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?

Please kindly give your advice on this issue. I am using Stata. Thank you in advance.

  • $\begingroup$ Are you not looking for a cross lagged panel model? $\endgroup$ – POC Mar 3 at 15:17
  • $\begingroup$ @POC Thank you so much for your comment. I have never heard of cross lagged panel model, and I have just looked through it. It seems very complicated than my idea as it also includes serial correlations, and T0 also affects Y1. In my data, T during period t only affects Y at t, and Y affects T at t+1. Do you think that it is still useful to use "cross lagged panel model" in this case? $\endgroup$ – Yendao Su Mar 3 at 19:21

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