Is a likelihood ratio test appropriate when checking for unnecessary lags in a VAR? If I trying to estimate the proper VAR to do a one period ahead forecast, but I am unsure of how many periods I should lag one of my variables, for example if I should lag it 2 periods instead of 3, can I do a likelihood ratio test to compare the fit of the model with 3 lags and 2 lags? Or does the fact that the variables contain previous information about each other affect how we are supposed to test for this?
If anybody could provide clarification on this I would really be appreciate it.
Thanks
 A: Yes. A likelihood ratio test of a VAR(2) vs. a VAR(3) is quite common for this sort of problem. Say you want to compare a VAR(2) against a VAR(3),
$Y_{t}=A_{0}+A_{1}Y_{t-1}+A_{2}Y_{t-2}+\varepsilon_{t}
 $ vs. $Y_{t}=A_{0}+A_{1}Y_{t-1}+A_{2}Y_{t-2}+A_{3}Y_{t-3}+\varepsilon_{t}$.
You can test this using the LR test,
$-2\ln Q\left(H_{VAR\left(2\right)}/H_{VAR\left(3\right)}\right)=T\left(\ln\left|\hat{\Omega}_{VAR\left(2\right)}\right|-\ln\left|\hat{\Omega}_{VAR\left(3\right)}\right|\right)\sim\chi^{2}\left(n^{2}\right)$.
Where $T$ is the effective sample, $\ln\left|\hat{\Omega}_{VAR\left(k\right)}\right|$ for $k=2,3$ is the (natural) log determinant of the residual covariance matrix and $n$ is the number of variables in your VAR (dimension of the VAR).
Alternatively you could use information criterions such as the Akaike information criteria (AIC), Hannan-Quinn information criteria (H-Q) or the Schwartz information criteria (SC),
$AIC=\ln\left|\hat{\Omega}\right|+\left(n^{2}k\right)\frac{2}{T}$, $SC=\ln\left|\hat{\Omega}\right|+\left(n^{2}k\right)\frac{\ln T}{T}$, $H-Q=\ln\left|\hat{\Omega}\right|+\left(n^{2}k\right)\frac{2\ln\ln T}{T}$, where $k$ is the number of lags in your model.
Remember to get a well specified model with no autocorrelation as this implies dynamic completeness (you have modelled the dynamics in the series of interest).
