The Projection-Matrix Formula for 2SLS

From an Instrument to a Projection

Two-stage least squares (2SLS) uses the part of an endogenous regressor that can be predicted from the instrument or instruments. The projection-matrix formula expresses that idea compactly: it first projects the regressor into the space spanned by the instruments, then uses that projection in the estimator. This notation is useful because it makes the fitted values, dimensions, and conditions for a unique estimate explicit.

The setup here has one endogenous regressor and no additional exogenous covariates. Let \(y\) be the outcome vector, \(x\) the endogenous-regressor vector, and \(Z\) the matrix whose columns are the instruments. For \(n\) observations and \(L\) instruments, \(y\) and \(x\) are \(n\times 1\), and \(Z\) is \(n\times L\). The structural equation is \(y=x\beta+u\), where \(\beta\) is a scalar and \(u\) is the structural error vector.

Here \(Z'\) is \(L\times n\), \(Z'Z\) is \(L\times L\), and \(Z(Z'Z)^{-1}Z'\) is \(n\times n\), as required to multiply an \(n\times 1\) vector. The projection is determined by the instrument space, not by the outcome. The same matrix projects both \(x\) and \(y\), when a projection of the outcome is useful for describing the estimator.

Two properties follow directly from the formula. First, \(P_Z\) is symmetric: \(P_Z'=P_Z\). Second, it is idempotent: \(P_Z^2=P_Z\). To verify the latter, use \(Z'Z\) as a single matrix product:

Idempotence means that projecting a vector that is already in the instrument space leaves it unchanged. Symmetry and idempotence also ensure that the quadratic expression involving \(P_Z\) below measures the squared length of the projected residual.

Deriving the Projection-Matrix Estimator

The projection-matrix form of 2SLS is

For one endogenous regressor, \(x'P_Zx\) is a scalar, so its inverse exists precisely when that scalar is nonzero. More generally, if \(X\) has \(K\) endogenous-regressor columns, the corresponding expression is \((X'P_ZX)^{-1}X'P_Zy\), provided that the \(K\times K\) matrix \(X'P_ZX\) is nonsingular. The examples here keep to one endogenous regressor.

A direct derivation starts by measuring the discrepancy between the observed outcome and the structural prediction after projecting that discrepancy onto the instrument space. Define the criterion \(Q(\beta)=(y-x\beta)'P_Z(y-x\beta)\). Since \(P_Z\) is symmetric, expanding and differentiating this scalar quadratic gives

Setting the derivative equal to zero gives \(x'P_Zx\,\hat{\beta}=x'P_Zy\). If \(x'P_Zx>0\), solving yields the stated estimator. The criterion is then strictly convex in \(\beta\), so this solution is its unique minimum. The condition \(x'P_Zx>0\) says that the instrument-space projection of \(x\) is not the zero vector: the instruments must contain some variation related to the endogenous regressor.

The fitted endogenous regressor is obtained separately from the first-stage projection:

This expression has \(n\) fitted values, one for each observation. It is a linear combination of the columns of \(Z\), so every fitted value vector lies in the instrument space. The projection formula does not by itself establish that the resulting coefficient has a causal interpretation. That interpretation requires the instrumental-variable assumptions established in earlier material, including relevance and a valid relationship between the instruments and the structural error. In moment notation, a key exogeneity condition is \(E[Z'u]=0\); a nonzero projection of \(x\) is also needed for identification.

Reading Fitted Values and Checking the Conditions

The fitted values are not generally the original values of \(x\). They are the components of \(x\) that lie in the instrument space. With one instrument \(z\), the formula simplifies to

Thus every fitted value is the same scalar multiple of the corresponding instrument value. If the instrument vector is rescaled by a nonzero constant, the projection does not change: the span of the instrument is unchanged. This is a useful check against interpreting the numerical size of an instrument's units as a measure of its relevance.

The matrix formula also provides practical diagnostics. Confirm that \(Z\) has full column rank before using \((Z'Z)^{-1}\). Then check that the projected regressor has nonzero variation, equivalently \(x'P_Zx>0\) in the one-regressor case. A value close to zero signals that the data contain little instrument-related variation in \(x\); although an inverse may technically exist, the estimate can be highly sensitive to small changes in the data. The formula alone does not supply an inference procedure or a test of the IV assumptions.

When \(Z\) has linearly dependent columns, \(Z'Z\) is singular and the displayed inverse is undefined. Redundant instrument columns do not add a new direction to the instrument space. One can represent that space using a full-rank basis, or define a projection with a generalized inverse, but the ordinary inverse formula as written requires full column rank. Similarly, if \(P_Zx=0\), then \(x'P_Zx=0\) and no unique 2SLS coefficient is available from this setup.

What the Formula Does—and Does Not—Establish

The projection-matrix representation is valuable because it separates three issues that are easy to conflate. The matrix \(P_Z\) defines which variation is retained; the rank condition determines whether the coefficient can be uniquely calculated; and the IV assumptions determine whether the coefficient has the intended causal interpretation. A successful matrix calculation addresses the first two, not automatically the third.

In particular, nonzero sample association between \(Z\) and \(x\) is not proof of instrument validity. The earlier material on instrumental-variable assumptions distinguishes relevance from independence and exclusion: the instruments must be informative about the endogenous regressor, and their relationship with the outcome must support the proposed causal interpretation. Those assumptions cannot generally be established from the projection formula itself.

A common mistake is to say, “The instrument predicts \(x\), so the 2SLS estimate is causal.” Prediction is only the relevance part of the argument. An instrument can strongly predict \(x\) and still be related to the structural error or affect \(y\) through a route other than \(x\). The correction is to report the calculation and the assumptions separately, and to avoid treating algebraic computability as evidence that those assumptions hold.

A second pitfall is to mistake fitted values for observed treatment values. The entries of \(\hat{x}=P_Zx\) are projections, often fractional even when \(x\) is binary. They represent instrument-space components, not a new observed treatment assignment. Finally, if the matrix to be inverted is nearly singular, a finite result should not be mistaken for a stable one. Inspect relevance and the conditioning of the relevant cross-product matrix before relying on the estimate.