This paper contains supplemental material to Andrews, Moreira, and Stock (2006a), hereafter AMS. Section 2 provides details concerning the sign-invariant power envelope for similar tests introduced in AMS. Section 3 does likewise for the LU power envelope for invariant similar tests. Section 4 reports additional numerical results to those in AMS. Section 5 establishes consistency of the covariance matrix estimator in AMS. Section 6 gives proofs of Lemmas 1 and 2 of AMS. Section 7 proves the claim made in Comment 2 to Corollary 1 of AMS that when k =1 the optimal invariant similar test in terms of two-point weighted average power is the Anderson-Rubin test (which is equivalent in this case to the LM and CLR tests). An Appendix describes numerical methods used in Section 4.