A stock's 36-month rolling beta has ranged from 0.7 to 1.5 and its 12-month rolling beta from 0.3 to 2.1. How much of that is estimation noise, how would you test for genuine change, and which window would you use?

Almost all of the swing is noise. A 12-month beta here has a standard error of about 0.38 and a 36-month beta about 0.21, so ranges of 0.3 to 2.1 and 0.7 to 1.5 are roughly what a constant beta of 1.1 would produce: simulated ranges are about 0.1 to 2.1 and 0.7 to 1.5. Test change on non-overlapping windows, and estimate on two years of weekly data shrunk towards 1.

Think of estimating a cricketer's average from one series. Three innings tell you little; thirty tell you a lot, and the improvement comes with the square root, not in a straight line. A regression beta has a standard error of about the stock's idiosyncratic volatility divided by the market's volatility, divided by the square root of the number of observations. Here the idiosyncratic volatility is about 5.4% a month against the market's 4.5%, a ratio of 1.2. With 12 months the standard errorHow far an estimate typically lands from the true value because of the luck of the sample. About two thirds of estimates fall within one standard error. is 1.2 over the square root of 10, about 0.38; with 36 months it is about 0.21.

Compare the swing with the standard errors. The 12-month range, 0.3 to 2.1, is 0.8 to 1.0 either side of 1.1, about two and a half standard errors. The 36-month range, 0.7 to 1.5, is 0.4 either side, about two. Over 15 years a rolling estimate visits its extremes many times, so a range of two to two and a half standard errors either side is what a perfectly constant beta produces. Simulating a constant beta of 1.1 with these volatilities confirms it: the median range is about 0.1 to 2.1 for the 12-month window and 0.7 to 1.5 for the 36-month window, close to what the review found.

Never test on overlapping windows: consecutive 36-month estimates share 35 months, so they look smooth and trending even when nothing changes. Split the history into non-overlapping blocks and test whether the betas differ by more than their combined standard error. Two 36-month blocks each have a standard error of about 0.21, so their difference has one of about 0.29; a change from 0.8 to 1.2 is only about 1.4 of those, not evidence. A formal version is a regression with an interaction term on a dummy for the later period, or a structural break test across candidate dates, adjusted for having searched over dates. Look also for a business reason: a large acquisition or a change in debt would make a genuine change plausible.

The 12-month monthly beta is the worst available choice: it hedges mostly noise and the hedge flips around from quarter to quarter, costing trading fees for nothing. Use more observations, not an older window: two years of weekly returns give 104 points and a standard error near 0.12, while still adapting within a couple of years. Then shrink the estimate part of the way towards 1, the average beta, because extreme estimates are more often extreme by luck. Weekly rather than daily returns avoid the bias from shares that trade less often than the index.

Say the limit. The square-root rule assumes the stock's own noise is stable and unrelated to the market; if idiosyncratic volatility jumps in crises, standard errors in those windows are larger still. And longer windows trade noise for staleness: if the business really did change, a five-year window will take years to notice.