On Inconsistency of the Overidentification Test for the Model-implied Instrumental Variable Approach

In the context of structural equation modeling, the model-implied instrumental variable (MIIV) approach has been shown to be more robust against model misspecification than the systemwide approaches (e.g., maximum likelihood and least squares). Besides the goodness-of-fit tests that test the fit of the entire hypothesized covariance structure, the overidentification tests for MIIV can be used to test model specification on an equation-by-equation basis. However, it is known in the econometrics literature that the overidentification tests are inconsistent against general misspecification, if it is used to test a zero correlation between the instrumental variables and the error terms. In this paper, we show that such inconsistency can also occur for the MIIV approach. Numerical examples where the powers of the tests converge to the size are presented. Theoretical results are proved to support the numerical findings. Implications on when the overidentification tests are consistent are also presented.