Why Two Different Restrictions Are Needed
An instrumental-variables argument uses a variable \(Z\) to learn about the effect of an endogenous regressor \(X\) on an outcome \(Y\). The instrument must do two distinct things: it must predict \(X\), and it must not be associated with the unobserved determinants of \(Y\) represented by the structural error. The first requirement is relevance; the second is exclusion in the linear model considered here.
These are assumptions about different relationships. An instrument can predict the regressor but also be related to the structural error, making it unsuitable. Conversely, an instrument could be unrelated to the structural error but fail to predict the regressor, leaving it uninformative about the effect of \(X\). Neither restriction implies the other.
The covariance statements are population properties. In data, a sample association can provide evidence about a population association, but a finite sample cannot establish either restriction with certainty. The restrictions also have substantive economic meaning: they concern how the instrument is generated and how it relates to the outcome equation, not just how variables are named in a data set.
Relevance: The Instrument Predicts the Endogenous Regressor
Relevance requires that variation in \(Z\) be associated with variation in \(X\). In the scalar linear setting, the condition is \(\operatorname{Cov}(Z,X)\ne 0\). Equivalently, because both variables have positive variance, their population correlation is nonzero. The sign indicates the direction of their linear association; relevance itself requires only that the association is not zero.
“Predictive” here does not mean that \(Z\) must perfectly determine \(X\), or that \(Z\) must cause every change in \(X\). It means that \(Z\) contains information about \(X\) in the population. A weak association may technically be nonzero while providing little useful variation in \(X\). The restriction is therefore conceptually binary, but the strength of the association matters in applications.
Worked Example: A Training Program and Eligibility
Suppose a hypothetical labor-market model studies the effect of training \(X\), measured in hours, on earnings \(Y\). Let \(Z\) indicate eligibility for a training subsidy, with 1 meaning eligible. For four hypothetical people, the observed values are:
| Person | Eligibility \(Z\) | Training hours \(X\) |
|---|---|---|
| A | 0 | 8 |
| B | 0 | 10 |
| C | 1 | 12 |
| D | 1 | 14 |
The sample means are \(\bar{Z}=0.5\) and \(\bar{X}=11\). The sample covariance, using denominator \(n-1\), is
In these hypothetical data, eligible people have more training on average, and the sample covariance is positive. This is evidence of predictive association in this small sample, not proof that the population covariance is nonzero. The example also says nothing by itself about exclusion: eligibility might predict training and still be related to earnings through another pathway.
Relevance is not the same as a causal effect of \(Z\) on \(X\). The population condition is an association condition. In a particular economic design, a causal mechanism—such as eligibility changing access to training—may explain why the association is plausible. But an observed association alone does not establish that mechanism or establish the other instrument restriction.
Exclusion: No Association With the Structural Error
The structural error \(U\) contains determinants of \(Y\) not represented by the included regressor \(X\), according to the specified equation. Exclusion requires \(\operatorname{Cov}(Z,U)=0\). In words, the instrument must not systematically vary with the unobserved part of the outcome equation.
A useful economic interpretation is that \(Z\) has no effect on \(Y\) except through \(X\). For example, if eligibility changes earnings by changing training, that pathway is consistent with exclusion. If eligibility also changes earnings through a separate benefit, access to job placement, or another channel omitted from the structural equation, that separate pathway is part of \(U\) and can make \(Z\) associated with \(U\). Exclusion can also fail even without a direct pathway: a common cause of \(Z\) and an omitted determinant of \(Y\) can create nonzero covariance.
The no-direct-effect description and the zero-covariance condition should not be treated as interchangeable in every setting. The precise restriction here is about the structural error in the stated linear model. A plausible story that \(Z\) has no direct causal effect does not, on its own, rule out common causes or other sources of association with \(U\).
Worked Example: Distance to College and Omitted Ability
Consider a hypothetical model in which \(Y\) is earnings and \(X\) is years of education. Distance \(Z\) from a person’s home to a college is proposed as an instrument. Suppose an illustrative population has four equally weighted locations with the following centered structural-error values. Here \(U\) represents unobserved determinants of earnings, such as ability and family resources, after accounting for education.
| Location | Distance \(Z\), km | Structural error \(U\), illustrative units |
|---|---|---|
| A | 1 | 3 |
| B | 2 | 1 |
| C | 3 | -1 |
| D | 4 | -3 |
The means are \(\bar{Z}=2.5\) km and \(\bar{U}=0\). The illustrative covariance is
The negative association means that, in this hypothetical setup, greater distance is associated with lower values of the unobserved earnings component. If this association arose because locations farther from college also had systematically different family resources or local opportunities, exclusion would fail. The values of \(U\) are shown only to make the logic transparent; in an actual application, the structural error is not directly observed and the restriction requires an economic argument rather than a direct calculation from a table of residual truth.
