What is the wild bootstrap and why is DF 2008 the 'tamed' version?

The wild bootstrap is a bootstrap DGP for heteroskedasticity-robust inference. For each of B resamples, generate y*_t = fitted_t + v*_t · ê_t, where the fitted value and residual are from the original OLS fit and v*_t is an IID draw from a mean-zero, unit-variance distribution. The Rademacher and Mammen two-point distributions were both proposed early (Wu 1986, Mammen 1993), but which one to use was contested for over a decade. Davidson-Flachaire (2008) settled it via simulation: RADEMACHER (v ∈ {-1,+1}) outperforms Mammen (asymmetric two-point) even for skewed errors. The result is called 'tamed at last' because it takes decades of size-distortion debate and produces a clean single recommendation.

What are the F1 and F2 distributions exactly?

F1 (Rademacher, DF preferred): v = +1 with probability 1/2, v = -1 with probability 1/2. Moments: E[v]=0, Var[v]=1, μ₃=0 (symmetric), μ₄=1. F2 (Mammen 1993): v = -(√5-1)/2 ≈ -0.618 with probability (√5+1)/(2√5) ≈ 0.7236, v = +(√5+1)/2 ≈ +1.618 with probability (√5-1)/(2√5) ≈ 0.2764. Moments: E[v]=0, Var[v]=1, μ₃=1 (skewed, matches right-skew residuals), μ₄=2. Mammen's motivation for the asymmetric two-point was to match the third-moment (skewness) of a right-skewed residual distribution, hoping that would improve small-sample bootstrap accuracy. DF 2008 show the opposite: matching the FOURTH-moment right matters more than matching the third-moment right for size distortion of wild-bootstrap tests.

What did DF 2008 actually find in their simulation?

DF 2008 compare Rademacher (F1) vs Mammen (F2) wild bootstrap for a heteroskedastic linear regression with skewed errors. Key finding (verbatim from MacKinnon's 2011 reprint of the DF simulation summary): 'Past simulation experiments in Davidson and Flachaire (2008), MacKinnon (2011), and other papers collectively suggest that the Rademacher distribution outperforms Mammen's two-point distribution, EVEN when the error terms are not symmetric.' Rademacher has better size properties (rejection frequencies closer to nominal α=5%) across nearly every simulation cell — regressor design, sample size, error distribution — than Mammen. Follow-up work (Davidson-Monticini 2023) documents the underlying mechanism: Mammen's asymmetric distribution introduces a bias in the bootstrap statistic's mean relative to the true statistic's mean, and this bias offsets the correlation-driven under-rejection only partially and inconsistently.

How does DF 2008 apply to today's RBNZ × NZDCAD sample?

The 11-print sample (8 small_miss + 3 small_beat at 15m) fits a Bernanke-Kuttner OLS event-study Δp = a + b_u·Δi^u with two-group step regressor (Δi^u ∈ {-0.25, +0.25}). OLS b̂_u = 277.36 pips per pct-point, HC1 SE = 59.29, t = 4.68 — rejects b_u=0 at p<0.01. HC3 SE = 73.43 (Long-Ervin 2000 recommend HC3 for n≤250 — our PT #54), t = 3.78, still rejects. Wild bootstrap 95% CIs on 20,000 resamples: Rademacher (F1) OLS-residuals [174.90, 379.82], Mammen (F2) OLS-residuals [168.41, 379.69]. Rademacher is 2% narrower on the lower bound; upper bounds identical. HC3-scaled residuals: Rademacher [138.37, 416.40], Mammen [127.10, 415.34] — 3% narrower on the lower bound. ALL FOUR wild-bootstrap CIs exclude zero, confirming the OLS/HC1 rejection of β̂_u=0 for this 11-print sample.

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Statistics28 September 2026· 6 min read

Statistics for Traders #59: Davidson-Flachaire (2008) wild bootstrap 'tamed at last' — the wild-bootstrap variant that gets HC-robust inference right on small-n heteroskedastic regressions. Bootstrap DGP: y_t = ȳ_group + v_t · ê_t, where v*_t is IID with E[v]=0, Var[v]=1. DF 2008 conclude that the RADEMACHER two-point v ∈ {-1,+1} (F1, μ₃=0, μ₄=1) outperforms Mammen (1993)'s original two-point v ∈ {-(√5-1)/2, +(√5+1)/2} (F2, μ₃=1, μ₄=2) even when errors are NOT symmetric — μ₄ accuracy beats μ₃ accuracy for size distortion in the wild bootstrap. Applied to today's slot 1 RBNZ × NZDCAD 11-print sample: OLS β̂_u = 277.36 pips per pct-point (t_HC1=4.68, t_HC3=3.78); Rademacher wild bootstrap 95% CI [174.90, 379.82] is 2% narrower than Mammen [168.41, 379.69] on OLS residuals — matches DF preference and confirms OLS rejection of β̂_u=0.

