Evidence of adverse selection in automobile insurance market: A seemingly unrelated probit modelling

Figures & data

Table

Variable	Definition
Occup 1	1 if the policyholder is an official (21.21%); 0 otherwise
Occup 2	1 if the policyholder is a senior executive (2.78%); 0 otherwise
Occup 3	1 if the policyholder is unemployed (8.01%); 0 otherwise
Occup 4	1 if the policyholder is a craftsman (11.96%); 0 otherwise
Occup 5	1 if the policyholder is an employee (48.61%); 0 otherwise
Occup 6	1 if the policyholder is retired (5.49%); 0 otherwise
Occup 7	1 if the policyholder is a middle manager (1.94%); 0 otherwise

Table

Variable	Definition
GZ1	1 if the policyholder lives in big Tunis and 0 otherwise, accounting for approximately 29% of the whole portfolio
GZ2	1 if the policyholder lives in the North and 0 otherwise, accounting for approximately 24% of the whole portfolio
GZ3	1 if the policyholder lives on the Coast and 0 otherwise, accounting for approximately 30%
GZ4	1 if the policyholder lives in the Central or the Southern regions (5.71%); 0 otherwise, accounting for approximately 17% of the whole portfolio

Table

Variable	Definition
Origin 1	1 if the car was made in France (58.81%); 0 otherwise
Origin 2	1 if the car was made in Italy (12.62%); 0 otherwise
Origin 3	1 if the car was made in Germany (17.31%); 0 otherwise
Origin 4	1 if the car was made in China or in Korea or in Japan (9.92%); 0 otherwise
Origin 5	1 if the car was made in other countries (0.70%); 0 otherwise

Table 1. Distribution of number of accidents

Number of accidents	Whole portfolio	Beginner drivers	Experienced drivers
0	87.68	89.18	86.28
1	6.42	5.46	7.32
2	4.39	4.12	4.64
3	0.88	0.76	1.00
4	0.43	0.37	0.49
5	0.09	0.07	0.11
6	0.06	0.03	0.09
7	0.02	0.00	0.04
8	0.03	0.01	0.04

Notes: The first column of the table displays the number of accident with values lying between 0 and 8. The three remaining columns show the relative frequency in percentage of the number of accidents for the entire portfolio, the beginner drivers and for the experienced drivers.

Table 2. Frequency table for coverage choice and accident occurrence

Whole portfolio		Accident occurrence
		0	1	Total	Accident %
Deductible choice	0	21.31	16.71	20.75	9.93
	1	78.69	83.29	79.25	12.95
	Total	100	100	100	12.32
Beginner drivers
Deductible choice	0	27.55	18.90	26.61	7.69
	1	72.45	81.10	73.39	11.97
	Total	100	100	100	10.83
Pearson χ^2 (1) = 55.737; p = 0.000
Experienced drivers
Deductible choice	0	15.27	15.10	15.25	13.59
	1	84.73	84.90	84.75	13.74
	Total	100	100	100	13.72
Pearson χ^2 (1) = 0.043; p = 0.835

Notes: The table displays the association between deductible choice and accident occurrence for the entire portfolio, for the beginner drivers and for the experienced drivers. In this table the accident occurrence is a dichotomous variable coded 0 if the policyholder has no accident and 1 if the policyholder has at least 1 accident. The deductible choice is a dichotomous variable coded 0 if the policyholder purchased a full coverage contract and 1 if he purchased a third party liability contract. Pearson χ² (1) correspond to the Pearson’s test of association between two categorical data. Finally, p is the p-value, usually, if p < 0.05 then reject the null hypothesis, and accept the alternate (There is a difference in the groups under study).

