Continuous Instruments And Covariance Ratios

Why Use a Covariance Ratio?

With a continuous instrument, the instrument and treatment need not split observations into two groups. Instead, the instrument varies across observations, and its association with the outcome can be compared with its association with the treatment. The scalar IV estimand is the ratio of those two associations.

This ratio is useful because the instrument may be associated with the treatment while the treatment itself is endogenous. Under the appropriate structural model and instrument moment condition, the outcome association divided by the treatment association identifies the structural coefficient. The ratio also makes clear why a relevant instrument is needed: if the instrument has no covariance with the treatment, the denominator is zero and the ratio cannot be calculated.

Definition: Let $Z$ be a scalar instrument, $D$ a scalar treatment or endogenous regressor, and $Y$ an outcome, with finite second moments. The population scalar IV estimand is $$ \beta_{IV}=\frac{\operatorname{Cov}(Z,Y)}{\operatorname{Cov}(Z,D)}, $$ provided $\operatorname{Cov}(Z,D)\ne 0$. The numerator is the instrument–outcome covariance, and the denominator is the instrument–treatment covariance.

Covariance measures how two variables vary together after subtracting their means: $\operatorname{Cov}(A,B)=E[(A-E[A])(B-E[B])]$. Thus, the definition uses centered variation, not the variables’ unadjusted levels. In a sample of size $n$, the corresponding just-identified IV estimate with an intercept is $\hat{\beta}_{IV}=\widehat{\operatorname{Cov}}(Z,Y)/\widehat{\operatorname{Cov}}(Z,D)$, when the sample denominator is nonzero. Because both sample covariances use the same divisor $n-1$, it cancels in the ratio.

How the Structural Model Gives the Ratio

To give the ratio a structural interpretation, consider a linear structural equation with an intercept:

Y=\alpha+\beta D+U.

Here $\alpha$ is an intercept, $\beta$ is the structural coefficient on $D$, and $U$ contains the unobserved determinants of $Y$. The treatment $D$ may be correlated with $U$; that endogeneity is why an instrument is being considered. The instrument assumptions relevant to this scalar ratio are:

Relevance: $\operatorname{Cov}(Z,D)\ne 0$.
Exogeneity moment: $\operatorname{Cov}(Z,U)=0$.
Exclusion in the structural equation: $Z$ does not enter the outcome equation separately from its relationship to $D$; the equation above contains no direct $Z$ term.

These are assumptions about the model and instrument, not conclusions that can be established merely by calculating the ratio. In particular, relevance is a condition for the ratio to be defined; it does not by itself establish exogeneity or exclusion.

Derivation: Take covariance with $Z$ on both sides of the structural equation. Covariance with a constant is zero, and covariance is linear, so $$ \operatorname{Cov}(Z,Y) =\operatorname{Cov}(Z,\alpha+\beta D+U) =\beta\operatorname{Cov}(Z,D)+\operatorname{Cov}(Z,U). $$ Under the exogeneity moment $\operatorname{Cov}(Z,U)=0$, this becomes $$ \operatorname{Cov}(Z,Y)=\beta\operatorname{Cov}(Z,D). $$ If relevance holds, divide both sides by the nonzero $\operatorname{Cov}(Z,D)$ to obtain $$ \frac{\operatorname{Cov}(Z,Y)}{\operatorname{Cov}(Z,D)}=\beta. $$

This derivation shows both what the ratio represents under the stated model and why it is not automatically a causal effect: the structural interpretation depends on the model and moment condition. Without them, the ratio is still a well-defined statistical quantity when its denominator is nonzero, but its structural interpretation is not established.

