Consider the three-equation model, y = x + u ; x = u + ; z = + v ,

Consider the three-equation model, $y=$ $\beta \mathit{x}+u;\mathit{x}=\lambda u+\epsilon ;\mathit{z}=\gamma \epsilon +v$, where the mutually independent errors $u,\epsilon$, and $v$ are iid normal with mean 0 and variances, respectively, ${\sigma }_u^2,{\sigma }_{\epsilon }^2$, and ${\sigma }_v^2$.

(a) Show that plim $({\hat{\beta }}_{\text{OLs }}-\beta )=\lambda {\sigma }_u^2/({\lambda }^2{\sigma }_u^2+{\sigma }_{\epsilon }^2)$.

(b) Show that ${\rho }_{XZ}^2=\gamma {\sigma }_{\epsilon }^2/({\lambda }^2{\sigma }_u^2+{\sigma }_{\epsilon }^2)({\gamma }^2{\sigma }_{\epsilon }^2+{\sigma }_v^2)$.

(c) Show that ${\hat{\beta }}_{\mathrm{I}\mathrm{V}}=m_{zy}/m_{zx}=\beta +m_{zu}/(\lambda m_{zu}+m_{zz})$, where, for example, $m_{zy}=$ $\sum _i{z}_i{y}_i$.

(d) Show that ${\hat{\beta }}_{\mathrm{I}\mathrm{V}}-\beta \to 1/\lambda$ as $\gamma ($ or ${\rho }_{xz})\to 0$.

(e) Show that ${\hat{\beta }}_{\text{IV }}-\beta \to \mathrm{\infty }$ as $m_{zu}\to -\gamma {\sigma }_{\epsilon }^2/\lambda$.

(f) What do the last two results imply regarding finite-sample biases and the moments of ${\hat{\beta }}_{\text{IV }}-\beta$ when the instruments are poor?