A dynamic model of car fuel-type choice and mobility

0191-2615/92 $S.CYJ+.OO 0 1992 Pergamon Press plc

Trons/m. Res.43, Vol. 26B. No. I. pp. 77-96, 1992 Printed in Great Britain.

A DYNAMIC MODEL OF CAR FUEL-TYPE CHOICE AND MOBILITY LEO VAN WISSEN* Netherlands Interdisciplinary Demographics Institute, The Hague, The Netherlands

Institute of Transportation

THOMAS F. GOLOB Studies, University of California, Irvine, CA, U.S.A.

Abstract-This study examines the relationship between car mobility and the choice of alternative-fuel versus gasoline cars in the Netherlands during the 1984-1988 period. One alternative fuel, liquified petroleum gas (LPG), is priced considerably lower than gasoline and is available at service stations throughout the Netherlands. Conversion costs lead to higher capital costs for LPG cars. A joint continuous/discrete multivariate demand model is applied to panel data to quantify the relationships among fuel-type choice, annual car usage and commuting distance, and to determine the effects of commuting subsidies, fixed and variable work locations, rail season tickets, and household socioeconomic characteristics. The model has lagged effects, individual-specific time-invariant terms, period effects, and compensation for panel conditioning and attrition. Results show that higher levels of car use favor choice of LPG cars, but the lower operating costs in turn lead to increases in car use. This latent demand for car travel is accentuated by travel reimbursements provided by employers. 1. OBJECTIVES AND SCOPE

In The Netherlands, households have the opportunity to operate cars on an alternative fuel, liquified petroleum gas (LPG). LPG is composed primarily of propane and butane. These are the heavier compounds in natural gas, so LPG can be produced as a byproduct of Dutch North Sea natural gas. It is priced much lower than gasoline (benzine) and is available at service stations throughout the country. Essentially any spark-ignition car engine can be converted to LPG use, adding fixed car ownership costs either in the form of a retrofit or a premium for LPG cars in the used-car market. LPG is stored under low pressure in liquid form and is burned in the engine as a gas with approximately 80 percent of the energy density of gasoline (Sperling, 1988). The other alternative fuel to benzene (gasoline) in the Netherlands is diesel fuel (petroleum distillate). The fixed versus operating costs of diesel cars are more similar to gasoline than LPG cars, but the lower fuel costs and slightly higher fixed costs of diesel cars relative to gasoline cars places diesel in between gasoline and LPG. According to the Dutch National Mobility Panel, the data source used in the present study, approximately 12 percent of the cars in general use in the Netherlands in 1984-88 were LPG, 8 percent were diesel, and the remaining 80 percent gasoline. For single-car households (about 60 percent of all Dutch households), the approximate fuel type breakdown is 11 percent LPG, 7 percent diesel, and 82 percent gasoline. The first question addressed in this research is: how is fuel-type choice related to car mobility, where mobility is measured in terms of overall usage (kilometers per year) and commuting distance? Causality can be anticipated in both directions: a high travel demand might explain the purchase of a car with lower fuel costs, but the ownership of such a car might result in more travel. The second question is: what are the influences of commuting subsidies, public transport season tickets, income, and other background socio-demographic variables on fuel-type choice and car mobility? A joint continuous/discrete choice demand model is specified in terms of a set of dynamic simultaneous equations. The endogenous (dependent) variables are car fuel type, car usage, and commuting distance, each measured at two points in time. Car fuel type is treated as a three-category discrete choice ordered in terms of fuel cost versus capital cost; usage is a continuous variable; and commuting distance is a censored continuous variable (censored at zero for households with no workers outside the home location). *Formerly, Department of Spatial Economics, Free University, Amsterdam, The Netherlands.

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The model is estimated on a pooled sample of the Dutch National Mobility Panel for the years 1984-1988. Elasticities are calculated for each endogenous variable as a function of the other endogenous variables and certain exogenous variables, All differences in vehicle characteristics between alternative-fuel and gasoline vehicles are subsumed in the choice-specific constants in the choice model. This is an obvious simplification that assumes uniform differences between LPG and gasoline vehicles across all makes and models. While just about any make or model automobile, light truck or van used as a personal vehicle in the Netherlands has been or can be converted to LPG use, there are likely to be differences in performance between the LPG and gasoline versions of the same vehicle, and such differences are likely to vary by vehicle power and weight. Also, cargo space is lost to the LPG fuel tank, and the relative loss is greater for smaller vehicles than for large ones with greater absolute cargo space. The model ignores potential influences of vehicle make and model class on car usage, and this might affect results if there are systematic differences in make and model class associated with the choice of LPG versus gasoline versions. Another simplification is that the analysis is restricted to single-car households; a joint ownership level, fuel type, and mobility model is outside the scope of this study. Single-car households represent approximately sixty percent of all Dutch households during the period of the study. Thus, the model ignores the causal influences of fuel type and car mobility on the level of car ownership. Fortunately, only about ten percent of households in the Dutch National Mobility Panel moved in or out of the “l-car” classification during the 1984 to 1988 period (Golob, 1990; pp. 446-447). Finally, choice utilities of the three discrete fuel types are assumed to be ordered according to the ratio of capital versus operating costs: LPG, diesel, and gasoline. This allows the use of ordered-response probit models for the fuel type choice variables at each point in time. Ordered probit models can be included in structural equation systems, provided that appropriate estimation methods are used. Results indicate that this ordering of alternatives was not a major problem. A joint car type, fuel type, and mobility model, while outside the scope of this study, is an extremely important topic for further research. Such future research can be based on an integration of the present research with a car-type holdings and utilization model, such as that provided by Berkovec and Rust (1985), Hensher and Smith (1990), Lave and Train (1979), Mannering and Winston (1985), and Train (1986) (see Mannering and Train, 1985 for a review of car-type models).

