From Instrument Assignment to a Treatment Effect
An instrument can shift treatment receipt even when not everyone follows their assignment or encouragement. Two-stage least squares (2SLS) uses this shift to estimate an effect of treatment on an outcome. In the setting here, the instrument \(Z\) and treatment \(D\) are binary, while the outcome \(Y\) is continuous. The key quantities are the first stage, which measures how much the instrument changes treatment receipt, and the reduced form, which measures how much it changes the outcome.
The ratio of these two differences is the just-identified 2SLS estimate when there is one instrument and one endogenous treatment. The ratio is not automatically a causal treatment effect: that interpretation requires assumptions about how the instrument is assigned, how it affects the outcome, and how it changes treatment.
In a sample, estimate the first stage by regressing \(D\) on an intercept and \(Z\), and the reduced form by regressing \(Y\) on an intercept and \(Z\). With a binary \(Z\), the coefficient of \(Z\) in each regression is the corresponding difference in group means. The 2SLS procedure first predicts treatment using the first-stage regression, then regresses \(Y\) on the predicted treatment and an intercept. With one instrument and one treatment, its treatment coefficient equals the reduced-form coefficient divided by the first-stage coefficient.
Here \(\widehat D_i\) is a fitted value, not a recoding of each person's observed treatment. It can be fractional: if estimated treatment rates are 0.40 and 0.60, the fitted values for the two instrument groups are those rates. The second-stage coefficient measures outcome units per unit change in treatment on the scale induced by the instrument.
Worked Example: Encouragement, Treatment, and Mean Outcome
Worked Example: Encouragement, Treatment, and Mean Outcome
Suppose a hypothetical randomized encouragement study assigns 200 people to \(Z=1\) and 200 to \(Z=0\). The outcome is a continuous score measured in points. The group summaries are:
| Instrument group | Number of people | Number treated | Treatment rate | Mean outcome |
|---|---|---|---|---|
| \(Z=1\) | 200 | 120 | 0.60 | 8.4 points |
| \(Z=0\) | 200 | 80 | 0.40 | 7.2 points |
First stage. The estimated change in treatment receipt is \(\widehat\pi=120/200-80/200=0.60-0.40=0.20\). Assignment to encouragement increases treatment receipt by 0.20, or 20 percentage points, in this sample.
As a descriptive precision check, if the two groups are independently sampled, the standard error for the difference in treatment proportions is \(\sqrt{0.60(0.40)/200+0.40(0.60)/200}=\sqrt{0.0024}=0.0490\). The estimated first stage is about \(0.20/0.0490=4.08\) standard errors from zero under this calculation. This is evidence of a nonzero first stage in this hypothetical sample; it does not establish the other IV assumptions.
Reduced form. The estimated change in mean outcome is \(\widehat\rho=8.4-7.2=1.2\) points. The group means and sample sizes alone do not give the outcome standard error; the within-group outcome variability would also be needed.
2SLS estimate. Dividing the reduced form by the first stage gives \(\widehat\beta_{\mathrm{2SLS}}=1.2/0.20=6.0\) points per treatment. Equivalently, the first-stage fitted treatment is 0.40 for \(Z=0\) and 0.60 for \(Z=1\), while the fitted outcome difference across those values is 1.2 points. The slope is \(1.2/(0.60-0.40)=6.0\) points per unit change in treatment.
The causal interpretation, if the assumptions below hold, is a local average treatment effect for people whose treatment receipt is changed by the encouragement—not necessarily the average effect for everyone in the study.
Assumptions and Why the Ratio Can Be Causal
To state the interpretation precisely, let \(D(z)\) be the treatment a person would receive under instrument value \(z\), and let \(Y(d)\) be the outcome under treatment value \(d\). Assume consistency (the observed treatment and outcome match the potential values under the received assignments), no interference between people, and well-defined instrument and treatment conditions. In addition, the usual binary-instrument interpretation requires:
- Relevance: \(E[D\mid Z=1]\ne E[D\mid Z=0]\). The instrument must change treatment receipt in the population.
- Independence: \(Z\) is independent of the potential treatments and outcomes, \(\{D(0),D(1),Y(0),Y(1)\}\). Random assignment can support this assumption; observational assignment needs substantive justification.
- Exclusion: the instrument affects the outcome only through treatment. In potential-outcome notation, \(Y(d,z)=Y(d)\): changing \(Z\) while holding treatment fixed does not change \(Y\).
- Monotonicity: \(D(1)\ge D(0)\) for every person. There are no people who move treatment in the direction opposite to the instrument for this coding.
- Instrument variation: both instrument values occur with positive probability, so the group comparisons are defined.
Under independence, exclusion, and monotonicity, people can be classified by their potential treatment responses. Compliers have \(D(1)=1,D(0)=0\); always-takers have \(D(1)=D(0)=1\); never-takers have \(D(1)=D(0)=0\). Monotonicity rules out defiers, who would have \(D(1)=0,D(0)=1\).
The derivation shows why the ratio targets compliers. By consistency and exclusion, the reduced-form difference is the mean of \(Y(D(1))-Y(D(0))\) across people, since independence makes the instrument groups comparable in their potential values. Always-takers and never-takers have no treatment change, so their contribution to that difference is zero. Compliers contribute \(Y(1)-Y(0)\). Thus:
where \(C\) denotes the complier group. The second equality follows because \(D(1)-D(0)=1\) for compliers and \(0\) for always-takers and never-takers. Dividing gives \(\rho/\pi=E[Y(1)-Y(0)\mid C]\). Relevance ensures \(P(C)>0\) under monotonicity, so the denominator is not zero.
