# 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?

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).

• Alecos, Many thanks for your post. Yes, most researchers use exogenous variables i.e. geographic, climate and other related variables as instruments for economic variables. However, some do use lagged values of X as instruments of current values of X. Given that lagged values of X correlated with current y and uncorrelated with current u, can they be valid instruments. Many thanks in advance! – mr.rox Sep 4 '13 at 16:48
• Yes, they are valid instruments, as long as they do correlate with the dependent variable of course (due to the still not well understood inertia of economic systems, they usually do). – Alecos Papadopoulos Sep 4 '13 at 17:33
• Be well - have you thought about alternative energy sources as possible instruments? – Alecos Papadopoulos Sep 4 '13 at 18:33
• Yep, I did, but their current values are correlated with u, the past values do not have sufficient correlation with the current X. Moreover, the data for renewables is not available for my sample. I have a set of LDCs and developing countries for which I am looking at income differences given the oil consumption. Renewables data is not availale for most of them if I use lags. I thought of using natural disasters in energy exporting countries which could lead to supply shocks. But, natural disaster dummies proved to be weak instruments too. – mr.rox Sep 4 '13 at 18:41
• Think about reversing the implied "causality": regress oil consumption as the dependent variable on income as the regressor. Intuitively, the case for endogeneity will be much less strong here. You can look up Maddala (2001) "Introduction to Econometrics" ch 3 pp 71-75, for a first exposition of the "reverse regression", and how the coefficients of the one way related to the coefficients of the reverse way. But there are many other resources available online. – Alecos Papadopoulos Sep 4 '13 at 19:20