2. METHODOLOGY

2.1. Non-normal endogenous variables Classical theory of simultaneous linear equations assumes that the set of endogenous variables is multinormally distributed. This is clearly not the case in the fuel-type choice and mobility model: Fuel type is a categorical variable, with an assumed ordering of the categories according to the fuel versus capital costs. Annual car travel is a continuous variable (in principle it is censored at zero, but the censoring does not take effect because all single-car households reported some car kilometers each year). Commuting distance is censored at zero; if there are no workers in the household this variable is zero. Hence, the endogenous variables consist of an ordered three-category probit (fuel type), a continuous variable (annual car usage) and a tobit variable (commuting distance). Under certain assumptions, these observed non-normal variables can be transformed into normal latent variables, and simultaneous equation systems of the transformed normal variables can then be estimated using appropriate distribution-free methods. The model results can be interpreted in terms of the latent variables, but this has limitations, because they cannot be observed; an elasticity calculated for unobserved latent variables is not very meaningful in itself. However, it is possible to evaluate the model results in terms of their observed non-normal counterparts. The transformation of non-normal to normal variables is performed in the so-called

19

Dynamic model of car fuel-type choice and mobility

measurement model (sometimes called the “outer” measurement model, to distinguish it from factorial models). For a continuous variable the measurement model linking the observed Y and the latent variable rC is simply the identity: Y = Y*. For an ordered probit discrete choice variable, it is assumed that there is a latent continuous variable r* which is normally distributed with mean zero and unit variance. The ordinal indicator Y is related to the unobserved Y* in the following way: Y=l Y=2

if 6, < Y* < 6, if 6, < Y* < i&

I;

* if6K_,

(1) =K

< Y* < AK

where 6, =

-oJ,6,

< 62 < 6,

c . . . . < 6,6,

=

+m

are the threshold values of the cumulative normal distribution corresponding to the marginal distribution of the population over the categories. A variable with K categories has K - 1 unknown thresholds. These are estimated as:

(2) for k = 1,2, . . . , K - 1, where Q,denotes the standard cumulative normal distribution function. Here, n, is the sub-sample size falling in category j of the ordinal variable and N is the effective sample size. In the transformation from ordinal to normal variables the ordinal score Y = k is replaced by the normal score zk, which is the mean of Y* in the interval Bk_] < Y* < &:

9(Sk-,) - dJ(6k)

zk= a(&)

-

@(a,_,)

(3)

where q5denotes the standard normal density function. In a similar fashion, a censored variable can be transformed to a normal variable. If Y is a censored variable that is observed only if it is positive, then it is assumed that Y is generated by a latent normal variable Y* that is uncensored. Thus: Y = Y*

ifY* > 0

Y=O

ifY* 5 0

The latent variable is assumed to be normally distributed with mean ~1and standard deviation u. For values below the threshold value, the mean of the normal score of the latent variable in this interval is assumed, given by:

d( -CL/a)

z=p- a+(-p/a)

These latent variables are multivariate normally distributed. A simple way of computing the variance-covariance matrix of the transformed variables is to use the normal scores from the marginal distributions of the variables. However, this is not an optimal solution. By using bivariate information of all pairs of variables, polychloric and polyserial correlations can be computed; these are consistent estimates of the underlying population statistics.

L. VAN WISSENand T. F. GOLOB

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2.2. The structural model The general form of any structural equation system can be given by:

Y* = BY* -trx + 6

(6)

where Y* is a column vector of p endogenous variables, X is a column vector of q exogenous variables, and 6 is a column vector of p disturbance terms. The B matrix, of order (p x p), contains the structural effects among the endogenous variables, and the P Matrix, of order (q x p), is the matrix of regression coefficients of all exogenous variables on the endogenous variables. The disturbance (or residual) terms are structural model is defined in terms of the latent variables, i.e. after the transformations performed in the measurement model. An important distinction in simultaneous equation systems is that between direct, indirect and total effects. Direct effects are given in the B- and P-matrices. Indirect effects exist if variable “a” is related to “b”, which is in turn related to “c”; thus, there is an indirect effect from “a” to “c” through the path involving “6”. Total effects are simply the sum of direct and indirect effects. The formulas for calculating these effects among the endogenous variables are: Yto Y xto Y Direct effects: Indirect effects: Total effects:

B (I -B)-’ - B - I (I - B)-’ - I

t; - B)-‘I’ - r (I - B)_T

Estimation of the parameters in the model is performed using the Generalized Weighted Least Squares method developed by Browne (1974; 1982; 1984). This method is described in more detail in Golob and van Wissen (1989), and van Wissen and Golob (1990). Asymptotically distribution free estimates are generated by minimizing the function: F = (s -

6)’ W-‘(s - 4)

(7)

where s is the vectorized set of all sample variances and covariances (the sample statistics generated in the measurement part of the model), 6 the vectorized set of estimated variances and covariances, and W the asymptotic variance-covariance matrix of the sample statistics. The value of F times the sample size N is an overall measure of goodness-of-fit. It is distributed asymptotically as X2 with degrees of freedom determined as follows: For p endogenous variables and q exogenous variables, there are ?h(p* + p) moments among the endogenous variables and pq regression slopes from exogenous to endogenous variables; thus, the number of degrees of freedom is ?h(p2 + p) + pq - r, where r is the number of free parameters in the model. However, with large sample sizes the X2-test will almost always result in a non-fitting model, according to this statistic (Bentler and Bonett, 1980). This does not necessarily imply a poorly fitting model. Alternative tests have been proposed for sample sizes larger than 200. One is the statistic given by the product of N and F divided by degrees of freedom. If this statistic is less than 3, this indicates a good fit (Carmines and McIver, 1981). Another test is to take the observed N times F value, and calculate the hypothetical sample size, given this statistic, that would be needed for non-rejection of the model. If this sample size is larger than 200, this indicates a good fit (Hoelter, 1983). 3. DATA

The model is estimated on a pooled sample of the Dutch National Mobility Panel, 1984-1989. This data set, described in Golob et al. (1986) and van Wissen and Meurs (1989), consists of annual (sometimes biannual) observations on approximately 1,800

