3.5 The Estimation Results
3.5.1 Estimation Results for the Four-Factor Continuous-Time Models
For the first stage twenty-four four-factor models are estimated, four models for each of the five LIBOR curves and another four models for the UK spot curve. The QMLE estimates of the parameters are grouped in the vector solution to the respective optimization problem of maximizing the objective function given in equation (3.25) and are presented together with their standard errors entered in the next column in Tables 3.20-3.25. The vector parameter to be estimated has thirty-four components under the general model CKLS and thirty under any of the restricted models. Given the high dimension of the vector of parameters, each table consists of two parts: part a) for the drift parameters and part b) for the diffusion parameters, respectively.
The estimation results are interpreted separately based on the money-market and bond market data sets, respectively. The parameters of interest are the vector of the level effects, the feedback matrix and the covariance matrix. For three out the five LIBOR time-series, namely GBP-LIBOR, USD-LIBOR and JPY-LIBOR, the level effect estimates are close to unity. This suggests a strong dependence of the volatility of the interest rate changes on the level of the interest rate itself. For the EUR-LIBOR and CAD-LIBOR multivariate time series the estimates regarding the elasticity of the volatility parameter are situated in the vicinity of 0.5. The restricted models are tested for their explanatory power against the general CKLS model using the likelihood ratio test (LR). Based on the corresponding
2(4df )
tests, under the null hypothesis of the validity of the nested model, the results indicate rejection at the 1% level of significance of all of the restricted models. According to the values of log-maximum likelihood functions, the best performing restricted specifications are: the BS model for GBP-LIBOR, USD-LIBOR and JPY-LIBOR rates, while the CIR model for EUR-LIBOR and CAD-LIBOR rates. The drift function of the models proposed is defined by the four-dimensional intercept vector
and the feedback matrix of sixteen components. The121 estimates of all intercept elements are almost zero, most of them being however statistically significant.
The majority of the feedback estimates are significant indicating evidence of feedback in most directions. For the GBP-LIBOR time series, the matrix can be re-specified by assuming
13
31 0
while all the other elements of the feedback matrix are significantly different from zero; in the case of the USD-LIBOR time series there is no feedback evidence from the six-month LIBOR rate to the twelve-month USD-LIBOR rate in either direction as
34
43 0
, both being statistically insignificant. For the EUR-LIBOR time series some elements can be considered zero:11 12 43 44
0
; for the JPY-LIBOR rates the inference suggests that23 24 41 42
0
; for the CAD-LIBOR rates only one feedback coefficient is insignificant
32 0
implying that there is no feedback from the one-month rate to the six- month rate. Finally, for the U.K. spot rates the estimation results for the feedback matrix imply
