Bayesian semiparametric multivariate stochastic volatility with application

Figures & data

Figure 1. Simulated variance processes (blue lines), posterior means (red lines), and 90% highest posterior density intervals (green lines).

Figure 2. Simulated correlation processes (blue lines), posterior means (red lines), and 90% highest posterior density intervals (green lines).

Table 1. Parameter values, posterior means, 90% highest posterior density intervals.

	True parameter	Posterior mean	90% HPDI
${\varphi }_{\mathbf{h}1}$	0.95	0.8694	(0.6001 0.9676)
${\varphi }_{\mathbf{h}2}$	0.95	0.9668	(0.9455 0.9874)
${\varphi }_{\mathbf{h}3}$	0.95	0.9786	(0.9623 0.9939)
${\varphi }_{\mathbf{h}4}$	0.95	0.9701	(0.9489 0.9908)
${\varphi }_{\mathbf{h}5}$	0.95	0.9886	(0.9779 0.9984)
${\sigma }_{\mathit{\eta }1}^2$	0.04	0.0822	(0.0158 0.3117)
${\sigma }_{\mathit{\eta }2}^2$	0.04	0.0438	(0.0209 0.0651)
${\sigma }_{\mathit{\eta }3}^2$	0.04	0.0514	(0.0219 0.1072)
${\sigma }_{\mathit{\eta }4}^2$	0.04	0.0340	(0.0180 0.0540)
${\sigma }_{\mathit{\eta }5}^2$	0.04	0.0238	(0.0117 0.0384)
		$\mathbb{E}({\alpha }_{ij,t})=\frac{c_{\mathit{\alpha }i}}{(1-{\varphi }_{\mathit{\alpha }i})}$
α12	–0.5	–0.4813	(–0.5411 –0.4220)
α13	–0.5	–0.4759	(–0.5568 –0.4032)
α23	–0.5	–0.4741	(–0.5388 –0.4134)
α14	–0.5	–0.5171	(–0.6016 –0.4416)
α24	–0.5	–0.4746	(–0.5430 –0.3975)
α34	–0.5	–0.4784	(–0.5476 –0.4076)
α15	–0.5	–0.4529	(–0.5411 –0.3538)
α25	–0.5	–0.5284	(–0.6095 –0.4415)
α35	–0.5	–0.4443	(–0.5199 –0.3646)
α45	–0.5	–0.5020	(–0.5636 –0.4302)

Notes: Simulated model according to Eqs. (3)–(9), with $m=5,T=1000,$ and a location-scale mixture of two normal distributions for the error term, as specified in Section 3.4.

Figure 3. Index values and daily returns (in %).

Table 2. Descriptive statistics of daily returns (in %).

	NIK	SHA	EU	DAX	DJ
Mean	0.0509	0.0177	0.0125	0.0302	0.0226
Median	0.0086	0.0000	0.0111	0.0614	0.0066
Variance	1.9457	2.7957	1.5524	1.4216	0.6229
Skewness	–0.2386	–0.8491	–0.1151	–0.2329	–0.1961
Kurtosis	6.3634	8.1768	4.4545	4.2553	4.7188
Sample correlation:
NIK	1.0000
SHA	0.2160	1.0000
EU	0.2158	0.1349	1.0000
DAX	0.2194	0.1390	0.9526	1.0000
DJ	0.1224	0.1418	0.5929	0.5743	1.0000

Note: The indices are abbreviated as NIK (Nikkei 225), SHA (Shanghai Shenzen CSI 300), EU (EuroStoxx), DAX (DAX 30 Performance), DJ (Dow Jones Industrial).

Table 3. Posterior means and standard deviations (in parentheses).

i	$c_{\mathit{\alpha }i}$	${\varphi }_{\mathit{\alpha }i}$	${\sigma }_{\mathbf{e}i}^2$	${\varphi }_{\mathbf{h}i}$	${\sigma }_{\mathit{\eta }i}^2$	${\overline{n}}_i$
1	–0.1455	0.0649	0.0474	0.9610	0.0510	9
	(0.0582)	(0.3532)	(0.0144)	(0.0158)	(0.0176)
2	–0.1614	0.0601	0.0657	0.9799	0.0364	10
	(0.0548)	(0.2965)	(0.0180)	(0.0087)	(0.0103)
3	–0.0662	–0.1444	0.0945	0.9323	0.0724	12
	(0.0338)	(0.1953)	(0.0234)	(0.0372)	(0.0499)
4	–0.0159	–0.0370	0.0261	0.9980	0.0330	5
	(0.0120)	(0.1403)	(0.0054)	(0.0013)	(0.0135)
5	0.0025	–0.0509	0.0310	0.9938	0.0674	10
	(0.0124)	(0.1679)	(0.0091)	(0.0039)	(0.0245)
6	–0.4111	0.5478	0.0254
	(0.1070)	(0.1182)	(0.0069)
7	0.0008	0.4161	0.0466
	(0.0127)	(0.1831)	(0.0121)
8	–0.0018	–0.4898	0.0277
	(0.0218)	(0.1796)	(0.0107)
9	–0.1832	0.2271	0.0498
	(0.0986)	(0.3241)	(0.0159)
10	–0.1745	–0.1649	0.0488
	(0.0922)	(0.2732)	(0.0136)

Note: ${\overline{n}}_i$ denotes the average number of nonempty mixture components rounded to the nearest integer.

