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no´mico: Economıa Cuantitativa, Universidad Auto´noma de Madrid, 28049 Madrid,. Spain (antonio.garcia@uam.es). Ara´nzazu de Juan and Pilar Poncela are ...
Journal of Transport Economics and Policy, Volume 40, Part 1, January 2006, pp. 45–67

Demand Forecast and Elasticities Estimation of Public Transport Antonio Garcı´ a-Ferrer, Marcos Bujosa, Ara´nzazu de Juan, and Pilar Poncela

Address for correspondence: Professor Antonio Garcı´ a-Ferrer, Dpto. de Ana´lisis Econo´mico: Economı´ a Cuantitativa, Universidad Auto´noma de Madrid, 28049 Madrid, Spain ([email protected]). Ara´nzazu de Juan and Pilar Poncela are also at Universidad Auto´noma de Madrid; Marcos Bujosa is at Universidad Complutense de Madrid. This research was financed by Comunidad Auto´noma de Madrid, ref. 06-0170/2000. The authors are grateful to Victoria Herna´ndez and the CTM for providing the data, to the Journal’s Editor and two anonymous referees for useful comments and suggestions. Any remaining errors are the authors’ responsibility.

Abstract This paper is related to the choice of alternative types of public transport modes and its incidence in the Madrid Metropolitan Area. When planning transport facilities, two conditions are needed: efficient estimation of the users’ response to changes in prices and in the characteristics of the services; and reliable predictions of demand. These two conditions are the main objectives of this paper. Given a monthly database, the authors address the first objective using a causal econometric model. As a baseline for forecasting comparisons, they also use new variants of univariate unobserved components models. Forecasting evaluation is based on a variety of accuracy measures to avoid misleading conclusions.

Date of receipt of final manuscript: June 2005 45

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1.0 Introduction Most studies on public transport demand consider travel as a derived demand associated not with its own consumption value, but rather to facilitate a complex and spatially varied set of activities such as work, recreation, shopping, and home life. Consequently, its analytical complexity is remarkable and most studies have adopted vastly simplified models that restrict their objectives to manageable dimensions. Following traditional literature (see, for example, Beesley and Kemp, 1987; and Small, 1992) many metropolitan transport planning agencies consider their decision setting as involving four choice dimensions: trip generation, trip distribution, modal choice, and trip assignment. Our interest in this paper is focused on one of these dimensions, namely the one related to the choice of the public transport mode and its impact on the Madrid Metropolitan Area (MMA). Our starting hypothesis is that when planning transport facilities, it is necessary to forecast how much they will be used. Also, to price them rationally and determine the best operating policies, it is necessary to know how users respond to changes in prices and service characteristics. Therefore, obtaining efficient predictions and estimating price elasticities are the main objectives of this paper. Public transport in the MMA encompasses four basic modes: the Metro System, the Municipal Bus Company (EMT), the RENFE suburban Train Service and the Interurban Buses. Because of data restrictions and service characteristics, the last two modes are not included in our analytical framework.1 The whole public transport system is managed by the Consorcio de Transportes de Madrid (CTM) that came into existence in 1986 to combine the efforts of public and private institutions related to public transport for the purpose of coordinating services, networks, and fares so as to offer consumers a higher-capacity, higher-quality service, with the aim of promoting public transport use and shifting demand away from private cars. Examples of similar policies in other European countries are well documented in Pucher and Kurth (1996). At the end of 2000, a total of 179 municipalities representing practically the entire population of Madrid region belonged to CTM. Despite decreasing population in recent years, the number of passengers using these services has grown from 951 million in 1986 to 1,549 million in 2001. This was not only a reflection of the recent investment and coordination effort, but also the result of a growing residential suburbanisation process in the Madrid region, rail and bus service improvement, and such economic incentives that travel cards entail for long distance travellers. 1

At the end of 2000, the Metro/Bus share represents 71 per cent of total passengers.

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Although all modes have been growing steadily throughout the past few years, the Metro system has been undergoing especially significant growth, having recorded a 9.6 per cent rise in demand in 1999, and a 9.3 per cent in 2000. As we will see later, this growth has been primarily induced by an important increase of the Metro supply services on the basis of the number of lines, stations, and rolling stock. Finally, as happens in many European cities, public transport fares in the Madrid region are based both on single-mode as well as multiple-mode tickets, the principal one being CTM’s travel card.2 In particular, the introduction of cheaper season tickets can provide a powerful incentive to shift transport modes, and is not without theoretical support (see, for instance, Carbajo, 1988; and FitzRoy and Smith, 1999). On the other hand, the evaluation of the consequences for revenue derived from changes on the fare system will depend upon how users are distributed among different type of tickets. Although the empirical evidence on this issue is mixed, most results indicate that this type of service improvement and integrated fare system will require substantial government subsidies. (See, among others, White, 1981; Pucher and Kurth, 1996; and Matas, 2002.) The paper is organised as follows. Section 2 presents the characteristics of the database and analyses the presence of many outliers that create considerable estimation instability and complicate the subsequent forecasting exercise. Section 3 describes the theoretical framework and presents the estimation results. Own-price and cross-price elasticities among different tickets are presented and discussed. Section 4 analyses the predictive performance of alternative models. Finally, Section 5 concludes.