Table 3. Seemingly unrelated probit model exclusive of the deductible choice as an endogenous dummy

	Coefficient	z		Coefficient	z
Equation 1. Accident number			Equation 2. Deductible choice
Gender	−.0340 (0.023)	−1.46	Gender	−0.254 (0.027)***	−9.35
Use of car	0.024 (0.012)***	1.99	Use of car	1.909 (0.030)***	62.49
No claim bonus rate	0.005 (0.004)	1.18	No claim bonus rate	−0.025 (0.005)***	−4.80
GZ1	0.004 (0.026)	0.18	GZ1	−0.197 (0.042)***	−4.69
GZ2	−0.0002 (0.017)	−0.01	GZ2	−0.272 (0.026)***	−10.20
GZ3	−0.035 (0.018)**	−1.86	GZ3	−0.216 (0.030)***	−7.05
Occup 1	0.034 (0.044)	0.78	Occup 1	0.260 (0.052)***	5.01
Occup 2	0.038 (0.052)	0.74	Occup 2	0.129 (0.063)**	2.05
Occup 3	0.038 (0.025)	1.53	Occup 3	0.252 (0.035)***	7.18
Occup 4	0.034 (0.018)**	1.84	Occup 4	−0.082 (0.026)	−3.17
Occup 5	0.014 (0.036)	0.40	Occup 5	0.065 (0.056)	1.15
Occup 6	−0.0053 (0.031)	−0.17	Occup 6	0.083 (0.043)*	1.92
Age1	0.048 (0.026)**	1.78	Age1	−0.149 (0.044)	−3.38
Age2	0.016 (0.017)	0.98	Age2	−0.069 (0.027)**	−2.55
Age3	0.103 (0.048)**	2.12	Age3	0.066 (0.097)	0.69
Origin 1	0.014 (0.021)	0.68	Origin 1	−0.021 (0.037)	−0.56
Origin 2	−0.002 (0.021)	−0.09	Origin 2	0.067 (0.023)***	2.84
Origin 3	0.015 (0.023)	−0.68	Origin 3	0.145 (0.041)***	3.53
Origin 4	0.145 (0.124)	1.17	Origin 4	0.352 (0.127)***	2.76
Origin 5	0.133 (0.125)	1.06	Origin 5	−0.062 (0.147)	−0.42
Constant1	−0.403 (0.044)***	−9.03	Constant2.	0.047 (0.054)	0.87
LL	−33755.021
WT	6620.94***
Athrho	−0.0277 (0.013)***
ρ	−0.0276 (0.013)***
LR	${\mathit{\chi }}^2(1)$ = 4.473
	$\left(P>{\mathit{\chi }}^2=0.0344\right)$

Notes: This table reports results of the seemingly unrelated probit model exclusive of the deductible choice as an endogenous dummy. LL denote the log likelihood value, WT is the value Wald test for the null hypothesis ρ = 0. Finally, LR is the Likelihood-ratio test of ρ = 0; In the maximum likelihood estimation, ρ is not directly estimated, but atanh ρ is 1: $\text{atanh}\mathit{\rho }=\frac{1}{2}ln\left(\frac{1+\mathit{\rho }}{1-\mathit{\rho }}\right)$

* The coefficients are significance at 10%.

** The coefficients are significance at 5%.

*** The coefficients are significance at 1%.

Table 4. Seemingly unrelated probit model with deductible choice as endogenous dummy