Dynamic model of car fuel-type choice and mobility

81

households originally clustered in twenty communities throughout the Netherlands. These observations involve one-week travel diaries and personal and household questionnaires. For the present study, annual measurements conducted in the spring of each of the five years 1984 through 1988 were used to construct a pooled year triplet sample. The sample was restricted to households that had a single car in each of the three years representing approximately 63 percent of all panel households. The composition of the pooled year triplet sample is shown in Table 1. Each subsample represents the observation of households for three consecutive years. In the model structure, these years are denoted by lo for year one, t, for year two, and t2 for year three. There were 494 single-car households in the panel for the three years 1984, 1985, and 1986. Similarly, there were 606 single-car households for 1985, 1986, and 1987; and 693 single-car households for 1986, 1987, and 1988. A random selection of 494 households was made for each of the larger two subsamples, resulting in a pooled sample size of 1482 observations of households over three adjacent years. This aids in comparing standard errors. Empirical comparisons of estimation results using the randomly selected sample (1482 observations) and the total sample (1793 observations) showed no noticeable differences in parameter values. Pooled panel sampling is discussed in van der Eijk (1987) and Golob (1990). One major advantage is that two types of effects can be separated and estimated with such a sample: (a) panel conditioning effects due to temporal biases in response, and (b) period effects due to conditions that influence all respondents uniformly at a given point in time. The second advantage is that attrition bias is less in a pooled sample than in a “stayers” sample of only those households that participated in all waves of the panel (in this case 1984 through 1988). This is because households added as panel refreshment (in 1985 and 1986) are included, as are households that dropped out of the panel after three or four years. The disadvantage is that there is sample redundancy (Chamberlain, 1980), but this disadvantage is outweighed by the advantages of including period effects and reducing sample attrition bias. In particular, the ability to estimate period effects, potentially due to fuel prices, other costs of living, and trends in taste, is central to this study. It is not possible to identify period effects in a cross-sectional model. The model variables are listed in Tables 2A (the endogenous variables) and 2B (the exogenous variables). The six endogenous variables (Table 2A) represent measurements of the same three travel behavior variables at two points in time, one year apart. Through use of the measurement model, the fuel type variables at tl and t2 are treated as ordered probits, the car usage variables at t, and t2 are treated as continuous variables, and the commuting distance variables at t, and tz are treated as tobits. The sixteen exogenous variables (Table 2B) are subdivided into three groups: (a) initial conditions, (b) time invariant background variables, and (c) dynamic background variables. The three exogenous variables in the initial conditions group are simply the three endogenous behavioral variables measured one year prior to year t = 1. These variables are taken as given because there is no way to explain these variables in terms of state dependence from a previous year, nor is there any way of specifying autocorrelated disturbance terms. If there are lagged effects, omitted variables, or serially correlated disturbances, these are not strictly the initial conditions in the dynamic process being modeled (Chamberlain, 1985). More complicated treatments of left-censored processes, such as those described in Hsaio (1986) or Heckman (1981), are outside the scope of this study. Table 1. Pooled year triplet sample Triplet

Year 1

1 1984 2 1985 3 1986 TOTAL OBSERVATIONS

Year 2

Year 3

1985 1986 1987

1986 1987 1988

Total Households 494 606 693 1793

Randomly Selected 494 494 494 1482

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Table 2A. The endogenous variables Variable Fuel Type T = 1 YearKMT = 1 Comm. Dis. T = 1 Fuel Type T = 2 YearKMT = 2 Comm. Dis. T = 2

Abbreviation FT, YK, CD, FT, YKz CD2

Description 3-category fuel type (benzine, Car usage in km/year in year Average commuting distance 3-category fuel type (benzine, Car usage in km/year in year Average commuting distance

diesel, LPG) in year t, t, of primary worker in year t, diesel, LPG) in year t2

t2 of primary worker in year t2

The time invariant exogenous variables are divided into three subgroups: The first subgroup (2.1) is comprised of four household economic and socio-demographic variable that remain constant for the vast majority of households over the three-year time horizon. The lack of temporal variation in these variables means that they must be treated the same as variables in a cross-sectional model. These include household size, one life-cycle dummy variable, and two income class dummy variables. Dummy variables for the remaining life-cycle and income categories (J. Golob et al., 1986) were tested, but were found to add no explanatory power to the model; they were either not related to the endogenous variable or were redundant with other variables in the model. In addition, the distinction in residential location (between major urban centers, regional centers, suburban cities and other municipalities) was determined to be unimportant. The second subgroup (2.2) consists only of the single variable, the natural logarithm of years-to-date in the panel. The use of this variable in reducing panel conditioning and attrition biases is discussed in Golob (1990). The logarithm form is consistent with the biases detected as a function of panel longevity by Meurs, Visser, and van Wissen (1989). Finally, subgroup (2.3) is comprised of two dummy variables capturing period effects, one for each of the first and last of the three year-triplet sub-samples listed in Table 1. With three pooled panel sub-samples it is possible to have two such variables. The variables measure uniform trends over the population that are not explained by the remaining variables (Golob, 1990), with the base period being 1985-1987: links from the PERIOD ‘84-‘86 variable to endogenous variables at C, measure period effects for 1985 relative to 1986; links from this same variable to endogenous variables at t2 variables measure period effects for 1987 relative to 1986, and those from PERIOD ‘86-‘88 to t2 variables measure period effects for 1988 relative to 1987. The final group (3) of dynamic exogenous variables consists of three variables measured at each of the two years tl and t,. These three variables are: the number of household workers receiving car commuting subsidies from their employers, the number of

Table 2B. The exogenous variables. Variable

Description

Initial conditions Initial conditions Initial conditions Time invariant Time invariant Time invariant Time invariant Time invariant Time invariant Time invariant Dynamic

Fuel Type T = 0 Year KM T = 0 Comm. Dis. T = 0 H size HD > 35OKD INC 24-38 K INC > 38K Ln (YRS PANEL) PERIOD ‘84-‘86 PERIOD ‘86-‘88 TRAVEL SUBS. T = 1

Dynamic Dynamic Dynamic

RAIL CARD T = 1 NON-FIX W.L.T = 1 TRAVEL SUBS. T = 2

Dynamic Dynamic

RAIL CARD T = 2 NON-FIX W.L.T = 2

3-category fuel type in year prior to year t, Car usage in year prior to year t, Average commuting distance in year prior to year t, Household size Life cycle dummy: head of household > 35, no children Income dummy: 24-38,000 fl/year Income dummy: > 38,000 fl/year Natural log of years in panel Dummy: year to = 1984 (t, = 1985, t2 = 1986) Dummy: year t,, = 1986 (t, = 1987, tl = 1988) No. of workers with car commuting subsidized by the t, employee No. of rail (NS or OV) season tickets in year t, No. of workers with non-fixed work locations in year f, No. of workers with car commuting subsidized by employer in f2 No. of rail (NS or OV) season tickets in year t2 No. of workers with non-fixed work locations in year t,

Variable Type

Dynamic model of car fuel-type choice and mobility

adult rail season tickets (including for a specific rail travel corridor), work locations. The first two of purposes. The third variable was variables.

