# Beta and Frequency of Data

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

• 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). Apr 12 '14 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. Apr 15 '14 at 6:24

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.
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