THE BOND MARKETS: ANALYTICAL METHODOLOGY 4 ESTIMATION

The analytics from Section 2 are applied to daily data on the Fed funds rate for the period January 1988 to December 1997. The total number of observations is 2609. The data is from the Federal Reserve web site and is plotted in Figure 1. The descriptive statistics for the data are in Table 1. An examination of the data reveals that changes in interest rates evidence a very high degree of kurtosis. a stylized fact that predicates the use of a jump model. О л-or the entire 10 year period, rates have quickly risen to a peak of 10% and then fallen to a low of 3%, finally stabilizing at a 6% level. Our estimation exercise uses (i) continuous time estimators, (ii) discrete approximation estimators, and (iii) method of moments techniques. Our jump-diffusion The following table presents descriptive statistics for the Fed Funds rate over the period January 1988 to December 1997. The data is daily in frequency. The statistics reported are for the interest rate level (r) and the change in interest rates (dr).

This table presents the results of the continuous-time jump-diffusion model. Estimation is undertaken using maximum-likelihood where the transition density function is obtain by numerical Fourier inversion at each point, in time. Optimization of the likelihood function is then undertaken numerically over the numerically obtained probability density. The parameter estimates for the continuous-time model are reoorted below.

Models are extended for ARCH effects. They allow for mean-reversion in jump processes, and also test for the impact of Federal Reserve actions and day-of-the-week effects.
Continuous-Time Estimation. The process was estimated using continuous time transition density functions for the jump-diffusion process. The log-likelihood function in equation 2.4 is used for the estimation. Since the Vasicek model is nested within the jump-diffusion framework of this paper, a comparison of log-likelihoods reveals the improvement, in fit obtained by adopting a jump-diffusion model. The results are provided in Table 2. The model finds a large number of jumps in the data, seen in parameter h.

The average size of each jump is given by i = 1/365.62, i.e. 27 basis points. From the descriptive statistics it is known that mild positive skewness exists, evidenced by the parameter ф = 0.54. Therefore, the results (i) confirm that the jump parameters are statistically significant, and (ii) that the jump comprises a reasonable component of the stochastic variation in interest rates.  