Instrumental Variables

State of the Union Address

The Selection Problem

• The headline is about State of the Union watchers, not a random sample.
• When the president is a Democrat, Democrats are more likely to watch.
• Controlling for party isn’t enough: Republicans who did watch differ from typical Republicans.

Visualizing the Problem (and Solution)

Randomizing the Encouragement

Compliance

• Compliers: take treatment when encouraged and do not take treatment when not encouraged. [D_i \mid Z_i = 1] = 1 \ \text{and} \ [D_i \mid Z_i = 0] = 0

• But…some people do not comply.

  • Always-takers: [D_i \mid Z_i = 1] = [D_i \mid Z_i = 0] = 1
  • Never-takers: [D_i \mid Z_i = 1] = [D_i \mid Z_i = 0] = 0
  • Defiers: [D_i \mid Z_i = 1] = 0 and [D_i \mid Z_i = 0] = 1

• The treatment effect can only be assessed for compliers.

We Don’t See the Counterfactual

Some Assumptions

• Ignorability of the instrument: (Y_i^1, Y_i^0) \perp Z_i
• Exclusion restriction: Z (instrument) only affects Y (outcome) through D (treatment). We cannot test the exclusion restriction. It must be defended theoretically.
• No weak instruments (relevance): \mathbb{E}[ D_i | Z_i = 1] - \mathbb{E}[ D_i | Z_i = 0] \neq 0
• No defiers (monotonicity).

Intent-to-Treat Effect

Why This “Works”

• Z (encouragement) is randomly assigned.
• Exclusion restriction ensures any change in Y caused by Z operates entirely through D (no other pathways).
• If Z is relevant/not weak, then some proportion of our sample consists of compliers.
  • Compliers drive the ITT. Always-takers and never-takers contribute nothing to the effect, but don’t distort its direction.
• There are no defiers (they cannot flip the sign of our estimate).

Two Questions, Two Estimands

• The ITT tells us: what happens if we run an encouragement campaign?
  • Useful for a policymaker deciding whether to send the mailer.
• The problem: most people’s behavior didn’t change (always-takers and never-takers are in the ITT).
• If you want to know what watching the speech actually does, the ITT underestimates.
• What we want: the effect for people who only watched because we encouraged them, aka the Local Average Treatment Effect (LATE).

Decomposing the ITT

Estimating Pr(compliers)

• We need \Pr(c) to recover the LATE, but we cannot observe types directly.

Calculating the LATE

Two-Stage Least Squares

Recover the LATE using two regressions chained together:

• First stage: regress D on Z to isolate variation in treatment driven by instrument.
• Second stage: regress Y on the “fitted values,” \hat{D}, from stage one.
• The stage-two coefficient on \hat{D} is the LATE.

First Stage

Second Stage

• Regress outcome Y on the fitted treatment values \hat{D} from the first stage.

• The \hat{D} coefficient (0.33) is the LATE.

Some Finer Points

• Causal effect measured only for compliers, not an average treatment effect.
• The complier population may be small, limiting the precision of the LATE.
• Individual compliance types cannot be identified.

Which Effect to Estimate?

Both!

• ITT: Because not everyone complies, it gives us an idea of how a policy works in the real world.
• LATE: A more targeted estimate of how the treatment works for those whose treatment status changes because of the instrument.

It’s Not Just Randomized Experiments

• Does having more children lead women to leave the labor force?
  • Instrument: having two same-sex children (Angrist and Evans 1998).
• Does police presence on the street reduce crime?
  • Instrument: mayoral elections (Levitt 1997).

Who Gets to Vote?

Florida 2018 and What Comes After

White (2019)

White (2019), Figure 2