  We consider a linear combination of jackknife Anderson-Rubin (AR), jackknife
Lagrangian multiplier (LM), and orthogonalized jackknife LM tests for inference
in IV regressions with many weak instruments and heteroskedasticity. Following
I.Andrews (2016), we choose the weights in the linear combination based on a
decision-theoretic rule that is adaptive to the identification strength. Under
both weak and strong identifications, the proposed test controls asymptotic
size and is admissible among certain class of tests. Under strong
identification, our linear combination test has optimal power against local
alternatives among the class of invariant or unbiased tests which are
constructed based on jackknife AR and LM tests. Simulations and an empirical
application to Angrist and Krueger's (1991) dataset confirm the good power
properties of our test.
