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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.

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$y_t =\alpha+\beta_1s_t+\beta_p p_t +\epsilon_t$

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: $E[\hat{\beta_p}]= \beta_p+\beta_s\frac{Cov(p_t,s_t)}{Var(p_t}$

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  • $\begingroup$ @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? $\endgroup$ Jul 20 at 2:25
  • $\begingroup$ Sorry I misunderstood. As you now have written, I think all you can do is estimate the effect of $p_t$ on $z_t$. $\endgroup$ Jul 20 at 11:52

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