Just-Identified IV as a Reduced-Form Ratio

Why a Ratio Appears in Just-Identified IV

An instrument can change an outcome through its effect on treatment. The reduced form measures how the instrument changes the outcome; the first stage measures how it changes treatment. When there is one instrument for one treatment, the just-identified IV coefficient compares these two changes: it divides the outcome change by the treatment change. This puts the result on the scale of an outcome change per unit of treatment.

The ratio is simple to calculate, but its meaning depends on the model and the instrument assumptions. In particular, relevance alone is not enough: a nonzero first stage makes the ratio calculable, but does not establish that it measures a causal effect.

Model and Population Ratio

Let $Y$ be an outcome, $D$ a treatment, and $Z$ a single instrument. Start with the linear structural equation

Y=\alpha+\beta D+u,

where $\alpha$ is an intercept, $\beta$ is the structural effect of a one-unit increase in $D$, and $u$ collects other influences on $Y$. With an intercept in each regression, define the population reduced-form slope $\gamma$ and first-stage slope $\pi$ as

\gamma=\frac{\operatorname{Cov}(Z,Y)}{\operatorname{Var}(Z)}, \qquad \pi=\frac{\operatorname{Cov}(Z,D)}{\operatorname{Var}(Z)}.

The standard just-identified IV conditions for the ratio are that the instrument is relevant, so $\operatorname{Cov}(Z,D)\ne 0$, and exogenous for the structural equation, so $\operatorname{Cov}(Z,u)=0$. Exclusion is represented here by the absence of a direct $Z$ term in the structural equation: $Z$ affects $Y$ only through $D$. These conditions are substantive assumptions, not conclusions that can be read off from a strong first stage.

Definition: In the just-identified linear IV model with one treatment and one instrument, the population IV coefficient is the reduced-form slope divided by the first-stage slope: $$ \beta_{\mathrm{IV}}=\frac{\gamma}{\pi}. $$ This ratio equals the structural coefficient $\beta$ when the stated structural equation and instrument-exogeneity condition hold and the first stage is nonzero.

The result follows directly from the structural equation. Taking covariance with $Z$ gives

\operatorname{Cov}(Z,Y) =\operatorname{Cov}(Z,\alpha+\beta D+u) =\beta\operatorname{Cov}(Z,D)+\operatorname{Cov}(Z,u).

The constant $\alpha$ has zero covariance with $Z$. Under instrument exogeneity, $\operatorname{Cov}(Z,u)=0$, so $\operatorname{Cov}(Z,Y)=\beta\operatorname{Cov}(Z,D)$. Dividing by $\operatorname{Var}(Z)$ gives $\gamma=\beta\pi$. Since relevance ensures $\pi\ne0$, division by $\pi$ yields $\gamma/\pi=\beta$. This is the population justification for the ratio.

Ratio of Sample Regression Slopes

In data, the ratio is calculated from two regressions that include intercepts: regress $Y$ on $Z$ for the reduced form, and regress $D$ on $Z$ for the first stage. Write their ordinary least squares slopes as $\hat{\gamma}$ and $\hat{\pi}$. If $\bar{Z}$, $\bar{Y}$, and $\bar{D}$ are sample means, define $S_{ZZ}=\sum_i(Z_i-\bar{Z})^2$, $S_{ZY}=\sum_i(Z_i-\bar{Z})(Y_i-\bar{Y})$, and $S_{ZD}=\sum_i(Z_i-\bar{Z})(D_i-\bar{D})$. The two slopes are $S_{ZY}/S_{ZZ}$ and $S_{ZD}/S_{ZZ}$. Thus, when $S_{ZD}\ne0$,

\hat{\beta}_{\mathrm{IV}} =\frac{\hat{\gamma}}{\hat{\pi}} =\frac{S_{ZY}/S_{ZZ}}{S_{ZD}/S_{ZZ}} =\frac{S_{ZY}}{S_{ZD}}.

