# When modelling ARCH/GARCH effects, do we use excess returns?

When modelling ARCH/GARCH effects, do we use excess returns?

Is it common in the literature to use excess returns when modelling volatility as opposed to raw return data?

## 1 Answer

GARCH models have little to do with the economics of the data generating process of the series you model, so both returns and excess returns (and log-returns, and inflation-adjusted ones, even ones measured in yen!) are valid input. However, there is usually the conditional mean equation besides the variance equation in a GARCH set-up, and your risk-free perfectly predictable component would in this case be part of the conditional mean.

You can have something like this: $$r_{t+1} = r_{f,t} + \mu + \varepsilon_{t+1}, \\ \varepsilon_{t+1} \sim N(0, \sigma_{t+1}^2), \\ \sigma_{t+1}^2 = \alpha + \beta \sigma_t^2 + \gamma \varepsilon_t^2,$$ where the first equation is the mean equation, and you estimate $$\{ \mu, \alpha, \beta, \gamma \}$$. In this case, ignoring the risk-free rate $$r_{f,t}$$ would lead to erroneous estimates. But again, it's up to you to assume or not this holds.

• Thanks for the answer. Could you explain specifically how the mean equation fits the risk-free rate in. – Tony Chivers Dec 3 '18 at 13:17