Q:

Understand the white noise condition in Vector Autoregression

In the following vector autoregression model with lag polynomial representation: 
$$\Phi (L) y_t= \epsilon_t$$
where $Y$ is the vector of endogenous variables,  $\Phi$ is the parameters matrix, $\epsilon$ is the error term, and $L$ is the lag polynomial factor. 
The basic assumption in the above model is that the residual follow a multivariate white noise, i.e.
$$E(\epsilon_t )=0$$
and $E(\epsilon_t \epsilon_s^{‚})$ equals either $0$ if $t \neq s$, or $\sum{\epsilon}$ if $t=s$.
My question is, what $t=s$ and $t \neq s$ really mean in the above condition and from where we get $s$? 

A:

I think the mistake is how to define $\ Y_t$. It is supposed to contain endogenous and exogenous variables. Hence, the multivariate white noise  in the VAR analysis should full fill the following conditions:  
$E(\epsilon_t )=0$ and $E(\epsilon_t \epsilon_s^{‚})$ equals either $0$ if $t \neq s$, or $\sum{\epsilon}$ if $t=s$.
In the case of  $t=s$, this refer to multivariate covariance stationary condition for the endogenous variable. In the case of $t \neq s$, this refer to the exogenous condition (if the model have an exogenous variables). In this case, $E(\epsilon_t \epsilon_s^{‚})$  should equal zero (ie. $t \neq s$ refer to the existence of an exogenous variable).  
reference: "Lecture note from Christopher F Baum"

