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

of free parameters is k - r for all cointegration vectors. As demonstrated by
Johansen (1995, p. 75), any linear hypothesis of the form Rjβj = rj can be
rewritten as in (10). Inserting the hypothesis in (5) yields a set of r equations

^zj,it = ^θj (^φj^yi,t-1) + ^vjit ^{,                           (11)}

where zj,_it is the j ’th element of z_it. Accordingly, under the alternative
the parameter vector θj can be estimated by a least squares regressions of
zbj_,it on (Φ⁰_jy_i,t-1). Since the system equations for j = 1, . . . , r do no longer
involve the same set of regressors, the SUR system should be estimated by
GLS in order to achieve asymptotic efficiency. It is interesting to note that
no “switching-algorithm” needs to be applied as in Johansen and Juselius
(1994).

Let e_tt and e*_t = [e↑_it_t,... ,v*_it]0 denote the residual vectors of the unre-
stricted regression (5) and the restricted regression (11), respectively. A test
statistic that is asymptotically equivalent to the ML statistic is

The asymptotic properties of such a test are considered in the following
theorem.

Theorem 3: Let yit be generated as in (1). Furthermore εit and εjt are
independent for i 6= j. Under the null hypothesis (10) the test statistic ξLR
defined in (10) is asymptotically χ² distributed with r(k-r) - _j^r₌₁ qj degrees
of freedom.

Alternatively, a Wald test procedure can be applied that is based on the
results of Theorem 1.

5 Extensions

So far we have considered the cointegrated VAR(1) model with E(εit) = 0
for all i and t. Although such a limitation is convenient for expositional
purposes, it is of course too restrictive for practical applications. A more
realistic model is the cointegrated VAR(p) model with individual specific

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