# Can a dummy variable be used to correct ARCH?

Is it possible to use a dummy variable to allow for a structural break, in order to correct Autoregressive Conditional Heteroscedasticity?

If you have a one-time jump in variance occuring due to a structural break and nothing but the error variance changes at that point, you can use a dummy in the variance equation of the model. If the other parameters may have changed at the same time, you can estimate two different models for the periods before and after the break and see how the models differ.

If you have autoregressive conditional heteroskedasticity (ARCH), thus autocorrelated squared errors, a dummy variable will not help; a model with a dummy simply cannot represent the ARCH kind of behaviour in the conditional variance.

Yes. A simple "regime-switching" GARCH model could be implemented by specifying the GARCH equation as

$$\sigma_{t+1}^2 = \alpha + \alpha^* I(t>T^*) + \beta \sigma_t^2 + \gamma \varepsilon_t^2$$

where $$I(t>T^*)$$ is one when time is larger than a given threshold. This model will allow the intercept (and thereby the baseline volatility) to be time-varying. A more general model could be obtained by multiplying the dummy onto all elements of the GARCH equation.

$$\sigma_{t+1}^2 = \alpha + \beta \sigma_t^2 + \gamma \varepsilon_t^2 + I(t>T^*)(\alpha^* + \beta^* \sigma_t^2 + \gamma^* \varepsilon_t^2 )$$

• Looks like we agree. I would not call a single structural break regime switching though, given the context of typical regime switching models. And the second model is essentially (if not exactly) the same as splitting the sample in two and estimating two separate GARCH(1,1) models, right? – Richard Hardy Apr 2 at 9:26
• @RichardHardy I will agree that a proper regime-switching model should have the switching dynamics as an integral part of the model. A dummy variable approach will be more like a regime-switching, and not only a model allowing for a structural break, if we replace time as a state variable with a macroeconomic variable (say the ADS index). Then we will have a type of macro-regime model. I believe it will be more efficient to estimate the model jointly - at least if only the intercept is changing. – Johan Stax Jakobsen Apr 2 at 12:58
• The second model has everything changing, which is why I asked about it, so then there should be no gains in efficiency from join estimation but some gains in clarity from estimating separately. Though if the conditional mean model is kept the same before and after the change, then you are right. But why would anyone expect that if the variance dynamics is changing completely... – Richard Hardy Apr 2 at 13:55
• A lot of models in the GARCH literature assume that the mean dynamics is the same, e.g. just constant mean and only look at the vol dynamics. – Johan Stax Jakobsen Apr 2 at 13:57
• Just a constant? I would call that "mean statics" then :) But surely, for financial returns this may be good enough. – Richard Hardy Apr 2 at 14:01