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Geostatistical spatial models are widely used in many applied fields to forecast data observed on continuous three-dimensional surfaces. We propose to extend their use to finance and, in particular, to forecasting yield curves. We present the results of an empirical application where we apply the proposed method to forecast Euro Zero Rates (2003–2014) using the Ordinary Kriging method based on the anisotropic variogram. Furthermore, a comparison with other recent methods for forecasting yield curves is proposed. The results show that the model is characterized by good levels of predictions’ accuracy and it is competitive with the other forecasting models considered.

The study of the evolution of interest rates through time represents one of the major challenges in quantitative finance literature. In recent years, with the growth of derivative contracts markets and with the evolution of portfolio and risk management techniques, studies on this issue have gained further importance from an empirical point of view. The increasing concern of policy makers, investors and market operators on accurate interest rate estimates and forecasts has further stressed the need of defining models that are not only theoretically sound, but also usable in practice. In particular, according to Hunt

The two major objectives of the statistical analyses on interest rates are (i) the estimation of the term structure at a given point in time and (ii) the prediction of future rates. Despite few remarkable exceptions, in most cases the two objectives have been treated separately. With reference to the second objective, when forecasting interest rates, two dimensions are relevant: the moment of time of observations and the length of the term structure (the maturities). While the naïve approach forecasts the two dimensions separately, an obvious way of forecasting them jointly consists in the application of a multivariate time series modeling framework (Hamilton [

This paper contributes to the existing literature by proposing a new, unified, approach for modeling interest rates as a joint function of their two fundamental dimensions (time and maturity) so as to achieve more accurate forecasts. Our approach thus deviates dramatically from the econometric strategies traditionally employed for modeling the term structure of interest rates as we rely on geostatistical methods and kriging forecasting techniques (Cressie [

Specifically, the interest rates considered in our empirical analyses are the Euro Zero Rates (EZR). EZR are determined as the zero-coupon yields which are implied in the term structure of Euro Swap Rates (ESR). Their relevance is due to the fact that the Euro interest rate swap market is one of the largest and most liquid financial markets in the world. In particular, the Euro swap curve has emerged as the preeminent benchmark yield curve in euro financial markets, against which even some government bonds are now often referenced (Remolona and Wooldridge [

The paper is organized as follows. As said,

Interest rate term structure modeling is a widely studied issue in the financial literature and important theoretical advances have been reached so far in this area. The large interest of the scientific community on term structure modeling is witnessed by the huge number of alternative models proposed in the literature (for a review, see Brigo and Mercurio [

The

In contrast to this line of research, the

Several term structure affine models have been proposed in the academic literature. Duffee [

With reference to the analysis of forward rates, an important contribution to modeling the term structure is represented by the Nelson and Siegel [

Conversely, despite their growing practical importance, relatively little work has been done on forecasting models of the yield curve. As evidenced by Diebold and Li [

One of the early contributions in interest rates forecasting is the work of Fama and Bliss [

Finally, a further alternative approach in this field is the Cochrane and Piazzesi [

The high number of models proposed so far by researchers evidences that the issue of estimating and forecasting the term structure of interest rates is still being explored, hence the debate for identifying the best models for in-sample and out-of-sample predictions is still open. However, there is a wide consensus among scholars and financial institutions on the primary role of the Nelson-Siegel [

Spatial statistical methods constitute a relatively new area of research that emerged in the last decades with application in very different scientific fields (Cressie [

In order to introduce the geostatistical approach, let us first define a random field Z

The function

One of the simplest models is the Gaussian semivariogram defined as:

Other models often used in the literature are the Exponential semivariogram, defined as:
_{v}

From the above definitions, the problem of prediction in geostatistics can be summarized as follows: given the random field _{0}_{i}

In the present paper, we adopted an ordinary kriging approach instead of a universal kriging for the sake of comparison with the other competing models. In fact, the LARX model used for updating the parameters of the DNS benchmarking model employs only the inflation rate as exogenous variable and there is no way of using this variable in a universal kriging approach; this is because we can only observe its evolution through time, and not also through the maturity axis. In this respect, our results reinforce the power of the proposed approach, which achieves very accurate forecasts compared with the DNS model even if it is a purely autopredictive model that does not use any regressors.

