Instrumental Variables

Part of: Econometrics Lecture 04 — Applied Econometrics, Dr. Aluma Dembo Key concepts: Instrumental Variables, Endogeneity, Two Stage Least Squares, Omitted Variable Bias, Instrument Validity, Instrument Relevance, First Stage, Second Stage, Weak Instruments

The Core Problem: Why OLS Breaks Down

OLS gives you the causal effect of X on Y only if X is truly exogenous — i.e. nothing else is simultaneously causing both X and Y.

In practice, this fails constantly. The three classic culprits are:

When any of these are present, $\text{cov}(x_i, u_i) \neq 0$ and OLS is biased and inconsistent — it's not measuring the causal effect you want, even in large samples.

OLS doesn't heal with more data

With an endogenous regressor, increasing your sample size doesn't fix the bias. You're consistently estimating the wrong thing. This is why IV exists — it's not just a sample-size fix.

The IV Intuition: Borrow Some Exogenous Variation

The idea behind instrumental variables is beautifully simple:

Find a variable Z that only moves X — and has no direct route to Y. Use that clean variation in X to estimate the effect on Y.

If X is contaminated by confounders, IV carves out the clean slice of X that moves for exogenous reasons, and looks at what Y does when that slice moves.

         Z ────────▶ X ────────▶ Y
                      ↑
                      u (confounder — does NOT connect Z to Y)

The instrument Z is essentially a natural experiment hidden inside your data.

type: causal-diagram
diagram: iv

Reading the diagram

The confounder $u$ poisons the direct X→Y comparison (it points into both). The instrument Z sidesteps this: it pushes X (relevance, the blue arrow) but has no arrow into Y except through X (the exclusion restriction — the greyed-out path). IV uses only the Z-driven movement in X, which the confounder can't touch.

The "lottery ticket" analogy

Imagine you want to know if military service affects lifetime earnings. You can't randomly assign service. But the Vietnam-era draft lottery randomly selected birth cohorts for service. That randomness (Z = draft lottery number) affects military service (X) but has no other reason to affect earnings (Y). So it's a valid instrument. This is the classic Angrist (1990) study.

The Two IV Conditions

For Z to be a valid instrument it must satisfy both of these:

Condition 1: Relevance — Z must actually move X

Z has to have a real, substantial effect on the endogenous variable X. If Z barely moves X, you have a weak instrument — see the warning below.

How to test: Run the first stage regression and check the F-statistic. If F < 10, the instrument is considered weak. This can be tested.

Condition 2: Validity (Exclusion Restriction) — Z affects Y only through X

Z has no direct effect on Y, and doesn't correlate with anything else that affects Y. The only pathway from Z to Y goes through X.

How to test: This cannot be directly tested — there's no statistical test because $u$ is unobserved. It's a theoretical / conceptual argument. You have to convince people with logic, not numbers.

The exclusion restriction is the hard part

Almost every IV debate in economics is about whether the exclusion restriction holds. Critics attack the instrument by arguing it has a direct effect on Y through some other channel. Always think hard: "Is there any way Z could be affecting Y without going through X?"

Classic Real-World Examples of IV

1. 🎓 Education & Earnings (Card, 1995)

Problem: Estimating the causal return to education is hard — smarter/more motivated people get more education and earn more. OLS overstates education's effect.

Instrument: Whether you grew up near a college (geographic proximity).

• Relevance: ✅ Living near a college increases the likelihood of attending (lower cost/distance).
• Validity: ✅ The distance from your childhood home to the nearest college shouldn't directly affect your adult earnings (assuming you don't stay in the same town for work).

Result: IV estimate of returns to education is higher than OLS — suggests the people who go to college because it's nearby are those who respond more to schooling (compliers).

2. 🎖️ Military Service & Earnings (Angrist, 1990)

Problem: Did Vietnam service hurt long-run earnings? Veterans might have different pre-war characteristics, making direct comparison invalid.

Instrument: Vietnam draft lottery number (randomly assigned by birthday).

• Relevance: ✅ Having a low draft number increased probability of serving.
• Validity: ✅ Your birthday is random with respect to your future earnings potential.

Result: IV found a significant negative effect of military service on earnings (~15% lower).

