<p>The theory is: <span class="math-container">$c_t=\beta_0+\beta_1b_t+e_t$</span> and <span class="math-container">$b_t=\alpha_0+\alpha_1a_t+v_t$</span>. This means your regression is effectively, <span class="math-container">$c_t=\gamma_0+\gamma_1a_t+u_t$</span> Where <span class="math-container">$\gamma_1=\beta_1\alpha_1$</span> If you only observe <span class="math-container">$\gamma_1$</span>, is not possible to identify any aspect of <span class="math-container">$\beta_1$</span>, even the sign.</p>
<p>You can run a regression <span class="math-container">$c_t=\alpha_0+\alpha_1a_t+\alpha_2b_t+u_t$</span>, but you'd estimate the effect of <span class="math-container">$a_t$</span> holding fixed <span class="math-container">$b_t$</span>, while part of <span class="math-container">$a_t$</span>'s effect presumably operates through <span class="math-container">$b_t$</span>.</p>

CC BY-SA 4.0
2021-08-14T20:25:02.033Z
2021-08-14T20:25:02.033Z
Michael Gmeiner
<p>Let's say I have some mechanism that theoretically says <span class="math-container">$a_t$</span> positively affects <span class="math-container">$b_t$</span> and <span class="math-container">$b_t$</span> positively affects <span class="math-container">$c_t$</span>. I have data on <span class="math-container">$a_t$</span> and <span class="math-container">$c_t$</span> but not <span class="math-container">$b_t$</span>. Thus, <span class="math-container">$b_t$</span> is unobserved. I run a regression: <span class="math-container">$c_t$</span> = <span class="math-container">$\alpha_0 + \alpha_1 a_t$</span> where <span class="math-container">$\alpha_1 &gt; 0$</span>. Can I say something about the empirical effect of <span class="math-container">$a_t$</span> on <span class="math-container">$b_t$</span> although <span class="math-container">$b_t$</span> is unobserved? In this example, may be I can say that if <span class="math-container">$a_t$</span> positively affects <span class="math-container">$c_t$</span>, then more likely it positively affects <span class="math-container">$b_t$</span>? Thus, I am speculating about the effect of <span class="math-container">$a_t$</span> on <span class="math-container">$b_t$</span> here given the theory and the regression model.</p>
<p>Lemme also ask a related question. If <span class="math-container">$a_t$</span> affects <span class="math-container">$b_t$</span> and <span class="math-container">$b_t$</span> affects <span class="math-container">$c_t$</span>, I can not run the regression <span class="math-container">$c_t$</span> = <span class="math-container">$\alpha_0 + \alpha_1 a_t + \alpha_2 b_t$</span> since <span class="math-container">$a_t$</span> and <span class="math-container">$b_t$</span> would be collinear, right? And I can only do that in a multiple equation model like the SVAR?</p>

CC BY-SA 4.0
2021-08-14T11:09:47.937Z
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.
Michael Gmeiner
CC BY-SA 4.0
2021-08-14T11:11:06.830Z
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 $b_t$.
Michael Gmeiner
CC BY-SA 4.0
2021-08-14T14:48:19.953Z
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.
Andrew M
CC BY-SA 4.0
2021-08-14T20:21:27.193Z
@MichaelGmeiner Please post answers as answers.
Giskard
CC BY-SA 4.0
2021-08-14T20:22:25.157Z
@AndrewM Please post answers as answers. Also, the suggested regression does not seem feasible as OP states $b_t$ is unobserved.
Giskard
CC BY-SA 4.0
2021-08-15T11:29:54.007Z
@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.
Michael Gmeiner
<econometrics>
Speculating about causal effects
Emmanuel Ameyaw
# Speculating about causal effects

(Asked by: username_0 on 2021-08-14)

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?

**Tags:** econometrics

> **Comments:**
> **username_1**: 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.  
> **username_1**: 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 $b_t$.  
> **username_2**: 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.  
> **username_3**: @MichaelGmeiner Please post answers as answers.  
> **username_3**: @AndrewM Please post answers as answers. Also, the suggested regression does not seem feasible as OP states $b_t$ is unobserved.  
> **username_1**: @username_3  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.  

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## 1 answers

### Answer by username_1 on 2021-08-14. Score: 3 | Accepted answer

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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