Causal inference with cross-temporal design.

When many participants in a randomized trial do not comply with their assigned intervention, the randomized encouragement design is a possible solution. In this design, the causal effects of the intervention can be estimated among participants who would have experienced the intervention if encouraged. For many policy interventions, encouragements cannot be randomized and investigators need to rely on observational data. To address this, we propose a cross-temporal design, which uses time to mimic a randomized encouragement experiment. However, time may be confounded with temporal trends that influence the outcomes. To disentangle these trends from the intervention effects, we replace the commonly used exclusion restrictions with temporal assumptions. We develop Bayesian procedures to estimate the causal effects and compare it to instrumental variables and matching approaches in simulations. The Bayesian approach outperforms the other 2 approaches in terms of estimation accuracy, and it is relatively robust to various violations of the common trends assumption. Taking advantage of the expansion of the Medicare Advantage (MA) program between 2011 and 2017, we implement the proposed method to estimate the effects of MA enrollment on the risk of skilled nursing facility residents being re-hospitalized within 30 days after discharge from the hospital.