When ordinary least squares is BLUE with correlated errors

Consider the linear model $X=Z\beta +\varepsilon$, where $Z$ is a known $n\times p$ matrix, $\beta$ is a $p$-vector of unknown parameters, and $\varepsilon$ is a random $n$-vector with $\mathbb {E}\left[\varepsilon \right]=0$ and finite $\operatorname {Var}\left[\varepsilon \right]=\Sigma$. Show the following statements are equivalent:

1. The LSE $l^\tau \widehat{\beta }$ is the best linear unbiased estimator (BLUE) of $l^\tau \beta$.

2. $\operatorname {Var}\left[\varepsilon \right]=Z\Lambda_1Z^\tau +U\Lambda_2U^\tau$ for some matrices $\Lambda_1$ and $\Lambda_2$, where $U$ is a matrix such that $Z^\tau U=0$ and $\mathcal R(U^\tau )+\mathcal R(Z^\tau )=\mathbb {R}^{n}$.

3. $\operatorname {Var}\left[\varepsilon \right] Z=ZB$ for some matrix $B$.

4. $\mathcal R(Z^\tau )$ is generated by $r$ eigenvectors of $\operatorname {Var}\left[\varepsilon \right]$, where $r$ is the rank of $Z$.

Suppose that

where $\mu \in \mathbb {R}$ is an unknown parameter, $J_n$ is the $n$-vector of $1$'s, $H$ is an $n\times p$ known matrix of full rank, $\xi$ is a random $p$-vector with $\mathbb {E}\left[\xi \right]=0$ and $\operatorname {Var}\left[\xi \right]=\sigma_\xi^2I_p$, $e$ is a random $n$-vector with $\mathbb {E}\left[e\right]=0$ and $\operatorname {Var}\left[e\right]=\sigma^2I_n$, and $\xi$ and $e$ are independent. Show that the LSE of $\mu$ is the BLUE if and only if the row totals of $HH^\tau$ are the same.

Consider a linear model

where $\mu$, $\alpha_i$'s, and $\beta_j$'s are unknown parameters, $\mathbb {E}\left[\varepsilon_{ij}\right]=0$, $\operatorname {Var}\left[\varepsilon_{ij}\right]=\sigma^2$, $\operatorname {Cov}\left[ \varepsilon_{ij}, \varepsilon_{i'j'} \right]=0$ if $i\ne i'$, and $\operatorname {Cov}\left[ \varepsilon_{ij}, \varepsilon_{ij'} \right]=\sigma^2\rho$ if $j\ne j'$. Show that the LSE of $l^\tau \beta$ is the BLUE for any $l\in \mathcal R(Z)$.

Consider the linear model $X=Z\beta +\varepsilon$, where $Z$ is a known $n\times p$ matrix, $\beta$ is a $p$-vector of unknown parameters, and $\varepsilon$ is a random $n$-vector with $\mathbb {E}\left[\varepsilon \right]=0$ and $\operatorname {Var}\left[\varepsilon \right]=$ a block diagonal matrix whose $i$th block diagonal $V_i$ is $n_i\times n_i$ and has a single eigenvalue $\lambda_i$ with eigenvector $J_{n_i}$ (the $n_i$-vector of $1$'s) and a repeated eigenvalue $\rho_i$ with multiplicity $n_i-1$, $i=1,\ldots ,k$, $\sum_{i=1}^kn_i=n$. Let $U$ be the $n\times k$ matrix whose $i$th column is $U_i$, where $U_1=(J_{n_1}^\tau ,0,\ldots ,0)$, $U_2=(0,J_{n_2}^\tau ,\ldots ,0)$,, $U_k=(0,0,\ldots ,J_{n_k}^\tau )$, and let $\widehat{\beta }$ be the LSE of $\beta$.