2.0 The Database Initially, we use CTM monthly data, from January 1987 to December 2000 for the main public transport variables: Single Trip Metro Tickets (SMT ), 10-Trip Metro Tickets (10MT ), Single Trip Bus Tickets (SBT ), 10-Trip Bus Tickets (10BT ), Regular Travel Card (TC ), and Junior Travel Card (JTC ). Precise definitions of the variables are given in Table 1, and their plots (in logs) are shown in Figure 1 where both non-stationarity as well as strong seasonality is clearly evident. The aberrant observations in both SBT and 10BT in the early months of 1992 correspond to a severe general strike on the bus network that took place during that period. 2

A detailed explanation of the characteristics of the current public transport fares in Madrid region can be seen in Matas (2002).

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Table 1 Definition of the main variables Name

SMT 10MT SBT 10BT TC JTC

Definition

Single Metro Tickets 10 Metro Tickets Single Bus Tickets 10 Bus Tickets Regular Travel Card Junior Travel Card

Sample

Estimation

Forecasting

Period

Size

Period

Size

Period

Size

1987:1–2001:12 1987:1–2001:12 1988:1–2001:12 1988:1–2001:12 1987:1–2001:12 1988:2–2001:12

180 180 168 168 180 167

1987:1–1999:12 1987:1–1999:12 1988:1–1999:12 1988:1–1999:12 1991:1–1999:12 1991:1–1999:12

156 156 144 144 108 108

2000:1–2001:12 2000:1–2001:12 2000:1–2001:12 2000:1–2001:12 2000:1–2001:12 2000:1–2001:12

24 24 24 24 24 24

The introduction of travel cards in 1987 caused a drastic change in the trends of use for the remaining tickets. This process has been the logical consequence of a price policy that penalises the users of single- and 10trip tickets through large price increases, while holding the price of travel cards constant from their introduction in 1987 until 1992. During this period, for instance, single ticket prices increased by 130 per cent and, although this price trend has been reversed, regular users found multimodal travel cards much more advantageous when purchasing tickets. As a result, the demand shares have shifted dramatically during the last 15 years. Single ticket use fell to just over 4 per cent for both Metro and EMT, while the market share of 10-ride tickets has dropped significantly among bus users and has almost remained unchanged in the case of Metro.3 Finally, the market share for travel cards in 2001 was over 60 per cent, which was about the figure that the CTM had in mind when the integrated system was established in 1986.

3.0 Methodologies and Estimation Results The plots of the main variables in Figure 1 indicate that the statistical characteristics of such series changed considerably over the sample interval, so that the series can be considered nonstationary in a statistical sense. All series exhibit clear upward or downward trends, together with pronounced annual periodicity. The trend behaviour is a classic example of a local mean value changing markedly over time. The nature of the seasonality varies 3

This different behaviour can be explained by the fact that subway transfers are free while bus transfers are penalised in monetary terms.

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Figure 1 Graph of the logs of the main variables

over the six series but, in general, there are signs of steady growth in most of them, indicating non-stationarity in the variance of the seasonality about the trend. These various kinds of nonstationarity are indicative of changes in the underlying statistical properties of the data. Therefore, we decided to use 49

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two alternative statistical approaches capable of characterising the nonstationary features in an acceptable manner. One is the well known Dynamic Transfer Function Causal Model obtained using the intervention IARIMA model developed by Box and Tiao (1975) as a starting model. The second alternative is the Dynamic Harmonic Regression (DHR) model, developed by Young et al. (1999). For the latter, however, both the identification and estimation stages will be carried out on the Bujosa–Garcı´ a-Ferrer (BGF) algorithm implemented in Bujosa et al. (2005). Although the previous approaches are, basically, univariate alternatives, they also allow for the possibility of including exogenous inputs associated with intervention effects (such as strikes, working days, Easter effects, and so on) as well as price and service changes in the system. However, their univariate frameworks imply that the time processes in each sub-market of the system are independently analysed. If independence is not a good assumption, demands in those sub-markets are correlated and their time processes should be modelled jointly within a multivariate set-up.4 3.1 The Dynamic Harmonic Regression model The DHR model developed by Young et al. (1999) belongs to the Unobserved Component (UC) type and is formulated within the state space framework. The model is based on a spectral approach under the hypothesis that the observed time series can be decomposed into several DHR components whose variances are concentrated around certain frequencies. This hypothesis is appropriate if the observed time series has well defined spectral peaks, which implies that its variance is distributed around narrow frequency bands. Basically, the method attempts to: (1) identify the spectral peaks, (2) assign a DHR component to each spectral peak, (3) optimise the hyper-parameters that control the spectral fit of each component to its corresponding spectral peak, and (4) estimate the DHR components using the Kalman Filter (KF) and the Fixed Interval Smoothing (FIS) algorithms. In the univariate case, the DHR model can be written as a special case of the univariate UC model, which has the general form: y t ¼ Tt þ St þ e t ;

4

t ¼ 0; 1; 2; . . . ;

ð1Þ

We thank an anonymous referee for pointing out this issue. In previous versions of this paper we have estimated some multivariate models to check this conjecture. Nevertheless, due to the superimposition of price changes, strikes and the various kinds of outliers, multicollinearity effects and the loss of degrees of freedom in the multivariate models, the estimates from those models are less precise than those from the single equation models.

