this is a follow-up to this question.

I wanted to estimate using Stata's cmp command a system of 2 equations: an ordered probit and a linear equation.

1 - linear model: $$y = \alpha + \beta z + \epsilon_1$$. 2 - ordered probit: $$z^* = \gamma x + \epsilon_2 \\ z = j \quad \alpha_{j-1} \leq z^* \leq \alpha_j \quad j \in \{-4, -3, \ \dots \ 3, \ 4\} $$

As the linked question asked, I wanted to derive marginal effects of $x$ on $y$. Since the margins command in Stata wouldn't account for this indirect link between the two variables ($x \to z \to y$), I asked the package's author for a possible alternative. He suggested using $z$'s linear predictor on the first equation. (in case you know cmp, it would be as in cmp(y = x#) (x = z), vce(robust) ind($cmp_cont $cmp_oprobit) nolr).

This seems like using $z$ as an instrument for $x$. I have then a few questions:

1- Is that so?

2- What would be the difference in intuition from the approaches? Is there a way to think about which one fits best?


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