The Newey–West Bartlett Estimator

Why the Bartlett Estimator Matters

With dependent errors, the score contributions from nearby observations can covary. An ordinary heteroskedasticity-robust covariance estimator accounts for variation in each observation’s score but omits these cross-observation terms. A heteroskedasticity-and-autocorrelation-consistent (HAC) estimator adds lagged score products, while giving less weight to more distant lags and setting sufficiently distant lags to zero.

The Newey–West estimator uses Bartlett weights: the weight declines linearly with lag until it reaches zero. This weighting rule has an important finite-sample property: the estimated covariance matrix is positive semidefinite, so no linear combination of estimated coefficients has a negative estimated variance. That property follows from the weights themselves, not from a large-sample approximation.

Definition and Construction

Consider an ordered sample of \(n\) observations from the linear regression \(y_t=x_t'\beta+u_t\), where \(y_t\) is a scalar outcome, \(x_t\) is a \(k\)-vector of regressors, and \(\beta\) is a \(k\)-vector of coefficients. Let \(\hat\beta=(X'X)^{-1}X'y\) be the OLS estimate, and let \(\hat u_t=y_t-x_t'\hat\beta\) be its residual. Assume \(X'X\) is nonsingular. The estimated score contribution at observation \(t\) is \(g_t=x_t\hat u_t\), a \(k\)-vector.

The two products at each positive lag are transposes of one another, so the meat matrix is symmetric. The lag-zero term uses each score’s outer product once; each positive-lag cross-product pair is weighted by \(w_\ell\). In particular, \(w_0=1\), \(w_L=1/(L+1)\), and the Bartlett sequence would assign weight zero at lag \(L+1\). The displayed formula is the unscaled finite-sample sandwich form; it does not automatically include a degrees-of-freedom multiplier.

The construction can also be expressed through lagged score cross-products \(\widehat\Gamma_\ell=\sum_{t=\ell+1}^n g_tg_{t-\ell}'\). Then \(\widehat S_L=\widehat\Gamma_0+\sum_{\ell=1}^Lw_\ell(\widehat\Gamma_\ell+\widehat\Gamma_\ell')\), with \(\widehat\Gamma_0=\sum_{t=1}^n g_tg_t'\). This form emphasizes that the estimated long-run score variation combines contemporaneous and lagged dependence.

Why the Finite-Sample Matrix Is Positive Semidefinite

A symmetric matrix \(A\) is positive semidefinite if \(a'Aa\geq0\) for every vector \(a\) of the appropriate dimension. To establish the property for the Bartlett meat, define \(w_h=1-|h|/(L+1)\) when \(|h|\leq L\), and \(w_h=0\) otherwise. The meat can be written as \(\widehat S_L=\sum_{t=1}^n\sum_{s=1}^n w_{t-s}g_tg_s'\). This double sum is exactly the lag-zero term plus the two weighted cross-product terms for each positive lag in the definition.

For any two integer indices \(t,s\), the number of consecutive windows of length \(L+1\) that contain both indices is \(L+1-|t-s|\) when \(|t-s|\leq L\), and zero otherwise. Consequently, the Bartlett weight has the representation

Substituting this identity into the double sum and collecting terms within each window gives the finite-sample representation

Only finitely many windows in this sum contain any observed score. For any vector \(a\), let \(G_r\) denote the sum of scores in window \(r,\ldots,r+L\), restricted to the observed sample. Then

Every term is a square multiplied by a positive constant. This proves that \(\widehat S_L\) is positive semidefinite for any finite score sequence, without requiring the errors to be independent or the sample to be large. Since \(X'X\) is nonsingular, for any coefficient vector \(c\), \(c'(X'X)^{-1}\widehat S_L(X'X)^{-1}c\geq0\) as well: apply the meat result to \(a=(X'X)^{-1}c\). Thus the sandwich covariance estimate is also positive semidefinite.

Worked Regression Calculation

The next example calculates the full sandwich for a short, hypothetical regression. It makes the ordering explicit because the lagged products depend on which observations are adjacent.

What the Estimator Does—and Does Not—Establish

The Bartlett estimator is a way to estimate coefficient uncertainty when score contributions may be heteroskedastic and serially dependent. The coefficient estimates remain OLS; it is the covariance estimate that changes. For asymptotic HAC inference, the regression and dependence process must satisfy appropriate regularity conditions, such as a nonsingular limiting regressor second moment and sufficiently weak dependence and finite moments. Positive semidefiniteness alone does not verify those conditions, establish that the estimate is close to the true covariance, or guarantee accurate finite-sample confidence intervals and tests.

In practice, the user must specify a maximum lag \(L\) for the calculation. The Bartlett rule determines how lags up to that value are weighted; it does not choose a suitable value for a particular application. The number of observations, the dependence pattern, and the inferential goal matter when making that choice. Also, the observation order must correspond to the sequence whose dependence is being modelled: reordering observations changes the lagged products and therefore changes the estimate.

A common mistake is to calculate only \(\sum_t g_tg_{t-\ell}'\) at each positive lag, without adding its transpose. That leaves an asymmetric meat matrix and does not implement the stated HAC covariance formula. The correction is to include \(w_\ell(\widehat\Gamma_\ell+\widehat\Gamma_\ell')\) at every positive lag. Another mistake is to infer that a positive semidefinite estimate must be accurate: the proof guarantees nonnegative estimated variances, not correct uncertainty quantification under every dependence process or sample size.