“using daily returns over rolling annual periods from the regression”

Question

I stumbled upon the following sentences many times in different papers, all describing an approach for the same experiment: They try to calculate sensitivities (Beta) for different stocks via regression to then sort them into different buckets (portfolios). The papers say:

"For each stock we estimate factor loadings at the individual stock level using daily returns over rolling annual periods from the regression“

"We estimate factor loadings for individual stocks using daily returns over rolling annual periods from the regression“

"For the regressions, we use daily returns over rolling annual periods to estimate the sensitivities/Betas"

I do not fully understand this approach. Lets say we have data from 2018 until 2019 for 2 stocks. Do they:

A. Calculate the returns over a rolling window and then perform ONE regression per year (so as a result have one Beta per stock for 2019)

or

B. Calculate the returns over a rolling window and then perform a regression also with a rolling window, speaking in the end they have 252 (=business days in a year) different Betas per stock?

Any help is highly appreciated, I am new to econometric analysis and struggling a bit

Thanks in advance

Answer 1

Don’t worry, you don’t need to compute returns over any period. You simply take your daily (percentage) returns and regress them on the market (and other factors). The excess returns of course. Your regression uses an estimation period the last 12 months. Hence it’s called rolling window. So your beta for month $t$ uses all daily returns from month $t-11$ (including) until month $t$ (including). Note that there are different conventions and some people may want to include an additional month or do similar small changes.

Using daily returns over one year is the standard nowadays. People used to use monthly returns of the last five years (also rolling window!) to compute market betas but is less common nowadays.

When computing the standard CAPM market beta, I tend to follow the approach from Lewellen and Nagel (2006, JFE) and include lagged market returns as regressors to address non synchronous trading issues, see Equation 7 in their paper.