So, in the study I am currently working on, I have the following (simplified) Poisson panel model with fixed-effects specification:

(1) y(it) = a + x1(it) + x2(it), where y is a non-negative count, x1 is binary and endogenous and x2 is exogenous. Noticeably, the theory suggests that the effect of x1 could be estimated using lagged effects specification, so I also have another model:

(2) y(it) = a + x1(it-1) + x2(it)

Further, there is an additional control -- m1 -- that clearly shows signs of a mediator (I personally rely on "Baron, R. M., & Kenny, D. A. (1986). The moderator–mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. Journal of personality and social psychology, 51(6), 1173." but there are other sources, of course). As such, I also estimate the following model:

(3) y(it) = a + x1(it) + x2(it) + m1(it)

As is supposed to be with mediation, the affect of x1 shrinks (and looses its significance) in the presence of a mediator (i.e., m1).

What concerns me is when I also estimate Equation (2) using lagged values of x1 with an included mediator (m1) -- (4) y(it) = a + x1(it-1) + x2(it) + m1(it) -- the coefficient of x1(it-1) does not loose it significance, neither I observe any shrinkage of the effect.

The questions I am looking for the answers are:

- What could be the plausible explanation for the result in Equation (4)?

- Do I even need to estimate Equation (4) to establish the mediation effect of m1 on x1? On the one hand the answer seems to be "no"; however on the other hand based on theory I'd somewhat expect m1 to act similarly with x1's t and t-1 values.

Thank you in advance for your comments