<p><span class="math-container">$y_t =\alpha+\beta_1s_t+\beta_p p_t +\epsilon_t$</span></p>
<p>If you don't observe a variable, I don't think you want to &quot;take it to the left-hand side&quot;, doing that is just unnecessarily complicated. It would be this:</p>
<p><span class="math-container">$y_t-s_t =\alpha+(\beta_1-1)s_t+\beta_p p_t +\epsilon_t$</span></p>
<p>Where <span class="math-container">$s_t$</span> is still unobserved on the right-hand side.</p>
<p>Rather, if a variable isn't observed you want to think about it being incorporated in your error term.</p>
<p><span class="math-container">$y_t =\alpha+\beta_p p_t +\epsilon^*_t$</span>   where <span class="math-container">$\epsilon^*_t=\epsilon_t+\beta_1s_t$</span>.</p>
<p>If you believe that <span class="math-container">$Cov(p_t, \epsilon^*_t)=0$</span>, then OLS is consistent. No problems at all!</p>
<p>Otherwise, you are in a situation of omitted variables bias. You should consider trying to find an instrument for <span class="math-container">$p_t$</span> and performing 2SLS.</p>
<p>If you aren't able to find a suitable instrument, you should use the well-known OVB formula to hypothesize the direction and size of the bias, and interpret your OLS results cautiously in light of the bias. OVB: <span class="math-container">$E[\hat{\beta_p}]= \beta_p+\beta_s\frac{Cov(p_t,s_t)}{Var(p_t}$</span></p>

CC BY-SA 4.0
2021-07-20T02:25:35.287Z
@Michael...Thanks!! But my problem is I don't observe $y_t$ in the real world. What I observe is $z_t = y_t - s_t$. So I want to estimate the effect of $p_t$ on $z_t$. $y$ is production of some item not observed, $s$ is part of $y$ that workers hide. What is observed is the difference, $z$ - amount recorded by managers. Both $y$ and $s$ depends on $p$ - price of the good. So if I am interested in the effect of p on z, then I don't have measurement issues, yeah?
Emmanuel Ameyaw
CC BY-SA 4.0
2021-07-20T11:52:39.587Z
Sorry I misunderstood. As you now have written, I think all you can do is estimate the effect of $p_t$ on $z_t$.
Michael Gmeiner
CC BY-SA 4.0
2021-07-18T11:22:56.220Z
2021-07-18T11:22:56.220Z
Michael Gmeiner
<p>Let's say I have a model</p>
<p><span class="math-container">$y_t = \alpha + \beta_1 s_t + \beta_2 p_t + \epsilon_t$</span></p>
<p>But <span class="math-container">$s_t$</span> depends on <span class="math-container">$p_t$</span> too, however, not observed. So I take it to the left-hand side.</p>
<p><span class="math-container">$y_t-s_t = \alpha + \beta_p p_t + \epsilon_t$</span></p>
<p>I don't observe <span class="math-container">$y_t$</span> and <span class="math-container">$s_t$</span> but I observe <span class="math-container">$z_t = y_t-s_t$</span>. So I guess I can estimate the following model, right?</p>
<p><span class="math-container">$z_t = \alpha + \beta_p p_t + \epsilon_t$</span></p>
<p>And I do not have measurement problems, I guess. That is if I care about the effect of <span class="math-container">$p_t$</span> on <span class="math-container">$z_t$</span>.</p>
<p>Sorry if my question does not fit the title.</p>

<econometrics>
How to deal with non-random measure error in dependent variable
Emmanuel Ameyaw
# How to deal with non-random measure error in dependent variable

(Asked by: username_0 on 2021-07-18)

Let's say I have a model

$y\_t = \alpha + \beta\_1 s\_t + \beta\_2 p\_t + \epsilon\_t$

But $s\_t$ depends on $p\_t$ too, however, not observed. So I take it to the left-hand side.

$y\_t-s\_t = \alpha + \beta\_p p\_t + \epsilon\_t$

I don't observe $y\_t$ and $s\_t$ but I observe $z\_t = y\_t-s\_t$. So I guess I can estimate the following model, right?

$z\_t = \alpha + \beta\_p p\_t + \epsilon\_t$

And I do not have measurement problems, I guess. That is if I care about the effect of $p\_t$ on $z\_t$.

Sorry if my question does not fit the title.

**Tags:** econometrics

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## 1 answers

### Answer by username_1 on 2021-07-18. Score: 1

$y\_t =\alpha+\beta\_1s\_t+\beta\_p p\_t +\epsilon\_t$

If you don't observe a variable, I don't think you want to "take it to the left-hand side", doing that is just unnecessarily complicated. It would be this:

$y\_t-s\_t =\alpha+(\beta\_1-1)s\_t+\beta\_p p\_t +\epsilon\_t$

Where $s\_t$ is still unobserved on the right-hand side.

Rather, if a variable isn't observed you want to think about it being incorporated in your error term.

$y\_t =\alpha+\beta\_p p\_t +\epsilon^\*\_t$ where $\epsilon^\*\_t=\epsilon\_t+\beta\_1s\_t$.

If you believe that $Cov(p\_t, \epsilon^\*\_t)=0$, then OLS is consistent. No problems at all!

Otherwise, you are in a situation of omitted variables bias. You should consider trying to find an instrument for $p\_t$ and performing 2SLS.

If you aren't able to find a suitable instrument, you should use the well-known OVB formula to hypothesize the direction and size of the bias, and interpret your OLS results cautiously in light of the bias. OVB: $E[\hat{\beta\_p}]= \beta\_p+\beta\_s\frac{Cov(p\_t,s\_t)}{Var(p\_t}$

> **Comments:**
> **username_0**: @Michael...Thanks!! But my problem is I don't observe $y_t$ in the real world. What I observe is $z_t = y_t - s_t$. So I want to estimate the effect of $p_t$ on $z_t$. $y$ is production of some item not observed, $s$ is part of $y$ that workers hide. What is observed is the difference, $z$ - amount recorded by managers. Both $y$ and $s$ depends on $p$ - price of the good. So if I am interested in the effect of p on z, then I don't have measurement issues, yeah?  
> **username_1**: Sorry I misunderstood. As you now have written, I think all you can do is estimate the effect of $p_t$ on $z_t$.  

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