A question about serial correlation I'm new to Time Series and I'm confused by some assumptions relating the classical regression model and time series. Econometrics, Fumio Hayashi's book, when introducing the classical regression model, makes some caveats for the case of time series. For example, in page 45:

But, as emphasized in Section 1.1, the strict exogeneity assumption is not satisfied in time-series models typically encountered in econometrics, and serial correlation is an issue that arises only in time-series models.

Moreover, the linear model is given by
$$y_i = x_i'\beta + \varepsilon_i, \quad i = 1,...,n$$
and the Assumption 1.4 (spherical error variance) is:
$$E[\varepsilon_i^2|X]=\sigma^2 > 0, \quad E[\varepsilon_i \varepsilon_j |X] =0 $$
where $X = [\mathbf{x}_1,...,\mathbf{x}_K]_{n\times K}$ matrix.  As said earlier, the most time series models do not satisfy this hypothesis, because of the serial correlation ($E(\varepsilon_i \varepsilon_j)\neq 0$). But for me, I can't see that most time series have serial correlation. For example, AR(1) processes are:
$$y_t =  \rho y_{t-1} + \varepsilon_t$$
with $(\varepsilon_t)$ being i.i.d.. So, I think that in this case, I don't have serial correlation, because the error is i.i.d. And in many other time series models (MA, ARMA) it seems to me that we have i.i.d errors.
So in what cases (examples) do we have serial correlation?

Hi: you do have serial correlation in the AR(1) but it's kind of masked by the way it's written.
$y_t = \phi y_{t-1} + \epsilon_t$ is one way of writing it. Using the lag operator, it can also be written as:
$y_t = \sum_{i=0}^{\infty} \phi^{i} \epsilon_{t-i}$.
Writing it this way, shows that the response, $y_t$, is dependent on all previous shocks in a geometrically declining manner. So, although the $\epsilon_{i}$ are independent, the $y_{t}$ are serially correlated so the model does not satisfy the usual OLS assumptions.