In the next section, we will exploit this analytical framework to forecast interest rates along the time and maturity dimension. To implement our procedure we build up a fictitious three-dimensional space where the value

In order to test on empirical data the accuracy of the kriging forecasting model proposed in the previous section, we consider an interest rate characterized by a plurality of maturities and with sufficiently long historical data. As said, considering their importance in the European context and the relative length of their time series, we selected the Euro Zero Rates (EZR). Specifically, in each time period, EZR represent the Euribor 12-months rate for the maturity one year, while they were derived by bootstrapping the Euro Swap Rates (ESR) for maturities from 2 to 50 years. For a description of swap contracts, swap rates and bootstrapping procedures see Hull [

In order to assess the accuracy of our kriging forecasting model we compared its performances with the results of two other prediction methods: an extended version of the dynamic Nelson-Siegel (DNS) model, as better described in

The observational period ranges between 1 January 2003 and 30 June 2014. In particular, the historical data used to estimate the empirical variogram ranges from the 1 January 2003 to 31 December 2013, while the last six months (from 1 January 2014 to 30 June 2014) have been used only as out-of-sample data to assess the accuracy of the forecasting procedure. The empirical data were collected on the 1 July 2014 on a daily basis, excluding non-trading days: as a consequence, for each maturity, we considered about 256 observations per year. On each day we record the closing value, as reported by the Bloomberg financial database.

Main features of the interest rates considered.

Maturities |
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 20, 25, 30, 40 and 50 | |||

Period | From 1 January 2003 to 30 June 2014 | |||

Reference curve | Euro (ID: S45) | |||

Data source | Bloomberg | |||

Type of rate | Swap | Zero | Forward (Zero) | |

Name | EUR Swap Annual | EUR Zero Rate | EUR Forward (Zero) Rate | |

Bloomberg ticker | EUSA |
S0045Z |
- | |

Symbol | ESR | EZR | EFR | |

Time horizon |
- | - | 3, 6 and 12 | |

Day count | Fixed | 30U/360 | Determined by bootstrapping the EUR Swap Annual curve for |
Determined on the basis of the EUR Swap Annual bootstrapped curve (rate type “Zc”, side “Mid”) using the “BCurveFwd” Bloomberg function. |

Floating | ACT/360 | |||

Payment frequency | Fixed | Annual | ||

Floating | Semi-annual | |||

Floating | Index | Euribor 6 months | ||

Index ticker | EUR006M Index | |||

Reset frequency | Semi-annual |

Note: Bloomberg tickers are specific for each maturity; therefore, given a maturity of (e.g.,)

Descriptive statistics of Euro Zero Rates (EZR).

Maturity |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 15 | 20 | 25 | 30 | 40 | 50 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

2003 | 2.33 | 2.64 | 2.96 | 3.26 | 3.51 | 3.73 | 3.92 | 4.09 | 4.23 | 4.35 | 4.46 | 4.56 | 4.79 | 5.04 | 5.14 | 5.15 | 5.09 | 5.00 |

2004 | 2.27 | 2.64 | 2.97 | 3.25 | 3.49 | 3.70 | 3.88 | 4.04 | 4.18 | 4.29 | 4.39 | 4.48 | 4.70 | 4.94 | 5.04 | 5.07 | 5.07 | 5.03 |

2005 | 2.33 | 2.54 | 2.71 | 2.85 | 2.99 | 3.12 | 3.24 | 3.35 | 3.45 | 3.54 | 3.62 | 3.69 | 3.86 | 4.03 | 4.11 | 4.13 | 4.14 | 4.12 |

2006 | 3.44 | 3.62 | 3.69 | 3.75 | 3.80 | 3.84 | 3.89 | 3.93 | 3.98 | 4.02 | 4.06 | 4.10 | 4.19 | 4.28 | 4.31 | 4.30 | 4.25 | 4.19 |

2007 | 4.45 | 4.44 | 4.43 | 4.43 | 4.45 | 4.46 | 4.48 | 4.51 | 4.54 | 4.57 | 4.60 | 4.63 | 4.70 | 4.75 | 4.74 | 4.70 | 4.60 | 4.49 |

2008 | 4.83 | 4.32 | 4.29 | 4.30 | 4.32 | 4.35 | 4.40 | 4.45 | 4.50 | 4.55 | 4.60 | 4.65 | 4.74 | 4.75 | 4.64 | 4.54 | 4.37 | 4.23 |

2009 | 1.61 | 1.88 | 2.28 | 2.59 | 2.85 | 3.07 | 3.25 | 3.40 | 3.53 | 3.65 | 3.75 | 3.85 | 4.07 | 4.19 | 4.05 | 3.89 | 3.57 | 3.43 |

