed of 338 RP individuals and
1,058 SP pseudoindividuals, was used for the model estimation.
Details on the data set are available elsewhere (7 ).
MODELS ESTIMATED
AND DEMAND FORECASTS
Several nested logit (NL) models with linear and nonlinear utility
functions, including allowance for correlation among RP options,
were estimated. As the SP design was a binary choice experiment
Transportation Research Record 1977
between car and rail for current car users and between bus and rail
for current bus users, correlation was tested only for the RP alternatives. Notwithstanding, a mixed RP-SP model is always a special
type of NL model, as the NL artificial structure comes from the
parameter used to scale one data set with respect to the other. The
models are summarized in Table 1. All the attributes have the usual
meaning, but “comfort” and “early/late” deserve remark. Comfort
was coded as two dummies in the public transport alternatives:
comf1 equals 1 if the level of comfort was poor, and comf2 equals 1
if comfort was sufficient, living the “good” level as reference. The
early/late variable, instead, measures by how many minutes a user
must anticipate or postpone her or his departure time to adjust to the
scheduled time of the train service. It was included for the train alternative alone to try to capture the differences between a scheduled
time system (train), a frequency (bus), and a continuous departure
time (car). Finally, all models in Table 1 were estimated by using an
expenditure rate specification (14), where the cost variable is divided
by the expenditure rate (g), which is equal to the ratio between
income and free time. As the data bank had only a small percentage
of self-employed, it was found (8) that this yielded superior results
to the more typical specification, where the cost variable is divided
by the wage rate (15).
When RP and SP data are used jointly to estimate models, it is
good practice to first estimate separate models for each data set to
detect which attributes are candidates to be generic in both sets. This
analysis suggested that all the linear and nonlinear variables (12, 16)
were good candidates for such a generic estimation, except walking
time. Table 1 reports the results of the joint RP-SP estimation. In particular, Models NL1 and NL2 clearly show that the specification
improves significantly [likelihood ratio (LR) = 99.99%] when walking time is specific, rather than generic, in RP and SP. However, the
marginal utility in the RP data is seven times larger than that in the SP
data. Although from a pure modeling-exercise point of view this is
quite a good model, using it to test policies implying a change in
walking time could be quite risky, because depending on whether the
RP or the SP parameter is used, very different predictions could be
obtained.
As for interaction terms, results in Table 1 show that they improved
significantly model results. In fact, Model NL2 is clearly superior to
Model NL3 (LR = 99.99%). Moreover, specifying the interactions
only in the SP alternatives clearly gave the best results. As illustrated
in Model NL4, interactions in the RP data do not enrich the data;
when specified as RP-SP generic, the interactions are not significant,
and the log likelihood also worsens for Model NL2.
Although Model NL2 appears to be very good from a statistical
point of view, from a microeconomic point of view it is quite poor.
Checking the microeconomic conditions on the marginal utility
shows that the number of individuals who do not satisfy them is quite
large. The first three columns in Table 2 show the number of observations that do not satisfy the microeconomic conditions for the estimated models. In all cases except Model NL4, only SP data do not
satisfy the microeconomic conditions, because the RP utility is linear
in the attributes. Model NL3 has not been included because the utility is linear in the attributes in both the RP and the SP environment.
The values in the first three columns of Table 2 have been calculated
by using the conditions presented in Equation 11.
To improve the model microeconomically, the reason these individuals had a counterintuitive marginal utility was carefully checked.
It was found that the problem arose mainly for individuals with a combination of high travel time and fare and low expenditure capability.
