valid instrument for oil consumption in IV model I want to run a gdp vs. oil consumption model where oil consumption is suspected to be endogenous - correlated with the error terms. Can a variable correlated with world oil price but not with the gdp of the country of interest be a valid instrument? 
 A: No. To look at the most simple case, consider the model (vactor-matrix notation)
$$ \mathbf y = \mathbf X\beta + \mathbf u$$
where we believe that the $\mathbf X$-regressors are correlated with the error term. We have an alternative set of regressors, $\mathbf Z$, which are correlated with $\mathbf X$ but not correlated with $\mathbf u$ (which is the reason why we consider using $\mathbf Z$ in place of $\mathbf X$). 
This gives us the moment condition
$$ E\Big(\mathbf Z'\mathbf u\Big ) = 0 \Rightarrow E\Big(\mathbf Z'(\mathbf y - \mathbf X\beta)\Big ) = 0 \Rightarrow E\Big(\mathbf Z'\mathbf y - \mathbf Z'\mathbf X\beta)\Big ) = 0$$
$$\Rightarrow E(\mathbf Z'\mathbf y) - E(\mathbf Z'\mathbf X)\beta = 0 \Rightarrow \beta = \Big (E(\mathbf Z'\mathbf X)\Big)^{-1}E(\mathbf Z'\mathbf y) $$
If $\mathbf Z$ is uncorrelated with $\mathbf y$, we will have $E(\mathbf Z'\mathbf y)=0 $ which would lead us to $\beta =0$, which contradicts our specification. So $\mathbf Z$ must be correlated with $\mathbf y$, while being uncorrelated with $\mathbf u$ - and this required combination of characteristics is what makes candidate instrumental variables so difficult to be convincing (and on top of that, the existence or not of this correlation / non-correlation combination cannot be statistically tested).
