Why Conditional Error Decompositions Matter
When a regressor is endogenous, its conditional association with the structural error prevents the outcome equation from being interpreted as though the regressor were conditionally exogenous. A control function addresses this dependence by representing the part of the error predictable from information about how the endogenous regressor was generated. The representation is useful only if that information captures the relevant conditional error mean.
Let \(Y\) be an outcome, \(V\) an endogenous regressor, \(X\) observed exogenous covariates, and \(Z\) observed first-stage variables such as instruments. Write the structural equation as \(Y=m(V,X)+U\), where \(m\) is the structural outcome component and \(U\) is its error. Suppose first-stage information also determines an observable variable \(R=r(V,Z,X)\). In a correctly specified first stage, \(R\) might be a disturbance or another summary of the information that generates \(V\). The key question is whether conditioning on \(R\) captures the relevant conditional mean of \(U\).
The phrase “almost surely” means that the equality holds except possibly on events with probability zero. The condition is about conditional means, not necessarily full statistical independence. It says that once \(R\) and \(X\) are known, additionally knowing \(V\) and \(Z\) does not change the conditional expectation of the structural error. Since \(R\) is calculated from observable first-stage information, the function \(c(R,X)\) can then be used to represent that conditional error mean.
Assume the relevant conditional expectations exist. The first-stage relation must make \(R\) observable from \((V,Z,X)\), and the structural outcome equation must exclude \(Z\) directly from \(m(V,X)\). The conditional-mean sufficiency condition is a separate requirement: observability alone does not guarantee that \(R\) captures the conditional error mean. Instrument validity, first-stage relevance, and other conditions needed for causal identification are also separate; a control-function decomposition by itself does not establish them.
Deriving the Decomposition
Define \(c(R,X)=E[U\mid R,X]\), then let \(\eta=U-c(R,X)\). By the definition of a conditional expectation, \(E[\eta\mid R,X]=0\). That fact alone is not enough for the outcome adjustment: the remainder must also have conditional mean zero given the variables relevant to the outcome comparison.
The third line uses conditional-mean sufficiency and the fact that \(R\) is determined by \((V,Z,X)\). Thus the structural equation can be written as
This representation explains the adjustment: \(c(R,X)\) accounts for the predictable part of the structural error, while \(\eta\) has zero conditional mean given the endogenous regressor, the first-stage variables, and \(X\). It does not imply that \(E[U\mid V,X]=c(R,X)\). By iterated expectations, the precise relationship is
In general, conditioning on \(Z\) as well as \(V\) is more informative than conditioning on \(V\) alone. The conditional mean given \(V,X\) averages the control function over the conditional distribution of the first-stage information given \(V,X\). It need not equal a particular value of \(c(R,X)\).
Worked Example: A Valid First-Stage Control
Consider a hypothetical population with \(X\) fixed. Let \(R\) take the values 2 and 3 with equal probability, and let \(Z\) independently take the values 0 and 1 with equal probability. Set \(V=R+Z\), and let \(U=2R+\varepsilon\), where \(E[\varepsilon\mid R,Z]=0\). The observable first-stage information recovers \(R=V-Z\).
Construct the control function. Given \(R\), the conditional error mean is \(c(R)=E[U\mid R]=2R+E[\varepsilon\mid R]=2R\). Moreover, \(E[U\mid V,Z]=2(V-Z)\), because \(R=V-Z\) and the conditional mean of \(\varepsilon\) is zero. This equals \(c(R)\), so the conditional-mean sufficiency condition holds in this example.
Evaluate it at an observation. If \(V=4\) and \(Z=1\), then \(R=V-Z=3\). The control is \(c(3)=2(3)=6\), and the conditional mean of \(U\) given \(V=4,Z=1\) is also \(2(4-1)=6\). The remainder \(U-c(R)\) therefore has conditional mean zero at those first-stage values.
Check the mean given \(V\) alone. When \(V=3\), there are two possible \((R,Z)\) pairs: \((2,1)\) and \((3,0)\). Each has probability \(1/4\), so conditional on \(V=3\) the two pairs have equal probability. Their conditional error means are \(2(2)=4\) and \(2(3)=6\). Thus \(E[U\mid V=3]=(4+6)/2=5\). Equivalently, \(E[c(R)\mid V=3]=(4+6)/2=5\), as the iterated-expectations identity requires.
What the Sufficiency Condition Does—and Does Not—Say
A proposed first-stage summary must capture the conditional error mean, not merely be correlated with the endogenous regressor or be easy to calculate. The condition \(E[U\mid V,Z,X]=E[U\mid R,X]\) is the substantive requirement. If it fails, subtracting \(E[U\mid R,X]\) may leave a nonzero conditional mean in the remainder. In that case the proposed adjustment does not deliver the stated decomposition.
