Chapter 34 of 57

Instrumental Variables and 2SLS, on One Page

Created Jul 5, 2026 Updated Oct 25, 2026

Your regressor X is endogenous — tangled up with the error term — so OLS gives a biased slope. Instrumental variables is the fix, and the idea is almost sneaky: if X is dirty, don't clean X. Find a third variable that jiggles X around for reasons unrelated to the structural error — it reaches y, but only by going through X — and use only that clean jiggle.

That third variable is an instrument Z, and it has to earn three things:

Relevance — Z genuinely moves X. $\mathrm{Cov}(Z, X) \neq 0$ . Weak movement is a real problem, more on that below.
Independence — Z is unrelated to the unobserved causes of y. A draft lottery qualifies because the draw was random; a variable that merely looks exogenous but shares a confounder with y does not.
Exclusion — Z reaches y only through X, with no direct path of its own.

The last two are different claims, and both are assumptions about the world that no test proves. In the linear model they combine into the one condition the algebra uses: $\mathrm{Cov}(Z, \varepsilon) = 0$ .

Two-stage least squares (2SLS) turns that into arithmetic:

Stage 1 — regress X on Z. Keep the fitted values X̂: the slice of X that moves with the instrument.
Stage 2 — regress y on X̂ instead of X. If the instrument is valid and strong enough, that slope consistently estimates the causal effect — the cleanliness comes from the instrument being independent and excluded, not from the act of fitting.

One practical trap if you run it by hand: the second regression's standard errors are wrong, because X̂ is estimated rather than observed. Use an IV routine (or the 2SLS formula) for inference, not the second OLS's output.

The left panel is stage 1 — X regressed on the instrument Z, giving the fitted X̂. The right panel is not the second regression but its result, drawn on the structural (X, y) plane: the OLS slope is dragged off the true β = 1 by the hidden confounder, while the 2SLS slope — built only from X̂ — lands on it. Pull the instrument π slider toward zero and watch the first-stage F collapse: the 2SLS estimate starts swinging wildly on every resample, the weak-instrument failure mode where IV becomes both more biased and noisier than OLS.

Two things the picture makes concrete:

Weak instruments bite. If Z barely moves X (a limp first stage, low F-statistic), stage 1 produces a noisy X̂ and stage 2 amplifies it: the estimate becomes unstable, with a heavy-tailed distribution and a finite-sample bias that pulls back toward OLS. A weak-instrument estimate can end up less useful than the biased OLS it was meant to replace.
What IV identifies depends on whether the effect is the same for everyone. If the causal effect is a single constant — the assumption the picture above quietly makes, with β = 1 for every unit — a valid instrument recovers it, and it is the average effect too. Once effects differ across units, a binary instrument and binary treatment identify the Local Average Treatment Effect under an added monotonicity assumption: the effect for the compliers, the units whose X actually responds to Z. That is not the whole-population ATE, and pretending otherwise is a common overreach.

Classic instruments: a lottery draft for military service, distance-to-college for years of schooling, a cost shifter (weather, input prices) for the price a firm charges. Each one is defended by a story, not a statistic.

The full derivation — LATE, the exclusion restriction, weak-instrument diagnostics — is in the Instrumental Variables (2SLS) note. The same identification, reparameterised as an extra control instead of a replacement, is the control function.