Generalized Linear Model

Study guides: Exam PA, Exam MAS-I, Exam 5, Exam PCPA, Exam 8

A Generalized Linear Model (GLM) extends ordinary linear regression to allow the response variable Y to follow any distribution in the exponential family and to be related to the predictors through a Link Function g.

g(\mu_i) = \eta_i

= \beta_0 + \beta_1 x_{i1} + \cdots + \beta_p x_{ip}

\text{where } \mu_i = E[Y_i \mid \mathbf{x}_i]

Three components of a GLM:

1. Random component: Y_i follows an exponential family distribution (Normal, Poisson, Gamma, Binomial, etc.)
2. Systematic component: linear predictor \eta_i = \mathbf{x}_i^\top \boldsymbol{\beta}
3. Link function: g(\mu_i) = \eta_i connects the mean to the linear predictor

Common GLM specifications:

• Parameters are estimated by maximum likelihood, not OLS
• Model fit is assessed via Deviance, AIC, BIC, and Residual Plots
• ANOVA (Analysis of Deviance) compares nested GLMs

In ratemaking, the GLM is the standard tool for classification work, because it estimates every rating variable simultaneously and so reports each one net of the others — solving the correlation problem that makes univariate relativities double-count. Standard practice:

• Frequency is modelled Poisson with a log link and exposures as an offset; severity is modelled Gamma with a log link, weighted by claim counts. Multiplying the two gives the pure premium. A single Tweedie model on pure premium is the one-step alternative.
• The log link makes the model multiplicative, so e^{\beta} is a relativity and the fitted coefficients drop straight into a multiplicative rating algorithm.
• Output is judged on more than significance: consistency of the pattern across levels, standard errors, the effect of removing the variable (analysis of deviance), and validation on a holdout period.
• Model relativities are still credibility-weighted and are still subject to the operational, social and legal criteria for a rating variable — statistical significance alone does not make a variable usable.

Poisson GLM for Claim Counts An actuary models claim counts N_i \sim \text{Poi}(\mu_i) as a function of driver age. The fitted model gives \ln\hat{\mu}_i = -2.1 + 0.04 \cdot \text{age}_i. Find the expected claim count for a 30-year-old driver.

Answer \ln\hat{\mu} = -2.1 + 0.04(30) = -2.1 + 1.2 = -0.9 \hat{\mu} = e^{-0.9} \approx 0.407 \text{ claims per year}

Reading GLM Output as Rating Relativities A frequency GLM (Poisson, log link, exposure offset) and a severity GLM (Gamma, log link) are fitted to a personal auto book. Coefficients for the territory variable, relative to the base territory:

Territory	Frequency \beta	Severity \beta
Base	0	0
Suburban	0.1823	0.0488
Urban	0.4055	0.1310

Derive the pure premium relativities.

Answer With a log link the relativity is e^{\beta}, and the pure premium relativity is the product of the two:

\begin{align*}
\text{Suburban freq} &= e^{0.1823} = 1.200 \\
\text{Suburban sev} &= e^{0.0488} = 1.050 \\
\text{Suburban PP} &= 1.200 \times 1.050 = 1.260 \\[6pt]
\text{Urban freq} &= e^{0.4055} = 1.500 \\
\text{Urban sev} &= e^{0.1310} = 1.140 \\
\text{Urban PP} &= 1.500 \times 1.140 = 1.710
\end{align*}

Equivalently, sum the coefficients before exponentiating: e^{0.4055 + 0.1310} = e^{0.5365} = 1.710.

Two things this shows that a one-way analysis would not. First, the urban loading is driven mostly by frequency (+50\%) rather than severity (+14\%) — a density and traffic effect rather than a repair-cost effect. Second, these relativities are net of every other variable in the model: if urban policyholders also skew toward higher limits or older vehicles, that effect sits with those variables, not here. Multiplying a 1.710 territory factor by the other fitted factors therefore does not double-count.