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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?

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    $\begingroup$ 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. $\endgroup$ Aug 14 at 11:09
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    $\begingroup$ 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 $b_t$. $\endgroup$ Aug 14 at 11:11
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    $\begingroup$ 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. $\endgroup$
    – Andrew M
    Aug 14 at 14:48
  • $\begingroup$ @MichaelGmeiner Please post answers as answers. $\endgroup$
    – Giskard
    Aug 14 at 20:21
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    $\begingroup$ @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. $\endgroup$ Aug 15 at 11:29
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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 $b_t$.

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