Spectral density bandwith choice and prewightening in the estimation of heteroskadasticity and autocorrelation consistent covariance matrices in panel data models

if λ_t is random, and {μ_i } to be spatially correlated if μ_i is random. We also allow a certain
degree of heterogeneity in panel data—{Yn , X¹_ii}' need not be stationary for each i, and the
errors {Vi_i} may have different variances across i. In particular, we allow some nonstationary
processes. One example of nonstationary panel time series is the deterministic trend process
(e.g., Kao and Emerson 1999)

Yt = ɑ + Ylw + 72w^{2 +} ’ ’ ’ ⁺ Ypw^{p +} μi ^{+ λ + v}it =

This is covered by (2.1) with X_it ≡ [t/т, (t∕T)², ...> (t∕T)^p]^z and θ ≡ (Tγ₁, ...,T^pγ_p)^z. Note
that Xn and β₀ depend on T. Another example is the panel cointegration process (e.g.,
Phillips and Moon 1999, Kao and Chiang 2000):

Yit = ɑ + γZ_it + μ_i + λt + Va,

where Zn = Zn-ɪ + %n, {ε* j is I(0) for each i, and {%i j may or may not be correlated with
{Vi j. This process is also covered by (2.1) with X_it ≡ T~¹Zn and β₀ ≡ Tγ. We will provide
regularity conditions on transformed variables {Xi j and transformed parameters β.

The parameter vector β₀ in (2.1) can be estimated by the popular within estimator

∑ ∑ (X« - ^v - ^v' X) (Xft - X - Xt + X)^z
i=l t=l

■ n T_i

∑ ∑ ( Xn - X_i - Xt + X ) ( Yt - Y - Y + γ )
i=l t=l

where X_t ≡ T_i ¹ S/ɪ X_tt, X_t ≡ n ¹ S"₌ ɪ X_it and X ≡ (nT_i) ¹ S"₌₁ S’ɪ X_it. The variables

Yi,Y_t and Y are defined in the same ways. Its asymptotic covariance matrix is

.       .              i-------- ,'^T         _   ,             /     .           ʌ ∖

AVAR TTY (β - β_fi ) = (p Iim M_nT]

where

₁ n τ_i

Mτ≡ Tf ∑∑ X^v-
n

i=l t=l

• n T_i T_i

^lnT ≡ ~TT_t ΣΣΣ^Xit^Vit^(vis^Xis') ^,
П i=l t=l s=l

Xn ≡ Xn - X7i - Xt + X7, and Vn ≡ Vn - Vβ - v_t + v. To estimate (2.3), pIim Ω_nr is more
challenging to estimate.

More generally, for many panel estimators β, we have

(M_nτΩ_nM'_nτ)-¹ YTt(e - β₀) $ N(0, I_r),     u ∈ Z⁺,

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