Heteroskedasticity-Robust First-Stage Diagnostics

Why Robust First-Stage Diagnostics Are Needed

A first-stage statistic asks whether excluded instruments predict an endogenous regressor after accounting for included exogenous controls. Under homoskedasticity, the familiar first-stage F statistic summarizes a joint test of zero coefficients on the excluded instruments. With heteroskedastic errors, the usual covariance formula for that test is generally wrong: the variance of the coefficient estimates depends on the pattern of error variances across observations. A heteroskedasticity-robust covariance estimate changes the test statistic accordingly.

This adjustment matters, but it has a precise scope. A robust Wald statistic tests the null that the first-stage coefficients are zero and provides a relevance diagnostic. Its large-sample null reference distribution does not, by itself, ensure that conventional two-stage least-squares inference is reliable when instruments are weak. The weak-instrument results and Stock-Yogo thresholds from the earlier material address different calibration questions and should not be transferred automatically to a heteroskedasticity-robust statistic.

Partialling Out Controls and Defining the Test

Let \(x_i\) be one endogenous regressor, \(z_i\) a \(q\)-vector of excluded instruments, and \(w_i\) a \(p\)-vector of included exogenous controls. Assume the first-stage linear projection is \(x_i=w_i'\gamma+z_i'\pi+v_i\), where \(\pi\) is the \(q\)-vector of coefficients whose relevance is being tested. The null hypothesis is \(H_0:\pi=0\). The intercept, if included, is part of \(w_i\).

Write \(W\) and \(Z\) for the sample matrices with rows \(w_i'\) and \(z_i'\). Define the projection residuals of the instruments on the controls by \(\hat r_i=z_i-\hat\Pi'w_i\), where \(\hat\Pi=(W'W)^{-1}W'Z\), assuming \(W'W\) is invertible. Thus, the vectors \(\hat r_i\) are the rows of the residualized-instrument matrix \(M_WZ\), with \(M_W=I-W(W'W)^{-1}W'\). Define \(\hat Q=n^{-1}\sum_i\hat r_i\hat r_i'\).

This is the HC0 sandwich version: there is no degrees-of-freedom correction multiplying the “meat” \(\hat\Omega\). Other HC versions and finite-sample scaling conventions exist, so a reported statistic should be read with its covariance and scaling convention in view. The essential distinction is between the unrestricted residuals \(\hat v_i\), which estimate the error variance in the sandwich, and the errors used to derive the sampling behavior of \(\hat\pi\).

Why the Robust Wald Statistic Has a Chi-Square Null Limit

The following result gives the large-sample justification. It is an asymptotic result, not an assertion that the statistic has an exact finite-sample F distribution.

Here heteroskedasticity is allowed: no requirement says that \(E[v_i^2\mid w_i,z_i]\) is constant. Positive definiteness of \(\Omega\) ensures that the limiting covariance is invertible. The stated moment and rank conditions rule out degenerate designs and support the approximations below.

Proof. Let \(\hat r_i=z_i-\hat\Pi'w_i\). The partial-regression identity gives the exact coefficient expansion

The \(v_i\) here are the reduced-form errors in \(x_i=w_i'\gamma+v_i\) under the null, not the residuals from the unrestricted first-stage regression. To obtain the limit, note that \(\hat\Pi-\Pi=O_p(n^{-1/2})\), while \(n^{-1/2}\sum_iw_iv_i=O_p(1)\) because \(E[w_iv_i]=0\). Therefore,

Also, \(\hat Q\xrightarrow{p}Q\). The multivariate central limit theorem gives \(n^{-1/2}\sum_i r_iv_i\xrightarrow{d}N(0,\Omega)\), so Slutsky’s theorem yields

It remains to justify the estimated covariance. The unrestricted OLS residuals satisfy \(\hat v_i=v_i-w_i'(\hat\gamma-\gamma)-z_i'\hat\pi\) under the null. OLS consistency and the moment conditions imply that replacing \(v_i^2\) by \(\hat v_i^2\) in the sample average changes it by \(o_p(1)\); this follows by expanding the square and applying the law of large numbers and Cauchy-Schwarz to the terms involving coefficient errors. Replacing \(r_i\) by \(\hat r_i\) likewise changes the average by \(o_p(1)\), using \(\hat\Pi-\Pi=o_p(1)\) and the finite moments. Consequently \(\hat\Omega\xrightarrow{p}\Omega\), and \(\hat Q\xrightarrow{p}Q\).

The estimated covariance of \(\sqrt n\,\hat\pi\) is \(\hat Q^{-1}\hat\Omega\hat Q^{-1}\), which converges in probability to \(Q^{-1}\Omega Q^{-1}\). The Wald statistic is the quadratic form of the asymptotically standard normal vector obtained by studentizing \(\sqrt n\,\hat\pi\). Its limit is therefore \(\chi_q^2\). Dividing by \(q\) gives the stated limit for \(F_{\mathrm{rob}}\). \(\square\)

Worked Calculations

Kleibergen-Paap-Style Interpretation

For a single endogenous regressor, a heteroskedasticity-robust joint Wald test of the excluded-instrument coefficients is the relevant robust rank/relevance test. A common reported form is called a Kleibergen-Paap rk Wald F statistic. With the convention used here, its F-scaled form is \(W_{\mathrm{rob}}/q\). Software can apply different finite-sample or normalization conventions, so the statistic’s label alone does not specify every numerical detail. The null being tested remains that the excluded instruments have zero first-stage coefficients.

This robustification replaces the homoskedastic covariance estimate with a sandwich estimate; it does not turn the diagnostic into a general weak-IV inference procedure. In particular, the chi-square null approximation describes behavior under the zero-relevance null with regular fixed-rank conditions. It does not establish that conventional 2SLS Wald confidence intervals have reliable coverage when the first stage is weak. Nor should a heteroskedasticity-robust statistic be compared mechanically with Stock-Yogo thresholds derived for a different covariance model and calibration objective.

What the Diagnostic Does—and Does Not—Establish

A small robust Wald statistic indicates that the data provide limited evidence against zero first-stage relevance under the stated robust test. A large value is evidence against that null. Neither conclusion certifies the exclusion restriction, proves that the instruments are exogenous, or establishes that 2SLS behaves well in the sample. Instrument validity is a separate modeling requirement, and weak-IV inference requires methods whose guarantees address weak identification rather than merely using a robust standard error.

A common mistake is to take a robust statistic, compare it to a conventional first-stage F threshold, and conclude that passing the threshold guarantees valid 2SLS inference. This reasoning mixes a heteroskedasticity-robust null test with a threshold calibrated under different assumptions and treats a diagnostic as an inference guarantee. The correction is to identify the covariance estimator and reference distribution, interpret the statistic as evidence about first-stage relevance, and separately assess whether the desired structural inference is valid under possible weak instruments.