In Section 3.2 of R. S. Tsay, Analysis of Financial Time Series, I read:
The basic idea behind volatility study is that the series {r_t} is either serially uncorrelated or with minor lower order serial correlations, but it is a dependent series.
and a little further:
…it seems that the returns are indeed serially uncorrelated, but dependent.
I'm very confused by these statements, because I thought that serial correlation (autocorrelation) and dependence were basically the same thing. Here, for instance, the coin tossing game is given as an example of independence, as a series where each throw has no memory of the previous throws. Therefore, it seems to me that instead a not independent variable should be serially correlated.
Can you give me an example of a time series that is not serially correlated but dependent, and another one that is serially correlated but independent? Is that possible?
Or perhaps am I wrongly assuming that causation implies correlation?
Best Answer
Just elaborating on Matthew Gunn answer, we have that, due to the independence of the various components,
$$E[x_tx_{t-1}] = E[z_t\sigma_tz_{t-1}\sigma_{t-1}] = E(z_t)E(z_{t-1})E(\sigma_t\sigma_{t-1})$$
$$=0\cdot0\cdot E(\sigma_t\sigma_{t-1}) = 0$$
Since the $X$-process has zero mean, the above implies that there is no autocorrelation.
But
$$E[x_t^2x_{t-1}^2] = E[z_t^2\sigma_t^2z_{t-1}^2\sigma_{t-1}^2] = E(z_t^2)E(z_{t-1}^2)E(\sigma_t^2\sigma_{t-1}^2)$$
$$=1\cdot 1 \cdot E(\sigma_t^2\sigma_{t-1}^2) = E(\sigma_t^2\sigma_{t-1}^2)$$
while
$$E[x_t^2]\cdot E[x_{t-1}^2] = ...=E(\sigma_t^2)\cdot E(\sigma_{t-1}^2)$$
Because the sigmas are autocorrelated, if one carries out the multiplications, one will find that
$$E[x_t^2x_{t-1}^2] \neq E[x_t^2]\cdot E[x_{t-1}^2]$$
which shows that there exists higher-order dependence in the $X$-process.