Table4.5 reports the logarithmic Bayes factors of the LRSV-SKT and LSRV-NCT models against the competing models.
Table 4.5: Logarithmic Bayes factors of the LRSV-SKT and LRSV-NCT models with against competing models evaluated in the TOPIX data set.
Returns distribution
LNCT LT L LNCT LT L
RV
TOPIX 2004–2007 TOPIX 2004-2011 RV1 26.56 32.09 39.86 13.49 18.64 30.61
- 5.53 13.30 - 5.15 17.12
RV5 14.66 20.37 30.22 27.00 33.18 37.58
- 5.71 15.56 - 6.18 10.58
BV1 13.48 23.33 41.25 33.49 39.83 51.93
- 9.85 27.77 - 6.34 18.44
TSRV5 19.11 33.37 44.47 17.01 29.85 40.22 - 14.26 25.36 - 12.84 23.21
We observe that in each period and RV estimator, the LRSV-SKT model ranks first, followed by the LRSV-NCT model and the LRSV-T model. In fact, the logarithmic Bayes factors for the SKT model against others and the LRSV-NCT model against both LRSV-T and LRSV models are very strongly in favor of the former model. It suggests that incorporating skewness and heavy-tailedness features into the error distribution in the returns is supported by the data. It is also consistent with the findings of Nakajima and Omori (2012) and Tsiotas (2012), who introduced similar assumptions into the error distribution in their LSV models.
fitted the TOPIX data, although the skewness parameter in some of model is not fully guaranteed by the 90% HPD interval.
This chapter extended the LRSV model by assuming generalized Student’s t-distributions for return errors. In the next chapter, the volatility process in these models are extended to non-linear function as inTsiotas(2009) by applying power transformations.
Appendices
4.A Summary of the Posterior Samples for the TOPIX 2004–2007 Data Set
Table 4.6: Summary of the posterior samples of the LRSV model for the TOPIX 2004–2007 data set.
Parameter Statistic Data
RV1HL RV5HL BV1HL TSRV5HL
ρ
Mean (SD) −0.413 (0.058) −0.409 (0.055) −0.321 (0.056) −0.392 (0.057) 90% HPD [−0.512,−0.319] [−0.503,−0.319] [−0.410,−0.227] [−0.486,−0.301]
IF (NSE) 16.1 (0.0026) 14.9 (0.0025) 12.1 (0.0021) 12.6 (0.0020) β0
Mean (SD) 0.025 (0.043) −0.035 (0.048) 0.040 (0.037) −0.049 (0.050) 90% HPD [−0.044,0.095] [−0.113,0.049] [−0.020,0.100] [−0.137,0.030]
IF (NSE) 36.5 (0.0026) 31.9 (0.0028) 38.7 (0.0022) 29.1 (0.0027) β1
Mean (SD) 0.847 (0.054) 0.979 (0.056) 0.841 (0.053) 1.006 (0.056) 90% HPD [0.757,0.932] [0.886,1.071] [0.750,0.923] [0.914,1.099]
IF (NSE) 67.4 (0.0045) 37.9 (0.0033) 51.0 (0.0036) 31.8 (0.0033) σy
Mean (SD) 0.277 (0.009) 0.362 (0.012) 0.233 (0.009) 0.373 (0.013) 90% HPD [0.262,0.294] [0.342,0.383] [0.218,0.248] [0.351,0.393]
IF (NSE) 10.8 (0.0003) 8.6 (0.0004) 12.6 (0.0003) 10.3 (0.0004) α
Mean (SD) −0.021 (0.134) −0.031 (0.133) −0.020 (0.132) −0.027 (0.129) 90% HPD [−0.239,0.194] [−0.252,0.182] [−0.230,0.199] [−0.237,0.184]
IF (NSE) 3.9 (0.0036) 3.5 (0.0035) 2.3 (0.0030) 3.3 (0.0026) φ
Mean (SD) 0.948 (0.010) 0.9427 (0.010) 0.949 (0.010) 0.940 (0.011) 90% HPD [0.932,0.966] [0.924,0.960] [0.933,0.966] [0.923,0.959]
IF (NSE) 6.3 (0.0002) 6.6 (0.0003) 4.9 (0.0002) 6.6 (0.0003) σh
Mean (SD) 0.205 (0.018) 0.226 (0.018) 0.200 (0.016) 0.230 (0.018) 90% HPD [0.176,0.237] [0.195,0.254] [0.173,0.229] [0.200,0.260]
IF (NSE) 69.3 (0.0015) 59.8 (0.0013) 45.4 (0.0010) 46.8 (0.0012)
Table 4.7: Summary of the posterior samples of the LRSV-T model for the TOPIX 2004–2007 data set.
