Binary-Instrument 2SLS Under Effect Heterogeneity

Why Effect Heterogeneity Changes the Interpretation

When treatment effects are constant, a valid and relevant instrument can identify that common effect. If effects vary across individuals, a single 2SLS coefficient still has an interpretation, but it need not be the average effect for the whole population. With a binary instrument, the key question is whose treatment status changes when the instrument changes?

For example, an encouragement might lead some people to receive treatment while leaving others’ decisions unchanged. Under specific assumptions, the binary-instrument 2SLS coefficient is the average treatment effect for those whose treatment is changed by that encouragement. It does not automatically describe people whose treatment would remain the same either way.

Potential Outcomes and the Instrument’s Assumptions

Let $Z$ be a binary instrument, with $Z=1$ denoting one instrument condition and $Z=0$ the other. Let $D$ be a binary treatment. Write $D(z)$ for the treatment a unit would receive under instrument value $z$, and $Y(d)$ for the outcome that unit would have under treatment value $d$. The individual treatment effect is $Y(1)-Y(0)$. These potential outcomes describe hypothetical possibilities; for each unit, only the outcome corresponding to the treatment actually received is observed.

Definition: A complier is a unit with $D(1)=1$ and $D(0)=0$: its treatment status responds to the instrument. An always-taker has $D(1)=D(0)=1$, and a never-taker has $D(1)=D(0)=0$. The monotonicity assumption rules out defiers, who would have $D(1)=0$ and $D(0)=1$.

The interpretation developed here requires the following conditions for the population being studied:

Consistency and no interference: a unit’s observed treatment and outcome agree with its potential treatment and outcome under the instrument value it actually receives, and one unit’s instrument or treatment does not change another unit’s outcome.
Independence: $Z$ is independent of the potential treatments $D(0),D(1)$ and potential outcomes $Y(0),Y(1)$. Random assignment of the instrument is one way to justify this condition.
Exclusion: the instrument affects the outcome only through treatment. In particular, the potential outcome under instrument value $z$ and treatment $d$ is $Y(d)$, with no additional dependence on $z$.
Monotonicity: $D(1)\geq D(0)$ for every unit. The instrument does not push some units toward treatment while pushing others away from it.
Relevance: the instrument changes the population treatment rate: $E[D\mid Z=1]\ne E[D\mid Z=0]$.

These are substantive assumptions, not consequences of fitting 2SLS. Independence, exclusion, and monotonicity are generally not established just by inspecting the observed instrument, treatment, and outcome data. Relevance is reflected in a nonzero first-stage difference, but a very small difference also makes the ratio unstable and calls for care.

The Binary-Instrument 2SLS Coefficient

Consider 2SLS with an intercept, one binary endogenous treatment $D$, and one binary instrument $Z$, with no additional regressors. In the population, its coefficient is the ratio of the outcome difference across instrument groups to the treatment difference across those same groups. This is also called the Wald ratio:

\beta_{\mathrm{2SLS}} = \frac{E[Y\mid Z=1]-E[Y\mid Z=0]} {E[D\mid Z=1]-E[D\mid Z=0]}.

For a finite sample in this just-identified setup, the corresponding 2SLS coefficient is the sample difference in outcome means across instrument groups divided by the sample difference in treatment means. The ratio is meaningful when the first-stage difference is nonzero. With covariates or additional instruments, the specification and interpretation require further care; the result here concerns the simple binary-instrument setup.

Theorem: Under consistency, no interference, independence, exclusion, monotonicity, and relevance, the population binary-instrument 2SLS coefficient equals the complier average treatment effect: $$ \beta_{\mathrm{2SLS}} = E[Y(1)-Y(0)\mid D(1)>D(0)]. $$ This is the average treatment effect among compliers, not necessarily the population average treatment effect $E[Y(1)-Y(0)]$.

Why the Ratio Averages Complier Effects

The key is to examine what changes between the two instrument groups. Under independence, each group represents the same distribution of potential outcomes and potential treatment types, in expectation. Under exclusion, any outcome difference between the groups must operate through treatment. Under monotonicity, only compliers change their treatment status when $Z$ changes.

Let $C$ denote the complier group. The first-stage difference is the increase in the probability of treatment. Always-takers are treated in both instrument conditions, and never-takers are untreated in both, so neither contributes to that increase. Compliers move from untreated to treated. Therefore:

E[D\mid Z=1]-E[D\mid Z=0]=P(C).

For outcomes, always-takers receive treatment in either condition and never-takers receive no treatment in either. Their treatment status, and hence their outcome under exclusion, does not change with $Z$. Compliers move from $Y(0)$ to $Y(1)$. Averaging over the population, the outcome difference is consequently the complier share multiplied by their average treatment effect:

E[Y\mid Z=1]-E[Y\mid Z=0] = P(C)E[Y(1)-Y(0)\mid C].

Dividing the outcome difference by the nonzero first-stage difference cancels $P(C)$, giving the theorem. This argument also explains why the coefficient is local to the units whose treatment responds to this particular instrument contrast.

Worked Example: A Complier Effect Different From the Population Average

Consider a hypothetical population of 1,000 units. Suppose 200 are always-takers, 500 are compliers, and 300 are never-takers. Their average treatment effects are 1, 4, and 2 outcome units, respectively. Assume the instrument is randomly assigned, exclusion holds, and no unit is a defier. To make the outcome means explicit, suppose average untreated outcomes are 10 for always-takers, 8 for compliers, and 6 for never-takers.

Check the assumptions for this illustration. Random assignment supplies independence; the stated exclusion condition means the instrument has no direct effect on outcomes; and the stated types include only always-takers, compliers, and never-takers, so monotonicity holds. Relevance follows because the compliers change treatment status and make up half the population.

