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 {\(y_{t}\)}.