2010 | 1.35 | 1.47 | 1.75 | 2.02 | 2.27 | 2.50 | 2.69 | 2.86 | 3.00 | 3.12 | 3.23 | 3.32 | 3.53 | 3.62 | 3.54 | 3.37 | 3.14 | 3.06 |

2011 | 2.01 | 1.85 | 2.06 | 2.28 | 2.50 | 2.67 | 2.83 | 2.95 | 3.06 | 3.16 | 3.25 | 3.34 | 3.52 | 3.57 | 3.47 | 3.34 | 3.20 | 3.18 |

2012 | 1.11 | 0.78 | 0.88 | 1.04 | 1.24 | 1.43 | 1.61 | 1.76 | 1.89 | 2.01 | 2.12 | 2.21 | 2.40 | 2.47 | 2.44 | 2.41 | 2.46 | 2.53 |

2013 | 0.54 | 0.52 | 0.68 | 0.89 | 1.10 | 1.30 | 1.49 | 1.66 | 1.82 | 1.96 | 2.09 | 2.20 | 2.44 | 2.59 | 2.62 | 2.61 | 2.66 | 2.73 |

2014 | 0.57 | 0.44 | 0.57 | 0.75 | 0.95 | 1.15 | 1.34 | 1.52 | 1.69 | 1.83 | 1.96 | 2.08 | 2.33 | 2.52 | 2.56 | 2.56 | 2.57 | 2.57 |

Observations | 2944 | 2944 | 2944 | 2944 | 2944 | 2944 | 2944 | 2944 | 2944 | 2944 | 2944 | 2944 | 2944 | 2944 | 2944 | 2944 | 2944 | 2944 |

Mean | 2.31 | 2.34 | 2.52 | 2.70 | 2.87 | 3.02 | 3.16 | 3.29 | 3.40 | 3.49 | 3.58 | 3.66 | 3.84 | 3.96 | 3.95 | 3.90 | 3.81 | 3.76 |

St. deviation | 1.34 | 1.32 | 1.26 | 1.21 | 1.15 | 1.10 | 1.06 | 1.02 | 1.00 | 0.97 | 0.95 | 0.93 | 0.91 | 0.92 | 0.94 | 0.96 | 0.94 | 0.90 |

Minimum | 0.47 | 0.31 | 0.39 | 0.50 | 0.65 | 0.82 | 1.00 | 1.17 | 1.33 | 1.47 | 1.60 | 1.71 | 1.89 | 1.88 | 1.85 | 1.81 | 1.83 | 1.85 |

Maximum | 5.53 | 5.48 | 5.40 | 5.27 | 5.19 | 5.14 | 5.10 | 5.08 | 5.07 | 5.09 | 5.10 | 5.12 | 5.15 | 5.28 | 5.38 | 5.40 | 5.43 | 5.37 |

This table shows the one-year means of the daily values of Euro Zero Rates (%) observed within each year of the reference period (1 January 2003–30 June 2014) and some descriptive statistics related to the whole dataset for each examined maturity.

Descriptive statistics of Euro Forward Rates (EFR).

Date |
Year | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Day/Month | 29/12 | 28/12 | 31/12 | 31/12 | 31/12 | 30/12 | 29/12 | 31/12 | 31/12 | 31/12 | 31/12 | |

Time horizon |
Maturity |
|||||||||||

3 month | 4.54 | 3.90 | 3.51 | 4.22 | 4.75 | 3.83 | 3.77 | 3.46 | 2.46 | 1.66 | 2.29 | |

6 months | 4.62 | 3.97 | 3.54 | 4.23 | 4.76 | 3.89 | 3.87 | 3.55 | 2.51 | 1.73 | 2.37 | |

12 months | 4.78 | 4.08 | 3.59 | 4.24 | 4.80 | 4.01 | 4.04 | 3.70 | 2.63 | 1.86 | 2.53 | |

Time horizon |
Maturity |
|||||||||||

3 month | 5.07 | 4.45 | 3.83 | 4.35 | 4.98 | 3.92 | 4.25 | 3.85 | 2.75 | 2.28 | 2.86 | |

6 months | 5.12 | 4.48 | 3.84 | 4.36 | 4.98 | 3.92 | 4.28 | 3.88 | 2.76 | 2.31 | 2.90 | |

12 months | 5.19 | 4.54 | 3.87 | 4.36 | 4.99 | 3.94 | 4.35 | 3.93 | 2.80 | 2.36 | 2.96 | |

