Instrumental Variables: Untangling a Confounded Cause

Data-Generating Process

True causal effect β 1.00

Confounding strength 1.20

Instrument strength 1.50

Noise level 0.60

Live Estimates

True β

1.00

Naive OLS β̂

—

2SLS β̂ (IV)

—

Instrument relevance r(Z,X)

—

Naive OLS fit (Y on X) — biased by U

2SLS fit (Y on X̂) — recovers true β

individual, instrument Z=1

individual, instrument Z=0

How it works

Each of the 260 points is a simulated individual with an unobserved confounder U, a binary instrument Z, a treatment X and an outcome Y:

U  ~ N(0, 1)                     (hidden — affects both X and Y)
Z  ~ Bernoulli(0.5)              (instrument — independent of U)
X  =  a·Z + c·U + noise·ε₁       (treatment)
Y  =  β·X + c·U + noise·ε₂       (outcome, true effect β)

Because U pushes X and Y in the same direction, a naive regression of Y on X picks up both the real effect β and a spurious share of U — the red "Naive OLS" line is biased even though every point obeys the same true β. Toggle "Reveal hidden confounder" to tilt the view onto the U axis and see the points fan out along exactly the dimension the naive fit cannot see.

A valid instrument Z moves X (it has "relevance", shown as r(Z,X)) but has no direct arrow into Y — it can only affect Y through X. Two-stage least squares exploits that:

Stage 1 — regress X on Z, keep the predicted values X̂ = â₀ + â₁Z (the part of X driven only by the instrument, with U's influence averaged out by Z's independence from U).
Stage 2 — regress Y on X̂ instead of X. Because X̂ carries none of U's variation, its slope is a consistent estimate of the true β — the teal "2SLS" line tracks the true-β line even as you crank up confounding.

Sliding confounding strength to 0 makes the naive and 2SLS lines coincide (no bias to correct). Sliding instrument strength toward 0 makes Z barely move X — r(Z,X) collapses toward 0 and the 2SLS estimate gets noisy and unreliable, illustrating the "weak instrument" problem economists and epidemiologists worry about in real IV studies (e.g. using distance-to-clinic as an instrument for treatment uptake).

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