Interpreting IV Coefficients Under Constant Effects

What an IV Coefficient Can Mean

An instrumental-variables coefficient is not automatically a causal effect simply because it was calculated using an instrument. Its interpretation depends on the structural model and on whether the instrument satisfies assumptions that connect its variation to the outcome only through the endogenous regressor. Under a linear model with a constant effect, a valid instrument can identify one common structural slope.

This interpretation differs from the ordinary least-squares (OLS) slope. OLS describes the linear association between the observed regressor and outcome; when the regressor is endogenous, that association need not equal the structural effect. IV instead uses variation in the regressor induced by the instrument, under assumptions that make this variation informative about the structural slope.

Definition: A constant-effect linear structural model is $$ Y = \alpha + \beta X + u, $$ where $Y$ is the outcome, $X$ is the regressor of interest, $\alpha$ is an intercept, $\beta$ is the structural slope, and $u$ contains other determinants of $Y$. The effect is constant in the sense that changing $X$ by one unit changes the structural outcome by $\beta$ units for every unit represented by the model. The model does not include effect heterogeneity in $X$.

For example, if $Y$ is annual earnings in dollars and $X$ is years of schooling, $\beta$ is measured in dollars per additional year of schooling. Calling $\beta$ a structural effect requires more than observing that earnings and schooling move together. In particular, the equation must describe the relevant outcome relationship, and the instrument must satisfy the conditions below. The constant-effect assumption is substantive: if effects vary across people or across values of $X$, a single linear slope may not describe every unit's effect.

Assumptions Behind the Structural Interpretation

Let $Z$ be an instrument for $X$. The key population moment condition is that $Z$ is uncorrelated with the structural error, while $Z$ is correlated with $X$. In the basic model without additional controls, these conditions are:

Instrument conditions:

Relevance: $\operatorname{Cov}(Z,X) \ne 0$. The instrument must generate variation in the regressor.
Exogeneity: $\operatorname{Cov}(Z,u) = 0$. The instrument must be uncorrelated with the unobserved determinants of the outcome in the structural equation.
Exclusion: The instrument affects $Y$ only through $X$; it has no additional direct path into the outcome equation. In the stated model, this means that $Z$ does not appear as a separate outcome determinant beyond its possible relationship with $X$.

Relevance is needed to distinguish the effect of $X$ from the instrument's influence: if $Z$ does not move $X$, its association with $Y$ cannot reveal the slope on $X$. Exogeneity and exclusion support the moment condition. They rule out, respectively, a relationship between $Z$ and omitted outcome determinants and a direct effect of $Z$ on $Y$. These are assumptions about the data-generating process, not conclusions established just by computing an IV coefficient.

The structural error $u$ may be correlated with $X$. That is the endogeneity problem IV is designed to address. The required condition is not that $X$ be uncorrelated with $u$, but that the instrument be uncorrelated with $u$ while being relevant for $X$. When a model includes exogenous controls, the corresponding conditions apply after accounting for those controls; the interpretation of the slope is then conditional on the specified model.

Why the IV Ratio Identifies the Slope

The population IV ratio for one instrument in the basic model is the covariance of $Z$ with $Y$, divided by the covariance of $Z$ with $X$. Its structural interpretation follows directly from the model and the instrument moment condition.

Start with the structural equation.
Substitute $Y = \alpha + \beta X + u$ into the covariance between the instrument and outcome.

Use covariance properties.
A constant has zero covariance with $Z$, so $\operatorname{Cov}(Z,Y) = \beta\operatorname{Cov}(Z,X) + \operatorname{Cov}(Z,u)$.

Apply exogeneity and relevance.
Exogeneity sets $\operatorname{Cov}(Z,u)=0$, and relevance ensures the denominator is nonzero. Dividing gives the structural slope.