83

a general rail pass, a public transport pass, or a pass and the number of household workers with non-fixed these variables are important for policy evaluation found to be important in explaining the endogenous

4. MODEL SPECIFICATION

4. I . Structural equations The goal is to develop and estimate a dynamic model of fuel-type choice and car mobility. In a dynamic context it is likely that fuel type and total mobility will show a more complicated pattern of causality than can be specified in a single equation model of fuel type as a function of total mobility. Moreover, commuting distance plays an important role in determining total mobility and fuel type, especially when travel reimbursements and subsidies are taken into account. Therefore, the following variables were defined to be endogenous: (a) fuel type (b) annual car usage, and (c) commuting distance. This implies a simultaneous model system with three endogenous variables per time period. A pooled household data file is used where each observation covers three time periods, each one year apart. Any dynamic model estimated using these data might be biased by a number of factors, including: initial conditions, panel attrition and conditioning, and period effects. The first time point is used as an approximation to initial conditions: fuel type, annual car usage, and commuting distance for year to are treated as exogenous variables, leaving time periods t, and t2 to be determined in the model. Thus, the model can estimate time lags of at most one year. A second factor controlled for is the panel conditioning effect. In previous research (Meurs et al., 1989) it has been shown that the number of years a respondent has participated in the panel is an important conditioning factor for mobility reporting. Consequently, this variable, in log transform, was included as an exogenous variable. Finally, due to the pooling of the sample, pure time effects can be distinguished from period effects. The longitudinal character of the data allows inclusion of individual-specific time invariant effects (see Meurs, 1990a, or Hensher, 1988, for a detailed exposition of these effects). These effects are included in terms of decompositions of the residual terms in the model. Suppose the model is of the form:

Y;, =

cj Pj&

+

Eir

(8)

This model might describe annual car usage for household i in year t. Of course, the set of predictors X will not be complete. Each household and time point has omitted variables that also determine its mobility level. Some of these omitted variables are time invariant (e.g. specific habits and tastes that characterize the household’s behavior at any point in time). Therefore, the residual error term can be decomposed in a household specific, time invariant part, oi, and a “truly random” part, ci,:

Ej,

=

(Yi

+

S;(

(9)

By decomposing the error term in this way, it is possible to control for potential spurious serial correlation or spurious state dependence effects that might obscure the true causal pattern underlying the data. Within this framework, the fuel type and mobility model can be specified in terms of the general structural equation system (6):

84

L. VAN WISSEN and T. F. GOLOB

where 0 denotes null matrices, and the following symbols have been used (see Table 2A for endogenous variable abbreviations): v, = [FT*,YK*,CD*]r

t = 0,1,2

a = bFT4QK4kJ1T E(C) = E(Lh) = 0 t = 1,2 E(5tfi) = *r E(S;Ii) E(1;CJ E(LC)

= 0 = 0 = Q,

There are six endogenous variables in total, three for each time period. The B matrix, broken down into nine sub-matrices, three of which are null sub-matrices, contains the causal effects of the endogenous variables upon each other. In B,,, and B2,2the contemporaneous, or instantaneous effects are contained, while in B2,1the lagged effects of period 1 to period 2 are given. The sub-matrix B,,z is relevant only if anticipatory effects are present. Furthermore, the B matrix contains the individual specific effects CY,one for each endogenous variable. The B,,, matrices are diagonal matrices. For pure individual-specific effects these matrices should be identity matrices. However, as will be explained in Section 5, it was necessary to release one diagonal element of B,,,. The I’ matrix contains the regression, or conditioning effects, of exogenous variables upon fuel type, annual car usage and commuting distance. Several submatrices can be discerned. First, there are the conditioning effects of period to, which correspond to V,, in the vector of regressors. Next, there are dynamic variables that have instantaneous effects on the endogenous variables. These coefficients are given in PI,, and P2,* and the corresponding regressors are given in X, and X,. A fourth set of regressors is time invariant. They have an influence through the individual specific effects. They explain part of the time-invariant individual specific dispersion in the data. These time-invariant regressors are denoted by U. Finally, there are the time-period dummy variables, R, as explained previously, that influence the endogenous variables at year t, and time year tz through the coefficient sub-matrices P,,4 and P2,+ It is also possible that the endogenous variables of system (10) have serially correlated disturbances, such as those found in Dutch National Mobility Panel data for travel times by car, public transport and non-motorized modes in Golob (1990). Such autocovariances are manifested as significantly non-zero diagonal terms in the covariance matrix of the 5; and r2 disturbance vectors. However, it has been shown that time-invariant individual-specific exogenous variables, such as those specified in the present model, can account for the serially correlated errors (Meurs, 1990a and 1990b; Golob, Kitamura, and Supernak, 1991). Internalizing the time-invariant heterogeneity of the sample is a solution preferred to adding error autocovariances. Providing that the more complicated model with disturbance-team autocovariances is identified, the significance of each disturbance-term autocovariance can be assessed by adding the appropriate term to the model of system (11) and comparing the chi-square statistics based on objective function (7) using standard nested-model criterion, all such tests were negative. 4.2. Elasticity formulas Model specifications are usefully translated into elasticities, defined as the percentage change in an endogenous variable due to a unit percentage change in an exogenous variable or another endogenous variable. For probit and tobit variables (that can occur

Dynamic model of car fuel-type choice and mobility

85

both as dependent and independent variables) some difficulties arise in applying the concept of elasticity. Therefore, each of the relevant causal links among the endogenous variables in the model will be discussed. The structural model defines relationships among the multivariate normal latent variables. These latent variables are equal to the observed variables only with continuous variables. For tobit and probit variables there is a nonlinear measurement relation between observed and latent variable, as described in Section 2.1. The model structure is among the latent variables. However, elasticities need to be defined in terms of observed variables. In developing observed-variable elasticities, it is convenient to introduce notation to distinguish the endogenous variables of the three different measurement type: yearly kilometers YK, fuel type FT and commuting distance CD. The expected value of the latent underlying variables is denoted by r](yK)for yearly kilometers; by 7(F7)for fuel type; and by 71W) for commuting distance. The expected values of the latent variables for the structural model system can be calculated using the reduced form of the general structural equation system (6), where endogenous variables Y are replaced by their latent variable counterparts: ?j = (I

-a-TX

(11)

Due to the nonlinearities it will be necessary to calculate the elasticities for each observation. Evaluation of the elasticity functions at sample mean points is not correct. Instead, the mean elasticities are calculated by evaluating the elasticity formulas for each observation in the sample. Thus, if ei is the elasticity evaluated for observation i, total elasticity is calculated as

