  We propose the double robust Lagrange multiplier (DRLM) statistic for testing
hypotheses specified on the pseudo-true value of the structural parameters in
the generalized method of moments. The pseudo-true value is defined as the
minimizer of the population continuous updating objective function and equals
the true value of the structural parameter in the absence of
misspecification.\nocite{hhy96} The (bounding) chi-squared limiting
distribution of the DRLM statistic is robust to both misspecification and weak
identification of the structural parameters, hence its name. To emphasize its
importance for applied work, we use the DRLM test to analyze the return on
education, which is often perceived to be weakly identified, using data from
Card (1995) where misspecification occurs in case of treatment heterogeneity;
and to analyze the risk premia associated with risk factors proposed in Adrian
et al. (2014) and He et al. (2017), where both misspecification and weak
identification need to be addressed.
