Supplemental Appendix to Conditional Superior Predictive Ability

This supplemental appendix contains additional technical details and numerical results. Section SA provides a detailed proof of Proposition 1 in the main text. Section SB provides proofs for Lemma A1 and Lemma A2 that are used for proving Theorem 1 in the main text. In Section SC, we describe the pre-whitened HAC estimator and establish its rate of convergence under the growing-dimensional setting, which can be used to verify the high-level condition Assumption 1. Finally, Section SD contains additional simulation results demonstrating the usefulness of the selection step, and robustness to an alternative data generating process. SA Detailed proof of Proposition 1 In this section, we provide a detailed proof for Proposition 1 in the main text. Section SA.1 introduces some notation and preliminary technical results. Section SA.2 presents the proof of that proposition. SA.1 Notation and preliminary results We need several technical lemmas. The first lemma concerns the uniform convergence rate of the estimator of the standard error function. We recall that V ≡ { (j, x) : 1 ≤ j ≤ J and x ∈ X} and σj,n (x) ≡ √ P (x)>Q−1 n An (j, j)Q −1 n P (x), where An (j, j) is the jth mn ×mn diagonal block of An. Lemma SA1. Under Assumptions 1 and 2, we have sup (j,x)∈V ∣∣∣∣σj,n (x) σ̂j,n(x) − 1 ∣∣∣∣ = Op(δQ,n + δA,n). Proof. Denote Ωn (j, j) = Q −1 n An (j, j)Q −1 n . By definition, Ω̂n (j, j)− Ωn (j, j) = Q̂−1 n Ân(j, j)Q̂−1 n −Q−1 n An(j, j)Q−1 n = (Q̂−1 n −Q−1 n )Ân(j, j)Q̂−1 n +Q−1 n (Ân(j, j)−An(j, j))Q̂−1 n +Q−1 n An(j, j)(Q̂−1 n −Q−1 n ). Under the maintained assumptions, the eigenvalues of Qn, An, Q̂n, and Ân are bounded from above and away from zero in probability. Hence, by the triangle inequality and the Cauchy– Schwarz inequality, we have uniformly in x, ∣∣σ̂2j,n (x)− σ j,n(x)∣∣ = ∣∣∣P (x)> (Ω̂n (j, j)− Ωn (j, j))P (x)∣∣∣ ≤ Op(1) ‖P (x)‖ (‖Q̂n −Qn‖S + ‖Ân −An‖S). We then deduce that, uniformly in (j, x), ∣∣∣σ̂2j,n (x)− σ2 j,n(x)∣∣∣ σ̂j,n(x) ≤ Op(1) · (‖Q̂n −Qn‖S + ‖Ân −An‖S) = Op(δQ,n + δA,n), and hence, ∣∣∣∣σj,n (x) σ̂j,n(x) − 1 ∣∣∣∣ ≤ ∣∣∣σ̂2j,n (x)− σ2 j,n(x)∣∣∣ σ̂j,n(x) = Op(δQ,n + δA,n). 2 This finishes the proof. To state Lemma SA2, we introduce a selection operator. We partition a Jmn-dimensional vector z as z = ( z> 1 , . . . , z > J )> , where each zj is mn-dimensional. The selection operator [ · ]j is defined as [z]j = zj for each j ∈ {1, . , J}. (SA.1) Lemma SA2. Let Dn denote the σ-field generated by the data and Ωn ≡ (IJ ⊗Qn)An (IJ ⊗Qn) . Under Assumptions 1 and 2, we have P  sup (j,x)∈V ∣∣∣∣∣ P (x)> [ Ω̂ 1/2 n ξn ] j σ̂j,n (x) − P (x)> [ Ω 1/2 n ξn ] j σj,n(x) ∣∣∣∣∣ > (log n)−3/2 ∣∣∣∣∣Dn  = op(1), where ξn is a Jmn-dimensional standard Gaussian random vector independent of Dn. Proof. By Assumptions 1 and 2, we have ∥∥∥Ω̂n − Ωn∥∥∥ S ≤ ∥∥∥(Q̂−1 n −Q−1 n )ÂnQ̂ n ∥∥∥ S + ∥∥∥Q−1 n (Ân −An)Q̂ n ∥∥∥ S + ∥∥∥Q−1 n An(Q̂ n −Q−1 n )∥∥∥ S = Op(δQ,n + δA,n). We can then use the same argument as in the proof of Lemma 5 in Chernozhukov, Lee, and Rosen (2013) to get P  sup (j,x)∈V ∣∣∣∣∣ P (x)> [ Ω̂ 1/2 n ξn ] j σ̂j,n (x) − P (x)> [ Ω 1/2 n ξn ] j σj,n(x) ∣∣∣∣∣ > (log n)−3/2 ∣∣∣∣∣Dn  = Op((δQ,n + δA,n)(log n)). Since both δQ,n and δA,n vanish at polynomial rates (as n → ∞) by assumption, the assertion of the lemma readily follows from the estimate above. We now need some additional notation. Consider a generic sequence of Gaussian vectors Ñn ∼ N (0, An) and set Ñj,n = [Ñn]j ; recall that the selection operator [ · ]j is defined in (SA.1). For each subset V ⊆ V, we denote κn,V (q) ≡ the q-quantile of sup (j,x)∈V P (x)>Q−1 n σj,n(x) Ñj,n. Recall the definition of γ̃n from Algorithm 1 and pick a sequence γn such that γ̃n ≥ γn ≥ γ̃n − o(1). We then set Vn ≡ { (j, x) ∈ V : hj(x) ≤ η∗ + nκnσj,n(x) } , where κn ≡ κn,V (γn) .