Figure 4. In-sample correlations: posterior means plus 50% and 90% highest posterior density intervals.

Figure 5. Sample correlations obtained from a rolling window of size 50 centered around the actual observation with the sample-correlation (horizontal line).

Figure 6. One-step-ahead density forecasts (of the elements of ${\mathbf{H}}_{T+1}^{\ast \mathrm{ }}$).

Figure 7. Contour plots of pairwise one-step-ahead density forecasts.

Table 4. Posterior summary of the elements of the one-step-ahead covariance matrix.

${\mathbf{H}}_{T+1}^{\ast }$	Mean	Median	90% HPDI
${\mathbf{H}}_{T+1}^{\ast \mathrm{ }\text{NIK}}$	7.2337	6.1811	(1.5248, 13.2226)
${\mathbf{H}}_{T+1}^{\ast \mathrm{ }\text{SHA}}$	3.2385	2.6356	(0.3246, 6.2110)
${\mathbf{H}}_{T+1}^{\ast \mathrm{ }\text{EU}}$	3.7878	3.1497	(0.6449, 6.9720)
${\mathbf{H}}_{T+1}^{\ast \mathrm{ }\text{DAX}}$	3.6246	2.9022	(0.4688, 6.8787)
${\mathbf{H}}_{T+1}^{\ast \mathrm{ }\text{DJ}}$	1.6232	1.1733	(0.0774, 3.3822)
${\mathbf{H}}_{T+1}^{\ast \mathrm{ }\text{SHA},\mathrm{ }\text{NIK}}$	0.9987	0.7307	(–1.8207, 4.0100)
${\mathbf{H}}_{T+1}^{\ast \mathrm{ }\text{EU},\mathrm{ }\text{NIK}}$	1.5025	1.1124	(–1.9305, 5.3189)
${\mathbf{H}}_{T+1}^{\ast \mathrm{ }\text{EU},\mathrm{ }\text{SHA}}$	0.4096	0.2605	(–1.6861, 2.7071)
${\mathbf{H}}_{T+1}^{\ast \mathrm{ }\text{DAX},\mathrm{ }\text{NIK}}$	1.5457	1.1560	(–1.9358, 5.4155)
${\mathbf{H}}_{T+1}^{\ast \mathrm{ }\text{DAX},\mathrm{ }\text{SHA}}$	0.4052	0.2579	(–1.7940, 2.9082)
${\mathbf{H}}_{T+1}^{\ast \mathrm{ }\text{DAX},\mathrm{ }\text{EU}}$	3.5458	2.9134	(0.5579, 6.7130)
${\mathbf{H}}_{T+1}^{\ast \mathrm{ }\text{DJ},\mathrm{ }\text{NIK}}$	0.1496	0.0923	(–2.9625, 3.1029)
${\mathbf{H}}_{T+1}^{\ast \mathrm{ }\text{DJ},\mathrm{ }\text{SHA}}$	0.0052	–0.0103	(–1.5690, 1.7853)
${\mathbf{H}}_{T+1}^{\ast \mathrm{ }\text{DJ},\mathrm{ }\text{EU}}$	1.2784	0.9443	(–0.9013, 3.6663)
${\mathbf{H}}_{T+1}^{\ast \mathrm{ }\text{DJ},\mathrm{ }\text{DAX}}$	1.2232	0.8752	(–0.9412, 3.5763)

Note: The indices are abbreviated as NIK (Nikkei 225), SHA (Shanghai Shenzen CSI 300), EU (EuroStoxx), DAX (DAX 30 Performance), DJ (Dow Jones Industrial).

Figure 8. Transformed posterior DP precision.

Table 5. Cumulative log predictive likelihoods (CPL).

Model	CPL	Predictive log Bayes factor
Cholesky DPM-SV	–685.5444
Gaussian Cholesky SV	–699.9647	14.4203
Student-t Cholesky SV	–690.6329	5.0885
Asymmetric Cholesky DPM-SV	–688.2255	2.6811

Note: The three benchmark models (Gaussian and Student-t Cholesky SV, asymmetric Cholesky DPM-SV) are estimated as described in Section 4.4.1. Out-of-sample period: February 22, 2016 – July 8, 2016 (100 observations).