# Reverse Causality with Additional Period

I have been struggling for a model to estimate related to sequential treatment effect and need a help desperately. I would greatly appreciate it if you guide me to the resources or advice me on this matter.

In some situation in reality, treatment affects outcome, and outcome affects treatment in the next period. For example, if we see the effect of construction of post box in the municipality and then increase the mail so that increase in the mail leads to more post box construction.

It can be interpreted "reverse causality", but, in the sense that outcome does not affect treatment in the past, it has a following sequential I guess. (Arrows means causal relationship).

$$T_0 \to Y_0 \to T_1 \to Y_1 \to T_2 \to Y_2 \to T_3 \to Y_3\dots$$

\begin{align*} Y_{i,t} &= a + B\,T_{i,t} + u_{i,t}\\ T_{i,t} &= c + b\,T_{i,t-1} + e_{i,t-1} \end{align*}

There are several periods $$t = 1,2,3,4,...,n,$$ and $$T$$ is a treatment/intervention variable (dummy) and $$Y$$ is outcome (continuous; it can be converted into a dummy if needed).

Does this model can be appropriately estimated by simply using lag term? For the causal inferences, what would be the best way to estimate this kind of mechanism?