12
24
34
41
44 0
. When analysed in comparison with the corresponding best restricted models, there is always a higher degree of significant elements in the feedback matrix in the more general CKLS model. Hence, we can argue that the increased flexibility provided by the CKLS specification by not restricting the elasticity of the variance parameter, may render higher degree of significance in the feedback matrix reflecting a stronger correlation among the factors explicitly modelled via a more complex deterministic drift.The estimates for the correlation coefficients are all positive under the CKLS model for all LIBOR currencies. A ranking in terms of the degree of correlation among the factors can be observed across all the LIBOR data sets. The estimation results for the correlation coefficients indicate that the six-month and twelve-month rates are most highly correlated with the value of the correlation coefficient
34 between 0.75 (JPY-LIBOR) and 0.98 (USD-(JPY-LIBOR). The other pairs of highly correlated short-term interest rates are for the maturities of one-month with six-month and one-week with one-month.In the case of the bond market data, the estimation results are rather different, with a much lower level effect estimates and another correlation structure. As can be seen in the Table 3.25, the components of vector are estimated within the range (0.000004, 0.22), suggesting a much weaker sensitivity of the conditional variance with respect to the level
122 of interest rate; and only
10.22
is statistically different from zero. The Vasicek model supports the data best with the highest value of the restricted log-likelihood functions LogLF 105,776.12. All the restricted models are rejected against the unrestricted CKLS model. Regarding the drift components, under the CKLS, the intercept estimates are very close to zero, whereas the feedback matrix has only five elements statistically insignificant and they degenerate to zero(
12
24
34
41
44 0)
. As expected, the correlation coefficients are higher between the spot rates corresponding to the flatter end of the term structure with
34 0.95,
23 0.94
and
24 0.82
.Table 3.20a) LIBOR-GBP, The Drift Coefficients Estimates, Four-Factor Models
Param. CKLS SE Vasicek SE TR CIR SE BR&SC SE
Alpha1 0 0 -0.00066 0.00011 -0.00007 0 0.00004 0
Alpha2 0.00003 0 -0.00011 0.00002 -0.0001 0 0 0
Alpha3 0.00003 0 -0.00003 0 -0.00011 0 -0.00002 0
Alpha4 0.00005 0 0.00003 0 -0.00008 0 -0.00004 0
B11 0.01438 0.00496 -0.0175 0.00948 -0.19935 0 0.05997 0.0067
B12 -0.01732 0.00644 -0.00322 0.0162 0.26642 0 -0.04241 0.00873
B13 -0.00395 0.00312 -0.08595 0.03195 -0.14338 0 -0.04315 0.00531
B14 0.00439 0.00148 0.11546 0.02602 0.07316 0 0.02383 0.00266
B21 0.02164 0.00352 0.03063 0.00207 0.01596 0.00229 0.03622 0.00212
B22 -0.01966 0.00457 -0.05358 0.00193 -0.00678 0.00308 -0.03477 0.00268 B23 0.00147 0.00215 0.03197 0.00445 -0.03782 0.00261 -0.00769 0.00149
B24 -0.00399 0.00095 -0.00712 0.00377 0.0301 0.00131 0.00594 0.00072