Parameter Statistic Data
RV1HL RV5HL BV1HL TSRV5HL
ν
Mean (SD) 23.21 (4.72) 23.73 (4.54) 23.25 (4.98) 24.66 (5.49) 90% HPD [15.89,30.82] [15.68,30.64] [15.47,31.39] [16.14,33.75]
IF (NSE) 88.4 (0.4194) 67.2 (0.3617) 95.5 (0.4582) 99.9 (0.5256) ρ
Mean (SD) −0.417 (0.054) −0.417 (0.056) −0.330 (0.058) −0.402 (0.059) 90% HPD [−0.508,−0.331] [−0.510,−0.323] [−0.429,−0.236] [−0.500,−0.305]
IF (NSE) 11.9 (0.0022) 11.8 (0.0018) 13.9 (0.0023) 14.0 (0.0023) β0
Mean (SD) 0.092 (0.043) 0.040 (0.051) 0.112 (0.045) 0.026 (0.053) 90% HPD [0.019,0.161] [−0.041,0.122] [0.040,0.185] [−0.060,0.115]
IF (NSE) 46.6 (0.0023) 33.3 (0.0028) 53.7 (0.0033) 38.4 (0.0026) β1
Mean (SD) 0.844 (0.059) 0.990 (0.065) 0.830 (0.055) 1.005 (0.062) 90% HPD [0.746,0.938] [0.886,1.099] [0.744,0.923] [0.9036,1.1065]
IF (NSE) 78.4 (0.0050) 58.4 (0.0048) 55.0 (0.0038) 42.1 (0.0038) σy
Mean (SD) 0.278 (0.009) 0.364 (0.012) 0.234 (0.009) 0.375 (0.013) 90% HPD [0.262,0.293] [0.343,0.384] [0.218,0.248] [0.354,0.397]
IF (NSE) 7.6 (0.0002) 9.0 (0.0003) 13.7 (0.0003) 10.5 (0.0004) α
Mean (SD) −0.096 (0.133) −0.105 (0.127) −0.103 (0.138) −0.099 (0.130) 90% HPD [−0.321,0.115] [−0.301,0.114] [−0.336,0.114] [−0.305,0.119]
IF (NSE) 3.5 (0.0029) 3.8 (0.0027) 3.8 (0.0040) 4.5 (0.0027) φ
Mean (SD) 0.949 (0.009) 0.943 (0.010) 0.950 (0.010) 0.942 (0.010) 90% HPD [0.933,0.965] [0.926,0.960] [0.934,0.966] [0.924,0.960]
IF (NSE) 5.4 (0.0002) 7.1 (0.0003) 5.5 (0.0002) 7.3 (0.0003) σh
Mean (SD) 0.205 (0.018) 0.221 (0.019) 0.201 (0.017) 0.227 (0.019) 90% HPD [0.175,0.235] [0.188,0.253] [0.171,0.228] [0.194,0.259]
IF (NSE) 84.3 (0.0016) 70.0 (0.0015) 49.9 (0.0010) 64.1 (0.0015)
Table 4.8: Summary of the posterior samples of the LRSV-NCT model for the TOPIX 2004–2007 data set.
