For the random one-way error components model given in (12.1) and (12.2), consider the OLS estimator of var(uit) = 2, which is given by s2

For the random one-way error components model given in (12.1) and (12.2), consider the OLS estimator of var(uit) = σ2, which is given by s2 = ee/(n−K), where e denotes the vector of OLS residuals, n = NT and K = K + 1.

(a) Show that E(s2) = σ2 + σ2

μ[K− tr(IN ⊗ JT )PX]/(n − K).

(b) Consider the inequalities given by Kiviet and Kr¨amer (1992) which state that 0 ≤ mean of n − K smallest roots of Ω ≤ E(s2) ≤ mean of n − K largest roots of Ω ≤ tr(Ω)/(n − K)

where Ω = E(uu). Show that for the one-way error components model, these bounds are 0 ≤ σ2

ν + (n − TK

)σ2

μ/(n − K

) ≤ E(s2) ≤ σ2

ν + nσ2

μ/(n − K

) ≤ nσ2/(n − K

).

As n → ∞, both bounds tend to σ2, and s2 is asymptotically unbiased, irrespective of the particular evolution of X, see Baltagi and Kr¨amer (1994) for a proof of this result.