Content uploaded by Josep Raymond
Author content
All content in this area was uploaded by Josep Raymond on Mar 28, 2022
Content may be subject to copyright.
1 23
Transportation
Planning - Policy - Research - Practice
ISSN 0049-4488
Volume 39
Number 1
Transportation (2011) 39:1-17
DOI 10.1007/s11116-011-9325-1
Traffic forecasts under uncertainty and
capacity constraints
Anna Matas, Josep-Lluis Raymond &
Adriana Ruiz
1 23
Your article is protected by copyright and
all rights are held exclusively by Springer
Science+Business Media, LLC.. This e-offprint
is for personal use only and shall not be self-
archived in electronic repositories. If you
wish to self-archive your work, please use the
accepted author’s version for posting to your
own website or your institution’s repository.
You may further deposit the accepted author’s
version on a funder’s repository at a funder’s
request, provided it is not made publicly
available until 12 months after publication.
Traffic forecasts under uncertainty and capacity
constraints
Anna Matas •Josep-Lluis Raymond •Adriana Ruiz
Published online: 24 February 2011
ÓSpringer Science+Business Media, LLC. 2011
Abstract Traffic forecasts provide essential input for the appraisal of transport invest-
ment projects. However, according to recent empirical evidence, long-term predictions are
subject to high levels of uncertainty. This article quantifies uncertainty in traffic forecasts
for the tolled motorway network in Spain. Uncertainty is quantified in the form of a
confidence interval for the traffic forecast that includes both model uncertainty and input
uncertainty. We apply a stochastic simulation process based on bootstrapping techniques.
Furthermore, the article proposes a new methodology to account for capacity constraints in
long-term traffic forecasts. Specifically, we suggest a dynamic model in which the speed of
adjustment is related to the ratio between the actual traffic flow and the maximum capacity
of the motorway. As an illustrative example, this methodology is applied to a specific
public policy that consists of suppressing the toll on a certain motorway section before the
concession expires.
Keywords Traffic forecast Uncertainty Toll motorways
Introduction
Traffic forecasts provide essential input for the appraisal of transport investment projects
and public policies. In spite of significant improvements to transport demand models over
the past few decades, there are still high levels of uncertainty in long-term forecasts. For
instance, a recent study by Flyvbjerg et al. (2006) concludes that accuracy in forecasting
traffic flow has not improved over time. Given that project profitability is highly dependent
on predicted traffic flow, uncertainty has to be quantified and accounted for in project
evaluation.
A. Matas (&)A. Ruiz
Departament d’Economia Aplicada, Universitat Auto
`noma de Barcelona, Barcelona, Spain
e-mail: anna.matas@uab.es
J.-L. Raymond
Departament d’Economia i Histo
`ria Econo
`mica, Universitat Auto
`noma de Barcelona, Barcelona, Spain
123
Transportation (2012) 39:1–17
DOI 10.1007/s11116-011-9325-1
Author's personal copy
This article quantifies uncertainty in traffic forecasts for the tolled motorway network in
Spain. We estimate a demand model using a panel data set covering 67 tolled motorway
sections between 1980 and 2008. Uncertainty is quantified in the form of a confidence
interval for the traffic forecast that takes account of both the variance of the traffic forecast
related to the stochastic character of the model (model uncertainty) and the uncertainty that
underlies the future values of the exogenous variables (input uncertainty). Furthermore, as
an illustrative example we apply this methodology to a specific public policy consisting of
suppressing the toll on a certain motorway sections before the concession expires. In this
case, the government has to compensate the private motorway concessionaire for the
revenue forgone up to the end of the concession period. We present a point estimate for the
present value of the forgone revenue, as if the result were certain, and then a set of
confidence intervals at different levels of significance that account for the variance of the
forecasting error.
The predictions are based on an aggregate demand equation, where traffic flow depends
on the following variables: Gross Domestic Product, toll per kilometre, petrol price and a
set of dummy variables that account for major changes in the road network. However, if
maximum infrastructure capacity is not allowed for in the model, it may well be that
predictions lie above this maximum value. To avoid this problem we should, ideally,
estimate an integrated demand–supply system. However, as is often the case, we are not
able to model the supply side of the system due to lack of data. Our article contributes to
this issue by proposing a new functional form for the demand equation that accounts for the
fact that the rate of growth of traffic flow diminishes as the volume approaches full
capacity. Specifically, as detailed in ‘‘The model’’ section, we suggest a modified partial
adjustment model with variable adjustment speed. In our case, and in terms of forecasting
capacity, this proposal is preferable to the traditional logistic functional form with a
saturation level equal to maximum capacity, given that we avoid the assumption that traffic
follows an S-shaped growth curve.
Literature review of uncertainty in traffic forecasting
Several recent studies confirm the inaccuracy of traffic predictions. Among them, the
extensive work by Flyvbjerg et al. (2006) based on 210 transport infrastructure projects in
14 nations, 27 of which correspond to rail projects and the rest to road projects. They
conclude that passenger forecasts for nine out of ten rail projects are overestimated, with an
average overestimation of 106%. The authors suggest that there is a systematic positive
bias in rail traffic forecasts. For road projects, forecasts are more accurate and balanced,
although for 50% of the projects the difference between actual and forecasted traffic was
more than ±20%.
