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)