Consider the model Where X 1 : n × p’, and X 2 : n × (t−p ) and ϵ ∼ N(0, Iσ 2 ). Suppose we estimate β1 and σ2 using the subset model Under what conditions is 2 unbiased for σ 2 ? (c) Let X = ( X 1 X...

Where X₁
: n × p’, and X₂: n × (t−p ) and ϵ ∼ N(0, Iσ²). Suppose we estimate β1 and σ2 using the subset model

Under what conditions is

²
unbiased for σ²?

(c) Let X = ( X₁
X₂
) be of full column rank. Show that

(d) Using (c), show that the least squares estimators of the elements in β₁, based on the subset model in (a), have smaller variance than the corresponding estimators of the full model.