Linear Regression and Generalised Linear Models
This domain covers regression as a modelling framework, from ordinary least squares through to generalised linear models, which are the standard tool for mortality and claims rate modelling in actuarial practice.
Linear regression
- Y = X beta + epsilon with independent normal errors of constant variance. The OLS estimator is beta-hat = (X'X)^-1 X'y, and Gauss-Markov makes it best linear unbiased under the weaker assumptions of zero mean, constant variance and uncorrelated errors.
- Assumptions to check: linearity, independence, homoscedasticity, approximately normal errors for small-sample tests, and no perfect collinearity. Residual plots, leverage and Cook's distance diagnose problems.
- Categorical covariates enter through dummy variables, omitting one baseline level.
Generalised linear models
- Three ingredients: a random component from the exponential family, a linear predictor eta = X beta, and a link function g with g(mu) = eta.
- Canonical links: normal-identity, Poisson-log, binomial-logit, gamma-reciprocal.
- Poisson GLMs with a log link and an offset log(E) model counts as rates, so that mu/E = exp(X beta).
- Estimation is by maximum likelihood, usually through iteratively reweighted least squares or Newton-Raphson. The deviance is twice the difference between the saturated and fitted log-likelihoods; the dispersion is deviance divided by residual degrees of freedom, and exceeds 1 when the data are overdispersed.
Worked example
A Poisson GLM of mortality counts has log(mu_x) = beta0 + beta1 x + log(E_x), giving force of mortality exp(beta0 + beta1 x). If beta1-hat = 0.05, each additional year of age multiplies mortality by exp(0.05) = 1.0513, a 5.13% increase. Because the link is logarithmic, coefficients act multiplicatively - not additively - on the rate.
Traps that cost marks
- Interpreting a log-link coefficient as an absolute increment.
- Omitting the offset and modelling counts instead of rates.
- Ignoring overdispersion when deviance far exceeds the degrees of freedom.