Heteroskedasticity-Robust 2SLS Standard Errors

Why 2SLS Needs a Robust Variance Estimate

Two-stage least squares (2SLS) estimates a structural relationship using variation in regressors predicted by instruments. Its coefficient estimate does not require the outcome error to have the same variance for every observation. But the usual homoskedastic variance formula does make that assumption. If the error variance differs across observations, that formula can misstate the sampling uncertainty of the 2SLS estimate.

The heteroskedasticity-robust variance estimator replaces the common error variance with a sum of observation-level contributions. Each contribution combines that observation’s squared 2SLS residual with its instrument values. The resulting estimator is often called a sandwich estimator because it has a middle “meat” matrix between two “bread” matrices.

Model and Assumptions

Let \(y\) be an \(n \times 1\) vector of outcomes, \(X\) an \(n \times K\) matrix of structural regressors, and \(Z\) an \(n \times L\) matrix of instruments, where \(L \geq K\). The structural model is

Here \(\beta\) is the \(K \times 1\) coefficient vector and \(u\) is the \(n \times 1\) error vector. The rows of \(X\) and \(Z\) are \(x_i'\) and \(z_i'\). The regressors may include endogenous variables; any included exogenous regressors should also be included among the instruments. Assume that \(Z\) has full column rank and that \(X'P_ZX\) is invertible, where

The 2SLS estimator is

For the robust variance estimator to be justified, the instruments must satisfy the relevant exogeneity moments, such as \(E[z_i u_i]=0\), and provide enough identifying variation for the coefficient vector. The observations are assumed independent, with suitable finite moments and a stable, nonsingular identification structure. Their error variances need not be equal: conditional on the instruments and regressors, \(\operatorname{Var}(u_i\mid Z,X)\) may differ from one observation to another. Independence matters here; allowing unequal variances is not the same as allowing arbitrary dependence across observations.

The Sandwich Formula

Let \(\hat u=y-X\hat{\beta}_{2SLS}\), with \(i\)th residual \(\hat u_i\). A common heteroskedasticity-robust estimator, often called the HC0 form, is

The middle matrix is the estimated “meat.” It sums the outer product \(z_i z_i'\) for each observation, weighted by that observation’s squared residual. The two surrounding parts are the “bread.” Because this is a matrix, its diagonal entries estimate the variances of the coefficient estimates, while its off-diagonal entries estimate their covariances.

The 2SLS estimating equation gives a short justification for this form. Substituting \(y=X\beta+u\) into the estimator yields

With \(X\) and \(Z\) treated as fixed, the middle random quantity is a linear combination of the errors. If the errors are independent but have different conditional variances, then

Premultiplying and postmultiplying this expression by the coefficient matrices in the equation for \(\hat{\beta}_{2SLS}-\beta\) gives the sandwich structure. Replacing the unknown error variances by squared residuals gives the displayed estimator. Under the stated regularity conditions, this estimates the sampling variance consistently as the sample grows; it is not generally an exactly unbiased finite-sample variance estimate.

Connection to the Homoskedastic Formula

If the errors instead have a common conditional variance, \(\operatorname{Var}(u_i\mid X,Z)=\sigma^2\), then the middle population quantity is \(\sigma^2 Z'Z\). Substitution into the sandwich expression reduces it to

This is the homoskedastic 2SLS variance formula. Its usual estimated version replaces \(\sigma^2\) with a residual variance estimate, commonly \(\hat{\sigma}^2=\hat u'\hat u/(n-K)\). The robust formula does not make that common-variance substitution: it retains the observation-specific squared residuals inside the meat. Even if the two estimators happen to be similar in a particular sample, they rely on different variance assumptions.

What Robust Standard Errors Do—and Do Not—Address

A heteroskedasticity-robust 2SLS standard error is appropriate when independent observations may have unequal error variances and the IV model’s identification and exogeneity conditions are credible. It describes estimated sampling variability under those conditions; it does not validate the instruments. In particular, changing the variance estimator cannot make an instrument exogenous or create identifying information that is absent.

The formula above also relies on independence across observations. If observations are dependent within groups, treating them as independent can still give an inappropriate standard error. Robustness to unequal individual variances should not be confused with robustness to such dependence.

A frequent mistake is to run the second stage using fitted values and then report the ordinary OLS standard error from that regression. The fitted regressor does not make the ordinary second-stage OLS variance formula appropriate for 2SLS. For this tutorial’s focus, the relevant calculation is the 2SLS sandwich: use the original instrument matrix, the structural residuals \(y-X\hat{\beta}_{2SLS}\), and the full bread matrices. A standard error is not chosen by looking at which one is smaller or more convenient; its assumptions must match the sampling structure and model.