THE INSTRUMENTAL VARIABLE

You cannot see the confounder, so you cannot adjust it away. Find instead a lever that jiggles only the cause — a valve upstream of X that never touches Y except through X. Its wobble carries a clean signal of the causal slope through the noise. OLS is fooled; the instrument is not. Rendered, not quoted.

source Angrist & Imbens, “Identification and Estimation of Local Average Treatment Effects,” Econometrica 62(2):467–475 (1994) · method after P. G. Wright, appendix B to The Tariff on Animal and Vegetable Oils (1928).

Blue team · builds & defends
3

The model

A linear-Gaussian structural causal model. U is an unobserved confounder feeding both X and Y. Z is the instrument: it moves X and reaches Y only through X.

Z ---a--> X ---β--> Y U --c--^ ^--d-- U (U hidden) X = a·Z + c·U + eₓ Y = β·X + d·U + eᵧ

True causal effect β = 2. Coefficients a=0.8, c=1.5, d=1.2; all noises unit-variance, independent. Everything below is computed from these — no baked numbers.

5

The lineage

Identification without adjustment. The neighbouring sphere the-backdoor-criterion closes confounding by conditioning on observed variables. But you cannot condition on the-confounder when it is unmeasured — the back door has no handle.

The instrument is the escape hatch: instead of blocking the path, find a Z that opens a front door you fully control. This is the tool econometrics leans on where the backdoor criterion cannot reach.

7

The witness

Live re-check of the identification claim. Re-derives IV from the current model and tests exclusion. Flips red the instant the instrument is tampered.

checking…
The machine
4

Data in in ↓

The analyst observes the joint (Z, X, Y) only. The confounder U and every structural noise term are never seen. All the engine may use are covariances of the three observed columns.

0

The panel lit

Three estimates of the same slope. OLS regresses Y on X directly; IV divides Cov(Z,Y) by Cov(Z,X). Drag the instrument strength.

Truth   do(X): dE[Y]/dx—
OLS(Y~X) biased by U—
IV = Cov(Z,Y)/Cov(Z,X)—
instrument strength a (Z→X)0.80
Cov(Z,X) relevance—
IV asymptotic Var ∝ 1/Cov(Z,X)²—

8

Data out out ↓

Proven: under a hidden confounder, OLS is biased to ≈2.463 while a valid instrument recovers β = 2 exactly (population / large N).

—
Red team · attacks & breaks
1

The adversary

wall IV buys identification with two untestable promises. Break either and the estimator lies with a straight face:
• Exclusion — any direct Z→Y path (even tiny) biases IV by g/a.
• Relevance — a weak first stage (small Cov(Z,X)) makes the ratio explode.
• Monotonicity — “defiers” break the LATE interpretation of a heterogeneous effect.
2

The graveyard

  • “IV needs no assumptions — it's model-free.” → It trades observed-confounding assumptions for relevance + exclusion, which no data can verify.
  • “Add more controls and OLS will reach the causal effect.” → Controls cannot touch an unobserved U; the bias term d·c·Var(U)/Var(X) survives.
  • “A significant first stage means a strong instrument.” → Significance ≠ strength; weak instruments (low F) inflate variance and bias toward OLS.
6

The tamper

Planted void (disclosed): add a direct edge Z→Y (g=0.5), violating exclusion. IV drifts from β to β+g/a. The witness (7) catches it live.