Multiple mediators and interaction effects- causal mediation effect Assume we have a structural equation system with treatment $T,$ mediator
$1$ (discrete) and mediator $2$ (continuous). We write this as:
$$
Y=\beta_{0}+\beta_{1}M_{1}+\beta_{2}M_{2}+\beta_{3}TM_{2}+\beta_{3}T+\epsilon
$$
$$
M_{1}=\alpha+\alpha_{1}T+v
$$
$$
M_{2}=\gamma+\gamma_{1}T+e
$$
For $M_{1},$ the causal mediation effect is easy to obtain- $\alpha_{1}\beta_{1}.$
However, given that we have an interaction effect with $M_{2}$, would
the causal mediation effect be:
$$
\gamma_{1}\beta_{2}+\gamma_{1}\beta_{3}M_{2}
$$
If this is the case, would a ``reasonable'' estimate of the causal
mediation effect be:
$$
\gamma_{1}\beta_{2}+\gamma_{1}\beta_{3}\mathbb{E}\left[M_{2}\right]
$$
where we take the average value of $M_{2}?$
 A: An important first step of an analysis is define the estimand, or estimation target. When you write, "causal mediation effect", the estimand is left ambiguous. There are several possible terms you could be describing. I'll take you to mean the indirect natural effect.
For convenience, this answer will use counterfactual notation, with an asterisk denoting a counterfactual. With a single mediator, this is defined as $$\mathbb{E}[Y_{t, M_{1}^*}^* - Y_{t, M_{0}^*}^*],$$ for a possible treatment value $t$. It is interpreted as giving the causal effect of the treatment that flows through a mediator $M$ when the treatment is held at $t$ (to block the direct effect).
Your problem has multiple mediators, but their analysis is simplified because neither is a descendant of the other. Thus we can study the effect of each mediator on its own.
For the first mediator, the indirect natural effect is
\begin{align*}
 & \mathbb{E}[Y_{t, M_{1,1}^*}^* - Y_{t, M_{1,0}^*}^*] \\
=\, &  \mathbb{E} \left[ \{\beta_0 + \beta_1(\alpha + \alpha_1 + v) + \beta_2M_2 + \beta_3tM_2 + \beta_4t + \epsilon\} - \\ \{\beta_0 + \beta_1(\alpha + v) + \beta_2M_2 + \beta_3tM_2 + \beta_4t + \epsilon\} \right] \\
=\, & \beta_1\alpha_1,
\end{align*}
for any $t$. This is the value you described.
For the second mediator, the indirect natural effect is
\begin{align*}
 & \mathbb{E}[Y_{t, M_{2,1}^*}^* - Y_{t, M_{2,0}^*}^*] \\
=\, &  \mathbb{E} \left[ \{\beta_0 + \beta_1 M_1 + \beta_2(\gamma + \gamma_1 + e) + \beta_3t(\gamma + \gamma_1 + e)+ \beta_4t + \epsilon\} - \\ \{\beta_0 + \beta_1M_1 + \beta_2(\gamma + e) + \beta_3t(\gamma + e) + \beta_4t + \epsilon\} \right] \\
=\, & \gamma_1 ( \beta_2 + \beta_3 t)
\end{align*}
for any $t$. Thus the natural indirect effect depends on the treatment value $t$. Holding the treatment at $t=0$ or at $t=1$ will lead to a different effect of the second mediator $M_2$.
Each indirect effect can be consistently estimated by plugging in a consistent estimator of the regression coefficients.
