Data analysis with GARCH modeling

I'm currently analyzing the relationship between stock, bonds, and real estate returns in Germany. I've gathered my data and am planning on estimating this equation:

$$\sigma_t = \beta_0 + \beta_1 R_{t+} + \beta_2 R_{t-} + y_t$$

through a GARCH model. In the article where I found equation 1 (Chan and Chang, 2014), they have delineted the parameters as so: $$R_{t+}= \max [0, R_t]$$, $$R_{t–} = \min [0, R_t]$$, and $$R_t$$ is the monthly return of stock, bond, or real estate. We use a three-month rolling return to calculate the standard deviation of the return." Does this mean they use the Max and Min of the series as parameters or the Max and Min of each rolling average across the series?

• This questions seem quite distinct: please post separate questions in separate posts. – Giskard Aug 26 '19 at 14:07

1. Compute $$R_t$$ - the rolling average for month t.
2. $$R_t > 0 \Rightarrow R_{t+} = R_t \wedge R_{t-} = 0$$ and$$R_t < 0 \Rightarrow R_{t-} = R_t \wedge R_{t+} = 0$$