Etude des marchés d'assurance non-vie à l'aide d'équilibres de ...

tel-00703797, version 2 - 7 Jun 2012
1.5. Testing asymmetry of information
endogeneous variable and Z a decision variable. The absence of adverse selection is equivalent
to the prediction of Z based on the joint distribution of X and Y coincides with prediction
with X alone. This indirect characterization leads to
l(Z|X, Y ) = l(Z|X), (1.3)
where l(.|., .) denotes the conditional probability density function.
One way to test for the conditionnal independence of Z with respect to Y is to regress the
variable Z on X and Y and see whether the coefficient for Y is significant. The regression
model is l(Z|X, Y ) = l(Z; aX + bY ). However, to avoid spurious conclusions, Dionne et al.
(2001) recommend to use the following econometric model
l(Z|X, Y ) = l Z|aX + bY + c
E(Y |X) , (1.4)
where E(Y |X), the conditionnal expectation of Y given the variable X, will be estimated by
a regression model initially. The introduction of the estimated expectation E(Y |X) allows to
take into account nonlinear effects between X and Y , but not nonlinear effects between Z and
X, Y ∗ .
Summarizing the testing procedure, we have first a regression Y on X to get E(Y |X).
Secondly, we regress the decision variable Z on X, Y , and E(Y |X). If the coefficient for Y is
significant in the second regression, then risk adverse selection is detected. The relevant choice
for Z is the insured deductible choice, with X rating factors and Y the observed number of
claims. E(Y |X) will be estimated with a Poisson or more sophisticated models, see below.
1.5.2 A deductible model
The deductible choice takes values in the discrete set {d0, d1, . . . , dK}. The more general
model is a multinomial model M(1, p0, . . . , pK), where each probability parameter pj depends
on covariates through a link function. If we assume that variables Zi are independent and
identically distributed random variables from a multinomial distribution M(1, p0, . . . , pK) and
we use a logit link function, then the multinomial regression is defined by
P (Zi = dj) =
e xT i βj
1 + K
e
l=1
xT i βl
for j = 1, . . . , K where 0 is the baseline category and xi covariate for ith individual, see, e.g.,
McFadden (1981), Faraway (2006) for a comprehensive study of discrete choice modelling.
When reponses (d0 < d1 < · · · < dK) are ordered (as it is for deductibles), one can also
use ordered logistic models for which
P (Zi = dj) =
eθj−xT i β
1 + eθj−xT −
i β eθj−1−x
,
T i β
1 + e θj−1−x T i
Note that the number of parameters substantially decreases since the linear predictor for
multinomial logit regression, we have ηij = x T i βj, whereas for the ordered logit, ηij = θj −x T i β.
∗. See Su and White (2003) for a recent procedure of conditional independence testing.
β .
63