Partialling Out Exogenous Regressors

Why Partial Out Included Controls?

An IV model often contains both endogenous regressors of interest and exogenous controls that should remain in the structural equation. The controls can account for systematic variation in the outcome, the endogenous regressors, and the instruments. The Frisch–Waugh–Lovell principle gives a precise way to remove that variation first: regress each relevant variable on the included controls, then work with the residuals. For IV and 2SLS, the instruments must be handled consistently with this residualization.

The benefit is computational and conceptual. It isolates the part of the outcome and endogenous regressors not linearly explained by the controls, and uses instrument variation not explained by those same controls. The coefficient on an endogenous regressor is unchanged by this re-expression, provided the model and rank conditions are maintained.

Model and Residualization

Definition: Write the structural equation as $y=W\gamma+X\beta+u$, where $y$ is the outcome vector, $W$ is the matrix of included exogenous controls (including an intercept when appropriate), $X$ is the matrix of endogenous regressors, and $Z$ is the matrix of excluded instruments. Let $W$ have full column rank. Define $P_W=W(W'W)^{-1}W'$ and $M_W=I-P_W$. For any vector or matrix $A$, its residualized version is $\tilde A=M_WA$. Thus $\tilde y=M_Wy$, $\tilde X=M_WX$, and $\tilde Z=M_WZ$.

The columns of $M_W A$ are the residuals from regressing each column of $A$ on all columns of $W$, by ordinary least squares. In particular, residualization removes both the intercept and the included controls if they are in $W$. The residuals are orthogonal to the controls: $W'M_WA=0$.

The structural assumptions give the residualization its IV interpretation. The controls are exogenous, and the excluded instruments satisfy the relevant orthogonality condition with the structural error, while providing variation related to $X$ after conditioning on $W$. In a sample, the algebra below requires the stated matrix ranks; those algebraic identities do not by themselves establish that the instruments are valid.

The Partialled-Out 2SLS Result

Theorem: Let the full instrument matrix be $A=[W,Z]$, and suppose the required inverses and identification ranks exist. The 2SLS coefficient on $X$ in the regression of $y$ on $[W,X]$, using $[W,Z]$ as instruments, is $$ \hat\beta_{\mathrm{2SLS}} =(\tilde X'P_{\tilde Z}\tilde X)^{-1} \tilde X'P_{\tilde Z}\tilde y, \qquad P_{\tilde Z}=\tilde Z(\tilde Z'\tilde Z)^{-1}\tilde Z'. $$

Here $P_{\tilde Z}$ projects onto the column space of the residualized excluded instruments. If the columns of $\tilde Z$ are redundant, the same projection is defined using any full-rank basis for that column space. For the displayed inverse involving $\tilde X$, the projected regressors must have sufficient rank to identify $\beta$.

A short derivation shows why all three objects are residualized. The full instrument space is the sum of the space spanned by $W$ and the space spanned by $M_WZ=\tilde Z$. These two spaces are orthogonal, since $W'\tilde Z=0$. Therefore

P_{[W,Z]}=P_W+P_{\tilde Z}.

2SLS minimizes the squared length of the outcome residual after projection onto the instrument space. For a proposed value of $\beta$, the fitted contribution of $W$ can first be removed. The remaining outcome residual is $M_W(y-X\beta)=\tilde y-\tilde X\beta$, which is orthogonal to $W$. Its projected sum of squares is therefore

(\tilde y-\tilde X\beta)'P_{\tilde Z}(\tilde y-\tilde X\beta).

Differentiating this quadratic with respect to $\beta$, setting the derivative to zero, and solving gives

-2\tilde X'P_{\tilde Z}\tilde y +2\tilde X'P_{\tilde Z}\tilde X\beta=0, \qquad \hat\beta_{\mathrm{2SLS}} =(\tilde X'P_{\tilde Z}\tilde X)^{-1} \tilde X'P_{\tilde Z}\tilde y.

This is the partialling-out result: the controls’ fitted components can be removed before the IV projection, as long as the instruments are residualized on those controls too. In a just-identified model, the same idea can be expressed through the sample IV moment equations; with more instruments than endogenous regressors, the projection formula specifies the 2SLS estimate.

Worked Examples

Worked Example: A Just-Identified IV Estimate With One Control

Suppose a hypothetical sample has one endogenous regressor $x$, one excluded instrument $z$, and an intercept and control $c$. The four observations are:

Observation	$c$	$y$	$x$	$z$
1	-3	9	3	9
2	-1	-1	-1	5
3	1	15	7	3
4	3	17	7	3

Let $W$ contain the intercept and $c$. To make the residualization transparent, define $a=(1,-1,-1,1)'$ and $e=(1,-3,3,-1)'$. Both vectors sum to zero and are orthogonal to $c=(-3,-1,1,3)'$, so they are orthogonal to $W$. The data can be written as $y=10\mathbf{1}+2c+3a+2e$, $x=4\mathbf{1}+c+a+e$, and $z=5\mathbf{1}-c+a$. Thus $\tilde y=3a+2e$, $\tilde x=a+e$, and $\tilde z=a$.

With one instrument and one endogenous regressor, the just-identified sample IV estimate is the ratio of the instrument–outcome cross-product to the instrument–regressor cross-product, after accounting for the controls:

\hat\beta_{\mathrm{IV}} =\frac{\tilde z'\tilde y}{\tilde z'\tilde x} =\frac{a'(3a+2e)}{a'(a+e)} =\frac{3(4)+2(0)}{4+0} =\frac{12}{4} =3.

