# Instrumental variables in a cointegrated regression

I have two variables, Y and X, and I have managed to find these to be cointegrated. Unfortunately, endogeneity is present yet I have access to data which has high correlation to the endogeneous regressor, X.

My question is whether it is valid to use these instrumental variables as instruments for the endogeneous regressor (X) in this cointegrated regression even if they are not cointegrated with the dependent variable, Y, or does this result in a spurious regression?

If the instruments (call them $Z$) for $X$ are not cointegrated with $Y$ but are integrated, then running a regression of $Y$ on $Z$ will yield spurious results. The regressand $Y$ will be diverging from any linear combination of the regressors $Z$, by the definition of absence of cointegration.
Regressing $\Delta Y$ on $\Delta Z$ is another matter as these are stationary (supposing that the order of integration of $Y$ and $Z$ is both 1). However, if $Y$ adjusts towards the equilibrium with $X$ (i.e. there is an error correction term in the equation for $Y$ as a function of $X$), this will not be captured in the regression of $\Delta Y$ on $\Delta Z$, and so there will be an omitted variable bias.
But before modelling $Y$ and $Z$, I would go back and question the choice of the instruments $Z$. The lack of cointegration between $Y$ and $Z$ implies also that $X$ and $Z$ are not cointegrated, which means $X$ will be diverging from any linear combination of $Z$. Hence, the instruments are poorly suited. This also suggests that what you describe by
• Welcome! As you know, high correlation by itself may be spurious. On the other hand, your economic-theoretical argument might be valid for selecting the instruments, which raises the question whether the cointegration tests might have failed in your particular sample. Perhaps $Z$ are indeed good instruments and are cointegrated with $X$ and subsequently with $Y$, but this just does not show up in the sample at hand. It is hard to know for sure, of course. – Richard Hardy Dec 13 '17 at 12:47