Minimum Return History for Three- and Five-Year Beta Estimates

# Period length and maximum data points on estimating the 5-year Beta-factor


# Period length and maximum data points on estimating the 5-year Beta-factor












I currently read chapter 8 Beta from Bali, Engle and Murray's book Empirical Asset Pricing: The Cross Section of Stock Returns and do not understand their estimation on the five-year Beta-factor (denoted as $\beta^{5Y}$) using monthly data. On p. 124 they write:

> We also calculate market beta using monthly excess return observations over the past one, two, three, and five years, requiring 10, 20, 24, and 24 valid monthly excess return observations, respectively. Our choice to require a maximum of 24 monthly data points to calculate beta, even for the five-year measure, follows common practice when using monthly data to estimate beta.

How would $\beta^{5Y}$ with 24 data points differ from $\beta^{3Y}$ also using a maximum of 24 monthly data-points?

The only interpretation would be, that $\beta^{5Y}$ requires a data-history of 5 years prior to the estimation date of $\beta$, but only using the latest 24 available monthly data for estimation. However, on table 8.1 they report on a CRSP-sample from June 1963 - November 2012 an average observation of 3,958 estimations of $\beta^{3Y}$ and 3,992 average observations of $\beta^{5Y}$ per year. Assuming that $\beta^{5Y}$ requires a longer observation history prior to the date of estimation, one would assume less observations of $\beta^{5Y}$ than of $\beta^{3Y}$.

Additionally, does one know a financial paper following this common practice on estimating $\beta^{5Y}$ with requiring a maximum of 24 prior data observations?

[1] Bali, Turan G., Robert F. Engle, and Scott Murray. Empirical asset pricing: the cross section of stock returns. John Wiley & Sons, 2016.

## Answer by Tim Wilding (score 2, accepted)

https://quant.stackexchange.com/a/39195

I don’t have access to the text, but I think I can see an alternative interpretation. I think the maximum requirement that they are referring to is actually the largest minimum required number of data points!

$\beta^{5Y}$ would typically be calculated from a data sample that goes back 5 years and would use up to 60 months of returns for calculation. However, securities may have fewer than 5 years of data for several reasons (suspended, recent IPO, illiquid). It is possible that the authors require at least 24 months of excess returns for the calculation of $\beta^{5Y}$ (although they would obviously prefer to have the full 60 month sample).

Similarly, $\beta^{3Y}$ would ideally be calculated from 36 months of excess returns. In situations without full returns, they require at least 24 months of excess returns for the calculation.

Given that, there might be more securities with $\beta^{5y}$ than $\beta^{3y}$ in their data sample because the 3 year calculation requires a far higher proportion of valid data to calculate $\beta^{3y}$ ( = 24 / 36) than we do to calculate $\beta^{5y}$ ( = 24 / 60). Hence, more securities are likely to pass the minimum data requirement.

I don’t know how typical Bali et al.'s requirements are, but papers usually impose some form of minimum data requirement. For example, Chordia, Goyal, and Shanken (2015, p6 at https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2549578 ) require at least 400 days from the last 2 years of data to calculate $\beta$.