Generalized Nelson–Siegel term structure model: do the second slope and curvature factors improve the in-sample fit and out-of-sample forecasts?

Abstract

The dynamic Nelson–Siegel (DNS) model and even the Svensson generalization of the model have trouble in fitting the short maturity yields and fail to grasp the characteristics of the Japanese government bonds yield curve, which is flat at the short end and has multiple inflection points. Therefore, a closely related generalized dynamic Nelson–Siegel (GDNS) model that has two slopes and curvatures is considered and compared empirically to the traditional DNS in terms of in-sample fit as well as out-of-sample forecasts. Furthermore, the GDNS with time-varying volatility component, modeled as standard EGARCH process, is also considered to evaluate its performance in relation to the GDNS. The GDNS model unanimously outperforms the DNS in terms of in-sample fit as well as out-of-sample forecasts. Moreover, the extended model that accounts for time-varying volatility outpace the other models for fitting the yield curve and produce relatively more accurate 6- and 12-month ahead forecasts, while the GDNS model comes with more precise forecasts for very short forecast horizons.

Keywords:

• term structure of interest rates
• latent factors model
• state-space model
• Kalman filter
• EGARCH
• forecasting
• bond market

JEL Classification:

• C32
• C53
• C51
• E43
• G12
• G17

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. For example, on 17 February 2009, the seven-year interest rate becomes relatively low compared with the six-year and eight-year rates [19]. Detail description of these features of JGBs yield curve is given in Ullah et al. [26], Ullah [25], Kim and Singleton [20], and Kikuchi and Shintani [19].

2. The parameter in NS spot rate function specifies the location of the hump or the U-shape on the yield curve. The small values of , which have rapid decay in regressors, tend to fit low maturities interest rates quite well and larger values of lead to more appropriate fit of longer maturities spot rates.

3. It is used at the Federal Reserve Board (see Gurkaynak et al. [13]), the European Central Bank (see Coroneo et al. [6]), and many other central banks (see Bank for International Settlements [1]). Detailed discussion is given in De Pooter [7].

4. Assuming a second-order differential equation, to describe the movements of the yield curve, with the assumption of real and un-equal roots, the solution will be the instantaneous implied forward rate function. The solution for the yield function can be found by integrating the forward rate function. The resulting yield curve function will consists of five factors (one level, two slopes and two curvatures factors) and two decay parameters, that is, and , which corresponds to two different roots of the second-order differential equation.

5. Besides its wide use, the Svensson [24] model in its dynamic form cannot be derived in the standard finance arbitrage-free affine term structure representation [4]. However, the model with two slopes and curvature can easily be derived in the arbitrage-free affine framework, by generalizing the method adopted in Christensen et al. [5].

6. Moreover, Diebold and Li [8] find that the time series of estimated factors of NS model are highly persistent, which implies that these can be modeled as AR(1) or VAR(1). Using the Japanese market data, Ullah et al. [26,28] find that the three latent factors of yield curve are highly persistent and VAR(1) specification is more appropriate than the AR(1) and random walk specifications.

7. Two typical examples in this line are Bjork and Christensen [2] and Svensson [24]. Bjork and Christensen [2] make progress in this direction by adding a fourth factor to the NS model. The fourth factor also affects short-term maturities like the second component and, therefore, can be interpreted as a second slope factor. The two slope factors are governed by the same loadings but with different decay rates. Svensson [24] suggests another kind of four-factor extended NS specification and he adds a second curvature factor.

8. The model with two slope factors (as in Bjork and Christensen [2]) or two curvatures (such as in Svensson [24]) may also serve the purpose of fitting curves with special shapes, such as twists, but Christensen et al. [4] shows that the model, which accounts for two slope and curvature factors simultaneously, outperforms the standard Bjork and Christensen [2] and Svensson [24] models. Secondly, the models with either two slopes or two curvatures cannot be derived in the affine framework (for detail see Christensen et al. [4]).

9. Koopman et al. [21] has used the GARCH specification to model the variance , but financial markets respond in different ways to positive and negative shocks and it is a common knowledge that volatility tends to increase quickly when negative news reaches to traders and investors, whereas positive news usually has a much less pronounced effect [27].

10. The time varying volatility component can also be included in the standard DNS model, but similar type of models have been estimated for the Japanese bond market in Ullah et al. [27] and for the USA in Koopman et al. [21]. Furthermore, in the context of the term structure modeling and forecasting, Ullah [25] empirically compares four models, the DNS model, the DNS model with EGARCH disturbances in the observation equation (DNS–EGARCH), the affine arbitrage free Nelson–Siegel model (AFNS), and the AFNS with EGARCH component for the volatility (AFNS–EGARCH). He concludes that the DNS–EGARCH outperforms the standard DNS model for almost all maturities in terms of in-sample fitting as well as out-of-sample forecasting (at least for long horizons forecasts). Therefore, the DNS–EGARCH model is skipped and the results are not reproduced here.

11. The conditional expectation at of the latent variables in Equation (10) gives: where the estimate of is the last element of from the update step.

12. It is worthwhile to note that the quasi-maximum likelihood procedure has been used to estimate the parameter vector for the EGARCH based model, because the state-space model with time-varying stochastic volatility is not conditionally Gaussian and the Kalman filter loses its optimality property in the sense of minimizing the mean square estimates – as pointed out by Harvey et al. [15].

13. The bonds of maturity less than two months have almost same prices because of the very low interest during the sampled period and it implies to some strange estimates of the zero-coupon rates, such as the rate for one-month maturity is higher than of the one and half month maturity bonds. Moreover, the inflation indexed bonds have floating rates (coupon is not fixed) that change in each period. Therefore, the bonds with maturity of less than two months and floating rates are omitted from the sample at the stage of calculating the zero-coupon rates.

14. The pairwise correlation of estimated factors across three models is as follows. The correlation of estimated level factor is and , for the estimated first slope factor is and , whereas for the first curvature across three models is and . The correlation of second slope and second curvature factors in the GDNS and GDNS–EGARCH models is 0.9193 and 0.9435, respectively. It shows that the estimated factors follows almost the same pattern and are closely correlated across models.

15. The data for the macroeconomic variables, the annualized growth of industrial production , the growth rate of money supply as an indicator of monetary policy; and inflation rate , measured as annualized monthly changes in the consumer price index is obtained from the International Financial Statistics.

16. One should be aware of two big fall in industrial production during late 2008 and early 2011. These two falls do not correspond to the domestic policy shocks, therefore, are not reflected by the yield curve slope. The former corresponds to the global financial crisis, during this period the Japanese exports fall sharply and hence the real activity slows down. The second fall refers to the impact of the great East Japan earthquake and tsunami of 2011.

17. It means that we estimate and forecast recursively. For the first forecast, the models are estimated by using data from 1996:01 to 2006:12. Then, one month data is added to re-estimate the models, and another forecast is constructed.

18. We estimated the random walk and AR(1) model of yield, but all three models, that is, DNS, GDNS and GDNS–EGARCH models outpace the benchmark AR(1) and random walk specifications of yields forecasts, therefore the results are not reported.

Additional information

Funding

This research is supported by Grant-in-aid [No. 25-03309] from Japan Society for the Promotion of Science (JSPS).