B31 -0.00293 0.0007 0.00492 0.00165 0.01713 0.0017 0.00726 0.0017
B32 0.01334 0.00061 -0.00347 0.00132 -0.00426 0.00224 -0.00761 0.00221 B33 -0.0094 0.00135 0.00027 0.00187 -0.04474 0.00215 -0.00342 0.00155 B34 -0.00157 0.00074 -0.00125 0.00151 0.03364 0.00107 0.00356 0.00075
B41 -0.01358 0.00133 0.0022 0.00213 0.01106 0.00209 -0.0168 0.0022
B42 0.0235 0.00134 -0.00726 0.00202 0.00295 0.0027 0.01639 0.00294
B43 -0.00556 0.0014 0.02532 0.00195 -0.04132 0.00216 -0.0051 0.00202
B44 -0.00528 0.00078 -0.02085 0.00133 0.02843 0.00083 0.00528 0.00087
Table 3.20b) LIBOR-GBP, The Diffusion Coefficients Estimates, Four-Factor Models
Param. CKLS SE Vasicek SE CIR SE BR&SC SE TR
Gamma1 1.59404 0.00696 0 N/A 0.5 N/A 1 N/A
Gamma2 1.22368 0.01377 0 N/A 0.5 N/A 1 N/A
Gamma3 1.03083 0.00288 0 N/A 0.5 N/A 1 N/A
Gamma4 1.39513 0.00476 0 N/A 0.5 N/A 1 N/A
123 Corr24 0.66319 0.00845 0.60368 0.00741 0.61929 0.00695 0.26443 0.01247 Corr34 0.93308 0.00145 0.92748 0.00139 0.92822 0.00135 0.87946 0.00238 LogLF 112,577.66 N/A 105,903.42 13,348.48+ 109,627.13 5,901.06+ 110,947.21 3,260.91+
Note The cells marked with * contain values smaller than 106; the cells marked with + contain the corresponding values of the LR test statistics (
crit2 (4 ,1%) 13.28df ).Table 3.21a) LIBOR-USD, The Drift Coefficients Estimates, Four-Factor Models
Param. CKLS SE VASICEK SE CIR SE BS SE Beta14 0.00443 0.00153 0.01642 0.00879 0.00546 0.00046 -0.00209 0.00043 Beta21 0.0228 0.00291 0.07734 0.00215 0.09286 0.00259 0.08553 0.00255 Beta33 -0.01592 0.00359 0.00812 0.00246 -0.01049 0.00205 0.00103 0.00132 Beta34 0.00134 0.00291 -0.00162 0.0019 0.00875 0.00101 -0.00379 0.00058 Beta41 -0.01595 0.00314 0.00829 0.00307 0.01634 0.00274 -0.01586 0.00514 Beta42 0.03148 0.00237 -0.01322 0.00267 -0.01401 0.00407 0.0183 0.00593 Beta43 -0.0006 0.00434 0.01727 0.00252 -0.00435 0.00338 0.00749 0.0026 Beta44 -0.01558 0.00348 -0.01304 0.00174 0.0012 0.00186 -0.01004 0.00112
Table 3.21b) LIBOR-USD, The Diffusion Coefficients Estimates, Four-Factor Models
Param. CKLS SE VASICE
124
Sigma1 0.01945 0.00047 0.00052 0.01421 0.00289 0.00403 0.02139 0.00605 Sigma2 0.007 0.00019 0.00031 0.00897 0.00187 0.00765 0.01461 0.00667 Sigma3 0.01557 0.00016 0.00031 0.00856 0.00183 0.00834 0.01223 0.00819 Sigma4 0.01704 0.00026 0.00043 0.00832 0.00252 0.00819 0.01718 0.00843 Corr12 0.55108 0.00999 0.57393 0.00974 0.55132 0.00746 0.48312 0.00855 Corr13 0.24234 0.01213 0.32574 0.01286 0.3246 0.01043 0.13464 0.01244 Corr14 0.19591 0.01238 0.20007 0.01293 0.19688 0.01143 -0.03379 0.01301 Corr23 0.5616 0.0087 0.68931 0.0061 0.66626 0.00629 0.4416 0.00999 Corr24 0.49847 0.00953 0.47818 0.00912 0.46016 0.0092 0.21256 0.01208 Corr34 0.98207 0.00037 0.91024 0.00183 0.90904 0.00173 0.89437 0.00211 Log LF 112,669.38 N/A 107,366.63 10,605.51+ 111,026.43 3,285.90+ 111,329.85 2,679.06+
Note: The cells marked with * contain values smaller than 106; the cells marked with + contain the corresponding values of the LR test statistics (
crit2 (4 ,1%) 13.28df ).Table 3.22a) LIBOR-EUR, The Drift Coefficients Estimates, Four-Factor Models
Param. CKLS S.E. VASICEK S.E. CIR S.E. BS S.E.
