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.
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1$\begingroup$ 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 $\hat\gamma_2$ ? $\endgroup$– UteCommented Jul 14, 2023 at 12:29
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$\begingroup$ Yes @Ute . I have fixed the typo. $\endgroup$– Kwame BrownCommented Jul 18, 2023 at 13:40
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$\begingroup$ 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". $\endgroup$– UteCommented Jul 20, 2023 at 16:28
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