BL-A-002: Statistical Appendix

Status: RELEASED

Classification: PUBLIC

Revision: 1

Date: 2026-09-07

Every statistical quantity used anywhere in this record, defined once. Notation follows BL-D-011.

1. The correlation correction

The single most consequential number on this substrate.

Given N instruments with mean pairwise return correlation rho, the effective number of independent observations per timestamp is

N_eff  =  N / ( 1 + ( N - 1 ) * rho )

Measured across the fifteen major perpetuals:

rho    =  0.737
N_eff  =  15 / ( 1 + 14 * 0.737 )  =  1.33

Fifteen instruments carry the information of roughly one and a third.

The consequence for a standard error is that treating N correlated observations as independent understates it by approximately

sqrt( N / N_eff )  =  sqrt( 15 / 1.33 )  =  3.36

A t-statistic computed per leg on this panel is therefore overstated by a factor of roughly three and a third. That is not a refinement. It is the difference between a result at t=4 and a result at t=1.2.

2. Portfolio collapse

The correction is applied by collapsing before resampling, not by adjusting afterwards.

R_t  =  sum over i of  w_{i,t} * r_{i,t,h}

One portfolio return per timestamp. All subsequent inference runs on the series R, of length T, rather than on the N by T panel.

3. Moving-block bootstrap

Given R of length T and block length b:
 
  n_blocks  =  ceil( T / b )
  For each of B replications:
      draw n_blocks starting positions uniformly from 1 .. T - b + 1
      concatenate the blocks, truncate to length T
      record the mean
 
  CI95  =  [ 2.5th percentile, 97.5th percentile ] of the B recorded means

Block length exceeds the holding horizon, so that overlapping positions fall inside a block rather than across a boundary. B is at least 1000 throughout this record.

4. Permutation nulls

Sign permutation, for directional strategies:

eps_t  ~  Uniform{ -1, +1 }, independent across t
s_t    ->  eps_t * s_t

Cross-sectional shuffle, for panel strategies:

within each timestamp t, apply a random permutation to the weights w_{.,t}

Reported quantities:

p_perm       =  ( 1 + |{ b : X_b >= X_obs }| ) / ( 1 + B )
sigma_above  =  ( X_obs - mean(X_null) ) / sd(X_null)

The 1 + in the numerator and denominator of p_perm is the standard finite-sample correction and prevents a reported p-value of exactly zero from a finite number of permutations.

The null mean is reported rather than assumed to be zero. On a cost-bearing strategy the null centres near minus the round trip, because a randomised strategy still transacts.

5. Multiple-testing correction

Benjamini-Hochberg, controlling the false discovery rate at level alpha across a family of m tests:

sort p_(1) <= p_(2) <= ... <= p_(m)
find the largest k such that  p_(k) <= k * alpha / m
reject hypotheses 1 through k

Applied across the stop-level sweep in BL-SD-040, where two of four levels survived correction as significantly negative.

Deflation for search. A Sharpe ratio selected as the best of M configurations is not the Sharpe ratio of a single test. A naked Sharpe is not reportable anywhere in this record, and the number of configurations tried is recorded alongside any Sharpe that is reported.

The hypothesis budget of twelve in Programme II is the same control applied at programme level rather than at test level.

6. Information coefficient, and its prohibition

IC(h)  =  Spearman rank correlation between  s_t  and  r_{t,h}

The coefficient is permitted for characterisation and forbidden in any decision path. R-07 states the rule and this is the evidence for it.

Converting an information coefficient into an expected profit over-predicted the realised figure by the following factors on three separate occasions:

The ratio is not merely biased. It is unstable across configurations, which means no correction factor rescues the conversion. A coefficient describes the shape of information. It does not price it.

7. Distributional measures

Skewness, used in the crowding hypothesis:

skew  =  E[ ( r - mu )^3 ] / sigma^3

The hypothesis predicted a more negative forward skew under crowding. Measured, crowded skew was +1.087 against a matched baseline of +0.412, a difference of +0.675 with a 95 percent interval of [-0.938, +2.020]. The sign is opposite to the prediction.

The mirror test. For any hypothesis claiming a conditioning variable separates outcomes, the same measurement is run on the opposite condition. If both sides produce the same sign, the variable is not separating; it is sorting on something correlated with return in both states.

This closed the positioning-drift hypothesis: forward return was +1.4 basis points after accumulation and +6.7 after release.

8. Brier skill

Used for the probabilistic regime gate.

BS      =  ( 1 / n ) * sum over i of  ( p_i - o_i )^2
BSS     =  1 - BS / BS_reference

where p_i is the forecast probability, o_i the realised binary outcome, and the reference is the base-rate forecast.

The score sits below the threshold, so the component emits a binary decision rather than a probability. A model not calibrated well enough to express a probability should not be permitted to express one.

9. Markout

markout(h)  =  side * ( mid_{t+h} - fill_price_t ) / fill_price_t * 1e4

Negative means the price moved against the passive side. Compared against the half-spread earned rather than read alone.

10. Anchor sensitivity

For any strategy with a periodic rebalance of period P:

n_positive     =  |{ i : r_i > 0 }|
n_significant  =  |{ i : |t_i| > 1.96 }|
rank(k)        =  1 + |{ i : r_i > r_k }|
sign_consistent =  all phases share the sign of the median

Reporting a single phase without disclosing that P phases exist is unacknowledged multiple testing. BL-IS-004 is the instrument and BL-IN-006 is why it exists.

11. Cost identity

E[r_net]  =  E[r_gross] - c_rt
c_rt      =  2 * fee + spread_cost + slippage + funding_accrual
 
funding_bps  =  direction * sum over settlements k in [t, t+h] of  rate_k * 1e4

For a hedged pair, the fee term scales with the hedge ratio:

c_rt_pair  =  fee_per_leg * ( 1 + beta )  +  spread_per_leg

12. Reporting conventions

Every quantity carrying a verdict is reported as a triple: point estimate, interval, sample size. Test statistics name their test. Intervals are 95 percent unless stated. Returns and costs are in basis points to one decimal place with a leading minus sign for negatives.

Nothing is described as significant without the interval that makes it so.

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