# Sargan–Hansen test

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The Sargan–Hansen test or Sargan's ${\displaystyle J}$ test is a statistical test used for testing over-identifying restrictions in a statistical model. It was proposed by John Denis Sargan in 1958,[1] and several variants were derived by him in 1975.[2] Lars Peter Hansen re-worked through the derivations and showed that it can be extended to general non-linear GMM in a time series context.[3]

The Sargan test is based on the assumption that model parameters are identified via a priori restrictions on the coefficients, and tests the validity of over-identifying restrictions. The test statistic can be computed from residuals from instrumental variables regression by constructing a quadratic form based on the cross-product of the residuals and exogenous variables.[4]:132–33 Under the null hypothesis that the over-identifying restrictions are valid, the statistic is asymptotically distributed as a chi-square variable with ${\displaystyle (m-k)}$ degrees of freedom (where ${\displaystyle m}$ is the number of instruments and ${\displaystyle k}$ is the number of endogenous variables).

This version of the Sargan statistic was developed for models estimated using instrumental variables from ordinary time series or cross-sectional data. When longitudinal ("panel data") data are available, it is possible to extend such statistics for testing exogeneity hypotheses for subsets of explanatory variables.[5] 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.