The elasticity formulas for the most important causal relations in the model are given below. The starting points for these calculations are the derivatives of the normal and truncated normal probability distributions given in Maddala (1983). These elasticities vary by time period; but, for clarity, the time period subscript is excluded in eqns (14) through (21). 1. The elasticity of annual car usage on fuel type. This represents the effect of a continuous variable on an ordered probit. The elasticity of this link is defined as: the percentage change in the probability of having a particular fuel type due to a percent change in annual car usage. Thus, for the three-category fuel type variable there are three elasticities, one for each fuel type. Normally one would expect to find opposite signs of elasticities among the cells, since the respective cell probabilities cannot all increase simultaneously. For example, greater annual car usage increases the probability of owning a car with lower fuel costs and decreases the probability of owning a gasoline car. The elasticity formula is given by:

or more compactly:

(14) where the following notational simplifications have been made:

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4i,k = 9 tsk +,,k = 4?(6, -

TicFT’)

qitFT))

(1%

The index k denotes fuel type category, and the thresholds 6 are those of eqns (1) through (4). 2. The elasticity of commuting distance on annual car usage. This is the effect of a censored (tobit) variable (commuting distance) on a continuous variable (annual car usage). Because the tobit variable is censored at zero, the elasticity of households without commuting distance will be zero. Therefore, the conditional elasticity is used: thepercentage change in annual car usage due to a one percent change in commuting distance, given the information of non-zero commuting distance. It is given by:

(16)

where the term between brackets in the numerator is the conditional expectation of commuting distance, given that it is non-zero. Further, v is the standardized latent variable of the tobit: Vi v

(CD)

=-

u

(17)

where u is the standard deviation of the latent variable. 3. The elasticity of commuting distance to fuel type. This is an indirect link in the model, since there is no direct causal relation from commuting distance to fuel type (Section 5). Again, if the household has no commuting distance, the elasticity will be zero. Therefore conditional elasticity is defined as: the percentage change in the probability of having a particular fuel type due to a one percent change in commuting distance, given the information of a non-zero commuting distance. Thus, there are three elasticities, one for each fuel type category:

where k is the index of the fuel type category. 4. The elasticity of fuel type at t, to annual car usage at t,. The elasticity of this dynamic link indicates the latent travel demand effect of lower fuel costs: thepercentage change in annual car usage due to a percent change of the probability of having a particular fuel type. There are three elasticities, one for each fuel type:

(19)

with 1 the index of the fuel type category. 5. The elasticity of fuel type at t, to commuting distance at t2. This is also a latent demand effect of fuel type. For zero commuting distance, this elasticity is not defined. A conditional elasticity is defined: the percentage change in commuting distance, due to a

Dynamic model of car fuel-type choice and mobility

81

one percent change in the probability of having a particular fuel type, given the information ofnon-zero commuting distance. The formula for this elasticity is:

5. RESULTS

The model X2 value was 212.78 with 76 degrees-of-freedom (df) and a sample size of 1482. The X2/df ratio is 2.8, indicating an acceptable goodness of fit (Carmines and McIver, 1981). The critical sample size for non-rejection of this model at the p = .05 level is 583. According to Hoelter (1983) this also indicates an acceptable model because the critical sample size is greater than 200. The R2 values were 0.35 for Year Km at period t,, 0.35 for Comm. Dis. at t,, 0.60 for Year Km at t2 and 0.37 for Comm. Dis. at t2. For the fuel-type probit variables, the variances are not identified, and thus no R2 statistic is available. With few exceptions, all parameter estimates are significant at the p = .05 level. 5.1. The causal structure The final model form is depicted in the flow diagrams of Figs. 1 through 4, and coefficient estimates are listed in Tables 3 and 4. The focus in this section is on the causal links between variable pairs. The scale of the coefficient of a link is only interpretable for continuous variables. Qualitative results are given in the next section in terms of elasticities. Turning first to the causal structural relating fuel type, annual car usage and commuting distance (Fig. 1. and Table 3), the interpretation of the contemporaneous structure is clear: choice of fuel type is a direct function of annual car usage, which, in turn is determined to a high degree by commuting distance. It is total car mobility, in terms of annual car usage, that has a direct influence on the fuel type. The coefficients have the correct sign: more travel leads to a higher probability of owning a car with lower fuel cost and higher capital cost. Commuting distance has an indirect, rather than direct, effect on fuel type choice.

‘\

“‘,;d

\,6

It Cc&&,j. D,S. _______________ .639“‘\

T=O

T=

1

INITIAL CONDITIONS

Fig. 1. Dynamic structure of fuel type model.

T=2

L. VANWISSEN~~~T.F.GOLOB

88

LdYRs

INC

PANL)

24-38K

Fig. 2. Structure of individual specific effects.

There is an important lagged influence of fuel-type choice on commuting distance in the next time period. Lower operating costs encourage longer or more frequent commuting trips, according to this model. There is also an effect of fuel-type choice on annual car usage, but this effect is not significant at the p = .05 level. These results imply that some households with alternative fuel vehicles may react in one or more of several ways: they may relocate to more desirable residences further from their jobs, they may take jobs at locations further from their home, or they make more frequent commuting trips to alternative work locations further from home. Of course, choice of alternative fuel vehicles (cars) could be made during a process of gradual increases in commuting requirements. Fuel-choice related increases in car use for non-commuting purposes are much less than those for commuting. Another important dynamic feature of the model is state dependence, expressed through the auto-lags present in the B-matrix. All three endogenous variables - fuel type choice, annual car usage and commuting distance- exhibit significant state dependency. Often, state dependency is the spurious result of time-invariant sample heterogeneity (Meurs, 1990a, 1990b), but these effects persist after partial control for heterogeneity by means of the individual-specific terms (the cyterms in the model). The individual-specific terms exhibit pure time-invariant structures for both fuel-type choice and annual car usage because the CY,and cry2terms have equal influence (unity coefficients) on the respective endogenous variables at the two points in time (Fig. 1). The individual-specific term for commuting distance (03) represents a more complex factor structure on the error terms, because the link from olj to commuting distance at time t2 was found to be significantly different from that at time c,. Reasons for this result are unknown. The explanation of the individual-specific terms in terms of the exogenous variables

Fig. 3. Exogenous variables in fuel type model.

Dynamic model of car fuel-type choice and mobility

89

m

COW

DE.