1
For both road and rail projects, the estimated standard deviation of the
forecasting error is high, showing a high level of uncertainty and risk.
Bain (2009) presents the results from a study that analyses the performance of traffic
forecasts for toll road traffic from a database including over 100 international toll road
projects. The research confirms a large range of error in traffic forecasting and the exis-
tence of systematic optimism bias. On average, toll road forecasts overestimated first-year
traffic by 20–30%.
1
The authors suggest reference class forecasting as an alternative methodology. This proposal is detailed in
Flyvbjerg (2008).
2 Transportation (2012) 39:1–17
123
Author's personal copy
Using data on 14 toll motorway concessions in Spain, Vassallo and Baeza (2007) found
that, on average, actual traffic during the first 3 years of operation was overestimated by
approximately 35%. They conclude that there is a substantial optimism bias in the ramp-up
period for toll motorway concessions in Spain.
The aforementioned studies suggest that the positive bias found for rail and toll mo-
torways appears when there is a strong will for the approval of the project.
In spite of the significant errors present in traffic forecasting, uncertainty is often a
neglected issue. Most of the predictions are presented as point estimates and the probability
distribution of the outcome is forgotten about. The most common way to deal with
uncertainty is to present alternative estimates based on different scenarios for the exoge-
nous variables. However, this approach does not recognise all sources of uncertainty and,
most importantly, does not provide the likelihood of each alternative forecast.
As stated by de Jong et al. (2007), the literature on quantifying uncertainty in traffic
forecasting is fairly limited. The author reviews a considerable amount of the literature on
that subject considering both the methodology employed and the results obtained. He
distinguishes between input uncertainty, associated with the fact that future values of the
exogenous variables are unknown, and model uncertainty, which includes random term
uncertainty and coefficient uncertainty. Given that the 21 studies reviewed use different
measures to express uncertainty and many of them do not present quantitative outcomes,
providing an order of magnitude for uncertainty is difficult. De Jong suggests that input
uncertainty is more important than model uncertainty; studies on input uncertainty or both
input and model uncertainty obtain 95% confidence intervals for the mean value of traffic
flow between ±18 and ±33%. The aforementioned paper also offers a methodology for
quantifying uncertainty for a case study in The Netherlands.
The literature shows that quantifying forecast uncertainty and its causes is an area that
deserves more attention. This article intends to contribute to this issue with new findings.
The model
Given that the demand equation is estimated in order to predict future traffic flow, when
specifying the equation we should take into account that as the volume of traffic increases,
costs related to congestion emerge and the rate of traffic growth diminishes as traffic
volume approaches maximum capacity of a road section. Toll motorways were introduced
in the early 1970s on the road network in Spain. Nowadays, some of these motorways are
close to their maximum capacity. This problem mainly affects those toll roads near urban
areas and the main corridor along the Mediterranean coast, where it is difficult and costly
to expand capacity. In these cases capacity constraints need to be considered when fore-
casting in order to avoid excessively optimistic results.
2
Ideally, congestion costs and capacity constraints should be accounted for through a
network assignment model, allowing a feedback between the various stages of the travel
demand forecasting process.
3
However, frequently such a model is unavailable. As an
2
The proposed functional form can be useful with other transport modes. The authors have applied it to
forecasting air transport. With respect to this transport mode, Riddington (2006) concludes that the air traffic
predictions in the United Kingdom are excessively optimistic and that no consideration is given to the
existing restrictions on the capacity to handle this forecast demand.
3
A sensible policy for motorway operators would be to increase tolls as demand approaches maximum
capacity. Nonetheless, this is not a feasible policy in Spain as toll rates are set by law and are, therefore,
Transportation (2012) 39:1–17 3
123
Author's personal copy
alternative approach, we suggest a functional form that can be considered as an implicit
reduced form for the demand function. Specifically, we estimate a modified partial
adjustment model, where the speed of adjustment is variable. The proposed equation can
be derived as follows:
The static equation of the partial adjustment model takes the standard form and shows
the logarithm of the equilibrium value of traffic Y
it on road section iin period tas a
function of a set of variables X
it
:
ln Y
it ¼aiþbln Xit ð1Þ
The dynamic of the adjustment is modified by introducing a variable adjustment parameter,
k
it
:
Dln Yit ¼ln Yit ln Yit1¼kit ðln Y
it ln Yit1Þþeit ð2Þ
We assume that the speed of adjustment decreases as traffic flow increases in the following
terms. Let us define the quality level of the motorway, s, as a function of the traffic flow
related to the maximum capacity of the infrastructure, Y
max
:
sit ¼Ymax
iYit1
Ymax
ið3Þ
Then, the adjustment parameter is assumed to be a function of s
it
:
kit ¼hYmax
iYit1
Ymax
i
¼hsit ð4Þ
where his a parameter that links the speed of adjustment and the level of use of the
motorway section.
This functional form accounts for the fact that the rate of traffic growth diminishes as
traffic volume approaches the capacity limit. Its implications can be best observed in two
extreme cases. When there is no traffic on the motorway, the speed of adjustment is
maximum:
Yit1!0)sit !1)kit !hð5Þ
In the opposite case, when traffic has reached capacity, the speed of adjustment is zero:
Yit1!Ymax
i)sit !0)kit !0ð6Þ
By substituting Y
it from Eqs. 1into 2, we get the first equation:
Dln Yit ¼kit ðaiþbln Xit ln Yit1Þþeit ð7Þ
Next, substituting from k
it
for Eq. 4we get the final equation:
Dln Yit
sit ¼ðhaiþhbln Xit hln Yit1Þþeit
sit ð8Þ
Footnote 3 continued
exogenous to the companies. Presently, toll increases are set according to the Consumer Price Index (CPI)
plus an efficiency term.
