Explaining stock returns' volatility with ARCH models

In this recipe, we approach the problem of explaining the conditional volatility of stock returns, with the Autoregressive Conditional Heteroskedasticity (ARCH) model.

The logic of the ARCH method can be represented by the following equations:

The first equation represents the return series as a combination of the expected return μ and the unexpected return _t. The latter one is also known as the mean-corrected return, error term, or innovations. _t has white noise properties—the conditional mean equal to zero and the time-varying conditional variance . Error terms are serially uncorrelated but do not need to be serially independent, as they can exhibit conditional heteroskedasticity.

A zero mean process implies that the returns are only described by the residuals, r_t = _t. Other popular options include constant...