Parameter Statistic Data
RV1HL RV5HL BV1HL TSRV5HL
µ
Mean (SD) 0.070 (0.032) 0.074 (0.032) 0.072 (0.032) 0.074 (0.032) 90% HPD [0.016,0.122] [0.022,0.127] [0.021,0.127] [0.021,0.127]
IF (NSE) 1.6 (0.0004) 1.5 (0.0004) 1.4 (0.0004) 1.5 (0.0005) ν
Mean (SD) 23.58 (4.77) 23.50 (4.56) 22.66 (4.51) 23.77 (4.74) 90% HPD [16.18,32.11] [16.25,30.48] [15.61,30.07] [15.73,30.51]
IF (NSE) 85.1 (0.4224) 65.1 (0.3307) 93.8 (0.4124) 97.0 (0.4333) ρ
Mean (SD) −0.416 (0.056) −0.409 (0.057) −0.324 (0.057) −0.391 (0.058) 90% HPD [−0.513,−0.330] [−0.500,−0.310] [−0.422,−0.233] [−0.484,−0.288]
IF (NSE) 16.3 (0.0022) 17.6 (0.0023) 12.4 (0.0021) 13.3 (0.0021) β0
Mean (SD) 0.091 (0.043) 0.039 (0.050) 0.116 (0.044) 0.029 (0.048) 90% HPD [0.018,0.162] [−0.041,0.128] [0.044,0.190] [−0.049,0.112]
IF (NSE) 58.2 (0.0029) 39.3 (0.0028) 56.6 (0.0033) 30.4 (0.0025) β1
Mean (SD) 0.843 (0.053) 0.976 (0.062) 0.834 (0.056) 0.998 (0.064) 90% HPD [0.752,0.928] [0.872,1.077] [0.740,0.925] [0.891,1.102]
IF (NSE) 54.1 (0.0037) 42.6 (0.0038) 53.0 (0.0042) 42.0 (0.0038) σy
Mean (SD) 0.278 (0.009) 0.362 (0.012) 0.233 (0.008) 0.373 (0.013) 90% HPD [0.262,0.293] [0.341,0.382] [0.219,0.248] [0.350,0.393]
IF (NSE) 9.9 (0.0003) 14.0 (0.0004) 11.6 (0.0002) 11.0 (0.0004) α
Mean (SD) −0.213 (0.142) −0.221 (0.139) −0.198 (0.142) −0.213 (0.135) 90% HPD [−0.440,0.022] [−0.446,0.005] [−0.414,0.044] [−0.448,−0.003]
IF (NSE) 3.5 (0.0036) 3.8 (0.0031) 3.8 (0.0039) 3.1 (0.0028) φ
Mean (SD) 0.949 (0.009) 0.942 (0.010) 0.950 (0.010) 0.941 (0.011) 90% HPD [0.933,0.966] [0.925,0.960] [0.932,0.966] [0.922,0.959]
IF (NSE) 4.9 (0.0002) 9.2 (0.0003) 4.7 (0.0002) 7.8 (0.0003) σh
Mean (SD) 0.205 (0.016) 0.226 (0.020) 0.201 (0.017) 0.231 (0.020) 90% HPD [0.176,0.232] [0.193,0.257] [0.172,0.228] [0.198,0.265]
IF (NSE) 57.9 (0.0012) 43.8 (0.0011) 65.5 (0.0013) 58.6 (0.0015)
Table 4.9: Summary of the posterior samples of the LRSV-SKT model for the TOPIX 2004–2007 data set.
Parameter Statistic Data
RV1HL RV5HL BV1HL TSRV5HL
β
Mean (SD) −0.505 (0.353) −0.465 (0.369) −0.180 (0.309) −0.417 (0.350) 90% HPD [−1.115,0.046] [−1.100,0.104] [−0.703,0.320] [−0.992,0.177]
IF (NSE) 55.7 (0.0255) 86.1 (0.0320) 39.9 (0.0197) 47.6 (0.0219) ν
Mean (SD) 25.12 (5.45) 25.27 (4.91) 23.00 (4.37) 25.31 (5.04) 90% HPD [15.91,33.29] [17.68,33.05] [15.79,30.02] [17.10,33.32]
IF (NSE) 97.5 (0.5085) 75.1 (0.3961) 66.1 (0.3365) 83.9 (0.4363) ρ
Mean (SD) −0.439 (0.057) −0.432 (0.060) −0.343 (0.057) −0.423 (0.061) 90% HPD [−0.530,−0.342] [−0.530,−0.330] [−0.438,−0.248] [−0.526,−0.322]
IF (NSE) 18.5 (0.0022) 24.6 (0.0032) 12.3 (0.0024) 21.5 (0.0027) β0
Mean (SD) 0.107 (0.0499) 0.060 (0.0630) 0.120 (0.0462) 0.045 (0.0585) 90% HPD [0.028,0.189] [−0.044,0.160] [0.039,0.190] [−0.051,0.139]
