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

Theorem 2: Let y_t be generated by a VAR(1) process with EC representation
(1) and 0 ≤ r < k - 1. Furthermore ε_it and εj_t are independent for i 6= j. Let
b_i,⊥ and β⊥ be √T-consistent estimates for some orthogonal complements of
αi,⊥ and β⊥, respectively. If T → ∞ is followed by N → ∞ we have

where

dWk-rWk⁰-r

E tr [ λi ( r ) — μr ]²

dWk-rWk⁰-r

λ(r) = N ¹ Pⁿ=₁ λ_i (r) and λ_i (r) denotes the LR, Wald or LM statistic of
the hypothesis δi = 0 in the regression

ubi_t = δ_i⁰wbi_,t-1 + ei_t , t = 1, . . . , T ,                       (9)

.—.

where ubit = αb⁰_i,⊥∆yit and wbit = β_⊥⁰ yit .

A convenient (Wald type) test statistic of the null hypothesis is

T              T                ^-1 T              T

^vit^wi⁰,t-1        ^wi,t-1^wi⁰,t-1            wi,t-1vi⁰t        vitvi⁰t

t=1             t=1                   t=1             t=1

The values μ_r and σ2 are computed by Lyhagen et al. (2001) for the model
without deterministic terms.

Hypotheses on the cointegration parameters can be tested by using a LR
statistic. Following Johansen and Juselius (1994) we consider the following
class of linear hypotheses on the cointegration vectors:

H0 :   β= [β1,...,βr] = [Φ1θ1,...,Φrθr] ,                (10)

where Φj is a known k × qj matrix with 1 ≤ qj ≤ k — r and θj is a qj × 1
vector for j = 1, . . . , r . Note that for the identification of the cointegration
vectors r normalization restrictions are required so that the maximal number

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