Worked Example: A Smaller First Stage
Worked Example: A Smaller First Stage
In another hypothetical study, each instrument group has 150 people. In the \(Z=1\) group, 78 receive treatment and the mean outcome is 5.6 units. In the \(Z=0\) group, 60 receive treatment and the mean outcome is 5.0 units.
The treatment rates are \(78/150=0.52\) and \(60/150=0.40\), so the first stage is \(\widehat\pi=0.52-0.40=0.12\), a 12-percentage-point increase. The reduced form is \(\widehat\rho=5.6-5.0=0.6\) outcome units. Therefore, \(\widehat\beta_{\mathrm{2SLS}}=0.6/0.12=5.0\) outcome units per treatment.
For an independent-groups precision check, the standard error for the first-stage difference is \(\sqrt{0.52(0.48)/150+0.40(0.60)/150}=\sqrt{0.003264}=0.0571\). The difference is about \(0.12/0.0571=2.10\) standard errors from zero under this calculation. The point estimate is defined, but a smaller first stage can make the ratio highly sensitive to sampling variation: dividing by a denominator near zero amplifies changes in the reduced form and first stage. This calculation is not a complete inferential analysis of the 2SLS effect.
Worked Example: Why Exclusion Matters
Worked Example: Why Exclusion Matters
Consider a hypothetical setting in which treatment truly increases the mean outcome by 4 units, but the instrument also directly increases the outcome by 2 units, apart from any treatment change. Suppose the first stage increases treatment by 0.25. In this simple additive description, the instrument's total outcome difference is \(\rho=4(0.25)+2=3\) units: 1 unit through the change in treatment and 2 units through the direct effect.
The ratio is \(\rho/\pi=3/0.25=12\) outcome units per treatment, not the true treatment effect of 4 units. The arithmetic is valid as a ratio, but the exclusion assumption fails, so the ratio does not have the stated complier-effect interpretation. Data showing a strong first stage cannot rule out this problem.
Diagnostics, Alternatives, and Common Pitfalls
Start by reporting the instrument-group sizes, treatment proportions, first-stage difference, outcome means, reduced-form difference, and ratio. Check that treatment coding and instrument coding have clear directions; reversing the instrument coding reverses both differences, so their ratio stays the same. If the first-stage difference is close to zero, the estimate can be unstable and conventional large-sample uncertainty summaries may be misleading. A detailed weak-instrument analysis is a separate inferential problem.
Diagnostics can reveal some problems but cannot verify the central assumptions. A near-zero first stage directly contradicts relevance. An unexpectedly strong association between the instrument and measured baseline characteristics may raise concern about independence, although balance on observed variables does not prove independence from unmeasured causes. Exclusion generally cannot be tested from the instrument-treatment-outcome data alone; it depends on subject-matter knowledge about possible direct pathways. Monotonicity is likewise an assumption about individual responses that cannot usually be checked by comparing only group averages.
Ordinary least squares regresses \(Y\) directly on observed \(D\). It may be more precise, but if unmeasured causes affect both treatment and outcome, its treatment coefficient can be confounded. 2SLS instead uses the part of treatment variation associated with \(Z\), and trades the need for treatment-outcome exchangeability for the IV assumptions. The reduced form is also useful: it reports the effect of the instrument on the outcome, but it is an assignment or encouragement effect, not an effect per treatment. In a propensity-score analysis, treatment comparisons rely on measured-confounder assumptions; that is a different identification strategy, not a way to validate an instrument.
- Mistake: “The encouragement raised the outcome by 1.2 points, so treatment raised it by 1.2 points.” Correction: 1.2 points is the reduced-form effect of the instrument. The 2SLS estimate divides it by the 0.20 treatment-rate increase, giving 6.0 points per treatment under the IV assumptions.
- Mistake: “A large first-stage statistic proves the instrument is valid.” Correction: It speaks to relevance, not independence, exclusion, or monotonicity.
- Mistake: “The IV estimate is the average treatment effect for every person.” Correction: Under the stated assumptions, the binary-instrument ratio identifies the average effect among compliers. Other groups may have different effects.
Check Your Understanding
Work through each question, then compare with the solutions below.
- A binary instrument raises the treatment rate from 0.30 to 0.45 and raises the mean outcome from 10 to 10.6 units. Calculate the first stage, reduced form, and 2SLS estimate.
- In a study, the estimated first stage is 0.05 and the reduced form is 0.20 outcome units. What is the ratio estimate, and what caution does the small first stage suggest?
- Which assumption is contradicted if the instrument can directly affect the outcome while treatment is held fixed?
- Under the binary-instrument assumptions, which group’s average treatment effect does the ratio identify?
Solutions. (1) The first stage is \(0.45-0.30=0.15\); the reduced form is \(10.6-10=0.6\) units; the ratio is \(0.6/0.15=4.0\) outcome units per treatment. (2) The ratio is \(0.20/0.05=4.0\) outcome units per treatment. Because the denominator is small, sampling variation can make the ratio unstable; the point estimate alone does not describe its uncertainty. (3) Exclusion is contradicted, because it requires no effect of the instrument on the outcome except through treatment. (4) The ratio identifies the average treatment effect among compliers, the people whose treatment changes with the instrument, under the stated assumptions.