Consider the linear regression model with time series errors in Section 12.5.

Assume that zt is an AR(p) process (i.e., zt = φ1 zt−1 +

··· + φpzt−p + at). Let φ = (φ1,…,φp)′ be the vector of AR parameters. Derive the conditional posterior distributions of f(β | Y, X, φ,σ 2), f(φ | Y, X, β,σ 2), and f(σ 2 | Y, X, β, φ) assuming that conjugate prior distributions are used: that is,

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B~ N(B, o), ~ N(O, A), (v2)/0 ~ xv. 2