4 Transportation (2012) 39:1–17
123
Author's personal copy
This is a heteroskedastic model, so we have estimated using weighted least squares. This
formulation does not need to be restricted to the partial adjustment model. It can be easily
generalised to slagged values, as shown in Appendix 1.
We estimate a standard demand equation where variables are expressed in logarithms.
4
The traffic volume in each section is a function of the level of economic activity (measured
by Gross Domestic Product, GDP), the toll rate per kilometre, the price of gasoline and a
set of dummy variables that capture major changes in the road network.
5
The full set of
dummy variables is detailed in Appendix 2. The demand function can be expressed as
follows:
Dln Yit
sit ¼ðhaiþhb1iln GDPtþhb2iln GPtþhb3iln Tit
þhciZit hln Yit1Þþeit
sit ð9Þ
where Y
it
is the traffic volume on motorway section iin period t; GDP
t
is the real GDP in
period t;GP
t
is the gasoline price in period tdeflated by Consumer Price Index, CPI; T
it
is
the motorway toll in section iperiod tdeflated by CPI; Z
it
is the dummy variables capturing
major changes in the network; a
i
is the individual fixed effects; e
it
is the error term; (h,a
i
,
b
1i
,b
2i
,b
3i
,c
i
) are the coefficients to be estimated.
The individual fixed effects explain the differences between motorway sections (cross-
section units) not captured by the variables included in the model. In our case, they may
capture generation and attraction effects that determine the magnitude of traffic in each
motorway section.
The data
To estimate the demand equation, we used a panel data set of 67 motorway sections
observed between 1980 and 2008, although not all cross-section units were observed for
this temporal span. The total number of observations was 1765. The cross-section obser-
vations correspond to the shortest motorway section allowed by the data collection pro-
cesses, with an average length of 20 km.
The dependent variable is the annual average daily traffic volume in each section. The
explanatory variables are: real GDP, gasoline price and toll per km. The last two deflated
by CPI. GDP and gasoline price are defined at the national level and take the same value
for all sections in the sample.
6
Finally, a set of 30 dummy variables captures the most
important changes in the road network. For example, improvements on a parallel free road
were captured by a dummy variable that takes value 1 since the opening year. The
4
We considered the three alternatives most widely used to estimate aggregate demand functions: the linear
model, the semi-log model and the log-linear model. According to the Schwarz criterion, based on the log of
the likelihood functions from each model, we selected the log-linear specification.
5
It must be noted that equation (9) could be reparametrised so that the relationship between traffic flow and
the explanatory variables can be expressed in levels. Matas and Raymond (2003) provide a justification for
this model specification.
6
In some preliminary estimations we used GDP and gasoline prices at the regional level. The estimated
coefficients and the degree of adjustment showed to be almost the same. Therefore, given that for series
defined at national level the available time span is much larger, we decided to use GDP and gasoline prices
at the national level in order to obtain better forecasting models for the input variables.
Transportation (2012) 39:1–17 5
123
Author's personal copy
advantage of working with a panel data set is the high variability observed in the sample.
See Table 1.
It is interesting to note that there are substantial differences in traffic volume among the
different sections of the motorway network. The daily average traffic flow ranges from
1689 vehicles in the section and year having the lowest volume to 90033 in the section and
year with the highest. Furthermore, we found an extensive price range for toll rates. For the
whole period, at 2006 prices, the lowest price paid per km was about 0.058 €, whereas the
highest was about 0.34 €. The reasons for this wide variation are twofold. Firstly, each
motorway has to cover its own construction costs, so the toll rates are higher on those
motorways with larger construction costs or lower traffic volume. Secondly, the changes in
toll policies during the last two decades have resulted in a wide variation of rates across the
country and over time. For instance, on some motorway sections tolls decreased as much as
40% in one year.
The maximum capacity of each motorway section was calculated according to the
number of lanes and types of vehicle.
Model estimation and results
Before estimating the model equation stated in (9), and in order to decide whether to
estimate in levels or differences, we analyzed the existence of unit roots and cointegration
of the series. The traffic volume and GDP variables were clearly non-stationary. So, both
variables could be considered as integrated which means that the expected value and the
variance are non constant. The evidence for motorway tolls and gasoline prices was more
doubtful. In any case, to justify an estimation using levels for all the variables, it is
necessary to guarantee that a cointegration relation exists among them. This means it is
possible to find a linear combination of the series that is stationary. In our case, according
to the Kao cointegration test for panel data, the null hypothesis of no cointegration was
clearly rejected.
7
Therefore, we proceeded to estimate the equation in levels.
As specified in Eq. 9, the estimation of the demand equation would require to estimate
400 coefficients. Given that the number of total observations was 1765, it seemed advisable
to introduce some constraints to the coefficients in order to allow for efficiency gains.