The condition is also distinct from the claim that \(Z\) is a valid instrument. In a structural equation that excludes \(Z\), the instrument may still convey information about \(U\) after conditioning on \(R,X\). If so, \(R\) is not sufficient for the conditional error mean. Conversely, a successful conditional error decomposition does not, by itself, prove that the instrument is relevant or that a causal parameter is identified.
Worked Example: A First-Stage Summary That Is Not Sufficient
Consider a hypothetical setting with \(X\) fixed. Let \(R\) take the values 2 and 3, let \(Z\) independently take the values 0 and 1 with equal probability, and define \(V=R+Z\). Suppose the structural error is \(U=2R+Z\), and the proposed control is based only on \(R\).
Calculate the proposed control. Conditional on \(R=3\), \(Z\) is still equally likely to be 0 or 1. Therefore \(c(3)=E[U\mid R=3]=2(3)+E[Z\mid R=3]=6+0.5=6.5\).
Check the required conditional mean. If \(R=3\) and \(Z=1\), then \(V=3+1=4\), and \(U=2(3)+1=7\). Thus \(E[U\mid V=4,Z=1]=7\), whereas \(c(3)=6.5\). The difference is \(7-6.5=0.5\), so the conditional-mean sufficiency condition fails. In fact, \(E[U\mid V,Z]=2(V-Z)+Z\), which is not generally \(2R+0.5\).
The proposed adjustment leaves conditional error behind. Here \(Z\) appears directly in \(U\), so it contributes information about the structural error beyond \(R\). Merely recovering \(R=V-Z\) from first-stage variables does not make \(R\) sufficient.
Worked Example: Conditioning on Covariates Correctly
Let \(X\) be fixed at a specified value. Suppose \(Z\) is Bernoulli with \(P(Z=1)=0.5\), \(Z\) is independent of \(V\), and \(\varepsilon\) satisfies \(E[\varepsilon\mid V,Z,X]=0\). Consider the hypothetical conditional error model \(U=2V+Z+\varepsilon\).
Condition on \(V,X\). Since \(X\) is fixed and \(Z\) is independent of \(V\), \(E[Z\mid V,X]=0.5\). Therefore \(E[U\mid V,X]=2V+0.5\). At \(V=4\), this conditional mean is \(2(4)+0.5=8.5\).
Condition on \(V,Z,X\). The conditional mean of \(\varepsilon\) is zero, so \(E[U\mid V,Z,X]=2V+Z\). At \(V=4,Z=1\), the conditional mean is \(2(4)+1=9\); at \(V=4,Z=0\), it is \(8\). Averaging these equally likely values gives \((9+8)/2=8.5\), consistent with \(E[U\mid V,X]\).
This example depends on \(X\) being fixed (or, more generally, on the stated condition \(E[Z\mid V,X]=0.5\)). Independence of \(Z\) and \(V\) alone does not establish this conditional mean when \(X\) is random and associated with \(Z\). Also, the two conditional means just calculated differ: knowing \(Z\) changes the conditional expectation of \(U\). Thus this calculation does not establish conditional-mean sufficiency for a control that omits \(Z\); it illustrates why the conditioning variables must be stated precisely.
A Practical Validity Check
Before using a control function, distinguish three questions. First, is the proposed first-stage information observable from the variables in the data? Second, does it satisfy the conditional-mean sufficiency condition for the structural error? Third, are the additional assumptions needed for the intended structural or causal interpretation satisfied? A positive answer to the first question does not imply a positive answer to the second, and neither answer alone settles the third.
A useful way to verify the algebra is to define the adjusted remainder explicitly as \(\eta=U-c(R,X)\), then calculate \(E[\eta\mid V,Z,X]\). If the result is not zero, the proposed control has not removed the conditional error mean under the stated model. This check also prevents a common mistake: using \(E[U\mid R,X]=c(R,X)\) to conclude, without further assumptions, that \(E[U\mid V,Z,X]=c(R,X)\). Those are different conditional expectations; their equality is exactly what sufficiency must justify.
Check Your Understanding
Answer each question by identifying the conditioning information and stating any assumption you use.
- Given \(Y=m(V,X)+U\), \(c(R,X)=E[U\mid R,X]\), and \(E[U\mid V,Z,X]=E[U\mid R,X]\), define the remainder and show its conditional mean given \(V,Z,X\).
- Why does \(E[\eta\mid R,X]=0\) alone not establish \(E[\eta\mid V,Z,X]=0\)?
- If \(E[U\mid V,Z,X]=c(R,X)\), what does iterated expectations imply about \(E[U\mid V,X]\)?
- In the example with \(U=2R+Z\), calculate the difference between \(E[U\mid V=4,Z=1]\) and the proposed control \(E[U\mid R=3]\). What does that difference indicate?
- Why is independence of \(Z\) and \(V\) not by itself enough to conclude \(E[Z\mid V,X]=E[Z]\) when \(X\) is random?