# Mediation analysis and no difference in explanatory variable's point estimate

I am attempting to perform mediation analysis to explain variable $$y$$ as a function of $$x$$, $$z_{1}.$$ and $$z_{2}$$. $$x$$ is the variable of interest, while $$z_{1}$$ and $$z_{2}$$ are the mediating variables. I run the following regression: $$y=\beta_{0}+\beta_{1}x+\epsilon$$ and I obtain an estimate of $$\beta,\hat{\beta}.$$ I then run another regression, augmenting the original regression with the variables $$z_{1}$$ and $$z_{2}$$ $$y=\gamma_{0}+\gamma_{1}x+\gamma_{2}z+\gamma_{3}z_{2}+v$$ The second regression delivers significant point estimates on $$z_{1},$$ and $$z_{2}$$ , namely $$\hat{\gamma}_{1}$$ and $$\hat{\gamma}_{2}$$ but also a value of $$\hat{\gamma}_{1}$$ which is very close to $$\hat{\beta}_{1}.$$ Specifically, I am unable to reject the null that: $$H_{0}:\hat{\gamma}_{1}-\hat{\beta_{1}}=0$$ Is it correct to interpret, therefore, that there is no mediating effect of $$z_{1}$$ and $$z_{2}$$ on $$x?$$ Could it be that the mediation effect of $$z_{1}$$ and $$z_{2}$$ oppose each other? It is indeed suprising, because as long as, for example, $$corr\left(x,z_{1}\right)$$ is not 0, and $$z_{1}$$ is significant, it is strange that the point estimate on $$x$$ does not change. Any help on this is much appreciated.

• Should it perhaps read $y=\gamma_{0}+\gamma_{\color{red}{1}}x+\gamma_{\color{red}{2}}z_{1}+\gamma_{3}z_{2}+v$ instead of $y=\gamma_{0}+\gamma_{2}x+\gamma_{1}z+\gamma_{3}z_{2}+v$ and ...significant point estimates on $z_{1}$ and $z_{2}$, namely $\hat\gamma_{\color{red}{2}}$ and $\hat\gamma_{\color{red}{3}}$ instead of ... namely $\hat\gamma_1$ and $\hat\gamma_2$ ?
– Ute
Commented Jul 14, 2023 at 12:29
• Yes @Ute . I have fixed the typo. Commented Jul 18, 2023 at 13:40
• I think you are right that it could mean that $z_1$ and $z_2$ have opposite effects. The answer depends on how you define "no mediating effect".
– Ute
Commented Jul 20, 2023 at 16:28