OveridentificationTests

Overidentification tests are statistical procedures used in instrumental variables (IV) and generalized method of moments (GMM) frameworks to assess the validity of instruments when there are more instruments than endogenous regressors. When the number of instruments exceeds the number of endogenous variables, the model is overidentified, meaning there are additional moment conditions beyond those needed for identification. These extra conditions can be used to test whether the instruments are valid, i.e., uncorrelated with the structural error term and correctly excluded from the estimated equation.

The Sargan test and Hansen J test are the most common overidentification tests. The Sargan test assumes homoskedasticity and tests whether the residuals from a two-stage least squares (2SLS) or IV estimation are uncorrelated with all instruments. The Hansen J test generalizes this idea to allow heteroskedasticity; it uses a robust covariance matrix estimate. Under the null hypothesis that all instruments are valid, both tests yield a chi-square distribution with degrees of freedom equal to the number of overidentifying restrictions, which is the number of instruments minus the number of endogenous regressors.

Calculation, in broad terms, involves the estimated residuals and the moment conditions implied by the instruments. The test statistic measures the extent to which these moment conditions fail to hold in the sample. A large statistic leads to rejection of the null, suggesting that some instruments are invalid or improperly excluded.

Practical considerations include small-sample sensitivity and weak instruments, which can affect test power and size. In robust settings, the Hansen J test is preferred due to its heteroskedasticity-robust nature, while the Sargan test remains a useful benchmark under homoskedasticity.