Model NL5 shows the results obtained by excluding those individuals. As can be seen, this is a very good model; although not superior
Cherchi and Ortúzar
31
TABLE 1
Model Estimation Results
Attribute
NL1
NL2
Walking time (RP)
−0.0439
(−1.8)
−0.1963
(−3.2)
—
Walking time (SP)a
—
NL3
NL4
NL5
−0.04575
(−2.1)
−0.09
(−1.9)
−0.2221
(−4.0)
−0.02621
(−1.7)
−0.04476
(−1.5)
−0.1191
(−2.0)
−0.2237
(−3.9)
−0.0227
(−1.6)
—
−0.06187
(−2.1)
−0.1699
(−2.9)
−0.2165
(−3.4)
−0.03998
(−2.0)
−0.03593
(−2.7)
0.4719
(3.3)
−3.106
(−3.6)
−1.541
(−3.3)
−1.148
(−2.4)
−0.1920
(−2.3)
12.19
(3.2)
0.00127
(2.9)
−0.00692
(−2.1)
—
−0.006689
(−0.8)
0.2334
(1.9)
−2.311
(−2.2)
−1.134
(−2.1)
−0.743
(−1.6)
−0.2944
(−3.3)
8.838
(2.5)
—
−0.0186
(−1.4)
0.2594
(1.9)
−2.028
(−2.2)
−0.9987
(−2.0)
−0.7488
(−1.7)
−0.2867
(−3.3)
9.444
(2.5)
—
−0.04833
(−3.0)
0.4745
(3.2)
−3.172
(−3.6)
−1.453
(−3.1)
—
—
Traveltime*fare
−0.0660
(−2.6)
−0.03278
(−2.9)
0.5944
(3.8)
−3.261
(−4.0)
−1.608
(−3.6)
−1.262
(−2.5)
−0.2203
(−2.7)
11.38
(3.2)
0.00119
(2.9)
−0.01026
(−3.0)
—
−0.0619
(−2.2)
−0.2043
(−3.2)
−0.2123
(−3.5)
−0.0369
(−2.0)
—
—
Traveltime*freq
—
—
—
K_car (RP+SP)
1.624
(1.8)
−0.9868
(−2.9)
0.4091
[3.93]
0.6179
(3.8)
[2.01]
−744.3223
1.677
(1.8)
−1.044
(−2.8)
0.2988
[6.74]
0.6704
(3.5)
[1.71]
−735.5511
0.1255
(0.2)
−0.7164
(−2.0)
0.2962
[1.99]
0.8827
(2.1)
[0.279]
−747.6844
0.000518
(1.7)
−0.002257
(−0.8)
0.7863
(1.0)
−0.6820
(−2.0)
0.2813
[6.53]
1.031
(2.1)
[0.06]
−744.8081
1.611
(1.7)
−0.9548
(−2.6)
0.3170
[6.15]
0.6355
(3.4)
[1.97]
−672.7029
0.1365
1,396
0.1467
1,396
0.1326
1,396
0.1360
1,396
0.2688
1,289
Travel time PT
Travel time car
Walking time
Cost/g
Frequency
Comfort 1
Comfort 2
Transfer
Early/late (RP)
Car/licenses (RP)
Traveltime*fare (SP)
Traveltime*freq (SP)
K_train (RP+SP)
f1 (EMU)b
f2 (SP scale factor)b
Log likelihood
ρ2(C)
Sample size
−0.9008
(−1.9)
−0.1816
(−2.2)
11.29
(3.2)
0.00097
(2.0)
−0.00597
(−1.8)
—
—
NOTE: Values in parentheses are t-statistics.
EMU = expected marginal utility.
a
Car walking time introduced only in the RP alternatives.
b
Values in brackets are t-statistics with respect to 1.
to Model NL2 for the t-test of some parameters, it is globally better
(the ρ2 value is higher). Moreover, the number of individuals who do
not satisfy the microeconomic conditions is significantly reduced.
The microeconomic conditions need to be verified also for the
model used in forecasting, and, as discussed previously, this puts a
limit to the policies that can be tested. Table 3 shows the limits that
are implicit in the estimated models, that is, the maximum value (or
better combination of values) that each attribute can assume to satisfy
the microeconomic conditions implicit in each specific model. These
values also define the range of policies that can be tested with each
model. As can be seen, Model NL2 also has the narrowest range of
policies that can be tested in forecasting.
Finally, the last three columns in Table 2 report the number of individuals in the RP environment whose marginal utility has an incorrect
sign when the models are used to forecast, that is, by using the conditions illustrated in Equation 12. It clearly appears that Model NL5 is
superior to all the other models. Moreover, it is important to note that
in nine of the 13 cases in which the marginal utility of the bus fare is
positive, the microeconomic condition is at the limit (i.e., the travel
time is equal to 50 min instead of 49.87, as in Table 3).
32
Transportation Research Record 1977
TABLE 2
Number of Observations Not Satisfying Microeconomic Conditions
RP+SP Observations
RP Observations
Conditions on
Estimated Models
Conditions on Models
Used in Forecasting
Number of Individuals with
NL2
NL3
NL4
NL2
NL3
NL4
MU(travel time by train) > 0
MU(travel time by car) > 0
MU(travel time by bus) > 0
MU(cost by train) >0
MU(cost by car) > 0
MU(cost by bus) > 0
MU(frequency by train) < 0
MU(frequency by bus) < 0
234
110
25
175
0
322
0
0
184
13
12
84
0
216
0
0
38
17
0
6
0
20
0
0
48
1
19
53
0
160
1
0
15
0
3
19
0
82
0
0
19
1
2
3
0
13
1
0
MU = marginal utility.
