Abstract
This note contains additional technical details of Cheng, Dou, and Liao (2020). Section A provides the proofs of several lemmas on the asymptotic convergence of the random components in the test statistic and the conditional critical value. Section B verifies the bounded Lipschitz properties of the test statistic and the conditional critical value, which are used to show their weak convergence in large sample. Section C includes some auxiliary lemmas. Section D derives the Euler equations that serve as the asset pricing moment conditions in the long-run risk model and the disaster risk model. Section E considers the long-run risk model and shows that the Gaussian limit is an innocuous assumption. Section F provides derivations for the time-varying disaster risk model in the empirical application of the paper. The full-text version of this paper can be found at: https://ssrn.com/abstract=3609627.
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