Actuarial Modelling of Claim Counts Risk Classification, Credibility ...

Risk Classification 63
Example 2.3 Figure 2.7 suggests that the variables Age and Gender interact in Portfolio
A. It is thus not possible to represent accurately the effect of being a male policyholder
(compared with being a female policyholder) in terms of a single multiplier, nor can the
effect of Age be represented by a single multiplier. The relevant explanatory variables are not
x i1 , x i2 , x i3 , and x i4 but rather x i1 x i4 , x i2 x i4 , x i3 x i4 , 1 − x i1 x i2 x i3 x i4 , x i1 1 − x i4 , x i2 1 − x i4 ,
and x i3 1 − x i4 (denoted henceforth as x
ij ′ ).
To reflect the situation accurately, it is thus necessary to consider multipliers dependent
on the combined levels of Age and Gender. To this end, a variable Gender ∗ Age is created,
with levels ‘Female 18–24’, ‘Female 25–30’, ‘Female 31–60’, ‘Female over 60’, ‘Male
18–24’, ‘Male 25–30’, ‘Male 31–60’ and ‘Male over 60’. This new variable possesses 8
levels, and is coded by means of 7 dummies, being all 0 for the reference level (taken as
‘Male 31–60’). Specifically, the explanatory variables x i1 to x i4 in Example 2.2 are replaced
with
{ 1 if policyholder i is a female less than 24
x ′ i1 = 0 otherwise
{ 1 if policyholder i is a female aged 25–30
x ′ i2 = 0 otherwise
{ 1 if policyholder i is a female aged 31–60
x ′ i3 = 0 otherwise
{ 1 if policyholder i is a female over 60
x ′ i4 = 0 otherwise
{ 1 if policyholder i is a male less than 24
x ′ i5 = 0 otherwise
{ 1 if policyholder i is a male aged 25–30
x ′ i6 = 0 otherwise
{ 1 if policyholder i is a male over 60
x ′ i7 = 0 otherwise
Rather than declaring Age and Gender as two explanatory variables coded with the help of
x i1 to x i4 , a combined Age ∗ Gender variable is declared and is coded with the help of the
covariates x
i1 ′ to x′ i7 .
The part of the linear predictor involving Age and Gender is 1 x i1 +···+ 4 x i4 without
interaction, and becomes ′ 1 x′ i1 +···+′ 7 x′ i7
if allowance is made for interaction. If Age and
Gender indeed interact then the values of these two linear combinations differ from each
other, whereas they collapse in the absence of interaction.
Allowing for interaction thus dramatically increases the number of explanatory variables.
Introducing Age and Gender separately requires 4 binary variables whereas allowing
for the Age–Gender interaction requires 7 dummies. As a consequence, the number
of parameters to be estimated increases accordingly. We will see below that grouping
some levels of the combined variable accounting for interaction is nevertheless often
possible.