Time horizon |
Maturity |
|||||||||||

3 month | 5.18 | 4.56 | 3.86 | 4.29 | 4.90 | 3.45 | 3.98 | 3.49 | 2.56 | 2.33 | 2.83 | |

6 months | 5.21 | 4.58 | 3.86 | 4.29 | 4.90 | 3.44 | 4.00 | 3.50 | 2.56 | 2.35 | 2.85 | |

12 months | 5.25 | 4.62 | 3.88 | 4.29 | 4.90 | 3.44 | 4.03 | 3.51 | 2.58 | 2.39 | 2.88 |

This table reports the values of Euro Forward Rates (%) with a time horizons

Three-dimensional representation of Euro Zero Rates sample data (see

In order to forecast the term structure of EZR for all the maturities and all time period, we adopt a three-steps procedure. At a first step, we calculate the empirical anisotropic variogram of the sample observations by choosing a cutoff parameter of 12 months. This value represents the limiting spatial distance to include pairs of observations in the semivariance estimates: outside this threshold pairs of observations are not considered. In a second step, we calibrate the best-fitting model among four different variograms, namely: Bessel, Exponential, Gaussian and Power (see

In order to test the accuracy of the proposed model, we forecast the values of the last working day of December of each year (2003 to 2013), for a total of 11 reference dates (see

The dynamic Nelson-Siegel model, proposed by Diebold and Li [

Nelson-Siegel factor loadings (λ = 0.3189).

The loadings of

Time evolution of Nelson-Siegel factors for Euro Swap Zero Rates.

The time evolution of the factors

A further refinement of the Nelson and Siegel [

However, since our objective is comparing the prediction ability of the proposed OK model with the accuracy of other forecasting methods, the same historical observations shall be considered to apply all the models at each forecasting date. Therefore, in this work, the optimization procedure of the ADNS model was not applied. As said, we focus on the DNS model, considering the extension proposed by Chen and Niu [

In order to test the accuracy of forecasts, for each examined model (OK and DNS) we calculate statistics based on the root mean squared error (RMSE), defined as the square root of the mean squared error. We considered the RMSE as it represents a standard measure in the financial literature for measuring and comparing the accuracy of interest rates prediction models (Diebold and Li [

In particular, in order to refine our comparative analysis, we consider alternative versions of the RMSE, based on statistics other than the mean, that are the minimum value, the maximum value and the quartiles of the models’ squared errors. Therefore, for each statistic

Furthermore, for the sake of comparison, we calculate the RSSE of EFR predictions as follows:

For each of the 31 combinations [

Theoretical anisotropic variogram models.

Dataset Length |
2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 |
---|---|---|---|---|---|---|---|---|---|---|---|

1 year | Exp | Bes | Exp | Gau | Bes | Bes | Pow | Bes | Bes | Bes | Bes |

3 years | - | - | Exp | Pow | Pow | Bes | Bes | Bes | Pow | Pow | Pow |

Max | mod (1 year) | Bes | mod (3 year) | Pow | Pow | Bes | Pow | Pow | Pow | Pow | Pow |

This table reports the theoretical anisotropic variogram models which best fit the empirical variogram for each combination of reference year

On the basis of the selected variogram models shown in

As it was reasonable to expect, forecasts are characterized by higher median errors and a higher level of uncertainty when a greater forecasts’ time horizon is considered. In particular, using the max-length dataset, the median

Prediction accuracy of the Ordinary Kriging (OK) model and the Euro Forward Rates (EFR) forecasting method (overall analysis).

Statistics of | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Data set amplitude |
1 year | 3 years | Max | - | |||||||||

Time horizon |
Maturity |
10 | 20 | 30 | 10 | 20 | 30 | 10 | 20 | 30 | 10 | 20 | 30 |

3 months | Max | 0.57 | 0.59 | 0.60 | 0.45 | 0.51 | 0.46 | 0.46 | |||||

Q3 | 0.35 | 0.34 | 0.36 | 0.36 | 0.31 | 0.34 | 0.35 | ||||||

Median | 0.27 | 0.23 | 0.21 | 0.26 | 0.20 | 0.21 | 0.30 | 0.19 | |||||

Q1 | 0.11 | 0.14 | 0.18 | 0.14 | 0.12 | 0.16 | 0.11 | 0.08 | |||||

Min | 0.05 | 0.10 | 0.03 | 0.03 | 0.05 | 0.05 | 0.04 | 0.02 | |||||

6 months | Max | 0.87 | 0.97 | 0.99 | 0.70 | 0.76 | 0.83 | 0.90 | 0.79 | 0.77 | |||

