HansenJTest

HansenJTest, commonly referred to as Hansen's J test, is a statistical test used in econometrics to assess the validity of instruments in generalized method of moments (GMM) estimation. Named for Lars Peter Hansen, the test evaluates whether the overidentifying restrictions implied by the instruments hold in the population.

In a GMM framework, the estimator uses moment conditions E[g(Z, theta)]=0. The Hansen J statistic measures departure from zero using the sample moments at the estimated theta: J = g_hat' W_hat^{-1} g_hat, where g_hat is the vector of sample moment conditions evaluated at theta_hat and W_hat is a consistent estimate of the covariance matrix of g_hat. Under the null hypothesis that the instruments are valid and the model is correctly specified, J is asymptotically chi-square distributed with degrees of freedom equal to the number of overidentifying restrictions (the number of instruments minus the number of endogenous variables).

Interpretation of the test is straightforward: a large J statistic leads to rejection of the null, suggesting that some instruments may be invalid or the model is misspecified. A small J value fails to reject, providing evidence that the moment conditions hold in large samples.

Variants and related concepts include the Sargan test, a special case that assumes homoskedasticity and uses an identity weighting matrix; Hansen's J generalizes to heteroskedasticity-robust implementations. In practice, the test depends on the chosen weighting matrix and may be sensitive to finite-sample properties, such as weak instruments.

Implementation is common after IV or GMM estimation, including applications following 2SLS or more general GMM procedures.