2SLS With Multiple Instruments

Why Use More Than One Instrument?

A single endogenous regressor may be associated with several proposed excluded instruments. Each instrument supplies a direction of variation that may help predict the endogenous regressor, but 2SLS does not simply calculate a separate estimate for each instrument and average them. It projects the endogenous regressor onto the full space spanned by the instruments, then uses that projected variation in the structural equation. Understanding that joint projection clarifies both the estimator and what it means for a model to be overidentified.

The discussion below concerns one endogenous regressor. Included exogenous covariates are allowed; the earlier material on partialling out exogenous covariates shows why they must be accounted for in both stages. The focus here is how several excluded-instrument directions work together, including the distinction between the number of instrument columns and their rank.

Model, Projection, and Identification

Definition: Let \(y\) be an \(n \times 1\) outcome vector, \(x\) an \(n \times 1\) endogenous regressor, \(W\) a matrix of included exogenous regressors (including an intercept if the model has one), and \(Z\) a matrix of excluded instruments. The structural equation is \(y = x\beta + W\gamma + u\). The instrument matrix for 2SLS is \(Q = [W\ Z]\). If \(Q\) has full column rank, its projection matrix is \(P_Q = Q(Q'Q)^{-1}Q'\). The 2SLS estimator is \(\hat{\theta}_{2SLS} = (X'P_QX)^{-1}X'P_Qy\), where \(X=[x\ W]\) and \(\theta=(\beta,\gamma')'\), provided \(X'P_QX\) is invertible.

The instrument matrix includes \(W\) because included exogenous regressors instrument themselves. The excluded instruments \(Z\) are the additional sources of variation used to instrument \(x\). If the columns of \(Q\) are not full rank, redundant columns can be removed, or the projection can be defined using a generalized inverse; the projection onto the column space is what matters, not the particular list of columns.

The model requires a correct structural specification and valid instruments. In the standard linear IV setup, the excluded instruments must satisfy the relevant exogeneity moments \(E[Z'u]=0\), and their exclusion means they do not enter the structural equation directly: their association with \(y\) is through the endogenous regressor, conditional on \(W\). The included exogenous regressors must also satisfy the model's exogeneity conditions. These are substantive assumptions; observing a strong association with \(x\) does not establish that an instrument is valid.

For the scalar \(x\), identification requires relevant instrument variation after accounting for \(W\). To state this precisely, let \(M_W=I-P_W\), where \(P_W\) projects onto the columns of \(W\). Define \(\tilde{x}=M_Wx\), \(\tilde{y}=M_Wy\), and \(\tilde{Z}=M_WZ\). The excluded instruments identify \(\beta\) when the population relationship between \(\tilde{Z}\) and \(\tilde{x}\) has nonzero rank—in this scalar case, at least one instrument direction must predict the residualized \(x\). The model is overidentified when the number of linearly independent excluded-instrument directions after accounting for \(W\) exceeds the number of endogenous regressors. Here that means \(\operatorname{rank}(\tilde{Z})>1\). Counting columns alone is not enough: duplicate or linearly dependent columns add no independent directions.

How Several Instruments Enter the Estimate

After removing the linear contribution of \(W\), let \(P_{\tilde Z}\) be the projection onto the columns of \(\tilde{Z}\). For full-column-rank \(\tilde{Z}\), \(P_{\tilde Z}=\tilde{Z}(\tilde{Z}'\tilde{Z})^{-1}\tilde{Z}'\). The coefficient on \(x\) can then be written as:

\hat{\beta}_{2SLS} = \frac{\tilde{x}'P_{\tilde Z}\tilde{y}} {\tilde{x}'P_{\tilde Z}\tilde{x}}.

The projection \(P_{\tilde Z}\tilde{x}\) is the component of residualized \(x\) that lies in the space spanned by the excluded instruments. With several instruments, this projected component can use a linear combination of all the instrument columns. The outcome is then related to this combined predicted variation, rather than to the original endogenous \(x\) alone.