Based on a previous work by Matas and Raymond (2003), we assumed that the demand
Table 1 Descriptive statistics
Mean Maximum Minimum Std. dev.
Traffic volume 16807 90033 1689 13523
GDP (millions of €)
a
733009 1063202 471466 177785
Gasoline price (€per l)
a
0.982 1.496 0.832 0.176
Toll (€per km)
a
0.126 0.343 0.058 0.050
Maximum capacity 78700 121192 59700 15407
a
The base year for variables expressed in €is 2006
7
The Kao test is based on the analysis of the residuals. The estimated statistic for the Augmented Dickey-
Fuller (ADF) test for the residuals is -4.677, which clearly rejects the null hypothesis of no cointegration
(p-value almost zero). For a clear presentation of unit roots and cointegration see Hamilton (1994).
6 Transportation (2012) 39:1–17
123
Author's personal copy
elasticity of GDP and gasoline prices were the same across all motorway sections.
Nonetheless, we maintained a specific toll coefficient for each motorway section.
Under these assumptions, we estimated Eq. 9using weighted least squares. The random
disturbance of the equation was modelled as a first order autoregressive process (rho) to
control for autocorrelation. The coefficients for GDP, gasoline price, and the lagged value
of the dependent variable take the expected sign and were estimated with a high degree of
precision. In relation to the toll coefficients, a significant variation across motorway sec-
tions was observed. A Chi-square test allowed us to clearly reject the null hypothesis of
equality of toll coefficients across all sections. However, the difference in the values of the
toll coefficients could be explained by certain motorway characteristics: contiguous sec-
tions on the same motorway present very similar elasticities; the more inelastic sections are
located on corridors with high traffic volumes, and demand is seen to be more elastic where
a good alternative free road exists.
The observed results suggested the possibility of re-estimating the model by introducing
the hypothesis of equality of toll coefficients across those motorway sections that showed
similar coefficients in the initial model. Hence, we proceed by testing equality constraints
among the toll coefficients for those motorways with similar coefficients in the original
estimation. Based on the results of the Wald test, the motorway sections were classified
into 3 groups as follows:
– Low toll elasticity: sections with toll coefficient between 0 and -0.2.
– Medium toll elasticity: sections with toll coefficient between -0.2 and -0.35
– High toll elasticity: section with toll coefficients larger than -0.35.
The final estimation results are detailed in Table 2and the coefficients for the dummy
variables in Appendix 2. As can be observed, the toll coefficients are estimated with a high
degree of precision. Given that the variables are log-transformed, the estimated coefficients
can be interpreted as short term elasticities. Demand is sensitive to toll variations, although
in the short term it is inelastic in all three groups.
Table 2 Summary of the esti-
mation results Dependent Variable: D(Ln (traffic volume))/tau
Estimation method: weighted least squares
Coefficient Std. Error t-Statistic
Ln (GDP) 0.7538 0.0403 18.72
Ln (gas price) -0.3802 0.0157 -24.19
Ln (traffic volume(-1)) -0.6059 0.0226 -26.82
Ln (toll_1) -0.1549 0.0159 -9.72
Ln (toll_2) -0.3403 0.0193 -17.62
Ln (toll_3) -0.4879 0.0276 -17.65
Rho 0.7347 0.0218 33.64
Dummy variables Yes
Fixed effects Yes
R
2
0.62
Observations 1668
Transportation (2012) 39:1–17 7
123
Author's personal copy
To provide an additional insight into the accuracy of our model we compared its
forecasting capacity to that of a logistic regression model. Using the same explanatory
variables, we estimated a logistic regression with a saturation level equal to maximum
capacity. According to the mean square error (MSE) for a dynamic forecast over the period
2000–2008, our approach was clearly preferable to the logistic approach.
8
An interesting property of the proposed functional form is that it makes it possible to
avoid the often unrealistic assumption of constant elasticity. As shown in Appendix 3,
demand elasticity with respect to an explanatory variable X
k
depends on the value of s
it
,
that is, it depends on the degree of motorway use. For s
it
=s
0
, the elasticity with respect to
variable X
k
in period J is given by:
eJ¼b
kð1cJþ1Þ
ð1cÞð10Þ
where b
k
*
=s
0
b
k
, being b
k
the coefficient associated to X
k
, and c
*
=(1 -s
0
h)
As an illustration, we compute the demand elasticity with respect to GDP for different
values of s
0
and for the first 6 years after the change in the exogenous variable. Elasticities
are detailed in Table 3. For s
0
=1, when the level of traffic approaches 0, short-term
elasticity is 0.8; after 5 years, the elasticity tends to the long-term value, 1.24. However, as
traffic increases and s
0
decreases, demand elasticity becomes less sensitive to GDP vari-
ations. For s
0
=0.1, when traffic flow approaches capacity, short-term elasticity is less
than 0.1. The elasticity values computed for s
0
=0.7, which correspond to the average
observed value our sample, are in line with those reported in the literature.
Figure 1displays the elasticity values for s
0
ranging from 0.1 to 1.
For the particular case where s
0
=1, the coefficients can be interpreted as those in the
standard partial adjustment model. The short- and long-term elasticities for all the
explanatory variables are reported in Table 4.
