dynamic panel models, one must be careful about how to choose the valid instruments when
the serial correlation has the unknown form. For example, the consistency of the GMM
in Arellano and Bond (1991) that use the lagged dependent variables be the instruments
requires no second-order serial correlation in the first difference residuals It seems that if
the error terms are serially correlated with the unknown form, then one cannot choose the
lagged dependent variables to be the valid instruments in the dynamic panel models. In
stead of using lagged dependent variables, we may have to use exogenous variables to be
instruments.
Usually, Mnτ is relatively simple to estimate, often by its sample analog. Our focus is
estimation of Ωnτ. When Vββ0) is a second order stationary process with mean O, we have
1 n 1 n
Iim lnτ = Iim — Ω, = Ω ≡ Iim — 2πfi (O) ,
iT→<x> n→<x> q ʃ n→∞ q ʃ J
(2.6)
i=l
i=l
where
∞
i(O) = (2π)-‘ [ Γ.(I)
l=-∞
is the p ×p spectral density matrix of Vt(βo) at frequency O, with Γi(I) = E[Vt(βo)Vht-ββ0)z].
Thus, Ωi can be consistently estimated by a nonparametric spectral density estimator at
frequency O, as suggested in Brillinger (1975), Hansen (1982) and Phillips and Ouliaris
(1988) among others. Newey and West (1987) propose a convenient positive semi-definite
kernel estimator for Ωj :
Bn
ʌ ⅛ lΛ ʌ
ω⅛nw = ∑ к(j∕Bir)Γr(I), (2.7)
l=-Bn
where K(x) = (1 — ∣x∣)1(∣x∣ ≤ 1) is the Bartlett kernel, 1(∙) is the indicator function, Biτ is
a lag truncation parameter depending on the sample size T,
γ (o)=f τ-S^l+1u(βe)vht√e)z,
1 τ- 1∑Γ=1 Y,vht+κ e) V β βγ ,
I = O , 1 ,...,T — 1 ,
I = —1,..., —( T — 1),
(2.8)
is the sample autocovariance matrix of Vt(β), and β is a consistent estimator of θ.
Andrews (1991) consider a general class of estimators
T-I
^ 4 lΛ ʌ
Ωa = ∑ K(j∕Bir)Γr(I),
(2.9)
I=I-T
where K : U → [—1, 1] is a general kernel, and Bn a bandwidth. Examples of K(∙) include
Bartlett, Parzen, QS, Tukey-Hanning, and truncated kernels (e.g., Andrews 1991). When
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