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Why are the betas of individual securities essentially the same whether we use daily or weekly data when calculating?

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This would only be true in population, the actual estimates would differ with probability 1 in an almost surely sense (since estimator distribution is over reals). –  user2763361 Apr 12 at 12:02
    
I see that you explicitly mentioned daily and weekly time scales, but more generally this is not the case because of microstructure contamination and issues like the Epp's Effect. This is particularly relevant for high frequencies. –  Jacob M. Morley Apr 15 at 6:24

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Suppose you have $$X\equiv\left(x_{1},\: x_{2}\right) $$ where $x_{1}$ are the daily log returns of the security and $x_{2}$ are the daily log returns of the market. Assume further that $X$ is iid multivariate normal $$X\sim N\left(\mu,\Sigma\right) $$ People frequently calculate beta as $$\beta_{1,2}\equiv\frac{\Sigma_{1,2}}{\Sigma_{2,2}} $$ If you convert $X$ from a daily series to a weekly series, you could say that the weekly variables are just the sum of the daily variables. Due to the properties of a normal distribution, this means you could write $$X_{weekly}\sim N\left(5\mu,5\Sigma\right) $$ This implies a weekly beta of $$\beta_{1,2}^{weekly}\equiv\frac{5\Sigma_{1,2}}{5\Sigma_{2,2}}=\frac{\Sigma_{1,2}}{\Sigma_{2,2}}$$ or that the beta is the same as the daily version.

There are a few wrinkles to this argument. First, returns may not be iid normally distributed, which would mean that the covariance of the weekly data may not be proportional to the covariance of the daily data. Second, the beta that really matters to an investor is the forward-looking beta on arithmetic returns. That beta is more complicated to calculate since it involves the conversion between the log returns and the arithmetic returns.

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It is only true that the beta would be unchanged under particular assumptions about the underlying process and what beta you're using. –  John Apr 11 at 18:34
    
There could be many different betas (e.g., Fama-French, APT), but I'm referring more to how to calculate beta. See papers.ssrn.com/sol3/papers.cfm?abstract_id=1619923 In reality, returns are not normally distributed, why should I assume they are? –  John Apr 11 at 18:44

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