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Get HAC (Newey-West) standard errors on a regression

Serially correlated errors do not bias OLS coefficients — they wreck the standard errors. In tsecon the whole robust ladder is one keyword on tsecon.ols: se_type="nonrobust", "hc0", "hc1", "hc2", "hc3", or "hac". This recipe is about the last one. The HAC rung is the Bartlett-kernel Newey-West estimator and matches statsmodels' cov_type="HAC" to 1e-10 when use_correction is matched — the two libraries default that flag opposite ways (see Gotchas); the hc* rungs fix heteroskedasticity only, and the section below shows why they are no help here and where they are indispensable instead.

The recipe

import numpy as np, tsecon

rng = np.random.default_rng(1)                     # persistent regressor, persistent errors
n = 300
x = np.zeros(n)
e = np.zeros(n)
for t in range(1, n):
    x[t] = 0.8 * x[t - 1] + rng.standard_normal()
    e[t] = 0.7 * e[t - 1] + rng.standard_normal()
y = 1.0 + 0.5 * x + e                              # true slope = 0.5
X = np.column_stack([np.ones(n), x])               # add your own constant column

for se in ("nonrobust", "hc1", "hac"):
    r = tsecon.ols(y, X, se_type=se)
    print(f"{se:>9}   beta {r['params'][1]:.4f}   se {r['bse'][1]:.4f}   t {r['tvalues'][1]:5.2f}")

nonrobust   beta 0.5298   se 0.0423   t 12.53
      hc1   beta 0.5298   se 0.0441   t 12.02
      hac   beta 0.5298   se 0.0681   t  7.78

Reading it

Three things to internalise, all visible in three lines of output.

  1. The coefficient never moves. Robust standard errors reweight uncertainty, not estimates. If a package changes your point estimate when you change se_type, something else is going on.
  2. hc1 does almost nothing here — and neither does any other HC rung. White-type corrections fix heteroskedasticity and assume independent errors. On time series data they are just as overconfident as nonrobust. Reaching for "robust standard errors" out of cross-sectional habit is the single most common inference mistake in applied time series.
  3. HAC is 60% wider, and the t-statistic falls from 12.5 to 7.8. In a borderline case that is the difference between a published finding and a null.

tsecon.ols returns params, bse, tvalues, and the echoed se_type — no p-values, deliberately, because the right reference distribution depends on the inference you are doing. 2 * scipy.stats.norm.sf(abs(t)) is the usual normal approximation if you want one.

Choosing the bandwidth

maxlags=None (the default) uses the Newey-West rule of thumb. Set it explicitly when you know the persistence in your data outruns the rule:

print("Newey-West rule of thumb :", round(tsecon.ols(y, X, se_type="hac")["bse"][1], 4))
print("maxlags=12               :", round(tsecon.ols(y, X, se_type="hac", maxlags=12)["bse"][1], 4))

Newey-West rule of thumb : 0.0681
maxlags=12               : 0.0756

Widening the window raises the standard error by another 11%. That sensitivity is real and it is why a HAC standard error without its kernel and bandwidth is not a reportable number — say "Newey-West, Bartlett kernel, 12 lags" in the table note. When you replicate someone else's HAC t-statistic and it does not match, the bandwidth is the first suspect, not the estimator.

If the problem is heteroskedasticity, not serial correlation

Everything above assumes you diagnosed the disease correctly. If the errors are independent but unequally scaled, HAC is the wrong tool and the leverage- corrected hc2/hc3 rungs are the right one. Run both claims on the same page:

print("hc3 on the recipe design :", round(tsecon.ols(y, X, se_type="hc3")["bse"][1], 4))

g = np.random.default_rng(7)                        # short sample, NO serial correlation
m = 25
xs = g.chisquare(1.0, m)                            # right-skewed -> a few high-leverage points
ys = 1.0 + 2.0 * xs + xs * g.standard_normal(m)     # sd(e|x) = x
Xs = np.column_stack([np.ones(m), xs])

for se in ("nonrobust", "hc1", "hc2", "hc3", "hac"):
    print(f"{se:>9}   se {tsecon.ols(ys, Xs, se_type=se)['bse'][1]:.4f}")

hc3 on the recipe design : 0.0446
nonrobust   se 0.2327
      hc1   se 0.4805
      hc2   se 0.5450
      hc3   se 0.6496
      hac   se 0.4662

The first line closes the loop on point 2 above: climbing to the top of the HC ladder on the serially correlated design moves the standard error from hc1's 0.0441 to 0.0446, about 1%, while hac needs 0.0681. HC is not a partial fix for autocorrelation.

The second block is the mirror image. hc1 scales every squared residual by the same \(n/(n-k)\); hc2 and hc3 divide each one by \(1-h_t\) and \((1-h_t)^2\), where \(h_t\) is that observation's leverage — so only they can tell that a handful of points are carrying the design. Here hc3 is 35% wider than hc1, and the coverage audit measures what that width is worth: on this design at \(n=25\), hc1 covers 0.68 against a nominal 0.95 while hc3 covers 0.86.

Note the last row. hac is narrower than hc1 here, because it carries no leverage correction at all — at zero lags it reduces to hc0, never to hc2 or hc3. Neither family contains the other. Pick by which assumption your data violates: serial correlation → hac; heteroskedasticity with a short sample or influential points → hc3; both → hac, and treat it as a floor.

Gotchas

  • x is used exactly as given. tsecon.ols adds no constant; build the design matrix yourself with np.column_stack([np.ones(n), ...]).
  • use_correction toggles the small-sample \(n/(n-k)\) factor, and the packages default it opposite ways: tsecon defaults it on (the finite-sample choice), statsmodels cov_type="HAC" defaults it off (its cov_hac_simple helper, confusingly, defaults on). So tsecon.ols(..., se_type="hac") and a bare statsmodels .fit(cov_type="HAC", ...) disagree by exactly \(\sqrt{n/(n-k)}\) — 4.3% at \(n=25, k=2\) — while matching either flag setting agrees to 1e-10. Pass use_correction=False to reproduce a default statsmodels call. It is the first thing to check, after the bandwidth, when two HAC numbers disagree.
  • HAC undercovers in small samples even when the bandwidth is right — see the coverage experiment and the fixed-b / EWC discussion in the guide chapter below.
  • use_correction does not apply to hc2/hc3: their correction is per-observation, through leverage, not a single scalar. And if some observation's leverage is numerically 1, HC2/HC3 are undefined — tsecon raises rather than handing back a near-infinite standard error.
  • Estimating a long-run variance on its own rather than a regression? That is tsecon.long_run_variance.

See also