Claim Frequency GLM: Offsets, Overdispersion and an Exposure Coefficient

Linear predictor: $\eta=-2.30+0.45(1)+0.30(1)=-1.55$.

The offset enters with a fixed coefficient of 1, so $\log\mu=\eta+\log(\text{exposure})$, i.e. $\mu=\text{exposure}\times e^{\eta}$. Thus the annual frequency is $e^{-1.55}=0.2122$ claims per policy-year, and with six months of exposure the expected count is $0.5\times0.2122=0.1061$.

Interpretation is multiplicative on the rate, not additive on the count:

• Age: $e^{0.45}=1.568$. Holding area and exposure fixed, a 17-24 year old has about 57% more claims per year than a 25+ driver.
• Area: $e^{0.30}=1.350$, so urban exposure carries about 35% more claims per year than rural.

The offset is not a convenience: it is the assumption that expected claims are exactly proportional to exposure. Fixing its coefficient at 1 is what makes $e^{\eta}$ a rate, which is why the answer can be quoted as an annual frequency. Quoting the number 0.1061 as a 'frequency' would be wrong - that is a count for half a year.

Under a correctly specified Poisson model the residual deviance is approximately $\chi^2$ with residual degrees of freedom, so the dispersion estimate is $\hat\phi=1840/1500=1.227$. A quick reference distribution uses $D\approx\chi^2_{1500}$, with mean 1500 and sd $\sqrt{2\times1500}=54.8$: $(1840-1500)/54.8=6.2$, an extremely small p-value. There is real overdispersion, though of modest size.

Consequences:

• Poisson maximum likelihood still gives consistent coefficient estimates; overdispersion is a problem for standard errors, not for the point estimates.
• Naive standard errors are too small by a factor $\sqrt{1.227}=1.108$.
• If the reported SE of the age coefficient is 0.10, the naive $z=0.45/0.10=4.50$ becomes $z=0.45/(0.10\times1.108)=0.45/0.1108=4.06$, giving $p\approx5\times10^{-5}$. The age effect survives comfortably.

I would refit under quasi-Poisson or negative binomial to get honest standard errors and use an F-test on the scaled deviance rather than the raw likelihood ratio test. Before settling for a distributional fix, I would also ask whether the overdispersion reflects a missing covariate - vehicle group, mileage, no-claims history - because absorbing genuine heterogeneity into a dispersion parameter leaves the model misspecified for pricing at the individual level.

The offset model imposes $\beta_{\text{exposure}}=1$. The covariate model estimates it. Testing $H_0:\beta_{\text{exposure}}=1$:

$$z=\frac{0.82-1}{0.19}=-0.95,\quad p\approx0.34.$$

There is no evidence against proportionality, so the constrained offset specification is supported by the data and should be preferred. Reasons: it encodes the actuarially required property that doubling exposure doubles expected claims; it saves a parameter; and it keeps the distinction between frequency and count clean. If the exposure coefficient were freely estimated, it would absorb variation correlated with exposure - for instance mid-term cancellations, policies written for part-years, or a portfolio that changed its underwriting mix - and the remaining factor coefficients would be contaminated.

Two practical points. First, AIC can be compared here because both models use the same response and the same records, and the offset model has one fewer parameter; on that basis it wins unless the deviance falls enough to justify the extra parameter, which it does not. Second, a coefficient materially different from 1 would be a diagnostic flag rather than an invitation to fit exposure as a rating factor: it usually signals a data problem - earned exposure not reconciled to earned premium, duplicate records, or a change in how exposure is recorded - and that should be investigated before the model is accepted.