IF (NSE) 78.6 (0.0040) 58.2 (0.0044) 64.0 (0.0036) 46.9 (0.0037) β1
Mean (SD) 0.848 (0.056) 0.993 (0.064) 0.853 (0.061) 1.027 (0.070) 90% HPD [0.756,0.941] [0.881,1.094] [0.748,0.948] [0.914,1.143]
IF (NSE) 67.9 (0.0043) 52.1 (0.0044) 64.3 (0.0047) 58.5 (0.0049) σy
Mean (SD) 0.278 (0.009) 0.364 (0.012) 0.235 (0.008) 0.377 (0.013) 90% HPD [0.262,0.293] [0.344,0.386] [0.220,0.249] [0.354,0.397]
IF (NSE) 9.2 (0.0002) 13.1 (0.0005) 10.4 (0.0003) 11.8 (0.0004) α
Mean (SD) −0.121 (0.134) −0.126 (0.134) −0.110 (0.137) −0.118 (0.131) 90% HPD [−0.344,0.091] [−0.351,0.082] [−0.339,0.106] [−0.339,0.087]
IF (NSE) 5.8 (0.0048) 6.6 (0.0042) 4.2 (0.0043) 5.6 (0.0040) φ
Mean (SD) 0.949 (0.009) 0.944 (0.010) 0.951 (0.010) 0.943 (0.010) 90% HPD [0.932,0.964] [0.926,0.961] [0.933,0.967] [0.927,0.962]
IF (NSE) 6.1 (0.0002) 9.5 (0.0003) 4.8 (0.0002) 7.9 (0.0003) σh
Mean (SD) 0.204 (0.018) 0.220 (0.019) 0.193 (0.017) 0.220 (0.019) 90% HPD [0.174,0.233] [0.185,0.251] [0.165,0.223] [0.188,0.253]
IF (NSE) 68.5 (0.0014) 60.2 (0.0015) 76.4 (0.0014) 69.7 (0.0015)
4.B Summary of the Posterior Samples for the TOPIX 2004–2011 Data Set
Table 4.10: Summary of the posterior samples of the LRSV model for the TOPIX 2004–2011 data set.
Parameter Statistic Data
RV1HL RV5HL BV1HL TSRV5HL
ρ
Mean (SD) −0.357 (0.040) −0.358 (0.040) −0.318 (0.037) −0.358 (0.040) 90% HPD [−0.425,−0.293] [−0.422,−0.290] [−0.382,−0.257] [−0.424,−0.290]
IF (NSE) 13.2 (0.0017) 11.4 (0.0012) 10.9 (0.0013) 11.8 (0.0013) β0
Mean (SD) 0.110 (0.031) 0.024 (0.033) 0.162 (0.030) 0.005 (0.034) 90% HPD [0.058,0.160] [−0.030,0.079] [0.112,0.211] [−0.050,0.061]
IF (NSE) 47.2 (0.0020) 25.4 (0.0016) 46.8 (0.0019) 30.4 (0.0018) β1
Mean (SD) 0.904 (0.033) 0.980 (0.037) 0.877 (0.030) 0.992 (0.036) 90% HPD [0.849,0.959] [0.914,1.040] [0.825,0.926] [0.930,1.048]
IF (NSE) 59.9 (0.0025) 56.7 (0.0029) 58.5 (0.0023) 48.1 (0.0025) σy
Mean (SD) 0.302 (0.008) 0.358 (0.009) 0.243 (0.007) 0.360 (0.009) 90% HPD [0.288,0.315] [0.342,0.372] [0.230,0.254] [0.345,0.376]
IF (NSE) 12.1 (0.0003) 9.9 (0.0002) 13.9 (0.0002) 10.2 (0.0003) α
Mean (SD) 0.257 (0.117) 0.251 (0.119) 0.290 (0.114) 0.260 (0.119) 90% HPD [0.066,0.448] [0.055,0.443] [0.096,0.471] [0.066,0.457]
IF (NSE) 1.8 (0.0020) 1.5 (0.0015) 1.8 (0.0022) 1.5 (0.0020) φ
Mean (SD) 0.955 (0.006) 0.955 (0.006) 0.954 (0.006) 0.955 (0.006) 90% HPD [0.944,0.966] [0.943,0.965] [0.942,0.965] [0.944,0.966]
IF (NSE) 5.1 (0.0001) 5.9 (0.0002) 3.6553 (0.0001) 4.8460 (0.0001) σh
Mean (SD) 0.233 (0.013) 0.238 (0.014) 0.229 (0.012) 0.237 (0.013) 90% HPD [0.211,0.256] [0.214,0.261] [0.209,0.249] [0.214,0.258]
IF (NSE) 81.6 (0.0011) 61.3 (0.0011) 50.3 (0.0008) 72.9 (0.0010)
Table 4.11: Summary of the posterior samples of the LRSV-T model for the TOPIX 2004–2011 data set.
