# Serially Uncorrelated but dependence in ARCH model

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

In a text I'm reading, the author gives the ACF plots on the return series of a stock.

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]

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+ \beta\sigma_{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 $$corr(\varepsilon_{t-1}^2,\varepsilon_{t-h}^2) = K(\alpha + \beta)^h$$ 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.

• I think you forgot to square $\varepsilon_{t-1}$ in the second equation. Later, in the expressions of expected values you are skipping some steps which could have been instructive if kept (but that is just a personal opinion). Jan 28 '18 at 15:51
• "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." Okay so what does this mean empirically? If I do find some significant correlation in the returns, then I simultaneously apply an ARMA-GARCH model to the squared returns, and if I don't, then I am free to apply a GARCH model to the squared returns. Am I correct there? Jan 28 '18 at 21:59
• Ok. Maybe the wording is a bit bad. One approach is to 1) estimate an ARMA model if you find serial correlation in returns 2) check (if no ARMA needed) squared returns or squared errors from ARMA model for serial correlation 3) If serial correlation in either squared returns or squarred error, estimate respectively an GARCH or ARMA-GARCH Jan 29 '18 at 15:30

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?

On the first chart (of raw returns), the ACF does not appear to be significantly non-0 anywhere (other than at lag 1 of course). In other words, there is no serial correlation of the returns.

On the second chart (of squared returns), the ACF does appear to be significantly non-0 at certain lags. So this gives us hope that we can use these squared returns to predict something by using them.

(Interestingly, if you plotted the ACF for absolute returns, you would find something similar to the second chart. This is because the absolute value and the square both discard the sign to measure some sort of "deviation", in the non-technical sense of the word.)

And in general, what is the intuition behind looking at the square of the series when we are searching for the "ARCH" effect?

It's not just any series, it's a series of raw returns. And as you saw in the first chart, raw returns are not serially correlated.

However, squared (and absolute) returns are, which is good news! This is because we can now use them to predict "volatility" (i.e. the conditional variance) using ARCH models. And this is what ARCH models do.

Incidentally, I wouldn't use the term "ARCH effect". What I would say is that ARCH-type models take us some way towards predicting the conditional variance of the returns of financial assets.

A good reference on this topic is Taylor.