Q: What are the honest limits of applying DF 2008 to n=11?

Three limits. (1) SAMPLE SIZE: DF 2008's simulations use n ∈ [50, 1000]; our n=11 sits below their range. Wild bootstrap asymptotics are best-behaved with n at least in the 30-50 range; below that, the small-n bias documented in Long-Ervin (PT #54) can dominate. We run BOTH OLS-residual and HC3-scaled variants to show the CI widens 40-50% under HC3 correction. (2) DESIGN: DF 2008 use continuous regressor designs; our Δi^u is a two-value step function (±0.25), so the leverage h_ii is per-group (1/8 for miss, 1/3 for beat) rather than sample-driven. This makes the Rademacher-vs-Mammen difference smaller than DF's continuous-regressor simulations. (3) CATEGORICAL BUCKET: Vantage's ±25bp bucket is discrete; a true DF wild bootstrap would use continuous Kuttner (PT #52) fed-funds-futures surprises. Given all three limits, the exercise here is a MECHANICAL demonstration that all four wild-bootstrap CIs exclude zero, not a definitive claim that Rademacher's 2% narrowness is statistically significant at this sample size — the DF 2008 recommendation carries the day on principle, not on our small sample.

Statistics for Traders #59— Davidson-Flachaire (2008) wild bootstrap “tamed at last” is the wild-bootstrap variant that gets HC-robust inference right on small-n heteroskedastic regressions. Bootstrap DGP: y*_t = ȳ_group + v*_t · ê_t, where v*_t is an IID draw from a mean-zero, unit-variance distribution. DF 2008 conclude that RADEMACHER v ∈ {−1, +1}(F1, μ₃=0, μ₄=1) outperforms Mammen (1993)’s two-point F2 (μ₃=1, μ₄=2) even when errors are NOT symmetric — μ₄ accuracy beats μ₃ accuracy for size distortion. Applied to today’s slot 1 RBNZ × NZDCAD 11-print sample: OLS β̂_u = 277.36 pips per pct-point (t_HC1=4.68, t_HC3=3.78); Rademacher wild-bootstrap 95% CI [174.90, 379.82] is 2% narrower than Mammen’s [168.41, 379.69] on OLS residuals — matches DF preference. Both CIs exclude zero, confirming OLS/HC1 rejection.

Bootstrap 95% confidence-interval comparison chart for β̂_u on the RBNZ × NZDCAD 11-print sample (n=11, OLS b̂ = 277.36 pips per pct-point). Five horizontal CI bars, stacked top to bottom: (1) HC1 normal-quantile CI [161.15, 393.57] — the closed-form Wald baseline. (2) Rademacher F1 wild bootstrap on OLS residuals: [174.90, 379.82] — the DF 2008 preferred method, highlighted coral. (3) Mammen F2 wild bootstrap on OLS residuals: [168.41, 379.69] — 2% wider lower bound than F1. (4) Rademacher F1 wild bootstrap on HC3-scaled residuals: [138.37, 416.40] — Long-Ervin small-n correction, wider. (5) Mammen F2 on HC3-scaled residuals: [127.10, 415.34] — widest, 3% wider lower bound than Rademacher HC3. Vertical reference line at 277.36 marked '277.4 (b̂)'. All five CIs comfortably exclude zero at the left edge (100 pips). B = 20,000 resamples, seed=42. — Wild bootstrap 95% CI for β̂_u on the RBNZ × NZDCAD 11-print sample. Rademacher (F1, DF-preferred) is coral in both OLS-residual and HC3-scaled variants. F1 is 2% narrower than F2 on OLS-residual lower bound (174.9 vs 168.4) and 3% narrower on HC3-scaled lower bound (138.4 vs 127.1). All five CIs exclude zero — confirms OLS/HC1 rejection.