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where yt is the observed time series, Tt is the trend or low-frequency component, St is the seasonal component, and et is an irregular component normally distributed with zero mean value and variance s2e . Equation (1) is appropriate for dealing with economic data exhibiting pronounced trend and seasonality as is the case with the monthly variables used in this paper. When set in state-space form, each component is modelled in a manner that allows yt to be represented as a set of discretetime equations that are the basis for recursive state-space estimation and p forecasting. Tt and St consist of a number of DHR components, st j , with pj the general form st ¼ ajt cosðoj tÞ þ bjt sinðoj tÞ, where pj and oj ¼ 1=pj are the period and the frequency associated with each jth DHR component 1 respectively. Tt is the zero frequency PR pj term ðTt  st ¼ a0t Þ, while the seasonal component is St ¼ j ¼ 1 st , being j ¼ 1; . . . ; R the seasonal frequencies. Hence, the complete DHR model is yt ¼

R X j ¼0

p

st j þ et ¼

R X fajt cosðoj tÞ þ bjt sinðojt Þg þ et :

ð2Þ

j ¼0 p

The oscillations of each DHR component, st j , are modulated by fajt g and fbjt g, which are AR(1) or AR(2) stochastic process with at least one unit root; therefore, nonstationarity is allowed in the various components. The variance of the innovations of the AR stochastic process ðs20 ; s21 ; . . . ; s2R Þ5 and s2e are the unknown hyper-parameters of the model. These hyper-parameters are estimated in the frequency domain. The noise variance ratios ðNVRj ¼ s2j =s2e Þ work as smoothing parameters; the smaller the NVR0 is, the closer to a linear deterministic trend the estimated trend is. In the limit, when the NVR0 ¼ 0 the estimated trend is linear. In the case of seasonal components, the smaller the NVRj is, the smoother are the changes in the amplitude of the oscillations of p the jth DHR component st j . The presence of important outliers, however, can easily affect both the estimation and the posterior identification and estimation stages of the DHR models. To avoid these effects, we have implemented the following iterative process: 1. 2.

5

Initial identification and estimation of DHR models is proposed. Using the KF and the FIS algorithms on the original series, we treated outliers as missing values and use variance intervention (Young and Ng, 1989) to handle level changes due to large variations in fares or the introduction of new types of tickets.

Each s2j is common in the innovations of fajt g and fbjt g.

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Table 2 NVRs estimates of the main variables NVR Series

T

S 12

S6

S4

S3

S 2:4

S2

s2e

SMT SBT 10MT 10BT TC JTC

0.091 0.001 0.029 0.001 0.008 0.050

0.359 0.023 0.948 0.010 0.041 3.779

0.117 0.013 0.434 0.007 0.006 1.818

0.109 0.024 0.302 0.026 0.081 1.199

0.343 0.017 0.357 0.107 0.023 0.278

0.071 0.013 0.490 0.034 0.026 0.026

0.052 0.011 0.301 0.022 0.006 0.012

0.00005 0.00038 0.00004 0.00036 0.00021 0.00004

3.

4.

New series were reconstructed where each outlier was substituted by its estimated value and the level shifts are accounted for through variance intervention. Finally, using the reconstructed series, return to step 1.

In this paper we use the BGF algorithm. This algorithm is an extension of Young’s estimation procedure by linear methods. This extension also provides an automatic identification of the complete DHR model.6 In all series, robust results were obtained after three iterations and the NVR estimation results, shown in Table 2, correspond to the third iteration. Also, in all cases, the identification stage suggests an IRW trend component (that is, a0t  Ið2Þ) and an RW model (that is, fajt g and fbjt g  Ið1Þ) for the seasonal component of period 12 and its harmonics (6, 4, 3, 2.4 and 2). In all cases, the estimated NVRT for the trends are different from zero, confirming the absence of deterministic linear trends but suggesting (apart from the breaks) very smooth long-term behaviour. On the other hand, the estimated seasonal components indicate a clear changing seasonality pattern confirming the inappropriateness of deterministic seasonal schemes for this dataset. 3.2 The dynamic transfer function causal model Computation of price elasticities has been the aim of a large number of transport demand studies. Good examples, for several countries including Spain, can be seen in Goodwin (1992), de Rus (1990), Dargay and Hanly (1999), Oum et al. (1992) and Matas (2002), among others. Although in general low-price elasticity is found in many applications, large differences in scope, data characteristics, and econometric methodologies make these 6

Detailed technical description of the BGF algorithm is available from the authors.

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results discouraging for the prospect of achieving optimal pricing of public transport. When using annual or quarterly data, demand equations can be based on causal models where the demand for transport services can be assumed to depend on the attributes of each mode (monetary costs and quality of service), the competing modes of transport, and on certain socio-economic and demographic variables (such as income, employment, and population). This approach, however, is not without problems when interpreting estimation results due to a large number of potential econometric problems. Issues of dynamic specification of the model, a small sample size, a large number of regressors (small number of degrees of freedom), and extreme multicollinearity, should be seriously considered before interpreting estimates as current elasticities. The multicollinearity problem is particularly harmful in this context, especially when models are specified in levels.7 It is well known that multicollinearity may affect estimated standard errors and signs of the coefficients, providing misleading results. Far from being a good solution, the often-used alternative of eliminating statistically insignificant regressors may be even worse in terms of interpreting results. Let us emphasise, however, that this is not a model problem (whether the model is causal or not) but a data problem that can equally affect other modelling alternatives. When using monthly data, as we did in this paper, the use of causal models is restricted by data availability. Lacking monthly income, employment, and even population data for the Madrid region, our causal model was based only on fare changes and service quality changes plus the corresponding deterministic intervention effects and stochastic seasonality variation. In the case of independent sub-markets for each type of tickets, our dynamic causal model can be written as: fðLÞðLs Þrd rD yt ¼

k X

vi ðLÞrd rD xit þ

i¼1

þ yðLÞðLs Þat ;

m X

@j ðLÞrd rD zjt

j ¼1

ð3Þ

where xit are intervention variables, vi ðLÞ includes the dynamic model for the ith intervention variable, yðLÞ and ðLs Þ are the regular and seasonal moving average operators, and fðLÞ and ðLs Þ are the regular and seasonal autoregressive operators. The process at is assumed to be white noise, and 7

This is acknowledged by Matas (2002) when estimating demands for bus and metro trips in the MMA using annual data from 1979 to 2001. All coefficients within the correlation matrix of the main economic indicators (GDP, population, suburbanisation and employment) are in the range of 0.895 and 0.984, indicative of a serious multicollinearity problem.

