The Supplementary Appendix contains proofs of some results stated in the paper "Uniform Inference in Autoregressive Models" by Anna Mikusheva. In particular, it provides a proof of a statement about strong approximation, proofs of Lemmas 11 and 12 from the paper about the asymptotic approximations for scheme of series. It also proves results stated in Remarks 2, 3 and 4 for AR(1) processes with a linear time trend. Section 5 proves the validity of parametric and non-parametric grid bootstrap procedures for AR(p) processes with at most one root close to the unit circle. Section 7 contains an extensive Monte-Carlo study of finite sample properties of discussed methods. We keep notations introduced in the paper.