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?

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#### Best Answer

Just elaborating on Matthew Gunn answer, we have that, due to the independence of the various components,

$$E[x_tx_{t-1}] = E[z_tsigma_tz_{t-1}sigma_{t-1}] = E(z_t)E(z_{t-1})E(sigma_tsigma_{t-1})$$

$$=0cdot0cdot E(sigma_tsigma_{t-1}) = 0$$

Since the $X$-process has zero mean, the above implies that there is no autocorrelation.

But

$$E[x_t^2x_{t-1}^2] = E[z_t^2sigma_t^2z_{t-1}^2sigma_{t-1}^2] = E(z_t^2)E(z_{t-1}^2)E(sigma_t^2sigma_{t-1}^2)$$

$$=1cdot 1 cdot E(sigma_t^2sigma_{t-1}^2) = E(sigma_t^2sigma_{t-1}^2)$$

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.

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