A more powerful subvector Anderson Rubin Test in Linear Instrumental Variables Regression with Heteroskedasticity

We study subvector inference in the linear instrumental variables model allowing for arbitrary forms of heteroskedasticity and weak instruments.
The subvector Anderson and Rubin (1949) test that uses chi square critical values with degrees of freedom reduced by the number of parameters not under test,
proposed by Guggenberger, Kleibergen, Mavroeidis, and Chen (2012), has correct asymptotic size under homoskedasticity but is generally conservative.
Guggenberger, Kleibergen, Mavroeidis (2019) propose a conditional subvector Anderson and Rubin test that uses data dependent critical values that adapt to the strength of identification of the parameters not under test. This test also has correct asymptotic size under homoskedasticity and strictly higher power than the subvector
Anderson and Rubin test by Guggenberger et al. (2012). Here we first generalize the test in Guggenberger at al (2019) to a setting that allows for a general Kronecker product structure which covers homoskedasticity and some forms of heteroskedasticity. To allow for arbitrary forms of heteroskedasticity, we propose a two step testing procedure. The first step, akin to a technique suggested in Andrews and Soares (2010) in a different context, selects a model, namely general Kronecker product structure or heteroskedasticty. If the former is selected, then in the second step the generalized version of Guggenberger et al. (2019) is used, otherwise a particular version of a heteroskedasticity robust test suggested in Andrews (2017). We show that the new two step test has correct asymptotic size and is more powerful and quicker to run than several alternative procedures suggested in the recent literature.