What beta to Bitcoin means for an alt

A regression produces one slope for the whole sample, and that slope is an average. If an asset tracked bitcoin closely for most of the window and then decoupled for a month, the single beta describes neither period: it is a weighted blend of the two. The number is arithmetically correct and descriptively misleading.

The problem is worst in exactly the conditions where a reader most wants the estimate. Sensitivity to a shared factor is not constant; it tends to rise when the factor moves sharply, because the same selling pressure hits every asset in the complex at once. A beta estimated over a calm window therefore understates the sensitivity that applies during a stress event, and a beta estimated over a stressed window overstates the sensitivity that applies in calm conditions.

The practical response is to report beta as a rolling series rather than a single number, and to state the window. A rolling beta shows when the relationship changed, which is the information a single coefficient destroys. Where a single figure is unavoidable, it should be accompanied by the window and by an explicit statement that the estimate is an average across whatever regimes the window contained.

Dataset. Daily closing prices for bitcoin and for the non-bitcoin asset under comparison, from CoinGecko's historical price endpoint, with Coin Metrics community data as a cross-check. Aggregated closes across venues.

Period. The window actually used must be stated with any estimate. Because the large alts have different listing dates, a fixed calendar window implies a different number of observations for each asset, and that difference should be disclosed.

Method. Ordinary least squares of the asset's daily log return on bitcoin's daily log return, over a trailing window, recomputed at each date for a rolling series. No adjustment is made for non-synchronous closes, which biases the estimate downward for assets whose deepest market closes at a different hour.

Limitations. Beta is an average across the window and hides regime change. It is sensitive to outliers, and a single extreme day can move it materially on a short window. It assumes a linear relationship, which is a poor description of an asset that tracks bitcoin in calm markets and decouples in stress. And it is estimated against a reference that is itself volatile, so the estimate inherits bitcoin's own measurement error.