A short derivation makes the combination explicit. Suppose the residualized instrument columns are orthonormal, so that \(\tilde{Z}'\tilde{Z}=I\). Then \(P_{\tilde Z}=\tilde{Z}\tilde{Z}'\). Write \(\hat{\pi}=\tilde{Z}'\tilde{x}\) and \(\hat{\rho}=\tilde{Z}'\tilde{y}\); their entries are the sample instrument associations with \(x\) and \(y\), respectively. Substitution gives:

\hat{\beta}_{2SLS} = \frac{\hat{\pi}'\hat{\rho}}{\hat{\pi}'\hat{\pi}} = \frac{\sum_j \hat{\pi}_j\hat{\rho}_j} {\sum_j \hat{\pi}_j^2}.

Thus, in this special orthonormal case, each instrument direction contributes according to its association with \(x\), and the denominator measures the total projected variation in \(x\). With nonorthogonal instruments, the projection accounts for their covariance rather than treating their columns as independent pieces. In neither case is the general estimator an unweighted average of separate single-instrument estimates.

Worked Example: Calculating 2SLS With Two Independent Instruments

Consider a hypothetical data set of four observations with an intercept in \(W\). The two excluded instrument columns and a vector \(v\) are:

z_1=(1,-1,1,-1)',\quad z_2=(1,1,-1,-1)',\quad v=(1,-1,-1,1)'.

Suppose the observed regressor and outcome are \(x=2z_1+z_2+v\) and \(y=3x+2v\). Thus \(x=(4,-2,0,-2)'\) and \(y=(14,-8,-2,-4)'\). Each of \(z_1,z_2,v\) has mean zero; \(z_1\) and \(z_2\) are orthogonal, and \(v\) is orthogonal to both. The two instruments are therefore independent directions after accounting for the intercept. The structural equation has \(\beta=3\) and error \(u=2v\), with \(z_1'u=z_2'u=0\) in this constructed sample. These calculations check the sample projection and moment conditions; instrument validity in an application still requires substantive justification.

Because \(z_1'z_1=z_2'z_2=4\) and the instruments are orthogonal, the projection of \(x\) onto their span is:

\hat{x}=P_Zx =\frac{z_1'x}{4}z_1+\frac{z_2'x}{4}z_2 =2z_1+z_2 =(3,-1,1,-3)'.

The 2SLS coefficient, with the intercept already accounted for by the centered vectors, is:

\hat{\beta}_{2SLS} =\frac{\hat{x}'y}{\hat{x}'\hat{x}} =\frac{3(14)+(-1)(-8)+1(-2)+(-3)(-4)} {3^2+(-1)^2+1^2+(-3)^2} =\frac{60}{20}=3.

The result is 3 outcome units per one-unit increase in \(x\), matching the constructed structural coefficient. Both instruments contribute to \(\hat{x}\); neither is used as a separate final estimator.

Overidentification Is About Rank, Not a Column Count

The difference between the number of listed instruments and the number of independent instrument directions matters. A duplicated column does not expand the projection space, so it cannot supply an additional independent moment condition. Conversely, genuinely distinct instrument directions can make a model overidentified, although that classification alone says nothing about whether those instruments are valid or sufficiently informative.

Worked Example: Duplicate Columns Do Not Create Overidentification

In another hypothetical four-observation example, let \(z=(1,-1,1,-1)'\), \(v=(1,1,-1,-1)'\), and include an intercept. Suppose the data have \(x=2z+v\) and \(y=4x+v\). An analyst lists two excluded-instrument columns, \(z_1=z\) and \(z_2=2z\).

Although the instrument matrix has two excluded columns, they are linearly dependent. After accounting for the intercept, its rank is one: \(\operatorname{rank}([z_1,z_2])=1\). The projection is unchanged by including the duplicate column. Since \(z'z=4\), \(P_Zx=2z\), and \(P_Zy=8z\). The scalar 2SLS calculation is:

\hat{\beta}_{2SLS} =\frac{(2z)'(8z)}{(2z)'(2z)} =\frac{16(z'z)}{4(z'z)} =\frac{64}{16}=4.

There is one endogenous regressor and only one independent excluded-instrument direction, so this model is exactly identified, not overidentified. The duplicate has changed neither the projection nor the amount of identifying information. Here \(v\) is orthogonal to the intercept and \(z\), so the constructed error \(u=v\) also has zero sample instrument moment.

