Past month volatility as predicting variable

I am doing a project about the predictability of stock returns. I am using following regression model: $$$$r_{t} = \alpha+\beta X_{t-1}+\epsilon_{t},$$$$ where $$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:

$$$$\sigma_{monthly}=\sqrt{(\mu_{t}-\overline{\mu})^2}$$$$ where $$\mu_t$$ is monthly stock return at the time $$t$$ and $$\overline{\mu}$$ is the average stock return defined as: $$$$\overline{\mu}=\frac{1}{1764}\sum_{j=1}^{1764} \mu{_j}.$$$$

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

• thank you so much for this, butterflyknife. I really appreciate this simplistic formula as modeling Garch model is above my competence. – Filip Pierzgalski May 24 at 13:22

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 $$X_{t-1} = \text{stdev}(r_{t-1}, r_{t-2}, ... r_{t-n})$$.