2.3.2 Variable structure cointegration model
Because of the effect of the external environment, variables may show
structural breaks. Hence, the long-term stable relationships between
variables may vary. The cointegration relationships before and after
structural breaks reflect the original and current long-term stable
relationships, respectively. Thus, when there are significant structural
variations between variables, cointegration analysis has to consider
structural breaks as well (Singh, 2015; Vicente, 2014). When economic
structures or policy systems are altered, parametric cointegration is
normally adopted.
The point of the structural break is first determined. To construct a
variable structure cointegration model, it is then assumed that the
structural break is mainly caused by series \(x_{t}\). A virtual
variable is introduced:
\(D_{t}=\left\{\par
\begin{matrix}0,t\leq T_{\tau}\\
1,t>T_{\tau}\\
\end{matrix}\right.\ \) (11)
where, \(T_{\tau}\) denotes the time of the structural break.
The cointegration parameter variations of a variable structure
cointegration model can be primarily divided into the following three
scenarios:
Scenario 1: Variable structure cointegration because of a constant term
shift
In this case, only the variation in the constant term c of the model is
considered. The following resulted:
\(y_{t}=c_{1}+D_{t}c_{2}+\alpha^{T}x_{t}+\varepsilon_{t},\ \ t=1,2,3\cdots T\)(12)
where, \(c_{1}\) is the constant term before the shift and \(c_{2}\) is
the amount of the shift.
Scenario 2: Variable structure cointegration because of shifts in both
the constant term and trend term
The variation in both the constant term and trend term is considered.
This gives the following:
\(y_{t}=c_{1}+D_{t}c_{2}+\beta t+\alpha^{T}x_{t}+\varepsilon_{t},\ \ t=1,2,3\cdots
T\)(13)
where, \(\beta\) denotes the coefficient of the time trend term.
Scenario 3: State switch variable structure cointegration model
In this case, the variation in the constant term, trend term, and
cointegration vector term are taken into consideration.
\(y_{t}=c_{1}+D_{t}c_{2}+\beta t+\alpha_{1}^{T}x_{t}+D_{t}\alpha_{2}^{T}x_{t}+\varepsilon_{t},\ \ t=1,2,3\cdots T\)(14)