D.B. Nugroho, T. Morimoto
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MA.49
measure of variance. The logarithmic transformation of RV is known to have better finite sample properties than RV. Goncalves and Meddahi 2011 provided a short overview of this
transformation and suggested that the log transformation can be improved upon by choosing values of BC parameter other than zero corresponding to the log transformation. Motivated
by this result, we apply the BC transformation to the RV measure to obtain a class of R-ASV model called the BCR-ASV model that takes the following form:
ü = exp -
› ”
ℎ .Y 7 =
¼
+ ℎ +
…
ℎ = m + † ℎ − m + ×
ℎ ~ •m,
Û
[ ”
o—
”
– N
K O
K P
, 1
where corr Y , ×
= , represents normal distribution, and 7 is the BC transformation for realized RV defined as
7 = 7 ü , G = ß
\ ]
o À
, if G ≠ 0, log ü
, if G = 0. 2
Notice that this transformation contains the log transformation for RV when G = 0 and the
raw statistic when G = 1 as special cases.
As in previous studies, in order to adopt a Gaussian nonlinear state space form where the error terms in both return and volatility equations are uncorrelated, we introduce a
transformation ×
= Y + Ý1 − ,
where ~ 0,1 and corr Y ,
= 0. A reparameterization of the model 1 yields ℎ
= m
¼
+ † ℎ − m
¼
+ Æü exp-−
› ”
ℎ . + T ,
ℎ ~ •m
¼
,
Û
[ ”
o—
”
–, where
Æ = and
T = 1 − .
3. MCMC simulation in the BCR-ASV model
Let = ü +
, ½ = ü +
, _ = ℎ +
, = -G,
¼
, ,
…
. , and = m, †, Æ, T . By Bayes theorem, the joint posterior distribution of , _ given R and V is
, , _| , ½ = , × |_ × ½| , _ × _| , ,
3 where
, is a joint prior distribution on parameter , . The MCMC method simulates a new value for each parameter in turn from its
conditional posterior distribution assuming that the current values for the other parameters are true values. We employ the MCMC scheme as follows:
Initialize , , _ .
1 Given the current values, draw
¼
, ,
…
, m, Æ, and T from their conditional posteriors
directly. 2
Given
¼
, ,
…
, and ½, draw G by HMC.
3 Given
m, Æ, T, _, and , draw † by RMHMC. 4
Given , , , and ½, draw _ by RMHMC.
For steps 1, 3, and 4, we implement the same sampling scheme proposed in Nugroho Morimoto 2014, 2015. In the following, we study the sampling scheme in step 2 only.
As far as the power parameter G ≠ 0 is only concerned to be sampled and by taking
the normal distribution Ž
À
,
À
for the prior of G, the logarithm of conditional posterior
distribution for G on the basis of Eq. 3 has the following expression:
Box–Cox realized asymmetric stochastic volatility model
SWUP
MA.50
G ∝ −
Û
B”
∑ È
\ ]
o À
−
¼
− ℎ É −
Ào•
] ” ]
, which is not of standard form, and therefore we cannot sample
G from it directly. In this case, we implement the HMC algorithm for sampling
G because the metric tensor required to implement the RMHMC sampling scheme cannot be explicitly derived from the above
distribution.
4. Empirical results on real data sets
In this section, the BCR-ASV model and MCMC algorithm discussed in the previous section are applied to the daily returns and RK data for six different international stock
indices: DAX, DJIA, FTSE100, Nasdaq100, Nikkei225, and SP500. These data sets were downloaded from Oxford-Man Institute ``realised library and range from January 2000 to
December 2013. Analyses are presented for the sampling efficiency, key parameters that build the extension models.
We run the MCMC algorithm for 15,000 iterations but discard the first 5,000 draws. From the resulting 10,000 draws which have passed the Geweke 1992s convergence test
for each parameter, we calculate the posterior mean, standard deviation SD, and 95 highest posterior density HPD interval. Summary of the posterior simulation results for
G are presented in Table 1.
Table 1. Summary of the posterior sample of parameter
G in the BCR-ASV model for six data sets from January 2000 to December 2013.
Statistic DAX
DJIA FTSE100
Nasdaq100 Nikkei225
SP500
Mean –0.0406
0.1226 0.1572
0.0420 0.0503
0.1149 SD
0.0104 0.0099
0.0096 0.0096
0.0128 0.0094
LB –0.0609
0.1022 0.1385
0.0242 0.0241
0.0968 UB
–0.0202 0.1409
0.1765 0.0617
0.0741 0.1340
Note: LB and UB denote lower and upper bounds of 95 HPD interval, respectively.
The posterior means of G are positive for DJIA, FTSE100, Nasdaq100, Nikkei225, and
SP500 and negative for DAX. In all cases, the 95 HPD interval of G exclude 0 the log
transformation or 1 the raw version. Although not reported, we find that even 99 HPD interval of
G exclude 0 or 1, providing significant evidence to transform the RV series using the BC transformation against both the logarithmic transformation and no transformation
cases. To understand the effect of BC transformation, we present the marginal densities of
the transformed RK implied by G = 0 and the estimated G for each stock index in Figure 1.
The marginal density of the BC RV is obviously different from that of the log RV. When the estimated
G is a positive value, the left tail of the former is lighter than that of the latter, while the right tail of former is heavier, and vice versa when the estimated
G is a negative value. Moreover the marginal density of the former has a heavier right tail than left tail as
that of the latter for both positive and negative values of G. Furthermore, the marginal
density of the former has a higher peak than, and a different location from, that of the latter.
D.B. Nugroho, T. Morimoto
SWUP
MA.51
Figure 1. Marginal densities of the log RV and BC-RV for six different international stock indices.
5. Conclusions and extensions