Alpha1 0 0* -0.00001 0.00002 0.00006 0* 0 0*
Alpha2 -0.00003 0* -0.00003 0.00001 0.00004 0.000001 0.00004 0*
Alpha3 -0.00003 0* 0 0 0.00002 0.000002 0 0*
Alpha4 -0.00002 0* 0.00007 0 0.00006 0.000004 0.00001 0
Beta11 0.00559 0.00406 -0.0032 0.00792 -0.00405 0.00364 0.01012 0.00293 Beta12 -0.00415 0.00459 0.00322 0.00819 -0.00369 0.0046 -0.00432 0.00314 Beta13 -0.0045 0.00194 0.00316 0.008 0.03634 0.00267 -0.01373 0.00099 Beta14 0.00236 0.00103 -0.00313 0.00548 -0.03042 0.0012 0.00767 0.00049 Beta21 0.02631 0.00316 0.04727 0.00267 0.02565 0.00229 0.04537 0.00218 Beta22 -0.0228 0.0035 -0.06318 0.00286 -0.03559 0.00283 -0.0525 0.00243 Beta23 -0.02009 0.00146 0.02516 0.0032 0.03275 0.00167 0.02346 0.00085 Beta24 0.01709 0.00082 -0.00825 0.00268 -0.02374 0.00089 -0.01696 0.00043 Beta31 0.01791 0.00115 0.00533 0.00137 0.00417 0.00206 0.00036 0.002 Beta32 -0.01345 0.00199 -0.00083 0.00233 -0.00016 0.00264 0.00176 0.00248 Beta33 -0.01827 0.00177 -0.00377 0.00285 0.00006 0.00179 -0.00016 0.00142 Beta34 0.01454 0.00092 -0.00057 0.00176 -0.00441 0.001 -0.00199 0.00075 Beta41 0.01306 0.0021 -0.00281 0.00235 0.00156 0.00309 -0.01864 0.00311 Beta42 -0.01182 0.00312 0.00315 0.00408 -0.00176 0.00398 0.01808 0.00404 Beta43 -0.00301 0.00249 0.02722 0.00433 0.02561 0.00272 0.01339 0.00277 Beta44 0.0021 0.00133 -0.02871 0.0023 -0.02672 0.00153 -0.01327 0.00146
Table 3.22b) LIBOR-EUR, The Diffusion Coefficients Estimates, Four-Factor Models
Param. CKLS S.E. VASICEK S.E. CIR S.E. BS S.E.
Gamma1 0.69498 0.007182 0 N/A 0.5 N/A 1 N/A
Gamma2 0.66565 0.015356 0 N/A 0.5 N/A 1 N/A
Gamma3 0.75755 0.001991 0 N/A 0.5 N/A 1 N/A
Gamma4 1.04215 0.015731 0 N/A 0.5 N/A 1 N/A
125
Sigma1 0.00526 0.01721 0.00037 0.02025 0.00236 0.00593 0.02429 0.00906 Sigma2 0.00249 0.02845 0.00023 0.00738 0.00132 0.00754 0.01214 0.00735 Sigma3 0.00257 0.00992 0.00018 0.00846 0.00103 0.00843 0.00675 0.00882 Sigma4 0.0098 0.02541 0.00028 0.00767 0.00148 0.00821 0.00934 0.01032 Corr12 0.6222 0.00803 0.4761 0.0149 0.58779 0.01107 0.67471 0.01204 Corr13 0.38564 0.01021 0.2897 0.012 0.37323 0.01032 -0.05124 0.01421 Corr14 0.24264 0.01133 0.1513 0.01233 0.23817 0.01144 -0.24354 0.01366 Corr23 0.61826 0.00776 0.5441 0.00877 0.61991 0.00712 0.26422 0.01395 Corr24 0.40621 0.01007 0.3471 0.01127 0.42326 0.00981 -0.00937 0.01477 Corr34 0.86406 0.00273 0.8584 0.00273 0.86823 0.00257 0.84559 0.00333 Log LF 115,424.04 N/A 111,188.22 8,471.63+ 114,586.98 1,674.13+ 112,955.44 4,937.20+
Note: The cells marked with * contain values smaller than 106; the cells marked with + contain the corresponding values of the LR test statistics (
crit2 (4 ,1%) 13.28df ).Table 3.23a) LIBOR-JPY, The Diffusion Coefficients Estimates, Four-Factor Models
Param. CKLS S.E. VASICEK S.E. CIR S.E. BS S.E.
