# 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?

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.