Consider the linear regression model for data independent over $i$ with $y_i=$ ${\mathbf{x}}_i^{\mathrm{\prime }}\mathit{\beta }+u_i$. Suppose $\mathrm{E}[u_i\mid {\mathbf{x}}_i]\ne 0$ but there are available instruments ${\mathbf{z}}_i$ with $\mathrm{E}[u_i\mid {\mathbf{z}}_i]=0$ and $\mathrm{V}[u_i\mid {\mathbf{z}}_i]={\sigma }_i^2$, where $\dim (\mathbf{z})>\dim (\mathbf{x})$. We consider the GMM estimator $\hat{\beta }$ that minimizes

\[ Q_{N}(\boldsymbol{\beta})=\left[N^{-1} \sum_{i} \mathbf{z}_{i}\left(y_{i}-\mathbf{x}_{i}^{\prime} \beta\right)\right]^{\prime} \mathbf{W}_{N}\left[N^{-1} \sum_{i} \mathbf{z}_{i}\left(y_{i}-\mathbf{x}_{i}^{\prime} \beta\right)\right] \]

(a) Derive the limit distribution of $\sqrt{N}(\hat{\mathit{\beta }}-{\mathit{\beta }}_0)$ using the general GMM result (6.11).

(b) State how to obtain a consistent estimate of the asymptotic variance of $\hat{\mathit{\beta }}$.

(c) If errors are homoskedastic what choice of ${\mathbf{W}}_N$ would you use? Explain your answer.

(d) If errors are heteroskedastic what choice of ${\mathbf{W}}_N$ would you use? Explain your answer.