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

tel-00703797, version 2 - 7 Jun 2012
1.2. GLMs, a brief introduction
What makes the present paper different from previous research on the topic is the fact
that we tackle the issue of price elasticity from various points of view. Our contribution is
to focus on price elasticity of different markets, to check the impact of distribution channels,
to investigate the use of market proxies and to test for evidence of adverse selection. We
have furthermore given ourselves the dual objective of comparing regression models as well as
identifying the key variables needed.
In this paper, we only exploit private motor datasets, but the methodologies can be applied
to other personal non-life insurance lines of business. After a brief introduction of generalized
linear models in Section 1.2, Section 1.3 presents a naive application. Based on the dubious
empirical results of Section 1.3, the Section 1.4 tries to correct the price-sensitivity predictions
by including new variables. Section 1.5 looks for empirical evidence of asymmetry of information
on our datasets. Section 1.6 discusses the use of other regression models, and Section
1.7 concludes. Unless otherwise specified, all numerical applications are carried out with the
R statistical software, R Core Team (2012).
1.2 GLMs, a brief introduction
The Generalized Linear Models (GLM ∗ ) were introduced by Nelder and Wedderburn
(1972) to deal with discrete and/or bounded response variables. A response variable on the
whole space of real numbers R is too retrictive, while with GLMs the response variable space
can be restricted to a discrete and/or bounded sets. They became widely popular with the
book of McCullagh and Nelder, cf. McCullagh and Nelder (1989).
GLMs are well known and well understood tools in statistics and especially in actuarial
science. The pricing and the customer segmentation could not have been as efficient in non-life
insurance as it is today, without an intensive use of GLMs by actuaries. There are even books
dedicated to this topic, see, e.g., Ohlsson and Johansson (2010). Hence, GLMs seem to be the
very first choice of models we can use to model price elasticity. This section is divided into
three parts: (i) theoretical description of GLMs, (ii) a clear focus on binary models and (iii)
explanations on estimation and variable selection within the GLM framework.
1.2.1 Theoretical presentation
In this section, we only consider fixed-effect models, i.e. statistical models where explanatory
variables have deterministic values, unlike random-effect or mixed models. GLMs are
an extension of classic linear models, so that linear models form a suitable starting point for
discussion. Therefore, the first subsection shortly describes linear models. Then, we introduce
GLMs in the second subsection.
Starting from the linear model
Let X ∈ Mnp(R) be the matrix where each row contains the value of the explanatory
variables for a given individual and Y ∈ R k the vector of responses. The linear model assumes
the following relationship between X and Y :
Y = XΘ + E,
∗. Note that in this document, the term GLM will never be used for general linear model.
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