Strict exogeneity and lagged variables I am confused why strict exogeneity must be violated when we have lagged time series variables. My understanding of strict exogeneity is that a variable must be uncorrelated with error terms in all periods. But isn't exogeneity always a necessary assumption for estimation? If $x_t$ and $u_t$ are uncorrelated, and $x_{t-1}$ and $u_{t-1}$ are uncorrelated, how would it violate strict exogeneity if we have a specification with $x_t$ and $x_{t-1}$?
 A: In the most cases it is assumed that $E[\epsilon_t]=0$. Then, strict exogeneity implies that the regressors are orthogonal to the error term for all observations $s$, i. e. $E[x_s \epsilon_t]=0$. For some time series models this is violated. Consider the AR(1) model $ \ y_t=\beta y_{t-1}+ \epsilon_t \ $ with $ \ \epsilon_t \sim N(0, \sigma^2) \ $  $ \ \forall \ $ $t$. Since you regress $y_t$ on $y_{t-1}$ the error term $\epsilon_t$ is orthogonal to $y_{t-1}$, i. e. 
$E[y_{t-1} \epsilon_t]=0$. 
However, strict exogeneity requires $y_t$ to be orthogonal to $all$ $\epsilon_t$. That does not hold for the considered model - as will be shown:
$E[y_t \epsilon_t]=E[(\beta y_{t-1}+ \epsilon_t)\epsilon_t] \qquad (by \ \ \ y_t=\beta y_{t-1}+ \epsilon_t)$
                    $ \quad \qquad =\beta E[y_{t-1} \epsilon_t]+E[\epsilon_t^2]$
                    $ \quad  \qquad =E[\epsilon_t^2] \qquad \qquad \qquad \quad (by \ \ \ E[y_{t-1} \epsilon_t]=0)$.
$ \quad  \qquad =\sigma^2 \qquad \qquad \qquad \quad \quad (by \quad  \epsilon_t \sim N(0, \sigma^2))$.
Therefore, $y_t$ is not orthogonal to all error terms but the regressor for $y_{t+1}$. Thus, strict exogeneity is violated. 
This implies, there is only strict exogeneity if $\epsilon_t = 0$ for all $t$. 
A: 
Strict exogeneity and lagged variables

The problem your refers on is about strict exogeneity and lagged dependent variables. The two things together are contradictory.
Indeed as shown in random guy answer if we have a model like
$y_t = \beta_1 y_{t-1} + \epsilon_t$
(we assume $[\epsilon_t]=0$)
We have that $E[y_t \epsilon_t]=\sigma^2$
However if other lagged variables as $x_{t-j}$ are included, and dependent are not, the contradiction disappear.
Moreover the main point, in my view, is that exogeneity is a causal concept (in any detailed form) and it must be write on a structural equation not a regression one. Indeed the equation above must be intended as structural. (Read here for more about: Under which assumptions a regression can be interpreted causally?)
The reply of random_guy is mathematically correct but terminology used is ambiguous and tend to mix concepts. Mainly regression equation and/or errors and exogeneity assumpion. Speaking about AR (autoregressive) model the conflation become even more evident. I read the the ref suggested in the comment (http://assets.press.princeton.edu/chapters/s6946.pdf), and It suffer from the same problem. This problem is quite common (read here: How would econometricians answer the objections and recommendations raised by Chen and Pearl (2013)?)