To test the effect of misusing RP-SP specifications in prediction,
the variation in aggregate market shares for various policy measures
was computed. The response to a change in the prediction was calculated as the percentage change in the aggregate share of mode j over
the initial situation (do-nothing):
ΔPj =
Pj − P j0
(13)
P j0
where Pj0, Pj are the aggregate probabilities of choosing mode j before
(do-nothing) and after introducing the measure, calculated by sample
enumeration.
By using the estimated models, two types of analyses aimed at
evaluating the effect of misusing RP-SP specifications on market
share predictions are described here. First, the effect of using the RP
or the SP parameter (when different RP and SP parameters are estimated) is investigated. Second, the effect of using only RP information in forecasting is tested, and, in particular, the effect of not
including the interactions in prediction and the effect of not satisfying the microeconomic conditions are tested. In all cases, the policy
tested implied a change in only one attribute (walking time or cost)
at a time.
Table 4 shows the variation in market shares for some changes in
walking time. In particular, in Case NL2(a) the RP walking time
parameter was used; in Case NL2(b) the SP walking time parameter
was used. Following the theoretical discussion in earlier sections,
the SP attributes were not scaled. The RP walking time parameter is
roughly six times larger than the SP walking time parameter. This
reflects on the market share variation, which is systematically bigger in Model NL2(a) than in Model NL2(b). As expected, Model
NL1 produces results that stay halfway between the two versions of
Model NL2.
Table 5 shows the variation in market shares for changes in the
travel costs by train and by car, and it allows one to appreciate the
effect of not accounting for interaction terms in predictions. By comparing Model NL3 with Models NL2 and NL5, it is clear that not
including interactions systematically reduces the effect of any policy.
Note that the interaction between travel time and cost reduces the
effect of the linear variables (because they have opposite signs; see
Table 1). However, in this case a comparison is made with a model
estimated without interactions (Model NL3), where the estimated linear parameter of the cost (which is a measure of the marginal utility
of the cost for an individual with an average travel time) is about three
times bigger than the value estimated when interactions are included
(Model NL5).
Regarding the effect of the microeconomic conditions on the marginal utility, results reported in Table 6 are interesting. Table 6 shows
for each policy tested the number of individuals who do not satisfy the
microeconomic conditions. Model NL5 confirms its superiority;
Model NL2 does not perform well (because the number of individuals
who have a counterintuitive behavior is high), even when the policy
implies a strong reduction in the travel cost.
TABLE 3 Maximum Values That Can Assume Attributes to Satisfy Microeconomic
Conditions on Marginal Utility
Attribute
MU(travel time PT) < 0
Cost/g <
MU(travel time car) < 0
MU(cost) < 0
MU(frequency) > 0
Cost/g <
Travel time <
Travel time <
Frequency
1
6
12
NL2
NL3
NL4
54.18
81.43
114.13
160.87
28.29
68.19
90.66
112.42
138.53
229.66
35.86
114.93
69.94
100.71
137.63
175.15
49.82
79.48
Cherchi and Ortúzar
TABLE 4
33
Effects in Prediction of Using Different Walking Time Specifications
% Variation in Demand Predictions by Mode
Model NL1
Attribute
% Change
Walking time
by train
Walking time
by bus
TABLE 5
−5
−10
−25
−50
−5
−10
−25
−50
Model NL2(a)
Model NL2(b)
Bus
Train
Car
Bus
Train
Car
Bus
Train
Car
−1.4
−2.8
−7.1
−14.1
1.1
2.2
5.6
11.1
3.1
6.3
15.8
31.8
−2.0
−4.0
−10.0
−19.7
0.0
−0.1
−0.2
−0.5
−0.1
−0.1
−0.3
−0.7
−2.9
−6.0
−15.8
−32.5
2.6
5.1
12.1
21,7
10.5
21.6
57.5
120.7
−7.6
−14.8
−34.3
−58.4
−0.1
−0.1
−0.4
−1.2
−0.2
−0.4
−1.1
−2.4
−0.9
−1.8
−4.6
−9.1
0.7
1.4
3.4
6.9
1.8
3.5
8.9
17.9
−1.1
−2.2
−5.4
−10.8
0.0
−0.1
−0.1
−0.3
0.0
−0.1
−0.2
−0.4
Effects in Prediction of Including SP Specific Interaction Terms
% Variation in Demand Predictions by Mode
Model NL2
Attribute
Fare by train
Cost by car
Model NL3
Model NL5
% Change
Bus
Train
Car
Bus
Train
Car
Bus
Train
Car
−5
−10
−25
−50
−5
−10
−25
−50
−0.1
−0.2
−0.5
−1.0
0.0
0.0
0.1
0.2
0.3
0.7
1.7
3.8
0.0
−0.1
−0.1
−0.3
0.0
0.0
0.0
0.0
0.0
0.0
0.0
−0.1
−0.1
−0.3
−0.6
−1.3
0.1
0.1
0.4
0.7
0.8
1.7
4.2
8.6
0.1
0.2
0.5
1.0
0.0
0.0
0.0
−0.1
−0.1
−0.1
−0.3
−0.6
−0.5
−1.1
−2.8
−5.7
0.2
0.4
1.0
2.0
2.3
4.6
11.9
24.4
0.2
0.4
1.1
2.4
0.0
0.0
−0.1
−0.2
−0.1
−0.3
−0.7
−1.4
have been reported, and some important issues (about moving
from estimation to prediction) do appear not to have been reported
before.
