Robust standard errors with Newey–West

When neighbouring errors are linked

Ordinary standard errors assume every observation brings new information. Financial data often breaks that. Returns measured over three months and sampled every month overlap, so each observation shares two months with its neighbour, and the errors move together. The same happens when volatility clusters. The fix keeps the coefficients and changes only the standard errors. The Newey–West estimator, also called HAC for heteroskedasticity and autocorrelation consistent, estimates the long-run variance by adding the lagged autocovariances $\gamma_\ell$ of the residuals, each multiplied by a Bartlett weight $1-\ell/(L+1)$ that shrinks to zero at the cut-off lag $L$. In statsmodels you only add cov_type='HAC' and cov_kwds={'maxlags': L} to .fit().

Real data shows the size of the effect. The average 3-month return of SPY, sampled monthly, is $2.78\%$ over $256$ overlapping observations. Ordinary standard errors give $0.456\%$ and $t=6.10$. The autocorrelations of the demeaned series are $0.68$, $0.35$ and $0.08$ at lags $1$, $2$ and $3$, the triangle that overlap of two months out of three creates. With Newey–West at $L=2$ the standard error is $0.664\%$ ($t=4.18$), at $L=4$ it is $0.732\%$ ($t=3.80$) and at $L=6$ it is $0.762\%$ ($t=3.65$), and the $86$ non-overlapping quarters give $t=3.55$. In a monthly three-factor regression for Apple ($319$ months) the change is smaller but real: the standard error of the value loading rises from $0.144$ to $0.170$, so its t-statistic moves from $-4.98$ to $-4.22$.

• Same coefficients, new standard errors: HAC changes only the uncertainty, never the estimate.
• Overlap sets the lag: with $h$-month returns sampled monthly, use at least $L=h-1$ and often a few more.
• Report the honest t: a t of 6 can be a t of 4 once neighbouring errors are accounted for.

Lab · what's graded

• The variance term10 pts
• The first autocovariance10 pts
• The long-run variance15 pts
• The HAC standard error15 pts
• The robust t-statistic10 pts
• The statsmodels call10 pts
• Match the vocabulary10 pts
• Why the ordinary t is too big20 pts