# Estimating Productivity Loss Cost according to Severity of Vehicle Crash Injury

Authors: Rong-Chang Jou; Tzu-Ying Chen
Journal: Journal of Advanced Transportation
 (2019)

---

## Abstract

This study estimates the productivity loss cost according to the severity of vehicle crash injury. A contingent valuation survey was conducted to estimate the willingness to pay (WTP) of vehicle crash offenders in Taiwan. In addition, a Double-Hurdle model was used to deal with the large number of zero WTP responses. The results show that the estimated productivity loss cost of vehicle crash ranges from 2,000 USD to 47,000 USD. In addition, as expected, the individuals’ WTP is positively related with education, average monthly income, share of vehicle crash responsibility, experience of moderate or worse vehicle crash injury, work days lost after a vehicle crash, and experience of receiving compensation for productivity loss from a vehicle crash. This study also demonstrates that the Double-Hurdle model was statistically superior to the Tobit model.


---

## Body

## 1. Introduction

Injuries and deaths from vehicle crashes are a global problem, and current trends show that by 2030, vehicle crash injuries will become the fifth leading cause of death. Such injuries burden low- and middle-income countries with an estimated US$100 billion per year [1]. Owing to this significant economic burden, in addition to the increasing numbers of injuries and fatalities, methods to estimate vehicle crash costs have increasingly become important.The economic costs of vehicle crashes have been extensively studied [2–6]. Although BRS and TRL [7] reports have indicated the crash costs of traffic road include property damage, medical costs, lost output, human costs, and administrative costs, most studies focus on the short-term medical costs of vehicle crash victims. Blincoe et al. [8] show that lost market and household productivity due to vehicle crashes in the U.S. accounted for $93 billion of the total $277 billion economic costs, but fewer studies discuss costs related to reduction in quality of life [9–12] and lost output, such as the temporary disability, long-term productivity loss, and productivity loss from death [13–16]. Our previous study show that productivity loss cost accounts for up to 50 per cent of vehicle crash costs in Taiwan [17]. In addition to injuries and deaths, vehicle crash victims can also incur a disability, which involves partial or total loss of an individual’s value, or ability to function. These outcomes mean that victims who die can no longer work for or offer their services to their families or companies, and victims who become disabled incur increased long-term care expenses. Therefore, if we cannot estimate the productivity loss from disability and death, we will underestimate the total vehicle crash costs. Since productivity loss from vehicle crashes is substantial, it should be estimated from vehicle crash costs appropriately and separately to give decision makers (e.g. policymakers, courts, insurance companies, and individuals involved) sufficient and clear information about the impact of different injury levels.Taiwanese courts have not yet clearly defined compensation for productivity loss, so the public is not familiar with this type of compensation and does not know whether it is possible to make a claim for this type of compensation. In order to monetise the productivity loss cost of vehicle crashes, this study estimates individuals’ Willingness To Pay (WTP) for productivity loss cost according to the severity of vehicle crash injury. We distributed 100,000 questionnaires to offenders and victims who had been involved in a vehicle crash in 2010 to inquire about the compensation they paid and received, respectively, for productivity loss in order to understand their perceptions of reasonable compensation levels.On the other hand, Contingent Valuation (CV) is a stated preference method that has been widely used to reveal information about the values associated with public and nonmarket goods. As individual preferences for the provision of nonpriced goods and services are not readily determined from observable behaviour, CV is a useful tool that allows values to be associated with such goods and services [18]. It is an assessment method based on hypothetical questions aimed at allowing the calculation of the monetary value of public goods or policies [19]. CV can be implemented through four formats: open-ended, iterative bidding, multiple-bounded dichotomous choice, and payment card. It has been broadly applied in different areas, such as environment [20–22] and transportation, including for vehicle crash costs [14, 23], carbon offsets [24–26], and air pollution [27, 28]. In this study, because sanctions for productivity loss of different severity caused by traffic vehicle crash have not been implemented in Taiwan, the economic cost of such measures to the parties involved cannot be evaluated. Thus, this study utilised the CV method to obtain interviewees’ subjective valuation of compensation for productivity loss caused by vehicle crashes; allows interviewees to express their WTP based on their experience or knowledge but determining the maximum amount of money that respondents are willing to pay as compensation for productivity loss caused by a vehicle crash as to obtain a more effective WTP.In our surveys on WTP, we have identified a large number of respondents who are not willing to pay (zero WTP) for vehicle crash, generating a large proportion of observed values censored at zero and resulting in samples that apparently do not follow normal distributions. That is, in determining the compensation amount, responses with a zero WTP price are considered to address the complications from many zero WTP price responses. The classic ordinary least squares regression could not provide consistent and unbiased estimates of parameters [29]. To address this issue, some studies have used the Tobit model [30], which was developed to recognise that WTP values are censored at zero and to minimise the bias and inconsistency of parameter estimates.Although the Tobit model is suitable for handling the problem of censored dependent variables, it has two limitations. First, it assumes that the zeros arise purely, that is, are a corner solution, since it also assumes that the zeros are the result of the respondent’s economic circumstances [31]. Second, it assumes that the individual’s zero response is generated from the same process by which the same variables affect the probability of a nonzero observation (the participation decision), as well as the level of a positive observation (the expenditure decision), and moreover, with the same sign. In order to address these shortcomings, this study uses the Double-Hurdle model developed by Cragg [32] and previously applied by Eakins [33]. The model postulates that two hurdles must be overcome before we can observe a positive expenditure amount. The first hurdle corresponds to factors affecting participation in the market for the good, and the second to the level of expenditure on the good. That is, the model may be more reasonable in allowing the factors of the two decisions to be different.The application of the Double-Hurdle model in empirical research began with the works of Jones [34], Yen and Su [35, 45], Goodwin et al. [36], and can be seen in more recent studies [37]. Their studies indicate that with the Double-Hurdle model, an individual has to overcome two hurdles in order to have a positive expenditure. First, the individual decides whether or not to participate in the market, a stage known as the participation stage (of a potential consumer). Next, the individual decides the level of expenditure, a stage referred to as the expenditure stage.This study aims to offer a comprehensive framework (Figure1) for estimating productivity loss cost and to compare the costs for different injury levels using two estimation methods. In this study, we apply CV to estimate the productivity loss incurred by vehicle crash victims. Apart from estimating productivity loss cost, this study considers that the samples may include a high proportion of zero responses by offenders, that is, a high proportion of offenders who are not willing to pay for productivity loss compensation. Therefore, two econometric models are applied. First, we apply the Tobit model, which assumes that all the respondents are willing to pay for productivity loss from a vehicle crash. Second, for comparison, we apply the Double-Hurdle model to deal with the issue of zero responses and take into account the many individuals who are willing to participate in this market and willing to pay for productivity loss from a vehicle crash.Figure 1
The comprehensive framework of the study.The rest of this paper is structured as follows. Section2 describes the construction of the Double-Hurdle model. Section 3 describes the data and analysis. Section 4 shows the results of the model estimation and the analysis of elasticities. Finally, Section 5 presents the conclusions and recommendations.


## 2. Econometric Methodology

### 2.1. Tobit Model

This study aims to discuss the WTP of parties for productivity loss caused by traffic accidents. It is very common that many observations are with zero values because not every party has the same experience of compensation for these types of loss, nor is there a prescribed fine, especially for the death party of a vehicle crash. The standard Tobit model assumes that all respondents have the same needs for compensating the cost caused by traffic crashes, and in the mind of respondenti, he/she is willing to pay Y
i
∗ to compensate for the damage caused by an crash. The function is described in Equation (1). If there are n respondents in total, the mean of errors (ε
i) is zero, σ
2 is the variance, and the samples follow Independent and Identical Distributed (IID) normal distribution.(1)
Y
i
∗
=
β
X
i
+
ε
i
,
ε
i
∼
N
0
,
σ
2
,whereY
i
∗ is the dependent variable and also a random variable (WTP to compensate for the productivity loss caused by a crash); β is a vector of coefficients to be estimated; X
i is a vector of explanatory variables (socioeconomic and accident related characteristics). When Y
i
∗ is greater than zero, the WTP of respondent i, Y
i, will be equal to the WTP in his or her mind, and in that case, the actual WTP in his or her mind can be observed. If the WTP in the mind of respondent i is less than or equal to zero, then WTP is zero, which is the lower limit of WTP. That is, the value of zero may be either an actual WTP or the observed value of a WTP with a negative value, as shown in the following Equation (2) (Since the designed scenarios provided to respondents were assumed that they were all perpetrators, there has no possibility that Y
∗<0. Nevertheless, the way equation (2) is formulated will not affect results since the cumulative probability of Y
∗<0 is always zero.):(2)
Y
i
=
Y
i
∗
,
if
Y
i
∗
>
0
0
,
if
Y
i
∗
≤
0
.When the WTP of respondents is greater than zero, the probability density function of the sample can be expressed as follows:(3)
Prob
Y
i
>
0
f
Y
i
Y
i
>
0
=
1
σ
ϕ
i
Y
i
−
β
X
i
σ
,whereϕ
i is the standard normal probability density function. When observed values are censored at zero, the function of the continuous random variables is not a probability density function. Therefore, the probability distribution of observed values of zero WTP follows a cumulative distribution function. That is, when respondents’ WTP (Y
i) is zero, the actual WTP in their minds may be either zero or less than zero, and thus, the probability function of the sample can be expressed as follows:(4)
Prob
Y
i
=
0
f
Y
i
≤
0
=
1
−
Φ
i
β
X
i
σ
.In Equation (4), Φ
i is the standard normal cumulative distribution function. Next, the likelihood function of the standard Tobit model can be derived from Equations (3) and (4), as shown in Equation (5):(5)
L
=
∏
Y
i
=
0
1
−
Φ
i
β
X
i
σ
×
∏
Y
i
>
0
1
σ
ϕ
i
Y
i
−
β
X
i
σ
.


