Standard error of product of coefficients

Suppose that I have a regression model: $$y=\beta+\beta_{1}T+\beta_{2}M+\epsilon$$ and $$M=\alpha+\alpha_{1}T+v$$ In this model, $$y$$ is the outcome variable, $$T$$ is treatment, and $$M$$ is the mediating variable. In the terminilogy of mediation analysis, we have that the \textbf{causal mediation }effect is defined as $$\alpha_{1}\beta_{2}$$. Using OLS on these two models, we obtain an estimate of this effect as $$\hat{\alpha}_{1}\hat{\beta}_{2}$$. Although getting the point estimate is straightforward, how would we obtain the standard error of this effect, that is $$SE\left(\hat{\alpha}_{1}\hat{\beta}_{2}\right)$$?

• What data did you use? Of importance is whether you used the same responses in both procedures.
– whuber
May 2 at 18:14
• Same responses- just different models for $y$. May 2 at 18:30
• Please edit your post to indicate that, because it has a huge influence on any correct solution. But it makes one wonder: you seem to be asking for the product of estimates of two contradictory models. How do you intend to interpret that??
– whuber
May 2 at 18:45
• @whuber I Have modified the problem to more accurately suit what I have in mind. May 6 at 16:26