T=2

Fig. 4. Time and period effects in fuel type model.

is shown in the flow diagram of Fig. 2, and the estimated coefficients of the links depicted in Fig. 2 are listed in Table 4. A significant covariance was found between (1~~ and (Ye,the time-invariant components of annual car usage and commuting distance. The timeinvariant component of fuel-type choice (aI) is influenced by four variables: Larger households tend to choose alternative fuel vehicles; while households with heads older than 35, without children at home, and households in the middle income category (Hfl 24,000-38,000) tend to choose gasoline vehicles. The link from panel tenure to CX,reveals that panel “stayers” are more likely to own gasoline vehicles, as opposed to households that drop out of the panel or are part of the panel refreshment, ceterisparibus. Households in the highest income category (Hfl 38,000+) have higher annual car usage (a positive link to factor CQ) and the primary worker in larger households has longer or more frequent commuting trips (a positive link to factor CQ). The significant link from panel tenure to CQindicates that panel “stayers” tend to have longer average commuting distances. This is consistent with the result reported by Kitamura and Bovy (1987), that less mobile persons tended to drop out of the Dutch National Mobility Panel. Decompositions of the variances of each endogenous variable are shown in Table 5 in terms of total variance, variance of the time-invariant component and residual variance. The time-invariant factors account for a large share of the total variance of each variable. For the four continuous variables with known sample variances, the estimated variance of the time-invariant individual-specific effects (CYfactors) accounts for between

Table 3. Parameter estimates and t-values of B matrix Sub-matrix:

From:

To:

Estimate:

T-value:

B I.1

Year KM Comm. Dis.

Fuel Type Year KM

0.191 0.100

8.428 12.385

BA?.

Year KM Comm. Dis.

Fuel Type Year KM

0.191 0.038

8.428 12.385

B2.1

Fuel Type Fuel Type Fuel Type Year KM Comm. Dis.

Fuel Type Year KM Comm. Dis. Year KM Comm. Dis.

0.714 0.014 0.153 0.816 0.751

26.476 1.136 5.048 36.254 30.340

B,,*

alpha,, alphavx alpha,, alpha, alphavx alpha,,

Fuel Type Year KM Comm. Dis.

1.000 1.000 1.000

-

Fuel Type Year KM Comm. Dis.

l.OOQ 1.000 0.251

B2,*

5.146

L. VAN WISSEN and T. F. GOLOB

90

Table 4. Parameter Sub-matrix:

estimates

and T-values of r matrix

From:

r 1.0 (initial conditions)

(instantaneous effects)

r (Ge invariant effects)

(period effects on t = 1 vars) I’,,,Period ‘84-‘86 (period effects on t = 2 vars)

To:

Estimate:

T-value:

Fuel Type Fuel Type Year KM Year KM Comm. Dis.

Fuel Type Year KM Year KM Comm. Dis. Comm. Dis.

1.611 0.182 0.442 0.314 0.639

21.973 8.689 36.895 10.410 39.748

Travel Subs. Travel Subs. Rail Card Rail Card Non-Fix W. Lot.

Year KM Comm. Dis. Fuel Type Year KM Comm. Dis.

0.137 0.682 -0.154 -0.032 0.573

5.709 10.256 - 2.260 -1.146 6.356

HH. Size HH. Size Head < 35 OK. Log(Y. in Panel) Log(Y. in Panel) Income 24-38K Income 38 + K

%T ffCD ffFT %T %D %r %K

0.090 0.318 -0.212 - 0.285 0.440 -0.178 0.047

4.110 7.922 - 2.692 - 1.791 1.737 -2.952 1.434

Period Period

Year KM Year KM

-0.273 - 0.096

-6.158 -2.277

‘84-86 ‘86-‘88

Fuel Type Period ‘84-‘86 Period ‘84-‘86

0.218 Year KM Comm. Dis.

2.153 0.195 0.350

5.053 3.102

16% and 37% of total variance. The sample variances of the categorical fuel type is not identified in a probit model, but the residual (unexplained) variances of these two variables was fixed at the conventional standardization value of 1.O. Comparing the estimated variance of the cr factor to this fixed residual variance, the variance of the (Yfactor is slightly more than half of the variances of the residuals. This is similar to the result for the four continuous endogenous variables; for these four variables, the estimated variance of the Q!factors ranges between 38% and 58% of the estimated residual (unexplained) variance of each variable. Next are the effects of the dynamic exogenous variables on the three endogenous variables (Fig. 3 and Table 4). There are three exogenous variables specified for two time periods. No lagged effects are present and the structure was equated for the two time periods t, and t2. An important exogenous variable is whether there is a commuting reimbursement by the employer. This variable has no direct effect on the fuel-type choice, but does have a positive effect both on commuting distance and car usage. Taking into account the different scales of annual car usage (thousands of kilometers per year) and average commuting distance (kilometers), the effect of travel subsidy on commuting distance is almost twice that on annual car usage. Households with persons having some form of season ticket or long-term discount ticket for the train or train and public transport combined are more likely to choose

Table 5. Variance

Variable

Total Sample Variance

decomposition Estimated Residual Variance

of endogenous

variables

Variance of Time-Invariant (Individual-Specific) Effects T-value Estimate

R2 _

Fuel Type Year KM Comm. Dis.

0.492 3.982

1.000 0.322 2.571

0.519 0.124 1.501

49.467 33.943

0.346 0.354

Fuel Type Year KM Comm. Dis.

