2.3.1 Cointegration model
The cointegration theory comes from the field of economics. Whether long-term cointegration relationships exist for non-linear series can be determined by applying this theory (Gao et al., 2019; Boukhelkhal and Bengana., 2018; Heberling et al., 2015). This theory was proposed by Engle and Granger in 1987 to process a non-stationary series. Only when the variables of the non-stationary series become integrated at the same order after differentiation, valid cointegration regression models can be constructed. In general, cointegration relationships are tested through two steps: (1) The time series are subjected to unit root tests (augmented Dickey–Fuller, ADF tests) to determine whether the variable series are stationary. (2) The Engle–Granger two-step method is adopted to analyze whether cointegration relationships exist between the variables. The method is illustrated below:
First, for observed data series {\(x_{t}\)} and {\(y_{t}\)}, ordinary least squares (OLS) are used to estimate α and β and to calculate the residual series \(\varepsilon_{t}\). A cointegration regression equation for the two series is given as follows:
\(\ y_{t}=\alpha+\beta x_{t}+\varepsilon_{t}\ \) (10)
Second, the stationarity of the residual series is tested.
The residual series \(\varepsilon_{t}\) is subjected to the ADF test. If it is a stationary series, then {\(x_{t}\)} and {\(y_{t}\)} exhibit a cointegration relationship. On the contrary, if the residual series is non-stationary (a unit root exists), then no cointegration relationship is present for {\(x_{t}\)} and {\(y_{t}\)}.