### 2.2. Double-Hurdle Model

Jones [38] was the first to define the structure of the Double-Hurdle model, but we, nevertheless, describe the model in this subsection to help ensure clarity. For observation i, let D
t
∗ be defined as an unobserved variable representing the decision whether to participate or not (willingness or unwillingness to pay for productivity loss). Let Y
t
∗ be the latent variable representing the value of an individual’s actual WTP. The two latent variables are described as follows:(6)
D
i
∗
=
x
1
i
α
i
+
u
i
,
u
i
∼
N
0
,
σ
2
Participation
decision
,
(7)
Y
i
∗
=
x
2
i
β
i
+
ν
i
,
ν
i
∼
N
0
,
σ
2
Expenditure
decision
,whereu
i
,
ν
i
~
B
V
N
0
,
Σ
,
Σ
=
1
σ
ρ
σ
ρ
σ
2, and x
i is a vector of variables representing a set of individual characteristics explaining the participation and expenditure decisions, respectively. In this study, we assume that the participation and expenditure decisions have the same set of explanatory variables, that is, x
1
i
=
x
2
i
=
x
i. α and β are vectors of parameters that enter the first and second hurdles. The observed expenditure, Y
i, relates to the latent variables Y
i
∗ and D
i
∗, which can be expressed as follows:(8)
Y
i
=
Y
i
∗
,
if
Y
i
∗
>
0 and
D
i
∗
>
0
=
0
,
if
Y
i
∗
≤
0 and
D
i
∗
>
0
or
Y
i
∗
>
0 and
D
i
∗
≤
0
or
Y
i
∗
≤
0 and
D
i
∗
≤
0
.Using 0 to denote zero observations and + to denote positive observations, this study writes the model’s sample likelihood as(9)
L
=
∏
0
1
−
F
u
i
>
−
x
1
i
α
,
ν
i
>
−
x
2
i
β
∏
+
F
u
i
>
−
x
1
i
α
,
ν
i
>
−
x
2
i
β
f
ν
i
u
i
>
−
x
1
i
α
,
ν
i
>
−
x
2
i
β
.According to Gao et al. [39], Equation (9) can be simplified as(10)
L
=
∏
0
1
−
Φ
−
x
1
i
α
i
,
−
x
2
i
β
i
σ
,
ρ
×
∏
+
Φ
x
1
i
α
i
+
ρ
σ
i
Y
i
∗
−
x
2
i
β
i
1
−
ρ
2
1
σ
ϕ
Y
i
∗
−
x
2
i
β
i
σ
i
,whereΦ
· and ϕ
· are the cumulative distribution function (cdf) and standard normal probability density functions (pdf), respectively, for a standard normal random variable.


### 2.3. Verification

Following Goodwin et al. [36], we compare the models by the values of the log-likelihood functions of the Tobit, Probit, and truncated models and determine whether the Double-Hurdle model outperforms the Tobit model (Equation (11)). Assuming the three equations have the same set of explanatory variables, λ will be viewed as a χ
2 distribution with degrees of freedom equal to the number of explanatory variables under the null hypothesis that the Tobit model is the correct specification:(11)
λ
=
−
2
L
T
−
L
P
−
L
TR
,whereL
T is the likelihood value for the Tobit model, L
P is the likelihood value for the Probit model, and L
TR is the likelihood value for the truncated model.


### 2.4. Elasticities

Following Anastasopoulos et. al. [40], the expectation of the standard Tobit model can be expressed as:(12)
E
Y
=
β
X
F
β
X
i
σ
+
σ
f
β
X
i
σ
,whereβ
X
i
/
σ is z-score for an area under, which follows the normal distribution, F
β
X
i
/
σ is the cumulative normal distribution function and is related to the proportion of events greater than zero, f
β
X
i
/
σ is the probability density function, and σ is the standard deviation of the error term. If E
Y
′ is the expectation of observed values that are greater than zero, then E
Y can be expressed as:(13)
E
Y
=
F
β
X
i
σ
E
Y
′
.Refer to the Amemiya [41, 42], the impact of explanatory variables on expectations can be calculated using the first derivative of Equation (12) and is shown as follows:(14)
E
Y
′
=
E
Y
Y
>  0
=
E
Y
ε
>
−  β
X
i
=
β
X
i
+
σ
f
β
X
i
/
σ
F
β
X
i
/
σ
.The first-order partial derivative of the specific explanatory variables (X
k) in Equation (13), derived by McDonald and Moffitt [43], is shown in Equation (15):(15)
∂
E
Y
∂
X
k
=
F
β
X
i
σ
∂
E
Y
′
∂
X
k
+
E
Y
′
∂
F
β
X
i
/
σ
∂
X
k
.∂
E
Y
/
∂
X
k is the change in the overall expectation with respect to the explanatory variable; ∂
E
Y
′
/
∂
X
k is the change in the expectation of observed values greater than zero with respect to the explanatory variable (by weighting the probability of observed values that are greater than zero); ∂
F
β
X
i
/
σ
/
∂
X
k is the change in the cumulative probability of observed values greater than zero with respect to the explanatory variables (by weighting the expectation of observed values that are greater than zero) [44]. Equation (15) can be derived by estimating ∂
E
Y
′
/
∂
X
k and ∂
F
z
/
∂
X
k using β
k and σ and then integrating with F
β
X
i
/
σ, f
β
X
i
/
σ and β
k
X
k (X
k is the mean of the observed values). Equation (16) shows that changes in the expectation of observed values greater than zero with respect to the explanatory variable is:(16)
∂
E
Y
′
∂
X
k
=
β
k
1
−
β
X
i
σ
×
f
β
X
i
/
σ
F
β
X
i
/
σ
−
f
β
X
i
/
σ
2
F
β
X
i
/
σ
2whenX
=
∞, F
β
X
i
/
σ
=
1, and F
β
X
i
/
σ
=
0 [43]. However, the range of WTP in our study is not infinite, and thus, Equation (16) should be used to calculate the marginal effect of observed values that are greater than zero.Secondly, following Yen and Su [35, 45], we rewrite the probabilities of participation and expenditure of Double-Hurdle model as(17)
P
D
i
∗
>
0
=
Φ
x
1
i
α
i
,
(18)
P
Y
i
>
0
=
Pr
D
i
∗
>
0
,
Y
i
∗
>
0
=
Φ
x
1
i
α
i
Φ
x
2
i
β
i
σ
i
.The marginal effects represent the probability change when the explanatory variable is the dummy variable, which shifts from zero to one, holding all the other variables constant. Based on the maximum likelihood estimated (MLE) parameters, several types of elasticities can be calculated: the so-called elasticity of participation, elasticity of the probability of expenditure, and finally, elasticity of the unconditional level of expenditure or the total elasticity [46].The derivative of the participation probability (17) with respect tox
i
j is(19)
∂
P
D
i
∗
>
0
∂
x
i
j
=
∂
Φ
x
1
i
α
i
∂
x
i
j
=
ϕ
x
1
i
α
i
α
j
.The marginal effect ofx
i
j on the probability of expenditure is(20)
∂
P
Y
i
>
0
∂
x
i
j
=
∂
Φ
x
1
i
α
i
Φ
x
2
i
β
i
/
σ
i
∂
x
i
j
=
Φ
x
2
i
β
i
σ
i
ϕ
x
1
i
α
i
α
j
+
Φ
x
1
i
α
i
ϕ
x
2
i
β
i
σ
i
σ
i
−
1
×
β
j
−
x
2
i
β
i
σ
i
×
∂
σ
i
∂
x
i
j
.From Equations (18) and (20), it is obvious that the probability of expenditure depends on both the first-hurdle (α) and second-hurdle parameters (β). According to Amemiya [42], the conditional mean of Y
i is(21)
E
Y
i
Y
i
∗
>
0
=
x
2
i
β
i
+
σ
i
ϕ
x
2
i
β
i
/
σ
i
Φ
x
2
i
β
i
/
σ
i
.Another elasticity, the conditional mean ofY
i with respect to x
i
j, can then be calculated as follows [35, 45]:(22)
∂
E
Y
i
Y
i
>
0
∂
x
i
j
=
β
j
+
ϕ
x
2
i
β
i
/
σ
i
Φ
x
2
i
β
i
/
σ
i
∂
σ
i
∂
x
i
j
−
ϕ
x
2
i
β
i
/
σ
i
Φ
x
2
i
β
i
/
σ
i
×
β
j
−
x
2
i
β
i
/
σ
i
∂
σ
i
∂
x
i
j
×
x
2
i
β
i
σ
i
+
ϕ
x
2
i
β
i
/
σ
i
Φ
x
2
i
β
i
/
σ
i
.


