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Suppose you want to calculate the beta of a stock to an index using weekly returns. If the stock is sufficiently volatile, and you use few enough observations, it is possible that the absolute value of your beta estimate will be high even if there is no relationship between the stock and the index (i.e. the real beta is zero). So if you cannot reject the null hypothesis that beta is zero with enough confidence, you should probably assume that beta is zero.

But it is also possible that the actual beta is in fact high. Even though the confidence intervals around your beta estimate are large, your estimate of beta is still the best unbiased estimate you have, even though the variance is large. So then you should probably use the estimate of beta that you have.

Let's say you are using the betas to hedge your concentrated portfolio. How should you manage this dilemma? Should you assume that the stock is uncorrelated with the index and that beta is 0 as there is no significant evidence that it is not 0, or should you use the best unbiased estimate that you have? It seems like a bias-variance tradeoff.

If it depends on a particular use-case, are there examples where either decision would be preferable?

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  • $\begingroup$ Does volatility of the returns of a hedged portfolio goes down significantly? $\endgroup$ – LazyCat May 16 '18 at 17:18
  • $\begingroup$ @LazyCat We can only know that after the fact. What I am asking is how do we know if a particular beta is going to "work" in the future, and was not just a spurious historical observation. $\endgroup$ – rinspy May 17 '18 at 8:17
  • $\begingroup$ As you just mentioned yourself, you can only know that a particular beta is going to "work" after the fact. In cases like this people usually fall back on some heuristics, that is specific to their problem. AlexC mentioned one way: by default, in US equities, beta=1. I'm suggesting another: get beta from whatever data you have, then hedge the portfolio for the same dates. Yes, you'll have lookahead bias, but if even with this bias, the variance is not reduced by much, then based on the data there's no point in hedging. $\endgroup$ – LazyCat May 17 '18 at 14:46
  • $\begingroup$ @LazyCat we could consider beta to specific themes too (for example, "value" stocks). There we can assume that beta is 0 by default. A better idea to the in-sample testing that you suggested would be to backtest the hedging out-of-sample (i.e. forecast beta based on previous N years, test it on the following week). Still, the estimate for beta that you get on the latest data can be very different from what you saw in the past. At the same time, if beta estimates changes a lot over time, it will not be a good predictor of future beta. $\endgroup$ – rinspy May 17 '18 at 15:14
  • $\begingroup$ @LazyCat point is, I still need to forecast the beta as well as I can. In my post I outlined to contradicting approaches - fail to reject the null hypothesis and assume default beta or accept the unbiased (but high variance) estimate. When and why would we use either approach? Surely there is a better way of doing it rather than using ad-hoc heuristics? $\endgroup$ – rinspy May 17 '18 at 15:22

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