Alpha1 -0.00001 0* 0* 0* 0* 0* 0 0*
Alpha2 -0.00001 0* 0* 0* 0 0* 0* 0*
Alpha3 0* 0* 0* 0* 0 0* -0.00001 0*
Alpha4 0 0* 0* 0* 0* 0* 0 0*
Beta11 0.04638 0.00519 -0.0493 0.0082 -0.10204 0.00568 0.00606 0.00579 Beta12 -0.03782 0.00502 -0.0018 0.0104 0.04321 0.00706 -0.00905 0.00569 Beta13 0.01662 0.00346 0.0746 0.0135 0.04654 0.00605 -0.02784 0.00397 Beta14 -0.00954 0.00183 -0.0414 0.0082 -0.02002 0.003 0.01993 0.00215 Beta21 0.05453 0.00473 0.0004 0.0036 0.02624 0.0038 0.08036 0.00446 Beta22 -0.03377 0.0048 -0.0299 0.0043 -0.05356 0.00447 -0.09728 0.00477 Beta23 -0.00067 0.00269 0.0448 0.0049 0.02751 0.00381 0.01564 0.00286 Beta24 -0.00209 0.00141 -0.0237 0.0025 -0.00847 0.00205 -0.0012 0.0015 Beta31 -0.00821 0.00189 0.0117 0.0012 0.00484 0.00145 0.00114 0.00213 Beta32 0.00977 0.00216 -0.0099 0.0015 -0.0051 0.00176 0.00714 0.00254 Beta33 0.00772 0.00237 0.0008 0.0018 0.0005 0.0019 0.00001 0.00245 Beta34 -0.0076 0.00129 -0.0022 0.001 -0.0002 0.00105 -0.00055 0.00128 Beta41 0.00169 0.00146 0.0032 0.0011 0.00046 0.00132 -0.01146 0.00176 Beta42 0.00154 0.00177 -0.0069 0.0014 -0.00314 0.00163 0.01221 0.00215 Beta43 -0.00669 0.00229 0.011 0.0017 0.00797 0.00189 0.0041 0.00245 Beta44 0.00321 0.0013 -0.0087 0.001 -0.00522 0.00107 -0.00282 0.00136
Table 3.23b) LIBOR-JPY, The Diffusion Coefficients Estimates, Four-Factor Models
Param. CKLS S.E. VASICEK S.E. CIR S.E. BS S.E.
Gamma1 1.30589 0.00614 0 N/A 0.5 N/A 1 N/A
Gamma2 1.20374 0.00643 0 N/A 0.5 N/A 1 N/A
Gamma3 0.87365 0.00817 0 N/A 0.5 N/A 1 N/A
Gamma4 0.80425 0.00698 0 N/A 0.5 N/A 1 N/A
126
Sigma1 0.54749 0.02519 0.0004 0 0.0047 0.00614 0.07925 0.00064
Sigma2 0.1552 0.00726 0.0002 0 0.00232 0.00643 0.04243 0.0003
Sigma3 0.00807 0.00044 0.0001 0 0.00085 0.00817 0.0181 0.00014
Sigma4 0.00413 0.00018 0.0001 0 0.00075 0.00698 0.01383 0.00012
Corr12 0.57337 0.00796 0.4966 0.00859 0.55043 0 0.60882 0.00773
Corr13 0.21099 0.01198 0.2129 0.01174 0.29174 0.00002 -0.04755 0.01362 Corr14 0.12127 0.01215 0.1209 0.01212 0.19937 0.00001 -0.16739 0.0134 Corr23 0.42502 0.01021 0.4625 0.00943 0.52402 0.00001 0.13845 0.01422 Corr24 0.34108 0.01088 0.3368 0.01082 0.42389 0.0062 -0.01224 0.01425 Corr34 0.75093 0.00511 0.8379 0.00292 0.81611 0.01093 0.72776 0.00532 LogLF 131,915.25 N/A 121,197.90 21,434.71+ 128,664.45 6,501.60+ 130,785.48 2,259.54+
Note: The cells marked with * contain values smaller than 106; the cells marked with + contain the corresponding values of the LR test statistics (
crit2 (4 ,1%) 13.28df ).Table 3.24a) LIBOR-CAD, The Drift Coefficients Estimates, Four-Factor Models
Param. CKLS S.E. VASICEK S.E. CIR S.E. BS S.E.