Parameter Statistic Data
RV1HL RV5HL BV1HL TSRV5HL
ν
Mean (SD) 29.36 (4.86) 29.84 (4.77) 27.17 (4.09) 29.40 (4.63) 90% HPD [21.36,36.94] [22.02,37.58] [20.75,33.69] [21.78,36.85]
IF (NSE) 90.6 (0.4437) 95.5 (0.4486) 82.9 (0.3586) 76.6 (0.3832) ρ
Mean (SD) −0.365 (0.039) −0.368 (0.040) −0.323 (0.038) −0.363 (0.042) 90% HPD [−0.431,−0.300] [−0.434,−0.302] [−0.384,−0.258] [−0.429,−0.291]
IF (NSE) 16.1 (0.0016) 12.1 (0.0015) 12.2 (0.0013) 15.5 (0.0013) β0
Mean (SD) 0.162 (0.033) 0.072 (0.035) 0.222 (0.032) 0.056 (0.036) 90% HPD [0.107,0.216] [0.014,0.131] [0.168,0.277] [−0.006,0.114]
IF (NSE) 57.8 (0.0024) 44.1 (0.0020) 57.8 (0.0023) 39.6 (0.0022) β1
Mean (SD) 0.903 (0.037) 0.980 (0.037) 0.875 (0.038) 0.991 (0.038) 90% HPD [0.842,0.967] [0.916,1.038] [0.817,0.940] [0.932,1.058]
IF (NSE) 70.4 (0.0029) 38.8 (0.0022) 74.5 (0.0032) 41.0 (0.0026) σy
Mean (SD) 0.303 (0.007) 0.360 (0.009) 0.243 (0.007) 0.361 (0.009) 90% HPD [0.291,0.316] [0.3458,0.375] [0.230,0.256] [0.346,0.377]
IF (NSE) 12.5 (0.0002) 12.1 (0.0003) 15.0 (0.0002) 11.0 (0.0003) α
Mean (SD) 0.202 (0.118) 0.205 (0.121) 0.224 (0.115) 0.208 (0.120) 90% HPD [0.014,0.401] [0.009,0.405] [0.034,0.412] [0.009,0.400]
IF (NSE) 1.832 (0.002) 1.950 (0.002) 1.887 (0.002) 1.787 (0.002) φ
Mean (SD) 0.956 (0.006) 0.956 (0.006) 0.954 (0.006) 0.955 (0.006) 90% HPD [0.945,0.967] [0.945,0.967] [0.943,0.966] [0.944,0.966]
IF (NSE) 5.5 (0.0001) 5.9 (0.0002) 4.5 (0.0001) 5.9 (0.0002) σh
Mean (SD) 0.231 (0.014) 0.234 (0.014) 0.230 (0.014) 0.237 (0.014) 90% HPD [0.207,0.254] [0.210,0.258] [0.205,0.254] [0.212,0.260]
IF (NSE) 79.3 (0.0011) 77.1 (0.0011) 76.6 (0.0012) 61.9 (0.0011)
Table 4.12: Summary of the posterior samples of the LRSV-NCT model for the TOPIX 2004–2011 data set.
