Instrumental Variables / 2SLS

Motivating Example: Colonial Origins of Comparative Development

Why are some countries rich and others poor? Acemoglu et al. (2001) proposed that institutions — the rules governing economic activity — are the key driver. But institutions are endogenous: rich countries invest in better institutions, creating a classic chicken-and-egg problem.

Their solution was an instrument: settler mortality in the colonial era. The argument runs as follows:

1. In places where European settlers faced high mortality (tropical diseases), colonizers set up extractive institutions designed to transfer wealth to the metropole.
2. In places where settlers could survive (temperate climates), they created inclusive institutions with property rights and rule of law.
3. These institutional differences persisted and shaped modern economic outcomes.
4. Critically, settler mortality from centuries ago affects current GDP only through its effect on institutions — it has no direct effect on economic performance today.

This claim is the exclusion restriction: the instrument (settler mortality) affects the outcome (GDP) only through the endogenous variable (institutions). If this holds, 2SLS can recover the causal effect of institutions on development. A related strategy that builds on this IV logic is the shift-share (Bartik) instrument, which interacts local exposure shares with national-level shocks to generate cross-sectional variation.

Whether the exclusion restriction actually holds in this case has been debated for two decades. That debate is itself a masterclass in IV methodology.

This strategy is the fundamental logic of instrumental variables: find an external source of variation that shifts the endogenous regressor without directly affecting the outcome, and use that variation to recover causal effects.

A. Overview

The Endogeneity Problem

Consider the regression:

If $\text{Cov}(D_i, \varepsilon_i) \neq 0$ — the treatment or key regressor is correlated with the error — then OLS is biased and inconsistent. This endogeneity arises from omitted variables, reverse causality, or measurement error. Standard sensitivity analysis techniques can quantify how severe confounding must be to explain the estimated relationship, but when confounding is clearly present, OLS adjustment alone is insufficient.

The IV Solution

An instrumental variable $Z_i$ solves this endogeneity problem by isolating the part of $D_i$ that is uncorrelated with $\varepsilon_i$. The instrument must satisfy three conditions for consistent estimation, plus a fourth for the LATE interpretation:

1. Relevance: $\text{Cov}(Z_i, D_i) \neq 0$ — the instrument must actually affect the endogenous variable. This condition is testable.
2. Independence (Exogeneity): $\text{Cov}(Z_i, \varepsilon_i) = 0$ — the instrument must be uncorrelated with the error term. This condition is not directly testable with a single instrument.
3. Exclusion Restriction: $Z_i$ affects $Y_i$ only through $D_i$ — there is no direct effect. This restriction is a maintained assumption that must be argued substantively.
4. Monotonicity (for LATE interpretation): The instrument affects treatment status in only one direction for all units — there are no "defiers." This assumption is required for the IV estimate to be interpretable as the average effect for compliers.

Two-Stage Least Squares (2SLS)

The estimation proceeds in two stages:

Stage 1: Regress the endogenous variable on the instrument(s) and controls:

Stage 2: Regress the outcome on the predicted values $\hat{D}_i$ from Stage 1 and the controls:

The coefficient $\hat{\beta}_1$ uses only the variation in $D_i$ that is driven by $Z_i$ — purging the endogenous component.

IV Estimates as LATE

When treatment effects are heterogeneous, the IV estimator does not recover the average treatment effect (ATE) for the entire population. Instead, it recovers the Local Average Treatment Effect (LATE) — the average effect for compliers, i.e., units whose treatment status is affected by the instrument.

This locality means the IV estimate may differ from the OLS estimate even if OLS were unbiased, because they estimate effects for different subpopulations. For settings where treatment effects are heterogeneous and researchers want to understand the full distribution of effects, methods like causal forests provide complementary tools.

Common Confusions

B. Identification

Formal Conditions

For the model $Y_i = \beta D_i + u_i$ (suppressing controls for clarity), the IV estimator is consistent if:

1. $E[Z_i u_i] = 0$ (exogeneity / exclusion)
2. $E[Z_i D_i] \neq 0$ (relevance)

Under these conditions:

The LATE Theorem

With a binary instrument $Z \in \{0,1\}$ and binary treatment $D \in \{0,1\}$, there are four types of units:

• Compliers: $D(1) = 1, D(0) = 0$ — take treatment when encouraged, do not when not
• Always-takers: $D(1) = D(0) = 1$ — always take treatment
• Never-takers: $D(1) = D(0) = 0$ — never take treatment
• Defiers: $D(1) = 0, D(0) = 1$ — do the opposite of encouragement

Under the monotonicity assumption (no defiers), the Wald estimator

identifies the LATE: the average treatment effect for compliers.

C. Visual Intuition

Picture a scatter plot of $Y$ vs. $D$ where $D$ is endogenous. The OLS line through this cloud is biased because unobserved factors push both $D$ and $Y$ in the same direction.

