Supplement to two papers on multiple bubbles [Online supplementary materials]
补充了两篇关于S&P 500指数多重泡沫检测论文的稳健性检验和理论证明，包括最小窗口参数选择和递归单位根检验的极限行为。
This paper provides a supplement to two companion papers by the authors: “Testing for Multiple Bubbles: Historical Episodes of Exuberance and Collapse in the S&P 500” (PSY1 hereafter); and “Testing for Multiple Bubbles: Limit Theory of Real Time Detectors” (PSY2 hereafter). Section 1 supplements the empirical application of PSY1 by examining the robustness of the bubble identification and dating results to the choice of the minimum window size parameter used in the rolling regression framework of PSY. Section 2 provides proofs of supplementary lemmas that facilitate analysis of the multiple bubble case, derives the limit behaviour of the recursive unit root and BDF test statistics discussed in PSY2 in a model with two bubble episodes, and gives complete proofs for Theorem 4-9 in PSY2 which describe the consistency properties of the PWY, PSY and sequential PWY dating procedures.