Parameter Statistic Data
RV1HL RV5HL BV1HL TSRV5HL
µ
Mean (SD) 0.026 (0.022) 0.027 (0.022) 0.026 (0.023) 0.028 (0.022) 90% HPD [−0.010,0.064] [−0.009,0.066] [−0.010,0.064] [−0.009,0.066]
IF (NSE) 1.4 (0.0002) 1.5 (0.0003) 1.3 (0.0002) 1.4 (0.0003) ν
Mean (SD) 28.35 (4.53) 30.12 (5.23) 28.19 (5.07) 29.15 (4.72) 90% HPD [20.82,35.47] [21.87,38.73] [19.89,35.94] [21.72,36.71]
IF (NSE) 96.2 (0.4337) 79.5 (0.4535) 95.8 (0.4768) 95.8 (0.4322) ρ
Mean (SD) −0.360 (0.040) −0.362 (0.041) −0.323 (0.039) −0.361 (0.041) 90% HPD [−0.427,−0.294] [−0.433,−0.296] [−0.386,−0.258] [−0.431,−0.292]
IF (NSE) 16.9 (0.0014) 14.5 (0.0014) 11.3 (0.0011) 13.4 (0.0018) β0
Mean (SD) 0.163 (0.032) 0.073 (0.036) 0.215 (0.031) 0.058 (0.035) 90% HPD [0.110,0.220] [0.016,0.133] [0.165,0.268] [0.001,0.118]
IF (NSE) 43.6 (0.0018) 30.4 (0.0021) 61.3 (0.0023) 42.3 (0.0023) β1
Mean (SD) 0.898 (0.035) 0.973 (0.035) 0.880 (0.034) 0.993 (0.038) 90% HPD [0.841,0.958] [0.914,1.030] [0.825,0.936] [0.929,1.056]
IF (NSE) 63.1 (0.0026) 62.2 (0.0026) 74.1 (0.0026) 53.4 (0.0028) σy
Mean (SD) 0.303 (0.008) 0.359 (0.009) 0.243 (0.007) 0.361 (0.009) 90% HPD [0.289,0.317] [0.343,0.374] [0.231,0.256] [0.346,0.377]
IF (NSE) 15.5 (0.0003) 13.4 (0.0003) 15.7 (0.0002) 12.9 (0.0004) α
Mean (SD) 0.152 (0.127) 0.153 (0.129) 0.186 (0.122) 0.154 (0.128) 90% HPD [−0.049,0.362] [−0.043,0.377] [−0.018,0.383] [−0.057,0.361]
IF (NSE) 1.9 (0.0023) 2.3 (0.0026) 2.5 (0.0031) 2.1 (0.0022) φ
Mean (SD) 0.956 (0.006) 0.955 (0.006) 0.954 (0.006) 0.955 (0.006) 90% HPD [0.945,0.967] [0.944,0.966] [0.943,0.965] [0.945,0.967]
IF (NSE) 6.6 (0.0002) 6.6 (0.0002) 4.5 (0.0001) 7.5 (0.0002) σh
Mean (SD) 0.233 (0.015) 0.238 (0.014) 0.228 (0.012) 0.235 (0.014) 90% HPD [0.207,0.257] [0.214,0.261] [0.207,0.249] [0.212,0.260]
IF (NSE) 63.5 (0.0011) 74.3 (0.0011) 56.7 (0.0009) 71.1 (0.0011)
Table 4.13: Summary of the posterior samples of the LRSV-SKT model for the TOPIX 2004–2011 data set.