This is an exact algebraic identity for the sample slopes. The ratio is an estimate of the population IV coefficient; it need not equal the unknown structural coefficient in a particular sample. As with any estimate, its causal interpretation also depends on the instrument assumptions.

Worked Example: A Binary Encouragement Instrument

Suppose a hypothetical randomized encouragement study records whether a person received an encouragement ($Z=1$) or not ($Z=0$), whether the person began a treatment ($D=1$ or $0$), and symptom-free days during the following month ($Y$). Among 200 people with $Z=0$, the treatment proportion is 0.40 and the mean outcome is 12.0 days. Among 200 people with $Z=1$, those values are 0.55 and 13.2 days. Calculate the sample reduced form, first stage, and ratio.

Reduced form. For a binary $Z$, the regression slope with an intercept is the difference in group means:

\hat{\gamma} =\bar{Y}_{Z=1}-\bar{Y}_{Z=0} =13.2-12.0 =1.2\text{ days per encouragement}.

First stage. The corresponding slope for treatment is the difference in treatment proportions:

\hat{\pi} =\bar{D}_{Z=1}-\bar{D}_{Z=0} =0.55-0.40 =0.15.

The encouragement increases treatment uptake by 0.15, or 15 percentage points, in this sample. The ratio is

\hat{\beta}_{\mathrm{IV}} =\frac{1.2\text{ days}}{0.15} =8.0\text{ days per treatment}.

The estimate is 8.0 symptom-free days per treatment uptake on the treatment scale used here. If the encouragement was randomized, had no effect on the outcome except through treatment, and satisfied the relevant monotonicity condition, the binary-instrument Wald ratio identifies the complier average causal effect under the usual IV assumptions. Without those assumptions, the calculation is still a ratio of observed group differences, but the causal interpretation does not follow.

Continuous Instruments and Their Units

For a continuous instrument, the reduced form and first stage are slopes rather than differences between two groups. The ratio still works because both slopes describe changes per unit of the same $Z$. Dividing cancels the instrument’s units, leaving outcome units per treatment unit. The structural-model assumptions are the same; a continuous instrument does not become valid merely because it has many possible values.

Worked Example: A Continuous Instrument from Sample Cross-Products

Consider a hypothetical sample relating a continuous distance measure $Z$, in kilometres, to treatment intensity $D$, measured in sessions, and an outcome $Y$, measured in points. The sample cross-products are $S_{ZZ}=160\text{ km}^2$, $S_{ZD}=60\text{ km}\cdot\text{sessions}$, and $S_{ZY}=120\text{ km}\cdot\text{points}$. These summaries are enough to calculate the regression slopes with intercepts.

The first-stage slope is

\hat{\pi} =\frac{S_{ZD}}{S_{ZZ}} =\frac{60\text{ km}\cdot\text{sessions}}{160\text{ km}^2} =0.375\text{ sessions per kilometre}.

The reduced-form slope is

\hat{\gamma} =\frac{S_{ZY}}{S_{ZZ}} =\frac{120\text{ km}\cdot\text{points}}{160\text{ km}^2} =0.750\text{ points per kilometre}.

Therefore,

\hat{\beta}_{\mathrm{IV}} =\frac{0.750\text{ points per kilometre}} {0.375\text{ sessions per kilometre}} =2.00\text{ points per session}.

The kilometre units cancel. Under the linear structural model and valid-instrument conditions, the sample ratio estimates a structural change of 2.00 outcome points per additional treatment session. If treatment effects vary, this ratio does not automatically represent the effect for every individual or a population-wide average; its interpretation depends on the assumptions and estimand appropriate to the instrument.

Worked Example: Reading and Checking Reported Slopes

A hypothetical report gives a reduced-form slope of $-0.60$ blood-pressure units per one-unit increase in an instrument and a first-stage slope of $-0.20$ treatment units per one-unit increase in that instrument. Calculate the IV ratio and interpret its sign.