Alpha1 -0.00012 0 -0.00011 0.00001 0.00005 0* -0.00003 0*
Alpha2 -0.00012 0 -0.00011 0.00001 0 0 -0.00005 0*
Alpha3 -0.00013 0* 0.00003 0.00001 -0.00008 0 -0.00001 0*
Alpha4 -0.00007 0* 0.00007 0.00001 -0.00003 0* 0.00005 0
Beta11 -0.03696 0.01663 -0.01253 0.00405 -0.01063 0.00493 0.02391 0.00491 Beta12 0.04398 0.01811 0.00589 0.00431 0.00581 0.00554 -0.0225 0.00521 Beta13 -0.02876 0.00295 -0.00535 0.00359 0.01239 0.00304 -0.01054 0.00156 Beta14 0.02321 0.00141 0.01364 0.00173 -0.00945 0.00135 0.00878 0.00078 Beta21 0.0265 0.01615 0.0671 0.00378 0.05256 0.00428 0.07785 0.0046 Beta22 -0.02453 0.01676 -0.07678 0.00541 -0.06578 0.00479 -0.07882 0.00484 Beta23 -0.02337 0.00264 -0.00961 0.00458 0.01758 0.00237 -0.01193 0.00151 Beta24 0.02328 0.00131 0.02094 0.00201 -0.00493 0.00115 0.01378 0.00075 Beta31 0.01834 0.00179 0.01831 0.00403 -0.00009 0.00482 -0.00002 0.00481 Beta32 -0.00615 0.00349 -0.02359 0.00463 -0.00706 0.00548 -0.00034 0.00546 Beta33 -0.04321 0.00336 0.00603 0.00336 -0.00106 0.00238 -0.00198 0.00245 Beta34 0.03307 0.00151 -0.00185 0.0017 0.00906 0.00091 0.00181 0.00108 Beta41 0.02556 0.00367 0.00374 0.00527 -0.00008 0.00621 -0.00443 0.00636 Beta42 -0.01484 0.00547 -0.01401 0.00599 -0.00703 0.00725 -0.0031 0.00744 Beta43 -0.03409 0.0046 0.02427 0.00389 0.00307 0.00459 0.01927 0.00352 Beta44 0.02402 0.0021 -0.01607 0.00194 0.00364 0.00258 -0.01362 0.00138
Table 3.24b) LIBOR-CAD, The Diffusion Coefficients Estimates, Four-Factor Models
Param. CKLS S.E. VASICEK S.E. CIR S.E. BS S.E.
Gamma1 0.65518 0.01285 0 N/A 0.5 N/A 1 N/A
Gamma2 0.59215 0.02102 0 N/A 0.5 N/A 1 N/A
Gamma3 0.67015 0.00389 0 N/A 0.5 N/A 1 N/A
Gamma4 0.77973 0.01837 0 N/A 0.5 N/A 1 N/A
127
Sigma1 0.0038 0.00025 0.00035 0 0.00209 0.00001 0.01625 0.00011 Sigma2 0.00247 0.00023 0.00031 0 0.00174 0.00001 0.01366 0.00008 Sigma3 0.00306 0.00006 0.00029 0 0.00165 0.00001 0.0103 0.00009 Sigma4 0.00534 0.00036 0.00037 0 0.00201 0.00002 0.01189 0.0001 Corr12 0.59118 0.00832 0.48123 0.00836 0.56093 0.00747 0.59482 0.00714 Corr13 0.33732 0.01092 0.29445 0.01084 0.29695 0.01086 0.02271 0.01258 Corr14 0.23498 0.0118 0.21001 0.01151 0.19524 0.01157 -0.03793 0.01268 Corr23 0.51084 0.01025 0.4683 0.00893 0.49047 0.00866 0.30095 0.01161 Corr24 0.40887 0.01039 0.36166 0.01013 0.38595 0.01002 0.22973 0.01203 Corr34 0.87807 0.00227 0.88707 0.00202 0.88185 0.00203 0.87228 0.00226 Log LF 109,831.29 N/A 107,760.40 4,141.78+ 109,639.74 383.1+ 108,492.75 2,677.08 + Note: The cells marked with * contain values smaller than 106; the cells marked with + contain the corresponding values of the LR test statistics (
crit2 (4 ,1%) 13.28df ).
