Weighted Two-Stage Least Squares

Why Weight Both Stages?

In ordinary 2SLS, instruments project the regressors onto the instrument space, and the second stage uses those projected regressors. When observations have different weights, both operations must use the same weighted geometry. Weighting only the first stage or only the second does not generally produce weighted 2SLS.

Weights can represent different things: inverse error variances, sampling adjustments, or chosen emphasis in an objective function. Those interpretations are not interchangeable. The matrix calculation defines an estimator, but the reason for the weights and the assumptions that make them appropriate must be established separately.

The model is \(y=X\beta+u\). For consistency, a central condition is that the weighted instrument moments are valid: in a population formulation, \(E[w_i z_i u_i]=0\), where \(z_i\) is the instrument vector for observation \(i\). This follows, for example, if weights are fixed or suitably exogenous and \(E[z_i u_i]=0\) holds conditionally on the variables determining the weights. Ordinary instrument validity alone does not ensure validity after weighting if the weights depend on the structural error.

The Weighted Projection and Estimator

The weighted projection of any vector \(v\) onto the columns of \(Z\) is

This projection minimizes \((v-Z\gamma)'W(v-Z\gamma)\) over \(\gamma\). It is idempotent, \(P_Z^W P_Z^W=P_Z^W\), but it is not generally symmetric in the ordinary Euclidean inner product. Instead, it is self-adjoint under the weighted inner product: \(W P_Z^W=(P_Z^W)'W\). Equivalently, premultiplying each row by \(\sqrt{w_i}\) turns the problem into ordinary projection using \(W^{1/2}Z\).

The second stage is a weighted least-squares regression of \(y\) on \(\widehat{X}\). Thus the estimator is

The rank conditions in the definition ensure these inverses exist. In a model with multiple endogenous regressors, the instruments must provide enough independent weighted variation to identify all coefficients; having a formally large instrument count does not by itself establish that rank condition.

Weights and Covariance Calculations

Changing the weights changes not only the contribution of each observation to the outcome fit but also the projection of each regressor onto the instruments. Covariance calculations must reflect this change. Define

With heteroskedastic errors, an observation-level sandwich estimate is

The \(w_i^2\) in the middle is essential: the weighted moment contribution of observation \(i\) is \(w_i z_i\widehat{u}_i\), whose outer product includes \(w_i^2\). This formula is the observation-level heteroskedasticity-robust sandwich; dependence across observations requires a covariance estimator that accounts for that dependence, rather than this independent-observation form.

A simpler covariance expression applies under a specific variance model. If \(\operatorname{Var}(u\mid X,Z)=\sigma^2W^{-1}\), then the weighted-transformed errors \(W^{1/2}u\) have covariance \(\sigma^2 I\), and the covariance of weighted 2SLS is \(\sigma^2G^{-1}\). In contrast, merely assuming constant variance \(\operatorname{Var}(u\mid X,Z)=\sigma^2I\) while using arbitrary \(W\) does not generally yield \(\sigma^2G^{-1}\). In that case the middle covariance factor involves \(Z'W^2Z\), not \(Z'WZ\). The weights must match the variance assumption before using the simpler expression.

Implementation, Diagnostics, and Alternatives

A practical check is to compare a direct weighted 2SLS calculation with a two-stage implementation using the same \(W\) at both stages. They should agree up to numerical rounding. Also inspect the range and concentration of weights, the conditioning of \(Z'WZ\) and \(G\), and whether estimates are driven by a few heavily weighted observations. A numerically invertible matrix can still be poorly conditioned, producing unstable estimates.

Weighted 2SLS is not the only approach. Unweighted 2SLS uses the same instrument logic with \(W=I\). Generalized method of moments can use a different weighting matrix for instrument moments; it is not the same operation as multiplying observations by a diagonal \(W\). If weights are inverse conditional error variances, weighted 2SLS can improve efficiency under the stated variance model. If weights are sampling weights, the target and variance calculation must reflect the sampling design. Neither kind of weighting repairs invalid instruments or insufficient identification.

Common Pitfalls and Limits

• Weighting only one stage. Projecting with \(W\) and then running unweighted least squares does not implement the estimator defined here. Use weighted least squares in both stages.
• Using an ordinary projection. The matrix \(Z(Z'Z)^{-1}Z'\) ignores unequal observation weights. The weighted projection uses \(Z'WZ\) and \(Z'W\).
• Assuming weights guarantee valid instruments. If weights are related to the structural error, \(E[w_i z_i u_i]=0\) may fail even when an unweighted moment condition holds.
• Using the wrong covariance formula. The simple \(\sigma^2G^{-1}\) form requires \(\operatorname{Var}(u\mid X,Z)=\sigma^2W^{-1}\). Arbitrary weights and constant error variance call for a different covariance calculation.

As with other 2SLS procedures, estimates may be unstable when instruments provide little relevant variation. Extreme weights can make the effective information depend heavily on a small subset of observations, and standard errors may then be sensitive to modeling and variance assumptions. Weighted 2SLS also does not by itself establish that the weighted population is the desired target.