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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?

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  • $\begingroup$ This questions seem quite distinct: please post separate questions in separate posts. $\endgroup$
    – Giskard
    Commented Aug 26, 2019 at 14:07
  • $\begingroup$ FYI: looking at the number of threads concerning GARCH, you will see that there is much more activity on Cross Validated and Quantitative Finance Stack Exchange. $\endgroup$
    – Richard Hardy
    Commented Jan 19, 2022 at 17:07
  • $\begingroup$ It is simply a recoding of the return series splitting up positive and negative values. If you are familiar with regression and dummy interaction note the max(0,r)=1(r>0)r so equal to 0 if r is non-positive otherwise simply r. $\endgroup$
    – Jesper Hybel
    Commented Sep 5 at 5:37

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Well I havent read the paper but this is what it looks like to me:

  1. Compute $R_t$ - the rolling average for month t.
  2. $R_t > 0 \Rightarrow R_{t+} = R_t \wedge R_{t-} = 0$ and$R_t < 0 \Rightarrow R_{t-} = R_t \wedge R_{t+} = 0$
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  • $\begingroup$ Yes it means that they model the effect of changes in returns differently depending on whether the returns are negative or positive. It is a well known empirical tendency that a lot of financial instruments respond asymmetrically to good and bad "news". The effect is asymmetric when beta values are different and usually you would expect the reaction to change in negative returns to be larger. $\endgroup$
    – Jesper Hybel
    Commented Sep 5 at 5:32

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