## 2.1. Tobit Model

This study aims to discuss the WTP of parties for productivity loss caused by traffic accidents. It is very common that many observations are with zero values because not every party has the same experience of compensation for these types of loss, nor is there a prescribed fine, especially for the death party of a vehicle crash. The standard Tobit model assumes that all respondents have the same needs for compensating the cost caused by traffic crashes, and in the mind of respondenti, he/she is willing to pay Y
i
∗ to compensate for the damage caused by an crash. The function is described in Equation (1). If there are n respondents in total, the mean of errors (ε
i) is zero, σ
2 is the variance, and the samples follow Independent and Identical Distributed (IID) normal distribution.(1)
Y
i
∗
=
β
X
i
+
ε
i
,
ε
i
∼
N
0
,
σ
2
,whereY
i
∗ is the dependent variable and also a random variable (WTP to compensate for the productivity loss caused by a crash); β is a vector of coefficients to be estimated; X
i is a vector of explanatory variables (socioeconomic and accident related characteristics). When Y
i
∗ is greater than zero, the WTP of respondent i, Y
i, will be equal to the WTP in his or her mind, and in that case, the actual WTP in his or her mind can be observed. If the WTP in the mind of respondent i is less than or equal to zero, then WTP is zero, which is the lower limit of WTP. That is, the value of zero may be either an actual WTP or the observed value of a WTP with a negative value, as shown in the following Equation (2) (Since the designed scenarios provided to respondents were assumed that they were all perpetrators, there has no possibility that Y
∗<0. Nevertheless, the way equation (2) is formulated will not affect results since the cumulative probability of Y
∗<0 is always zero.):(2)
Y
i
=
Y
i
∗
,
if
Y
i
∗
>
0
0
,
if
Y
i
∗
≤
0
.When the WTP of respondents is greater than zero, the probability density function of the sample can be expressed as follows:(3)
Prob
Y
i
>
0
f
Y
i
Y
i
>
0
=
1
σ
ϕ
i
Y
i
−
β
X
i
σ
,whereϕ
i is the standard normal probability density function. When observed values are censored at zero, the function of the continuous random variables is not a probability density function. Therefore, the probability distribution of observed values of zero WTP follows a cumulative distribution function. That is, when respondents’ WTP (Y
i) is zero, the actual WTP in their minds may be either zero or less than zero, and thus, the probability function of the sample can be expressed as follows:(4)
Prob
Y
i
=
0
f
Y
i
≤
0
=
1
−
Φ
i
β
X
i
σ
.In Equation (4), Φ
i is the standard normal cumulative distribution function. Next, the likelihood function of the standard Tobit model can be derived from Equations (3) and (4), as shown in Equation (5):(5)
L
=
∏
Y
i
=
0
1
−
Φ
i
β
X
i
σ
×
∏
Y
i
>
0
1
σ
ϕ
i
Y
i
−
β
X
i
σ
.


## 2.2. Double-Hurdle Model

Jones [38] was the first to define the structure of the Double-Hurdle model, but we, nevertheless, describe the model in this subsection to help ensure clarity. For observation i, let D
t
∗ be defined as an unobserved variable representing the decision whether to participate or not (willingness or unwillingness to pay for productivity loss). Let Y
t
∗ be the latent variable representing the value of an individual’s actual WTP. The two latent variables are described as follows:(6)
D
i
∗
=
x
1
i
α
i
+
u
i
,
u
i
∼
N
0
,
σ
2
Participation
decision
,
(7)
Y
i
∗
=
x
2
i
β
i
+
ν
i
,
ν
i
∼
N
0
,
σ
2
Expenditure
decision
,whereu
i
,
ν
i
~
B
V
N
0
,
Σ
,
Σ
=
1
σ
ρ
σ
ρ
σ
2, and x
i is a vector of variables representing a set of individual characteristics explaining the participation and expenditure decisions, respectively. In this study, we assume that the participation and expenditure decisions have the same set of explanatory variables, that is, x
1
i
=
x
2
i
=
x
i. α and β are vectors of parameters that enter the first and second hurdles. The observed expenditure, Y
i, relates to the latent variables Y
i
∗ and D
i
∗, which can be expressed as follows:(8)
Y
i
=
Y
i
∗
,
if
Y
i
∗
>
0 and
D
i
∗
>
0
=
0
,
if
Y
i
∗
≤
0 and
D
i
∗
>
0
or
Y
i
∗
>
0 and
D
i
∗
≤
0
or
Y
i
∗
≤
0 and
D
i
∗
≤
0
.Using 0 to denote zero observations and + to denote positive observations, this study writes the model’s sample likelihood as(9)
L
=
∏
0
1
−
F
u
i
>
−
x
1
i
α
,
ν
i
>
−
x
2
i
β
∏
+
F
u
i
>
−
x
1
i
α
,
ν
i
>
−
x
2
i
β
f
ν
i
u
i
>
−
x
1
i
α
,
ν
i
>
−
x
2
i
β
.According to Gao et al. [39], Equation (9) can be simplified as(10)
L
=
∏
0
1
−
Φ
−
x
1
i
α
i
,
−
x
2
i
β
i
σ
,
ρ
×
∏
+
Φ
x
1
i
α
i
+
ρ
σ
i
Y
i
∗
−
x
2
i
β
i
1
−
ρ
2
1
σ
ϕ
Y
i
∗
−
x
2
i
β
i
σ
i
,whereΦ
· and ϕ
· are the cumulative distribution function (cdf) and standard normal probability density functions (pdf), respectively, for a standard normal random variable.


## 2.3. Verification

Following Goodwin et al. [36], we compare the models by the values of the log-likelihood functions of the Tobit, Probit, and truncated models and determine whether the Double-Hurdle model outperforms the Tobit model (Equation (11)). Assuming the three equations have the same set of explanatory variables, λ will be viewed as a χ
2 distribution with degrees of freedom equal to the number of explanatory variables under the null hypothesis that the Tobit model is the correct specification:(11)
λ
=
−
2
L
T
−
L
P
−
L
TR
,whereL
T is the likelihood value for the Tobit model, L
P is the likelihood value for the Probit model, and L
TR is the likelihood value for the truncated model.


## 2.4. Elasticities

Following Anastasopoulos et. al. [40], the expectation of the standard Tobit model can be expressed as:(12)
E
Y
=
β
X
F
β
X
i
σ
+
σ
f
β
X
i
σ
,whereβ
X
i
/
σ is z-score for an area under, which follows the normal distribution, F
β
X
i
/
σ is the cumulative normal distribution function and is related to the proportion of events greater than zero, f
β
X
i
/
σ is the probability density function, and σ is the standard deviation of the error term. If E
Y
′ is the expectation of observed values that are greater than zero, then E
Y can be expressed as:(13)
E
Y
=
F
β
X
i
σ
E
Y
′
.Refer to the Amemiya [41, 42], the impact of explanatory variables on expectations can be calculated using the first derivative of Equation (12) and is shown as follows:(14)
E
Y
′
=
E
Y
Y
>  0
=
E
Y
ε
>
−  β
X
i
=
β
X
i
+
σ
f
β
X
i
/
σ
F
β
X
i
/
σ
.The first-order partial derivative of the specific explanatory variables (X
k) in Equation (13), derived by McDonald and Moffitt [43], is shown in Equation (15):(15)
∂
E
Y
∂
X
k
=
F
β
X
i
σ
∂
E
Y
′
∂
X
k
+
E
Y
′
∂
F
β
X
i
/
σ
∂
X
k
.∂
E
Y
/
∂
X
k is the change in the overall expectation with respect to the explanatory variable; ∂
E
Y
′
/
∂
X
k is the change in the expectation of observed values greater than zero with respect to the explanatory variable (by weighting the probability of observed values that are greater than zero); ∂
F
β
X
i
/
σ
/
∂
X
k is the change in the cumulative probability of observed values greater than zero with respect to the explanatory variables (by weighting the expectation of observed values that are greater than zero) [44]. Equation (15) can be derived by estimating ∂
E
Y
′
/
∂
X
k and ∂
F
z
/
∂
X
k using β
k and σ and then integrating with F
β
X
i
/
σ, f
β
X
i
/
σ and β
k
X
k (X
k is the mean of the observed values). Equation (16) shows that changes in the expectation of observed values greater than zero with respect to the explanatory variable is:(16)
∂
E
Y
′
∂
X
k
=
β
k
1
−
β
X
i
σ
×
f
β
X
i
/
σ
F
β
X
i
/
σ
−
f
β
X
i
/
σ
2
F
β
X
i
/
σ
2whenX
=
∞, F
β
X
i
/
σ
=
1, and F
β
X
i
/
σ
=
0 [43]. However, the range of WTP in our study is not infinite, and thus, Equation (16) should be used to calculate the marginal effect of observed values that are greater than zero.Secondly, following Yen and Su [35, 45], we rewrite the probabilities of participation and expenditure of Double-Hurdle model as(17)
P
D
i
∗
>
0
=
Φ
x
1
i
α
i
,
(18)
P
Y
i
>
0
=
Pr
D
i
∗
>
0
,
Y
i
∗
>
0
=
Φ
x
1
i
α
i
Φ
x
2
i
β
i
σ
i
.The marginal effects represent the probability change when the explanatory variable is the dummy variable, which shifts from zero to one, holding all the other variables constant. Based on the maximum likelihood estimated (MLE) parameters, several types of elasticities can be calculated: the so-called elasticity of participation, elasticity of the probability of expenditure, and finally, elasticity of the unconditional level of expenditure or the total elasticity [46].The derivative of the participation probability (17) with respect tox
i
j is(19)
∂
P
D
i
∗
>
0
∂
x
i
j
=
∂
Φ
x
1
i
α
i
∂
x
i
j
=
ϕ
x
1
i
α
i
α
j
.The marginal effect ofx
i
j on the probability of expenditure is(20)
∂
P
Y
i
>
0
∂
x
i
j
=
∂
Φ
x
1
i
α
i
Φ
x
2
i
β
i
/
σ
i
∂
x
i
j
=
Φ
x
2
i
β
i
σ
i
ϕ
x
1
i
α
i
α
j
+
Φ
x
1
i
α
i
ϕ
x
2
i
β
i
σ
i
σ
i
−
1
×
β
j
−
x
2
i
β
i
σ
i
×
∂
σ
i
∂
x
i
j
.From Equations (18) and (20), it is obvious that the probability of expenditure depends on both the first-hurdle (α) and second-hurdle parameters (β). According to Amemiya [42], the conditional mean of Y
i is(21)
E
Y
i
Y
i
∗
>
0
=
x
2
i
β
i
+
σ
i
ϕ
x
2
i
β
i
/
σ
i
Φ
x
2
i
β
i
/
σ
i
.Another elasticity, the conditional mean ofY
i with respect to x
i
j, can then be calculated as follows [35, 45]:(22)
∂
E
Y
i
Y
i
>
0
∂
x
i
j
=
β
j
+
ϕ
x
2
i
β
i
/
σ
i
Φ
x
2
i
β
i
/
σ
i
∂
σ
i
∂
x
i
j
−
ϕ
x
2
i
β
i
/
σ
i
Φ
x
2
i
β
i
/
σ
i
×
β
j
−
x
2
i
β
i
/
σ
i
∂
σ
i
∂
x
i
j
×
x
2
i
β
i
σ
i
+
ϕ
x
2
i
β
i
/
σ
i
Φ
x
2
i
β
i
/
σ
i
.


