THE BOND MARKETS: Estimation 5

We minimize the usual objective function (denoted H). In contrast to Chan. Karolyi, Longstaff and Sanders, we use the exact continuous time moments rather than moments from a discretization of the short rate process. The jump-diffusion model is also estimated. One of the difficulties with the method of moments is that some parameters are not identifiable separately from the others. In the case of this model, the first jump moment E[J\ enters only as a sum with в in the first moment. In addition, the second jump moment E[J‘2\ always enters as a sum with v2 in the second, third and fourth moments, and hence, is not separately identified. The values of these two variables are subsumed into в and v2 respectively. We relabel these parameters в’ = в + and v’2 = v2 + hE(J2). Also, we estimate the composites hE(J3), and hE{J4) since E(J3), E(J4) do not appear except as
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Estimation results in Table 12 indicate a better fit for the jump-diffusion model versus the pure diffusion model, though neither model offers a very good statistical match. The difference in the objective functions (tested by x2 statistics) between the two models is significant. Specializations of the GMM approach used here may be achieved by simply choosing varied distributions for the jump. We have chosen here to retain a general form for the jump distribution.

Going beyond the method of moments analysis, it is instructive to examine the empirical moments over different data intervals. While the estimation results in Table 12 dealt with data only at daily intervals, we can use the theoretical results in Section 2.3 over multiple intervals with a view to understand wrhether the jump model is a-priori justified. Define the time interval between observations in the data as T. From Section 2.3. the variance of the jump-diffusion process is:
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kurtosis decline monotonically as T increases. We will use this theoretical property shortly. best online payday loans

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As a first check we compute the moments of the conditional distribution of interest rates using the estimated parameters for the jump-diffusion model in Table 3. Given the estimated values for the jump distribution (/i, 72), we can compute the following values: E( J) = f.i,E(J2) = f.12 + 72, E( J3) = /г3 -f З/172, and E(J4) = 4- 6/i272 + З74. Since our data is daily, the horizon T is ^ = 3- 846 2 x 10-3, given the number of trading days in a year. In order to make a rough comparison, the moments of the change in interest rates (dr) in Table 1 will correspond to the computed moments at T — 1/262. In fact they correspond well. The values are (values from Table 1 are in brackets): standard deviation—0.0029 (0.0029), skew’ness=0.3553 (0.3950), and kurtosis=13.36 (19.86).