Overidentifying

Overidentifying refers to a situation in econometrics in which a model has more instruments than the number of endogenous variables to be identified. This creates overidentifying restrictions: additional instruments impose extra conditions on the parameters beyond those needed for identification.

In instrumental variable estimation and generalized method of moments, the typical cases are underidentified (too few instruments), exactly identified (the number of instruments equals the number of endogenous variables), and overidentified (more instruments than endogenous variables). Overidentification allows the researcher to test the validity of the instruments as a group, rather than relying on a single instrument.

Overidentifying restrictions are commonly tested with statistics such as the Sargan test or the Hansen J test. These tests assess whether the instruments are uncorrelated with the structural error term and correctly excluded from the estimated equations. A failure to reject the null hypothesis supports the plausibility of instrument exogeneity and model specification, while rejection suggests that some instruments may be invalid or that the model is mis-specified.

Caveats include that rejection does not pinpoint which instrument is problematic, and tests can have low power in finite samples. Weak instruments can also affect the reliability of overidentification tests. In practice, researchers consider both the strength and validity of instruments, and may use robust or heteroskedasticity-robust versions of the tests depending on the econometric framework. Overall, overidentification provides a diagnostic tool for instrument validity in models with multiple instruments.