## 3. Data Analysis

Generally speaking, the crash analyses in Taiwan usually use the crash database of the National Police Administration. The traffic accident records contain information regarding the crash, vehicle, and driver characteristics such as the crash times and locations, weather conditions, and road geometry features. Unfortunately, in the database, there is no driving behaviour and habit of perpetrators. In 2014, we had a special opportunity to sample and investigate the incident (including accident and the driving behaviour) of perpetrator in the crash event. It’s a large-scale mailing questionnaire survey over the year of crashes for perpetrators in Taiwan. Of the 218,814 suitable samples, we mailed 100,000 questionnaires that asked offenders and victims about their accident history, along with the amount of compensation paid and received, respectively. From the responses, incomplete answers on driver behaviour and habit questions and erroneous records were discarded, resulting in a final sample of 4,089 valid questionnaires; the overall recovery rate of questionnaire was approximately 4%. Excluding the respondents whose questionnaires had incomplete responses to WTP questions (did not provide WTP data), the final effective questionnaires consisted of 2,122 respondents. The process of questionnaire survey is shown in Figure2. The questionnaire was divided into three parts: the respondents’ socioeconomic characteristics, traffic crash history, and WTP. In part 3, in those scenario questions, we assume that the respondent is the perpetrator, and the crash caused a certain level of bodily injury, we then query the respondent’s WTP for productivity loss compensation. The respondents answered questions based on injury severity in hypothetical scenarios: minor injury, moderate injury, severe injury, disability, and death (Minor injury: includes general head, chest, abdominal, lumbar, upper limb, and lower limb injuries, as well as multiple traumas. Moderate injury: includes head fractures; thoracic, abdominal and lumbar fracture, and dislocation; upper limb fracture and dislocation; and lower limb fracture and dislocation. Serious injury: includes serious head injuries; major chest, abdominal and lumbar organ injuries; and injury to the central nervous system and spinal cord. Disability: includes serious injury but qualifies for disability certificate or disability. Death: loss of life). The open CV survey allowed the respondents to specify monetary expenditure for productivity loss. The respondents’ WTP was elicited through the part 3 questionnaire framework. For example, the minor injury scenario is described as follows.Figure 2
The process of a questionnaire survey.Assume that you are responsible for a vehicle crash that caused a victim’s minor injury. Tobit and Double-Hurdle modelsy for the victim’s productivity loss (the reference values of court verdicts for minor injuries are US$2,305 for productivity loss).(a)
Would you be willing to pay for the productivity loss?□ NO
□ YES
(b)
If ‘YES’, how much would you be willing to pay?□ Willing to pay, but zero amount (cannot afford the expenditure).
□ Willing to pay, and the highest amount is US$________.The questionnaire asked the respondents whether they were willing to pay for the productivity loss. Figure3 presents the total sample size and subsample sizes according to the different scenarios. The respondents with a zero WTP fell into two categories: with and without protest. Figure 4 presents the sample distribution in the decision market according to existence of protest (with and without protest) and average WTP according to injury severity. Figure 4(a) shows the distribution for the entire sample, Figure 4(b) for the without protest sample within WTP ≥ 0, and Figure 4(c) for the without protest sample within WTP > 0 in the market. Figure 4(a) shows that the protest sample had a higher percentage of minor and moderate injuries (17.201% and 19.934%, respectively), but Figures 4(b) and 4(c) show that the without protest sample had a higher percentage of a severe or worse injury. We expect that respondents will be more willing to pay if they are aware of the injury severity that caused the significant productivity loss.Figure 3
Sample sizes according to different scenarios (injury severity; with and without protest).Figure 4
Participation rates and average WTP by injury severity. Note: WTP stands for willingness to pay.The average WTP increased with severer injury levels. The average positive WTP of the without protest sample is US$1,644, US$5,806, US$23,269, US$47,372, and US$58,579, respectively, for each. The average WTP of the without protest sample for severe injury, disability, and death increased significantly (by over 40%), from US$16,142, US$33,181, and US$41,633, respectively, in the full sample to US$23,269, US$47,372, and US$58,579, respectively, in the positive WTP sample. However, the average WTP was higher for the sample without protest. Even though there have not been actual cases of productivity loss compensation in Taiwanese courts, it is still necessary to discuss the monetary value that vehicle crash perpetrators place on their victims’ injuries that result in productivity loss.The analysis results for the respondents’ WTP for productivity loss according to their socioeconomic characteristics are shown in Table1. First, we verified the procedure of questionnaires to be effective and assumed to follow a population distribution. The homogeneity test according to the characteristics of age, gender, and injury experience were following the population distribution and did not reject the null hypothesis. That is, the distribution of received back of mailed questionnaires was consistent with the sample population. In order to distinguish between the results for with and without protest, the table divides the samples into two categories, one for the sample of respondents who did not want to participate in the market (with protest) and another for the sample of respondents who wanted to participate (without protest), including both respondents with a zero WTP and those with a positive WTP. From the table, it can be observed that the majority of the total sample was female (55.1%), of whom a relatively high proportion had a positive WTP for productivity loss. The 20 or younger age group had a high proportion of respondents who protested (28.7%), while the other age groups had high proportions of respondents with a zero WTP and a positive WTP. Respondents with a university degree or above had a higher proportion of those with a positive WTP (71.8%), compared to those without such a degree. Meanwhile, respondents with an average individual monthly income of US$1,460 or higher had a greater proportion of those with a positive WTP (22.2%), compared to those with a lower monthly income. Most of those who protested were concentrated in the under-US$1,460 individual monthly income bracket (85.8%). The proportions of those with a positive WTP for productivity loss increased with the level of household monthly income, with the highest proportion in the over-US$2,305 household monthly income bracket (37.0%). The proportions of those who protested decreased with the level of household monthly income.Table 1
WTP according to socioeconomic characteristics.
Item
Nonparticipation
Participation
Total sample
(%)
Protest
(%)
Zero
(%)
Positive
(%)
Genderpearson chi-square test =
2.85
<
χ
1
,
0.05
2
Female
183
(55.3)
229
(54.0)
768
(56.2)
1169
(55.1)
Male
148
(44.7)
195
(46.0)
599
(43.8)
953
(44.9)
Agepearson chi-square test =
8
.73
<
χ
5
,
0.05
2
20 or younger
95
(28.7)
101
(23.7)
222
(16.2)
410
(19.3)
21–30
103
(31.1)
151
(35.6)
505
(36.9)
755
(35.6)
31–40
53
(16.0)
74
(17.5)
296
(21.7)
428
(20.2)
41–50
36
(10.9)
41
(9.7)
181
(13.2)
260
(12.3)
51–60
24
(7.3)
33
(7.8)
106
(7.8)
166
(7.8)
61 or older
20
(6.0)
24
(5.7)
57
(4.2)
103
(4.8)
Education
Below university
155
(46.8)
282
(66.5)
385
(28.2)
719
(33.9)
University or above
176
(53.2)
142
(33.5)
982
(71.8)
1403
(66.1)
Individual monthly income
1,459 or lower
284
(85.8)
400
(94.3)
1064
(77.8)
1748
(82.4)
1.460 or higher
47
(14.2)
24
(5.7)
303
(22.2)
374
(17.6)
Household monthly income
1,317 or lower
114
(34.5)
169
(39.9)
367
(26.8)
645
(30.4)
1,318–2,305
111
(33.5)
153
(36.1)
495
(36.2)
755
(35.6)
2,306 or higher
106
(32.0)
102
(24.0)
505
(37.0)
722
(34.0)
Total sample
331
424
1367
2122
Note: The percentage of respondents in each category is given in parentheses.Table2 presents the characteristics of vehicle crash injuries. Respondents who had third-party liability insurance for their car had the highest proportion of positive WTP (83.0%). In addition, among respondents who wanted to participate in the market, those who had a zero WTP and those who did not have third-party liability insurance had higher proportions of respondents with a positive WTP (59.8% and 63.2%, respectively). Respondents who took full responsibility for a vehicle crash had a higher proportion of positive WTP (36.5%) than zero WTP (17.2% and 10.4%), while respondents who took no responsibility for a vehicle crash had a large proportion of the protest sample (36.0%). The respondents were asked to state the severity of the injury they sustained in a previous vehicle crash, and the results show that most of the injuries were either minor or moderate (93.2%). However, respondents who indicated a severe or worse injury had the highest proportion of positive WTP (7.5%), and respondents who were unable to work for more than 30 days had higher proportions of participation in the market (16.8% and 16.2%) than non-participation in the market (6.3%). Although the percentage of respondents who received no compensation was still quite high (86.7%), the results show that respondents who received compensation in a previous vehicle crash had a higher proportion of positive WTP (16.5%). This finding suggests that these respondents were willing to pay for productivity loss because they experienced receiving compensation in a previous vehicle crash and therefore, had a better understanding of the importance of compensation.Table 2
Respondents’ accident history.
Item
Nonparticipation
Participation
Total sample
(%)
Protest
(%)
Zero
(%)
Positive
(%)
Third-party liability insurance
No
198
(59.8)
268
(63.2)
232
(17.0)
698
(32.9)
Yes
133
(40.2)
156
(36.8)
1135
(83.0)
1424
(67.1)
Vehicle crash responsibility
0%
119
(36.0)
169
(39.8)
523
(38.3)
811
(38.2)
25%
58
(17.5)
85
(20.0)
184
(13.5)
327
(15.4)
50%
54
(16.3)
66
(15.6)
82
(6.0)
202
(9.5)
75%
43
(13.0)
60
(14.2)
78
(5.7)
181
(8.5)
100%
57
(17.2)
44
(10.4)
500
(36.5)
601
(28.4)
Injury experiencepearson chi-square test =
2
.86
<
χ
1
,
0.05
2
Minor or moderate
316
(95.5)
397
(93.6)
1265
(92.5)
1978
(93.2)
Severe or worse
15
(4.5)
27
(6.4)
102
(7.5)
144
(6.8)
Work days lost
0
121
(36.6)
207
(48.8)
500
(36.6)
867
(38.0)
1–30
188
(56.8)
146
(34.4)
645
(47.2)
979
(46.1)
31 or more
22
(6.3)
58
(16.8)
222
(16.2)
315
(14.9)
Experience receiving compensation
No
305
(92.1)
393
(92.7)
1142
(83.5)
1840
(86.7)
Yes
26
(7.9)
31
(7.3)
225
(16.5)
282
(13.3)
Total sample
331
424
1367
2122
Note: The percentage of respondents in each category is given in parentheses.