Now imagine the instrument $Z$ as a lever that shifts $D$ left or right. Some units get a "push" (high $Z$) and others do not (low $Z$). The IV estimator looks at how much $Y$ changes per unit of $Z$-induced change in $D$. It ignores the endogenous part of $D$ entirely.

Think of it as plumbing. The ordinary relationship between $D$ and $Y$ is contaminated — dirty water (endogeneity) flows in. The instrument $Z$ provides a clean source of variation in $D$. By isolating the variation in $D$ that comes from $Z$ (the clean water), you can estimate the effect of $D$ on $Y$ without contamination.

When confounding is zero, IV and OLS agree. As confounding increases, OLS drifts away from the truth while IV stays on target — but with wider confidence intervals. When instrument strength approaches zero, IV becomes erratic (the weak instrument problem).

D. Mathematical Derivation

E. Implementation

1library(fixest)
2library(ivreg)
3
4# 2SLS with fixest
5iv_fit <- feols(gdp_pc ~ controls | 0 | institutions ~ settler_mortality,
6              data = df, vcov = "HC1")
7summary(iv_fit)
8
9# First-stage diagnostics
10fitstat(iv_fit, type = "ivf")    # First-stage F
11fitstat(iv_fit, type = "sargan") # Over-ID test (if applicable)
12
13# 2SLS with ivreg (more diagnostics)
14iv_fit2 <- ivreg(gdp_pc ~ institutions + controls | settler_mortality + controls,
15               data = df)
16summary(iv_fit2, diagnostics = TRUE)
17
18# LIML (via the ivmodel package)
19library(ivmodel)
20iv_mod <- ivmodel(Y = df$gdp_pc, D = df$institutions, Z = df$settler_mortality,
21                X = as.matrix(df[, "controls", drop = FALSE]))
22LIML(iv_mod)

F. Diagnostics

First-Stage F-Statistic

A central diagnostic for IV. The rule of thumb from Staiger and Stock (1997) is $F > 10$.

More recent guidance from Lee et al. (2022) shows that the standard first-stage F must exceed 104.7 for the conventional $t$-ratio critical value of 1.96 to control size at 5% in the just-identified case. Below that threshold, the authors provide an adjusted critical-value function ($tF$ procedure) that remains valid. In practice, F-statistics between 10 and 104.7 warrant the use of weak-instrument-robust inference methods such as the Anderson-Rubin test or the $tF$ procedure.

Reduced Form

It is recommended to report the reduced-form regression: regress $Y$ directly on $Z$ (and controls). If the reduced form is insignificant, IV will be imprecise (even if the first stage is strong). The reduced form is the ITT analog in the IV framework.

Over-Identification Test (Sargan-Hansen J-Test)

When you have more instruments than endogenous variables, the J-test checks whether the instruments are consistent with each other. A rejection suggests at least one instrument violates the exclusion restriction. But beware: the test has low power when all instruments are invalid in the same direction.

Weak Instrument Robust Inference

When the first-stage F is low, use the Anderson-Rubin (AR) test, which is valid regardless of instrument strength. The AR confidence set inverts a test of the reduced-form null and has correct size even with arbitrarily weak instruments.

Hausman/Durbin-Wu-Hausman Test (OLS vs. IV)

Compare OLS and IV estimates formally. Under the null that OLS is consistent, OLS and IV should give similar estimates. If they differ significantly, OLS is likely biased.

Interpreting Results

• Sign and magnitude: IV estimates are often larger than OLS (in absolute value). Two common explanations: (1) measurement error in $D$ attenuates OLS estimates, and IV corrects this attenuation; (2) IV identifies the LATE for compliers, who may respond more strongly to treatment than the average person. These explanations are not mutually exclusive, and distinguishing between them requires additional analysis.
• Precision: IV estimates are typically much less precise than OLS. Wide confidence intervals are the norm, not the exception.
• Compliers: Think carefully about who the compliers are. In the Acemoglu et al. example, compliers are countries whose institutional quality was determined by settler mortality. In the Angrist and Krueger (1991) quarter-of-birth example, compliers are people who would have gotten more education if not for compulsory schooling laws.

G. What Can Go Wrong

H. Practice

I. Swap-In: When to Use Something Else

• OLS with controls: When conditional exogeneity (selection on observables) is more credible than the exclusion restriction, and a rich set of covariates is available.
• Regression discontinuity: When treatment is assigned by a threshold on a running variable — RDD provides a more transparent and locally randomized design.
• Difference-in-differences: When a policy change provides before/after and treated/untreated variation without requiring an instrument.
• Matching: When selection into treatment is primarily on observables and the overlap condition is satisfied.
• Reduced form only: When the instrument is valid but weak (F < 10), reporting the reduced-form effect of the instrument on the outcome avoids the bias amplification of 2SLS.

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