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- P̃K0 P̃K , Ã2 ≡ infx E[ε2 i |xi = x], Ã̄2 ≡ supx E[ε2 i |xi = x]. We first provide useful lemmas which will be used in the proof of Theorem 3.1. Versions of proof of Lemma 1 are available in the literature, such as Newey (1997), Belloni et al. (2015) and Chen and Christensen (2015b), among others. For completeness, we provide the results of Lemma 1. Note that different rate conditions can be used in Assumption 3.2, but lead to different bounds in (A.1)-(A.3) in the following Lemma 1. Lemma 1. Under Assumptions 3.1 and 3.2, for any K ∈ Kn, following holds || b QK − IK|| = Op( r ζ2 Kλ2 K log K n ), (A.1) R1(K) ≡ r nVK P̃K(x)0 b Q−1 K − IK P̃K0 (ε + rK) = Op( r λ2 Kζ2 K log K n (1 + `KcK √ K)), (A.2) R2(K) ≡ r
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