3.3 Multi-temporal runoff-sediment discharge cointegration relationships
3.3.1 ADF test
First, ADF tests were performed for the raw runoff and sediment discharge series. The results are summarized in Table 2.
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From the table, unit roots exist for X and Y before differentiation at the 1%, 5%, and 10% significance levels. Thus, the series are non-stationary. Contrarily, after the first-order differentiation, there are no unit roots for both series, and they are now stationary series. Therefore, X and Y are first-order integrated series, i.e.,\(X\sim I(1)\) and \(Y\sim I(1)\).
Subsequently, the multi-temporal runoff and sediment discharge components were subjected to stationarity tests. The ADF test results are listed.
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According to the above test results, unit roots do not exist for the undifferentiated series of the IMF1, IMF2, and RES components of the runoff and sediment discharge at the 1%, 5%, and 10% significance levels. The series of these components are stationary. Additionally, relatively satisfactory goodness of fit and higher R2values were noted for the double cumulative curves of the IMF1, IMF2, and RES components. In these plots, the cumulative runoff and sediment discharge gave a straight line, and they showed consistent variation. It is believed that the equations describing the runoff-sediment discharge relationships of the IMF1, IMF2, and RES components can be determined using linear regression. Furthermore, structural breaks are noted in the double cumulative curves of IMF3 and IMF4. The series of these components are found to be first-order integrated series according to the ADF tests. Thus, for these components, OLS was employed to estimate the regression parameters and to determine the cointegration equations.
3.3.2 Multi-temporal regression equation
Linear regression was used to determine the runoff-sediment discharge regression equations for IMF1, IMF2, and RES. The corresponding scatter plots and linear regression equations are illustrated below.
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From the scatter plots, high goodness of fit of the linear regression lines are noted for the components of runoff and sediment discharge measured at Tangnaihai station. The R2 values for the IMF1, IMF2, and RES components were 0.8105, 0.8357, and 0.9975, respectively. These indicated that the runoff and sediment discharge were significantly correlated with these components. At these scales, linear regression equations can be used to express the runoff-sediment discharge correlations. Therefore, the Engle–Granger two-step method was necessary only for the cointegration regression for the raw series and the IMF3 and IMF4 components. After that, the residual series of these series were subjected to the ADF tests. The results are listed in Table 5.
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Table 5 shows that all the residual series were stationary. Thus, it was concluded that the raw, IMF3, and IMF4 series all show cointegration relationships.
By employing linear regression and the cointegration theory, regression equations were obtained for the raw series and the multi-temporal component series. They are summarized in Table 6.
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In general, higher goodness of fit was noted for the regression model for the raw series. This indicated that the runoff and sediment discharge were significantly correlated at the macroscopic scale. Yet, their relationships at different time scales varied slightly. Among the series subjected to linear regression, the RES component demonstrated the highest goodness of fit. This suggested that the runoff and sediment discharge showed a relatively strong correlation in terms of their overall variation. Among those subjected to cointegration regression, the IMF3 component series (indicative of a medium/long-time scale) and the IMF4 component series (long-time scale) showed lower goodness of fit. This demonstrated weak runoff-sediment discharge cointegration relationships at these time scales.
When the cointegration test results were integrated with the double cumulative curves, the runoff and sediment discharge generally observed at Tangnaihai station consistently varied with each other proportionally. However, the relationship variation of the raw runoff and sediment discharge series was caused by multi-temporal variations. The weak runoff-sediment discharge relationships on the medium/long and long-time scales would affect that of the raw series.
3.3.3 Variable structure cointegration model
Because structural breaks were observed for the IMF3 and IMF4 runoff-sediment discharge relationships, the corresponding cointegration relationships were weaker. Hence, variable structure cointegration models were constructed to determine reasonable models by considering structural breaks. The most prominent break was noted for the IMF3 component in 2005 and that for the IMF4 component was observed in 2001. In model construction, these points were treated as structural breaks and virtual variables \(D_{1t}\) and \(D_{2t}\) were introduced.
For the IMF3 component, a virtual variable was introduced as follows:
\(D_{1t}=\left\{\par \begin{matrix}0,t\leq 2005\\ 1,t>2005\\ \end{matrix}\right.\ \) (15)
Three models corresponding to different scenarios (section 2.3.2) were examined to determine the most reasonable model. The models considering three different variation scenarios are summarized in Table 7.
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The R2 values of the models corresponding to scenarios 1, 2, and 3 were 0.606516, 0.788878, and 0.800332, respectively. These values were significantly higher than that of the original cointegration model (0.410943). The highest goodness of fit was achieved by model 3, therefore, it is believed that this model can more accurately depict the IMF3 runoff-sediment discharge relationship.
The virtual variable introduced into the model for the IMF4 component is given as follows:
\(D_{2t}=\left\{\par \begin{matrix}0,t\leq 2001\\ 1,t>2001\\ \end{matrix}\right.\ \) (16)
Table 8 shows the models considering the three scenarios:
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Under the three scenarios, the models give R2 values of 0.792911, 0.807911, and 0.841282, respectively. They were much higher than that of the original cointegration model for the IMF4 component (0.342336). More specifically, model 3 demonstrated the highest goodness of fit. Therefore, it could more accurately describe the runoff-sediment discharge relationship variation for IMF4 component.
3.3.4 Model accuracy
With the help of the runoff-sediment discharge regression models established for the components, the runoff during the research period was simulated. The results are illustrated in Figure 7. The cointegration model constructed using the raw series was called an original model. The model built based on the multi-temporal components was named composite model 1. Lastly, composite model 2 refers to the variable structure cointegration model constructed using the multi-temporal components and considering structural breaks.
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Figure 7 reveals that, in general, the models satisfactorily simulated the raw runoff. In particular, near the end of the research period, the composite model 2 showed higher accuracy. Its results showed the same variation trend as that of the raw series. From Figure 8, the average relative errors by the original model, the composite model 1, and the composite model 2 were 11.43%, 10.76%, and 7.82%, respectively. According to the Standard for Hydrological Information and Hydrological forecasting (GBT 22482-2008), an error tolerance of 20% is appropriate in hydrological forecasting. Hence, the relative errors of all the models fall within the acceptable range. In particular, the greatest average relative error resulted from the original model, whereas the smallest was generated by composite model 2 considering structural breaks. Figure 9 illustrates that this model yields relative errors less than 10% for 41 years out of the 54 years simulated. The relative errors by this model were 10%–20% for 9 years and they were greater than 30% for the remaining 4 years. Hence, most of the relative errors by this model were low, while few were high. Therefore, the resulting average relative error by composite model 2 was comparatively small. From Figure 8, at the beginning of the research period, the three models give the same relative error. However, the errors in the original model and the composite model 1 increased after 2000. The errors in 1997, 2002, and 2009 were the greatest. Furthermore, the composite model 2 provided more satisfactory runoff simulation results and smaller relative errors for these years. Because this model considered the local variation characteristics and structural breaks of the multi-temporal components, its overall simulation accuracy was higher. The model could more accurately reflect the long-term equilibrium and short-term fluctuating relationships between the runoff and sediment discharge in the source region of the Yellow River.