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.