0. Introduction

With the global economy continuous development, vehicle population is increasing. Traffic accident has become an important accident type because of the most death numbers and the biggest economic loss. According to historical data, in the past 30 years, traffic accident mortalities in the developing countries increase greatly, while traffic accident mortalities in the developed countries decrease in varying degrees. For instance, from 1980 to 2005, the mortality per 100 000 population of traffic accident increased by 3.4 times, the data in India increased by nearly 2 times, while the data in USA decreased by 34.5%, and the data in EU countries decreased by 20%-30%. Compared with the developed countries, per capita vehicle ownership in the developing countries is much lower, but the increase trend and absolute values of accident data such as mortality per 10 000 vehicles are higher. There is a great gap between the developing countries and the developed countries in economic development level and road condition. In addition, traffic accident mortality is also affected by relatively stable macroscopic factors such as national system.

At present, more attention is paid to microscopic road factors, the research of macroscopic factors is fewer, such as economic development level, population and vehicle ownership. Jia, et al analyzed the survey data of traffic accident, and got the nonlinear regreession model of traffic accident death numbers^[1], the regression model only considers factors such as population and vehicle, with the increase of vehicle motorization level in the developed countries, the mortality per 100 000 population of traffic accident decreased, so the model is no longer suitable for the situation in the developed countries. Zhang, et al calculated the mortality per 10 000 vehicles and vehicle ownership in China from 1980 to 2002 by using the regression model, and found that there was a good fitting between calculation values and actual values^[2]. But from 2003, the mortality numbers of traffic accidents decreased continuously in the next four years. Therefore, the model is also no longer suitable for the situation in China.

In 2001, Lassarre used macroscopic effects such as per capita GDP, the proportion of urban population and per capita road length to develop macroeconomics accident model for OECD countries^[3]. But in the model, some indicators are only available in OECD countries, the application ability of the model is poor.

In 2003, Kopits, et al completed the World Bank Group's report named traffic and economic development, elaborated the relation between economic growth and traffic accident mortality. Based on Cobb-Douglas function, the model of traffic accident mortality and per capita GDP was set up, the regression analysis was divided into two modes, such as the mode of the developing countries and the mode of the developed countries. But the model only emphasizes the relation between per capita GDP and traffic accident mortality, the other effects such as road condition and vehicle population are not considered.

From the points of economic development level, vehicle ownership, population number, road condition, the regularities of traffic accidents at home and abroad are analyzed, the internal relation between traffic accident mortality and macroscopic effects can be found, and the macroscopic econometrics model can be set up by using the historical data in the seven countries.

1. Macroscopic law of traffic accident

Traffic accident mortality (F₀) is an interaction result of motorization level (F₂) and mortality per 10 000 vehicles (F₃). F₂ and F₃ are effected by economic development level, road condition, driving habit and other macroscopic factors, where economic development level has the largest impact on F₂ and F₃. The typical historical data from 1963 to 2009 in USA, China, Japan, France, UK, Italy and Brazil are analyzed, and the relations among F₀, F₂, F₃ and per capita GDP are explored^[5-7].

1.1 Relation between per capita GDP and vehicle ownership per 1 000 population

The relations between per capita GDPs and vehicle ownerships per 1 000 population for seven countries are shown in Fig. 1, vehicle ownership and the elasticity cofficient of per capita GDP are shown in Fig. 2. From the points of development experiences in the developed countries, the vehicle demand always can be divided into four stages. In stage Ⅰ, vehicle ownership increases slowly with the increase of income; in stage Ⅱ, vehicle ownership increases rapidly with the increase of income; in stage Ⅲ, vehicle ownership increases slowly, and is followed by a stable state; in stage Ⅳ, vehicle ownership no longer increases with the increase of income.

Figure  1.  Relations between per capita GDPs and vehicle ownerships per 1 000 population

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Figure  2.  Vehicle ownership and elasticity coefficient

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In order to set up the quantitative relation between vehicle ownership and per capita GDP, the Gompertz function is used, regression equation is

$V=\mu \exp (\omega \text{e}{}^{\sigma G})(1)$

Where V is vehicle ownership per 1 000 population; G is per capita GDP; μ is saturation coefficient; ω and σ are negative parameters.

Through the regression analysis of historical data from 1963 to 2009 in China, the relation model between vehicle ownership per 1 000 population and per capita GDP is

$V=333\exp (-4.67692581\text{e}{}^{-0.00003683G})(2)$

From the regression result, the inflexion point of elasticity coefficient for per captia GDP in China is 3 870 dollars. In 2010, the per capita GDP in China was about 4 382 dollars, vehicle ownership per 1 000 population in China was about 58 veh, it means that vehicle market in China has changed from stage II to stage III.

