Identification of Causal Effects
Using Instrumental Variables

Jonas Dehning

A confounder is a common cause of treatment and outcome

  • Example: baseline health before treatment
  • Baseline health may affect both treatment X and outcome Y.

The observed \(X\)–\(Y\) correlation mixes \(X\rightarrow Y\) with \(X\leftarrow C\rightarrow Y\).

Common-cause structure

\[ C\longrightarrow X, \qquad C\longrightarrow Y. \]

Backdoor path

\[ X\longleftarrow C\longrightarrow Y. \]

Observed versus potential outcome

Let \(Y(x)\) be the outcome a person would have under treatment \(x\). In general,

\[ \mathbb E[Y\mid X=x] \neq \mathbb E[Y(x)]. \]

If the confounder is observed, adjustment can block its path

    Either:
  • Compare patients with the same C.
  • Add C to a regression of Y on X.

This is adjustment for \(C\): it “blocks’’ \(X\leftarrow C\rightarrow Y\).

Adjustment logic

If all relevant common causes are observed, treatment \(X\) is independent of the potential outcome \(Y(x)\) within groups with the same \(C\):

\[ Y(x)\perp X\mid C, \]

Here, \(\perp\) means independent of, and \(\mid C\) means within the same \(C\) group. Then

\[ \mathbb E[Y(x)] = \int \mathbb E[Y\mid X=x,C=c]\,dP(c). \]

Regression version

\[ Y=\beta X+\gamma^\top C+\varepsilon. \]

With exchangeability, overlap, and a correct model, \(\beta\) can represent the causal effect of \(X\) on \(Y\).

If the confounder is hidden, adjustment cannot block its path

Ability, motivation, frailty, or disease severity may affect both \(X\) and \(Y\) yet go unmeasured (an omitted variable).

We now need variation in \(X\) that does not originate from \(C\).

Why regression mixes two signals

Suppose

\[ Y=\beta X+\delta C+\varepsilon. \]

Then

\[ \operatorname{Cov}(X,Y) =\beta\operatorname{Var}(X) +\delta\operatorname{Cov}(X,C). \]

The observed association combines the causal effect \(\beta\) with selection through \(C\).

But: No observed control blocks X ← C → Y.

An instrument is an extra variable that may move \(X\) cleanly

Find a so-called instrument \(Z\) that can be used to measure the independent variation of \(X\).

For Z to be a valid instrument, three requirements must hold:
1

Relevance

Z changes treatment X.

Cov(Z, X) ≠ 0
2

Independence

Z does not share the hidden causes C.

Z independent of C
3

Exclusion

Z reaches outcome Y only through X.

no direct Z → Y effect

Why the variation can be clean

Use the route

Z → X → Y

and require

Z independent of C
no direct Z → Y effect

Example
A physician’s prescribing preference Z changes treatment X, but should not otherwise affect Y.

IV compares how the instrument moves \(Y\) with how it moves \(X\)

1
First stage: How much does Z move X?
ΔX
2
Reduced form: How much does Z move Y?
ΔY
3
IV ratio: response per induced unit of X
βIV=ΔYΔX

Only the variation in X induced by Z enters the IV estimate.

Why the ratio works

For

\[ Y=\beta X+\delta C+\varepsilon, \]

take covariance with \(Z\):

\[ \operatorname{Cov}(Z,Y) =\beta\operatorname{Cov}(Z,X) +\delta\operatorname{Cov}(Z,C) +\operatorname{Cov}(Z,\varepsilon). \]

Independence makes the last two terms zero. Thus

\[ \operatorname{Cov}(Z,Y) =\beta\operatorname{Cov}(Z,X). \]

If \(Z\) changes \(X\), divide by \(\operatorname{Cov}(Z,X)\):

\[ \boxed{\beta_{IV} =\frac{\operatorname{Cov}(Z,Y)} {\operatorname{Cov}(Z,X)}}. \]

Read the ratio

Numerator: variation in \(Y\) associated with \(Z\).

Denominator: variation in \(X\) associated with \(Z\).

The ratio converts the movement in \(Y\) into an effect per instrument-induced unit of \(X\).

IV requires a specific causal structure, but has been used in varied settings

Physician prescribing preference1
Z
physician’s tendency to choose one drug
X
drug received
Y
adverse outcome
C
hidden patient risk
Patients at higher risk may receive different drugs. Practice style can supply additional treatment variation.
Other classic examples
Distance to intensive cardiac care2
Z differential distance→X treatment intensity→Y survival
Nearby hospital capabilities changed treatment intensity for heart-attack patients.
Vietnam draft lottery3
Z lottery eligibility→X military service→Y lifetime earnings
Random lottery numbers changed the chance of military service.

The recurring design: \(Z\) moves \(X\), is unrelated to \(C\), and has no direct effect on \(Y\).

1 Brookhart et al. (2006), “Evaluating short-term drug effects using a physician-specific prescribing preference as an instrumental variable,” Epidemiology 17:268–275.
2 McClellan, McNeil & Newhouse (1994), “Does More Intensive Treatment of Acute Myocardial Infarction in the Elderly Reduce Mortality?”, JAMA 272:859–866.
3 Angrist (1990), “Lifetime Earnings and the Vietnam Era Draft Lottery,” American Economic Review 80:313–336.