in my understanding, SEM should be applied in a case such as:
\begin{equation}
Y= \alpha_1 + \beta_1 X + \epsilon
\end{equation}
\begin{equation}
X= \alpha_2 + \beta_2 Y + \epsilon
\end{equation}
as $X$ and $Y$ are jointly determined, while SURE in a case such as:
\begin{equation}
Y_1= \alpha_1 + \beta_1 X + \epsilon
\end{equation}
\begin{equation}
Y_2= \alpha_2 + \beta_2 X + \epsilon
\end{equation}
to account for correlation across the error terms of the equations.
My question is: I have this case
\begin{equation}
Y = \alpha_1 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 + \epsilon
\end{equation}
\begin{equation}
X_1 = \alpha_2 + \beta_4 X_2 + \beta_5 X_4 + \epsilon
\end{equation}
So we have that $X_1$ and $X_2$ determine $Y$, but $X_2$ determines $X_1$ too.
Which method should I use in this case?
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$\begingroup$ Yes sorry, they are $\alpha_1$ and $\alpha_2$. I edit the question. Thanks! $\endgroup$– AndreaSep 11, 2021 at 16:27
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$\begingroup$ What parameters do you want to estimate? $\endgroup$– dimitriySep 11, 2021 at 20:22
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$\begingroup$ @DimitriyV.Masterov I need to estimate $\beta_1$, $\beta_2$ and $\beta_4$ $\endgroup$– AndreaSep 12, 2021 at 9:17
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