A parametric approach to the estimation of cointegration vectors in panel data

regression is rewritten as

zb_i⁺_t = By_i⁽_,²_t⁾_-1 + vbit ,                                (6)

where z + = y(1^l-1 - (biΣ-^{1 b}i)^{-1 α}iΣ-¹∆yit^, yit = [y(¹0^,yi20]⁰ and y(¹ (y(² )
are r × 1 (k —r × 1) subvectors of y_it. It is interesting to note that z + adopts an
endogeneity correction similar as the estimator of Phillips and Moon (1999).
The important difference is, however, that the latter approach employs a
nonparametric estimate of the endogeneity effect, whereas z + is based on a
parametric endogeneity correction based on a VAR(p) model.

Based on a sequential limit theory, the following theorem states that the
two-step estimator has a normal limiting distribution.

Theorem 1: Let yit be generated as in (1) and B^b2S denotes the least-squares
estimator of B in the regression (6). Furthermore εit and εjt are independent
for i 6= j . If T → ∞ is followed by N → ∞ we have

T √^rNvec ( Bb₂ S — B) —→ N(0, Ω₂ ¹ ® Σ_v) ,

where Σv is defined in (4),

1^N

^ω2 = J^im Tt ∑^α2(^α0⊥^β⊥)^{-1 α}i⊥⊥^ςi^αi,⊥(^β⊥^αi,⊥)^{-1 β}⊥2 ^,
N→∞ N

i=1

αi,⊥    and β⊥    are orthogonal complements of αi andβ    and β⊥,2 is the lower

(n — r) × r block ofβ_⊥    .

From this theorem it follows that the long-run parameters are asymptotically
normally distributed and, therefore, the usual tests on the cointegration pa-
rameters involve the usual limiting distributions. In particular, the second-
step regression (6) can be treated as an ordinary regression equation, that
is, the nonstationarity of the regressors and the fact that zb_i⁺_t is estimated
can be ignored. Furthermore, it is interesting to note that for finite N and
T → ∞, the estimator are mixed normal, that is, normally distributed with
a stochastic covariance matrix. Therefore, the normal limiting distribution
is expected to yield a reliable approximation even if N is small.

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