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

where M_nʃ is a nonstochastic r × p matrix, I_r is a r × r identity matrix, and

ι ^{n Ti Ti}

p Iim Ù_nT = — ΣΣ∑Y.( ∕⅞ ) Y.( (⅛ )'
^г i=l t=l s=l

for some stochastic p × 1 vector process Y_i(β₀). For example, the function Y_i(β₀) can be the
moment function in GMM estimation, e.g., Arellano and Bond (1991), Arellano and Bover
(1995), Ahn and Schmidt (1995, 1997), Hahn (1997), Blundell and Bond (1998), and Im et
al. (1999). Also the HAC estimators proposed in this paper can be used in many other panel
models. For example, the panel cointegration tests and panel cointegration estimation (e.g.,
Kao, 1999; Kao and Chiang, 2000, Phillips and Moon, 1999) require the HAC estimation for
the long-run variance matrix of the error terms in the models. In an extensive simulation
study, Kao and Chiang (2000) pointed out the panel fully modified (FM) estimator and
t-statistic based on FM estimator are severely downward biased due to the failure of the
kernel-based HAC estimation for the long-run variance covariance matrix. More seriously,

Kao and Chiang also pointed out that the FM t-steatitic become more negatively biased as
the cross-sectional dimension, n, increases. All these indicate there is much to be done on

the HAC estimation in panel data models. For the panel cointegration test estimation (e.g.,
Kao and Chiang, 2000, p. 187; Phillips and Moon 1999, p. 1084), Y_i(β₀) usually takes the

Once the estimates of Yββ₀) , Viββ₀) were estimated, the HAC estimator of the long run
covariance matrix was estimated by

ι -ⁿ f ι _ ∣                       i YL /                                           ∖

Ω = ɪ Σ i T Σ Y.(β)Y'*^e) + T Σ ' ∑ (Y.(β)⅛- β) + ^v“-Λ^e)VEβ))
i=l t=l                    τ=l t=τ⅛l

where β is the within estimator or the FM estimator and w_τi is a weight function or a kernel.

The distribution results for the FM estimator in Kao and Chiang (2000) and Phillips and
Moon (1999) require _vA ^Ω — Ω^ does not diverge as n grows large. However, Ω — Ω may
not be small when T is fixed. It follows that ʌ/n ^Ω — lɔ may be non-negligible in panel
data with finite samples. It may be one of the reasons for the poor performance for panel

FM estimator. For GMM estimator in panel data, Yββ₀) usually takes the form:

Y<(β₀) = Z_it β_it — [√⅞)

where Zn is a vector of instruments with E Zn \ Yn — [βoj and Zn \Ец — Xβ^J may have
serial correlation and heteroskedasticity of unknown form. However, for the GMM is the

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