The ratio is

\hat{\beta}_{\mathrm{IV}} =\frac{-0.60\text{ blood-pressure units per instrument unit}} {-0.20\text{ treatment units per instrument unit}} =3.00\text{ blood-pressure units per treatment unit}.

Both slopes are negative, so their ratio is positive. A one-unit increase in treatment corresponds to an estimated increase of 3.00 blood-pressure units under the structural interpretation. It would be incorrect to call the estimated treatment effect negative just because the reduced-form slope is negative: the first-stage direction matters too. The ratio is only causally interpretable if the instrument assumptions hold and the first stage is nonzero.

Binary and Continuous Instruments Compared

The regression-ratio form applies to either instrument type. For a binary instrument, the slopes reduce to differences between the $Z=1$ and $Z=0$ groups. For a continuous instrument, they are linear projection slopes. In both cases the units and assumptions should be stated.

Instrument	Reduced form	First stage	Just-identified ratio
Binary $Z$	Difference in mean $Y$ between $Z=1$ and $Z=0$	Difference in mean $D$ between $Z=1$ and $Z=0$	Outcome difference divided by treatment difference
Continuous $Z$	Slope from regressing $Y$ on $Z$	Slope from regressing $D$ on $Z$	Reduced-form slope divided by first-stage slope

What the Ratio Does—and Does Not—Establish

The ratio separates the instrument’s association with the outcome from the amount by which the instrument shifts treatment. It is useful because it puts the reduced-form change on a treatment scale. But that algebra alone does not verify the model. If $Z$ directly affects $Y$, is associated with unmeasured causes of $Y$, or fails to generate a nonzero change in $D$, the usual causal interpretation is not justified. A strong first stage addresses relevance, not exclusion or exogeneity.

A small first-stage slope also deserves attention. When $\hat{\pi}$ is close to zero, dividing by it can produce a large ratio, and small changes in either estimated slope can change the ratio substantially. The ratio remains the correct just-identified calculation, but it may be imprecise or unstable. Assessing uncertainty and conducting formal inference require additional methods; the point estimate alone does not show how precisely the effect has been estimated.

Common mistake: “The instrument changes the outcome by 1.2 units, so the treatment effect is 1.2 units.” This confuses the reduced form with the IV coefficient. The instrument changes treatment by some amount as well, so the IV ratio is the reduced-form change divided by the first-stage change. Also, a calculated ratio is not automatically causal: the instrument assumptions are needed for that interpretation.

Calculate the reduced form.
Estimate the slope of $Y$ on $Z$, or for binary $Z$, subtract the two group means of $Y$.

Calculate the first stage.
Estimate the slope of $D$ on $Z$, or for binary $Z$, subtract the two treatment means.

Form and interpret the ratio.
Divide the reduced form by the first stage, report the resulting outcome-per-treatment units, and justify any causal interpretation with the instrument assumptions.

Check Your Understanding

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

A binary instrument changes mean treatment from 0.30 to 0.45 and mean outcome from 6.0 to 6.9. Calculate the reduced form, first stage, and sample IV ratio with units if the outcome is measured in days.
For a continuous instrument, the reduced-form slope is 2.4 outcome units per instrument unit and the first-stage slope is 0.8 treatment units per instrument unit. What is the IV ratio, and what are its units?
In one sentence, explain why the intercept in the structural equation does not affect the covariance derivation of the ratio.
Why does a nonzero first stage not establish that the instrument is valid?
If both the reduced-form and first-stage slopes are negative, what can be said about the sign of their ratio?

Instrument	Reduced form	First stage	Just-identified ratio
Binary \(Z\)	Difference in mean \(Y\) between \(Z=1\) and \(Z=0\)	Difference in mean \(D\) between \(Z=1\) and \(Z=0\)	Outcome difference divided by treatment difference
Continuous \(Z\)	Slope from regressing \(Y\) on \(Z\)	Slope from regressing \(D\) on \(Z\)	Reduced-form slope divided by first-stage slope