## 4. Model Estimation Results and Discussion

### 4.1. Variable Definition

The definitions of variables used, and the means for the entire sample, and the positive expenditure samples are presented in Table3. The dependent variable is the WTP stated by the respondent in the open-ended question, while the independent variables include an educational level of university or above, an average individual monthly income of higher than US$1,460, an average household monthly income of higher than US$2,305, third-party liability insurance, experience of moderate or worse vehicle crash injury, over 50% share of vehicle crash responsibility, work days lost after a vehicle crash, experience of receiving compensation for productivity loss, and eight dummy variables that are equal to one if the interview was conducted in the sections called ‘EDU’, ‘GIN’, ‘HIN’, ‘INST’, ‘EXP’, ‘RESP’, ‘DAY’, and ‘COMP’ (please refer to Table 3 definition). The mean of the entire sample’s WTP is US$782–US$41,633. However, the mean for the subsample that indicated a positive WTP for productivity loss according to injury level and fatality is US$1,644–US$58,579.Table 3
Summary statistics.
Variable
Description
Injury level
Entire sample
Participation sample
Ave.
Min.
Max.
Ave.
Expend
Individual’s WTP for productivity loss (US$)
1
782
0
21,404
1,644
2
3,541
0
46,101
5,806
3
16,142
0
1,136,064
23,269
4
33,181
0
987,882
47,372
5
41,633
0
658,588
58,579
Dummy variables (yes = 1; 0 otherwise)
EDU
1 If individual had university degree or above, 0 otherwise
1
0.59
0
1
0.85
2
0.61
0
1
0.82
3
0.65
0
1
0.71
4
0.63
0
1
0.74
5
0.80
0
1
0.88
GIN
1 If individual’s average monthly income was higher than US$1,459, 0 otherwise
1
0.16
0
1
0.18
2
0.15
0
1
0.24
3
0.16
0
1
0.18
4
0.14
0
1
0.15
5
0.27
0
1
0.35
HIN
1 If household’s average monthly income was higher than US$2,305, 0 otherwise
1
2.02
0
1
2.05
2
2.13
0
1
2.24
3
2.24
0
1
2.27
4
2.25
0
1
2.36
5
2.33
0
1
2.48
INST
1 If individual has third-party liability insurance, 0 otherwise
1
0.50
0
1
0.92
2
0.58
0
1
0.83
3
0.64
0
1
0.71
4
0.77
0
1
0.90
5
0.82
0
1
0.82
EXP
1 If individual had experience of moderate or worse vehicle crash injury, 0 otherwise
1
0.13
0
1
0.16
2
0.06
0
1
0.07
3
0.05
0
1
0.06
4
0.05
0
1
0.06
5
0.07
0
1
0.05
RESP
1 If individual’s share of vehicle crash responsibility was higher than 50%, 0 otherwise
1
0.21
0
1
0.28
2
0.17
0
1
0.21
3
0.16
0
1
0.22
4
0.14
0
1
0.15
5
0.15
0
1
0.16
DAY
1 If individual’s work days lost after a vehicle crash was higher than 30 days, 0 otherwise
1
0.39
0
1
0.39
2
0.40
0
1
0.38
3
0.63
0
1
0.69
4
0.49
0
1
0.52
5
0.40
0
1
0.38
COMP
1 If individual had experience of receiving compensation for productivity loss from a vehicle crash, 0 otherwise
1
0.12
0
1
0.18
2
0.15
0
1
0.20
3
0.13
0
1
0.14
4
0.12
0
1
0.14
5
0.15
0
1
0.17
Note: 1: Minor injury, 2: Moderate injury, 3: Severe injury, 4: Disability, 5: Death.


