Consider the following abstract for an econometrics paper:
"The method of instrumental variables (IV) and the generalized method of moments (GMM), and their applications to the estimation of errors-in-variables and simultaneous equations models in econometrics, require data on a sufficient number of instrumental variables that are both exogenous and relevant. We argue that, in general, such instruments (weak or strong) cannot exist."This is, in fact, the abstract for a recent paper by Hall et al. (2014), and when I first read it I was definitely intrigued!
Recall that when we look for instruments we need to find variables that are, on the one hand, (asymptotically) uncorrelated with the errors of our regression model; but are, on the other hand, highly correlated (asymptotically) with the random regressors. The abstract, and the paper itself (of course) suggests that usually this objective is not achievable.
Why is this?
The difficulty arises if we view the error term in our regression equation as arising from various mis-specifications in the model. The authors argue that this interpretation is generally appropriate in econometric applications. Building on earlier work by Pratt and Schlaifer (1988), they show that in this case it's generally the situation that the error is a function of the very regressors that we're trying to "instrument". That being the case, legitimate instruments will be unattainable.
Food for thought!
Hall, S. G., P. A. V. B. Swamy, and G. S. Tavlas, 2014. On the interpretation of instrumental variables in the presence of specification errors. Working Paper 14/19, Department of Economics, University of Leicester.
Pratt, J. W. and R. Schlaifer, 1988. On the Interpretation and observation of laws. Journal of Econometrics, 39, 23-52.