In a model with endogenous regressors, heteroskedastic and autocorrelated (HAC) errors and weak instruments, tests that depend on the data only through the Anderson-Rubin (AR) and Lagrange Multiplier (LM) statistics ignore important information on the regression coefficients. This is in contrast to the homoskedastic case, where these statistics, together with the rank statistic, are one-to-one with the maximal invariant. The information loss with heteroskedastic and/or autocorrelated errors can be so extreme that the LM and conditional quasi-likelihood ratio (CQLR) tests have power close to size when it is trivial to distinguish the null from the alternative hypothesis. The severe loss of power can occur if the Hermitian part of the reduced-form covariance matrix has eigenvalues of opposite signs. The conditional invariant likelihood (CIL) test proposed by Moreira and Ridder (2018) does not suffer this power loss. On the contrary, when the CQLR and LM tests fail, the CIL test can have power close to 1 under the alternative, even if its size is close to 0. This implies that the total variation distance between the null and subsets of the alternative is large, so that it is actually easy to distinguish between these hypotheses. We also show that in the HAC-IV model, there are invariant statistics beyond the triad of AR, LM and rank statistics, so that the latter are not maximal invariant in the HAC case. We conclude that the popular LM and CQLR tests use data inefficiently if the equation errors are HAC.
Endogenous regressor Instrumental variable Score test Invariant test HAC errors