When should I use Poisson regression and how do I interpret it?

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AcadiFi · Accepted Answer

Poisson regression models count data — non-negative integers like number of defaults, claims filed, transactions completed, or events per period. It's a GLM with Poisson distribution and log link: log(E[Y|X]) = Xβ, or equivalently E[Y|X] = exp(Xβ).

The Poisson distribution has one parameter λ (both mean and variance). For Y ~ Poisson(λ): P(Y=k) = e^(−λ) λ^k / k! for k = 0, 1, 2, ...

Example: Consider Marston Insurance modeling annual auto claims per policyholder. Predictors include age, miles driven, years of driving experience. Fitted model: log(λ) = −2.5 + 0.02 × miles_per_year/1000 − 0.01 × experience_years. A policyholder driving 15,000 miles with 10 years experience has log(λ) = −2.5 + 0.3 − 0.1 = −2.3, so λ = e^(−2.3) ≈ 0.100, meaning expected 0.1 claims per year.

Coefficient interpretation: exp(βⱼ) is the rate ratio or incidence rate ratio. Here exp(0.02) ≈ 1.0202, meaning each additional 1,000 miles driven multiplies expected claim rate by 1.02 (a 2% increase).

Key assumption: mean = variance (equidispersion). Financial count data often violates this — defaults cluster (contagion) creating overdispersion where variance exceeds mean. Diagnostic: compare Pearson chi-square / degrees of freedom to 1. If >> 1, overdispersion is present and standard errors are underestimated.

Solutions when overdispersed: (1) quasi-Poisson — same estimates, corrected standard errors; (2) negative binomial regression — explicitly models overdispersion; (3) zero-inflated Poisson — for excess zeros.

Offsets are critical when exposure varies: if policyholders are observed for different periods, include log(exposure) as offset in log(λ) = Xβ + log(exposure). This converts counts to rates per unit time.

Poisson regression is foundational for insurance pricing and credit modeling.

When should I use Poisson regression and how do I interpret it?

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