The Just-Identified IV Ratio

From an Instrument Moment to a Slope

A structural slope describes how an outcome changes with a regressor in the structural equation. When that regressor is endogenous, its association with the outcome can reflect both the slope and the relationship between the regressor and the structural disturbance. An instrumental variable isolates a source of variation in the regressor that, under appropriate conditions, is unrelated to that disturbance.

With one endogenous regressor and one instrument, the population moment condition has a particularly direct implication: if the instrument covaries with the regressor, the slope is the ratio of the instrument's covariance with the outcome to its covariance with the regressor. This is a population identification result. It says which value of the structural slope is compatible with the stated model and moments; it does not, by itself, establish that an instrument is credible in an application or describe a finite-sample estimator.

Model and conditions: Let $Y$ be an outcome, $X$ a scalar regressor, $Z$ a scalar instrument, and $U$ the structural disturbance. Suppose the structural equation is $Y=\beta X+U$, with an intercept allowed. The instrument is valid for this equation when $E[ZU]=E[Z]E[U]$, equivalently $\operatorname{Cov}(Z,U)=0$, with finite second moments. It is relevant when $\operatorname{Cov}(Z,X)\ne 0$.

Allowing an intercept means the structural equation could instead be written $Y=\beta_0+\beta X+U$. Covariance with $Z$ eliminates the constant: $\operatorname{Cov}(Z,\beta_0)=0$. Thus the intercept does not affect the slope ratio. The moment condition is about the instrument and the structural disturbance, not about $X$ and $U$ being uncorrelated. Indeed, endogeneity means that $\operatorname{Cov}(X,U)$ may be nonzero.

The word just-identified refers here to the scalar case with one instrumental moment for one unknown structural slope. The moment supplies exactly the restriction needed to determine that one coefficient, provided the instrument has a nonzero relationship with the regressor. The result can be stated as follows.

Theorem: Under $Y=\beta X+U$, finite second moments, the validity condition $\operatorname{Cov}(Z,U)=0$, and the relevance condition $\operatorname{Cov}(Z,X)\ne 0$, the structural slope is uniquely determined by $$ \beta=\frac{\operatorname{Cov}(Z,Y)}{\operatorname{Cov}(Z,X)}. $$

The numerator and denominator are both population covariances. No distributional assumption such as normality is required for this identity. Its key assumptions are structural linearity in the scalar slope, the instrument moment condition, finite moments, and nonzero relevance.

Why the Ratio Identifies the Slope

Covariance is linear in each argument, and the covariance of a variable with itself is its variance. Apply covariance with $Z$ to both sides of the structural equation. This gives the full identification argument:

\begin{aligned} Y &= \beta X+U,\\ \operatorname{Cov}(Z,Y) &= \operatorname{Cov}(Z,\beta X+U)\\ &= \beta\operatorname{Cov}(Z,X)+\operatorname{Cov}(Z,U)\\ &= \beta\operatorname{Cov}(Z,X). \end{aligned}

The last equality uses validity. Relevance permits division by $\operatorname{Cov}(Z,X)$, yielding the ratio in the theorem. It also establishes uniqueness: if two slopes $\beta_1$ and $\beta_2$ both satisfied the same moment equation, then

(\beta_1-\beta_2)\operatorname{Cov}(Z,X)=0.

Since the covariance is nonzero, $\beta_1-\beta_2=0$, so the two slopes must be equal. Without relevance, that uniqueness argument does not work. The moment may then place no restriction on the slope at all.

Worked Example: Recovering a Slope From a Small Hypothetical Population

Consider a hypothetical population with four equally likely types. The values of $Z$, $X$, and $Y$ are as follows. Suppose the structural equation is $Y=3X+U$, with $U=Y-3X$.

Type	$Z$	$X$	$Y$	$U=Y-3X$
1	-1	-3	-8	1
2	-1	-1	-4	-1
3	1	1	4	1
4	1	3	8	-1

Each variable has mean zero in this population, so its covariance with $Z$ is the average of the four products. In particular,

\begin{aligned} \operatorname{Cov}(Z,X) &= \frac{(-1)(-3)+(-1)(-1)+(1)(1)+(1)(3)}{4}\\ &= \frac{8}{4}=2,\\ \operatorname{Cov}(Z,U) &= \frac{(-1)(1)+(-1)(-1)+(1)(1)+(1)(-1)}{4}\\ &= \frac{0}{4}=0,\\ \operatorname{Cov}(Z,Y) &= \frac{(-1)(-8)+(-1)(-4)+(1)(4)+(1)(8)}{4}\\ &= \frac{24}{4}=6. \end{aligned}

The instrument is valid because $\operatorname{Cov}(Z,U)=0$, and relevant because $\operatorname{Cov}(Z,X)=2\ne 0$. Therefore the ratio is $6/2=3$, which recovers the structural slope of 3 outcome units per unit of $X$. The calculation uses population covariances: the four equally likely types define the population in this illustration.

How the Ratio Appears in Reduced Forms

The ratio also follows by comparing the instrument's relationship with each side of the structural equation. For clarity, define the reduced-form slopes on $Z$ as the covariance of each variable with $Z$, divided by the variance of $Z$. This is a way to express the population linear association with the instrument; no separate causal interpretation of these slopes is needed for the identity.

\pi_X=\frac{\operatorname{Cov}(Z,X)}{\operatorname{Var}(Z)}, \qquad \pi_Y=\frac{\operatorname{Cov}(Z,Y)}{\operatorname{Var}(Z)}.