Q3 | 0.60 | 0.64 | 0.67 | 0.61 | 0.67 | 0.68 | 0.72 | 0.68 | 0.66 | ||||

Median | 0.44 | 0.58 | 0.44 | 0.38 | 0.54 | 0.40 | 0.48 | 0.40 | 0.57 | ||||

Q1 | 0.29 | 0.25 | 0.25 | 0.32 | 0.23 | 0.23 | 0.23 | 0.24 | |||||

Min | 0.14 | 0.14 | 0.01 | 0.12 | 0.18 | 0.02 | 0.03 | 0.02 | |||||

12 months | Max | 1.02 | 1.37 | 0.93 | 0.95 | 1.37 | 0.98 | 1.27 | 1.18 | 1.45 | |||

Q3 | 0.79 | 0.70 | 0.64 | 0.75 | 0.72 | 0.71 | 0.97 | 0.73 | 0.72 | ||||

Median | 0.42 | 0.50 | 0.54 | 0.45 | 0.46 | 0.37 | 0.62 | 0.58 | 0.58 | ||||

Q1 | 0.35 | 0.29 | 0.38 | 0.34 | 0.32 | 0.25 | 0.53 | 0.48 | 0.45 | ||||

Min | 0.12 | 0.24 | 0.15 | 0.17 | 0.18 | 0.20 | 0.32 | 0.29 | 0.26 | ||||

No. underlined—All ^{(1)} |
12 | 14 | - | ||||||||||

No. underlined—Median ^{(2)} |
3 | 2 | - | ||||||||||

No. bold—All ^{(3)} |
10 | 11 | 7 | ||||||||||

No. bold—Median ^{(4)} |
3 | 1 | 1 | ||||||||||

^{(1)} Number of underlined statistics (total = 45, plus eventual repetitions for ^{(2)} Number of underlined median values (total = 9, plus eventual repetitions for ^{(3)} Number of statistics in bold (total = 45, plus eventual repetitions for ^{(4)} Number of median values in bold (total = 9, plus eventual repetitions for |

The statistics shown in this table are presented in terms of root square of each statistic

Summary statistics of the prediction accuracy of the Ordinary Kriging (OK) model and the Euro Forward Rates (EFR):

Statistics of | |||||||||
---|---|---|---|---|---|---|---|---|---|

Data set amplitude |
1 year | 3 years | Max | - | |||||

Time horizon |
2003–2007 | 2008–2013 | 2003–2007 | 2008–2013 | 2003–2007 | 2008–2013 | 2003–2007 | 2008–2013 | |

3 months | Max | 0.60 | 0.49 | 0.43 | 0.45 | 0.46 | |||

Q3 | 0.45 | 0.25 | 0.42 | 0.24 | 0.26 | ||||

Median | 0.23 | 0.27 | 0.21 | 0.20 | 0.19 | 0.18 | |||

Q1 | 0.18 | 0.19 | 0.23 | 0.11 | 0.12 | ||||

Min | 0.10 | 0.18 | 0.06 | 0.05 | 0.02 | ||||

6 months | Max | 0.99 | 0.69 | 0.83 | 0.70 | 0.77 | 0.76 | ||

Q3 | 0.83 | 0.61 | 0.76 | 0.59 | 0.64 | ||||

Median | 0.60 | 0.68 | 0.49 | 0.61 | 0.65 | 0.50 | |||

Q1 | 0.25 | 0.26 | 0.48 | 0.23 | 0.30 | ||||

Min | 0.01 | 0.21 | 0.02 | 0.14 | 0.02 | 0.23 | |||

12 months | Max | 1.37 | 1.37 | 0.81 | 0.83 | 1.45 | 0.96 | ||

Q3 | 0.68 | 0.60 | 1.07 | 0.52 | 0.74 | 0.54 | |||

Median | 0.44 | 0.39 | 0.67 | 0.37 | 0.65 | 0.50 | |||

Q1 | 0.39 | 0.54 | 0.34 | 0.62 | 0.43 | ||||

Min | 0.24 | 0.26 | 0.37 | 0.18 | 0.42 | 0.26 | |||

No. underlined—All ^{(1)} |
5 | 2 | 4 | 6 | - | - | |||

No. underlined—Median ^{(2)} |
0 | 0 | 1 | 1 | - | - | |||

No. bold—All ^{(3)} |
2 | 2 | 5 | 4 | 2 | ||||

No. bold—Median ^{(4)} |
0 | 0 | 1 | 1 | 0 | 0 | |||

^{(1)} Number of underlined statistics (total = 15, plus eventual repetitions for ^{(2)} Number of underlined median values (total = 3, plus eventual repetitions for ^{(3)} Number of statistics in bold (total = 15, plus eventual repetitions for ^{(4)} Number of median values in bold (total = 3, plus eventual repetitions for |

