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Controls, Offsets, and Omitted Variable Bias in GLMs

When to estimate a variable as a control, when to use an offset, and why omitted variables can bias insurance models.

GLM Controls Offsets

Controls and offsets are concepts that were not taught at school for me, and something that I realized is quite important especially when working with GLMs. One of the main purposes of using a GLM is interpretability I think, and we want to make sure that the predictors of my interest are not picking up any unintended signals. It is quite common when you want to estimate the effect of certain variables, while making sure to partiallying out the impact of a known factor that we would not want to add to the model. This is

Stated more formally, in regression analysis, control variables are included to account for factors that might influence the dependent variable but are not the primary focus of the study. Their inclusion helps ensure that the relationship between the primary independent variables (variables of interest) and the dependent variable is accurately estimated, without being distorted by these other factors.

Offsets are similar to controls except that you fix a coefficient of 1. Using log link function, this means we can include in the denominator of the target (exposure * offset).The difference is that a control is something the model is allowed to learn, and an offset is something the model is told to accept (by fixing coefficient at 1).

The omitted variable problem

Suppose I am modeling claim frequency, and the candidate predictors include driver age, vehicle use, multiple-car status, territory, and symbol.

Now imagine I drop territory from the model. If a particular territory has both higher claim frequency and a larger share of young drivers, driver age may start acting partly as a proxy for territory. The age coefficient is no longer just about age. It is absorbing some of the territory effect too.

In a simple linear setup, omitted variable bias is often written this way:

True model:
Y = beta_0 + beta_1 X + beta_2 Z + error

Fitted model, after omitting Z:
Y = alpha_0 + alpha_1 X + error

Bias in alpha_1 depends on:
beta_2 * relationship between X and Z

So the omitted variable has to matter for Y, and it has to be related to X. If both are true, the coefficient on X gets contaminated.

This is the part that feels especially important in pricing work. If a coefficient is going to influence a rating decision, it needs to represent the effect we think it represents.

What a control does

A control variable is included in the model and estimated alongside the variables we care about.

For a log-link GLM, that might look like:

log(E[Y]) = beta_0 + beta_1 age + beta_2 territory + beta_3 symbol + ...

If I include territory as a control, the model estimates the age effect while accounting for territory. That helps isolate the age effect from the territory effect.

The tradeoff is that controls can fight with other variables for the same signal. Large categorical controls can be especially tricky because they may overlap with many other predictors. The model can become harder to interpret, standard errors can inflate, and the coefficient of interest may look weaker than expected.

That does not mean controls are bad. It just means they are not free.

What an offset does

An offset is different because the model does not estimate its coefficient.

If territory and symbol factors were estimated somewhere else, I can treat those effects as known. For example, I might adjust exposure by multiplying it by known territory and symbol factors:

u_adjusted = u * tau_i * sigma_j

where:

  • u is the original exposure
  • tau_i is the territory factor
  • sigma_j is the symbol factor

In a log-link model, the same idea can be written as an offset:

log(E[Y]) = X beta + log(tau_i * sigma_j)

If I am also modeling claim counts with exposure (frequency), the offset may include exposure too:

log(E[claim_count]) = X beta + log(exposure) + log(tau_i * sigma_j)

The key is that the offset enters with a fixed coefficient of 1.

By adding the offset, which by algebra we are taking the offset factor’s effect away from the target, we let the model focus on explaining the remaining signal.

Control vs. offset

I think of controls as conditional adjustment.

Estimate:
log(E[Y]) = beta_0 + beta_1 X + beta_2 control

The model estimates the control and the main predictor together. If they share signal, they compete for it.

Offsets are structural adjustment.

Estimate:
log(E[Y]) = beta_0 + beta_1 X + offset

The control effect has already been removed or built in before estimating the remaining coefficients.

That distinction is why offsets can be useful when a control variable would create severe multicollinearity. Since the offset coefficient is not estimated, it does not fight with the other predictors in the same way.

How I would decide

To be honest, since the purpose for using controls/offsets are so that my predictors do not mix signals from the control/offset variables, I think either method give the same solution. But I think that controls may be easier to implement if you have already created the dataset with offsets baked into the target. I think offset is a more clean approach, but someone can argue otherwise…