HAC Standard Errors for 2SLS

Why 2SLS Needs HAC Standard Errors

When observations have a meaningful order—such as consecutive time periods—errors or other features of the data may be dependent across nearby observations. Heteroskedasticity-robust 2SLS inference allows the error variance to differ across observations, but it does not account for serial dependence. HAC (heteroskedasticity-and-autocorrelation-consistent) covariance estimates extend the 2SLS sandwich to account for both unequal variances and dependence across the ordered observations.

The essential object is not the residual sequence alone. It is the sequence of instrument-residual scores. HAC estimation measures how those scores co-vary at different lags, then maps their long-run covariance into the coefficient covariance using the 2SLS bread. The ordering, lag weights, and bandwidth are therefore part of the estimator, not cosmetic settings.

Model, Scores, and Assumptions

For ordered observations \(t=1,\ldots,n\), write the linear structural equation as \(y_t=x_t'\beta+u_t\). Here \(y_t\) is the outcome, \(x_t\) is a \(p\)-vector of regressors that may include endogenous variables and exogenous controls, \(\beta\) is the \(p\)-vector of coefficients, and \(u_t\) is the structural error. Let \(z_t\) be a \(k\)-vector of instruments, including any required exogenous regressors. Assume \(k\ge p\), and that the instruments identify the coefficients.

The standard conditions include instrument validity, \(E[z_tu_t]=0\), and identification: the population matrix \(Q_{XZ}=E[x_tz_t']\) has sufficient rank for the 2SLS coefficient vector to be identified. For the usual HAC approximation, the score sequence must also have suitable weak-dependence and moment properties, and its long-run covariance must be finite. Stationarity is a common sufficient framework, though other asymptotic frameworks can be used. These conditions justify a central limit approximation for sums of scores; they do not require independent or homoskedastic errors.

HAC does not repair a violation of instrument validity or identification. Nor does it make an invalid time ordering meaningful: observations must be arranged according to the dependence being modeled. In particular, validity of each instrument for its own observation does not automatically establish all assumptions needed in a specific time-series IV design.

Building the HAC 2SLS Covariance

Define the sample moment matrices and the 2SLS bread as

Assuming the required inverses exist, define \(H=G^{-1}Q_{XZ}Q_{ZZ}^{-1}\). The 2SLS estimation error can be written as \(H\) times the sample average of the instrument-residual scores. In population notation, the corresponding first-order representation is \(\hat\beta-\beta\approx H(n^{-1}\sum_t g_t)\). Thus the long-run covariance of the scores determines the coefficient covariance.

For lag \(\ell\ge0\), estimate score cross-products by

Choose a maximum lag \(L\) and weights \(w_\ell\) from a kernel. The Bartlett, or Newey–West, weights are \(w_\ell=1-\ell/(L+1)\) for \(1\le\ell\le L\). The estimated long-run covariance is

The transpose is essential: \(\hat\Gamma_\ell\) records covariance in one lag direction, and adding its transpose accounts for the reverse direction and makes the result symmetric. The resulting HAC covariance estimate for 2SLS is

The standard error for coefficient \(j\) is the square root of the \(j\)th diagonal element of this matrix. Units follow the corresponding coefficient: for example, if an outcome is measured in days and a regressor in dollars, the coefficient and its standard error are measured in days per dollar.

Why the Score Sandwich Is Appropriate

The justification follows directly from the 2SLS normal equations. Substituting \(y_t=x_t'\beta+u_t\) into the estimator gives the sample error as the bread times the average instrument-error product:

Under the moment, rank, and dependence conditions, the normalized sum of scores has an asymptotic covariance given by their long-run covariance \(\Omega\), which includes contemporaneous variation and all lagged cross-covariances. Therefore,

Replacing unknown quantities by sample estimates and truncating the lag sum with kernel weights yields the HAC estimator above. For consistency of a standard kernel estimate, the maximum lag typically grows with sample size but more slowly than \(n\), so that \(L\to\infty\) and \(L/n\to0\), along with regularity conditions on dependence and moments. In a finite sample, software often uses a fixed or data-selected bandwidth; that choice is an approximation, and sensitivity to it should be assessed.

Choosing a Bandwidth and Checking the Result

The bandwidth controls how many lagged score relationships contribute and how strongly they contribute. A bandwidth that is too short can omit meaningful dependence, leaving standard errors understated or otherwise distorted. A bandwidth that is too long can add noisy estimates of distant-lag covariances, making inference unstable in a limited sample. Bartlett weights taper contributions toward zero at the cutoff; other standard kernels use different weighting profiles, but the same score-sandwich logic applies.

Useful diagnostics and checks include:

• Verify the order. Confirm that adjacent rows really are adjacent in the process of interest. Arbitrarily sorting observations changes the lag products.
• Inspect score dependence. Examine the ordered score sequence or its lagged cross-products. Residual autocorrelation alone does not completely describe dependence in \(z_t\hat u_t\).
• Compare plausible bandwidths. Report whether standard errors and substantive conclusions are sensitive to reasonable bandwidth choices, rather than treating one choice as definitive.
• Check matrix behavior. A covariance estimate should be symmetric, and its diagonal entries should be nonnegative up to numerical precision. Unexpected behavior can indicate coding, scaling, or implementation problems.
• Assess identification separately. HAC standard errors do not solve weak-instrument problems. A seemingly ordinary standard error is not proof that the instruments provide strong identifying information.

HAC inference is also not a substitute for cluster-robust inference when dependence is primarily within a known set of clusters. HAC typically models dependence by distance in an ordered sequence; cluster-robust covariance allows arbitrary within-cluster dependence under its own asymptotic framework. The heteroskedasticity-robust 2SLS covariance is the corresponding no-cross-observation-covariance alternative. These methods answer different dependence questions, so the design—not which option produces the smallest standard error—should guide the choice.

Common Mistakes and Limitations

• Using \(x_t\hat u_t\) as the IV score by default. The moment conditions for IV use \(z_tu_t\). The instrument-score sequence must be propagated through the 2SLS mapping \(H\); an ordinary OLS score does not generally give the 2SLS covariance.
• Ignoring the transpose term. For each positive lag, include both \(\hat\Gamma_\ell\) and \(\hat\Gamma_\ell'\). Omitting one direction can yield an asymmetric covariance estimate.
• Treating the bandwidth as a harmless setting. The lag cutoff changes the estimated long-run covariance and can materially change standard errors, especially in shorter samples.
• Assuming HAC makes the instruments valid. HAC changes the covariance estimate, not the identifying moment condition \(E[z_tu_t]=0\).
• Confusing robust with exact. The usual HAC result is an asymptotic approximation. Its reliability depends on sample size, dependence, moments, bandwidth, and identification strength.

The main practical limitation is that long-run covariance estimation itself can be noisy. No kernel or bandwidth can recover reliable information that the sample does not contain, and an estimated HAC covariance does not address model misspecification or invalid instruments. Interpret results alongside the IV design, identification evidence, and bandwidth sensitivity.