This table shows summary statistics of the prediction accuracy of the Ordinary Kriging (OK) model and the Euro Forward Rates (EFR) method used for predicting 30-year maturity Euro Zero Rate, distinguishing between the

Furthermore, we study the performances of the model for different

The data reported in

Similarly to the analysis shown in

Generally speaking, as the maturity increases,

Our results are in line with the RMSEs determined in previous studies that are based on the DNS model. In particular, it is interesting to compare our RSSE median statistics with the RMSE calculated by Diebold and Li [

Similarly to the analysis shown in

Prediction accuracy of the extended dynamic Nelson-Siegel (DNS) model and the Euro Forward Rates (EFR) forecasting method.

Statistics of | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Data set amplitude |
1 year | 3 years | Max | - | |||||||||

Time horizon |
Maturity |
10 | 20 | 30 | 10 | 20 | 30 | 10 | 20 | 30 | 10 | 20 | 30 |

3 months | Max | 1.57 | 1.61 | 1.64 | 0.59 | 0.60 | 0.61 | 0.51 | 0.46 | 0.46 | |||

Q3 | 0.60 | 0.54 | 0.48 | 0.38 | 0.40 | 0.40 | 0.35 | ||||||

Median | 0.32 | 0.34 | 0.37 | 0.26 | 0.25 | 0.21 | 0.30 | 0.18 | 0.19 | ||||

Q1 | 0.27 | 0.28 | 0.29 | 0.20 | 0.20 | 0.15 | 0.11 | 0.12 | |||||

Min | 0.05 | 0.05 | 0.07 | 0.08 | 0.10 | 0.03 | 0.04 | 0.02 | |||||

6 months | Max | 1.53 | 1.46 | 1.40 | 1.32 | 1.34 | 1.37 | ||||||

Q3 | 1.09 | 1.10 | 1.11 | 0.96 | 0.95 | 0.93 | |||||||

Median | 0.90 | 0.85 | 0.80 | 0.96 | 0.53 | 0.48 | 0.57 | ||||||

Q1 | 0.51 | 0.52 | 0.53 | 0.52 | 0.34 | 0.26 | 0.23 | 0.23 | 0.24 | ||||

Min | 0.16 | 0.12 | 0.07 | 0.40 | 0.21 | 0.14 | 0.03 | ||||||

12 months | Max | 2.12 | 1.99 | 1.88 | 2.30 | 2.14 | 2.00 | 1.45 | |||||

Q3 | 1.15 | 1.20 | 1.51 | 1.42 | 1.35 | ||||||||

Median | 0.73 | 0.74 | 0.71 | 0.88 | 0.77 | 0.70 | 0.62 | 0.58 | 0.58 | ||||

Q1 | 0.33 | 0.37 | 0.34 | 0.36 | 0.35 | 0.35 | 0.53 | 0.48 | 0.45 | ||||

Min | 0.03 | 0.09 | 0.04 | 0.09 | 0.04 | 0.32 | 0.29 | 0.26 | |||||

No. underlined—All ^{(1)} |
8 | 11 | - | ||||||||||

No. underlined—Median ^{(2)} |
0 | 4 | - | ||||||||||

No. bold—All ^{(3)} |
4 | 8 | 16 | ||||||||||

No. bold—Median ^{(4)} |
0 | 3 | 1 | ||||||||||

^{(1)} Number of underlined statistics (total = 45, plus eventual repetitions for ^{(2)} Number of underlined median values (total = 9, plus eventual repetitions for ^{(3)} Number of statistics in bold (total = 45, plus eventual repetitions for ^{(4)} Number of median values in bold (total = 9, plus eventual repetitions for |

This table shows summary statistics of the prediction accuracy of the extended dynamic Nelson-Siegel (DNS) model and the Euro Forward Rates (EFR) forecasting method. Results are presented in terms of root square of each statistic

The results shown in

Summary statistics of the prediction accuracy of the extended dynamic Nelson-Siegel (DNS) model and the Euro Forward Rates (EFR):

