Understand and apply control and offset variables in GLMs.

A pricing actuary fits a Poisson GLM for claim frequency. Some predictors (driver age, vehicle class) are charged for in the rate. Others (territory loss-control programs, prior-year base rates) belong in the model to remove bias but should not flow to the customer's premium. Control variables and offsets are the two mechanisms that let you keep those effects in the math without letting them out of the actuarial back room.

The core distinction. Both controls and offsets are variables the analyst includes during fitting. They differ in two ways: whether the coefficient is estimated or fixed, and whether the term survives into the scored prediction or not.

KEY: "Control" is a use decision, not a math decision. The fitting code treats a control like any predictor. What makes it a control is that the actuary refuses to vary it (or drops it) when computing the customer-facing prediction.

Why controls exist. Insurance rating plans face regulatory, competitive, or social constraints that forbid charging on certain variables (gender in some jurisdictions, credit in others, ethnicity universally).

Common mistakes

• Using $E_i$ instead of $\ln (E_i)$ as the offset under a log link. The link puts the predictor on the log scale, so the offset must match. Raw exposure as offset produces a model that scales catastrophically wrong.
• Estimating a coefficient on $\ln (\text{exposure})$ "to see what it is." The coefficient will land near 1 with sampling noise. You burn a degree of freedom and risk a fitted scaling that contradicts the contract. Use an offset unless you have a specific reason to question linearity.
• Treating gender or credit as a regular predictor in a jurisdiction that forbids it. The fitted model still works, but you cannot file it. Refit with the restricted variable as a control if you want the bias correction without the regulatory violation.

Bottom line

• A control variable carries an estimated coefficient: you fit it to strip confounding, then drop it or hold it constant when scoring prices or predictions.
• An offset carries a coefficient fixed at 1 on the linear-predictor scale and stays in the prediction, baking a known effect (exposure, prior base rate, known elasticity) into the model.
• For a log-link Poisson frequency model with exposure $E$, the offset is $\ln (E)$, forcing expected claims to scale linearly with exposure.
• Offsets enter as $\eta ={\mathbf{x}}^{\top }\mathbf{\beta }+\text{offset}$; controls enter as ordinary terms ${\beta }_k{x}_k$ you simply ignore at scoring time.

Exam shortcut

When a problem mentions "earned exposure" or "policy-months" alongside a Poisson frequency model with a log link, the answer almost always uses $\ln (\text{exposure})$ as an offset; default to that unless the question explicitly contradicts it. When two candidate coefficients differ between a "with-gender" and "without-gender" model, the with-gender (control) coefficient is the unbiased one to file.

The full lesson (about 2,760 words, 18 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• C6

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