2 Estimating the continuous time stochastic volatility models
2.4 Numerical illustrations
With the model (2.9) and (2.10) for a numerical illustration of parameter estimation, ten datasets of length 500 and time between observations as 1 are generated according to the Euler approximation (2.11) and (2.12) with Δ=0.001. The initial value of Xt is
X
0=0 and V0 is generated from the stationary distribution of Vt:From the results shown in section 2.4, the approximating GARCH process corresponding to the model specified by (2.9) and (2.10) is
Δ
An approximation for the transition density of this type of GARCH model can be found in Duan et al. (1999). Their result also implies that Assumption 5 is satisfied for this model.
The parameters used in this simulation study are shown in Table 2.1. And this setting is compliant with the estimates for the NGARCH model with the NYSE composite index returns in Duan (1997).
Summary statistics for each dataset are listed in Table 2.2. Figure 2.2 shows the trend chart for the simulated values of Xt
-X
t-1 and Vt from the first 4 datasets. QQ-plots for the same 4 datasets are shown in Figure 2.3. The phenomena of volatility clustering can be easily observed. With the results of B-J tests, normality for Xt-X
t-1 is seen to be very different through datasets. Datasets 1, 6 and 10 are very close to being normally distributed, while others differ from normality significantly. On the other hand, the Ljung-Box test indicates that in principle the increments Xt-X
t-1 are statistically independent.Table 2.2: Summary statistics for the increments Xt-Xt-1.
dataset mean variance skewness kurtosis BJ stat p value Box- Ljung statistic
p-value
1 0.0075 0.7930 -0.0149 3.0649 0.11 0.948 2.3751 0.1233 2 -0.0260 0.7256 -0.2144 3.5662 10.51 0.005 0.0115 0.9146 3 0.0270 1.0351 -0.5145 6.5380 282.84 0 3.8780 0.0489 4 -0.0158 0.8316 -0.3243 4.6880 68.13 0 0.0000 0.9993 5 0.0236 0.7856 -0.0366 3.7860 12.98 0.002 0.1225 0.7263 6 0.0152 0.7894 -0.0426 3.1772 0.81 0.669 0.1910 0.6621 7 -0.0023 0.8312 -0.0689 3.6905 10.33 0.006 1.4088 0.2352 8 -0.0705 1.0778 -0.6887 6.8981 356.10 0 0.7640 0.3821 9 0.0421 0.8233 0.1272 5.3566 117.05 0 0.0659 0.7974 10 0.0333 0.6941 0.0116 3.4174 3.64 0.162 0.8251 0.3637
Figure 2.2: Trends of Xt-Xt-1of the first 4 datasets.
Figure 2.3: QQ-plots of Xt-Xt-1of the first 4 datasets.
Parameter estimation based on (2.7) with these datasets proceeds with Δ=0.2. By the method in Raftery and Lewis (1992), a small size experiment is conducted for determining the iteration times required to achieve convergence in each step.
For the Metropolis algorithm used in step 2, 150 burn-in iterations are exercised before taking one sample for each*X~t(n). Similarly, sample paths for *X~t(n) are taken every two iterations from the 200 iterations following 20 burn-in iterations.
Table 2.3: Parameter estimates. A comparison between the simulated likelihood, GARCH approximation and EMM.
Method Parameter True Value Mean Median Max Min STD κ 0.0699 0.0637 0.0587 0.1287 0.0423 0.0241 ξ 0.7928 0.8098 0.8063 0.9379 0.6994 0.0678 σ 0.1772 0.1928 0.1863 0.2255 0.1685 0.0218 Simulated
Likelihood
ρ -0.6500 -0.6955 -0.7079 -0.5164 -0.8524 0.0974 κ 0.0699 0.1174 0.0732 0.6060 0.0160 0.1734 ξ 0.7928 0.8321 0.8187 0.9587 0.7212 0.0801 σ 0.1772 0.1500 0.1523 0.2505 0.0781 0.0502 GARCH
ρ -0.6500 -0.9213 -0.9661 -0.5747 -1.0000 0.1314 κ 0.0699 0.0670 0.0557 0.1176 0.0161 0.0369 ξ 0.7928 0.8364 0.8135 1.1231 0.6964 0.1274 σ 0.1772 0.1899 0.1738 0.3357 0.0972 0.0853 EMM
ρ -0.6500 -0.7727 -0.7575 -0.6358 -0.9855 0.1237
Table 2.4: Parameter estimates. Estimates with the likelihood function approximated through the Euler approximation and Vt assumed to be observed.
