Multiple Instruments in Two-Stage Least Squares

Why Use Several Instruments Together?

When a treatment is endogenous, an instrumental-variable analysis uses variation in the treatment associated with instruments to estimate its relationship with an outcome. With several instruments, two-stage least squares (2SLS) does not generally calculate a separate IV estimate for each instrument and average them. It first projects treatment onto the space spanned by all the instrument columns, then uses that projected treatment in the second stage.

This distinction matters both computationally and conceptually. The joint projection incorporates the instruments’ scales, their associations with treatment, and any overlap between instrument columns. The resulting coefficient is a particular combination of the information in the instruments, not an automatic average of separate ratios.

Definition: With one endogenous treatment, let $D$ be the $n \times 1$ vector of observed treatment values, $Y$ the $n \times 1$ outcome vector, and $Z$ the $n \times K$ matrix whose columns are the $K$ instruments. If $Z$ has full column rank, the projection matrix onto the column space of $Z$ is $$ P_Z = Z(Z'Z)^{-1}Z'. $$ The projected treatment is $\widehat{D}=P_ZD$. In the no-covariate, centered-data setting used here, the 2SLS coefficient is $$ \widehat{\beta}_{2SLS} = (D'P_ZD)^{-1}D'P_ZY = \frac{D'P_ZY}{D'P_ZD}, $$ provided $D'P_ZD>0$.

The columns of $Z$ need not be mutually orthogonal. The matrix $P_Z$ projects onto their joint span; it does not, in general, equal the sum of separate one-column projection matrices. The full-rank condition ensures that $(Z'Z)^{-1}$ exists. The denominator condition ensures that the instruments’ joint projection contains some variation in treatment.

For clarity, calculations below use centered variables and no additional covariates. Centering removes the intercept from the displayed algebra; equivalently, one can include an intercept consistently in both stages. These examples explain the estimator’s algebra, not whether the instruments are valid. A causal interpretation still requires appropriate instrumental-variable assumptions, including relevance and the relevant independence and exclusion conditions. Multiple instruments do not establish those conditions merely by producing a coefficient.

How the Joint Projection Produces 2SLS

The first stage projects the observed treatment vector onto the columns of $Z$. Thus $\widehat{D}$ is the fitted treatment vector from a regression of $D$ on all the instruments. The second stage regresses $Y$ on that fitted treatment. For one treatment and centered data, the coefficient from that regression is $\widehat{D}'Y/(\widehat{D}'\widehat{D})$.

The projection matrix is symmetric and idempotent: $P_Z'=P_Z$ and $P_ZP_Z=P_Z$. Since $\widehat{D}=P_ZD$, these properties show why the usual matrix formula and the projected-treatment regression give the same coefficient.

\frac{\widehat{D}'Y}{\widehat{D}'\widehat{D}} = \frac{(P_ZD)'Y}{(P_ZD)'(P_ZD)} = \frac{D'P_ZY}{D'P_ZP_ZD} = \frac{D'P_ZY}{D'P_ZD}.

This is the key justification for calculating 2SLS with the projection matrix. In practice, one can compute $\widehat{D}$ and run the second-stage regression, or calculate the matrix expression directly. Both methods implement the same coefficient in this setting.

Worked Example: Constructing a Projection Matrix

Suppose four hypothetical observations have two centered instrument columns. Write the columns as $z_1=(1,-1,0,0)'$ and $z_2=(1,1,-1,-1)'$, so that $$ Z = \begin{bmatrix} 1&1\\ -1&1\\ 0&-1\\ 0&-1 \end{bmatrix}. $$ The columns are orthogonal: $z_1'z_2=0$. Their squared lengths are $z_1'z_1=2$ and $z_2'z_2=4$. Therefore $$ Z'Z = \begin{bmatrix} 2&0\\ 0&4 \end{bmatrix}, \qquad (Z'Z)^{-1} = \begin{bmatrix} 1/2&0\\ 0&1/4 \end{bmatrix}. $$

Substituting these matrices into the projection formula gives $$ P_Z = \begin{bmatrix} 3/4&-1/4&-1/4&-1/4\\ -1/4&3/4&-1/4&-1/4\\ -1/4&-1/4&1/4&1/4\\ -1/4&-1/4&1/4&1/4 \end{bmatrix}. $$ For treatment values $D=(3,-1,0,-2)'$, the projected treatment is $$ \widehat{D}=P_ZD=(3,-1,-1,-1)'. $$ The projection keeps the component of treatment that lies in the span of the two instrument columns and removes the part outside that span. In this case, $D-\widehat{D}=(0,0,1,-1)'$.

A Joint Estimate Is Not Generally an Average of Ratios

For a single instrument $z_j$, the centered, just-identified IV ratio is $z_j'Y/(z_j'D)$, when $z_j'D\ne 0$. With multiple instruments, these ratios can differ. The 2SLS coefficient instead uses the projection onto their joint span. Even when instruments are orthogonal, the joint coefficient weights their ratios according to their contributions to projected treatment; it does not assign each instrument equal weight.

Worked Example: Comparing 2SLS With Separate IV Ratios

Continue with the instruments from the previous calculation and let the hypothetical outcome vector be $Y=(13,5,-9,-9)'$. The instrument-specific ratios are computed separately. For $z_1$, $$ \frac{z_1'Y}{z_1'D} = \frac{13-5}{3-(-1)} = \frac{8}{4} = 2. $$ For $z_2$, $$ \frac{z_2'Y}{z_2'D} = \frac{13+5-(-9)-(-9)}{3+(-1)-0-(-2)} = \frac{36}{4} = 9. $$ Their unweighted average is $(2+9)/2=5.5$.

