Consider the linear regression model with time series errors.

Assume that \(z_{t}\) is an \(\operatorname{AR}(p)\) process (i.e., \(z_{t}=\phi_{1} z_{t-1}+\cdots+\phi_{p} z_{t-p}+a_{t}\) ). Let \(\boldsymbol{\phi}=\left(\phi_{1}, \ldots, \phi_{p}\right)^{\prime}\) be the vector of AR parameters. Derive the conditional posterior distributions of \(f\left(\boldsymbol{\beta} \mid \boldsymbol{Y}, \boldsymbol{X}, \boldsymbol{\phi}, \sigma^{2}\right), f\left(\boldsymbol{\phi} \mid \boldsymbol{Y}, \boldsymbol{X}, \boldsymbol{\beta}, \sigma^{2}\right)\), and \(f\left(\sigma^{2} \mid \boldsymbol{Y}, \boldsymbol{X}, \boldsymbol{\beta}, \boldsymbol{\phi}\right)\) using the conjugate prior distributions, that is, the priors are

\[ \boldsymbol{\beta} \sim N\left(\boldsymbol{\beta}_{o}, \boldsymbol{\Sigma}_{o}\right), \quad \boldsymbol{\phi} \sim N\left(\boldsymbol{\phi}_{o}, \boldsymbol{A}_{o}\right), \quad(v \lambda) / \sigma^{2} \sim \chi_{v}^{2} \]