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

This derivation justifies the population interpretation when the model and assumptions hold. A sample IV coefficient estimates the corresponding population quantity; it will not generally equal $\beta$ exactly in a finite sample. With multiple valid instruments, the same constant-effect model implies that each instrument's population covariance ratio identifies the same $\beta$, provided each is relevant and satisfies the required conditions. That common interpretation relies on the constant structural slope and valid instruments; it does not follow from the numerical procedure alone.

Worked Interpretations

Worked Example: A Continuous Instrument and an OLS Association

Suppose a hypothetical analysis relates years of schooling $X$ to annual earnings $Y$, and uses distance to a college $Z$ as an instrument. Assume the constant-effect model is appropriate, and suppose the hypothetical sample summaries are $\widehat{\operatorname{Cov}}(Z,Y)=6.0$, $\widehat{\operatorname{Cov}}(Z,X)=2.0$, $\widehat{\operatorname{Cov}}(X,Y)=8.0$, and $\widehat{\operatorname{Var}}(X)=4.0$. The covariance units are those implied by the variables: for example, the first has units of distance times dollars per year.

The sample IV ratio is $$ \hat{\beta}_{IV} = \frac{\widehat{\operatorname{Cov}}(Z,Y)} {\widehat{\operatorname{Cov}}(Z,X)} = \frac{6.0}{2.0} = 3.0. $$

The estimate is $3.0$ thousand dollars per year of schooling if earnings are recorded in thousands of dollars. Under the constant-effect model and the relevance, exclusion, and exogeneity assumptions, it estimates a structural increase of $3.0$ thousand dollars in annual earnings per additional year of schooling. This is an estimate, not a claim that the finite-sample value is exactly the population slope.

The OLS slope from the same summaries is $$ \hat{\beta}_{OLS} = \frac{\widehat{\operatorname{Cov}}(X,Y)} {\widehat{\operatorname{Var}}(X)} = \frac{8.0}{4.0} = 2.0. $$ It describes an observed linear association of $2.0$ thousand dollars in annual earnings per year of schooling. The two estimates differ. The IV interpretation is not that IV must be larger than OLS; it is that, under its assumptions, IV targets the structural slope, while OLS describes the observed association. In this hypothetical setting, the instrument is relevant because its estimated covariance with schooling is nonzero. Its exogeneity and exclusion cannot be verified from these covariance summaries; they must be defended substantively.

Worked Example: A Binary Instrument and a Constant Effect

Suppose, hypothetically, that $Z=1$ denotes eligibility for a training program and $Z=0$ denotes ineligibility. Let $X$ be the number of months of training and $Y$ be monthly earnings in hundreds of dollars. The group means are:

For $Z=0$, mean training is $0.30$ months and mean earnings are $10.00$ hundreds of dollars.
For $Z=1$, mean training is $0.50$ months and mean earnings are $10.60$ hundreds of dollars.

The instrument changes mean training by $0.50-0.30=0.20$ months, so there is a nonzero first stage in these summaries. The reduced-form difference in mean earnings is $10.60-10.00=0.60$ hundreds of dollars. The binary-instrument ratio is $$ \hat{\beta}_{IV} = \frac{10.60-10.00}{0.50-0.30} = \frac{0.60}{0.20} = 3.0 \quad\text{hundreds of dollars per month of training.} $$

Under a constant-effect linear model, and assuming eligibility affects earnings only through training and is uncorrelated with the structural error, the estimate means that one additional month of training raises monthly earnings by $3.0$ hundreds of dollars, or $300$ dollars, according to the model. The estimate is not merely the difference in earnings between eligible and ineligible people: that difference is $0.60$ hundreds of dollars, and it corresponds to the instrument's change in training of only $0.20$ months. The ratio scales the outcome difference by that first-stage change.

The group means show relevance but cannot establish the other assumptions. For instance, if eligibility also changes access to job placement services that directly affect earnings, exclusion would fail, and the ratio would not have the stated structural-slope interpretation.

What Goes Wrong When an Assumption Fails

A ratio can be calculated even if an instrument is invalid. In that case, the arithmetic remains correct, but the structural interpretation does not. A direct instrument effect makes this especially clear. Suppose the actual outcome equation contains a direct effect of $Z$:

Y = \alpha + \beta X + \delta Z + u.