1.2 Relation between per capita GDP and mortality per 10 000 vehicles

The relations between per capita GDPs and mortalities per 10 000 vehicles in six countries are shown in Fig. 3. In the most countries, there is an inverse proportion relation between per capita GDP and mortality per 10 000 vehicles. With the increase of per capita GDP, vehicle ownership per 1 000 population becomes more and more, it makes government pay more attention to the improvement and advancement of traffic safety. The whole quality of population becomes higher, it also can play an important role on the improvement of traffic safety. Therefore, with the increase of per capita GDP, mortality per 10 000 vehicles has a downward trend.

Figure  3.  Relations between per capita GDPs and mortalities per 10 000 vehicles

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Mortalities per 10 000 vehicles in USA, China, Japan, France, UK and Italy are shown in Fig. 4. Mortality per 10 000 vehicles in China is 3-5 times of that in the developed countries, and there is a big gap of per capita GDP between China and the developed countries. It shows that there is a negative correlation between per capita GDP and mortality per 10 000 vehicles in China.

Figure  4.  Mortalities per 10 000 vehicles

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1.3 Relation between per capita GDP and mortality per 100 000 population

The relations between per capita GDPs and mortalities per 100 000 population in the seven countries are shown in Fig. 5. In most countries, mortality per 100 000 population firstly increases and then decreases, and the inflection point of mortality per 100 000 population appears while the per capita GDP is in 2 000-5 000 dollars. In the early stages of development, with the increase of income, vehicles increase rapidly, and traffic accidents begin to deteriorate, the governments begin to adopt various measures to improve traffic status, so mortality per 100 000 population decreases.

Figure  5.  Relations between per capita GDPs and mortalities per 100 000 population

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2. Mocroscopic econometrics model

2.1 Model basic

The first important step in describing road traffic safety condition is to analyze the influence factors of traffic accident. Therefore, those factors chosen to develop models should be independent with each other, and various mathematical models must be compared during modeling process.

Rules for selection factors affecting road traffic safety are as follows.

(1) It should be higher correlation between macroeconomics variables and dependent variables, and lower relevances among the macroeconomics variables.

(2) While the independent variables are chosen, it is possible to select variables with uniform unit to the independent variables, and use absolute value and percentage to avoid mathematical derivation.

(3) As a whole, time series model is better than segmentation model, because the segmentation model does not take geographical factor and cultural factor into account.

(4) Large data gaps may not be suitable for regression analysis.

(5) The selected variables are easy to access, as well as to guarantee reliability.

Several principles mentioned above are combined, four macroeconomics factors are chosen, mortolity per 100 000 population is chosen as dependent variable, per capita GDP, road condition and vehicle ownership per 1 000 population are chosen as indepentdent variables. In order to describe other factors, such as geography and culture, the surplus variable panel data model can be used.

2.2 Model construction

2.2.1   Panel data model

Up till now, macroeconomics data model is not widely used to describe accidents in China. It is common to adopt Cobb-Douglas function for regression analysis at abroad. Different macroeconomics variables are used to set up the model, the function equation form is^[4-8]

$F{}_{it}=\text{e}{}^{\beta {}_0}{\displaystyle \sum _{j=1}^{k}X}{}_{jit}^{\beta {}_j}\text{e}{}^{u{}_{it}}(3)$

Where F_it is mortality per 100 000 population; i is country series(i=1, 2, …, N); t is year series(t=1, 2, …, T); X_jit is the factor of F_it; u_it is surplus factor; k is the total number of factors; β₀, β_j are panel data coefficients.

Taking a logarithmic for Eq.(3)

$\ln (F{}_{it})=\beta {}_0+{\displaystyle \sum _{j=1}^k\beta }{}_j\ln (X{}_{jit})+u{}_{it}(4)$

2.2.2   Panel data model theory

Panel data model is known as two-dimensional data model including longitudinal section and cross section. From the cross section, multiple-entity at a certain moment constitutes section observations.From the longitudinal section, it is a time sequence^[9]. Panel data is shown in Fig. 6.

Figure  6.  Panel data

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The form of panel data model is

$y{}_{it}=a{}_i+X{}_{it}\beta +u{}_{it}(5)$

Where y_it is dependent variable; β is undetermined coefficient; a_i is individual effect.

There are two ways to deal with the individual effect. One is dependent variable which changes over time, the corresponding model can be called fixed effect model. The other is random effect, the model can be called random effect model. The fixed effect model reflects individual differences as particular intercept. The random effect model assumes that all individuals have the same intercept, and individual differences are mainly reflected in the set of random factors.

2.2.3   Hausman test

When panel data model is constructed, it is hard to make a choice between fixed effect model and random effect model. Therefore, it could make a choice between fixed effect model and random effects model by testing a_i.

Based on the Hausman test, the original hypothesis is that a_i is not related to other variables, OLS estimation from fixed effect model and GLS estimation from random effect model are unbiased and consistent. If the original hypothesis is not existed, the parameter estimation of fixed effect model keeps consistent, but the parameter estimation of random effect model is not suitable. Therefore, in the original hypothesis, parameter estimation should not be significant difference. Test equation is

Where W is the statistic value of Hausman test; χ²(k-1) is the chi-square distribution while freedom is k-1; L is the OLS estimation of fixed effect model; M is the GLS estimation of random effect model; V(·) is variance function.

