I have a low success rate in finding good instruments to use in the GMM system and GMM difference estimation methods for growth models. These methods assume that some equations connect the parameters and observed values in a model. Even if the data seems to have a reasonable fit to a model, the equations are more demanding than just a good visual fit, and subsequent testing can reject the application of the method to the model. It is important to note, as I have in recent posts, that what is being tested is not just whether the model is acceptable, but whether the method conditions and model jointly apply.
Why bother with the methods, if they impose extra conditions which are not intrinsically in the model? Well, less demanding methods such as OLS may incorrectly estimate the parameters, so even though the model is OK according to the method and the method is applicable, the output is not good. What would be best is a method with low intrinsic demands in addition to the model, and which produces accurate results, but for growth models the method does not seem to have been devised yet.
And so to instruments. These are observed data used in the equations alongside the parameters, and can be just about anything. We can test whether the instruments are satisfying the equations, but generally I find that they are unlikely to satisfy them. The rejection is probabilistic; any model can generate data which could satisfy the equations, but for most models the chances of getting the data by chance is extremely remote. I try to ensure that the instruments would be compatible with the data around 90 percent of the time. Common tests are known as the Sargan and Hausman tests.
The problem is frequent among researchers. The theoretical literature reports how difficult it is to find instruments which fulfil the conditions, and many applied researchers avoiding reporting the Sargan or Hausman tests at all when using the GMM estimators. I looked at two of the few empirical growth research papers which report in full their statistics, and which discuss in depth their instrument selection. Neither of them ensure, even across a small number of specifications, that the 90 percent condition is met; in fact even a 95 percent condition is not met.
The rejection indicates misspecification of the model-method. It is not surprising, as the underlying growth models and method conditions are linearisations of the true, complex, and probably unknowable economic generators. So I think that Sargan and Hausman rejection at 90 percent is not the end of model-methods.