Technical Appendix for Generalized Least Squares Inference in Panel and Multilevel Models with Serial Correlation and Fixed Effects

or, in vector notation, Cs = Xsβ1 + Zsβ s 2 + Vs, where Cs = [Cs1, . . . , CsT ] ′ is T × 1, Xs = [xs1, . . . , xsT ] ′ is T ×k1, Zs = [zs1, . . . , zsT ]′ is T ×k2, and Vs = [vs1, . . . , vsT ]′ is T ×1. Also, let xsth be the h th element of xst so that x ′ st = [xst1, . . . , xstk1 ], and define zsth similarly. Define v̈st = vst − z′ st(Z ′ sZs)Z ′ sVs, ẍst = xst − z′ st(Z ′ sZs)Z ′ sXs, V̈s = [v̈s1, . . . , v̈sT ]′, and Ẍs = [ẍs1, . . . , ẍsT ] ′. Let v st be a p × 1 vector with v st = [vs(t−p), . . . , vs(t−1)], and define v̈ st similarly. Assumption 1 (S → ∞, T fixed). Suppose the data are generated by model (1) and N1. vst = v − st α + ηst, where ηst is strictly stationary in t for each s, E[η 2 st] = σ 2 η , E[ηstηsτ ] = 0 for t 6= τ , and the roots of 1 − α1z − α2z − . . . − αpz = 0 have modulus greater than 1. N2. {Xs, Vs, ηs} are iid across s. {Zs} are nonstochastic and identical across s. N3. (i) Rank( ∑T t=1 E[ẍstẍ ′ st]) = Rank(E[Ẍ ′ sẌs]) = k1. (ii) Rank(Z ′ sZs) = k2 ∀ s. N4. E[Vs|Xs] = 0, E[VsV ′ s |Xs] = Γ(α). N5. E[η4 st] = μ4 <∞ and E[xsth] ≤ ∆ <∞ ∀ s, t, h.