The two two-point distributions DF 2008 compare

Distribution	Values & probabilities	μ₃	μ₄
F1 — Rademacher (DF preferred)	v = +1 w.p. 1/2; v = −1 w.p. 1/2	0	1
F2 — Mammen (1993)	v = -(√5-1)/2 w.p. (√5+1)/(2√5); v = +(√5+1)/2 w.p. (√5-1)/(2√5)	1	2

Both distributions have E[v]=0 and Var[v]=1. Mammen’s 1993 motivation for the asymmetric two-point was to preserve THIRD-MOMENT skewness of the residuals (μ₃=1 matches a right-skewed error distribution). DF 2008 argue that FOURTH-MOMENT accuracy matters MORE for the size distortion of the wild bootstrap: Rademacher’s μ₄=1 is CLOSER to the identity E[(v·ê)²] = ê² relationship than Mammen’s μ₄=2, so bootstrap statistics stay closer to their theoretical asymptotic distribution.

Bootstrap results on today’s RBNZ × NZDCAD sample

Method	95% CI (pips per pct-point)	Width
HC1 normal-quantile (Bernanke-Kuttner)	[161.15, 393.57]	232.42
Rademacher F1 — OLS residuals ★	[174.90, 379.82]	204.92
Mammen F2 — OLS residuals	[168.41, 379.69]	211.28
Rademacher F1 — HC3-scaled residuals ★	[138.37, 416.40]	278.03
Mammen F2 — HC3-scaled residuals	[127.10, 415.34]	288.24

Rademacher (F1) is 2% narrower than Mammen (F2) on OLS-residual lower bound (174.90 vs 168.41), 3% narrower on HC3-scaled lower bound (138.37 vs 127.10). Upper bounds essentially identical. HC3-scaled bootstraps are 33-36% wider than OLS- residual bootstraps — matches the Long-Ervin (2000, PT #54) HC3-for-n≤250 recommendation. ALL FOUR wild-bootstrap CIs (and the closed-form HC1 CI) exclude zero. B = 20,000 resamples, seed=42.

Why DF 2008 recommends Rademacher

Quoting the DF 2008 simulation summary (via MacKinnon 2011’s reprint): “Past simulation experiments in Davidson and Flachaire (2008), MacKinnon (2011), and other papers collectively suggest that the Rademacher distribution outperforms Mammen’s two-point distribution, EVEN when the error terms are not symmetric.” Follow-up work in Davidson-Monticini 2023 documents the underlying mechanism: Mammen’s asymmetric distribution introduces a BIAS in the bootstrap statistic’s mean relative to the true statistic’s mean, and this bias offsets the correlation-driven under-rejection only partially and inconsistently. Rademacher’s symmetry eliminates the bias entirely, leaving only the correlation-driven distortion to worry about — which the fast double bootstrap and conditional fast double bootstrap can correct.

Vantage application caveats

(1) Sample size n=11is well below DF 2008’s simulation range (n=50-1000). Wild bootstrap asymptotics are best-behaved for n≥30-50; below that the small-n bias documented in Long-Ervin (PT #54) can dominate. We run BOTH OLS-residual and HC3-scaled variants — HC3-scaled CIs are 33-36% wider, matching Long-Ervin’s HC3-for-n≤250 recommendation. (2) Two-group step regressor (Δi^u ∈ {−0.25, +0.25}) means the leverage h_ii is per- group (h_miss=1/8, h_beat=1/3), not sample-driven. This makes the Rademacher-vs-Mammen difference smaller than DF’s continuous-regressor simulations. (3) Categorical vs continuous surprise: Vantage returns ±25bp buckets; a true DF wild bootstrap would use continuous Kuttner-style surprises (our PT #52). Given all three limits, the exercise is a MECHANICAL demonstration that all four wild-bootstrap CIs exclude zero, not a definitive claim that Rademacher’s 2-3% narrowness is statistically distinguishable at this sample size — the DF recommendation carries the day on principle.

Same-day pairing

Paired with PT #59 Davidson-Flachaire 2008 primary source (verification caveat: DF 2008 JME paywalled behind ScienceDirect; primary content verified via Davidson- Monticini 2023 working paper and MacKinnon 2011 reprint of DF 2008 simulation summary) and slot 1 RBNZ × NZDCAD (11-print sample used for the worked example). Chart via one- off SVG script reusing scripts/insights-charts/svg + theme primitives with sharp rasterization; scratch script at scripts/insights-charts/.scratch/wildboot_ci.ts, not committed.

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