Modeling the Estimated Factors

In the previous section, we have estimated the three factors in our model by ordi-nary least squares and analyzed most of its statistical properties. Observed that if we were only interested in fitting the term structure, the measurement equation in the state space representation will be sufficient.

However, the second step in the two step approach proposed by Diebold and Li (2006) suggested that we model the estimated factors as well. We therefore pro-ceeded by modeling the estimated factors using the state equation in section 6.4.

In Figures 6.17, 6.18 and 6.19 we plotted the empirical factors against the OLS factor estimates and the state representation factor estimates. The figure shows

Section 6.5. Analysis 45

Figure 6.16. The residual of the Nelson Siegel model fitted to Swedish government bonds.

that the factors obtained by using the state representation are almost similar to those obtained by the simple OLS method.

However, it is important to note that modeling the estimated factors enables us to construct forecast of the term structure. For example, given a fixed decay parameter in the three-factor Nelson Siegel model, the spot-rates depends only on the model factors i.e. level, slope and curvature. Thus, having a forecast of the factors

X_t+h|t= (I − A)µ + AX_t,t (6.5.1)

enables us to forecast the term structure as follows ˆ

y_t+h|t(τ ) = ZX_t+h|t (6.5.2)

The estimated mean µ, ar-coefficient matrix A and the variance σ, are shown in Table 6.4.

Observed that the co-variations between the factors are not reported because we are modeling the factors as independent univariate processes. The residuals au-tocorrelation of of our state space model fitted to the estimated factors are shown

Table 6.3. Descriptive Statistics, residuals

Maturity Mean Std minimum maximum ρ(1)ˆ ρ(12)ˆ ρ(30)ˆ (months)

1 0.0275 0.0573 -0.1098 0.2004 0.6377 0.0899 -0.0958 3 -0.0114 0.0436 -0.2534 0.0585 0.6145 0.1228 -0.0131 6 -0.0219 0.0538 -0.1835 0.1625 0.5966 -0.0731 0.0200 12 -0.0147 0.1123 -0.3654 0.4697 0.7578 0.0263 -0.1350 24 0.0262 0.0828 -0.2150 0.2964 0.7348 -0.1067 0.0315 60 0.0039 0.0782 -0.1423 0.2883 0.8683 0.1948 -0.0709 84 -0.0161 0.0603 -0.1787 0.1504 0.9235 0.0525 -0.0605 120 0.0066 0.0555 -0.1362 0.1531 0.8419 -0.0201 -0.1028

The table shows the residuals of the Nelson Siegel three-factor model with λ = 17.8348 fitted to Swedish government bonds at various maturities.

Table 6.4. Estimated Dynamic Factors Estimated Factor Mean AR-Coeff Var

level 4.6271 0.9942 0.0748

Slope -1.9436 0.9151 0.0652

Curvature -1.0886 0.9386 0.3766

The estimated mean, AR-coefficient, variance and the prediction error of the estimated Nelson Siegel factors modeled as independent factors using the state space representation in section 6.4

in Figure 6.20, 6.21 and 6.22. From these figures, we could observed that the auto-correlations are very small, thus indicating that the model captures the conditional means of the level-, slope and curvature factors.

Section 6.5. Analysis 47

Figure 6.17. The empirically de-fined level plotted against the OLS and state equation representation

Figure 6.18. The empirically de-fined slope plotted against the OLS and state equation representation

Figure 6.19. The empirically de-fined curvature plotted against the OLS and state equation representa-tion

Figure 6.20. The residual ACF for our state space model fitted to ˆβ1,t.

Figure 6.21. The residual ACF for our state space model fitted to ˆβ2,t.

Figure 6.22. The residual ACF for our state space model fitted to ˆβ_3,t

Section 6.6. Conclusion 49

6.6 Conclusion

In this thesis, I have shown that the term structure of Swedish government bonds can be fitted by applying the dynamic representation of the three-factor Nelson Siegel model. This was achieved by applying Diebold and Li (2006) two step ap-proach, where the model factors, i.e the level, the slope and the curvature, were first estimated using ordinary least squares and thereafter model as stable independent AR(1) processes. I have therefore shown that the, with only three unknown param-eters, the two-step approach can be used to estimate the term structure of Swedish government bonds. The estimates gave a good in-sample fit for short maturities but they display difficulties in fitting Swedish government bonds with long maturities and at dates where the term structure exhibits multiple minima/maxima.

I therefore instead to further study how these drawbacks can be improve upon in a masters thesis. I intend to follow Svensson (1997) and extend the three-factor model by adding a second curvature in the model. I also intend to used both the three-factor model and the Svensson extension to predict and forecast the term structure of Swedish government bonds. The performance of these models in-sample and out-in-sample fit will be contrasted. I also intend to lift the arbitrage-free assumption made on the Swedish government bonds throughout this thesis.

BIBLIOGRAPHY

[1] Interest Rate Models, Theory and Practice, Damiano Brigo and Fabio Mercurio, August 2001.

[2] Time Series Analysis, Henrik Madsen .

[3] Arbitrage Theory in Continuous Time, Bj¨ork 3th Edition.

[4] Finance Market,Instruments and Investments, Hans Bystr¨om, 2nd Edition.

[5] Modeling Non-Linear and Non-Stationary Time Series, Hendrik Madsen and Jan Holst, December 2000.

[6] Stationary Stochastic Processes,Georg Lindgren, Holger Rootz´en and Maria Sand-sten, September 2009.

[7] Derivative Pricing part I and II, Sebastian ˙Aberg, 2010.

[8] R¨anteb¨arande Instrument, V¨ardering och Riskhantering, Hossien Asgharian and Lars Nord´en.

[9] De Finansiella Marknaderna i ett Internationellt Perspektiv, Leif H¨assel, Marie Norman and Christian Andersson .

[10] Estimating Forward Interest Rates with the Extended Nelson and Siegel Mothod, Lars E.O. Svensson, (1994).

[11] Interest Rate Dynamics and Consistent Forward Rate Curves. Bj¨ork T. and Christensen B. J. (1999).

[12] Forecasting the term structure of government bond yields, Francis X. Diebold and Canlin Li, (2006) .

[13] Estimating the Yield Curve Using the Nelson-Siegel Model, A Ridge Regression Approach, Jan Annaert, Anouk G.P Claes, Marc J.K De Ceuster, Hairui Zhand, 2000.

50

Bibliography 51 [14] An Arbitrage-Free Generalized Nelson-Siegel Term Structure Model, Jens H.E.

Christensen, Francis X. Diebold, Glen D. Rudebusch, November 2008.

[15] Calibrating the Nelson-Siegel-Svensson model, M. Gilli, Stefan Grobe and E.Schumann. march 2010.

[16] Examining the Nelson-Siegel Class of Term Structure Models, Michiel De Pooter, June 2007.