3. Calculation result analysis

The historical data of mortalities per 100 000 population, per capital GDPs, per capita road lengths from 1963 to 2009 in USA, China, Japan, France, UK, Italy and Brazil are used in regression analysis^[10-13].

According to references [14, 15, 16, 17, 18], the macroscopic econometrics model of road traffic accident is

$\begin{array}{l}\ln (F{}_{it})=a+b\text{l}\text{n}(G{}_{it})+c\text{l}\text{n}(R{}_{it})+\\ d\text{l}\text{n}(V{}_{it})+u{}_{it}(8)\end{array}$

Where R_it is per capita road length; b, c, d are regression coefficients; a, G_it, V_it correspond to a_i, G, V.

Regression results are shown in Tabs. 1-3, in Tabs. 1-3, R, u correspond to R_it, u_it.

Table  1.  Regression result of fixed effect model

Parameters		t test value	p value
ln(G)	-0.644 60	-25.68	0.000
ln(R)	-0.293 15	-3.28	0.027
ln(V)	1.035 67	24.69	0.000
a	0.977 37	1.95	0.000
u	0.476 64

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Table  2.  Regression result of random effect model

Parameters		t test value	p value
ln(G)	-0.642 87	-25.69	0.000
ln(R)	-0.302 62	-3.57	0.001
ln(V)	1.033 13	24.85	0.000
a	0.919 29	24.85	0.072
u	0.471 88

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Table  3.  Hausman test

Parameters
ln(G)	-0.644 60	-0.642 87	-0.001 73
ln(R)	-0.293 15	-0.302 63	0.009 48
ln(V)	1.035 67	1.033 13	0.002 54

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From the Hausman test, W is 3.91. While p value is bigger than W, the probability is 0.02, it is appropriate to use fixed effect model to describe the relation between traffic accident and economic effects.

Therefore, the formula of fixed effect model is

Taking a logarithm for Eq.(9)

At the same time, using each national intercept, the traffic macroscopic accident econometrics models of USA, China, Japan, France, UK, Italy and Brazil respectively are

$\{\begin{array}{l}F{}_{1t}=\text{e}{}^{1.281773}G{}_{1t}^{-0.64460}R{}_{1t}^{-0.29315}V{}_{1t}^{1.03567}\\ F{}_{2t}=\text{e}{}^{1.491024}G{}_{2t}^{-0.64460}R{}_{2t}^{-0.29315}V{}_{2t}^{1.03567}\\ F{}_{3t}=\text{e}{}^{0.7446328}G{}_{3t}^{-0.64460}R{}_{3t}^{-0.29315}V{}_{3t}^{1.03567}\\ F{}_{4t}=\text{e}{}^{1.390349}G{}_{4t}^{-0.64460}R{}_{4t}^{-0.29315}V{}_{4t}^{1.03567}\\ F{}_{5t}=\text{e}{}^{0.5328025}G{}_{5t}^{-0.64460}R{}_{5t}^{-0.29315}V{}_{5t}^{1.03567}\\ F{}_{6t}=\text{e}{}^{0.7380679}G{}_{6t}^{-0.64460}R{}_{6t}^{-0.29315}V{}_{6t}^{1.03567}\\ F{}_{7t}=\text{e}{}^{0.2564355}G{}_{7t}^{-0.64460}R{}_{7t}^{-0.29315}V{}_{7t}^{1.03567}\end{array}(11)$

From the above analysis, the results are as follows.

(1) The confidence levels of ln(G), ln(R), ln(V) and a are less than 0.05, the value of F test is 157.75, so the regression effect and the degree of fit are good.

(2) Mortality per 100 000 population, per capita GDP, per capita road length and vehicle ownership per 1 000 population all exceed 0.05, it shows that independent variables and dependent variables are positive correlation.

(3) Due to the increase of GDP and the improvements of population quality, investment and related policy support, per capita GDP is negative correlation with traffic accident. Per capita road length is negative correlation with traffic accident mortality. The higher the motorization level is, the higher the chance of traffic accident is. In the most developed countries, the traffic accident mortality firstly increases and then decreases, but in China, the inflexion point of traffic accident mortality is not obvious, so the traffic accident mortality is still to increase if the vehicle ownership per 1 000 population increases rapidly.

(4) There are large differences among individual fixed effects, so the development trends between the developing and the developed countries are different.

4. Conclusions

Mortality per 100 000 population, per capita GDP, vehicle ownership per 1 000 population are analyzed, the relation between motorization level and traffic accident is analyzed. Fixed effect model and random effect model are used. Hausman test result shows that the fixed effect model is more suitable for the description of relation. According to regression results, Chinese government should pay more attention to traffic safety in the future.