How the Joint Projection Combines Information

Worked Example: The Combined Estimate Is Not a Simple Average

Suppose a hypothetical, intercept-adjusted sample has two orthonormal instrument directions. The first-stage associations of those directions with \(x\) are \(\hat{\pi}_1=2\) and \(\hat{\pi}_2=1\). Their reduced-form associations with \(y\) are \(\hat{\rho}_1=5\) and \(\hat{\rho}_2=2\). These are sample summaries, not claims that the instruments are valid in a population.

Using the orthonormal-instrument expression derived above:

\hat{\beta}_{2SLS} =\frac{2(5)+1(2)}{2^2+1^2} =\frac{12}{5}=2.4.

The separate ratios would be \(5/2=2.5\) and \(2/1=2\); their unweighted average is \(2.25\), not 2.4. 2SLS combines the directions through their joint projected variation in \(x\). The difference is not an error: the estimator is defined by the projection formula, not by averaging instrument-specific ratios. In real data, instrument correlations and scale also affect the projection, so the simple orthonormal expression does not apply unchanged.

Diagnostics, Alternatives, and Limitations

Several checks help identify a problematic instrument set, but none can prove the exclusion restriction. First check the rank and structure of the instrument matrix after accounting for \(W\): exact duplicates, near-linear dependence, or incorrectly coded columns can make the apparent instrument count misleading. Then examine whether the excluded instruments predict residualized \(x\). If they provide little relevant variation, the denominator \(\tilde{x}'P_{\tilde Z}\tilde{x}\) is small relative to the scale of the data, and the estimate can be unstable. A large number of instrument columns does not guarantee a strong first stage.

Finally, assess instrument validity using the design and substantive knowledge: explain why each excluded instrument is related to treatment or exposure, why it has no direct pathway to the outcome, and why it is not associated with unmeasured causes of the outcome after conditioning on \(W\). Overidentification gives additional restrictions implied by the model, but having more such restrictions does not make the assumptions true. A formal overidentification test and its interpretation belong to inferential material; even a non-significant result would not establish validity.

The main alternative is to use a single instrument, if one can be justified and identifies the effect. With multiple instruments, 2SLS uses their joint instrument space; choosing a subset changes that space and may change the estimate. Ordinary least squares is not a substitute when \(x\) is endogenous, because it does not use the instrument-based projection to address that problem. The choice among these approaches depends on the design, relevance, and validity arguments—not merely on which calculation produces a preferred coefficient.

Common mistake: “There are two excluded-instrument columns and one endogenous regressor, so the model is overidentified.” This conclusion can be wrong if the columns are duplicates or otherwise linearly dependent after accounting for included exogenous regressors. Determine the rank of the excluded-instrument directions conditional on \(W\); for one endogenous regressor, overidentification requires that rank to exceed one.

Key takeaway: Multiple instruments enter 2SLS through their joint projection space. Independent relevant directions can provide additional identifying information, but redundant columns do not; and neither rank nor relevance establishes instrument validity.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

There is one endogenous regressor and three excluded-instrument columns. After removing the included exogenous regressors, their rank is two. Is the model overidentified, and why?
With two orthonormal instruments, the first-stage associations are 3 and 1, and the reduced-form associations are 7 and 2. Calculate the scalar 2SLS estimate using the joint projection formula.
Why does adding a second column equal to twice the first instrument leave the projection space unchanged?
Name one diagnostic for instrument relevance and one reason it cannot establish instrument validity.

Solutions. (1) Yes. The rank of the independent excluded-instrument directions is two, which exceeds the one endogenous regressor. (2) \(\hat{\beta}_{2SLS}=[3(7)+1(2)]/[3^2+1^2]=23/10=2.3\). (3) A column that is a scalar multiple of another is linearly dependent on it, so their span—and thus the projection onto that span—is unchanged. (4) Check whether the excluded instruments predict \(x\) after accounting for included covariates. Relevance concerns association with \(x\); it does not show that an instrument has no direct effect on \(y\) or is otherwise exogenous.

What You'll Learn