**6. Descriptive Analysis**

This work used 10 years (2007–2017) of public historical data of US Daily Treasury Yield Curve Rates2. The data were partitioned so 2007–2016 was the sample data set and 2017 was the test data set. A US federal holidays dataset was built for auxiliary calculations of business days.

The sample data set had 27,544 data points in 2504 reference dates spanning between 2 January 2007 and 30 December 2016. The term structure horizon spans between 22 and 7920 business days. The sample data set originated 5008 yield curves, with 2504 AFNS and 2504 DCOBS.

By the nature of parametric models, there is no challenge in relating the coefficients in a time-dependent process. Thus, Arbitrage-Free Nelson–Siegel yield curves are fitted using the raw sample data.

On the other hand, nonparametric B-Splines models depend on its knots and data points position. Therefore, a two-step normalization procedure was applied and so the coefficients could be related in a time-dependent process. The first step normalizes the horizon length. The second step normalizes the data point positions.

In the first step, the Nelson–Siegel model is applied to extrapolate the horizon and calculate the yields on the boundaries of the term structure. The largest curve was picked from the dataset with a horizon of 7920 days.

For the second step, an auxiliary DCOBS curve was built with knots being equally distributed across the horizon. With the resulting fitted curve, we calculated the normalized term structure by evaluating the auxiliary curve at the points (0,*s*(132),*<sup>s</sup>*(594),*<sup>s</sup>*(1320),*<sup>s</sup>*(7920)). Theses knots were selected based on observed data and the overall fitting quality it produced3.

In this analysis, 2504 yield curves were generated by our computational program using both methods AFNS and the DCOBS. For each curve, the AFNS method produced three coefficients while

<sup>2</sup> For more information: https://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/TextView.aspx? data=yield.

<sup>3</sup> For more information: https://www.gnu.org/software/gsl/manual/html\_node/Evaluation-of-B\_002dspline-basisfunctions.html.

DCOBS produced five coefficients. The resulting DCOBS yield curves had a better performance compared to AFNS considering Root Mean Square Error in every year of a sample data set as shown in Table 1. The difference of fitting both methods can be seen in Figure 1 for the yield curve on 2 January 2008.


**Table 1.** Root of Mean Squared Errors for the AFNS and DCOBS.
