Let's say I have some mechanism that theoretically says $$a_t$$ positively affects $$b_t$$ and $$b_t$$ positively affects $$c_t$$. I have data on $$a_t$$ and $$c_t$$ but not $$b_t$$. Thus, $$b_t$$ is unobserved. I run a regression: $$c_t$$ = $$\alpha_0 + \alpha_1 a_t$$ where $$\alpha_1 > 0$$. Can I say something about the empirical effect of $$a_t$$ on $$b_t$$ although $$b_t$$ is unobserved? In this example, may be I can say that if $$a_t$$ positively affects $$c_t$$, then more likely it positively affects $$b_t$$? Thus, I am speculating about the effect of $$a_t$$ on $$b_t$$ here given the theory and the regression model.

Lemme also ask a related question. If $$a_t$$ affects $$b_t$$ and $$b_t$$ affects $$c_t$$, I can not run the regression $$c_t$$ = $$\alpha_0 + \alpha_1 a_t + \alpha_2 b_t$$ since $$a_t$$ and $$b_t$$ would be collinear, right? And I can only do that in a multiple equation model like the SVAR?

• The theory is: $c_t =\beta_0 +\beta_1 b_t +e_t$ and $b_t =\alpha_0 +\alpha_1 a_t +v_t$ This means your regression is effectively, $c_t =\gamma_0 +\gamma_1 a_t +u_t$ Where $\gamma_1 =\beta_1 \alpha_1$ If you only observe $\gamma_1$, is not possible to identify any aspect of $\beta_1$, even the sign. Aug 14 at 11:09
• You can run a regression $c_t = \alpha_0 +\alpha_1 a_t +\alpha_1b_t+u_t$, but you'd estimate the effect of $a_t$ holding fixed $b_t$, while part of $a_t$'s effect presumably operates through $b_t$. Aug 14 at 11:11
• If you run the regression $c_t = a_0 + a_1 a_t + a_2 b_t$ and $a_t$ only relates to $c_t$ through $b_t$ then the coefficient $a_1$ should be zero after controlling for $b_t$. They are not collinear since presumably $a_t$ and $b_t$ have their own noise terms. Aug 14 at 14:48
• @Giskard Andrew and I didn't suggest $c_t =\alpha_0 +\alpha_1 a_t +\alpha_2 b_t$. OP did in his second paragraph. Andrew and I responded to that. Aug 15 at 11:29
The theory is: $$c_t=\beta_0+\beta_1b_t+e_t$$ and $$b_t=\alpha_0+\alpha_1a_t+v_t$$. This means your regression is effectively, $$c_t=\gamma_0+\gamma_1a_t+u_t$$ Where $$\gamma_1=\beta_1\alpha_1$$ If you only observe $$\gamma_1$$, is not possible to identify any aspect of $$\beta_1$$, even the sign.
You can run a regression $$c_t=\alpha_0+\alpha_1a_t+\alpha_2b_t+u_t$$, but you'd estimate the effect of $$a_t$$ holding fixed $$b_t$$, while part of $$a_t$$'s effect presumably operates through $$b_t$$.