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rD and rd allow for seasonal and non-seasonal differencing. The usual stationarity and invertibility conditions apply for the AR and MA operators. As indicated by Pen˜a (2001), most outlier specifications in the literature (additive, innovational, level shifts, and transitory changes) are particular cases of equation (3) under proper parametrisations of the polynomial operator vi ðLÞ. On the other hand, zjt is a vector of exogenous variables that includes price and service changes of different tickets, and @j ðLÞ represents the dynamic model for the jth exogenous variable. This model’s specification allows the estimation of mean own-price elasticities, as well as the estimation of cross-elasticities among different tickets.

Table 3 Definitions of intervention variables Name

Definition

Facts

Affected variables

EASTER

1 in Easter months, 0 otherwise

Easter effects

DAYS

Trading days per month

Trading day effects

ALL

MAR 89

1 in 1989:3, 0 otherwise

Strike in Bus mode and large increase in single ticket prices

SMT, 10MT

MAR 90

1 in 1990:3, 0 otherwise

Strike in Bus mode

SMT, SBT, 10BT

APR 90

1 in 1990:4, 0 otherwise

Strike in Bus mode and large increase in single ticket prices

10MT, 10BT

JAN 91

1 in 1991:1, 0 otherwise

Strike in Metro mode and large increase in single ticket prices

SMT

APR 91

1 in 1991:4, 0 otherwise

Strike in Metro mode

SMT

FEB 92

1 in 1992:2, 0 otherwise

Strike in Bus mode

SMT, SBT, 10BT

MAR 92

1 in 1992:3, 0 otherwise

Strike in Bus mode

SBT, 10BT

APR 92

1 in 1992:4, 0 otherwise

Consequences of strikes in buses

SBT

JAN 98

1 in 1998:1, 0 otherwise

Introduction of the METROBUS ticket

SMT, SBT

FEB 93

1 in 1993:2, 0 otherwise

See footnote 9

JTC

APR 93

1 in 1993:4, 0 otherwise

Service Interruption in Metro Temporary closing of certain lines

JTC

MAR 94

1 in 1994:3, 0 otherwise

?

TC, JTC

JUL 95

1 in 1995:7, 0 otherwise

?

JTC

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3.2.1 Assessing the effects of the intervention variables Detailed definitions of the different intervention variables are included in Table 3. Estimation results by exact maximum likelihood are shown in Tables 4 and 5, but we will discuss each model separately. This leads us to a first issue related to the choice of the estimation period for each variable. Although most of the database covers the period 1987.1–2001.12, there are some problems that preclude using this whole period as a generalised database. First, monthly data on SBT and 10BT are not available before 1988. Second, the TCs from 1987 to 1990 include information that was inconsistent with the posterior 1991–2001 data. Due to the gradual introduction of different travel card options from 1987 to 1990, the data generation process cannot be considered identical before and after 1990. Third, in the case of JTC, the early period 1987–1990 data base suffers from identical problems and, consequently, has been discarded. As regards the Metro estimates (Table 4), the following results are worth mentioning: 1.

2.

3.

8

The effect of trading days is positive for both types of tickets. A 1 per cent increase in the number of working days will have a marginal increase of 1.10 per cent in the case of SMT and a 1.26 per cent increase for 10MT. On the other hand, the Eastern effect is negative for the two Metro variables, ranging from 7:3 per cent for SMT to 11:7 per cent for the 10MT variable. The remaining intervention variables are related either to strikes or to large changes in public transport fares. Positive coefficients in MAR89, APR91, and FEB92 are explained by strikes on the competing mode (bus).8 The negative effect in JAN91 is explained by a Metro strike, while the mixed effects in MAR90 coefficients correspond, respectively, to a bus strike (þ0:181), and to a large price increase in SMT the following month (0:376 and 0:102 coefficients), which explain why this effect does not show up in the 10MT equation. The JAN98 variable corresponds to the introduction of a new Metrobus ticket that has negatively affected single ticket sales permanently after its introduction. The aggregate effect of this intervention (that is, the sum of the three negative coefficients) results in a 27 per cent drop in single tickets sales. Finally, the positive sign of APR90 for 10MT is explained by a large price increase (38 per cent) of single tickets.

The negative signs of the MAR89 coefficient in the case of 10MT must be interpreted with care, given the contemporaneous coincidence of several factors: an EMT’s strike, an 8.3 per cent price increase of single tickets, and a simultaneous 10.8 per cent price increase of 10BT.