Checking the Restrictions Separately
A disciplined instrument argument asks two separate questions. First, is there a defensible reason to expect \(Z\) to predict \(X\)? Second, is there a defensible reason to expect \(Z\) to be unrelated to \(U\), including through direct effects and shared causes? The answers may differ. Evidence for relevance is not evidence for exclusion.
The distinction can be seen by substituting the structural equation into the covariance with \(Z\). Since a constant has zero covariance with \(Z\), covariance algebra gives
If exclusion holds, the second term is zero, so the association between \(Z\) and \(Y\) in this equation comes through the association between \(Z\) and \(X\). If exclusion fails, the association between \(Z\) and \(Y\) can also contain the term \(\operatorname{Cov}(Z,U)\). This algebra does not prove exclusion; it shows why the two restrictions matter separately.
Worked Example: Rainfall, Irrigation, and Crop Yield
Suppose a hypothetical agricultural model studies the effect of irrigation \(X\) on crop yield \(Y\). Rainfall \(Z\) is proposed as an instrument. To illustrate both restrictions, consider four equally weighted hypothetical seasons. The \(U\) values below represent the part of yield not accounted for by irrigation; in this example they include a direct rainfall contribution.
| Season | Rainfall \(Z\), units | Irrigation \(X\), hours | Structural error \(U\), illustrative units |
|---|---|---|---|
| A | 1 | 4 | 1 |
| B | 2 | 3 | 2 |
| C | 3 | 2 | 3 |
| D | 4 | 1 | 4 |
The means are \(\bar{Z}=2.5\), \(\bar{X}=2.5\), and \(\bar{U}=2.5\). Using denominator \(n-1=3\), the sample covariances in these illustrative data are
Rainfall is negatively associated with irrigation in these data, so the relevance condition is illustrated. But rainfall is also positively associated with the structural error. If rainfall directly affects crop yield, that effect is not mediated by irrigation and belongs in \(U\) under the stated equation; exclusion therefore fails in the illustration. The fact that rainfall predicts irrigation cannot repair this failure.
Scope and Common Pitfalls
The covariance formulation is the standard linear statement used here. Zero covariance is weaker than statistical independence: variables can have zero covariance while still being dependent in other ways. For the linear IV restriction, the relevant claim is the stated zero association with the structural error; stronger claims such as independence require additional assumptions.
- “The instrument predicts \(X\), so it is valid.” This checks relevance only. A direct effect on \(Y\), or a shared cause of \(Z\) and \(U\), can still violate exclusion.
- “The instrument has no direct effect, so it is exogenous.” A lack of a direct pathway does not rule out common causes that make \(Z\) associated with the structural error.
- “The sample covariance is nonzero, so population relevance is proven.” A sample covariance describes observed data. Population relevance is an assumption about the data-generating process, and sample evidence is subject to sampling variation.
- “Zero covariance means no relationship of any kind.” Zero covariance rules out linear association as specified, but does not generally imply independence.
The restrictions are not symmetric in their evidential status. Relevance concerns the relationship between observed \(Z\) and \(X\), so sample data can directly inform its plausibility. Exclusion concerns \(Z\)'s relationship with the structural error, which is unobserved and depends on the structural model. A persuasive exclusion argument therefore needs economic knowledge about assignment, pathways, and potential common causes, not only a favorable pattern in observed variables.
Check Your Understanding
Try these questions before moving on.
- In the model \(Y=\beta_0+\beta_1X+U\), state the relevance and exclusion restrictions using covariances.
- A proposed instrument predicts education but also changes earnings through access to job placement. Which restriction is threatened, and why?
- Why can a sample association between \(Z\) and \(X\) provide evidence about relevance without proving the population condition?
- Explain how a common cause of an instrument and an omitted determinant of the outcome can violate exclusion, even if the instrument has no direct effect on the outcome.
- Does zero covariance between \(Z\) and \(U\) generally imply that they are independent? Briefly explain.