# How to deal with non-random measure error in dependent variable

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.

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

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

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