A dynamic leverage stochastic volatility model

Figures & data

Table 1. Summary statistics of the three stock returns (1 January 1990 – 31 December 2017)

	Mean	Std.	Skewness	Kurtosis	Minimum	Maximum	J-B test	Ljung-Box(12)	ARCH(12)
DJI	0.031	1.057	−0.170	8.394	−8.200	10.508	20.729***	65.453***	1741.386***
FTSE	0.016	1.101	−0.127	6.067	−9.266	9.384	10.874***	78.307***	1581.783***
NIKKEI	−0.008	1.522	−0.130	5.397	−12.111	13.235	8.381***	24.844***	1202.434***

***,**,* denote significant at 1%, 5%, 10% levels, respectively.

Table 2. Posterior estimates for the in-sample data (1 January 1990 – 31 December 2017)

Model	${\mu }_h$	$\varphi$	${\sigma }_{\eta }$	$\rho$	$a$	$b$	LLH
(a) DJI
SV	−0.387	0.976	0.044				−8973.318
	(−0.514;-0.258)	(0.972;0.981)	(0.037;0.051)
LSV	−0.364	0.975	0.042	−0.650			−8852.364
	(−0.462;-0.265)	(0.971;0.979)	(0.036;0.049)	(−0.688;-0.609)
DLSV	−0.371	0.977	0.039	−0.639	0.486	0.983	−8844.172
	(−0.456;-0.286)	(0.973;0.981)	(0.033;0.045)	(−0.695;-0.547)	(0.172;0.787)	(0.946;0.999)
(b) FTSE
SV	−0.231	0.980	0.029				−9468.208
	(−0.356;-0.106)	(0.976;0.984)	(0.025;0.034)
LSV	−0.237	0.983	0.025	−0.685			−9359.464
	(−0.352;-0.123)	(0.980;0.986)	(0.021;0.029)	(−0.726;-0.640)
DLSV	−0.246	0.985	0.024	−0.702	0.700	0.990	−9347.305
	(−0.354;-0.139)	(0.983;0.988)	(0.020;0.028)	(−0.750;-0.646)	(0.259;1.203)	(0.980;0.999)
(c) NIKKEI
SV	0.469	0.966	0.044				−11792.990
	(0.372;0.565)	(0.961;0.972)	(0.038;0.051)
LSV	0.467	0.964	0.047	−0.563			−11692.408
	(0.392;0.543)	(0.958;0.969)	(0.040;0.054)	(−0.607;-0.519)
DLSV	0.482	0.965	0.045	−0.565	0.516	0.973	−11686.520
	(0.395;0.565)	(0.959;0.969)	(0.039;0.052)	(−0.612;-0.510)	(0.141;0.950)	(0.932;0.997)

Figure 1. The filtered volatility and the filtered dynamic leverage effect

Figure 2. The contour plot shows the impact of the temporal return innovations (${\epsilon }_t,{\epsilon }_{t-1}$) on the dynamic leverage correlation ${\rho }_{t+1}$ for the DJI data. The current GAS process $f_t$ is assumed to stay at its long-term mean $\omega$ and ${\zeta }_t=0$