Two Reduced-Form Slopes in the Ratio

The covariance ratio can also be expressed as a ratio of linear-projection slopes. Define the population slope from projecting $Y$ on an intercept and $Z$ as $b_{Y,Z}=\operatorname{Cov}(Z,Y)/\operatorname{Var}(Z)$, and the corresponding slope for $D$ as $b_{D,Z}=\operatorname{Cov}(Z,D)/\operatorname{Var}(Z)$. If $\operatorname{Var}(Z)>0$ and $\operatorname{Cov}(Z,D)\ne0$, then

\beta_{IV} =\frac{b_{Y,Z}}{b_{D,Z}} =\frac{\operatorname{Cov}(Z,Y)/\operatorname{Var}(Z)} {\operatorname{Cov}(Z,D)/\operatorname{Var}(Z)}.

The numerator slope describes the outcome’s linear association with the instrument; the denominator slope describes the treatment’s linear association with the instrument. Their ratio is measured in outcome units per treatment unit. These slopes describe associations with $Z$, not effects caused by a one-unit increase in $Z$.

Worked Example: Calculating the Ratio With Centered Data

Suppose four hypothetical observations have the following centered instrument values and recorded treatment and outcome values. Let $D$ be measured in treatment units and $Y$ in outcome points.

Observation	$Z$	$D$	$Y$
1	-1.5	-1	-1
2	-0.5	-1	1
3	0.5	1	3
4	1.5	1	5

The instrument and treatment means are both zero, and the outcome mean is $2$. Thus the centered outcome values are $-3,-1,1,3$. The sums of cross-products are

\sum_{i=1}^4 Z_iD_i =(-1.5)(-1)+(-0.5)(-1)+(0.5)(1)+(1.5)(1)=4, $$ $$ \sum_{i=1}^4 Z_i(Y_i-\bar{Y}) =(-1.5)(-3)+(-0.5)(-1)+(0.5)(1)+(1.5)(3)=10.

Since each sample covariance divides its centered cross-product sum by $n-1=3$, the sample covariance ratio is

\hat{\beta}_{IV} =\frac{10/3}{4/3} =\frac{10}{4} =2.5.

The corresponding reduced-form sample slope is $10/5=2.0$, because $\sum_i Z_i^2=5$. The first-stage sample slope is $4/5=0.8$. Their ratio is $2.0/0.8=2.5$, as required. Under the IV assumptions, this estimate is in outcome points per treatment unit. The four observations alone do not verify those assumptions.

Nonzero Means: Center Before Taking the Ratio

The covariance formula subtracts means before multiplying. This centering matters when the variables have nonzero means. With an intercept in the structural equation, the constant contributes nothing to covariance, but it would contribute to uncentered cross-products. Consequently, the ratio of raw sums of $Z_iY_i$ and $Z_iD_i$ is generally not the covariance ratio.

Worked Example: A Covariance Ratio With Nonzero Means

Consider a hypothetical sample with a continuous instrument $Z$, treatment $D$, and outcome $Y$. Their sample means are $13$, $22$, and $55.5$, respectively.

Observation	$Z$	$D$	$Y$
1	10	20	50
2	12	21	53
3	14	23	58
4	16	24	61

The centered values are $Z-\bar{Z}=(-3,-1,1,3)$, $D-\bar{D}=(-2,-1,1,2)$, and $Y-\bar{Y}=(-5.5,-2.5,2.5,5.5)$. The centered cross-product sums are

\sum_i (Z_i-\bar{Z})(D_i-\bar{D}) =(-3)(-2)+(-1)(-1)+(1)(1)+(3)(2)=14, $$ $$ \sum_i (Z_i-\bar{Z})(Y_i-\bar{Y}) =(-3)(-5.5)+(-1)(-2.5)+(1)(2.5)+(3)(5.5)=38.

Both covariances divide their sums by $3$, so

\hat{\beta}_{IV} =\frac{38/3}{14/3} =\frac{38}{14} \approx 2.714.

Equivalently, the reduced-form slope is $(38/3)/(20/3)=1.9$, since $\sum_i(Z_i-\bar{Z})^2=20$. The first-stage slope is $(14/3)/(20/3)=0.7$, and $1.9/0.7\approx2.714$. This sample ratio has outcome units per treatment unit.

