  This paper uses model symmetries in the instrumental variable (IV) regression
to derive an invariant test for the causal structural parameter. Contrary to
popular belief, we show that there exist model symmetries when equation errors
are heteroskedastic and autocorrelated (HAC). Our theory is consistent with
existing results for the homoskedastic model (Andrews, Moreira, and Stock
(2006) and Chamberlain (2007)). We use these symmetries to propose the
conditional integrated likelihood (CIL) test for the causality parameter in the
over-identified model. Theoretical and numerical findings show that the CIL
test performs well compared to other tests in terms of power and
implementation. We recommend that practitioners use the Anderson-Rubin (AR)
test in the just-identified model, and the CIL test in the over-identified
model.