If $\operatorname{Cov}(Z,u)=0$, then $$ \frac{\operatorname{Cov}(Z,Y)}{\operatorname{Cov}(Z,X)} = \beta + \delta\frac{\operatorname{Var}(Z)}{\operatorname{Cov}(Z,X)}. $$

Unless $\delta=0$, this ratio need not equal $\beta$. The instrument's relationship with the outcome includes both the effect working through $X$ and the direct effect. A nonzero first stage cannot remove that extra contribution.

Worked Example: A Relevant Instrument With a Direct Effect

Consider hypothetical population moments for the model $Y=\alpha+3X+1Z+u$: $\operatorname{Var}(Z)=4$, $\operatorname{Cov}(Z,X)=2$, and $\operatorname{Cov}(Z,u)=0$. The structural slope is $\beta=3$, and the direct effect is $\delta=1$. The instrument is relevant because $\operatorname{Cov}(Z,X)=2\ne0$, and it is uncorrelated with $u$ by assumption, but exclusion fails because $\delta\ne0$.

The covariance of $Z$ with the outcome is $$ \operatorname{Cov}(Z,Y) = 3\operatorname{Cov}(Z,X)+1\operatorname{Var}(Z)+\operatorname{Cov}(Z,u) = 3(2)+1(4)+0 = 10. $$

Consequently, the IV ratio is $$ \frac{\operatorname{Cov}(Z,Y)}{\operatorname{Cov}(Z,X)} = \frac{10}{2} = 5, $$ not the structural slope of $3$. The ratio is shifted by the instrument's direct effect. Interpreting $5$ as the effect of $X$ would therefore be incorrect, even though the first stage is nonzero and the calculation is exact.

Common Mistakes and Practical Interpretation

Mistake: “The IV coefficient is the causal effect because IV was used.” The correction is to state the model and assumptions. Without a suitable structural equation, relevance, exclusion, and exogeneity, the coefficient need not identify the structural slope.
Mistake: “The IV and OLS coefficients are both effects, just estimated differently.” OLS is a population linear association unless additional conditions justify a structural interpretation. IV targets the structural slope under its own assumptions.
Mistake: “A nonzero first stage proves the instrument is valid.” It establishes relevance in the relevant population or sample, but says nothing by itself about exclusion or exogeneity.
Mistake: “A constant-effect assumption is just a convenient way to summarize the estimate.” It is a substantive restriction: the model uses one slope for all units and relevant changes in $X$. If effects differ, that single-slope interpretation may be inappropriate.

A complete interpretation should therefore identify the outcome and regressor, give the coefficient's units, describe the structural change represented by one unit of $X$, and explicitly condition that interpretation on the model and instrument assumptions. If those assumptions are uncertain, describe the IV coefficient as an estimate whose causal interpretation depends on them rather than as an established causal fact.

Key takeaway: Under a constant-effect linear model, a relevant instrument satisfying exclusion and exogeneity identifies the structural slope through its covariance ratio with the outcome and regressor. OLS instead describes the observed linear association when regressor endogeneity is present.

Check Your Understanding

Answer each question by referring to the model, assumptions, and units in the situation.

In $Y=\alpha+\beta X+u$, what does the constant-effect assumption say about the structural change in $Y$ associated with a one-unit increase in $X$?
Why can the OLS slope differ from the structural slope when $X$ is correlated with $u$?
If $\operatorname{Cov}(Z,Y)=12$ and $\operatorname{Cov}(Z,X)=3$, what is the IV covariance ratio, and what assumptions are required to interpret it as the structural slope?
Does evidence that $\operatorname{Cov}(Z,X)\ne0$ establish that $Z$ is a valid instrument? Explain.
If an instrument has a direct effect on $Y$, why can the IV ratio fail to identify the coefficient on $X$, even when the instrument is relevant?