Forecast results and uncertainty
From the estimated demand model, we proceeded to forecast traffic flow for the 2009–2025
period. The first step was to predict the explanatory variables in the model. GDP and
gasoline price are predicted according to a time series model and motorway tolls are
Table 3 Elasticities with respect to GDP
Tau
j (years) 0.1 0.5 0.7 1
0 0.075 0.377 0.528 0.754
1 0.146 0.640 0.832 1.051
2 0.213 0.823 1.006 1.168
3 0.275 0.950 1.107 1.214
4 0.334 1.039 1.165 1.232
5 0.389 1.101 1.199 1.239
8
The results showed a value for the MSE of 5,979,872 for the logistic approach and 2,732,930 for our
proposal. A ‘‘t’’ test for the equality of both MSE clearly enabled to reject the null hypothesis (t=6.19).
8 Transportation (2012) 39:1–17
123
Author's personal copy
assumed to remain constant in real terms given that the toll revision formula is linked to
CPI. We applied univariate distributions for the exogenous variables given that no cor-
relations were observed among them.
Figure 2displays the forecasted traffic flow for two representative motorway sections
according to both a non-restricted model (standard partial adjustment model) and a
capacity restricted model (modified partial adjustment model). In the first one, traffic flow
is well below maximum capacity in the year 2025, whereas the second has reached
capacity by approximately 2019. As can be observed, the effect of the capacity constraint is
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
tau=0.1
tau=0.5
tau=0.7
tau=1
Lags in years
Elasticity
Fig. 1 Elasticities with respect
to GDP for different tau values
Table 4 Estimated demand
elasticities Tau =1
Short term Long term
GDP 0.754 1.244
Gasoline price -0.380 -0.628
Toll 1 -0.155 -0.256
Toll 2 -0.340 -0.562
Toll 3 -0.488 -0.805
0
20,000
40,000
60,000
80,000
100,000
120,000
140,000
160,000
80 85 90 95 00 05 10 15 20 25
Model with capacity constraint
Model without capacity constraint
Year
10,000
20,000
30,000
40,000
50,000
60,000
70,000
80,000
80 85 90 95 00 05 10 15 20 25
Model with capacity constraint
Model without capacity constraint
Year
Traffic
Maximum capacity
Maximum capacity
Fig. 2 Forecasted traffic flow for two motorway sections
Transportation (2012) 39:1–17 9
123
Author's personal copy
almost unnoticeable when traffic volume is below maximum capacity. However, the effect
is clear for the second motorway section. The standard partial adjustment predicts an
unrealistic level of traffic flow; whereas our suggested functional form forces traffic flow to
remain below capacity.
Finally, we proceeded to quantify uncertainty in the traffic forecasts. It is well known
that there are three possible sources of error in traffic forecasting. The first one is input
uncertainty, due to the fact that the future values of exogenous variables are unknown. The
second one is random term uncertainty that accounts for the random disturbance in the
demand equation. The third is coefficient uncertainty, due to using parameter estimates
instead of true population values. The sum of the last two corresponds to model
uncertainty.
To fix ideas, let us consider the following non-linear model:
y¼UðX;b;eÞð11Þ
in which the dependent variable is, in general, a non-linear function of a set of explanatory
variables, of a set of unknown bcoefficients and of a random term e. The forecasted values
of the dependent variable are obtained by substituting the unknown terms by their
respective estimates.
^
y¼Uð^
X;^
b;^
eÞð12Þ
In case we are dealing with a deterministic simulation, ^
eis fixed in the expected value of e,
that is zero, ^
bis the estimated value of b, and ^
Xis the assigned value of the explanatory
variables.
In a stochastic simulation we assume that each of the elements of Eq. 11 follows a
certain distribution. This is:
XDistð^
X;R^
XÞ
bDistð^
b;R^
bÞ
eDistð0;R^
eÞ
ð13Þ
Mrandom realizations of such distributions are generated using a bootstrap methodology.
The model is solved for each realization of those distributions. So, Mforecasted values of
the dependent variable are obtained. The empirical distribution of the forecasted values
enables an expected value to be computed that is the arithmetical average. Using the
empirical distribution, for a certain confidence level, it is also possible to compute upper
and lower limits. The contribution to total uncertainty derived from the components could
be calculated by subtraction. In this study all three types of uncertainty have been obtained
through a stochastic simulation process.
To evaluate total forecast uncertainty we consider the distribution of ^
yafter generating
Mrealizations of X,b,e.
To evaluate model forecast uncertainty we consider the distribution of ^yafter generating
Mrealizations of b,e; but holding the values of the explanatory variables Xfixed at ^
X.
Finally, input uncertainty can be computed from the difference between total forecast
uncertainty and model forecast uncertainty.
9
9
In general, when dealing with non-linear models the additive property is not fulfilled. This means the total
forecast uncertainty is not exactly equal to the sum of model forecast uncertainty plus input uncertainty
because the existence of interactions or mixed terms uncertainty. Although it would be possible to
10 Transportation (2012) 39:1–17
123
Author's personal copy
Because the model is non-linear it should be noted that the empirical average of the
stochastic simulations, in general, will not coincide with the deterministic simulation.
Therefore, in non-linear models the deterministic simulation will offer a biased forecast.
In this study the model has been solved repeatedly for 1000 random draws of various
components by using bootstraping method.