Under $Z=1$, all always-takers and compliers are treated, while never-takers are untreated. The treatment rate is $ (200+500)/1000=0.700 $. Under $Z=0$, only always-takers are treated, giving a rate of $200/1000=0.200$. Thus the first-stage difference is $0.700-0.200=0.500$.

The average outcomes under $Z=1$ are 11 for always-takers, 12 for compliers, and 6 for never-takers: their untreated means plus their stated effects when treated. Therefore, $E[Y\mid Z=1]=(200(11)+500(12)+300(6))/1000=(2200+6000+1800)/1000=10.00$. Under $Z=0$, always-takers remain treated, compliers are untreated, and never-takers remain untreated, so $E[Y\mid Z=0]=(200(11)+500(8)+300(6))/1000=(2200+4000+1800)/1000=8.00$.

The outcome difference is $10.00-8.00=2.00$, so the binary-instrument 2SLS coefficient is $2.00/0.500=4.00$ outcome units. This equals the average treatment effect among compliers by construction. The population average treatment effect, however, is $ (200(1)+500(4)+300(2))/1000=(200+2000+600)/1000=2.80 $ outcome units. The two averages differ because compliers have a different average effect from the population as a whole.

Worked Example: Averaging Heterogeneous Effects Within the Complier Group

Suppose a second hypothetical population has 20% always-takers, 40% compliers with an average treatment effect of 2 outcome units, 20% compliers with an average effect of 8 outcome units, and 20% never-takers. Assume independence, exclusion, and monotonicity, and suppose the untreated outcome is zero for every group. The two complier subgroups together are the complier group.

Check relevance. The complier share is $0.40+0.20=0.60$, so the instrument raises the treatment rate by 0.60. The assumptions stated for this hypothetical setup ensure the required independence, exclusion, and no-defiers conditions.

The average effect among compliers weights the subgroup effects by their shares within the complier group: $ (0.40(2)+0.20(8))/0.60=(0.80+1.60)/0.60=4.00 $ outcome units. Equivalently, the instrument-induced outcome difference is $0.40(2)+0.20(8)=2.40$: the always-takers’ outcome contribution is unchanged across instrument values, as is the never-takers’ contribution. Dividing by the first-stage difference gives $2.40/0.60=4.00$, as required.

The population average treatment effect includes all four groups: $0.20(1)+0.40(2)+0.20(8)+0.20(4)=0.20+0.80+1.60+0.80=3.40$ outcome units. Thus even though the treatment effect varies within the complier group, the IV coefficient is its share-weighted average, not the population average of 3.40 units.

Worked Example: Calculating the Wald Ratio From Sample Means

In a hypothetical randomized encouragement study, suppose a balanced sample of eight units in each instrument group has the following observed treatment and outcome summaries. The outcomes are measured in points.

Instrument group	Units	Treated	Mean outcome
$Z=1$	8	6	7.50 points
$Z=0$	8	2	4.50 points

The sample first-stage difference is $6/8-2/8=0.750-0.250=0.500$. The sample outcome difference is $7.50-4.50=3.00$ points. The just-identified sample 2SLS coefficient is therefore $3.00/0.500=6.00$ points per treatment.

To give the result a complier interpretation, suppose the hypothetical design also satisfies exclusion and monotonicity, and the instrument was randomized. These conditions justify the corresponding population interpretation; the observed summary table alone cannot verify all of them. Under those assumptions, the estimated 6.00 points is an estimate of the average effect for units whose treatment responds to the encouragement—not an estimate that every unit gains 6 points. Its sampling uncertainty is also not represented by the ratio calculation alone.

What the Coefficient Does—and Does Not—Say

A binary-instrument 2SLS estimate under these assumptions answers a specific question: what is the average treatment effect for the units induced to change treatment status by this instrument contrast? It does not by itself identify the effects for always-takers or never-takers, because their treatment status does not change with the instrument. Nor does it establish that the complier effect applies to a different population, a different encouragement, or a different treatment decision.

The distinction matters when treatment effects are heterogeneous. If effects are constant across all units, the complier average and population average coincide. They can also coincide by coincidence if the groups’ average effects balance. But without such conditions, the population average $E[Y(1)-Y(0)]$ and the complier average $E[Y(1)-Y(0)\mid C]$ are different estimands. A numerical 2SLS coefficient should therefore be described with its instrument-defined target group, alongside the assumptions that support that interpretation.

Common Mistake

Common mistake: “The instrument is valid, so 2SLS estimates the average treatment effect for everyone.”

Correction: In the binary-instrument setting, validity plus the stated monotonicity and other assumptions yields the average effect among compliers. The instrument changes treatment only for that group. Extending the estimate to the full population requires additional justification, such as constant effects or a defensible argument that the complier average equals the population average.

Key takeaway: With a binary instrument, one treatment, and an intercept, the 2SLS coefficient is the outcome difference across instrument groups divided by the treatment difference. Under independence, exclusion, monotonicity, and relevance, this Wald ratio is the average treatment effect among compliers—not necessarily the population average effect.

Check Your Understanding

Use the assumptions and definitions in this tutorial to answer the following questions.

What does monotonicity rule out, and why does that matter for identifying compliers?
If the outcome difference across instrument groups is 1.2 points and the treatment-rate difference is 0.30, what is the Wald ratio in points per treatment?
Why do always-takers and never-takers contribute nothing to the first-stage difference under the stated assumptions?
Can the complier average treatment effect differ from the population average treatment effect when all assumptions hold? Explain.
Which additional claim would be needed to interpret a complier effect as the effect for every unit when treatment effects are heterogeneous?