Statistics of | |||||||||
---|---|---|---|---|---|---|---|---|---|

Data set amplitude |
1 year | 3 years | Max | - | |||||

Time horizon |
2003–2007 | 2008–2013 | 2003–2007 | 2008–2013 | 2003–2007 | 2008–2013 | 2003–2007 | 2008–2013 | |

3 months | Max | 0.40 | 1.64 | 0.61 | 0.47 | 0.46 | 0.39 | ||

Q3 | 0.35 | 1.12 | 0.34 | 0.52 | 0.20 | 0.26 | |||

Median | 0.29 | 0.43 | 0.27 | 0.40 | 0.19 | 0.18 | |||

Q1 | 0.26 | 0.36 | 0.19 | 0.30 | 0.13 | 0.12 | |||

Min | 0.22 | 0.05 | 0.10 | ||||||

6 months | Max | 1.11 | 1.24 | 1.40 | 1.37 | 0.77 | |||

Q3 | 1.10 | 1.06 | 0.96 | 1.04 | 0.67 | ||||

Median | 1.09 | 0.71 | 0.65 | 0.65 | 0.50 | ||||

Q1 | 0.78 | 0.55 | 0.53 | 0.22 | 0.30 | ||||

Min | 0.52 | 0.53 | 0.14 | 0.23 | |||||

12 months | Max | 1.20 | 1.88 | 2.00 | 1.20 | 1.45 | |||

Q3 | 1.00 | 1.35 | 1.53 | 0.88 | 0.74 | ||||

Median | 0.76 | 0.68 | 0.37 | 0.77 | 0.65 | ||||

Q1 | 0.54 | 0.41 | 0.42 | 0.26 | 0.62 | 0.43 | |||

Min | 0.04 | 0.28 | 0.04 | 0.21 | 0.42 | 0.26 | |||

No. underlined—All ^{(1)} |
2 | 2 | 1 | 4 | - | - | |||

No. underlined—Median ^{(2)} |
0 | 1 | 0 | 1 | - | - | |||

No. bold—All ^{(3)} |
1 | 1 | 1 | 1 | 3 | 6 | |||

No. bold—Median ^{(4)} |
0 | 0 | 0 | 1 | 0 | 1 | |||

^{(1)} Number of underlined statistics (total = 15, plus eventual repetitions for ^{(2)} Number of underlined median values (total = 3, plus eventual repetitions for ^{(3)} Number of statistics in bold (total = 15, plus eventual repetitions for ^{(4)} Number of median values in bold (total = 3, plus eventual repetitions for |

This table shows summary statistics of the prediction accuracy of the extended dynamic Nelson-Siegel (DNS) model and the Euro Forward Rates (EFR) method used for predicting 30-year maturity Euro Zero Rate, distinguishing between the

Both the OK and the DNS models were applied to predict the EZR considering the same subset of historical observations at each forecasting date. Therefore, the statistics of RSSE shown in

Comparative analysis of prediction accuracy between the Ordinary Kriging (OK) model and the extended dynamic Nelson-Siegel (DNS) model.

Model | Model with the Lowest Statistic of |
|||||||||
---|---|---|---|---|---|---|---|---|---|---|

Data set amplitude |
1 year | 3 years | Max | |||||||

Time horizon |
Maturity |
10 | 20 | 30 | 10 | 20 | 30 | 10 | 20 | 30 |

3 months | Max | OK | OK | OK | OK | DNS | DNS | OK | OK | OK |

Q3 | OK | OK | OK | DNS | OK | OK | OK | OK | OK | |

Median | OK | OK | OK | DNS | OK | OK | DNS | DNS | DNS | |

Q1 | OK | OK | OK | OK | OK | DNS | DNS | OK | DNS | |

Min | DNS | DNS | OK | OK | OK | OK | OK | DNS | DNS | |

6 months | Max | OK | OK | OK | OK | OK | OK | OK | OK | OK |

Q3 | OK | OK | OK | OK | OK | OK | OK | OK | OK | |

Median | OK | OK | OK | OK | OK | DNS | OK | OK | DNS | |

Q1 | OK | OK | OK | OK | OK | OK | DNS | DNS | DNS | |

Min | OK | OK | OK | OK | OK | OK | DNS | DNS | OK | |

12 months | Max | OK | OK | OK | OK | OK | OK | OK | OK | DNS |

Q3 | OK | OK | OK | OK | OK | OK | OK | OK | OK | |

Median | OK | OK | OK | OK | DNS | OK | OK | OK | OK | |

Q1 | OK | OK | OK | OK | DNS | OK | OK | OK | DNS | |

Min | DNS | DNS | DNS | DNS | DNS | DNS | OK | DNS | DNS | |

No. OK—All | ||||||||||

No. DNS—All | 5 | 11 | 17 | |||||||

No. OK—Median | ||||||||||

No. DNS—Median | 0 | 3 | 4 |

This table shows the results of a comparative analysis of prediction accuracy between the Ordinary Kriging (OK) model and the extended dynamic Nelson-Siegel (DNS) model. The level of prediction accuracy is measured in terms of root square of each statistic