To illustrate the impact of uncertainty, we computed the 70% confidence interval for the
traffic forecast of one of the motorway sections. As can be observed in Fig. 3, model
uncertainty (dashed line) is relatively low and almost constant over time. However, once
input uncertainty is added (solid line) the confidence interval widens and clearly increases
over time. The second part of Fig. 3shows the expected value of traffic for a deterministic
forecast (dotted line), model uncertainty (solid line) and total uncertainty (dashed line). It
can clearly be observed that the deterministic simulation will underpredict the average
level of traffic flow.
10
As previously mentioned and shown in Eq. 11, the model is non-linear and stochastic.
Under these conditions, in general, the deterministic solution of a stochastic model will
offer a biased estimate of the expected traffic value. Nonetheless, the expected value of the
traffic forecast can be approximated by using the average of a set of stochastic simulations.
Applying this approach to all the motorway sections in the sample, we found that the
stochastic forecast for the year 2025 was on average 8.8% higher than the deterministic
forecast.
Table 5offers an order of magnitude of uncertainty for the same motorway section
featured in Fig. 3. The coefficient of variation for total uncertainty ranges from 0.03 in the
first forecasted year to 0.24 in the last. In the first few years, uncertainty is low and mainly
explained by model uncertainty. However, as time goes by, total uncertainty increases due
to lower precision in predicting the unknown values of exogenous variables.
10,000
12,000
14,000
16,000
18,000
20,000
22,000
24,000
26,000
2010 2012 2014 2016 2018 2020 2022 2024
Model and input uncertainty
Model uncertainty
Year
Traffic
70% Confidence interval for traffic forecast in a motorway section
10,000
12,000
14,000
16,000
18,000
20,000
22,000
24,000
26,000
2010 2012 2014 2016 2018 2020 2022 2024
Simulation with model and input uncertainty
Simulation with model uncertainty
Deterministic simulation
Expected value of traffic in a motorway section
Year
Fig. 3 Confidence intervals and expected traffic flow for one motorway section
Footnote 9 continued
differentiate between the three components of uncertainty, we have preferred to add the mixed terms to input
uncertainty. The reason for proceeding this way is that mixed terms are relatively unimportant compared
with the other two components. Therefore, we have computed input uncertainty as total uncertainty minus
model uncertainty.
10
The deterministic solution of a stochastic non-linear model will be biased. When variables are expressed
in logarithms, as in our case, the bias will be negative. However, with other functional forms the bias can
have a different sign.
Transportation (2012) 39:1–17 11
123
Author's personal copy
Uncertainty effects on forecasting forgone revenue
An issue on the political agenda of the Spanish government is to remove tolls on certain
motorways before the concession expires. In these cases, the government has had to
compensate the private motorway concessionaire for the revenue forgone up to the end of
the concession period. We selected one motorway section in the sample in order to
compute the effect of uncertainty on the revenue to be forgone. The selected section was 20
kilometres in length with an average traffic value of around 12800 vehicles per day. We
assumed that the concession period would expire in 2025.
The annual revenue was obtained by multiplying the predicted traffic by the average toll
paid by 365 days a year.
11
This value is computed for each forecasted year from 2009 to
2025 and for each of the 1000 random draws. Next, we worked out the results by calcu-
lating the Net Present Value (NPV) of the revenue to be forgone along these 17 years at a
discounting rate of 5%.
Finally, we analysed the empirical distribution of the NPV, which enabled us to cal-
culate the mean and the confidence intervals for different significance levels. For the
selected motorway section, the expected NPV of revenue is 123 million €. The minimum
and maximum values for the confidence interval at 70% significance are 107 million €and
138 million €; when we compute the interval at 95% the figures are 94 million €and 155
million €. In the first case, the difference between the two extremes is 29%, whereas in the
second it rises to 65%.
Figure 4presents the empirical distribution of the NPV.
Quantifying uncertainty provides evidence that using point estimates to assess invest-
ments or public policies can lead to errors in the decision-making process. In this example,
Table 5 Coefficient of variation
for total uncertainty and %
explained by model and input
CV Model (%) Input (%)
2009 0.032 80.3 19.7
2010 0.059 69.8 30.2
2011 0.082 59.3 40.7
2012 0.103 51.7 48.3
2013 0.121 47.2 52.8
2014 0.136 42.7 57.3
2015 0.150 38.9 61.1
2016 0.163 36.4 63.6
2017 0.174 34.8 65.2
2018 0.183 33.3 66.7
2019 0.191 31.8 68.2
2020 0.200 30.4 69.6
2021 0.209 29.1 70.9
2022 0.217 28.2 71.8
2023 0.224 27.1 72.9
2024 0.232 26.1 73.9
2025 0.242 25.3 74.7
11
To obtain the compensation to be paid to the concessionaire, we should deduct from the revenue to be
forgone any taxes or other costs related to toll operation.
12 Transportation (2012) 39:1–17
123
Author's personal copy
the negotiation process between government and concessionaire should include the
probabilities associated with the different forecasted revenue values.
Conclusions
This article contributes to the literature on transport demand forecasting in three different
ways: The proposal of a new methodology to account for capacity constraints in long
term forecasting, the analysis of the role played by the different components of uncertainty,
and the importance of using stochastic simulation techniques to avoid forecasting bias in
non-linear models.