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Table 4 Estimated Causal models for the Metro and Bus variables (in logs) rr12 SMT

rr12 10MT

rr12 SBT 

rr12 10BT

Constant



0:0004 ð0:014Þ



0:0017 ð0:0027Þ

0:00004 ð0:0015Þ

DAYS

0:0110 ð0:0020Þ

0:0126 ð0:0031Þ

0:0074 ð0:0026Þ

0:0164 ð0:0021Þ

EASTER

0:0728 ð0:0099Þ

0:1174 ð0:0159Þ

0:0185 ð0:0152Þ

0:1065 ð0:0141Þ

MAR89

0:0496 ð0:0340Þ

0:0820 ð0:0497Þ

MAR90

0:1815 0:3767B 0:1024B2 ð0:0352Þ ð0:0389Þ ð0:0306Þ

0:6110 0:3281B ð0:0462Þ ð0:0435Þ

0:0003 ð0:0002Þ

0:6597 0:2083B ð0:0401Þ ð0:0391Þ

0:1844 ð0:0470Þ

APR90 JAN91

0:4401 ð0:0260Þ

APR91

0:0492 ð0:0340Þ

FEB92

0:1369 ð0:0301Þ

JAN98

0:1642 0:0353B 0:0703B ð0:0135Þ ð0:0314Þ ð0:0335Þ

PSM

1:0325 ð0:0874Þ

0:6335 ð0:1424Þ

P10M

0:617 ð0:3668Þ

2:1687 ð0:6358Þ

ð1:5972 0:3435B 0:1261B2 Þ2:0599 0:3987B ð0:0448Þ ð0:0542Þ ð0:0448Þ ð0:0384Þ ð0:0405Þ 0:1334 0:1117B ð0:0431Þ ð0:0429Þ

PSB

1:0607 ð0:1172Þ

0:2207 ð0:0679Þ

P10B

0:3364 ð0:2809Þ

0:5184 ð0:1833Þ

0:7360 ð0:4575Þ

0:9074 ð0:2515Þ

0:2546 ð0:0932Þ

0:4088 ð0:0768Þ

PTC

0:4166 ð0:4635Þ

f1

0:2611 ð0:0436Þ

y1

0:3350 ð0:0913Þ

2:3594 ð0:8392Þ

0:5831 ð0:0775Þ

y2 12

0:2955 ð0:0797Þ

0:5896 ð0:0656Þ

0:5856 ð0:0652Þ

0:2596 ð0:0809Þ

24 sa LBQ(12, 24, 36) T

0:7986 ð0:0871Þ

0.034922

0.064547

0.0459999

0.0412699

(5.0, 12.4, 22.1) 156

(7.9, 14.2, 26.8) 156

(9.7, 15.3, 29.6) 144

(7.8, 19.7, 36.2) 144

Standard errors in parenthesis. ( ) not significant coefficient at 5 per cent. r ¼ ð1  BÞ: regular differences; r12 ¼ ð1  B12 Þ: seasonal differences; B: lag operator, LBQ: Ljung-Box Q statistics; sa ¼ residual standard error; T ¼ estimation size.

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As regards the EMT estimates the following results are worth mentioning (see Table 4). 1.

2. 3.

The effect of trading days is always positive and statistically significant in all cases. However, the quantitative impact is greater in the 10-trip variable. Easter holidays do not seem to affect single ticket sales (negative, nonsignificant coefficient) but strongly affect (11 per cent) 10-trip tickets. As happens in Metro, the remaining variables are associated with strikes or fare changes. However, the quantitative impacts of the MAR90 strike, for both SBT and 10BT are much larger than in the similar Metro tickets. The long bus network strike seen in the FEB92 variable shows temporary fluctuations of 80.2, 28.9, and 12.2 per cent in February, March and April 1992, respectively, with respect to the previous month in the same year; and an even large change (82.7 per cent in February and 36.6 per cent in March) in the case of 10BT. These figures show that fewer bus ticket purchases were only compensated by a mild increase in single Metro tickets purchases (13.7 per cent) during that period, as a result of less Metro coverage at that time. Finally, the JAN98 variable corresponds to the introduction of a new Metrobus ticket that negatively affected single ticket sales permanently after its introduction.

Finally, for travel cards estimates (Table 5), we should mention the following: 1. 2.

3.

4.

9

The estimation period is shorter (108 observations), given the anomalies mentioned earlier at the beginning of both series. The effect of trading days is almost negligible for TC and nonsignificant for JTC. This is a logical result, given the (monthly card) characteristics of these tickets. The EASTER effect is statistically significant and has the expected negative signs for both variables. The quantitative impacts (6:3 and 7:5 per cent) are similar to the ones observed in single- and 10-trip tickets. The remaining intervention variables are additive outliers that had not appeared earlier. The negative sign of APR93 is associated to service interruption and temporary closing of certain lines. The positive sign of FEB93 in the JTC equation is artificially caused by the use of the rr12 difference operator and the need to estimate a seasonal

This implies losing 25 observations from our original sample that started in January 1991. Therefore, our first residual corresponds to February 1993 and shows a large increase in comparison with the previous year due to the large February 1992 strike.

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Table 5 Estimated Causal models for TC and JTC variables (in logs)

Constant DAYS EASTER

r12 TC

rr12 JTC

0:0517 ð0:0050Þ 0:0045 ð0:0019Þ 0:0625 ð0:0069Þ

0:0002 ð0:0004Þ 0:0017 ð0:0011Þ 0:0748 ð0:0046Þ 0:0464 ð0:0178Þ 0:0723 ð0:0177Þ

FEB 93 APR 93 MAR 94

0:0823 ð0:0179Þ

JUL 95 PSM P10M PTC

0:2213 ð0:2375Þ 0:1800 ð0:3075Þ 0:0132 ð0:2308Þ

0:5557 ð0:3165Þ

PJTC MRLi

0:2517B4 0:418B5 ð0:1344Þ ð0:1665Þ 1 þ 0:9897B ð0:0072Þ

12 f1 12 sa LBQ(12, 24, 36) T

0:1800B4 0:2087B5 ð0:0861Þ ð0:0977Þ

0:9053 ð0:0549Þ

y2 y3

0:0615 ð0:0082Þ 0:4075 ð0:3257Þ 0:3498 ð0:3748Þ

0:4318 ð0:0956Þ 0:7009 ð0:0804Þ 0:3200 ð0:1064Þ 0:5187 ð0:0475Þ 0.020640 (6.1, 11.0, 16.9) 108

0:7942 ð0:1063Þ

0:3264 ð0:0781Þ 0.027637 (10.7, 18.0, 24.4) 108

Standard errors in parenthesis. ( ) not significant coefficient at 5 per cent. r ¼ ð1  BÞ: regular differences; r12 ¼ ð1  B12 Þ: seasonal differences; B: lag operator, LBQ: Ljung-Box Q statistics; sa ¼ residual standard error; T ¼ estimation size.

