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I am working on a project that has 4680 observations (18 years, 265 regions), in which one variable is endogenous. We tried various combinations of two IVs, but always end up with a significant Sargan-Hansen J Test. Recently I have found this quote:

"Testing of over-identifying assumptions is less important in longitudinal applications because realizations of time varying explanatory variables in different time periods are potential instruments, i.e., over-identifying restrictions are automatically built into models estimated using longitudinal data."

Does it mean that for panel data that contains large number of observations the Sargan-Hansen J Test is irrelevant as it almost always ends up significant?

The precondition of the test is that at least one of the IVs is a relevant IV. Should we therefore restrict ourselves to using one IV?

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I disagree with the quote. For example, in dynamic panel data models $$ y_{it}= \alpha_i+\rho y_{it-1}+u_{it} $$ higher lags of $y_{it}$ are invalid instruments if $u_{it}$ is itself autocorrelated, i.e., if the dynamics of the system have not been specified sufficiently richly.

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