3. 🌍 Institutions & Economic Growth (Acemoglu, Johnson & Robinson, 2001)

Problem: Good institutions cause growth, but rich countries can afford better institutions — reverse causality.

Instrument: Settler mortality rates in former colonies (how deadly was the colony for European settlers?).

• Relevance: ✅ Where settlers died in large numbers, they set up extractive institutions rather than settling. This led to worse institutions today.
• Validity: ✅ 16th/17th century disease environments don't directly affect modern GDP except through the institutional path they created.

Result: Institutions have a huge causal effect on GDP — one of the most influential IV papers ever written.

4. 🍻 Alcohol & Wages (Kenkel & Ribar, 1994)

Problem: Does drinking hurt earnings? People who drink heavily may have other traits (time preference, health) that also reduce earnings.

Instrument: State-level beer taxes or alcohol prices.

• Relevance: ✅ Higher prices reduce alcohol consumption.
• Validity: ✅ Beer tax is set by state governments for fiscal reasons, not to target individual earnings.

5. 📡 TV & Political Participation (Gentzkow, 2006)

Problem: Does watching TV reduce civic participation? Hard to isolate — TV adoption correlated with many community-level factors.

Instrument: Timing of TV introduction by city (driven by FCC licensing delays, which were largely arbitrary).

• Relevance: ✅ Earlier FCC approval → more TV ownership in that city.
• Validity: ✅ The bureaucratic timing of FCC hearings is unrelated to pre-existing civic engagement.

Lecture Case Study: Cai, De Janvry & Sadoulet (2013)

Setting

People's Insurance Company of China rolls out a new weather insurance policy for rice farmers. The researchers want to know: Does your social network (friends' behaviour) affect whether you buy insurance?

The endogenous regression would be:

where $\text{pre.takeup.rate}_i$ is the fraction of $i$'s friends who bought insurance in the first round.

Why OLS fails here

Friends who have similar risk preferences will both buy or both skip insurance. The shared preferences are the confounder — they affect both your friends' decision (X) and your own decision (Y). OLS can't separate "I was influenced by my friend" from "we just both have the same risk attitude."

     shared preferences
          ↓          ↓
  friends' decision ──→ my decision

The Instrument: Default Assignment

The experiment randomly assigned villages to either:

• Default Buy: farmers are opted in to insurance and must actively opt out
• Default Not Buy: farmers must actively opt in

The key move: your friend's default assignment is used as an instrument for your friend's take-up decision.

• Relevance: ✅ Being assigned to "Default Buy" substantially increases take-up (default bias is well-documented in behavioural economics — people stick with whatever option is presented as the default).
• Validity: ✅ The village's default assignment was randomised — it cannot correlate with the shared preferences between you and your friend. The assignment was bureaucratic, not based on friendship characteristics.

  Friend's Default Assignment (Z)
          │
          ▼
  Friend's Take-Up Decision (X) ────→ My Take-Up Decision (Y)
                                              ↑
                                    shared preferences (u) — NOT connected to Z

The 2SLS Procedure Applied

First Stage — regress the endogenous variable on the instrument:

This produces $\widehat{\text{pre.takeup.rate}}_i$ — the predicted friends' take-up rate that is explained only by the random default assignment.

Second Stage — use those predicted values in the main regression:

Because $\widehat{\text{pre.takeup.rate}}$ only moves due to the randomised default, it can't be correlated with the shared preferences confounder. The $\hat{\beta}_1$ we get is causal.

Result

A 20% increase in friends' take-up rate (i.e. 1 more friend out of 5 buying insurance) causes approximately a 23% increase in your probability of buying insurance. Social influence is real and large.

What if friends could choose their default?

If participants could self-select into which default option they received, the instrument breaks: people with stronger risk preferences might actively choose the "Default Buy" group. Now $\text{cov}(z_i, u_i) \neq 0$ — the instrument is correlated with the shared-preferences confounder. The second-stage estimate would be biased again.

How 2SLS Works: Step by Step

2SLS in plain English

You have a noisy, endogenous X. You use Z to find the clean, exogenous fraction of X. Then you run the regression using only that clean fraction. You're throwing away information (the endogenous part) to get something trustworthy.

