From Instrument Assumptions to Population Moments
An instrumental variable is useful not merely because it is associated with a regressor. It must also supply a restriction on the structural error. The central step in this tutorial is to express that restriction as an expectation equal to zero, then use the structural equation to turn it into a population equation involving the unknown coefficient.
This is a population argument: it concerns expectations under the data-generating distribution, not a calculation from a particular sample. It explains what information a valid instrument contributes and what additional condition is needed for that information to pin down a coefficient.
Centering means that the variables are expressed as deviations from their population means. In particular, \(E[Z]=E[U]=0\), so \(E[ZU]=0\) is equivalent to zero covariance between \(Z\) and \(U\). Likewise, \(E[ZX]\) is the population covariance between the instrument and regressor. The centered formulation avoids an intercept term; it does not assert that an uncentered structural equation must have zero intercept.
The orthogonality condition is a property of the instrument and the structural error, not a claim that the instrument is unrelated to the outcome. Indeed, a relevant instrument can be associated with the outcome through its association with \(X\). Nor does zero covariance generally assert independence: the moment condition requires zero product expectation, while independence is a stronger condition that can imply it when the relevant expectations exist.
Deriving the Instrument Moment Equation
Substitute the structural equation into the orthogonality condition. This converts a restriction involving an unobserved error into a restriction involving the outcome, regressor, instrument, and coefficient.
Validity supplies \(E[ZU]=0\), with \(U\) the error in the structural equation.
From \(Y=\beta X+U\), the structural error is \(U=Y-\beta X\).
Thus \(E[Z(Y-\beta X)]=0\), or, by linearity of expectation, \(E[ZY]-\beta E[ZX]=0\).
If \(E[ZX]\ne 0\), the equation has a unique solution for \(\beta\). If \(E[ZX]=0\), this moment has no coefficient-dependent term.
More formally, for a candidate value \(b\), define the population moment function \(m(b)=E[Z(Y-bX)]\). Under the structural model and instrument orthogonality, \(m(\beta)=0\). Also, \(m(b)=E[ZY]-bE[ZX]\). If \(E[ZX]\ne 0\), this function changes with \(b\), and at most one value can make it zero. The moment condition then identifies the structural coefficient, provided the model and orthogonality restriction are correct.
Worked Population Examples
Worked Example: Verifying a Population Moment
Suppose a hypothetical centered population satisfies \(Y=3X+U\), \(E[ZU]=0\), and \(E[ZX]=0.8\). Calculate the implied outcome-instrument cross-moment and check whether the moment identifies the structural coefficient.
Multiply the structural equation by \(Z\) and take expectations: \(E[ZY]=E[Z(3X+U)]=3E[ZX]+E[ZU]\). Substituting the stated population moments gives \(E[ZY]=3(0.8)+0=2.4\).
The units matter. \(E[ZY]=2.4\) has units of outcome times instrument; \(E[ZX]=0.8\) has units of regressor times instrument. The coefficient \(3\) has units of outcome per regressor, so \(3E[ZX]\) has outcome-instrument units, matching \(E[ZY]\). Since \(E[ZX]\ne 0\), the moment depends on the coefficient and uniquely identifies it in this model. In particular, the equation \(2.4-b(0.8)=0\) is satisfied by \(b=3\).
Worked Example: Computing Orthogonality From a Discrete Population
Consider a hypothetical population with four equally likely states. Let \(Z\) take values \(-1\) and \(1\), let \(V\) also take values \(-1\) and \(1\), and let the four combinations occur with probability \(1/4\). Define \(X=2Z+V\), \(U=V\), and \(Y=1.5X+U\). Calculate \(E[ZU]\), \(E[ZX]\), and \(E[ZY]\), then check the moment equation.
| \(Z\) | \(V\) | \(X=2Z+V\) | \(U=V\) | \(Y=1.5X+U\) |
|---|---|---|---|---|
| -1 | -1 | -3 | -1 | -5.5 |
| -1 | 1 | -1 | 1 | -0.5 |
| 1 | -1 | 1 | -1 | 0.5 |
| 1 | 1 | 3 | 1 | 5.5 |
The four products \(ZU\) are \(1,-1,-1,1\), so \(E[ZU]=(1-1-1+1)/4=0\). The products \(ZX\) are \(3,1,1,3\), giving \(E[ZX]=(3+1+1+3)/4=2\). The products \(ZY\) are \(5.5,0.5,0.5,5.5\), giving \(E[ZY]=(5.5+0.5+0.5+5.5)/4=3\).
Thus the instrument is orthogonal to the error in this constructed population and is relevant because \(E[ZX]=2\ne0\). The implied moment is \(3-1.5(2)=0\), as required. The cross-moment \(E[ZY]\) has outcome-instrument units, while \(E[ZX]\) has regressor-instrument units; multiplying the latter by the coefficient, measured in outcome per regressor, gives the former units.
