Consider IV estimation (Section 4.8) of the model y = x + u using instruments

Consider IV estimation (Section 4.8) of the model $y={\mathbf{x}}^{\mathrm{\prime }}\mathit{\beta }+u$ using instruments $\mathbf{z}$ in the just-identified case with $\mathbf{Z}$ an $N\times K$ matrix of full rank.

(a) What essential assumptions must $\mathbf{z}$ satisfy for the IV estimator to be consistent for $\beta$ ? Explain.

(b) Show that given just identification the 2SLS estimator defined in (4.53) reduces to the IV estimator given in (4.51).

(c) Give a real-world example of a situation where IV estimation is needed because of inconsistency of OLS, and specify suitable instruments.

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B2SLS = [X'Z(Z'Z)-'Z'X]'[X'Z(Z'Z)-'Z'y], (4.53)