# Effective breadth

> Twenty-two correlated assets are not twenty-two independent tests. M_eff measures how many you actually have.

## The problem

Cross-sectional testing is the best habit on this server: one specification
across many assets counts as one trial, and the assets are the sample.

But "22 assets" is not "22 independent observations" if the assets move
together. Crypto on a single venue moves together a great deal.

```
stats_effective_tests(universe="coinbase-liquid-22")
```

## What it measures

M_eff is computed from the eigenspectrum of the asset return correlation
matrix. For the 22 liquid Coinbase USD pairs with full history:

| Estimator | M_eff |
|---|---|
| Participation ratio | **3.0** |
| Entropy | **6.8** |
| Nominal asset count | 22 |

Mean pairwise correlation is **0.52**, and the first principal component
explains **57%** of variance.

Three independent tests. Not twenty-two.

## Both numbers are reported, and you need both

The server reports every estimator rather than reducing them to one, because
**the conservative direction is opposite for the two uses**:

- **Raising a multiplicity hurdle** — use the **larger** value. A hurdle
  based on the maximum over trials grows with the effective number of tests;
  understating it makes the hurdle too easy.
- **Widening a confidence interval** — use the **smaller** value. Fewer
  independent observations means a wider interval; overstating M_eff makes
  the interval too narrow.

Taking the smallest M_eff was documented as "conservative" in an earlier
version. For a maximum-over-trials hurdle it is *anti*-conservative, because
E[max SR] rises with N. The two directions are now reported separately.

## Why not the Kaiser criterion

M_eff no longer uses the Kaiser criterion (counting eigenvalues greater
than 1). It reports 1 for perfectly independent assets: with correlation
matrix `C = I` every eigenvalue is exactly 1, so none exceeds 1.

Participation ratio and entropy both satisfy the properties you want:

```
M_eff(independent assets) = k
M_eff(identical assets)   = 1
```

## What this means for your research

Adding coins on one venue buys far less breadth than the count suggests.
Going from 10 pairs to 22 pairs on Coinbase does not roughly double your
evidence; it adds a little.

Genuine breadth comes from sources that are not the same bet: different
venues, different asset classes, different time zones, different mechanisms.
Within a single crypto venue, most of what looks like diversification is one
factor wearing 22 hats.

Pooled standard errors on this server use M_eff, not the asset count.

## Cross-sectional dispersion is descriptive

The spread of per-asset results within a cross-sectional run is reported as a
**descriptive** statistic, not as a standard error. The assets are
correlated; treating their dispersion as if it came from independent draws
would understate uncertainty by roughly the square root of the ratio between
the asset count and M_eff.