### 4.2. Model Estimation Results

We compare the performances of the Tobit and Double-Hurdle models, where the same set of explanatory variables was used. (The coefficients of the variables used in our previous study [12] were insignificant and incorrectly signed in this study. In addition, the resulting WTP from our models is either overestimated or underestimate. For these reasons, the variables of the Double-Hurdle model were not used to develop the Tobit model in this study.)Table4 presents the estimation results from the Tobit model and Table 5 from the Probit and Truncated regressions of the Double-Hurdle model. As expected, the results for the Tobit and Double-Hurdle models indicate that the WTP was significant and positive and the magnitude of the regression coefficients was influenced by their respective variables. The results show that for an educational level of a university degree or higher, an average individual monthly income was higher than US$1,459, an average household monthly income was higher than US$2,305, third-party liability insurance, experience of moderate or worse injury in a vehicle crash, over 50% share of vehicle crash responsibility, work days lost after a vehicle crash was overhigher than 30 days, and experience of receiving compensation for productivity loss from a vehicle crash had a positive impact on WTP for productivity loss according to injury level.Table 4
Estimation results for WTP for productivity loss (Tobit model).
Variable
Minor injury
Moderate injury
Severe injury
Disability
Death
Coef. (t-value)
Coef. (t-value)
Coef. (t-value)
Coef. (t-value)
Coef. (t-value)
Constant
−1.33 (−8.52∗
∗)
−5.23 (−10.95∗
∗)
−1.70 (−6.88∗
∗)
−1.13 (−4.05∗
∗)
−0.15 (−4.59∗
∗)
EDU
1.19 (7.60∗
∗)
2.44 (4.49∗
∗)
0.33 (1.73∗)
0.90 (2.92∗
∗)
0.15 (4.69∗
∗)
GIN
—
1.92 (5.01∗
∗)
0.56 (2.23∗
∗)
1.27 (3.07∗
∗)
0.12 (4.38∗
∗)
HIN
—
—
—
1.09 (2.85∗
∗)
0.07 (2.66∗
∗)
INST
—
2.73 (5.17∗
∗)
0.42 (2.06∗
∗)
—
—
EXP
0.57 (2.87∗
∗)
4.20 (6.59∗
∗)
0.80 (3.72∗
∗)
2.60 (3.94∗
∗)
0.08 (2.73∗
∗)
RESP
—
1.34 (3.99∗
∗)
0.83 (4.30∗
∗)
—
—
DAY
0.42 (2.56∗
∗)
1.29 (2.99∗
∗)
0.67 (2.50∗
∗)
1.33 (3.27∗
∗)
0.11 (2.86∗
∗)
COMP
0.59 (3.16∗
∗)
1.28 (3.23∗
∗)
0.89 (3.20∗
∗)
1.58 (3.83∗
∗)
0.09 (2.742∗
∗)
σ
1.05 (18.22∗
∗)
2.58 (21.08∗
∗)
1.76 (24.02∗
∗)
2.74 (25.17∗
∗)
0.24 (24.37∗
∗)
Log-likelihood function
−338.43
−580.53
−648.49
−896.69
−783.22
LM test [df] for Tobit
235.00 [5]
181.44 [8]
668.48 [8]
350.44 [7]
256.48 [7]
Observations
365
423
431
464
439
Note:∗
∗indicates a significance level of 0.05; ∗indicates a significance level of 0.1.Table 5
Estimation results for WTP (Double-Hurdle model).
Variable
Minor injury
Moderate injury
Severe injury
Disability
Death
Probit
Truncated
Probit
Truncated
Probit
Truncated
Probit
Truncated
Probit
Truncated
Coef. (t-value)
Coef. (t-value)
Coef. (t-value)
Coef. (t-value)
Coef. (t-value)
Coef. (t-value)
Coef. (t-value)
Coef. (t-value)
Coef. (t-value)
Coef. (t-value)
Constant
−1.27 (−8.80∗
∗)
−7.26 (−3.90∗
∗)
−2.04 (−10.67∗
∗)
−17.07 (−3.94∗
∗)
−0.85 (−5.10∗
∗)
−0.54 (−10.59∗
∗)
0.16 (1.33)
−22.0 (−6.22∗
∗)
−0.59 (−3.65∗
∗)
−3.67 (−7.88∗
∗)
EDU
1.54 (9.61∗
∗)
2.17 (1.55)
1.61 (7.44∗
∗)
3.14 (1.02)
0.37 (−5.10∗
∗)
4.06 (4.69∗
∗)
0.37 (2.68∗
∗)
7.49 (3.06∗
∗)
0.86 (5.13∗
∗)
0.72 (−7.88∗
∗)
GIN
—
—
1.59 (2.86∗
∗)
6.49 (4.38∗
∗)
1.14 (4.10∗
∗)
1.20 (1.28)
0.39 (1.79∗)
6.76 (3.47∗
∗)
0.75 (3.79∗
∗)
0.88 (2.83∗
∗)
HIN
—
—
—
—
—
—
0.34 (1.67)
3.34 (1.78∗)
0.87 (4.34∗
∗)
0.25 (1.13)
INST
—
—
1.18 (5.89∗
∗)
2.57 (0.63)
0.45 (2.88∗
∗)
2.00 (2.22∗
∗)
—
—
—
—
EXP
0.51 (2.00∗
∗)
2.72 (2.51∗
∗)
1.01 (2.04∗
∗)
10.35 (5.30∗
∗)
0.69 (3.77∗
∗)
3.29 (0.25)
1.00 (1.89∗)
6.80 (3.19∗
∗)
0.57 (2.77∗
∗)
0.51 (1.66)
RESP
—
—
0.88 (3.52∗
∗)
2.74 (2.03∗
∗)
1.01 (6.71∗
∗)
0.20 (3.70∗
∗)
—
—
—
—
DAY
0.64 (3.11∗
∗)
0.86 (0.85)
0.72 (2.41∗
∗)
4.65 (2.78∗
∗)
1.27 (3.69∗
∗)
2.51 (2.43∗
∗)
0.45 (2.01∗
∗)
6.98 (3.86∗
∗)
1.18 (2.96∗
∗)
0.99 (2.62∗
∗)
COMP
0.84 (3.33∗
∗)
1.51 (1.48)
1.02 (3.52∗
∗)
2.52 (1.69)
0.22 (0.86)
4.67 (5.13∗
∗)
0.35 (1.61)
8.27 (4.77∗
∗)
0.39 (1.72)
0.71 (2.14∗
∗)
σ
—
1.81 (10.70∗
∗)
—
4.29 (11.31∗
∗)
—
2.30 (19.07∗
∗)
—
4.45 (14.65∗
∗)
—
0.66 (9.06∗
∗)
Log-likelihood function
−83.21
−55.31
−120.05
−350.55
−261.30
−138.09
−253.77
−420.05
−200.81
−244.43
PseudoR
2
0.67
—
0.59
—
0.29
—
0.06
—
0.21
—
Expected WTP excluding 0 (US$)
—
1,940
—
4,781
—
19,985
—
45,513
—
47,129
Observations
365
423
431
464
439
Note:∗
∗indicates a significance level of 0.05; ∗indicates a significance level of 0.1.The Double-Hurdle model can be divided into two parts. The first part, the Probit model, represents an individual’s decision to participate (pay for the expenditure), and the second part, the truncated model, represented the individual’s WTP. Table5 shows that as expected, the minor injury scenario yielded the least number of significant variables, perhaps because a minor injury does not significantly affect an individual’s income. In the minor and moderate injury scenarios, the coefficients of an educational level of a university degree or higher were 1.54 (t-value = 9.61) and 1.61 (t-value = 7.44), respectively, showing that the variable had the greatest impact on the decision to participate. Meanwhile, in the level of expenditure model, the coefficients of experience of moderate or worse vehicle crash injury were 2.72 (t-value = 2.51) and 10.35 (t-value = 5.30), respectively. In the severe injury and disability scenarios, the coefficients of experience of receiving compensation for productivity loss from a vehicle crash were 4.67 (t-value = 5.13) and 8.27 (t-value = 4.77), respectively, indicating that this variable had the greatest impact on the level of expenditure. In the fatality scenario, in both the participation and expenditure models, the coefficients of work days lost were 1.18 (t-value = 2.96) and 0.99 (t-value = 2.62), respectively, indicating that this variable had the greatest impact on the models for this scenario.According to Goodwin [36], ‘a comparison of the maximized log-likelihood function values allows formal statistical testing of the alternative estimators’ (p. 8). For this reason, we found that the log-likelihood function value of the Tobit (−338.43, −580.53, −648.49, −896.67, and −783.22, respectively) estimates was much smaller than that of the Double-Hurdle (−55.3, −350.55, −138.09, −420.05, and −244.43, respectively) estimates, an outcome that overwhelmingly rejects the Tobit specification. Moreover, we can compare the specification testing results of the Tobit and Double-Hurdle models. Table6 presents the estimated WTP for the two models according to injury level and shows that the specification testing of the Tobit model against the Double-Hurdle model can be accomplished using the statistic given by Equation (9). The table shows that the λ value for the minor injury, moderate injury, severe injury, disability, and fatality scenarios were 178.13, 219.86, 652.30, 454.35, and 240.40, respectively. The critical value of the chi-square for the different injury levels showed values between 18.48 and 20.09 at the α
=
0.01 level of significance, which strongly rejected the Tobit specification in favour of the Double-Hurdle model. The results suggest that individuals’ decisions to participate in the market and their WTP for productivity loss are generated by different processes. that is, Double-Hurdle models consider whether or not to participate; the willingness to pay for productivity loss is indeed superior to Tobit that only determines the positive WTP values as well as fits the sample data realistically.Table 6
Compare with Double-Hurdle versus Tobit models.
Scenario
χ
2 test result
Tobit model WTP
Double-Hurdle model WTP
Minor injury
λ
=
178.13
>
χ
2
7
=
18.48
2,865
1,940
Moderate injury
λ
=
219.86
>
χ
2
8
=
20.09
5,752
4,781
Severe injury
λ
=
652.30
>
χ
2
8
=
20.09
22,591
19,985
Disability
λ
=
454.35
>
χ
2
7
=
18.48
108,405
45,513
Death
λ
=
240.40
>
χ
2
7
=
18.48
60,185
47,129
∗Estimated WTP excluding 0 (US$).Table7 shows the marginal probability results for the Double-Hurdle model according to injury severity. The first column presents the marginal probability effects of participation and nonparticipation indicated by the changes in the explanatory variables, which are calculated from the means of the first-stage model’s explanatory variables. The second column presents the marginal probability effects for the second-stage model’s explanatory variables on the probability of the expenditure. Finally, the last column presents the marginal probability effects indicated by the changes in the explanatory variables for each scenario (injury level), with the positive WTP as the means of the explanatory variables.Table 7
Marginal probability results from Double-Hurdle model.
Variable
Minor injury
Moderate injury
Severe injury
Disability
Death
MP1
MP2
MP3
MP1
MP2
MP3
MP1
MP2
MP3
MP1
MP2
MP3
MP1
MP2
MP3
EDU
0.614
0.016
404
0.595
0.518
4,003
0.105
0.123
12,916
0.118
0.0148
23,112
0.343
0.371
21,879
GIN
—
—
—
0.585
0.715
8,260
0.320
0.324
3,809
0.123
0.0135
20,883
0.298
0.334
26,680
INST
—
—
—
0.435
0.395
3,266
0.126
0.135
6,376
—
—
—
—
—
—
HIN
0.107
0.0068
10,304
0.345
0.352
7,545
EXP
0.201
0.016
505
0.372
0.831
13,169
0.193
0.207
10,478
0.318
0.0143
20,992
0.226
0.246
15,324
RESP
—
—
—
0.324
0.345
3,493
0.284
0.284
650
—
—
—
—
—
—
DAY
0.254
0.006
160
0.264
0.427
5,919
0.356
0.366
7,997
0.141
0.0140
21,545
0.468
0.507
29,926
COMP
0.334
0.010
281
0.375
0.359
3,208
0.062
0.082
14,840
0.112
0.0163
25,521
0.154
0.184
21,581
∗
MP1
:
represent
∂
P
D
i
∗
>
0
/
∂
x
i
j
,
MP
2
:
represent
∂
P
Y
i
>
0
/
∂
x
i
j
,
MP
3
:
represent
∂
E
Y
i
|
Y
i
>
0
/
∂
x
i
j
.Table7 also presents the effects of significant dummy variables. These effects suggest that in the minor injury scenario, an individual with a university degree or above are about 61.4% more likely to participate in the market, 1.6% more likely to pay for the productivity loss, and if the individual is indeed willing to pay, they spend about US$404 more than those without a university degree. The effects of other dummy variables can be interpreted in the same manner.In the moderate injury scenario, an individual who has third-party liability insurance is 58.5% more likely to participate in the market and 71.5% more likely to be willing to pay for productivity loss. The result is similar to that for the educational level variables. The results for this scenario also suggest that an individual with an average monthly income higher than US$1,416 pays more for productivity loss (by US$8,260). An individual who has experience of moderate vehicle crash injury is 83.1% more likely to be willing to pay for productivity loss and pays the most for productivity loss (US$13,169). In the severe injury scenario, an individual who has previously received compensation for productivity loss from a vehicle crash is willing to pay more (by US$14,840) for productivity loss.In the disability and death scenarios, WTP is significantly influenced by the same explanatory variables. An individual’s experience with moderate or worse vehicle crash injury has a significant impact on the marginal probability effects on the individual’s decision to participate. An individual who has experience of receiving compensation for productivity loss is 1.63% more likely to be willing to pay for productivity loss and is willing to pay the highest amount (US$25,521). In the death scenario, an individual with an average household monthly income higher than US$2,305 is 34.5% more likely to participate in the market. Moreover, in both the severe injury and death scenarios, an individual who has lost over 30 work days after a vehicle crash is 35.6% and 46.8% more likely to participate in the market, and 36.6% and 50.7% more likely to be willing to pay for productivity loss. Likewise, an individual in either of these scenarios is willing to pay more (US$29,926).


