## THE BOND MARKETS: ANALYTICAL METHODOLOGY 6

A comparison of the pure-Gaussian model and the Poisson-Gaussian model reveals a sharp drop in Gaussian volatility (v) when jumps are introduced into a pure-Gaussian model. For example, in Table 3, the Gaussian volatility drops to one-third its prior level suggesting that jumps account for a substantial component of volatility.

Once again, the parameter 9 appears downward biased, but is actually so because of skewness from the jump. The unconditional mean of the interest rate under the discretized process is given by в + hfi = в + q{i/A, and computations using the values in Table 3 arrive at a value of 0.0557 or 5.57%, once again close to the mean value in Table 1.

In the Poisson-Gaussian model (Table 3) we find that q — 0.2162, which under our Bernoulli model is simply the probability of a jump on any day. Thus, we find that jumps occur once every five days over our sample period. In contrast., the ARCH-Poisson-Gaussian model provides a jump probability of only 0.1564, evidence of the fact that stochastic volatility will account for some of the jumps. We conclude by noting that pure-Gaussian models do not capture the features of the data. Moreover, Poisson-Gaussian and ARCH-Gaussian models as well fall short of the efficacy of the ARCH-Poisson-Gaussian model. This has implications that theoretical work be driven in the direction of a combined ARCH-Poisson-Gaussian model.

Observe that the coefficient of mean reversion drops from 2.88 to 0.85 when jumps are added to the diffusion model. This may imply that jumps provide a source of mean reversion. This happens when the skewness of the jump distribution depends on the level of the interest rate in such a way as to induce mean reversion i.e. there is a greater chance of a positive jump at low interest rate levels, and a higher chance of a negative jump at high interest rate levels. Thus, we may find that the jump size distribution is positively skewed at low levels of r and negatively skewed at high levels of r. This can be modelled by allowing the mean of the jump size to depend on the level of r. For example, we may use a specification such as fit = «о + ai(9 — rt). When cvi > 0, we obtain mean reversion of the short rate through the jump component of the process.

Table 4 reports the results of the time-varying mean reverting model when jumps inject mean reversion. The mean reversion in the process is now attributable to

both the drift term and the jump term. Since jump arrivals are uncertain, the rate of mean reversion is now time-varying, and the drift in the interest rate becomes stochastic. Ait-Sahalia [2] and Stanton [39] demonstrate that the drift term displays non-linear behavior, which may be partially explained if jumps inject ‘extra’ mean reversion at interest rates far away from the long run mean of the short rate. In fact these papers find that the mean reversion pull is far stronger when the interest rate lies outside the range 4%-17%, which is consistent with the phenomenon suggested here. We extend our empirical model to estimate the parameters (qo^i)- We estimated the Poisson-Gaussian and ARCH-Poisson-Gaussian model with time-varying jump means (Table 4). Table 4 can be compared with Table 3. Notice that the coefficient of mean reversion к is lower, as the mean reverting component has been redistributed partly to the jump component of the process. The T-statistic for a\ is significant for the jump model indicating that the mean of the jump process is time-varying. However, when an ARCH effect is added to the model, the time-varying drift coefficient becomes insignificant. The joint evidence of these two models appears to suggest that different specifications of the volatility and jump may result in a linear drift model. Wre explore this issue in greater detail in a later subsection.