The variables are centered: the four values of \(Z\) sum to zero, and so do those of \(X\), \(U\), and \(Y\). This makes the zero product expectation a zero covariance condition in this example.
Worked Example: Relevance Without Orthogonality
Now consider another hypothetical population with four equally likely combinations of \(Z\in\{-1,1\}\) and \(V\in\{-1,1\}\). Set \(X=Z+V\), \(U=0.5Z\), and \(Y=2X+U\). Determine whether the instrument moment correctly identifies the structural coefficient.
| \(Z\) | \(V\) | \(X\) | \(U\) | \(Y\) |
|---|---|---|---|---|
| -1 | -1 | -2 | -0.5 | -4.5 |
| -1 | 1 | 0 | -0.5 | -0.5 |
| 1 | -1 | 0 | 0.5 | 0.5 |
| 1 | 1 | 2 | 0.5 | 4.5 |
The products \(ZX\) are \(2,0,0,2\), so \(E[ZX]=(2+0+0+2)/4=1\); the instrument is relevant. But each product \(ZU\) equals \(0.5\), so \(E[ZU]=(0.5+0.5+0.5+0.5)/4=0.5\), not zero. The validity condition fails.
The outcome cross-moment is \(E[ZY]=(4.5+0.5+0.5+4.5)/4=2.5\). If one nevertheless imposes the zero moment equation, it becomes \(2.5-b(1)=0\), whose solution is \(b=2.5\), not the structural coefficient \(2\). The discrepancy is exactly the instrument-error moment: from the model, \(E[ZY]=2E[ZX]+E[ZU]=2(1)+0.5=2.5\). Relevance alone cannot repair a failure of orthogonality.
Here \(Z\) and \(U\) are centered, but their covariance is positive. The computation therefore illustrates why a nonzero association between an instrument and a regressor is not enough: the instrument must also satisfy the error restriction for the structural moment equation to be valid.
What the Moment Condition Does—and Does Not—Establish
A population moment condition is a restriction on the distribution implied by a model and its assumptions. It is not itself evidence that those assumptions hold. In an application, the claim \(E[ZU]=0\) depends on substantive knowledge about how the instrument is generated and how it could be related to unobserved determinants of the outcome. Since \(U\) is generally unobserved, the population condition cannot be established merely by calculating a sample average of \(ZU\).
The distinction between validity and relevance is especially important. Orthogonality gives \(E[ZY]=\beta E[ZX]\); relevance says that the right-hand side changes when \(\beta\) changes. If \(E[ZX]=0\), then the moment equation reduces to \(E[ZY]=0\) under the model, and does not distinguish among coefficient values. If \(E[ZX]\ne0\) but \(E[ZU]\ne0\), the equation based on a zero moment is not the structural equation. These are different failures and require different reasoning.
Units provide a useful consistency check. The product expectation \(E[ZY]\) is measured in outcome-times-instrument units, and \(E[ZX]\) in regressor-times-instrument units. The coefficient multiplying \(E[ZX]\) must therefore have outcome-per-regressor units. Writing the moments with these units prevents an invalid interpretation of a cross-moment or a coefficient.
Common Mistakes
- Using relevance as if it implied validity. \(E[ZX]\ne0\) says the instrument is associated with the regressor. It says nothing by itself about \(E[ZU]\). Both restrictions matter for the argument.
- Claiming that orthogonality proves independence. The condition \(E[ZU]=0\) is a zero-moment restriction. It does not generally imply that \(Z\) and \(U\) are independent.
- Calling a moment equation an empirical fact about the sample. The equation \(E[Z(Y-\beta X)]=0\) is a population implication under assumptions. A sample calculation is not, by itself, proof of those assumptions.
- Ignoring the units of cross-moments. \(E[ZY]\) has outcome-instrument units, whereas \(E[ZX]\) has regressor-instrument units. The coefficient bridges those units.
- Concluding identification from a zero equation alone. The moment equation identifies the coefficient only when its coefficient on \(\beta\), \(E[ZX]\), is nonzero.
Check Your Understanding
Use the population-moment argument to answer the following questions.
- Starting from \(Y=\beta X+U\) and \(E[ZU]=0\), derive the moment equation in terms of \(E[ZY]\) and \(E[ZX]\).
- If \(E[ZX]=0\), what does the moment equation imply, and why does it fail to identify \(\beta\)?
- Explain why \(E[ZU]=0\) is not generally the same as independence of \(Z\) and \(U\).
- Suppose \(E[ZY]=6\) outcome-instrument units and \(E[ZX]=1.5\) regressor-instrument units. What coefficient value solves the moment equation, and what are its units?
- An instrument is relevant but has \(E[ZU]\ne0\). Explain why the zero-moment equation need not recover the structural coefficient.