0.763 4.134

1.000 0.307 2.591

0.519 0.124 1.501

_ 31.628 33.900

0.598 0.373

_

Dynamicmodelof car fuel-typechoice and mobility

91

gasoline vehicles, and have lower annual car usage. As expected, there is a clear relationship between regular train and public transport use and the tradeoff between car operating cost and capital costs. Finally, the third dynamic exogenous variable is the number of household workers with a non-fixed work location. Also as expected, this variable has a high positive influence on commuting distance. The exogenous period effects are shown in Fig. 4 and Table 4. These results indicate that, for annual car usage, there was an unexplained increase from 1985 to 1986, a flattening of growth from 1986 to 1987, and an accelerated increase from 1987 to 1988. For fuel-type choice, there was an unexplained universal increase in the probability of owning alternative fuel cars over the 1985-1986 interval. And there was a similar unexplained increase in commuting distance in the 1985-1986 period. The total effects among the endogenous variables are given in Table 6. All variables at time point 1, have an effect on variables at time t2. Commuting distance affects fueltype choice indirectly through annual car usage. Such indirect effects show up in the calculations of total elasticities. 5.2. Elasticities In applying the elasticity formulas of Section 4.2 to calculate the mean elasticities, it is necessary to distinguish between direct elasticities, calculated from the direct effects between the endogenous variables and total elasticities, calculated from the reduced form equation system (Section 2.2). Total elasticities are more relevant for policy evaluation. The elasticities are parameterized by the dynamic exogenous variables-employer reimbursements of car commuting costs, rail season tickets, and non-fixed work locationsall of which are relevant for transportation policy. The mean direct and total elasticities for each pair of endogenous variables is listed in Table 7. Each relevant elasticity will be discussed separately. 5.2.1. Car usage to fuel type. As discussed in Section 5.1, the model results indicate that increasing annual car usage increases the probability of choosing an alternative fuel car with higher capital cost and lower operating cost. The three elasticities from car usage in the B,,, sub-matrix in Table 7 quantify this relationship. Both diesel and LPG have positive elasticities (0.16 and 0.45 respectively), but LPG is much more sensitive to usage changes than is diesel. A one percent increase in kilometers per year decreases the probability of owning a gasoline (benzine) car by 0.21 percent. Total elasticities are equal to the direct elasticities, and the computed elasticities at time t2 are somewhat lower than those at t,. Parameterizing by the dynamic exogenous variables, car usage elasticities for households with and without travel employer travel cost reimbursements are graphed in

Table 6. Total effects matrix E among the endogenous variables Sub-matrix:

From:

To:

Estimate:

E 1.1

Year KM Comm. Dis. Comm. Dis.

Fuel Type Year KM Fuel Type

0.191 0.100 0.100

E2.2

Year KM Comm. Dis. Comm. Dis.

Fuel Type Year KM Fuel Type

0.191 0.038 0.100

E2.1

Fuel Type Fuel Type Fuel Type Year KM Year KM Year KM Comm. Dis. Comm. Dis. Comm. Dis.

Fuel Type Year KM Comm. Dis. Fuel Tyue Year KM Comm. Dis. Fuel Type Year KM Comm. Dis.

0.714 0.014 0.153 0.816 0.816 0.816 0.751 0.751 0.751

92

L. VAN WISSEN and T. F. GOLOB Table 7. Direct and total elasticities

of fuel-type

model

Direct Elasticity Sub-matrix:

From:

To:

Value

B I.1

Year KM Year KM Year KM Comm. Dis. Comm. Dis. Comm. Dis. Comm. Dis.

Fuel Fuel Fuel Fuel Fuel Fuel Year

Type: Type: Type: Type: Type: Type: KM

benzine diesel L.P.G. benzine diesel L.P.G.

-.21 .16 .45 -

B 2.2

Year KM Year KM Year KM Comm. Dis. Comm. Dis. Comm. Dis. Comm. Dis.

Fuel Fuel Fuel Fuel Fuel Fuel Year

Type: Type: Type: Type: Type: Type: KM

benzine diesel L.P.G. benzine diesel L.P.G.

-.18 .12 .34 -

B 2.1

Fuel Type: benzine Fuel Type: diesel Fuel Type: L.P.G. Fuel Type: benzine Fuel Type: diesel Fuel Type: L.P.G. Year KM Year KM Year KM Year KM Comm. Dis. Comm. Dis. Comm. Dis. Comm. Dis.

Year KM Year KM Year KM Comm. Dis. Comm. Dis. Comm. Dis. Fuel Type: benzine Fuel Type: diesel Fuel Type: L.P.G. Comm. Dis. Fuel Type: benzine Fuel Type: diesel Fuel Type: L.P.G. Year KM

-.08 .Ol .12 - .05 .Ol .32 -

.61

.24

-

Total Elasticity

Standard Deviation

Value

Standard Deviation

.40 .46 .18 .18

-.21 .16 .45 -.12 .11 .28 .61

.40 .46 .18 .16 .23 .I4 .18

.34 .35 .14 .07

-.18 .12 .34 - .05 .02 .09 .24

.34 .35 .14 .06 .08 .05 .07

.08 .02 1.91 .05 .04 2.63 -

-.ll .Ol .I7 - .os .Ol .32 - .30 .23 .63 .Ol -.21 .18 .46 .78

.12 .03 2.71 .05 .04 2.63 .55 .60 .25 .OO .23 .34 .21 .26

Fig. 5. Travel subsidy leads to a stronger negative elasticity for choice of gasoline cars and much lower positive elasticity for choice of diesel cars. There is also a slightly higher positive elasticity for choice of LPG cars. The effect of a non-fixed work location on choice elasticities is graphed in Fig. 6. Households with primary workers with variable work location are much more sensitive to annual car usage with respect to ownership of gasoline cars. There is also a negative effect on the probability of choosing diesel, whereas in households with fixed work locations this elasticity is positive. Finally, Fig. 7 gives the mean elasticities for households with and without rail season tickets. Season ticket holding indicates regular train use, and this has a profound effect of fuel type choice. The absolute values of the choice elasticities for households with a rail season ticket are greater for diesel and LPG, but less for gasoline, due to the existing choice distribution for these households; households with rail season tickets own fewer alternative fuel vehicles. 5.2.2. Commuting distance to fuel type. There is no direct elasticity involved in this link at either point in time, but there is an indirect influence of commuting distance on fuel type choice through annual car usage. The elasticities are therefore roughly equal to the product of the direct elasticity of commuting distance on annual car usage (0.61) and the direct elasticity of annual car usage on fuel type choice at time t,. At time t2, the direct elasticity of commuting distance on annual car usage is only 0.24, and therefore the elasticities of commuting distance on fuel type choice is much lower than at time t,. Thus, the model indicates an elasticity of commuting distance on choice of gasoline in range of -0.05 to -0.12; on choice of diesel in the range 0.02 to 0.11; and on choice of LPG in the range 0.09 to 0.28. 5.2.3. Fuel type at t, to car usage at tZ.This dynamic effect represents the travel generating effect of the lower operating costs associated with alternative fuels in the

Dynamic model of car fuel-type choice and mobility

Year

KM t=l

to Fuel Type

t=l

and travel subsidy No Subsidy Slrbsidized

/

benzine

diesel Fuel Type

1.p.g.

Fig. 5. Total elasticities of car usage at t = 1 to fuel types at t = 1 with and without travel subsidies.

