Supplement to Empirical Bayes Regression with Many Regressors
Thomas Knox, James H. Stock · 2003
This part of the supplement contains the proofs omitted from the paper and appendix. All the proofs shown here use the simplifying assumption that K = ρT, where 0 < ρ < 1, that is, the proofs are for K = ρT δ with δ = 1. The integer constraint on K is ignored. Modifications of these proofs for δ < 1 are straightforward, and these modifications are discussed in the final section of this part. First a word about terminology. References to “Assumption 1, ” “Lemma 1, ” and “Theorem 1 ” are references to the indicated assumption, lemma, or theorem in the paper. Theorems 7 through 10 are stated and proven in this supplement; Theorem 7 is an extension of our results, while Theorems 8 through 10 are foundations needed for our results. This supplement also has its own lemmas, which are called “Lemma S-1, ” etc. A.1 Preliminary Results We start by collecting some additional definitions. Some of these repeat definitions in the paper and are included for completeness. Definition 1 fijK dK ˆ b−i σ2 ε log K