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

Risk Classification 103
Modelling Dependence with the ‘Working Correlation Matrix’
The ‘working’ correlation matrix R i takes into account the dependence between the
observations corresponding to the same policyholder. The form of this matrix must be
specified and depends on the parameter vector .
If R i = I, (2.18) gives exactely the likelihood equations (2.16) under the assumption
of independence. The SAS R /STAT procedure GENMOD supports the following structures
of the ‘working’ correlation matrix: fixed (user-specific correlation matrix, not estimated
from the data but specified by the actuary), independent (R i = I, giving the estimates
of obtained under serial dependence), m-dependent (correlation equal to t for lags
t = 1m, and 0 for higher lags), exchangeable (constant correlation , whatever the
lag), unstructured (each correlation coefficient jk , between observations made at times j
and k, is estimated from the data), and AR1 (autoregressive of order 1, with a correlation
coefficient equal to t at lag t).
Numerical Example
The GEE approach can be performed thanks to the procedure GENMOD of SAS R . The
‘Repeated’ statement of GENMOD invokes the GEE method, specifies the correlation structure,
and controls the displayed output from the GEE model. Initial parameter estimates for iterative
fitting of the GEE model are computed as in a Poisson regression model, as described
previously. Results of the initial model fit are displayed as part of the generated output
of SAS R . Statistics for the initial model fit such as parameter estimates, standard errors,
deviances, and Pearson Chi-squares do not apply to the GEE model, and are only valid
for the initial model fit. The SAS R parameter estimates table contains parameter estimates,
standard errors, confidence intervals, Z-scores, and p-values for the parameter estimates.
The ‘Repeated’ statement specifies the covariance structure of multivariate responses for
GEE model fitting in the GENMOD procedure. In addition, the ‘Repeated’ statement controls
the iterative fitting algorithm used in GEE and specifies optional output. Other GENMOD
procedure statements are used in the same way as they are for ordinary Poisson regression
models to specify the regression model for the mean of the responses.
The statement ‘SUBJECT = subject-effect’ identifies subjects in the input data set. The
subject-effect can be a single variable, an interaction effect, a nested effect, or a combination.
Each distinct value, or level, of the effect identifies a different subject, or cluster. Responses
from different subjects are assumed to be independent, and responses within subjects are
assumed to be correlated. A subject-effect must be specified, and variables used in defining
the subject-effect must be listed in the CLASS statement. In actuarial applications, the policy
number is typically used as subject-effect.
The same variables as for the model where the serial independence was assumed are kept
in the final model. The results are given in Table 2.12 that is similar to the SAS R outputs
for GEEs. The estimation of the ‘working correlation matrix’ gives
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1 00493 00460
00493 1 00493
00460 00493 1
and ̂ = 13419.
Since the GEE apprach is not based on a likelihood function, we cannot use the large
sample approximations for the estimated variance-covariance matrix ̂̂ of the estimated
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