Online Supplement Tests of Policy Interventions in DSGE Models by M . Hashem Pesaran and R . P . Smith
نویسندگان
چکیده
Proof. The result follows by direct manipulation of the terms. Lemma S2 Suppose that the k×k matrices A and B have bounded spectral norms ‖A‖ ≤ λ and ‖B‖ ≤ λ, for some fixed positive constant λ. Then ∥∥∥Ah −Bh∥∥∥ ≤ hλh−1 ‖A−B‖ , for h = 1, 2, .... (A.1) Proof. We establish this result by backward induction. It is clear that it holds for h = 1. For h = 2, using the identity A −B = A(A−B) + (A−B)B, the result for h = 2 follows ∥∥A2 −B2∥∥ ≤ (‖A‖+ ‖B‖) ‖A−B‖ = 2λ ‖A−B‖ . More generally, we have the identity A −B = A(A−B) + (A−B)B + A(Ah−2 −Bh−2)B. Now suppose now that (A.1) holds for h− 2, and using the above note that ∥∥∥Ah −Bh∥∥∥ ≤ ∥∥∥Ah−1∥∥∥ ‖A−B‖+ ‖A−B‖ ∥∥∥Bh−1∥∥∥+ ‖A‖ ∥∥∥Ah−2 −Bh−2∥∥∥ ‖B‖ ≤ ‖A‖h−1 ‖A−B‖+ ‖A−B‖ ‖B‖h−1 + ‖A‖ ∥∥∥Ah−2 −Bh−2∥∥∥ ‖B‖ ≤ 2λh−1 ‖A−B‖+ λ ∥∥∥Ah−2 −Bh−2∥∥∥ ≤ 2λh−1 ‖A−B‖+ λ [ (h− 2)λh−3 ‖A−B‖ ] ≤ hλh−1 ‖A−B‖ .
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