# Thread: Causal mediation analysis with lagged endogeneous regressors

1. ## Causal mediation analysis with lagged endogeneous regressors

Hello dear forum members!

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.

2. ## Re: Causal mediation analysis with lagged endogeneous regressors

It seems that sometimes one needs to talk to himself in order to see the hidden caveats, especially those in stats Mediation, including that with longitudinal data, is quite a tricky topic. Here are a couple of good readings for those interested:

1. Preacher, K. J. (2015). Advances in mediation analysis: A survey and synthesis of new developments. Annual review of psychology, 66, 825-852.
2. Selig, J. P., & Preacher, K. J. (2009). Mediation models for longitudinal data in developmental research. Research in Human Development, 6(2-3), 144-164.
3. http://davidakenny.net/cm/mediate.htm#DIA

Peace

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