(a) Verify (5.52) for the MA(1) model. Hint: Use the fact that \(\lim E\left(u^{\prime} uight) /\) \((N T)=\lim \operatorname{tr}(\Omega) /(N T)\) for deriving plim \(Q_{0}\). Similarly, use the fact that

\(\lim E\left(u^{\prime}\left(I_{N} \otimes G_{1}ight) uight) / N(T-1)=\lim \operatorname{tr}\left[\Omega\left(I_{N} \otimes G_{1}ight)ight] / N(T-1)\)

for deriving \(\operatorname{plim} Q_{1}\). Also,

\(\lim E\left(u^{\prime}\left(I_{N} \otimes G_{S}ight) uight) / N(T-s)=\lim \operatorname{tr}\left[\Omega\left(I_{N} \otimes G_{S}ight)ight] / N(T-s)\)

for deriving \(\operatorname{plim} Q_{s}\) for \(s=2, \ldots, S\).

(b) Verify (5.53) for the AR(1) model.

plim Qs = o for s = 2, ..., S (5.52)