The Naive Second-Stage Standard Error Pitfall

Why the Second-Stage Standard Error Is a Separate Problem

In two-stage least squares (2SLS), first-stage fitted values are used to isolate the variation in regressors explained by the instruments. Regressing the outcome on those fitted values can reproduce the 2SLS coefficient. That algebraic equivalence does not make the ordinary OLS standard error from that regression a valid 2SLS standard error.

The distinction matters because the original regressor may be endogenous: it may be correlated with the structural error. The fitted regressor is estimated from the instruments, and the outcome’s residual in a regression on fitted values is generally not the structural residual. A valid 2SLS covariance estimate must account for the instrument-based estimation and use the structural residuals in its variance calculation.

This setup presumes instrument validity and identification: instruments must satisfy the required orthogonality conditions with the structural error, and \(X'P_ZX\) must be invertible. For the usual independent-observation asymptotic arguments, the observations must also satisfy suitable sampling and moment conditions. These conditions concern the IV model; computing a standard error does not establish that the instruments are valid or sufficiently informative.

Why the Point Estimates Agree but the OLS Standard Errors Do Not

Since \(P_Z\) is symmetric and idempotent, \(\hat{X}'\hat{X}=X'P_ZX\) and \(\hat{X}'y=X'P_Zy\). Thus an OLS regression of \(y\) on all columns of \(\hat{X}\), with the same treatment of included controls and intercepts as the 2SLS specification, has coefficient $$ (\hat{X}'\hat{X})^{-1}\hat{X}'y=(X'P_ZX)^{-1}X'P_Zy. $$ This equality explains why the reported coefficient can look right. It says nothing by itself about the OLS regression’s estimated variance.

An ordinary OLS covariance calculation treats \(\hat{X}\) as the design matrix for an OLS model and uses the residuals \(\tilde{u}\). But the IV model is \(y=X\beta+u\), so the residuals estimating the structural errors are \(\hat{u}=y-X\hat{\beta}_{2SLS}\). The valid IV covariance also reflects the instrument-weighted estimating equation. Merely substituting fitted regressors into OLS does not generally accomplish either task.

Valid Covariance Estimates for 2SLS

Under homoskedasticity, the conventional 2SLS covariance estimate is $$ \widehat{\operatorname{Var}}_{\mathrm{hom}}(\hat{\beta}_{2SLS}) =\hat{\sigma}^2(X'P_ZX)^{-1}, $$ where a common estimate is \(\hat{\sigma}^2=\hat{u}'\hat{u}/(n-k)\). This is the conventional homoskedastic 2SLS formula, not a claim that the exact finite-sample variance conditional on \(Z\) equals that expression. Because \(X\) can be endogenous, treating it as fixed when deriving a conditional variance is generally unjustified. The formula is used under homoskedastic IV conditions as a conventional estimate and has the corresponding large-sample justification under standard IV regularity conditions.

For independent observations with heteroskedastic errors, a sandwich estimate is $$ \widehat{\operatorname{Var}}_{\mathrm{rob}}(\hat{\beta}_{2SLS}) =(X'P_ZX)^{-1} \left[ X'Z(Z'Z)^{-1} \left(\sum_{i=1}^{n}z_i z_i'\hat{u}_i^2\right) (Z'Z)^{-1}Z'X \right] (X'P_ZX)^{-1}, $$ where \(z_i'\) is row \(i\) of \(Z\), and \(\hat{u}_i\) is the \(i\)th structural residual. Equivalently, the middle matrix is \(X'P_Z\operatorname{diag}(\hat{u}_i^2)P_ZX\). This form shows both essential ingredients: structural residuals and instrument-weighted scores. The square roots of the diagonal entries of the estimated covariance matrix are the standard errors.

The example’s robust variance can also be checked directly from the instrument-score form. Here \(z_i\hat{u}_i\) equals \((-2,2,0)'\), so \(\sum_i z_i^2\hat{u}_i^2=8\). Since \(z'z=6\), the scalar middle term is \(8/6^2=2/9\); multiplying by the two outer factors of \(6\) again gives \(8\). Such checks are useful for catching a common computational error: using instrument values in a fitted-design HC0 formula, or fitted-design residuals in the IV sandwich.

How to Choose and Check the Calculation

The covariance estimator should match the error dependence structure and sampling design. The conventional homoskedastic formula relies on a homoskedastic specification. If error variance may differ across independent observations, use the heteroskedasticity-robust 2SLS sandwich. If observations are clustered or serially dependent, use the corresponding cluster-robust or HAC 2SLS covariance estimate rather than treating observations as independent. A finite-sample multiplier can modify a suitable covariance estimate; it cannot turn the wrong OLS covariance formula into a valid IV one.

Before interpreting an estimated standard error, check that the calculation uses the intended model and covariance estimator. In particular:

• Verify that \(X\) includes all structural regressors and \(Z\) includes the instruments and included exogenous controls.
• For the robust 2SLS sandwich, verify that residuals are \(y-X\hat{\beta}_{2SLS}\), not \(y-\hat{X}\hat{\beta}_{2SLS}\).
• Check that the “meat” is based on instrument-weighted residual scores, not the ordinary OLS HC0 meat for the fitted-values regression.
• Confirm that the chosen covariance estimator reflects heteroskedasticity, clustering, or serial dependence as appropriate.

A large discrepancy between an OLS second-stage standard error and a correctly constructed 2SLS standard error is a reason to inspect the formulas, not a reason to select whichever answer appears more plausible. Conversely, agreement does not validate the naive calculation: different formulas can coincide in special cases or by chance.

Common Mistake and the Main Alternatives

Wrong reasoning: “The regression on first-stage fitted values gives the 2SLS coefficient, so its usual OLS standard error—or its OLS HC0 standard error—must also be the 2SLS standard error.” The conclusion does not follow. Equality of coefficient formulas does not establish equality of their variance estimators. The correction is to use the 2SLS covariance formula appropriate to the sampling assumptions, with the structural residuals and the IV score structure.

The main alternative is to estimate the same IV model using software that reports 2SLS covariance estimates directly, specifying the desired homoskedastic, heteroskedasticity-robust, clustered, or HAC option. This avoids manually reconstructing the sandwich, but the analyst must still verify which estimator the software reports and whether its assumptions fit the data. Ordinary least squares on the original \(X\) is not a substitute when endogeneity is the reason for using instruments. The familiar material on homoskedastic and robust 2SLS inference gives the corresponding covariance choices; here the central issue is why the fitted-values OLS shortcut does not reproduce them.