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autoregressive coefficient.9 The CTM does not have any reasonable explanation for the positive coefficients associated with MAR94 and JUL95. Note that Ljung-Box statistics at 12, 24 and 36 lags, in Tables 4 and 5 show no signs of serial correlation in the estimated models. 3.3 Analysis of elasticities Before going into the estimation details, some comments regarding the components of the zjt vector are needed. First, rates of growth of real prices for all tickets are included in zjt . Price changes are identical for SMT and SBT, very similar for 10BT and 10MT, and there is also a high correlation (r ¼ 0:81) between price changes in 10MT and TC. This severe collinearity will have important consequences in assessing posterior robustness on elasticities estimation. Second, service quality can be measured using several indicators: route length, number of stations, number of trains or buses, vehicle-kilometres, and so on. Since all indicators are highly collinear, we have decided to use route length (MRL) as the main indicator of service quality. Estimated elasticities for bus and Metro tickets are shown in Table 4 (rows 12 to 16). Own-price elasticities appear in the main diagonal of the table, while cross-elasticities are the off-diagonal figures. A common finding in the literature (see, for example, de Rus, 1990; Hensher, 1998; and Matas, 2002) is the decrease of price elasticity as we move from single tickets to travel cards. This is only partially true in our case due the high elasticity shown by 10MT (2:17). In general, users are highly sensitive with respect to singleand 10-ride fares with elasticities values ranging from 0:52 to 2:17. However, while elasticities estimation in the cases of SMT, SBT, and 10BT proved to be robust to alternative model specifications, this is not the case for 10MT as a consequence of the high multicollinearity between price changes of 10MT and TC. As a matter of fact, when PTC is removed from the equation, the own-price elasticity for 10MT goes to 1:07. This is a well known effect of multicollinearity that Leamer (1978) coined as ‘weak evidence’ when interpreting estimated coefficients. In any case, our results are in line with those obtained by de Rus (1990) for other intermediate size Spanish cities, and Matas (2002) using 1979–2001 annual data for the Madrid region. In spite of site and time periodicity differences, de Rus (1990) obtained a price elasticity for single tickets of between 0:73 and 1:16, and for multi-ride fares between 0:27 and 2:25. Matas (2002) obtained a 1:48 for single fares and 1:07 for 10-ride fares. As regards as cross-elasticities, we did not find any evidence of cross-effects in the cases of single tickets and travel cards since all the 59

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cross-elasticities coefficients are not statistically significant. The only significant evidence is found in the case of the two 10-trip tickets. For 10MT, positive cross-effects are found with single tickets (0:62) and with travel cards (2:36).10 Similar results are found for 10BT although the size of the cross-effects is smaller than in the Metro case: 0:22 for single tickets and 0:90 for travel cards. In general, Metro users are more sensitive to price variations than EMT users as a consequence of the shorter coverage of the Metro network and the availability of alternative bus routes. Also, Metro users seem only sensitive to changes in the Metro services and place a high value on variables such as trip time and service reliability that cannot be equalled by the EMT network. Finally, TC and JTC (Table 5, rows 9 to 12) have not been affected either by their own price increases or by the price increases of competing tickets. A similar result is also found by Matas (2002), and both results are hardly surprising given the price policy followed by the CTM. When reviewing other countries’ experience, Matas (2002) notes that when the price of travel cards is raised, travel card holders do not reduce their trip frequency unless the price rises above a critical threshold at which the user switches to single- or 10-trip tickets at a much lower trip rate. Although part of the available empirical evidence tends to support very low values for moderate travel cards’ price increases, there is still a wide range of variation among empirical studies (Preston, 1998). Given their different historical periods, data periodicity, model specifications, and different definitions of the main variables, inter-country comparison of empirical results has to be made with extreme care. In any case, the estimated own and cross-price elasticities for the MMA provide some interesting hints for policy. Given the high elasticity values for single tickets, further price increases will probably affect demand negatively. Similar comments apply for the 10-ride tickets given its price differential with respect to travel cards. For the latter, however, the trifling sensitivity found to historical price increases indicates the possibility of moderate price increases without an important effect on travel demand. Therefore, there seems to be room for an alternative (more efficient) pricing policy with positive effects on demand, while minimising the negative effect on revenues.

10

Again, this last result should be interpreted with care given the collinearity problem between the parameters estimates of the 10MT equation.

