**7. Results**

The time series for AFNS coefficients can be seen in Figure 2. Coefficients *β*1,*t* and *β*3,*t* may be cointegrated, so we run a two-step Engle–Granger cointegration test Engle and Granger (1987). The linear regression of *β*3,*t* explained by *β*1,*t* returned an intercept of −0.10 and a coefficient of 0.84. Applying the Augmented-Dickey–Fuller Unit Root Test on regression residuals yielded a statistic of −2.68. Such statistics, confronting the critical values for the co-integration test of Engle and Yoo (1987) leads us to reject the unitary root hypothesis because the residuals are stationary. The conclusion is that there is cointegration between *β*1,*t* and *β*3,*t*.

In economic terms, the cointegration describes a strong relationship between long- and medium-term contracts, which can be a result of a political measure or some market characteristics that stimulated the emission of long-term contracts based on the price of medium-term contracts and vice versa.

The time series for *β*1,*t* and *β*2,*t* in (7) are modeled as AR(1) processes with one differentiation. The estimated *φ* for *β*1,*t* is −0.04 and for *β*2,*t* is 0.13.

As seen in AFNS coefficients time series, DCOBS estimated coefficients *<sup>a</sup>*2,*t* and *<sup>a</sup>*3,*t* seem to cointegrate in Figure 3, as well as coefficients *<sup>a</sup>*4,*t* and *<sup>a</sup>*5,*t*, so we run a two-step Engle-Granger cointegration test. The linear regression of *<sup>a</sup>*3,*t* explained by *<sup>a</sup>*2,*t* returned an estimated intercept of −7.61 and a coefficient of 8.60. The linear regression of *<sup>a</sup>*5,*t* explained by *<sup>a</sup>*4,*t* returned an estimated intercept of 0.11 and an estimated coefficient of 0.43. Applying the Augmented-Dickey–Fuller Unit Root Test on regression residuals yielded the statistics −5.48 for estimated coefficients *<sup>a</sup>*2,*t* and *<sup>a</sup>*3,*t*. The same test yielded the statistics −2.41 for estimated coefficients *<sup>a</sup>*4,*t* and *<sup>a</sup>*5,*t*. Confronting these statistics with the critical values in Engle and Yoo (1987) implied the unitary root hypothesis because the residuals are stationary. The conclusion is that there is a cointegration between *<sup>a</sup>*2,*t* and *<sup>a</sup>*3,*t* as well as between *<sup>a</sup>*4,*t* and *<sup>a</sup>*5,*t*.

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**Figure 3.** DCOBS estimated coefficient series.

Since B-Splines coefficients have a more local specific behavior, these cointegrations give a more detailed analysis than the AFNS model. It reveals the binding between short- and medium-term contracts on one side and medium- and long-term contracts on the other side. Like the economic interpretation of the AFNS model, this is an important feature of the model because it shows to investors and policy makers the magnitude of how the supply and demand on a type of contract can influence the price of another type of contract.

The time series for *<sup>a</sup>*2,*t* and *<sup>a</sup>*4,*t* are modeled as AR(1) processes with one differentiation. The estimated *φ* of *<sup>a</sup>*2,*t* is 0.02 and for *<sup>a</sup>*4,*t* is −0.03. Then, *<sup>a</sup>*3,*t* and *<sup>a</sup>*5,*t* are linear functions of *<sup>a</sup>*2,*t* and *<sup>a</sup>*4,*t*, respectively.

The time series modeled as AR(1) processes above were used to make the out-of-sample forecast with a horizon of 250 business days, the amount of business days in a test dataset of 2017. Three reference dates were considered for evaluation: 1 month (short-term), 6 months (medium-term), and 12 months (long-term). Figure 4 shows short-, medium- and long-term forecasts for AFNS and DCOBS curves.

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**Figure 4.** Short-, medium- and long-term forecasts for AFNS and DCOBS curves.

In a 1-month forecast, DCOBS performs a good fit to the term structure both in the short-term as in a long-term horizon, as the curve follows the data points. AFNS shows a heavy instability in the beginning of the curve in all forecasting, although, in the long-term, it performs well.

We compared both forecast techniques using the Diebold–Mariano accuracy test Diebold and Mariano (1995) with an alternative hypothesis being DCOBS outperforming AFNS prediction. As stated before, DCOBS outperforms AFNS in the short-term prediction. The absolute value of Diebold–Mariano statistics for a one-month forecast is greater than 1.96, so the null hypothesis that both techniques have the same accuracy is rejected. On the other hand, for 6-month and 12-month forecasts, the absolute value of Diebold–Mariano statistics stays lower than 1.96, which means that both techniques may have the same predictive accuracy. Table 2 shows forecasts' root mean square errors, and Table 3 shows Diebold–Mariano statistics results.


**Table 2.** Forecast root mean square errors.

**Table 3.** Diebold–Mariano Test Statistics.