## 4.1. Variable Definition

The definitions of variables used, and the means for the entire sample, and the positive expenditure samples are presented in Table3. The dependent variable is the WTP stated by the respondent in the open-ended question, while the independent variables include an educational level of university or above, an average individual monthly income of higher than US$1,460, an average household monthly income of higher than US$2,305, third-party liability insurance, experience of moderate or worse vehicle crash injury, over 50% share of vehicle crash responsibility, work days lost after a vehicle crash, experience of receiving compensation for productivity loss, and eight dummy variables that are equal to one if the interview was conducted in the sections called ‘EDU’, ‘GIN’, ‘HIN’, ‘INST’, ‘EXP’, ‘RESP’, ‘DAY’, and ‘COMP’ (please refer to Table 3 definition). The mean of the entire sample’s WTP is US$782–US$41,633. However, the mean for the subsample that indicated a positive WTP for productivity loss according to injury level and fatality is US$1,644–US$58,579.Table 3
Summary statistics.
Variable
Description
Injury level
Entire sample
Participation sample
Ave.
Min.
Max.
Ave.
Expend
Individual’s WTP for productivity loss (US$)
1
782
0
21,404
1,644
2
3,541
0
46,101
5,806
3
16,142
0
1,136,064
23,269
4
33,181
0
987,882
47,372
5
41,633
0
658,588
58,579
Dummy variables (yes = 1; 0 otherwise)
EDU
1 If individual had university degree or above, 0 otherwise
1
0.59
0
1
0.85
2
0.61
0
1
0.82
3
0.65
0
1
0.71
4
0.63
0
1
0.74
5
0.80
0
1
0.88
GIN
1 If individual’s average monthly income was higher than US$1,459, 0 otherwise
1
0.16
0
1
0.18
2
0.15
0
1
0.24
3
0.16
0
1
0.18
4
0.14
0
1
0.15
5
0.27
0
1
0.35
HIN
1 If household’s average monthly income was higher than US$2,305, 0 otherwise
1
2.02
0
1
2.05
2
2.13
0
1
2.24
3
2.24
0
1
2.27
4
2.25
0
1
2.36
5
2.33
0
1
2.48
INST
1 If individual has third-party liability insurance, 0 otherwise
1
0.50
0
1
0.92
2
0.58
0
1
0.83
3
0.64
0
1
0.71
4
0.77
0
1
0.90
5
0.82
0
1
0.82
EXP
1 If individual had experience of moderate or worse vehicle crash injury, 0 otherwise
1
0.13
0
1
0.16
2
0.06
0
1
0.07
3
0.05
0
1
0.06
4
0.05
0
1
0.06
5
0.07
0
1
0.05
RESP
1 If individual’s share of vehicle crash responsibility was higher than 50%, 0 otherwise
1
0.21
0
1
0.28
2
0.17
0
1
0.21
3
0.16
0
1
0.22
4
0.14
0
1
0.15
5
0.15
0
1
0.16
DAY
1 If individual’s work days lost after a vehicle crash was higher than 30 days, 0 otherwise
1
0.39
0
1
0.39
2
0.40
0
1
0.38
3
0.63
0
1
0.69
4
0.49
0
1
0.52
5
0.40
0
1
0.38
COMP
1 If individual had experience of receiving compensation for productivity loss from a vehicle crash, 0 otherwise
1
0.12
0
1
0.18
2
0.15
0
1
0.20
3
0.13
0
1
0.14
4
0.12
0
1
0.14
5
0.15
0
1
0.17
Note: 1: Minor injury, 2: Moderate injury, 3: Severe injury, 4: Disability, 5: Death.


## 4.2. Model Estimation Results

We compare the performances of the Tobit and Double-Hurdle models, where the same set of explanatory variables was used. (The coefficients of the variables used in our previous study [12] were insignificant and incorrectly signed in this study. In addition, the resulting WTP from our models is either overestimated or underestimate. For these reasons, the variables of the Double-Hurdle model were not used to develop the Tobit model in this study.)Table4 presents the estimation results from the Tobit model and Table 5 from the Probit and Truncated regressions of the Double-Hurdle model. As expected, the results for the Tobit and Double-Hurdle models indicate that the WTP was significant and positive and the magnitude of the regression coefficients was influenced by their respective variables. The results show that for an educational level of a university degree or higher, an average individual monthly income was higher than US$1,459, an average household monthly income was higher than US$2,305, third-party liability insurance, experience of moderate or worse injury in a vehicle crash, over 50% share of vehicle crash responsibility, work days lost after a vehicle crash was overhigher than 30 days, and experience of receiving compensation for productivity loss from a vehicle crash had a positive impact on WTP for productivity loss according to injury level.Table 4
Estimation results for WTP for productivity loss (Tobit model).
Variable
Minor injury
Moderate injury
Severe injury
Disability
Death
Coef. (t-value)
Coef. (t-value)
Coef. (t-value)
Coef. (t-value)
Coef. (t-value)
Constant
−1.33 (−8.52∗
∗)
−5.23 (−10.95∗
∗)
−1.70 (−6.88∗
∗)
−1.13 (−4.05∗
∗)
−0.15 (−4.59∗
∗)
EDU
1.19 (7.60∗
∗)
2.44 (4.49∗
∗)
0.33 (1.73∗)
0.90 (2.92∗
∗)
0.15 (4.69∗
∗)
GIN
—
1.92 (5.01∗
∗)
0.56 (2.23∗
∗)
1.27 (3.07∗
∗)
0.12 (4.38∗
∗)
HIN
—
—
—
1.09 (2.85∗
∗)
0.07 (2.66∗
∗)
INST
—
2.73 (5.17∗
∗)
0.42 (2.06∗
∗)
—
—
EXP
0.57 (2.87∗
∗)
4.20 (6.59∗
∗)
0.80 (3.72∗
∗)
2.60 (3.94∗
∗)
0.08 (2.73∗
∗)
RESP
—
1.34 (3.99∗
∗)
0.83 (4.30∗
∗)
—
—
DAY
0.42 (2.56∗
∗)
1.29 (2.99∗
∗)
0.67 (2.50∗
∗)
1.33 (3.27∗
∗)
0.11 (2.86∗
∗)
COMP
0.59 (3.16∗
∗)
1.28 (3.23∗
∗)
0.89 (3.20∗
∗)
1.58 (3.83∗
∗)
0.09 (2.742∗
∗)
σ
1.05 (18.22∗
∗)
2.58 (21.08∗
∗)
1.76 (24.02∗
∗)
2.74 (25.17∗
∗)
0.24 (24.37∗
∗)
Log-likelihood function
−338.43
−580.53
−648.49
−896.69
−783.22
LM test [df] for Tobit
235.00 [5]
181.44 [8]
668.48 [8]
350.44 [7]
256.48 [7]
Observations
365
423
431
464
439
Note:∗
∗indicates a significance level of 0.05; ∗indicates a significance level of 0.1.Table 5
Estimation results for WTP (Double-Hurdle model).
Variable
Minor injury
Moderate injury
Severe injury
Disability
Death
Probit
Truncated
Probit
Truncated
Probit
Truncated
Probit
Truncated
Probit
Truncated
Coef. (t-value)
Coef. (t-value)
Coef. (t-value)
Coef. (t-value)
Coef. (t-value)
Coef. (t-value)
Coef. (t-value)
Coef. (t-value)
Coef. (t-value)
Coef. (t-value)
Constant
−1.27 (−8.80∗
∗)
−7.26 (−3.90∗
∗)
−2.04 (−10.67∗
∗)
−17.07 (−3.94∗
∗)
−0.85 (−5.10∗
∗)
−0.54 (−10.59∗
∗)
0.16 (1.33)
−22.0 (−6.22∗
∗)
−0.59 (−3.65∗
∗)
−3.67 (−7.88∗
∗)
EDU
1.54 (9.61∗
∗)
2.17 (1.55)
1.61 (7.44∗
∗)
3.14 (1.02)
0.37 (−5.10∗
∗)
4.06 (4.69∗
∗)
0.37 (2.68∗
∗)
7.49 (3.06∗
∗)
0.86 (5.13∗
∗)
0.72 (−7.88∗
∗)
GIN
—
—
1.59 (2.86∗
∗)
6.49 (4.38∗
∗)
1.14 (4.10∗
∗)
1.20 (1.28)
0.39 (1.79∗)
6.76 (3.47∗
∗)
0.75 (3.79∗
∗)
0.88 (2.83∗
∗)
HIN
—
—
—
—
—
—
0.34 (1.67)
3.34 (1.78∗)
0.87 (4.34∗
∗)
0.25 (1.13)
INST
—
—
1.18 (5.89∗
∗)
2.57 (0.63)
0.45 (2.88∗
∗)
2.00 (2.22∗
∗)
—
—
—
—
EXP
0.51 (2.00∗
∗)
2.72 (2.51∗
∗)
1.01 (2.04∗
∗)
10.35 (5.30∗
∗)
0.69 (3.77∗
∗)
3.29 (0.25)
1.00 (1.89∗)
6.80 (3.19∗
∗)
0.57 (2.77∗
∗)
0.51 (1.66)
RESP
—
—
0.88 (3.52∗
∗)
2.74 (2.03∗
∗)
1.01 (6.71∗
∗)
0.20 (3.70∗
∗)
—
—
—
—
DAY
0.64 (3.11∗
∗)
0.86 (0.85)
0.72 (2.41∗
∗)
4.65 (2.78∗
∗)
1.27 (3.69∗
∗)
2.51 (2.43∗
∗)
0.45 (2.01∗
∗)
6.98 (3.86∗
∗)
1.18 (2.96∗
∗)
0.99 (2.62∗
∗)
COMP
0.84 (3.33∗
∗)
1.51 (1.48)
1.02 (3.52∗
∗)
2.52 (1.69)
0.22 (0.86)
4.67 (5.13∗
∗)
0.35 (1.61)
8.27 (4.77∗
∗)
0.39 (1.72)
0.71 (2.14∗
∗)
σ
—
1.81 (10.70∗
∗)
—
4.29 (11.31∗
∗)
—
2.30 (19.07∗
∗)
—
4.45 (14.65∗
∗)
—
0.66 (9.06∗
∗)
Log-likelihood function
−83.21
−55.31
−120.05
−350.55
−261.30
−138.09
−253.77
−420.05
−200.81
−244.43
PseudoR
2
0.67
—
0.59
—
0.29
—
0.06
—
0.21
—
Expected WTP excluding 0 (US$)
—
1,940
—
4,781
—
19,985
—
45,513
—
47,129
Observations
365
423
431
464
439
Note:∗
∗indicates a significance level of 0.05; ∗indicates a significance level of 0.1.The Double-Hurdle model can be divided into two parts. The first part, the Probit model, represents an individual’s decision to participate (pay for the expenditure), and the second part, the truncated model, represented the individual’s WTP. Table5 shows that as expected, the minor injury scenario yielded the least number of significant variables, perhaps because a minor injury does not significantly affect an individual’s income. In the minor and moderate injury scenarios, the coefficients of an educational level of a university degree or higher were 1.54 (t-value = 9.61) and 1.61 (t-value = 7.44), respectively, showing that the variable had the greatest impact on the decision to participate. Meanwhile, in the level of expenditure model, the coefficients of experience of moderate or worse vehicle crash injury were 2.72 (t-value = 2.51) and 10.35 (t-value = 5.30), respectively. In the severe injury and disability scenarios, the coefficients of experience of receiving compensation for productivity loss from a vehicle crash were 4.67 (t-value = 5.13) and 8.27 (t-value = 4.77), respectively, indicating that this variable had the greatest impact on the level of expenditure. In the fatality scenario, in both the participation and expenditure models, the coefficients of work days lost were 1.18 (t-value = 2.96) and 0.99 (t-value = 2.62), respectively, indicating that this variable had the greatest impact on the models for this scenario.According to Goodwin [36], ‘a comparison of the maximized log-likelihood function values allows formal statistical testing of the alternative estimators’ (p. 8). For this reason, we found that the log-likelihood function value of the Tobit (−338.43, −580.53, −648.49, −896.67, and −783.22, respectively) estimates was much smaller than that of the Double-Hurdle (−55.3, −350.55, −138.09, −420.05, and −244.43, respectively) estimates, an outcome that overwhelmingly rejects the Tobit specification. Moreover, we can compare the specification testing results of the Tobit and Double-Hurdle models. Table6 presents the estimated WTP for the two models according to injury level and shows that the specification testing of the Tobit model against the Double-Hurdle model can be accomplished using the statistic given by Equation (9). The table shows that the λ value for the minor injury, moderate injury, severe injury, disability, and fatality scenarios were 178.13, 219.86, 652.30, 454.35, and 240.40, respectively. The critical value of the chi-square for the different injury levels showed values between 18.48 and 20.09 at the α
=
0.01 level of significance, which strongly rejected the Tobit specification in favour of the Double-Hurdle model. The results suggest that individuals’ decisions to participate in the market and their WTP for productivity loss are generated by different processes. that is, Double-Hurdle models consider whether or not to participate; the willingness to pay for productivity loss is indeed superior to Tobit that only determines the positive WTP values as well as fits the sample data realistically.Table 6
Compare with Double-Hurdle versus Tobit models.
Scenario
χ
2 test result
Tobit model WTP
Double-Hurdle model WTP
Minor injury
λ
=
178.13
>
χ
2
7
=
18.48
2,865
1,940
Moderate injury
λ
=
219.86
>
χ
2
8
=
20.09
5,752
4,781
Severe injury
λ
=
652.30
>
χ
2
8
=
20.09
22,591
19,985
Disability
λ
=
454.35
>
χ
2
7
=
18.48
108,405
45,513
Death
λ
=
240.40
>
χ
2
7
=
18.48
60,185
47,129
∗Estimated WTP excluding 0 (US$).Table7 shows the marginal probability results for the Double-Hurdle model according to injury severity. The first column presents the marginal probability effects of participation and nonparticipation indicated by the changes in the explanatory variables, which are calculated from the means of the first-stage model’s explanatory variables. The second column presents the marginal probability effects for the second-stage model’s explanatory variables on the probability of the expenditure. Finally, the last column presents the marginal probability effects indicated by the changes in the explanatory variables for each scenario (injury level), with the positive WTP as the means of the explanatory variables.Table 7
Marginal probability results from Double-Hurdle model.
Variable
Minor injury
Moderate injury
Severe injury
Disability
Death
MP1
MP2
MP3
MP1
MP2
MP3
MP1
MP2
MP3
MP1
MP2
MP3
MP1
MP2
MP3
EDU
0.614
0.016
404
0.595
0.518
4,003
0.105
0.123
12,916
0.118
0.0148
23,112
0.343
0.371
21,879
GIN
—
—
—
0.585
0.715
8,260
0.320
0.324
3,809
0.123
0.0135
20,883
0.298
0.334
26,680
INST
—
—
—
0.435
0.395
3,266
0.126
0.135
6,376
—
—
—
—
—
—
HIN
0.107
0.0068
10,304
0.345
0.352
7,545
EXP
0.201
0.016
505
0.372
0.831
13,169
0.193
0.207
10,478
0.318
0.0143
20,992
0.226
0.246
15,324
RESP
—
—
—
0.324
0.345
3,493
0.284
0.284
650
—
—
—
—
—
—
DAY
0.254
0.006
160
0.264
0.427
5,919
0.356
0.366
7,997
0.141
0.0140
21,545
0.468
0.507
29,926
COMP
0.334
0.010
281
0.375
0.359
3,208
0.062
0.082
14,840
0.112
0.0163
25,521
0.154
0.184
21,581
∗
MP1
:
represent
∂
P
D
i
∗
>
0
/
∂
x
i
j
,
MP
2
:
represent
∂
P
Y
i
>
0
/
∂
x
i
j
,
MP
3
:
represent
∂
E
Y
i
|
Y
i
>
0
/
∂
x
i
j
.Table7 also presents the effects of significant dummy variables. These effects suggest that in the minor injury scenario, an individual with a university degree or above are about 61.4% more likely to participate in the market, 1.6% more likely to pay for the productivity loss, and if the individual is indeed willing to pay, they spend about US$404 more than those without a university degree. The effects of other dummy variables can be interpreted in the same manner.In the moderate injury scenario, an individual who has third-party liability insurance is 58.5% more likely to participate in the market and 71.5% more likely to be willing to pay for productivity loss. The result is similar to that for the educational level variables. The results for this scenario also suggest that an individual with an average monthly income higher than US$1,416 pays more for productivity loss (by US$8,260). An individual who has experience of moderate vehicle crash injury is 83.1% more likely to be willing to pay for productivity loss and pays the most for productivity loss (US$13,169). In the severe injury scenario, an individual who has previously received compensation for productivity loss from a vehicle crash is willing to pay more (by US$14,840) for productivity loss.In the disability and death scenarios, WTP is significantly influenced by the same explanatory variables. An individual’s experience with moderate or worse vehicle crash injury has a significant impact on the marginal probability effects on the individual’s decision to participate. An individual who has experience of receiving compensation for productivity loss is 1.63% more likely to be willing to pay for productivity loss and is willing to pay the highest amount (US$25,521). In the death scenario, an individual with an average household monthly income higher than US$2,305 is 34.5% more likely to participate in the market. Moreover, in both the severe injury and death scenarios, an individual who has lost over 30 work days after a vehicle crash is 35.6% and 46.8% more likely to participate in the market, and 36.6% and 50.7% more likely to be willing to pay for productivity loss. Likewise, an individual in either of these scenarios is willing to pay more (US$29,926).


