The Linear Model View: Good controls, overlap, and the bit the regression cannot see

The promise of “as‑if random” becomes tangible when you translate it into the language most of us actually use: a linear regression. The model looks innocuous: an outcome on the left, a constant, a treatment indicator, a bundle of pre‑treatment variables, and a leftover term on the right. The entire causal reading lives or dies with that leftovers term. If, after you condition on the right pre‑treatment information, the remainder is not systematically related to who received treatment, then the treatment coefficient isolates the effect of interest in the region of the data where treated and untreated truly overlap. If not, the coefficient mixes treatment with whatever forces leaked into the remainder, no matter how large the sample.

Write the model as:

\[\text{Outcome} = \text{constant} + (\text{effect} \times \text{treatment}) + \text{pre‑treatment information} + \text{leftover}.\]

The “as‑if random” claim is that, once the pre‑treatment information is in the model, the leftover no longer co‑moves with treatment. That is exactly the sense in which the remaining treatment variation behaves like chance for the purpose of this comparison. When that condition holds and there is genuine overlap between treated and untreated across the relevant covariates, the treatment coefficient can be read as a causal effect for the part of the population where you actually have a comparison.

Two features in the previous sentence are easy to gloss over and then forget (and regret). The first is pre‑treatment. The second is overlap. Everything that follows is about taking those two words literally.

Good controls are variables that sit upstream of both treatment and outcome in your story of the world. Timing helps you decide. If a variable can plausibly be measured before the treatment decision, and if in your setting it is a driver of both who gets treated and how outcomes evolve, then it belongs in the adjustment set. Prior earnings, age, sector, education, distance or time costs, baseline test scores, firm size, school type, cohort, and local labour‑market conditions are the usual suspects in applied work. When in doubt, ask whether you could have recorded the variable before anyone decided to take the programme.

Bad controls are variables that sit downstream of the treatment, or that are only measured because of the treatment. Adjusting for them can wash away part of the effect you are trying to measure or even fabricate a relationship that is not there. “Post‑treatment employment stability” in a training evaluation is a bad control; so is “current sales” in a policy that affects sales. Another class to avoid are variables that are created by the treatment or jointly caused by treatment and together with other factors; conditioning on such a variable can create a path you were trying to close.

Remember the rule: include genuine pre‑treatment drivers; leave consequences of treatment alone.

The second word that carries weight is overlap. The linear model can only compare like with like where both treated and untreated units actually exist. If all high‑experience, high‑education workers took the programme and no one else did, your regression line has to extrapolate across a gap. In that region, the line is not an adjustment; it is a guess. The “as‑if” story lives in the part of the covariate space where you can place treated and untreated units side by side with similar characteristics.

You can see overlap with simple pictures and summaries. Plot the distributions of key pre‑treatment variables by treatment status. If they barely touch, you do not have a comparison there. Look at the joint picture too: it is common to have decent overlap on each variable separately and still have poor overlap once you consider them together. When overlap is weak on the margins that matter for your story, trimming the sample to a region of common support is often more honest than relying on a functional form to bridge the gap. No specification can manufacture chance where none operated.

Even when you choose controls thoughtfully and restrict attention to the region of overlap, the linear model can be undone by structure it does not represent. Three issues account for most surprises.

The first is functional form. If the relationship between a pre‑treatment variable and the outcome is curved, or if the effect of treatment varies with the covariates, a straight line with no interactions pushes systematic structure into the leftovers term. If that structure is also related to treatment, you have re‑created the very link you wanted to sever. In practice this is a plea for flexible forms and interactions where the setting calls for them. Logs for positive outcomes, spline terms for age or tenure, and interactions between baseline performance and treatment are routine examples.

The second is measurement. Mis‑measured treatment status or key covariates pull the problem back into the leftovers term. If programme take‑up is recorded with error, the treatment coefficient is dragged towards zero and the residual can pick up whatever pattern mis‑measurement creates. If a crucial pre‑treatment driver is only observed noisily or intermittently, the “as‑if random” claim conditional on that variable is weaker than it looks on paper. Timing helps here as well: record baseline variables before the treatment decision and define treatment in a way that is hard to misclassify.

The third is simultaneity and feedback. When treatment and outcome are mutually influencing within the period you observe, a simple contemporaneous regression treats a two‑way street as a one‑way path. In those settings, design devices—staggered timing, cut‑offs, instruments, before‑after contrasts—do more than any functional form can.

Because they are easy to produce, balance tables and long control lists are often used as evidence for an “as‑if” claim. They should be read as necessary but not sufficient. Balance on observables is a good sign, but selection into treatment is usually driven by information that is hard to measure: motivation, networks, informal costs, local enforcement, managerial quality. A thick control set is not a causal model. If it contains post‑treatment variables or mechanically related measures, you can make things worse. The safe habit is to name, in words, the small set of pre‑treatment forces that plausibly sort units into treatment and drive outcomes in your setting, show that treated and untreated overlap on those forces, and then let the functional form work hard without inventing comparisons.

Even when the “as‑if” story holds, it helps to remember what the linear treatment coefficient actually averages. With constant effects and good overlap, the coefficient is the effect you have in mind. With heterogeneous effects and uneven overlap, ordinary least squares places more weight where the covariate distribution of treated and untreated overlap and where residual variance is low. In practice that means the estimate can lean towards the sub‑populations where comparisons are easiest and outcomes less volatile. This is not a flaw; it is a reminder to say for whom your estimate speaks and to avoid extrapolating beyond the region where you had data to compare.

A disciplined way to make the claim is as follows. Start by stating the treatment decision in your setting: who had the option, what costs and eligibility rules mattered, when the decision was taken. List the small set of truly pre‑treatment variables that capture those costs and rules in your context, and make a sentence for each explaining why it belongs. Show that treated and untreated occupy common territory on those variables; if they do not, restrict your attention to where they do. Then fit models that respect that design: flexible forms for important covariates, interactions where moderation is plausible, and a specification that does not adjust for consequences of treatment. Finally, keep your uncertainty statements honest about what they refer to: in this observational setting they are about samples not drawn, conditional on your “as‑if” story being true in the region of overlap. They are not guarantees against a bad story.