Supplement to “ Endogeneity in High Dimensions ”

This supplementary material contains the proof of Theorem 7.1 in the main paper. The following lemma is useful. Lemma 0.1. For a general loss function Ln, suppose it is twice differentiable in a neighborhood of β0. Assume max l / ∈S ∣∣∣∣∂Ln(β0) ∂βl ∣∣∣∣ = op(P ′ n(0)), (0.1) and there is a neighborhood U ⊂ R of β0S such that sup βS∈U max l / ∈S,j∈S ∣∣∣∣∂2Ln(βS, 0) ∂βl∂βj ∣∣∣∣ = op(P ′ n(0) skn ) . (0.2) where we denote P ′ n(0 ) = lim inft→0+ P ′ n(t) and kn = an+ √ sP ′ n(dn) as given in Theorem B.1 of the main paper. In addition, assume √ skn = o(P ′ n(0 )). Then Condition A in Theorem B.2 of the main paper is satisfied. Proof. If Ln is continuously differentiable in a neighborhood of β0, by the mean value theorem, there exists λ ∈ (0, 1) such that for h = λβ + (1− λ)Tβ, Qn(Tβ)−Qn(β) = ∑ l / ∈S ∂Ln(h) ∂βl (−βl)− ∑ l / ∈S P ′ n(|hl|)|βl| ≤ ∑ l / ∈S (∣∣∣∣∂Ln(h) ∂βl ∣∣∣∣− P ′ n(|hl|)) |βl|, (0.3) where we used the fact that sgn(hl) = sgn(βl) for l / ∈ S. Suppose we have (which we will prove later) max l / ∈S ∣∣∣∣∂Ln(β̂) ∂βl ∣∣∣∣ = op(P ′ n(0)), (0.4)