Page 27 - Contributed Paper Session (CPS) - Volume 1

CPS658 Sagaren P.
where Π = −( − −. . . − )and = −( +1 +. . . + )
1

Since all the variables are assumed I(1) or I(0),  −1 must be I(0) and
stationary.
Therefore det ( − ) = 0 which implies that = 1 and  is singular.

Suppose  has rank r :Let  and  be two k  r matrices of rank r Since  is

stationary, The linear transformation ( ) Π is also stationary
−1 ′
′

. ( ) ( )
′
−1 ′
′

= ( ) ( )
′
−1
′
′

=
′

Hence the rows of ′ are co-integrated vectors
′
Thus Δ = −1 + ∆ −1 +. . . + −+1 + (3)

1
After substituting = in (2)
′

2. The deterministic components on a co-integration model
Following Hansen and Juselius (1995) the deterministic components can
be modelled in five different ways. Case 1 does not allow for any deterministic
components in the data. This is unlikely since for stochastic co-integrated
variables in the Var model, deterministic terms should appear in the VECM via
the error correction term or as independent terms. In Case 2 the model does
not allow for a separate drift (constant) but allows for a constant in the Co-
integration space. In Case 3, there are no trends in the Cointegration space;
the linear trends enter the VECM as drift terms. In Case 4 there is a separate
drift in the VECM and a linear trend in the Co-integration space. Finally, in Case
5, there is a separate linear trend in the VECM.
For the constants or trends in Johansen’s Co-integration model to be
meaningful, they must be related to the Co-integration space or excluded
from it. The consequences of mis-specifying the deterministic components in
empirical studies is not well known (Ahking, 2002).

3. Data: South African Gross Earnings and Employment costs (2009-
2017)
The Quarterly Financial Statistics (QFS) survey produces employment costs
estimates. These estimates are based on information as defined by the
International Accounting Standards from an accounting perspective. The
Quarterly employment statistics (QES) survey produces estimates for
employment and earnings from a payroll perspective. Conceptually both gross
earnings are closely related and follow similar trends with level differences.
The similarities in the QFS employment costs and the QES gross earnings
provide a good basis to investigate both the Co-integration as well as the mis-
specification of the deterministic components in the Co-integration model.
When Constructing a VECM(p) from a Var(p) models, the deterministic terms
in the VECM(p) model can differ from those in the Var(p) model. (1) When

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