THE BOND MARKETS: ANALYTICAL METHODOLOGY 5

Estimation using Discrete-Time Approximations. Estimation using the continuous-time method of the previous section is an exceedingly intensive numerical process. It requires numerical optimization over a density function that is itself obtained by numerical Fourier inversion. In this section a simpler discrete-timc approach allows us to estimate a model where the jumps are normally distributed.

We estimate the Poisson-Gaussian interest rate model using a Bernoulli approximation first introduced in Ball &; Torous . The assumption being made here is that in each time interval either only one jump occurs or no jump occurs. This is tenable for short frequency data, and may be debatable for data at longer frequencies. Since we use daily data, this approximation is justifiable. As Ball and Torous found, it makes the estimation procedure highly tractable, stable and convergent. Since the limit of the Bernoulli process is governed by a Poisson distribution, we can approximate the likelihood function for the Poisson-Gaussian model using a Bernoulli mixture of the normal distributions governing the diffusion and jump shocks.6 In discrete time, we express the process in equation (2.1) as follows: We present results for the estimation of pure-Gaussian, Poisson-Gaussian, ARCH-Poisson-Gaussian and ARCH-Gaussian processes on daily data covering the period January 1988 to December 1997. The total number of observations is 2609. Estimation is carried out using maximum-likelihood incorporating the transition density function in equation (3.3). The discretized ARCH-Poisson-Gaussian Drocess estimated is specified as follows Maximum likelihood estimation results are presented in Table 3. In order to compare different processes for the short rate, we estimated four nested models on the data set. Using data from different sampling frequencies enables us to examine whether the stochastic process used is sensitive to this criterion. The models estimated are (i) a pur^Gaussian model (h — 0), (ii) the Poisson-Gaussian model of equation (2.1), (iii) an ARCH-Poisson-Gaussian model, which consists of the Poisson-Gaussian model with the variance of the Gaussian component following an ARCH(l) process,7 and (iv) a pure ARCH-Gaussian model. This parallels to a large extent the analyses carried out by Jorion  for the equity and foreign exchange markets.

Since the ARCH-Poisson-Gaussian model subsumes the other three models, likelihood ratio tests may be applied to compare nested models. Comparison of nested log-likelihoods via a x2 statistic with degrees of freedom equal to the difference in the number of parameters between two models reveals in Tables 3 that the ARCH-Poisson-Gaussian model outperforms the rest. The Poisson-Gaussian process fits the data significantly better than the pure-Gaussian one. Whereas the Poisson-Gaussian and ARCH-Gaussian models are not nested, the likelihood for the Poisson-Gaussian model is greater, suggesting that Poisson-Gaussian processes provide a better fit than ARCH volatility models.