Here $a'a=4$ and $a'e=0$. The residualized instrument has a nonzero sample cross-product with the residualized regressor, so the just-identified sample denominator is not zero. The calculation gives an estimate of 3 outcome units per unit of $x$, conditional on the included intercept and control. It does not, by itself, verify the instrument’s exogeneity.

Worked Example: 2SLS With Two Residualized Instruments

Use the same control space, with residual vectors $a$ and $e$ as above. Consider a hypothetical sample whose residualized variables are $\tilde x=2a+e$, $\tilde y=5a+3e$, and whose two residualized instruments are the columns of $\tilde Z=[a,e]$. The 2SLS projection onto these instruments is

P_{\tilde Z}=\frac{aa'}{4}+\frac{ee'}{20},

because $a'a=4$, $e'e=20$, and $a'e=0$. In this example the two instrument columns span the entire space orthogonal to the intercept and $c$, so $P_{\tilde Z}\tilde y=\tilde y$ and $P_{\tilde Z}\tilde x=\tilde x$. The scalar 2SLS estimate is

\hat\beta_{\mathrm{2SLS}} =\frac{\tilde x'P_{\tilde Z}\tilde y}{\tilde x'P_{\tilde Z}\tilde x} =\frac{(2a+e)'(5a+3e)}{(2a+e)'(2a+e)} =\frac{2(5)(4)+1(3)(20)}{2^2(4)+1^2(20)} =\frac{100}{36} \approx 2.778.

The numerator is 100 outcome-regressor cross-product units and the denominator is 36 regressor-squared units, giving an estimate of about 2.778 outcome units per unit of $x$. In samples where the residualized instruments do not span the full residual space, the projection retains only the components in their column space; the same 2SLS formula applies.

Worked Example: Raw Instruments With Residualized Variables

A common source of confusion is whether the instruments must be residualized in every cross-product. In the first example, the raw instrument is $z=5\mathbf{1}-c+a$, while $\tilde y=3a+2e$ and $\tilde x=a+e$. Since the residualized variables are orthogonal to $W$, the control components of $z$ contribute zero:

z'\tilde y=(5\mathbf{1}-c+a)'(3a+2e)=12 =\tilde z'\tilde y, \qquad z'\tilde x=(5\mathbf{1}-c+a)'(a+e)=4 =\tilde z'\tilde x.

So raw $z$ paired with residualized $y$ and $x$ gives the same IV moment cross-products as residualized $z$. More generally, $M_W$ is symmetric and idempotent, which gives $\tilde Z'\tilde y=Z'M_Wy=Z'\tilde y$, and likewise $\tilde Z'\tilde X=Z'\tilde X$. Residualizing $Z$ is still the appropriate way to form the residualized-instrument projection $P_{\tilde Z}$.

What Is and Is Not Equivalent

The equivalence of cross-products does not mean that all choices of raw or residualized variables can be mixed freely. For instance, in the first example, using raw $z$ with raw $y$ and raw $x$ gives

z'y=9(9)+5(-1)+3(15)+3(17)=172, \qquad z'x=9(3)+5(-1)+3(7)+3(7)=64.

The ratio $172/64=2.688$ is not the control-adjusted IV estimate of 3. Those raw cross-products include variation associated with the intercept and $c$, rather than isolating the IV relationship conditional on them. The correct choices are to use the full 2SLS system with controls and instruments $[W,Z]$, or to partial out $W$ consistently and use $\tilde y,\tilde X,\tilde Z$ in the projection. For the moment cross-products alone, raw $Z$ paired with residualized variables is also equivalent, as the preceding example demonstrates.

This distinction matters in practice because residualization is not a new instrument-validity assumption and does not repair an invalid instrument. It is an algebraic re-expression of the specified model. The controls remain part of the structural equation, while the IV relationship is identified from variation left after accounting for them.

Common Mistakes

Residualizing only the outcome and endogenous regressors, then projecting on raw instruments. The projection for partialled-out 2SLS is onto $\tilde Z=M_WZ$, not generally onto the raw $Z$ space. Use $P_{\tilde Z}$ in the residualized formula.
Claiming raw instruments cannot be used in any cross-product after partialling. That is too strong. Because $W'\tilde y=W'\tilde X=0$, $Z'\tilde y=\tilde Z'\tilde y$ and $Z'\tilde X=\tilde Z'\tilde X$. The equivalence applies to these moment cross-products, not to replacing the residualized projection with $P_Z$.
Using raw $y$ and $X$ in a ratio or moment equation while ignoring controls. This generally includes variation attributable to $W$ and need not equal the control-adjusted estimate. Either include the controls in the full 2SLS setup or residualize consistently.
Assuming residualization establishes instrument validity. It removes linear dependence on the included controls; it does not show that the remaining instrument variation is uncorrelated with the structural error or satisfies the exclusion restriction.

Key takeaway: For control-adjusted 2SLS, residualize $y$, $X$, and $Z$ on the same included controls, then project the residualized regressors and outcome onto the residualized instruments. Raw instruments paired with residualized variables yield equivalent IV moment cross-products, but raw variables used without accounting for controls generally do not.

Check Your Understanding

Use the definitions and calculations above to answer these questions.

What does $M_WA$ represent, and what orthogonality property does it satisfy?
Why does the residualized 2SLS projection use $P_{\tilde Z}$ rather than $P_Z$?
State the condition that makes $Z'\tilde y$ equal to $\tilde Z'\tilde y$.
In the first worked example, what are the numerator and denominator of the control-adjusted just-identified IV estimate?
Does partialling out controls establish that the excluded instruments are exogenous? Explain briefly.

Observation	\(c\)	\(y\)	\(x\)	\(z\)
1	-3	9	3	9
2	-1	-1	-1	5
3	1	15	7	3
4	3	17	7	3