Firstly, the proposed functional allows handling existing restrictions on the capacity in those
cases where it is not possible to jointly estimate the demand and supply side of the model. Our
approach makes it possible to account for capacity constraints in long term forecasting without
imposingan arbitrary functional form. This is achieved by specifyinga dynamic model in which
the speed of adjustment is related to the ratio between the actual traffic flow and the maximum
capacity of the motorway. Furthermore, with this functional form, demand elasticity is not
constant but depends on the degree of motorway use. As traffic increases, and approaches
maximum capacity, demand becomes less sensitive to changes in the explanatory variables.
With respect to uncertainty, this article outlines the importance of developing stochastic
simulations based on bootstrapping methodologies in order to obtain confidence intervals
for the forecast. The results confirm that in the first few years model uncertainty explains
most of the range of variation for the forecast traffic flow. However, as time goes by,
whereas model uncertainty remains almost constant, input uncertainty steadily increases so
that at the end of the forecasting period the last one accounts for almost 75% of total
variability. Based on previous experiences and on the results of this article, it can be
concluded that more effort must be made to improve model specification and to implement
the necessary mechanisms to avoid bias in forecasting. Nevertheless, input uncertainty has
proved to be the main factor for explaining uncertainty in the long run. Consequently, our
study shows that forecasting explanatory variables deserves special attention. So it would
be advisable to avoid introducing explanatory variables difficult to predict, although these
variables might increase the level of adjustment of the model.
0
20
40
60
80
100
120
140
160
60 80 100 120 140 160 180 200 220
Frequency
Revenue foregone (million )
Fig. 4 Distribution of the NPV of revenues foregone
Transportation (2012) 39:1–17 13
123
Author's personal copy
Finally, for non-linear models this article calls attention to the inadequacy of the
deterministic simulation to forecast future traffic volumes. When dealing with non-linear
models, the expected future traffic value can be approximated by averaging the different
realizations of the variable using stochastic simulations. As an illustration, this article
shows that the deterministic simulation at the end of the forecasting period underpredicts
expected traffic flow across all motorway sections in the sample by on average 9% with a
maximum difference of 12%.
Acknowledgments This work has benefit from a research grant from the Spanish Ministerio de Fomento-
CEDEX (PT2007-001-IAPP) as well as from the project ECO2009-12234. We would like also to thank the
researchers of the CEDEX project and two anonymous reviewers for their valuable comments.
Appendix 1. Generalisation to slags
The dynamic partial adjustment model can be easily generalised to account for slags as
follows:
The static equilibrium equation takes the standard form:
ln Y
it ¼aþbln Xit ð14Þ
whereas for the dynamic part we assume that the adjustment process is a weighted function
of slags:
Dln Yit ¼kit hw1ðln Y
it ln Yit1Þþw2ðln Y
it ln Yit2Þ
þwsðln Y
it ln YitsÞiþeit
X
s
i¼1
wi¼1
ð15Þ
That is, the correction of the disequilibrium between the actual value and the optimal or
desired value of the dependent variable at period tdepends on the disequilibrium in the
previous periods. Substituting the expression for Y
*
defined in (14) into Eq. 15, we obtain:
DYit ¼kit w1ðaiþbXit Yit1Þþw2ðaiþbXit Yit2ÞþwsðaiþbXit YitsÞ½þeit
¼kit aiþbXit w1Yit1w2Yit2w3Yit3ws1Yitsþ1Yits
½
þeit
¼kit aiþkit bXit kit w1Yit1kit w2Yit2kit ws1Yitsþ1kit Yitsþeit
Given that
kit ¼hYmax
iYit1
Ymax
i
¼hsit
and substituting:
Dln Yit ¼hsit aiþhsit bln Xit hsit w1ln Yit1hsit w2ln Yit2
hsit ws1ln Yitsþ1hsit ln Yitsþeit
That is:
Dln Yit
sit ¼haiþhbln Xit hw1ln Yit1hw2ln Yit2...
hws1ln Yitsþ1hln Yitsþeit
sit
14 Transportation (2012) 39:1–17
123
Author's personal copy
Appendix 2
See Table 6
Table 6 Definition of dummy variables and their estimated coefficients
Dummy variables Period Comment Coefficient
Z
1
2006–2008 It reflects the positive impact on traffic on
AP-1 motorway after removing the toll on
the connecting city ring-road.
0.1039
Z
2
–Z
4
2004–2008 They reflect the negative impact on traffic on
3 AP-2 motorway sections as a
consequence of the extension of an
alternative toll free road.
-0.1989
-0.2187
-0.2142
Z
5
–Z
6
1992 They account for the positive impact on the
2 AP-4 motorway sections, derived from
the Seville World Exhibition in 1992.
0.1828
0.1934
Z
7
2005–2008 It reflects the positive impact on traffic on
the 2 AP-4 motorway sections after
removing the toll in the final section of the
motorway.
0.1371
Z
8
–Z
9
2003–2008 They reflect the negative impact on traffic on
the 2 AP-68 motorway sections, derived
from quality improvements on the
alternative toll free road.
-0.0337
-0.0700
Z
10
–Z
13
1994–2008 They reflect the negative impact on traffic on
4 AP-7 motorway sections, derived from
the extension of an alternative tollway.
-0.0735
-0.0990
-0.1048
-0.0660
Z
14
2000–2008 It reflects the positive impact on traffic on 1
AP-7 motorway section after removing the
toll for the connecting city’s ring road.
