Omitted Variable Bias in a VAR-Model I am concerned about the following issue.
One big problem in OLS regression is omitted variable bias, which is normally reflected with explanatory variables being collinear with the error term. 
Now, I am modelling a VAR-Model with lag length $p=3$ and 6 variables. The model is stable and there is no autocorrelation in the residuals. The conclusion according to LM-test is that residuals are white noise. By definition this would mean that there is no information "left" in the error term and the model is well specified. I then conclude with an analysis of impulse response functions. 
When evaluating the validity of the model and explaining limitations, one point I have been thinking of has been confounding variables / omitted variable bias. However, my Time Series Econometrics text-books solely speak about omitted variable bias with regards to taking first differences and lag mis-specification.
Now, I was wondering if it could be that a variable outside the model exists(unobserved variable), which correlates with two or more variables in the model. If this would be the case, I conclude that my results from the VAR Model are biased. In a standard OLS framework this could be tested by for example the Ramsey test. But also in standard models this variable would have been incorporated in the error term. But in the VAR-Model the error term is a white noise process. According to my understanding a white noise process cannot contain any series and thus I would exclude the possibility of an omitted variable bias based on this reasoning. So now, I am asking myself if this is correct or if a VAR-Model can still be subject to an omitted variable bias through other forms and if so why?   
 A: 
I am asking myself if <...> a VAR-Model can still be subject to an omitted variable bias [even if the error term is a white noise process]

Yes, it can, because the omitted variable might be lurking inside the model error even if the error is a white noise process. Let me provide a constructive proof. Take one equation of a trivariate VAR(1) model for $(x_t,y_t,z_t)'$,
$$
x_t=\varphi_{10}+\varphi_{11}x_{t-1}+\varphi_{12}y_{t-1}+\varphi_{13}z_{t-1}+u_{1,t}.
$$
Now omit $z_{t-1}$ to get
$$
\begin{aligned}
x_t&=\varphi_{10}+\varphi_{11}x_{t-1}+\varphi_{12}y_{t-1}+\varepsilon_{1,t}, \\
\varepsilon_{1,t}&=\varphi_{13}z_{t-1}+u_{1,t}. \\
\end{aligned}
$$
Nothing prevents $z_{t-1}$ from being a white noise process AND $z_{t-1}$ being correlated with a linear combination of $x_{t-1}$ and $y_{t-1}$. (This was the crucial note.) If $u_{1,t}$ is also white noise and independent of $z_{t-1}$, then $\varepsilon_{1,t}$ is white noise, too.
This way we have constructed an equation of a VAR(1) model in which 


*

*there is an omitted variable,

*there is omitted variable bias (as the variable is correlated with a linear combination of the right-hand-side variables in the model), and

*the error term is white noise.