For 2SLS, the projected treatment from the preceding example is $\widehat{D}=(3,-1,-1,-1)'$. The outcome lies in the span of these instruments, so its projection is $P_ZY=Y$. Applying the joint 2SLS calculation, $$ \widehat{\beta}_{2SLS} = \frac{\widehat{D}'Y}{\widehat{D}'\widehat{D}} = \frac{3(13)+(-1)(5)+(-1)(-9)+(-1)(-9)} {3^2+(-1)^2+(-1)^2+(-1)^2} = \frac{52}{12} = 4.333. $$ The joint estimate is therefore $4.333$ outcome units per treatment unit, not the unweighted average $5.5$.

The unequal contributions can also be seen directly. Because these instrument columns are orthogonal, $D'P_ZY=16+36=52$, while $D'P_ZD=8+4=12$. Thus the combined coefficient is $52/12$. The separate ratios describe each instrument on its own; the joint calculation uses the instruments’ contribution to the projected treatment and outcome.

When Instrument Columns Are Correlated

If the columns of $Z$ are correlated, the off-diagonal entries of $Z'Z$ account for their overlap. Inverting $Z'Z$ is essential: simply adding separate projections would count shared directions more than once and would generally fail to project onto the joint span.

Worked Example: Projecting With Nonorthogonal Instruments

Consider four hypothetical observations with centered instruments $z_1=(1,-1,0,0)'$ and $z_2=(1,0,-1,0)'$. These columns are not orthogonal because $z_1'z_2=1$. The instrument matrix and its cross-product matrix are $$ Z = \begin{bmatrix} 1&1\\ -1&0\\ 0&-1\\ 0&0 \end{bmatrix}, \qquad Z'Z = \begin{bmatrix} 2&1\\ 1&2 \end{bmatrix}, \qquad (Z'Z)^{-1} = \frac{1}{3} \begin{bmatrix} 2&-1\\ -1&2 \end{bmatrix}. $$ Consequently, $$ P_Z = \begin{bmatrix} 2/3&-1/3&-1/3&0\\ -1/3&2/3&-1/3&0\\ -1/3&-1/3&2/3&0\\ 0&0&0&0 \end{bmatrix}. $$

Let $D=(4,0,-1,-3)'$ and $Y=(8,-3,-5,0)'$. First calculate the instrument cross-products: $$ Z'D = \begin{bmatrix} 4\\ 5 \end{bmatrix}, \qquad Z'Y = \begin{bmatrix} 11\\ 13 \end{bmatrix}. $$ The coefficients from projecting each vector onto $Z$ are $$ (Z'Z)^{-1}Z'D = \begin{bmatrix} 1\\ 2 \end{bmatrix}, \qquad (Z'Z)^{-1}Z'Y = \begin{bmatrix} 3\\ 5 \end{bmatrix}. $$ Thus $\widehat{D}=Z(1,2)'=(3,-1,-2,0)'$, and $P_ZY=Z(3,5)'=(8,-3,-5,0)'=Y$. The 2SLS estimate is $$ \widehat{\beta}_{2SLS} = \frac{\widehat{D}'Y}{\widehat{D}'\widehat{D}} = \frac{3(8)+(-1)(-3)+(-2)(-5)+0(0)} {3^2+(-1)^2+(-2)^2+0^2} = \frac{37}{14} = 2.643. $$ This calculation uses the inverse of the full cross-product matrix. It accounts for the instruments’ shared direction rather than treating them as independent columns.

What the Calculation Does—and Does Not—Establish

The projection formula answers a specific estimation question: given the chosen instrument columns, what coefficient results from 2SLS for one treatment and one outcome in this setup? It does not show that the instruments are relevant enough for reliable estimation, nor that they satisfy the conditions required for a causal interpretation. Those are substantive and statistical questions about how the instruments relate to treatment, outcomes, and unobserved causes.

It is also important to keep the one-treatment setup in view. Here, $D'P_ZD$ is a scalar, so the estimate is a single ratio. The instruments jointly generate the fitted treatment. Their count does not mean that each produces a separate coefficient in the final 2SLS regression.

Key takeaway: With several instruments, form $P_Z=Z(Z'Z)^{-1}Z'$, project treatment with $\widehat{D}=P_ZD$, and calculate the second-stage coefficient. The result is generally not an unweighted average of single-instrument IV ratios.

Common Mistakes

Averaging the individual IV ratios. Different instruments can have different associations with treatment and outcome. 2SLS combines their joint projected variation, not their ratios with equal weights.
Adding separate projections when instruments overlap. If instrument columns are correlated, their individual projection matrices overlap. Use $Z(Z'Z)^{-1}Z'$ to project onto their joint span.
Using $Z'Z$ without checking rank. The displayed inverse exists only when the instrument columns are linearly independent. Redundant columns make $Z'Z$ singular, so the formula as written cannot be used without addressing that redundancy.
Treating a calculated 2SLS coefficient as proof of validity. The formula computes an estimator. It does not establish the assumptions that justify a causal interpretation.
Regressing the outcome on observed treatment in the second stage. The second-stage regressor in this 2SLS calculation is the first-stage fitted treatment, not the original $D$.

Check Your Understanding

Use the definitions and calculations above to answer these questions.

What does $P_ZD$ represent when $Z$ has several instrument columns?
Why does the formula for $P_Z$ include $(Z'Z)^{-1}$ when the instruments are not orthogonal?
In the orthogonal-instrument example, what were the two separate IV ratios and the joint 2SLS estimate?
What condition on $Z$ ensures that $(Z'Z)^{-1}$ exists?
Why does calculating a multiple-instrument 2SLS estimate not, by itself, establish that the estimate is causal?