Consider the simultaneous equations model, where (x) is exogenous. Assume that (Eleft(e_{i 1} mid mathbf{x}_{1} ight)=Eleft(e_{i 2}

Consider the simultaneous equations model, where \(x\) is exogenous.

Assume that \(E\left(e_{i 1} \mid \mathbf{x}_{1}\right)=E\left(e_{i 2} \mid \mathbf{x}_{1}\right)=0, \operatorname{var}\left(e_{i 1} \mid \mathbf{x}_{1}\right)=\sigma_{1}^{2}, \operatorname{var}\left(e_{i 2} \mid \mathbf{x}_{1}\right)=\sigma_{2}^{2}\), and \(\operatorname{cov}\left(e_{i 1}, e_{i 2} \mid \mathbf{x}_{1}\right)=\sigma_{12}\).

a. Substitute the second equation into the first and find the reduced-form equation for \(y_{i 1}\).

b. Multiply the reduced-form equation for \(y_{i 1}\) from part (a) by \(e_{i 2}\) and find \(\operatorname{cov}\left(y_{i 1}, e_{i 2} \mid \mathbf{x}_{1}\right)=\) \(E\left(y_{i 1} e_{i 2} \mid \mathbf{x}_{1}\right)\).

c. Show that \(\operatorname{cov}\left(y_{i 1}, e_{i 2} \mid \mathbf{x}_{1}\right)=0\) if \(\alpha_{1}=0\) and \(\sigma_{12}=0\). Such a system is said to be recursive.

d. Is the OLS estimator of the first equation consistent under the conditions in (c)? Explain.

e. Is the OLS estimator of the second equation consistent under the conditions in (c)? Explain.

Transcribed Image Text:

= Yil 20xi1 + eil