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:

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.

Read this page as Markdown: /docs/concepts/breadth.md