Parameter Statistic Data
RV1HL RV5HL BV1HL TSRV5HL
β
Mean (SD) −0.509 (0.299) −0.456 (0.323) −0.211 (0.274) −0.406 (0.307) 90% HPD [−0.995,−0.007] [−0.999,0.059] [−0.648,0.249] [−0.931,0.081]
IF (NSE) 49.2 (0.0196) 88.2 (0.0277) 50.4 (0.0179) 32.9 (0.0154) ν
Mean (SD) 28.91 (4.78) 30.78 (5.04) 27.95 (4.93) 30.38 (4.77) 90% HPD [21.23,36.67] [22.68,39.05] [19.77,35.62] [22.37,38.00]
IF (NSE) 97.7 (0.4524) 82.5 (0.4361) 93.2 (0.4711) 85.3 (0.4260) ρ
Mean (SD) −0.382 (0.041) −0.381 (0.042) −0.331 (0.038) −0.374 (0.042) 90% HPD [−0.448,−0.314] [−0.450,−0.310] [−0.396,−0.268] [−0.442,−0.302]
IF (NSE) 16.3 (0.0015) 17.7 (0.0019) 10.4 (0.0013) 15.0 (0.0014) β0
Mean (SD) 0.184 (0.037) 0.091 (0.040) 0.221 (0.033) 0.071 (0.042) 90% HPD [0.123,0.247] [0.026,0.161] [0.166,0.275] [0.000,0.137]
IF (NSE) 56.0 (0.0025) 53.7 (0.0027) 90.0 (0.0029) 66.6 (0.0031) β1
Mean (SD) 0.908 (0.035) 0.977 (0.037) 0.881 (0.032) 1.003 (0.039) 90% HPD [0.848,0.964] [0.916,1.035] [0.830,0.937] [0.942,1.069]
IF (NSE) 50.4 (0.0024) 48.1 (0.0022) 56.7 (0.0023) 45.5 (0.0025) σy
Mean (SD) 0.304 (0.008) 0.361 (0.008) 0.244 (0.007) 0.362 (0.009) 90% HPD [0.290,0.317] [0.347,0.376] [0.231,0.257] [0.347,0.378]
IF (NSE) 12.2 (0.0002) 10.8 (0.0002) 15.4 (0.0003) 11.2 (0.0003) α
Mean (SD) 0.170 (0.118) 0.181 (0.124) 0.219 (0.117) 0.189 (0.120) 90% HPD [−0.020,0.369] [−0.028,0.374] [0.027,0.410] [0.000,0.393]
IF (NSE) 2.3 (0.0029) 2.4 (0.0031) 2.2 (0.0034) 2.6 (0.0034) φ
Mean (SD) 0.956 (0.006) 0.956 (0.006) 0.954 (0.006) 0.956 (0.006) 90% HPD [0.945,0.967] [0.945,0.967] [0.944,0.966] [0.944,0.966]
IF (NSE) 5.8 (0.0001) 5.3 (0.0001) 4.6 (0.0001) 5.8 (0.0002) σh
Mean (SD) 0.228 (0.013) 0.235 (0.014) 0.227 (0.012) 0.232 (0.014) 90% HPD [0.205,0.251] [0.211,0.258] [0.206,0.248] [0.208,0.255]
IF (NSE) 50.3 (0.0009) 53.8 (0.0009) 54.9 (0.0008) 69.9 (0.0011)
Power Transformations for the LRSV Models
This chapter extends the LRSV model with generalized Student’st-distribution by applying three families of power transformations to lagged log volatility. The model will be analysed using the same data as in the previous chapter.
5.1 Non-linearities in SV Models
In the context of SV models, Yu et al. (2006),X. Zhang and King (2008), and Tsiotas (2009) proposed new classes of non-linear version of SV (NSV) models based on the Box–Cox and Yeo–Johnson transformations using a different setup.
Yu et al.(2006) andX. Zhang and King(2008) transformed volatility by following a pure autoregressive process, and Tsiotas (2009) applied a transformation only to the lagged log volatility. Their empirical results provide evidence that the NSV model is a better model fit than the log-normal SV model.
Following Tsiotas’ (2009) setup, in this chapter, we investigate the usability of three modified power transformation families (exponential, modulus, and Yeo–
Johnson) for transformation of lagged volatility in the LRSV models. These trans-formations are indexed by the parameter λ and were selected on the bases of the fact that these families permit transformed data to be non-positive and contain a λ value that does not correspond to transformation because the main idea is to transform log volatility ht. Notice that in the case of Yeo–Johnson transfor-mation (Yeo and Johnson, 2000), interpretation of the transformation parameter is difficult as it has a different function for negative and non-negative values of transformed data.
As explained in literature, it is important to sample the latent log volatility ht in an efficient manner. In the context of NSV modelling, Yu et al. (2006) and
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X. Zhang and King (2008) employed the single-move random-walk Metropolis–
Hastings algorithm within the MCMC algorithm that may encounter a slow con-vergence. Tsiotas(2009) applied the Metropolis–Hastings algorithm, where a can-didate density is generated from an extended Kalman filter with two approxima-tions: the first around log volatility and the second around the transformation parameter value not equal to zero. Considering a different algorithm, our MCMC simulation employs the HMC sampler for updating the entire log volatility at once and for updating the transformation parameter. To update the other parame-ters that cannot be sampled directly, the implementation is able to sample these parameters using the RMHMC sampler as discussed in the previous chapter.