This report considered the problem of using mixed RP-SP models
in forecasting and analyzed the effect in predictions of using nonlinearities estimated in a mixed RP-SP model as specific variables of
the SP data. The potentiality of the partial data enrichment approach
in accounting for nonlinearities in the attributes was demonstrated; it
CONCLUSIONS
The mixed RP-SP approach has received a great deal of attention
over the years, and many major advances have been experienced
both in theory and in practice. Joint RP-SP estimation has also
been used in many applications, including cases with complex
utility functions and a large number of options. However, not
many applications of mixed RP-SP models as prediction tools
TABLE 6 Number of Observations Not Satisfying Microeconomic Conditions
for Different Policies
% Variation in Demand Predictions by Mode
Model NL2
Attribute
Fare by train
Cost by car
Model NL5
% Change
Bus
Train
Car
Bus
Train
Car
−5
−10
−25
−50
−5
−10
−25
−50
19
19
19
19
19
19
19
19
42
38
25
5
48
48
48
48
1
1
1
1
3
6
8
18
2
2
2
2
2
2
2
2
18
16
5
1
1
1
1
1
1
4
8
20
20
20
20
34
allows use of each data source to capture those aspects of the choice
process for which it is superior.
However, the partial preference homogeneity approach poses
some theoretical problems of whether to use in forecasting the RP
or the SP parameters and whether the SP parameters should be
scaled. This problem was analyzed in depth. It was concluded that
scaling an SP parameter corresponds to measuring the individual
preferences in the SP environment. Since any parameter estimated
in a mixed RP-SP model is deflected by the unknown RP scale
parameter, specific SP parameters should not be scaled when used
in forecasting. Moreover, as scaling is required because the different data show different market equilibriums, the problem of using
parameters estimated specific only for SP in the RP probability
should be regarded as a problem of having consistency between the
new RP environment and the trade-offs implied in the SP data.
Also discussed in depth was the problem of microeconomic conditions on the marginal utilities, specifically in the case of nonlinear
utilities and partial preference homogeneity. Nonlinear utilities
imply that the effect of a unit change in an attribute is a function of
the parameters involved in the nonlinearity. Thus the microeconomic
conditions should be verified for each individual and for each policy
tested, as they limit the range of the scenarios that can be tested.
Moreover, the partial homogeneity approach requires that the microeconomic conditions be verified for both the estimated model and the
model to be used in forecasting, because in the case of partial preference homogeneity these two models are not the same.
The empirical results demonstrate that the partial data enrichment
approach is particularly suitable to account for nonlinearities and
allows improving consistently the estimation results. In particular,
the estimation of interaction terms specific only for the SP data provided a significant improvement in model fit. However, the microeconomic conditions play a crucial role in evaluating the goodness
of a specification, as it was found that the model with the best fit was
quite poor from a microeconomic point view, with a large number
of individuals having a counterintuitive behavior.
It was found that the potential errors in predicting demand for reasonably sensible policies can be quite high. Although the magnitude
of these effects certainly depends on the specific application context,
this raises an alarm about a problem that merits further examination.
ACKNOWLEDGMENTS
The authors thank David Hensher and Joffre Swait for their comments. The authors also thank three anonymous referees for their
constructive comments. The second author acknowledges the support of the Chilean Fund for Scientific and Technological Research,
FONDECYT.
Transportation Research Record 1977
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The Transportation Demand Forecasting Committee sponsored publication of
this paper.