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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?

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    $\begingroup$ I am trying to understand the setup. Could you expand how the thereby works in a variable outside the model exists that correlates with two values in the model and thereby causes the structural shock and the impulse response? Also, by values do you actually variables? Also, how is the last sentence of the second paragraph relevant here? Should it belong to the last paragraph? $\endgroup$
    – Richard Hardy
    Commented Nov 18, 2016 at 17:15
  • $\begingroup$ Sorry for the unclear formulation and thanks for the comment Richard. I have adjusted the question and hope my problem becomes clear. $\endgroup$
    – Raphael
    Commented Nov 18, 2016 at 17:31
  • $\begingroup$ I still don't understand the setup and the mechanism that hides in thereby. Also, VAR can definitely be subject to omitted variable bias in general, but omitted variable bias need not be visible. Similarly as in univariate regression, you do not know whether there is omitted variable bias unless you have the "omitted" variable and can measure its correlation with the fitted values of the model. $\endgroup$
    – Richard Hardy
    Commented Nov 18, 2016 at 17:38
  • $\begingroup$ I understand, thank you for the comments. As I understand, in a standard OLS the "omitted variable" is contained in the error term. Since, in VAR Models the error term is white noise, I am confused how variables can still be omitted. Hope my third try to clarify in the question above makes it clear. $\endgroup$
    – Raphael
    Commented Nov 18, 2016 at 18:08

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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.
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  • $\begingroup$ @Raphael, good that you find it instructive. It might seem a bit contrived to think that the omitted regressor is white noise, but probably one could formulate an example where this is not the case. But even a white-noise regressors can be a meaningful variable, so this is not that big of a problem. $\endgroup$
    – Richard Hardy
    Commented Nov 18, 2016 at 19:12
  • $\begingroup$ Technically this is a reduced VAR(1) model. VAR(1) model should include all the lags, i.e. $z_{t-1}$ can be excluded only if $\phi_{13}=0$. And you can test that hypothesis. $\endgroup$
    – mpiktas
    Commented Nov 21, 2016 at 7:00

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