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