Consider the linear regression model y i = x i + u i with nonstochastic regressors x i and error u i that

Consider the linear regression model $y_i={\mathbf{x}}_i^{\mathrm{\prime }}\mathit{\beta }+u_i$ with nonstochastic regressors ${\mathbf{x}}_i$ and error $u_i$ that has mean zero but is correlated as follows: $\mathrm{E}\left[u_i{u}_j\right]={\sigma }^2$ if $i=j,\mathrm{E}\left[u_i{u}_j\right]=\rho {\sigma }^2$ if $|i-j|=1$, and $\mathrm{E}\left[u_i{u}_j\right]=0$ if $|i-j|>1$. Thus errors for immediately adjacent observations are correlated whereas errors are otherwise uncorrelated. In matrix notation we have $\mathbf{y}=\mathbf{X}\mathit{\beta }+\mathbf{u}$, where $\mathbf{\Omega }=\mathrm{E}[$ uu']. For this model answer each of the following questions using results given in Section 4.4.

(a) Verify that $\mathrm{\Omega }$ is a band matrix with nonzero terms only on the diagonal and on the first off-diagonal; and give these nonzero terms.

(b) Obtain the asymptotic distribution of ${\hat{\mathit{\beta }}}_{\text{ols }}$ using (4.19).

(c) State how to obtain a consistent estimate of $\mathrm{V}\left[{\hat{\mathit{\beta }}}_{\mathrm{O}\mathrm{L}\mathrm{S}}\right]$ that does not depend on unknown parameters.

(d) Is the usual OLS output estimate $s^2{\left({\mathbf{X}}^{\mathrm{\prime }}\mathbf{X}\right)}^{-1}$ a consistent estimate of $\mathrm{V}\left[{\hat{\mathit{\beta }}}_{\mathrm{O}\mathrm{L}\mathrm{S}}\right]$ ?

(e) Is White's heteroskedasticity robust estimate of $\mathrm{V}\left[{\hat{\beta }}_{\mathrm{O}\mathrm{L}\mathrm{S}}\right]$ consistent here?

BOLS N[(XX) XX(XX)'], (4.19)