## 5. Conclusions

In this study, several conclusions are obtained. First, the results provide important information regarding the calculation of productivity loss compensation for vehicle crash victims, especially in five injury levels were designed to analyse the compensations of productivity loss. As the results showed compensations were increased as injury severity increased. Secondly, the open-ended contingent valuation bids are employed to estimate parties’ productivity loss expenditure in Taiwan; therefore, the results of the survey can well capture the nature zero bids.Third, the compensation for productivity loss for different injury levels is estimated using Tobit and Double-Hurdle models, and the results showed compensations were increased as injury severity increased. We summarize the detail results of the Double-Hurdle model as follows: educational level, individual monthly income, household monthly income, third-party liability insurance, experience of severe or worse injury from a vehicle crash, individual’s share of vehicle crash responsibility, work days lost from a vehicle crash, and experience of receiving compensation for productivity loss from a vehicle crash are all significant variables affecting the WTP for productivity loss. Of the 2,122 individuals in the sample, 331 (15.6%) reported that they were not willing to pay. Excluding these respondents with a zero WTP, the estimated amounts the individuals were willing to pay for productivity loss were US$1,940, US$4,781, US$19,985, US$45,513, and US$47,129, for minor injury, moderate injury, severe injury, disability, and death, respectively.Fourth, a comparison of the results from the two models shows that the Double-Hurdle model was more flexible compared to the Tobit model in that it distinguishes between the process for discrete decision (determining whether individuals are willing to pay in the market) and how much they are willing to pay for productivity loss from a vehicle crash. Fifth, in addition to estimating the effects of participation and WTP, we calculated the significant elasticities of the variables. The overall elasticities show that the experience of severe injury had the highest positive WTP means of the explanatory variables in the minor and moderate injury scenarios. For the severe injury and death scenarios, a change in lost workdays of more than 30 days after a vehicle crash had a strong effect on the mean of a positive WTP. For the moderate or worse injury and fatality scenarios, a change in an individual’s monthly income had a much larger effect on the probability of a positive WTP.Finally, the study’s methodology differs from that of civil court verdicts in Taiwan in that it estimates productivity loss compensation not only based on an individual’s monthly income, but also on injury level (minor, moderate, and severe injuries), disability, and death; moreover, it identifies the significant factors affecting such compensation. The results highlight the importance of precise information in determining the individual’s actual WTP and in turn, in capturing the individual’s perceptions of reasonable compensation for productivity loss from vehicle crashes in Taiwan.The study has two recommendations. First, Taiwanese courts do not have an actual framework for calculating compensation for vehicle crash death. In the study, we constructed the hypothesis scenario and estimated the productivity loss value of death. Moreover, the study captures the individuals’ productivity loss value of vehicle crash victims’ rights according to different type injury severity. That is, for the victim fairness and justice, not only the government of Taiwan, but international countries should better consider the productivity loss of different injury level in the future similar studies and related policies. Secondly, according to the methodology of the study, D-H model was provided for the representative references for accident compensation. We suggest that different countries could refer to the model and update the study’s vehicle crash questionnaire to use it to estimate the overall economic losses caused by productivity loss according to injury severity.



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*Source: 7219047-2019-12-12.xml*