Focusing on the distinction between the

Comparative analysis of prediction accuracy between the Ordinary Kriging (OK) model and the extended dynamic Nelson-Siegel (DNS) model:

Statistics of | |||||||||
---|---|---|---|---|---|---|---|---|---|

Data set amplitude |
1 year | 3 years | Max | - | |||||

Time horizon |
2003–2007 | 2008–2013 | 2003–2007 | 2008–2013 | 2003–2007 | 2008–2013 | 2003–2007 | 2008–2013 | |

3 months | Max | DNS | OK | DNS | DNS | OK | OK | DNS | OK |

Q3 | DNS | OK | DNS | OK | OK | DNS | DNS | OK | |

Median | OK | OK | DNS | OK | OK | DNS | OK | OK | |

Q1 | OK | OK | DNS | OK | OK | DNS | OK | OK | |

Min | OK | DNS | DNS | OK | OK | DNS | OK | DNS | |

6 months | Max | OK | OK | DNS | OK | OK | OK | OK | OK |

Q3 | OK | OK | DNS | OK | OK | OK | OK | OK | |

Median | OK | OK | DNS | OK | DNS | DNS | OK | OK | |

Q1 | OK | OK | OK | DNS | OK | DNS | OK | OK | |

Min | OK | OK | OK | DNS | OK | DNS | OK | OK | |

12 months | Max | DNS | OK | DNS | OK | DNS | OK | DNS | OK |

Q3 | OK | OK | DNS | OK | OK | OK | OK | OK | |

Median | OK | OK | DNS | OK | DNS | OK | OK | OK | |

Q1 | OK | OK | DNS | OK | DNS | OK | OK | OK | |

Min | DNS | DNS | DNS | OK | DNS | OK | DNS | DNS | |

No. OK—All | 2 | ||||||||

No. DNS—All | 4 | 2 | 3 | 5 | 7 | 4 | 2 | ||

No. OK—Median | 0 | 1 | 1 | ||||||

No. DNS—Median | 0 | 0 | 0 | 0 | 0 |

This table shows the results of a comparative analysis of prediction accuracy between the Ordinary Kriging (OK) model and the extended dynamic Nelson-Siegel (DNS) model. The analysis is focused on the 30-year maturity Euro Zero Rate, distinguishing between the

To facilitate the visual comparison of the efficacy of each forecasting method,

We observe that for all the scenarios of dataset length—1 year, 3 years and

Box plots of the statistics shown in

Relating to the

This paper contributes to the existing literature by adopting an innovative approach to analyze the term structure of interest rates for short-term forecasting purposes. Our work is the first to develop a spatial statistical analysis of interest rates time series, outlining a procedure to calculate the empirical anisotropic variogram of Euro Zero Rates, to fit and select a theoretical variogram model and to use it to forecast the term structure up to 12 months using Ordinary Kriging.

The empirical results referring to Euro Zero Rates in the period 2003–2014 show that, for long-term maturities, the proposed model is characterized by good levels of forecasting accuracy, with prediction errors that increase as a greater time horizon is considered. From a comparative point of view, our model proves to be more accurate than using the forward rates implicit in the Euro Zero Rates curve as proxies of the market expectations of future interest rates and the best results are achieved when all the historical observations are used to build up the model. Furthermore, the comparison with the dynamic Nelson-Siegel model (based on Chen and Niu [

In future studies, our approach could be refined in various directions. First of all, future studies could extend the comparative analysis of efficacy to other interest rate forecasting models. Secondly, it would be interesting to test the performances of the proposed model in longer-term predictions going beyond the 12 months horizon to which we limited the present study. Finally, the proposed method could be applied and tested also for other yield curves and historical periods.

The authors contributed equally to this work.

The authors declare no conflict of interest.

All computations were performed using the software R.

Using the “EstimateAnisotropy” function of the software R (package “Intamap”).

Using the “

The information of date is expressed as (Day/Month/Year).