# Cluster RCT with different timing and exposure to treatment

We are currently evaluating a government Microenterprise program using a cluster RCT design. The treatment involves the provision of a grant to a poor household to start a micro-enterprise. We are looking at outcomes such as household income and expenditures.

An issue is that program households received their grant at different times. Implementation is devolved to the municipalities, so the timing of the release of the grants depend on whether the local program officers start recruitment early or late. Some households received the grant as early as July 2018, while some as late as December 2018. The baseline survey took place in August 2018 - December 2018, and the endline survey will take place in May-June 2019. This would mean exposure of households to the treatment would vary in length when the endline comes. Some would have had their microenterprises running longer than others.

Our estimation of the treatment effect involves a simple OLS regression with municipal fixed effects and cluster robust errors. It has the following specification:

$$y_{ij} = a + \delta T_j+ \gamma_k y_{ij_{-1}} + \Sigma_h \gamma_{h} x_{hij_{-1}} + \Sigma_m \gamma_{m}c_m + \epsilon_{ij}$$

where

• $$y_{ij}$$ is the outcome for household $$i$$ in municipality $$j$$
• $$T_j$$ is the treatment dummy
• $$y_{ij_{-1}}$$ if the baseline value of the outcome variable
• $$x_{hij_{-1}}$$ are $$h$$ household-specific covariates measures at baseline
• $$c_m$$ is a municipal dummy for the $$m^{th}$$ cluster
• $$\epsilon_{ij}$$ is the random error.

However I am not sure how we should take into account the varying exposure lengths of the households to the treatment. I have come across papers on how to do so when using a DID strategy but not in the context of an RCT.