Step 1: Use Z to explain X (the first stage regression). Step 2: Take only the part of X that Z explains — call it $\hat{X}$. Step 3: Regress Y on $\hat{X}$ — this is the second stage. Step 4: $\hat{\beta}_{1}^{IV}$ is the causal effect — it's the relationship between the Z-driven movement in X and the corresponding movement in Y.

Always use feols or ivreg in R — don't run 2SLS manually

Running two separate lm() calls gives the correct point estimates but wrong standard errors in the second stage (because the SEs don't account for the first-stage estimation uncertainty). Use feols(y ~ controls | x ~ z) from the fixest package.

With Multiple Instruments

If you have multiple instruments $z_1, z_2, \ldots$, include all of them in the first stage. This uses maximum information to produce the best $\hat{X}$. Any other exogenous controls in your model should also go in the first stage automatically.

What Can Go Wrong: Weak Instruments

If $\text{cov}(z_i, x_i)$ is very small — Z barely moves X — you have a weak instrument.

Weak instruments amplify bias

From the IV estimator formula: $\hat{\beta}_1 = \frac{\text{cov}(y,z)}{\text{cov}(x,z)}$. If the denominator is tiny, any small violations of the exclusion restriction get massively amplified in the estimate. Weak instruments can produce estimates worse than OLS.

Rule of thumb: First-stage F-statistic should be > 10. If it's below 10, the instrument is weak and results should be treated with scepticism.

type: weak-instrument

What the two panels show

On the left, Z has a steep, clear relationship with X — the first stage is strong (F ≈ 58) and gives a reliable $\hat{X}$. On the right, the cloud is nearly flat: Z barely moves X (F ≈ 2). Since $\hat\beta^{IV}=\mathrm{cov}(y,z)/\mathrm{cov}(x,z)$, that tiny denominator blows up both the variance and any small violation of the exclusion restriction — which is why a weak instrument can be worse than plain OLS.

When Should You Use IV?

IV is the right tool when:

1. You suspect your key regressor X is endogenous (correlated with the error term)
2. You can argue for an exogenous variable Z that moves X but has no direct effect on Y
3. The first stage is strong (F > 10)

The hardest part is always finding a convincing instrument. Good instruments typically come from:

• Natural experiments (lotteries, policy roll-outs, arbitrary bureaucratic decisions)
• Geographic variation (distance to a college, proximity to a border)
• Historical quirks (settler mortality, timing of infrastructure investment)
• Randomised within an experiment (like the default assignment in Cai et al.)

IV vs OLS: When Do They Differ?

IV is always less precise than OLS

You're throwing away information (the endogenous variation in X), so IV standard errors are always larger. This is the cost of consistency — you get unbiased estimates but with less precision. If your instrument is very weak, the precision loss can be enormous.

IV estimates a LATE, not ATE

IV doesn't estimate the effect for everyone — it estimates the Local Average Treatment Effect (LATE): the effect for "compliers", i.e. people whose X changes because of Z. People whose X doesn't change with Z ("never-takers" and "always-takers") are not informative for IV.

Summary

1. OLS breaks when X is endogenous — the coefficient is biased because confounders or reverse causality are contaminating the estimate.
2. IV finds a clean slice of variation in X using an instrument Z that only moves X for exogenous reasons.
3. Two conditions must hold — relevance ($\text{cov}(z,x) \neq 0$, testable) and validity ($\text{cov}(z,u) = 0$, not testable — requires argument).
4. 2SLS is the practical implementation — first stage regresses X on Z to get $\hat{X}$; second stage uses $\hat{X}$ in the main regression.
5. Weak instruments are dangerous — check the first-stage F-stat > 10.
6. IV gives LATE not ATE — the estimate applies to compliers only.
7. Good instruments come from natural experiments — lotteries, arbitrary policy roll-outs, geographic variation, historical accidents.

Related Notes

• Previous: Lec_03-Logit & Probit Models — when outcomes are binary
• Hub: Econometrics
• Foundational: OLS Estimation, Omitted Variable Bias, Endogeneity, Causal Inference
• Coming up: Regression Discontinuity, Difference-in-Differences
• Key concepts here: Two Stage Least Squares, Instrument Validity, Instrument Relevance, Weak Instruments, Local Average Treatment Effect