I am having some trouble trying to identify the parameters in the following structural model that I am trying to estimate.

$$ y = a'x_1 + \beta\eta + \epsilon_1 $$ $$ \eta = b'x_2 + \delta T+\epsilon_2 $$

$y$ is, say, any observable human behavior, $\eta$ is any unobservable human trait, $T$ is any treatment that affects the trait, $x_1,x_2$ are any individual-level covariates, and $\epsilon_1,\epsilon_2$ are errors. The reduced form of this model is $$ y = a'x_1+b'\beta x_2+\beta\delta T+ \epsilon_1 +\beta\epsilon_2 $$

The parameters I am interested in are $\beta,\delta,b'$, and I cannot figure out how I can identify them or estimate them.

I have been looking into Structural Equation Models (SEM), but was wondering if someone could direct me as to how to identify it or estimate this model.

  • $\begingroup$ Please say more about your exogenous manifest variables: ¿are $x_1$ and $x_2$ scalar variables or are they vectors? $\endgroup$ – Gregg H Dec 13 '18 at 19:27
  • $\begingroup$ The variables $x_1$ and $x_2$ are vectors of covariates that literature suggests to have some causal effect on $y$ and $\eta$. There will atleast one covariate in $x_2$ that is not in $x_1$. $\endgroup$ – FightMilk Dec 14 '18 at 19:56
  • $\begingroup$ Next query: ¿do you have any variables in the model to serve as measurement variables for the latent trait? $\endgroup$ – Gregg H Dec 15 '18 at 20:11
  • $\begingroup$ Also, as an aside, if there is overlap in $x_1$ and $x_2$, you might consider relabeling as $x_{1'}$, $x_{b}$, $x_{2'}$, where you have the unique variables to $y$, the shared variables, and the unique variables to $\eta$, respectively. $\endgroup$ – Gregg H Dec 15 '18 at 20:13

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