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

A consistent estimator (as T → ∞) of α_i can be obtained from estimating
separate models for all N cross sections. If r = 1, one may use the two-step
estimator suggested by Engle and Granger (1987), whereas for r > 1 the
ML estimator of Johansen (1988, 1991) can be used. However, in the latter
case it is important to re-normalize the cointegration vectors so that they
do not depend on individual specific parameters. Let β^b_i^{M L} denote the ML
estimator of the cointegration matrix suggested by Johansen (1988, 1991).

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The estimator is normalized such that β^b_i^ML0S_11,iβ^b_i^ML = I_r , where S_11,i =
^P_t^T₌₁ y_i,t-1y_i⁰_,t-1. Since the distribution of S_11,i depends on α_i and Σ_i, the
ML estimator applies an individual specific normalization. To obtain the

same normalization for all cointegration matrices β^b₁^{M L} , . . . , β^b_N^{M L}

apply the normalization β^b_c^M_,i^L = β^b_i^{M L} (β^b_i^M_,1^L)

one may
'---- T T

■----H r τ ■---■

'---'H Г т      ■-----

^-1 = [I, —B^b_i^ML]⁰, where B^b_i^ML0

-β^b_i^M_,2^L(β^b_i^M_,1^L)^-1 and β^b_i^M_,1^L (β^b_i^M_,2^L) denotes the upper (lower) r × r (n - r × r)
block of β^b_i^{M L} .

A problem with such a normalization is that β_i^M_,1^L needs not to be in-
vertible and, thus, the normalization may be invalid (see Boswijk 1996 and
Saikkonen 1999). To avoid such problems an estimator can be used that is
based on an eigenvalue problem not depending on nuisance parameters. Such
an estimator is obtained by solving the eigenvalue problem ∖λ_iI — S 11 J = 0.
The eigenvectors corresponding to the r smallest eigenvalues are called the
Principal Component (PC) estimator of the cointegration vectors (e.g. Har-
ris, 1997). The estimated cointegration matrices β^b_i^{P C} are normalized as

β^b_i^PC0β^b_i^PC = I_r and, thus, the normalization does not depend on individual
specific parameters.

At the first estimation stage, the restriction that the cointegration vectors
are the same for all cross section units is ignored, but this does not affect
the asymptotic properties of the estimator. For the asymptotic properties of
the two-step estimator it is only required that the parameters are estimated
consistently as T → ∞.

At the second stage, the system is transformed such that the cointegration
matrix β can be estimated by ordinary least-squares of the pooled regression

zbit = β⁰yi,t-1 + vbit   i = 1, . . . , N; t = 1, . . . , T,

(5)

where zbit = (αb⁰_iΣ^b _i^-1αbi)^-1αb⁰_iΣ^b _i^-1∆yit and vbit is defined analogously.

If the cointegration vectors are normalized as β = [I, —B]⁰, then the