I am doing a project about the predictability of stock returns. I am using following regression model: \begin{equation} r_{t} = \alpha+\beta X_{t-1}+\epsilon_{t}, \end{equation} where $r_{t}$ is the stock return in month ${t}$, $\beta$ is a slope of regression line, $X_{t-1}$ is the predictor of stock returns at month ${t}-1$, $\alpha$ is a constant and $\epsilon_{t}$ is the error term.
I have data set on monthly returns for 1764 months and I am trying to develop past month volatility as the predicting variable.
I tried to do it by following method:
\begin{equation} \sigma_{monthly}=\sqrt{(\mu_{t}-\overline{\mu})^2} \end{equation} where $\mu_t$ is monthly stock return at the time $t$ and $\overline{\mu}$ is the average stock return defined as: \begin{equation} \overline{\mu}=\frac{1}{1764}\sum_{j=1}^{1764} \mu{_j}. \end{equation}
However, I am not sure if this is done correctly as I can't find any sources on exactly this issue.
Thank you very much in advance for the help
That formula that you've written for $\sigma_{\text{monthly}}$ is just the absolute value of the distance from the historical mean: it's not a volatility.
A simplistic method of doing what you're trying to do is to let $X_{t-1}$ be a rolling standard deviation, so you pick an $n$, and then let $X_{t-1} = \text{stdev}(r_{t-1}, r_{t-2}, ... r_{t-n})$.
I think usual practice is to fit the returns series to a GARCH model, but that's a lot more complicated.