Analyzing the Term Structure of Interest Rates Using the Dynamic Nelson-Siegel Model with Time-Varying Parameters

Abstract

In this paper we introduce time-varying parameters in the dynamic Nelson-Siegel yield curve model for the simultaneous analysis and forecasting of interest rates of different maturities, known as the term structure. The Nelson-Siegel model has been recently reformulated as a dynamic factor model where the latent factors are interpreted as the level, slope and curvature of the term structure. The factors are modelled by a vector autoregressive process. We propose to extend this framework in two directions. First, the factor loadings are made time-varying through a simple single step function and we show that the model fit increases significantly as a result. The step function can be replaced by a spline function to allow for more smoothness and flexibility. Second, we investigate empirically whether the volatility in interest rates across different time periods is constant. For this purpose, we introduce a common volatility component that is specified as a spline function of time and scaled appropriately for each series.Based on a data-set that is analysed by others, we present empirical evidence where time-varying loadings and volatilities in the dynamic Nelson-Siegel framework lead to significant increases in model fit. Improvements in the forecasting of the term structure are also reported. Finally, we provide an illustration where the model is applied to an unbalanced dataset. It shows that missing data entries can be estimated accurately.

Dynamic functional data analysis with non-parametric state space models

“…For example, in modelling the yield curve the inclusion of an SV component captures the changes in the volatility of the yield curve in time. Latent factor models with SV applied to the modelling of yield curves can be found in [17,24].…”

“…Assuming that the decay parameter λ is constant this model can be estimated in a two-step procedure, the first step corresponding to estimating the equation for each day observed in the yield curve by ordinary least squares, collecting parameters β it , and the second stage involving modelling the dynamics of the latent factors by an autoregressive process. Several other extensions of this model have been proposed in the literature, such as models with more latent factors [43] and stochastic volatility [17,24,25]. The SV in the measurement error is defined as follows:…”

“…The database is composed of unsmoothed rates obtained by the procedure of Fama and Bliss [11], for US Treasury Bonds , with fixed maturities of 1, 3,6,9,12,15,18,21,24,30,36,48,60,72,84,96, 108 and 120 months, using monthly observations for the period 1985:01-000:12. Figure 2 shows a graph of this data set.…”

Dynamics of the term structure of interest rates and monetary policy: is monetary policy effective during zero interest rate policy?

“…Numerical optimization of the log-likelihood function (19) yields maximum-likelihood estimates of the hyper-parameters. The process to find the latent factors and consistent estimates of the hyper-parameters is recursive one.…”

“…The procedure is started by initiating the recursion using certain starting values for the hyper-parameters (θ 0 ) that enable the Kalman filter to obtain estimates of the latent factors (α 0 t ), conditional on the initial choice for the parameters. Subsequently, given (α 0 t ), the likelihood function (19) is maximized in the optimization step to obtain new estimates of the hyper-parameters, (θ 1 ), that yield a higher likelihood. These estimates are used in the Kalman filter again to obtain new estimates of latent factors, (α 1 t ) and the corresponding likelihood value and so on.…”