Netherlands, particularly the relatively low cost of LPG. The coefficient of the causal link from fuel type to annual car usage is not as strong as that from fuel type to commuting distance. There is a slight negative elasticity from choice of gasoline, indicating that a one percent increase in the probability of choosing gasoline at time t, has a negative direct effect of 0.08 percent on annual car usage at t2. The total elasticity is larger in magnitude (-0.1 l), because there is an additional indirect link through commuting distance. The direct elasticity reflects the effect on non-work related travel, which is higher than on commuting distance. The elasticity of choice of diesel cars is almost zero, but there is a positive elasticity of choice of LPG on annual car usage of 0.12 (direct) and 0.17 (total). Caution is necessary in interpreting this result, because the standard deviation of the LPG elasticities are very high, in part a consequence of the relatively low percentage of LPG cars in the sample. The effects of employer travel reimbursements and rail season tickets on the travel generating effects of alternative fuels are graphed in Figs. 8 and 9. Travel reimbursements

Year

KM t=l

to Fuel Type

and Non-Fixed

Work

t=l

Locahbn --c

Only Fixed Location t Non-Fixed Location

I

-0.5 J benzine

diesel Fuel Type

1.p.g.

Fig. 6. Total elasticities of car usage at t = I to fuel types at t = 1 for fixed and non-fixed work locations. 1RlBl

26:1-o

L. VAN WISSEN and T. F. GOLOB

0.6 -I

I

benzine

dies4 Fuel Type

1.p.g.

Fig. 7. Total elasticities of car usage oft = 1 to fuel types at t = 1 with and without train season ticket.

(Fig. 8) have a substantial impact on the influences of choice of LPG versus gasoline on annual car usage. For subsidized households, the positive effect of choice of LPG on car usage is much stronger (an elasticity of 0.33) than for the non-subsidized households (0.05). And for subsidized households, choice of gasoline implies less reduction in travel (- .05) than for non-subsidized households (- .14). Thus, employer reimbursements of travel costs mitigate the travel-reducing effects of the more expensive fuel, gasoline, and reinforce the travel-generating effect of the cheaper fuel, LPG. For rail season tickets, it is reversed (Fig. 9). Rail tickets reinforce the negative effects of travel demand of the more expensive fuel (an elasticity of -0.24 for regular train users, versus -0.10 for other households), and remove completely the latent demand effect of the cheaper fuel. 5.2.4. Fuel type at t, to commuting distance at t2. Here the pattern is similar to that for annual car usage, but with some differences in the levels of the effects. The elasticity of gasoline choice on commuting distance (-0.05) is lower than that on annual car usage

Fuel Type

benrine

t=l

to Year

and travel

subsidy

diesel Fuel Type

KM t=2

1.p.g.

Fig. 8. Total elasticities of fuel types at t = 1 to car usage at t = 2 with and without travel subsidy.

Dynamic

model of car fuel-type

Fuel Type

t=l

and Train

choice and mobility

to Year Season

95

KM t=2

Tickets

0.2 O.lS-

No Ticket holder

0.1

-0.2 -0.25 benzine

Fig. 9. Total elasticities

of fuel types at

diesel Fuel Type

1.p.g.

t = 1 to car usage at t =

2 with and without

train season ticket.

(- 0.08),

a difference which is statistically significant (Table 7). Non-work travel is more sensitive than work travel to choice of gasoline. However, choice of LPG has a relatively greater influence on commuting distance than on annual car usage. Caution must be taken in interpreting this result since the standard deviation of the LPG elasticity is very high. 5.2.5. Car usage at t, to fuel type at t2. These “dynamic” or lagged elasticities are much higher than their contemporaneous equivalents. For instance, the dynamic total elasticity of annual car usage on the probability of choosing gasoline in the next period is -0.30, whereas it is only in the range from -0.18 to -0.21 for contemporaneous linkages. Because a change of fuel type generally implies a change of car (as opposed to retrofit), it is likely that lagged effects of mobility play an important role. These elasticities are generated solely by direct effects: according to the model, high annual usage at t, has a negative impact on choosing a gasoline car at t,, which in turn has a diminishing effect on travel at time t2, which in turn negatively affects the probability of choosing benzine at time t,. The total result is a relatively large negative elasticity at time t,. For diesel and LPG a similar structure leads to a high probability of having these fuel types as a result of high car usage in previous years. For LPG this lagged elasticity is particularly high (0.63). 5.2.6. Commuting distance at t, to fuel type t = 2. This is again an indirect effect and the story is similar to the previous lagged elasticity. However, the elasticities of Commuting Distance are somewhat lower than those of annual car usage. Choice of an LPG car is again the most sensitive to changes in mobility: the total elasticity of commuting distance of future choice of LPG is 0.46. Clearly, commuting needs were a major determinant of the choice of LPG cars in the Netherlands in 1984-1988.

6. CONCLUSIONS

The model captures dynamic relationships among car fuel-type choice and car mobility, conditional on the effects of commuting subsidies, fixed and variable work locations, rail season tickets, and certain household socio-demographic and income variables. Car mobility is defined in terms of overall usage and commuting distance. The model is a joint continuous/discrete multivariate demand model with lagged effects, individualspecific terms, period effects, and compensation for panel attrition bias. The non-normal endogenous variables are treated as ordered probit and tobit variables. The elasticity estimates calculated for the observed non-normal variables could be useful in policy evaluation and forecasting.

L.

96

VAN

WISSENand T. F. GOLOB

Acknowledgements-This research was conducted for Bureau Goudappel Coffeng, Deventer, the Netherlands, ultimately for the Traffic Engineering Division, Public Works Administration, Dutch Ministry of Transport and Public Works. However, the views expressed are solely those of the authors. The participation of Leo van Wissen was made possible in part by a fellowship of the Koninkijke Nederlandse Academic van Wetenschappen. The authors would like to thank Henk Meurs of MuConsult, Utrecht, the Netherlands, and Toon van der Hoorn of the Dutch Ministry of Transport and Public Works for their support and encouragement. Co Holzapfel and his colleagues at Bureau Goudappel Coffeng expertly prepared the data. The authors also benefitted from critical comments from several anonymous referees. Naturally, the authors are solely responsible for any remaining errors of commission or omission. An earlier version of this paper was presented at the Annual Meeting of the Transportation Research Board, Washington, DC, January 13-17, 1991.

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