What Is The Effect When The Treatment Is Confounded? Two-Stage Least Squares

How it works

The first stage regresses the treatment on the instruments plus any controls with base least squares, and the F statistic comparing that fit against the same regression WITHOUT the instruments measures instrument strength on the excluded instruments alone. The second stage regresses the outcome on the fitted treatment from the first stage plus the same controls. The reported standard error is the two-stage least squares standard error: the second-stage regression's own figure is wrong because it measures residuals against the fitted treatment, so the residual variance is rebuilt from the structural equation (the outcome minus the fitted coefficients applied to the ACTUAL treatment) and used to scale the second-stage cross-product inverse. An ordinary least squares regression is run alongside for contrast, a Sargan over-identification test is computed when there are more excluded instrument parameters than confounded regressors, and a Wu-Hausman test asks whether the two estimates differ by more than sampling noise.

Use it when the treatment was not randomly assigned, you believe something unmeasured moves both the treatment and the outcome, and you have a variable that plausibly shifted the treatment for reasons unrelated to the outcome: a lottery, a distance, an eligibility cutoff, an administrative rule, a supply shock.

Not when you have no credible instrument, in which case matching or a difference-in-differences design may fit better. Not when the instrument fails the first-stage strength check. Not for more than one confounded treatment column at a time, and not when the design already gives you a comparison group directly, such as a randomized experiment or a sharp cutoff.

What data do you need?

One row per unit, with an outcome, a confounded treatment, at least one instrument, and any controls. For example, returns to schooling instrumented by distance to the nearest college:

Minimum 50 rows · Best with 300-10,000 rows, one confounded treatment, one to three instruments

What's in the report?

Standard-library analysis: what is the causal effect when the treatment is confounded? Map an outcome, a confounded treatment, and at least one instrument (a variable that shifts the treatment but has no other route to the outcome) and get two-stage least squares built stage by stage: the first-stage F on the excluded instruments and the instrument-strength verdict it implies, the ordinary least squares estimate and the instrumented estimate side by side with the difference between them made explicit, an over-identification (Sargan) test when more than one instrument is supplied, an endogeneity (Wu-Hausman) test, and a plain statement of the exclusion restriction the whole estimate rests on.