Stable Central Limit Theorems for Discrete-Time Lag Martingale Difference Arrays: Applications to Dynamic Causal Inference

Abstract

Recent work in dynamic causal inference introduced a class of discrete-time stochastic processes that generalize martingale difference sequences and arrays as follows: the random variates in each sequence have expectation zero given certain lagged filtrations but not given the natural filtration. We formalize this class of stochastic processes and prove stable central limit theorems (CLTs) via martingale-coboundary decomposition, leveraging the classical martingale CLT. We develop a variety of sufficient conditions, including conditions under which the limiting variance has a simple form that depends on variances and covariances of neighboring variates. We demonstrate the application of these results to inference for time-averaged treatment effects in switchback designs and present a simulation study supporting their validity. The CLTs enable various extensions to existing methodology for design-based approaches to dynamic causal inference, including time-lagged effects, random limiting variances, cross-unit dependence, and vector-valued estimands.