0.0653
Z
15
1999–2000 It reflects the positive impact on traffic on 1
AP-7 motorway section, due to the toll
exemption because of construction works
on the motorway.
0.1347
Z
16
–Z
17
1998–2008 They reflect the negative impact on traffic on
2 AP-7 motorway sections, derived from
the opening of an alternative toll
motorway.
-0.1453
-0.1203
Z
18
1998–2008 It reflects the positive impact on traffic on 1
AP-7 section after removing the toll in the
city ring-road.
0.1082
Z
19
–Z
20
2006 They reflect the positive impact on traffic on
2 AP-7 motorway sections, due to the road
works on the alternative toll free road
0.0636
0.1468
Z
21
–Z
22
2007–2008 They reflect the negative impact on traffic on
2 AP-7 motorway sections as a
consequence of the extension of an
alternative toll motorway.
-0.1540
-0.1578
Transportation (2012) 39:1–17 15
123
Author's personal copy
Appendix 3. Demand elasticity with a capacity constraint
This appendix shows how demand elasticity depends on the level of traffic when a con-
straint on infrastructure capacity is in force.
Starting from Eq. 8in the main text:
Dln Yit
sit ¼ðhaiþhbln Xit hln Yit1Þþeit
sit
Defining betas and taking expected values, we get:
Dln Yit
sit ¼b0þb1ln Xit hln Yit1
where b0¼haiand b1¼hb
It can be rewritten as:
ln Yit ¼sit b0þsit b1ln Xit þð1sit hÞln Yit1
ln Yit ¼b
0it þb
1it ln Xit þc
it ln Yit1
where
b
kit ¼sit bk;c
it ¼ð1sit hÞ
For a fixed level of traffic flow, s
it
=s
0
, we have:
ln Yit ¼b
0þb
1ln Xit þcln Yit1
b
k¼s0bk;c¼ð1s0hÞ
Taking into account the dynamic structure of the model and following a recursive process
of substitution (this is, substituting ln Yit1¼b
0þb
1ln Xit1þcln Yit2), the elasticity
in period J will be:
Table 6 continued
Dummy variables Period Comment Coefficient
Z
23
2004–2008 They reflect the negative impact on traffic on
1 AP-7 motorway section, derived from
the opening of an alternative toll free
motorway.
-0.2321
Z
24
–Z
26
1993–2008 They reflect the negative impact on traffic on
3 AP-7 motorway sections, derived from
the opening of an alternative toll free
motorway.
-0.0821
-0.0533
-0.0555
Z
27
2004–2007 It reflects the negative impact on traffic on 1
A-8 motorway section, derived from the
opening of a connection with the
alternative toll free motorway.
-0.2125
Z
28
–Z
30
2003–2007 They reflect the negative impact on traffic on
3 A-8 motorway sections as a consequence
of the extension of an alternative toll free
motorway.
-0.0915
-0.1212
-0.1274
16 Transportation (2012) 39:1–17
123
Author's personal copy
eJ¼b
kð1cJþ1Þ
ð1cÞ
References
Bain, R.: Error and optimism bias in toll road traffic forecasts. Transportation 36, 469–482 (2009)
de Jong, G., Daly, A., Pieters, M., Miller, S., Plasmeijer, R., Hofman, F.: Uncertainty in traffic forecasts:
literature review and new results for The Netherlands. Transportation 34, 375–395 (2007)
Flyvbjerg, B.: Curbing optimism bias and strategic misrepresentation in planning: reference class fore-
casting in practice. Eur. Plan. Stud. 16, 3–21 (2008)
Flyvbjerg, B., Skamris, M., Buhl, S.: Inaccuracy in traffic forecasts. Transp. Rev. 26, 1–24 (2006)
Hamilton, J.D.: Time Series Analysis. Princeton University Press, Princeton (1994)
Matas, A., Raymond, J.L.: Demand elasticity on tolled motorways. J. Transp. Stat. 6, 91–108 (2003)
Riddington, G.: Long range air traffic forecasts for the UK: a critique. J. Transp. Econ. Policy 40(2),
297–314 (2006)
Vassallo, J., Baeza, M. Why traffic forecasts in PPP contracts are often overestimated? Research Paper,
Final Draft, EIB University Research Sponsorship Programme,European Investment Bank, Luxem-
bourg (2007)
Author Biographies
Anna Matas is professor of Applied Economics at the Universitat Auto
`noma de Barcelona. Her main
research interests are in the areas of transport economics, regulation and impact of infrastructure on location.
She has several articles published in academic journals in the field of transport economics. She has also
coauthored some books on topics related to the Spanish Economy.
Josep-Lluis Raymond is professor of Econometrics at the Universitat Autonoma de Barcelona. His main
fields of interest are econometrics, human capital, transport and infrastructures. He is author of several books
and articles published in academic journals. He has also done consultancy work.
Adriana Ruiz is a PhD candidate at the Universitat Autonoma de Barcelona. Her doctoral dissertation
focuses on analyzing several economic aspects of transport infrastructure, such as the impact of toll roads on
consumer welfare, and the effects of transport infrastructure investment on the location of economic
activities in Spain.
Transportation (2012) 39:1–17 17
123
Author's personal copy
