Efficiency Loss of Asymptotically Efficient Tests in an Instrumental Variables Regression
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In an instrumental variable model, the score statistic can be stochastically bounded for any alternative in parts of the parameter space. These regions involve a constraint on the first-stage regression coefficients and the reduced-form covariance matrix. As a consequence, the Lagrange Multiplier (LM) test can have power close to size, despite being efficient under standard asymptotics. This loss of information limits the power of conditional tests which use only the Anderson-Rubin (AR) and the score statistic. In particular, the conditional quasi-likelihood ratio (CQLR) test also suffers severe losses because its power can be bounded for any alternative. A necessary condition for drastic power loss to occur is that the Hermitian of the reduced-form covariance matrix has eigenvalues of opposite signs. These cases are denoted impossibility designs or impossibility DGPs (ID). This restriction cannot be satisfied with homoskedastic errors, but it can happen with heteroskedastic, autocorrelated, and/or clustered (HAC) errors. We show these situations can happen in practice, by applying our theory to the problem of inference on the intertemporal elasticity of substitution (IES) with weak instruments. Out of eleven countries studied by Yogo (2004) and Andrews (2016), the data in nine of them are consistent with impossibility designs at the 95 confidence level. For these countries, the noncentrality parameter of the score statistic can be very close to zero. Therefore, the power loss is sufficiently extensive to dissuade practitioners from blindly using LM-based tests with HAC errors.
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