Dynamic Global Games of Regime Change: Learning, Multiplicity and Timing of Attacks

This supplementary document contains a formal analysis of some of the extensions briefly discussed in Section 5 of the published version. Section A1 considers the game in which agents receive signals about the size of past attacks. Section A2 considers the game with observable shocks to the fundamentals. Section A3 considers the variant in which agents observe the shocks with a one-period lag. Section A4 considers the game with short-lived agents in which the fundamentals follow a random walk. Finally, Section A5 collects the proofs of the formal results contained in this document. A1. Signals about past attacks For some applications, it might be natural to assume that agents collect information– either private or public– not only about the underlying fundamentals but also about the size of past attacks. To capture this possibility, we extend the game with public news examined in Section 5.1 as follows. In every period t ≥ 2, agents receive private and public signals about the size of the attack in the previous period. These signals are, respectively, X̃it = S(At−1, ξ̃it) and Z̃t = S(At−1, ε̃t), where ξ̃it is idiosyncratic noise, ε̃t is common noise, and S : [0, 1]× R→ R. To preserve Normality of the information structure, we adopt a specification similar to that in Dasgupta (2002): ξ̃it ∼ N (0, 1/γ t ) , ε̃t ∼ N (0, 1/γ t ) , and S (A, υ) = { Φ−1 (A) + υ if A ∈ (0, 1),