# Solved – Serially Uncorrelated but dependence in ARCH model

The title is a bit shaky, but it sums up the question I have in volatility models.

He says that if we look at the sample ACF of the returns (The first plot), there are no signs of significant serial correlation.
However, if we look at the ACF of the returns squared (The second plot), it tells us that the return series is indeed serially uncorrelated, but dependent. Could someone explain how he came to this conclusion? And in general, what is the intuition behind looking at the square of the series when we are searching for the "ARCH" effect? [Because we also sometimes do so for the residual of a series after applying an ARMA model]

Contents

This is a common observations for daily returns series. The level is often found to be unpredictable (if not, then we would be able to make a lot of money with a simple ARMA model), while we are able to predict volatility.

To be a bit more explicit, assume a GARCH model:

begin{align} r_t &= varepsilon_t = sigma_t z_t \ sigma_t^2 &= omega + alpha varepsilon_{t-1}^2+ betasigma_{t-1}^2 end{align} where \$z_t\$ is iid with zero mean and unit variance. We have \$E[r_t]=E[sigma_t]E[z_t]=E[sigma_t]cdot 0 = 0\$. Thus, we have that the autocorrelation of returns \$E[r_t r_{t-h}] = E[z_t]E[sigma_t r_{t-h}] = 0\$. However, it is possible to show that begin{equation} corr(varepsilon_{t-1}^2,varepsilon_{t-h}^2) = K(alpha + beta)^h end{equation} Hence, the correlation is proportional to \$(alpha + beta)^h\$ – this also explains why \$alpha + beta\$ is refered to as the persistence in a GARCH process.

The ACF of squared returns shows us that we have higher order dependence that we may model with a GARCH model.

Note that if the ACF of returns are not zeros, then we should employ some dynamics to filter this out, but if not the case one simply proceeds with zero or constant mean.

Rate this post