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Kleibergen, F. (2005). Testing Parameters in GMM without assuming that they are identified Econometrica, 73(4):1103--1124.

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We propose a generalized method of moments (GMM) Lagrange multiplier statistic, i.e., the K statistic, that uses a Jacobian estimator based on the continuous updating estimator that is asymptotically uncorrelated with the sample average of the moments. Its asymptotic ¿2 distribution therefore holds under a wider set of circumstances, like weak instruments, than the standard full rank case for the expected Jacobian under which the asymptotic ¿2 distributions of the traditional statistics are valid. The behavior of the K statistic can be spurious around inflection points and maxima of the objective function. This inadequacy is overcome by combining the K statistic with a statistic that tests the validity of the moment equations and by an extension of Moreira's (2003) conditional likelihood ratio statistic toward GMM. We conduct a power comparison to test for the risk aversion parameter in a stochastic discount factor model and construct its confidence set for observed consumption growth and asset return series.