For comparison, the raw sums are $\sum_i Z_iY_i=2924$ and $\sum_i Z_iD_i=1158$, giving $2924/1158\approx2.525$, not $2.714$. That raw ratio includes level and intercept contributions; it is not the just-identified IV ratio with an intercept.

Using Summary Statistics and Reading the Result

A covariance can also be calculated from standard deviations and a correlation: $\operatorname{Cov}(A,B)=\operatorname{Corr}(A,B)\sigma_A\sigma_B$, where $\sigma_A$ and $\sigma_B$ are population standard deviations. The sample analogue uses sample standard deviations and sample correlation. The covariance ratio may therefore be computed without listing every observation, provided the required summary statistics refer to the same variables and sample or population.

Worked Example: Computing the Ratio From Summary Statistics

Suppose a hypothetical population has an instrument with standard deviation $3$, an outcome with standard deviation $12$, and a treatment with standard deviation $2$. Suppose $\operatorname{Corr}(Z,Y)=0.4$ and $\operatorname{Corr}(Z,D)=0.6$. Their means may be nonzero; the covariance calculation uses the centered variation.

First compute the two covariances:

\operatorname{Cov}(Z,Y)=0.4(3)(12)=14.4, \qquad \operatorname{Cov}(Z,D)=0.6(3)(2)=3.6.

The denominator is nonzero, so the ratio is defined:

\beta_{IV}=\frac{14.4}{3.6}=4.

Under the structural and instrument assumptions, the ratio identifies a structural coefficient of $4$ outcome units per treatment unit. The positive ratio follows because both covariances are positive. The correlations and standard deviations establish the arithmetic, not the validity of the instrument.

Common Mistakes and Practical Checks

A frequent error is to calculate $\sum_i Z_iY_i/\sum_i Z_iD_i$ and call it the IV estimate, even when the model includes an intercept and the variables have nonzero means. The reasoning is that the formula should be a direct comparison of instrument–outcome and instrument–treatment products. The correction is to use covariances: subtract each variable’s mean before forming the cross-products, or equivalently use centered sums. Raw cross-products coincide with centered cross-products only under special conditions, such as zero sample means for the relevant variables.

Another mistake is to read the ratio as the effect of increasing $Z$ by one unit. It is a ratio of associations: outcome association per unit of instrument variation divided by treatment association per unit of instrument variation. Under the assumptions, this ratio identifies the structural coefficient on $D$; it does not turn the instrument–outcome association into a direct effect of $Z$.

Before reporting a scalar IV ratio, check that the numerator and denominator use the same instrument, outcome, treatment, and observations; that the denominator is nonzero; and that the coefficient’s units are clear. Also distinguish a calculated ratio from a justified structural interpretation: relevance alone does not guarantee the exogeneity moment or exclusion restriction.

Key takeaway: For a scalar continuous instrument, the IV estimand is the centered instrument–outcome covariance divided by the centered instrument–treatment covariance. Under the structural equation and instrument moment condition, this ratio equals the structural coefficient; with an intercept, do not replace covariances by uncentered cross-products.

Check Your Understanding

Try these questions using the definitions and calculations above.

State the population scalar IV estimand and the relevance condition required for it to be defined.
Why does taking covariance with $Z$ remove the intercept from the structural equation?
If the sample covariance of $Z$ with $Y$ is $12$ and its sample covariance with $D$ is $3$, what is the sample IV ratio, and what are its units if $Y$ is measured in points and $D$ in hours?
Why can $\sum_i Z_iY_i/\sum_i Z_iD_i$ differ from the covariance ratio when the variables have nonzero means?
Does a nonzero instrument–treatment covariance by itself establish the instrument’s exogeneity or exclusion? Explain.

Observation	\(Z\)	\(D\)	\(Y\)
1	10	20	50
2	12	21	53
3	14	23	58
4	16	24	61