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4.0 Forecasting There is a large body of literature on the subject of travel demand forecasts. (See, for example, Gillen, 1977; Lave, 1994; Baumgartner, 1995; Slavin, 1996; and Lythgee and Wardman, 2002, among others.) The level of detail and the forecasting horizons change considerably among different studies as a function of the travel demand model, the characteristics of the data and the number of input variables. Broadly speaking, these external influences can be separated into two components: those related to demographic and economic changes as well as other external variables, and those directly related to the public transport system. When long-run planning is the goal, both components need to be predicted and both involve economic assumptions about their future behaviour. However, when using monthly data and when (as in our case) short- and mediumterm forecasts are the objective, we can assume that the first set of factors is reasonably incorporated in the stochastic long-term trends. As we will show later, it is the second set of factors (basically, abrupt changes in travel supply and changing seasonality patterns) that is responsible for large variations in public travel demand in the MMA and the presence of forecast errors in the vicinity of certain months.11 Twelve periods’ ahead forecasts for the six variables in this paper have been obtained for 2000 and 2001. In the first case, estimation ends in December 1999, while in the second the estimation period is expanded one more year. The forecasting for these periods was particularly difficult given the large increase in Metro services over this time period, particularly during 1999. This meant a mixture of effects that influenced not only total demand but also temporary passenger shifts from Metro to EMT. This effect is very remarkable in the case of SBT. After four consecutive years of negative growth it shows a 3.7 per cent growth in 2000, only to return to its usual negative path in 2001 (1:9 per cent). Besides the obvious turning points experienced by SBT (both in 2000 and 2001) and 10MT (in 2001), the remaining series show a strong acceleration in their growth rates in 2000 only to return to their mean growth values the following year. Consequently, we should expect large forecast errors during those months when these events have occurred. As we will see later, these events have different effects on the alternative forecast accuracy measures used in this paper. 11

Among the variables associated with service changes, the Metro route length is primarily responsible for the demand changes experienced in the MMA, particularly in 2000. This variable remains unchanged from 1988 to 1993 and it changes very little (4 per cent) from 1994 to 1997. However, it shows a huge increase (43 per cent) during the last months of 1998 and throughout 1999, and zero growth during 2000 and 2001.

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Table 6 Mean absolute value (MAPE), root mean squared error (RMSE) of 12 steps-ahead forecasts for Metro and Bus tickets and travel cards for alternative models. Forecasting periods: 2000 and 2001 Metro Variables SMT

10MT

Bus Variables SBT

10BT

Travel Card Variables TC

JTC

Causal DHR Causal DHR Causal DHR Causal DHR Causal DHR Causal DHR 2000 MAPE 0.020 0.028 0.033 0.040 0.069 0.053 0.023 0.035 0.025 0.048 0.028 0.076 RMSE 0.026 0.034 0.044 0.046 0.073 0.049 0.031 0.042 0.034 0.053 0.039 0.150 2001 MAPE 0.159 0.059 0.052 0.039 0.044 0.031 0.030 0.036 0.021 0.034 0.019 0.063 RMSE 0.174 0.069 0.056 0.058 0.052 0.034 0.036 0.042 0.025 0.040 0.026 0.101

Forecast results for 2000 and 2001 are presented in Tables 6 and 7, where four accuracy criteria are used. Aggregate root mean square error (RMSE) and mean absolute prediction error (MAPE) are shown in Table 6, while annual percentage errors (APE) and forecasted annual growth Table 7 Annual percentage errors (APE), forecast annual growth rates (FGR), observed annual growth rates (OGR) of 12 steps-ahead forecasts for Metro, Bus tickets and travel cards for alternative models. Forecasting periods: 2000 and 2001 Metro Variables SMT

10MT

Bus Variables SBT

10BT

Travel Card Variables TC

JTC

Causal DHR Causal DHR Causal DHR Causal DHR Causal DHR Causal DHR 2000 APE 0.96 0.20 FGR 12.1 13 OGR 13.2

1.38 0.19 5.85 2.41 0.93 2.38 7.10 5.44 2.32 1.24 3.37 4.79 5.64 3.75 2.47

0.66 4.60 0.94 1.99 9.32 4.23 4.56 5.56 9.26 3.65

2001 APE 16.19 5.30 FGR 21.6 10.2 OGR 4.65

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5.31 3.51 5.05 3.26 0.24

4.40 0.18 2.46 0.18 1.86

2.26 0.52 0.43 2.85 0.009 1.13 1.17 2.85 5.58 9.06 2.87 3.98 3.36 6.04 2.88

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rates (FGR) are shown in Table 7. The adoption of a variety of error measures has been repeatedly stressed in the literature (see, for example, among others Armstrong and Collopy, 1992; and Fildes, 1992) to avoid the evaluation of different forecasting techniques depending upon the choice of a particular accuracy measure. Conflicts among them (although not desirable) only indicate different goals of alternative prediction exercises. However, for longer forecasting horizons (beyond the usual one step-ahead period) the use of RMSE or MAPE may be not only inappropriate but misleading (Garcı´ a-Ferrer and Queralt, 1997). In this case, we contend that FGR and APE become more relevant criteria. Table 6 presents the results for the RMSE and MAE, while the APE and FGR are shown in Table 7. As might be expected, no model dominates the other under all the accuracy criteria and forecasting horizons. However, the following tentative conclusions can be drawn from these tables. (1) As regards the Metro variables, the 2000 forecasting results for both models are excellent. The MAPE and RMSE figures are somehow inflated as a consequence of large errors around the summer months, but the other two criteria indicate an excellent forecasting performance. As regards the APE, for instance, the largest annual percentage error corresponds to 10MT (1.38 per cent) while this figure goes below 0.2 per cent in the case of the DHR model for both types of tickets. Also, predicted annual growth rates are very close to the observed ones. Predictive results for 2001 are considerably worse, in particular in the case of the causal model for single tickets. During this forecasting period, the DHR compares very favourably with its competitor regardless of the accuracy criteria used, in spite of a large forecasting error in November (16.1 per cent) in the case of 10MT. For this variable, however, no model is capable of detecting the turning point that takes place in 2001, although the DHR behaves slightly better than its competitor. (2) Similar comments apply to the case of the Bus variables, except that the causal model’s forecast deterioration for 2001 does not take place. For SBT, the DHR model outperforms the causal model under all criteria. Also, it correctly identifies the turning point that takes place in SBT in 2000 after four consecutive years of negative growth. Also, the DHR almost successfully predicts the posterior SBT turning point that happens again in 2001, confirming its potential in dealing with turning point situations (Garcı´ a-Ferrer and Queralt, 1998). For 2000, the largest APE is 2:41 per cent in the case of the DHR model for SBT. As happens with the Metro variables, the 2001 forecasting performance of the DHR model is excellent and the largest APE is 0.52 per cent for 10BT. 63

