Endogeneity and the AR model? an AR(p) process is given by:

$$y_t=\phi_0+\phi_1y_{t-1}+\phi_2y_{t-2}+...+\phi_{t-p}y_{t-p}+u_t$$

In such a model we have endogenous variables. My question is, why is this not an issue when dealing with time series data.
It is not necessarily true that lagged $y$ are endogenous. That depends entirely on the structure you assume for $u_t$. If $u_t$ is white noise, there is no endogeneity, in the sense that 

$$ E(u_t|\mathbf{y}) = 0 $$

where $\mathbf{y}=\{y_{t-1}, \cdots , y_{t-p}\}$.

This is trivial to prove, once you realise that the above equality can also be expressed as 

$$E(u_t y_{t-i})=0 \,\,\quad, i=\{1, \cdots, p\} $$

using the Law of Total Expectations.

Conversely, it is trivial to show that if $u_t$ follows an autoregressive process, the above conditions does not hold anymore, in which case there is endogeneity, and OLS estimates are inconsistent.