Parameter True Value Mean Median Max Min STD κ 0.0699 0.0632 0.0601 0.0927 0.0379 0.0164 ξ 0.7928 0.8065 0.7924 0.9047 0.7236 0.0678 σ 0.1772 0.1696 0.1689 0.1735 0.1656 0.0026 Δ=1
ρ -0.6500 -0.6507 -0.6507 -0.6165 -0.6812 0.0182 κ 0.0699 0.0659 0.0622 0.1031 0.0421 0.0181 ξ 0.7928 0.8121 0.8011 0.8996 0.7318 0.0676 σ 0.1772 0.1767 0.1769 0.1786 0.1726 0.0017 Δ =0.2
ρ -0.6500 -0.6504 -0.6508 -0.6365 -0.6600 0.0070
Estimates via NGARCH approximations and EMM methods for each dataset are also calculated for comparisons. The AR-NGARCH and AR-EGARCH processes are considered as the score generator for the EMM method. As in Anderson and Lund (1997), each dataset is first fitted with the two classes of processes. By the AIC
criterion the model that fits better is selected. It is found that AR terms in the mean are generally not necessary, and the EGARCH process fits 6 of 10 datasets better than the NGARCH process although the latter has this diffusion models (2.9) and (2.10) as its limit.
The SNP densities are then set with KX=0 and KZ=1 and thus have the form
( )
( )
∫
+= +
t t t
t t t
K z z dz
z S z
f 1 ( )
) ( ) 1
|
( 2
1 2 1
φ α
φ
η α .
On computing the expectation of the score, a sequence of length 100,000 is simulated under the Euler scheme in which
Δ
=0.04.Figure 2.4: Box plots for estimates from different methods. The dash lines indicate the true values.
The summary of estimation results is listed in Table 2.3 and graphically represented by box plots in Figure 2.4. The estimates for
ξ
,σ
, andρ
from theapproximate likelihood are concentrated around the true value. However,
κ
seems to be systematically underestimated. This may be explained as follows. First it should be noted that the GARCH processes only provide as an approximation but not an exact distribution. Sinceκ
is related to mean reversion, estimatingκ
precisely requires more information about the “events” of mean reversion. However, taking discreteobservations implies that those very quick mean reverse events will be dropped. In other words, the sampled “events” for estimating
κ
is biased under the discrete time scheme. An indirect evidence is shown in Table 2.4. The estimates are obtainedEul er s= 1 Euler s =0 .2 Simul GAR C H EM M
0.00.10.20.30.4
Estima tes of kappa
Eul er s= 1 Euler s =0 .2 Simul GAR C H EM M
0.60.70.80.91.01.1
Estima tes of theta
Eul er s= 1 Euler s =0 .2 Simul GAR C H EM M
0.100.150.200.250.300.35
Estima tes of sig ma
Eul er s= 1 Euler s =0 .2 Simul GAR C H EM M
-1.0-0.9-0.8-0.7-0.6-0.5
Estima tes of rho
through maximizing the likelihood of all Xt and Vt under the Euler scheme for Δ =1 and 0.2 respectively. It is seen that
κ
also tends to be underestimated, and estimates forκ
with Δ=1 is even lower than those with Δ=0.2.It can be seen that estimates from NGARCH approximations are much poorer than the estimates from the simulated likelihood. In fact, there is almost at least one parameter wildly estimated for each dataset. This result conforms to Wang (2002) and should not be too surprising since NGARCH approximations are indeed based on normally distributed innovations, while the bivariate diffusion models generate much more complicated distributions.
The EMM method provides more reasonable estimates than those from the NGARCH approximations. However, the EMM estimator is seen to be less efficient than that from the simulated likelihood. This result is conformable to the nature of EMM estimators discussed in Gallant and Tauchen (1996), since the diffusion models (2.9) and (2.10) is not embedded in either the NGARCH model or the EGARCH model.