Choosing Returns or Excess Returns for GARCH Volatility Models

Summary

The document asks whether volatility modeling with ARCH or GARCH methods should use raw returns or excess returns. The accepted answer says either can serve as input, since the model’s variance dynamics do not require a particular economic interpretation of the series. The choice should be consistent with how the conditional mean is specified.

In the example, the mean equation includes the risk-free rate, an additional mean component, and an error term; the variance equation models the error’s conditional variance using a constant, lagged variance, and lagged squared error. If the predictable risk-free component is relevant but omitted, parameter estimates may be affected. This is a modeling choice rather than a universal rule: the document gives no empirical comparison of specifications, and its illustrative normal-error setup is only one possible assumption.

Key ideas

• GARCH volatility models can be fitted to raw returns or excess returns.
• The choice of return series should match the conditional mean equation.
• A predictable risk-free component can be included in the mean specification.
• Omitting a relevant risk-free component may affect estimated model parameters.
• The example illustrates one specification and does not establish a universal modeling rule.

Tags

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


# 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?

## Answer by Igor Pozdeev (score 6, accepted)

https://quant.stackexchange.com/a/42888

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.