Why do some researchers include oil price in a VAR model? Why do some researchers include international oil prices in a VAR model for small open economies? My intuition is that international oil prices cannot be a endogenous variable in the context of a small open economy. For example, I don't see how one would justify a two-variable VAR model (say, for international oil price and GDP) for a country like Ghana. Because oil prices are not endogenous for small open economies, right? I mean, Ghana's GDP or its lags do not affect the international price of oil. However, I see in the literature that people still use a VAR model with oil prices as one of the endogenous variables in the model. I am finding it hard to understand how they can do that, and these papers do not state how.
Any help is appreciated.
Output and oil price can be endogenous even for relatively small and open country if the country is oil producer. Actually, in international trade whether economy is small depends on whether in particular case country can affect world price by varying its output. For example Kuwait is generally small open economy, but when it comes to oil production they are (at the time of writing) ranked 10th in the world so in oil market they are not small economy. For Kuwait there would be definitely an endogeneity between oil prices and output.
Ghana (at the time of writing) it is ranked 49th in world oil production which is somewhere in the middle between being significant and trivial so I can imagine someone arguing that there could be some endogeneity between oil prices and output. I suppose it is possible that if Ghana would shut down its oil production world price would move slightly.

VAR model actually does not require for all variables to be endogenous. It is completely fine to estimate VAR model where there are also exogenous control variables. Doing so does not violate any assumption of VAR. VAR allows for endogeneity to be present, it does not require it. However, if you are completely sure those variables are exogenous, and you are only using two variables, running VAR would be equivalent to just running two ARDL models where one of those ARDL models will be superfluous. It would not be the most efficient modeling choice but I also can't think of reason why it would introduce bias.