Provided $\operatorname{Var}(Z)>0$, the covariance result gives $\pi_Y=\beta\pi_X$. If $\pi_X\ne 0$, division yields $\beta=\pi_Y/\pi_X$, which is the same covariance ratio because the common variance of $Z$ cancels. This connects the ratio to reduced forms: the instrument's reduced-form association with the outcome equals its association with the regressor multiplied by the structural slope.

Worked Example: Using Reduced-Form Slopes

In a hypothetical population, let $Z$ be a policy eligibility indicator, $X$ be years of training, and $Y$ be hourly earnings in dollars. Suppose the structural model is $Y=\beta X+U$, the instrument is valid, and the instrument variance and reduced-form slopes are

\operatorname{Var}(Z)=0.25,\qquad \pi_X=0.8\text{ training years per unit of }Z,\qquad \pi_Y=2.0\text{ dollars per hour per unit of }Z.

The corresponding covariances are

\begin{aligned} \operatorname{Cov}(Z,X)&=\pi_X\operatorname{Var}(Z) =(0.8)(0.25)=0.20,\\ \operatorname{Cov}(Z,Y)&=\pi_Y\operatorname{Var}(Z) =(2.0)(0.25)=0.50. \end{aligned}

The first covariance is nonzero, so the instrument is relevant in this hypothetical population. The ratio is $0.50/0.20=2.5$ dollars per hour for each additional year of training. Equivalently, $\pi_Y/\pi_X=2.0/0.8=2.5$. This interpretation is a structural one only conditional on the stated model and validity condition; the arithmetic alone cannot establish those conditions.

Relevance and Validity Are Different Requirements

Validity and relevance play separate roles. Validity removes the term $\operatorname{Cov}(Z,U)$ from the covariance equation. Relevance makes the remaining coefficient on $\beta$ nonzero, so the moment determines one slope. A variable may be unrelated to the structural disturbance but still fail to identify the slope if it does not covary with $X$.

Worked Example: Zero Relevance Leaves the Slope Undetermined

Suppose a hypothetical population has $\operatorname{Cov}(Z,X)=0$ and $\operatorname{Cov}(Z,U)=0$. The structural equation then implies

\operatorname{Cov}(Z,Y) =\beta\operatorname{Cov}(Z,X)+\operatorname{Cov}(Z,U) =\beta(0)+0=0.

Thus the instrument-outcome covariance is zero for every value of $\beta$. The moment condition is compatible with $\beta=1$, $\beta=5$, or any other slope; it does not distinguish among them. The ratio would have a zero denominator, so it is undefined. Validity alone is not enough for identification: nonzero relevance is essential.

Conversely, relevance alone is not enough. If the instrument is related to the disturbance, the ratio generally combines the structural slope with that disturbance relationship. The algebra makes the problem explicit whenever $\operatorname{Cov}(Z,X)\ne 0$:

\frac{\operatorname{Cov}(Z,Y)}{\operatorname{Cov}(Z,X)} = \beta+ \frac{\operatorname{Cov}(Z,U)}{\operatorname{Cov}(Z,X)}.

The second term vanishes under validity. If it does not vanish, the covariance ratio is not in general the structural slope.

Common Pitfall: Treating the Ratio as Proof of Instrument Validity

A tempting but incorrect argument is: “The instrument is relevant, so the covariance ratio must be the causal slope.” Relevance establishes only that the denominator is nonzero. It does not establish that $Z$ is uncorrelated with $U$, and a nonzero covariance with the disturbance changes the ratio.

For example, suppose a hypothetical structural slope is $4$ outcome units per unit of $X$, while $\operatorname{Cov}(Z,X)=2$ and $\operatorname{Cov}(Z,U)=3$. Then

\operatorname{Cov}(Z,Y) =4(2)+3=11, \qquad \frac{\operatorname{Cov}(Z,Y)}{\operatorname{Cov}(Z,X)} =\frac{11}{2}=5.5.

The instrument is relevant, but the ratio is $5.5$, not the structural slope $4$, because the validity condition fails. The correction is to separate the algebraic ratio from its interpretation: the ratio equals the structural slope only when the model and instrument moment conditions hold, including both relevance and validity.

Key takeaway: With one scalar regressor and one scalar instrument, the valid-instrument moment gives $\operatorname{Cov}(Z,Y)=\beta\operatorname{Cov}(Z,X)$. Nonzero relevance makes the solution unique, yielding $\beta=\operatorname{Cov}(Z,Y)/\operatorname{Cov}(Z,X)$. The identity does not make an invalid or irrelevant instrument suitable.

Check Your Understanding

Use the population model and conditions from this tutorial to answer each question.

Starting from $Y=\beta X+U$, write the covariance equation that leads to the just-identified ratio.
Why does $\operatorname{Cov}(Z,X)\ne 0$ matter for identifying a unique slope?
If $\operatorname{Cov}(Z,X)=1.5$, $\operatorname{Cov}(Z,U)=0$, and $\operatorname{Cov}(Z,Y)=6$, what is the identified slope, including its interpretation in units?
If $\operatorname{Cov}(Z,X)=2$ and $\operatorname{Cov}(Z,U)=1$, by how much does the covariance ratio differ from $\beta$?
Explain why a zero instrument-regressor covariance prevents the ratio from identifying the structural slope, even when the instrument is valid.

Type	\(Z\)	\(X\)	\(Y\)	\(U=Y-3X\)
1	-1	-3	-8	1
2	-1	-1	-4	-1
3	1	1	4	1
4	1	3	8	-1

The Just-Identified IV Ratio

What You'll Learn

From an Instrument Moment to a Slope