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(3) As regards travel cards, the predictive performance of the causal model is excellent, outperforming the DHR model for both TC and JTC under all criteria and forecasting horizons. The poor result of the DHR model in the JTC case, RMSE ¼ 0:150 in 2000, is a consequence of large forecasting errors in the vicinity of summer months. In spite of these outliers, the APE (1:98 per cent) indicates a reasonable performance in terms of the total number of cards sold during this period. Nevertheless, the superior performance of the causal model in the case of travel cards is not surprising at all, given the amount of exogenous information it includes in comparison with the univariate DHR. Among the service variables included in its information set, the Metro route length (MRLt ) plays an important role in the posterior forecasting exercise, particularly in the case of travel cards. As a matter of fact, MRLt is never statistically significant in four out of the six causal models. It only becomes significant for TC and JTC when the estimation period ends in December 1999 and later, due to the abrupt changes in MRLt during this last period. The estimated dynamic response of this variable indicates a long run effect that goes up to eight months, implying a very long and smooth demand response to changes in supply. (4) Although it is useful to report forecast performance based on various measures, it might be misleading to synthesise the performance of different methods by aggregating results of various measures. In other words, policy makers have to decide which particular accuracy measure better suits their preferences and forecasting objectives. If total annual demand is the goal, aggregate RMSE and MAPE may provide misleading conclusions. A large forecasting error in a single month will raise their values considerably, as we have witnessed in several occasions in this paper. In these circumstances, both FGR and APE provide a better picture of overall performance. For this particular dataset and forecasting period, the causal model only outperforms the univariate DHR model in the case of the two travel cards. For the remaining variables, the DHR outperforms the causal model and confirms its potential as an alternative forecasting tool.

5.0 Conclusions As has been witnessed in other European cities, the recent experience of the Madrid Metropolitan Area shows that it is possible to reverse declining historical trends in public transport ridership. This was achieved through an integrated fare scheme based on low-cost travel tickets and improvements in the quality of service. Therefore, adequate planning of future 64

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public transport facilities requires two basic inputs: (a) reliable predictions of transport demand, and (b) efficient estimation of the users’ response to changes in prices and the characteristics of the service. This paper aims at providing these two inputs for the public transport system of the Madrid Metropolitan Area. Using recent monthly data, we address the problem of elasticities estimation and forecasting for a large number of tickets that are subject to the types of multiple, complex calendar effects, and superimposition of outliers, changing supply service, and changing seasonality. Two different approaches have been used to deal with these issues. The first one is a causal model based on a transfer function dynamic model that allows the incorporation of intervention and exogenous variables in a flexible way. The other is the Dynamic Harmonic Regression model, a new variant of unobserved component models with time varying parameters that allows the adaptability of the trend and the seasonal components as soon as the new information becomes available. Both methodologies are capable of dealing with the nonstationarity and strong seasonality features that characterise the database. Additionally, all series exhibit the presence of important outliers which considerably complicates the forecasting exercise. The estimation results indicate that the effects of these input variables have the expected signs and are highly significant from a quantitative point of view. However, their effects change considerably among different type of tickets and transport modes. The historical public transport fares scheme of the Madrid Metropolitan Area provides considerable help when designing a causal scheme of expected price- and cross-elasticities among different type of tickets. With the exception of travel cards, the remaining tickets show significant negative own-price elasticities. The range of estimated values, however, indicates large differences among estimates with the 10-trip metro tickets showing the highest sensitivity to price increases. We have also found evidence of moderate estimated cross-elasticities, in particular between bus and Metro 10-trip tickets and travel cards. Nevertheless, the high subsidies that characterise the Madrid public transport system make these results discouraging for the prospects of achieving optimal pricing of public transport. In spite of this, however, our empirical results indicate that there is room for an alternative pricing policy (based on moderate price increases of travel cards while maintaining single- and 10-ride prices nearly constant) aimed at having positive effects on demand while minimising the negative effects on revenues. As regards forecasting results, we have obtained two sets of twelve periods’ ahead forecasts for the six transport variables analysed in this paper for 2000 and 2001. The forecasting period is particularly difficult given the important increase of the Metro supply services just before this period. This meant a mixture of effects that affected not only total 65

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demand but also temporary passenger shifts between different modes. To avoid evaluation of different forecasting techniques depending upon the choice of a particular accuracy measure, the models’ forecasting performance has been appraised according to several accuracy criteria. Although no model entirely dominates the other for all the variables and forecasting horizons, the general predictive results indicate that both models are good alternatives in providing reliable forecasts. However, analysing the changes in seasonal patterns as well as exploring forecast combinations should be areas of important future research.

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