	Coefficient	z	Coefficient	z
Equation 1. Accident number			Equation 2. Deductible choice
Deductible	−0.039 (0.011)***	−3.53	–	–
Gender	−0.038 (0.023)	−1.63	(-0.254) (0.027)***	−9.37
Use of car	0.118 (0.029)***	4.05	1.908 (0.030)***	62.49
No claim bonus rate	0.009 (0.004)***	1.86	−0.025 (0.005)***	−4.81
GZ1	0.004 (0.026)	0.19	−0.197 (0.042)***	−4.69
GZ2	0.004 (0.017)	0.28	−0.272 (0.026)***	−10.19
GZ3	−0.034 (0.018)**	−1.85	−0.216 (0.030)	−7.03
Occup 1	0.036 (0.044)	0.82	0.260 (0.052)***	5.01
Occup 2	0.039 (0.052)	0.76	0.129 (0.063)***	2.05
Occup 3	0.026 (0.025)	1.05	0.253 (0.035)***	7.20
Occup 4	0.032 (0.018)*	1.71	−0.082 (0.026)***	−3.17
Occup 5	0.014 (0.036)	0.40	0.065 (0.056)	1.16
Occup 6	−0.008 (0.031)	−0.26	0.082 (0.043)**	1.91
Age1	0.052 (0.027)**	1.95	−0.149 (0.044)***	−3.38
Age2	0.018 (0.017)	1.10	−0.069 (0.027)**	−2.55
Age3	0.098 (0.048)**	2.01	0.066 (0.097)	0.68
Origin 1	0.019 (0.021)	0.88	−0.021 (0.037)	−0.57
Origin 2	0.004 (0.021)	0.22	0.067 (0.023)***	2.83
Origin 3	−0.028 (0.023)	−1.21	0.145 (0.041)***	3.53
Origin 4	0.160 (0.124)	1.28	0.351 (0.127)***	2.75
Origin 5	0.137 (0.125)	1.09	−0.063 (0.148)	−0.43
Constant	−0.422 (0.045)***	−9.38	0.048 (0.054)	0.88
LL	−33752.65
WT	6630.26***
Athrho	−0.0013 (0.015)***
ρ	−0.0013 (0.015)**
LR	${\mathit{\chi }}^2\left(1\right)$ = 0.0075
	$\left(P>{\mathit{\chi }}^2=0.9306\right)$

Notes: This table reports results of the seemingly unrelated Probit model with deductible choice as endogenous dummy. LL denote the log likelihood value, WT is the value Wald test for the null hypothesis, ρ = 0. Finally, LR is the Likelihood-ratio test of ρ = 0.

* The coefficients are significance at 10%.

** The coefficients are significance at 5%.

*** The coefficients are significance at 1%.

Table 5. Marginal effects of the deductible choice

	Model 1		Model 2
	dy/dx	z	dy/dx	z
Deductible
Gender	−0.0231 (0.006)***	−3.59	−0.022 (0.006)***	−3.46
Use of car	0.111 (0.003)***	33.47	0.114 (0.0035)***	31.96
No claim bonus rate	0.0001 (0.0013)***	0.10	0.0005 (0.0013)	0.41
GZ1	−0.009 (0.007)	−1.30	−0.0095 (0.0073)	−1.30
GZ2	−0.015 (0.004)***	−3.02	−0.014 (0.0049)***	−2.89
GZ3	−0.021 (0.005)***	−4.01	−0.0207855 (0.0053)***	−3.91
Occup 1	0.023 (0.012)**	1.94	0.022 (0.012)**	1.82
Occup 2	0.017 (0.014)	1.22	0.017 (0.014)	1.20
Occup 3	0.024 (0.006)***	3.46	0.021 (0.007)***	3.12
Occup 4	0.004 (0.005)	0.91	0.0042 (0.0052)	0.81
Occup 5	0.007 (0.010)	0.74	0.0074 (0.0101)	0.74
Occup 6	0.0031 (0.0088)	0.35	0.0023 (0.0088)	0.26
Age1	0.0047 (0.0076)	0.62	0.0054 (0.0076)	0.71
Age2	0.0006 (0.0048)	0.14	0.0010 (0.0048)	0.22
Age3	0.031 (0.014)	2.23	0.0309 (0.0140)***	2.20
Origin 1	0.0027 (0.0061)	0.45	0.0029 (0.0061)	0.49
Origin 2	0.0031 (0.0059)	0.53	0.0031 (0.0059)	0.53
Origin 3	0.0037 (0.0066)	0.55	0.0014 (0.0067)	0.22
Origin 4	0.058 (0.034)*	1.71	0.057 (0.034)*	1.70
Origin 5	0.032 (0.034)	0.94	0.032 (0.034)	0.93

Notes: This table reports results of the marginal effects of the deductible choice for the two estimated models. In a probit model, the marginal probability effects are the partial effects of each explanatory variable on the probability that the observed dependent variable Yi = 1, where in probit models.

* Significance at the 10% level.

** Significance at the 5% level.

*** Significance at the 1% level.