Instantaneous Error Term and Yield Curve Estimation 1

There exist many variations in yield curve modeling based on the Nelson and Siegel’s (1987) exponential components framework, but most of them do not care about the generating process of the error term. We often estimate parameters of the Nelson and Siegel’s model using an estimation equation that consists of the spot rate model with an independently and identically distributed error term as the traditional method. Usually, nonlinear least squares estimation or maximum likelihood estimation is applied to the model. In this specification, the error term may not be only heteroskedastic but also serially correlated because true covariance matrix of the error term is unknown. In this paper, to account for the possibility of misspecification, we propose a simple and natural method of yield curve estimation using an instantaneous error term which is generated by a standard Brownian motion process. First, we add the instantaneous error term to the Nelson and Siegel’s instantaneous forward rate model. Second, differencing the multiperiod spot rate models transformed from the Nelson and Siegel’s instantaneous forward rate model, we obtain a model with serially uncorrelated error terms because they have a property of the independent increment. In our specification, the error term is not serially correlated but heteroskedastic, so we apply weighted least squares estimation to the data. That is, this specification about the error term does not lead to incorrect estimation methods. We compare the estimated parameters and the shapes of the instantaneous forward rate and yield curves using our specification of the error term and the traditional method in empirical analysis. We use the Japanese Yen Tokyo interbank offered rates (TIBOR) as the spot rate. Some empirical examples in almost periods show that estimated parameters using the model proposed in this paper are similar to those from traditional method. And the probabilities of not rejecting the null hypothesis that the parameter is equal to zero in the proposed modeling are totally larger than those from the traditional estimation. However, we find that the estimates of the Nelson and Siegel’s model are possibly much different when Table 1. An example of different estimated parameters. β0 β1 β2 τ Our estimation method 2000/07/03 0.492 −0.371 −0.053 0.297 (11.71) (-6.60) (-0.09) (0.61) Traditional estimation method 2000/07/03 0.436 −0.361 0.000 0.200 (25.65) (-24.01) (0.01) (2.63) Note: β0, β1, β2 and τ are parameters of the Nelson and Siegel’s model defined later to be estimated from the interest rate data. The values in the parentheses represent estimates of t-statistics. 0.0 10.0 0.10 0.20 0.30 0.40 Percent Month−to−settlement July 3, 2000 our method traditonal actual Fitted yield curve 0.0 10.0 0.00 0.20 0.40 Percent Month−to−settlement July 3, 2000 our method traditional Fitted foward rate curve Figure 1. Fitted yield and forward rate curves by our estimation method and traditional estimation method. the fluctuations of TIBOR used in the estimation are relatively volatile. Table 1 represents the estimated parameters and Figure 1 plots the yield and forward rate curves in the sample period. We find that the fitted yield curve of our method is better/worse than that of traditional method when the number of months to settlement is long/short. From these results, we find that the shapes of the instantaneous forward rate curves change depending on different specification of the error term such as our specification and the other when the fluctuations of the interest rate are volatile. In other words, it is important to consider the specification of the error term carefully when we investigate the statistical properties of the yield curves or instantaneous forward rate curves.