Instrument Conditions With Exogenous Controls

Why Included Controls Change the Instrument Conditions

An instrument is not assessed in isolation from the other regressors in a structural equation. If a model includes exogenous controls, the instrument must provide relevant variation in the endogenous regressor apart from the variation accounted for by those controls. Likewise, exclusion concerns the instrument’s relationship with the structural error once the controls are held fixed. Writing these conditions precisely helps separate conditional assumptions about the data-generating process from the unconditional moments used to represent them.

Let \(Y\) be an outcome, \(X\) a \(K\)-dimensional vector of endogenous regressors, \(W\) a \(q\)-dimensional vector of included exogenous controls, and \(Z\) an \(L\)-dimensional vector of excluded instruments. Consider the structural model

where \(\beta\) and \(\gamma\) are coefficient vectors and \(U\) is the structural error. Assume the variables have finite second moments. The controls must have enough variation to identify their linear coefficients; in particular, assume \(E[WW']\) is nonsingular. An intercept, when required, is included as a component of \(W\).

Conditional Restrictions and Population Moments

A common conditional-mean formulation of validity is \(E[U\mid Z,W]=E[U\mid W]\), together with exogeneity of the controls, \(E[U\mid W]=0\). Under these restrictions, \(E[ZU\mid W]=0\), and averaging over \(W\) gives \(E[ZU]=0\). A weaker formulation used directly in linear IV models is the unconditional moment restriction \(E[ZU]=0\), alongside \(E[WU]=0\). These statements should not be confused: the conditional restrictions imply the unconditional moments, but unconditional moments alone need not imply the conditional restrictions.

For a linear model, relevance can be stated using the population partial covariance between \(Z\) and \(X\). It measures the association remaining after each variable’s linear relationship with \(W\) is removed. With multiple endogenous regressors, the corresponding rank requirement is that this partial covariance matrix have rank \(K\). In the scalar case, relevance requires a nonzero partial covariance.

Removing the Controls’ Linear Contribution

The population linear projection gives a precise way to describe what “after accounting for the controls” means. For any vector \(V\) with finite second moments, define its linear projection coefficient on \(W\) as \(\Pi_V=E[WW']^{-1}E[WV']\). The residual from this projection is \(\tilde V=V-\Pi_V'W\). It is orthogonal to the controls: \(E[W\tilde V']=0\). This is a linear residualization, not necessarily subtraction of the exact conditional expectation \(E[V\mid W]\).

Apply this definition to \(Z\), \(X\), and \(U\). The population IV moment after removing the controls is \(E[\tilde Z\tilde U]=0\). To see how this follows from the original moments, expand the residualized expression:

By construction, \(E[\tilde ZW']=0\), so the second term vanishes. Also, \(\tilde Z=Z-E[ZW']E[WW']^{-1}W\). Therefore,

Consequently, if \(E[ZU]=0\) and \(E[WU]=0\), then \(E[\tilde Z\tilde U]=0\). In fact, under \(E[WU]=0\), the linear projection coefficient of \(U\) on \(W\) is zero, so \(\tilde U=U\). Residualizing the structural equation also gives

This equality follows because linear projection is linear and the term \(W'\gamma\) lies in the linear span of the controls. The corresponding relevance matrix is

Thus, raw covariance \(E[ZX']\) is not by itself the relevant quantity when controls are included. The adjustment subtracts the component of the instrument–regressor association attributable to their separate linear relationships with \(W\).

Conditional Validity Is Stronger Than an Aggregate Moment

Residualized unconditional moments are useful, but they should not be mistaken for proof that exclusion holds in every control-defined subpopulation. The distinction can be seen in a simple hypothetical construction. Let \(W\) and \(Z\) each take values \(-1\) and \(1\), independently with equal probability. Set \(X=Z\) and \(U=WZ\). Here the instrument predicts the regressor even after controlling linearly for \(W\), but the conditional relationship between \(Z\) and \(U\) changes with \(W\).

What the Conditions Do—and Do Not—Say

The control-adjusted conditions have a clear interpretation. Exogeneity of \(W\) means the included controls are orthogonal to the structural error under the moment formulation. Exclusion means the excluded instruments are orthogonal to that error, either through a stated conditional restriction or through the particular unconditional moments assumed by the model. Relevance means the instruments explain some of the endogenous regressors’ variation that is not explained linearly by the controls.

These are assumptions about the population data-generating process, not conclusions established by computing a covariance from a sample. Nor does a valid instrument automatically imply that every coefficient has a causal interpretation under every model. The structural equation, the instrument conditions, and the interpretation of the coefficients must be considered together.

Common Mistakes

• Using raw covariance as conditional relevance. A nonzero \(E[ZX']\) can be entirely due to common linear relationships with \(W\). Use the partial covariance \(E[ZX']-E[ZW']E[WW']^{-1}E[WX']\).
• Treating an unconditional moment as a conditional guarantee. \(E[ZU]=0\) permits positive and negative conditional moments to cancel. It does not, on its own, establish \(E[ZU\mid W]=0\) or the conditional mean exclusion restriction.
• Confusing controls with excluded instruments. \(W\) appears directly in the structural equation; \(Z\) is excluded from it. The instrument’s role is to provide predictive variation in \(X\) beyond the controls.
• Calling linear residualization exact conditioning. Subtracting a linear projection removes the linear contribution of \(W\). Unless further assumptions hold, it need not equal subtracting \(E[V\mid W]\).