(SUR) equations. Such a procedure requires that the number of time series
observations (T) is substantially larger than the number of cross section units
(N ). In typical country studies, however, N and T are of the same order of
magnitude. In such cases the inverse of the estimated covariance matrix may
behave poorly and, therefore, a robust estimator in the spirit of Newey and
West (1987) is preferable. Let b = vec(B0) and
Xit = (Ir ® ei,t-1 ) ,
so that the second step regression model can be written as
zei+t = Xeitb + veit .
Furthermore, we stack the cross section observations and define
|
+ zet+ = |
ze1+t . . |
x4^ , Xet = |
x4^ . . |
, vet = |
ve1t . . . |
|
+ |
XeNt |
veNt |
so that the regression can be written as
zet+ = Xetb + vet .
(14)
In this regression the error vector vet is assumed to be (asymptotically) un-
correlated with ves for t 6= s. The asymptotic covariance matrix of the least
squares estimator of b = vec(B) is given by
lim
T,N→∞
T
Σ∙~∙∙-'
E(Xet0Xet)
t=1
-1
T
Σ∙~∙∙-'
E(Xet0vtvt0Xet)
t=1
T
Σ∙~∙∙-'
E(Xet0Xet)
t=1
-1
Therefore, a consistent estimator for the covariance matrix of the least squares
estimator of b can be constructed as
where vbt denotes the residual vector from the regression (14). This approach
is similar to the robust estimator of the covariance matrix suggest by Arellano
(1987). However Arellano’s estimator assumes that the errors are contem-
poraneously but not serially uncorrelated. Our robust estimator therefore
results from interchanging the role of i and t. Since the cointegration tests
suggested in Theorems 2 and 3 are based on similar least-squares regressions,
analog procedures can be used for the Wald test of the cointegration rank.
Veb =
T
Σ∙~∙∙-'
Xet0Xet
t=1
-1
T
X Xet0vbtvbt0Xet
t=1
T
Σ∙~∙∙-'
Xet0Xet
t=1
-1
(15)
11