Table 3.25a) U.K. Spot Rates, The Drift Coefficients Estimates, Four-Factor Models
Param CKLS S.E. VASICEK S.E. CIR S.E. BS S.E.
Table 3.25b) U.K. Spot Rates, The Diffusion Coefficients Estimates, Four-Factor Models
Param CKLS S.E. VASICEK S.E. CIR S.E. BS S.E.
Gamma1 0.21811 0.04274 0 N/A 0.5 N/A 1 N/A
Gamma2 0.02441 0.34311 0 N/A 0.5 N/A 1 N/A
Gamma3 0* 28.65801 0 N/A 0.5 N/A 1 N/A
Gamma4 0.09501 0.26365 0 N/A 0.5 N/A 1 N/A
128
Sigma1 0.00083 0.03155 0.00037 0.00739 0.0026 0.0077 0.02843 0.01235 Sigma2 0.00055 0.0295 0.00049 0.00864 0.0029 0.0092 0.01762 0.01221 Sigma3 0.00048 0.00907 0.00046 0.00867 0.00241 0.0092 0.00914 0.00666 Sigma4 0.00061 0.07252 0.00044 0.00858 0.00221 0.009 0.01013 0.01081 Corr12 0.66703 0.00649 0.63917 0.0067 0.65498 0.0119 0.61379 0.01016 Corr13 0.51899 0.00891 0.47226 0.00927 0.54301 0.0123 0.29423 0.0106 Corr14 0.42898 0.0101 0.37866 0.0104 0.4678 0.0123 -0.09382 0.01903 Corr23 0.93557 0.00151 0.92943 0.00158 0.94274 0.0117 0.67929 0.00932 Corr24 0.82032 0.00385 0.80745 0.00396 0.83423 0.0116 0.07866 0.02137 Corr34 0.94505 0.00113 0.94214 0.00114 0.94757 0.0106 0.74792 0.00749 Log LF 105,776.12 N/A 105,661.29 229.66+ 104,941.82 1,668.60+ 100,376.30 10,799.62+
Note: The cells marked with * contain values smaller than 106; the cells marked with + contain the corresponding values of the LR test statistics (
crit2 (4 ,1%) 13.28df ).The estimates for the level effect parameters in the unrestricted CKLS model across all the data sets are also presented in Table 3.26 below. Within the money-market context there are similarities between the U.K and Japan on one side, and between the U.S., the Eurozone and Canada on the other side. For the U.K and Japan, the shortest maturity rates (one-week and one-month) exhibit the highest dependence of the volatility on the level of interest rates, while for the U.S, Japan and Canada this happens for the six- and twelve-month LIBOR rates. In the case of the U.K. spot rates, only the first component of the level effect parameter significant, indicating that from the restricted models then Vasicek specification may explain the dynamics of the data as close as the general CKLS model.
The level effect parameter for the 15-year U.K spot rates is 0.000004 suggesting a constant conditional volatility for the process of these time-series.
Table 3.26 The Estimates for the Level - Effect Parameter for Four-Factor CKLS models.
CKLS GBP-LIBOR USD-LIBOR EURLIBOR JPY-LIBOR CAD-LIBOR UK Spot Gamma1 1.5940388 0.9751338 0.6949800 1.3058910 0.6551755 0.218114 Gamma2 1.2236800 0.8091528 0.6656462 1.2037392 0.5921455 0.024406 Gamma3 1.0308315 0.9944530 0.7575482 0.8736540 0.6701456 0.000004 Gamma4 1.3951253 0.9880993 1.0421511 0.8042529 0.7797263 0.095013