Prewhitening for HAC Covariance Estimation

Why Prewhiten a HAC Estimate?

HAC inference estimates uncertainty while allowing regression errors to be heteroskedastic and serially dependent. Its accuracy can depend on how well a finite bandwidth and kernel capture the serial dependence in the regression scores. If those scores are highly persistent, a finite-bandwidth estimate may capture their long-run covariance poorly. Prewhitening offers one adjustment: fit a short autoregressive filter to the scores, estimate a HAC covariance for the filtered sequence, and then recolor that covariance to put it back on the original scale.

Prewhitening is a covariance-estimation step, not a new regression model for the outcome. The regression coefficient estimate is unchanged; the procedure modifies the estimated covariance used for standard errors and inference. Its benefit depends on the filter and the remaining dependence being estimated well. Filtering does not automatically improve an estimate, and the recoloring step is essential.

Scores and the Prewhitening Filter

For a linear regression with \(T\) ordered observations and \(k\) regressors, let \(x_t\) be the \(k\)-vector of regressors for observation \(t\), and let \(\hat u_t\) be its fitted regression residual. The estimated score is the \(k\)-vector \(\hat g_t=x_t\hat u_t\). The long-run covariance of the scores is the object used in the HAC sandwich covariance estimator. It accounts for both contemporaneous score variation and covariation across time.

The autoregressive filter is commonly fitted by least squares, regressing each component of \(\hat g_t\) on the lagged score vector \(\hat g_{t-1}\). This is a multivariate regression through the origin; an intercept is not part of the stated filter. Because one lag is needed, the filtered sequence has \(T-1\) observations. A HAC estimator with a chosen kernel and bandwidth is then applied to that sequence, using a consistent normalization for its length.

A standard stability condition is that all eigenvalues of \(A\) have absolute value less than one. It ensures that the autoregressive filter has a stable long-run multiplier \((I-A)^{-1}\). The corresponding estimated condition is checked using the eigenvalues of \(\hat A\). If an estimated eigenvalue is very close to one, recoloring can amplify estimation error substantially.

Deriving the Recoloring Formula

Let \(\Omega_g\) denote the long-run covariance matrix of the stationary score process \(g_t\), and let \(\Omega_v\) denote the long-run covariance matrix of the filtered process \(v_t\). A long-run covariance is the sum of the lag covariances over all integer lags, when this sum exists. Under the stable VAR(1) relation, the score can be written as a linear filter of current and past innovations:

At long-run frequency, the matrices multiplying the innovations sum to \(I+A+A^2+\cdots=(I-A)^{-1}\). Equivalently, summing the VAR equation over a long stretch of observations gives \((I-A)\) times the accumulated scores equal to the accumulated innovations, up to boundary terms. For a stable process, those boundary terms do not contribute to the covariance per observation in the long run. Thus the long-run score variation is the innovation long-run variation transformed by \((I-A)^{-1}\) on the left and its transpose on the right.

The transpose on the right is important: covariance matrices transform as \(B\Omega B'\) when a vector is multiplied by a matrix \(B\). In an empirical procedure, replace \(A\) by \(\hat A\) and \(\Omega_v\) by a HAC estimate \(\hat\Omega_v\) computed from the filtered residual scores. The resulting estimate is the recolored HAC estimate \(\hat\Omega_g\). The filter need not eliminate every serial correlation; the HAC step still estimates the remaining long-run covariance of the filtered scores.

For example, using the usual lag-covariance convention, define \(\hat\Gamma_h\) for \(h\geq 0\) as the lag-\(h\) covariance estimate for the filtered scores, with \(\hat\Gamma_h\) formed from products \(\hat v_t\hat v_{t-h}'\). A kernel HAC estimate with bandwidth \(L\) has the form

where \(w_h\) is the selected kernel weight at lag \(h\). The kernel and bandwidth govern the residual HAC calculation; recoloring is a separate matrix transformation applied afterward. With these definitions, if \(\hat Q=X'X/T\), the estimated covariance matrix of the OLS coefficient vector is

Here \(X\) is the regression design matrix, and the formula assumes \(\hat Q\) is nonsingular. It is equivalent to using the unnormalized HAC “meat” in the usual sandwich form; keeping track of the \(1/T\) normalization avoids accidentally multiplying or dividing the covariance by \(T\) twice.

From Long-Run Covariance to Coefficient Uncertainty

Recoloring estimates uncertainty in the original score sequence, not directly in each regression coefficient. To obtain coefficient uncertainty, place the recolored estimate into the regression sandwich. Both the score covariance and the design matrix matter: a high-variance direction in the scores can have a different effect on coefficients depending on the scale and collinearity structure of the regressors.

When Prewhitening Helps—and When It Can Mislead

Prewhitening can help when score dependence is persistent and a short stable autoregression captures an important part of that persistence. The remaining score dependence may then be easier for a finite-bandwidth HAC estimator to represent. Recoloring preserves the long-run scale targeted by the original score sequence rather than treating the filtered residuals as though they were the original scores.

The correction is not guaranteed to improve finite-sample inference. The estimated autoregressive coefficients are uncertain, and that uncertainty can matter when \(T\) is small or the fitted filter is close to unstable. In that case, \((I-\hat A)^{-1}\) can have large entries, magnifying small estimation errors in both \(\hat A\) and \(\hat\Omega_v\). A misspecified filter may also leave substantial residual dependence. The residual HAC step is important for this reason; prewhitening is not a license to assume that filtered scores are independent.

The underlying long-run covariance target still requires appropriate regularity conditions, such as a stable dependence process with finite second moments and a well-defined long-run covariance. Standard consistency of the HAC step also relies on appropriate time-series conditions and bandwidth behavior. In applications, inspect the fitted filter's stability and the filtered scores' remaining dependence, and consider whether the result is sensitive to reasonable filtering and bandwidth choices. These checks inform judgment but do not establish that a model or covariance estimate is correct.

Prewhitening is also distinct from transforming the regression observations to estimate coefficients under a specified error covariance model. Here the fitted coefficients remain the OLS coefficients; only the estimated covariance matrix used for inference is adjusted. The practical question is therefore not whether the filter makes the regression “whiter,” but whether the complete filter–HAC–recolor procedure provides a credible estimate of the scores’ long-run covariance.