Supplement to ” Dynamic Spatial Panel Models , Common Shocks , and Sequential Exogeneity ” by

mq,r(δ) = plimn u1 (δ) Au1 (δ) = plimn −1u+′ 1 A u1 (D.2) + 2 (δ0 − δ)′ plimn−1W+′ 1 A u1 + (δ0 − δ) ′ plimn−1W+′ 1 A W 1 (δ0 − δ) = 2 (δ0 − δ)′ ΓWAru + (δ0 − δ) ′ ΓWArW (δ0 − δ) . Observe that Ql(δ) ≥ 0, Qq(δ) ≥ 0, Q(δ) ≥ 0 with Ql(δ0) = Qq(δ0) = Q(δ0) = 0, since ml(δ0) = 0 and mq(δ0) = 0. Thus any minimizer, say δ∗, of Q(δ) not only satisfies ∂Q(δ∗)/∂δ = 0, but also ∂Ql(δ∗)/∂δ = 0, and ∂Qq(δ∗)/∂δ = 0. Case 1: ΓHW has full column rank: In this case δ = δ0 is obviously the unique solution of ml(δ) = ΓHW (δ0 − δ) = 0. Of course, since ∂Ql(δ) ∂δ = 2ml(δ) ′(V h)−1 ∂ml(δ) ∂δ = −2ml(δ)(V )ΓHW