# 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?

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