Does the fear gauge predict downside risk more accurately than econometric models? Evidence from the US stock market

Figures & data

Table 1. Descriptive statistics of stock return, volatilities, and control variables for the out-of-sample period from 3 January 2006 to 28 February 2014

	dlogsp	iv	${\mathit{\sigma }}_{dlogsp}^{gf}$	${\mathit{\sigma }}_{dlogsp}^{egf}$
Mean	0.0196	21.5143	18.4034	17.2417
Median	0.0823	18.5450	14.6747	14.4128
Maximum	10.9572	80.8600	72.4703	68.4571
Minimum	−9.3537	9.8900	7.4181	6.3010
Standard deviation	1.3722	10.3913	11.1986	9.7996
Skewness	−0.2517	2.1062	2.3684	2.2419
Kurtosis	12.0416	8.7195	9.4006	9.2608
	${\mathit{\sigma }}_{dlogsp}^{pgf}$	${\mathit{\sigma }}_{dlogsp}^{tgf}$	def	dex
Mean	17.7252	18.3918	1.2219	−0.0043
Median	14.3023	14.6651	1.0300	−0.0026
Maximum	77.1682	78.6558	3.5000	1.6957
Minimum	6.5858	7.0698	0.6200	−3.3854
Standard deviation	11.0921	11.6382	0.5669	0.3813
Skewness	2.4604	2.4066	2.3484	−0.5417
Kurtosis	10.2981	9.7124	8.1271	9.3019
	term	dff	ivf^ARMA(2,1)	ivf^ARMA(4,4)
Mean	1.9605	−0.0020	21.4841	21.4842
Median	2.1500	0.0000	18.5561	18.5510
Maximum	3.8300	1.0500	78.8185	79.0233
Minimum	−0.4900	−0.9500	10.0573	10.0556
Standard deviation	1.1951	0.0898	10.2161	10.2217
Skewness	−0.5126	−0.3736	2.0836	2.0886
Kurtosis	2.1299	54.8551	8.5284	8.5680
	${\mathit{\sigma }}_{iv}^{tgf}$	${\mathit{\sigma }}_{iv}^{egf}$	${\mathit{\sigma }}_{dlogiv}^{tgf}$	${\mathit{\sigma }}_{dlogiv}^{egf}$
Mean	6.2343	5.4589	6.1464	5.9255
Median	3.7707	3.9702	5.5621	5.5795
Maximum	53.6336	36.4244	19.2418	13.8233
Minimum	1.2219	0.6919	4.2068	3.6799
Standard deviation	7.4033	5.0387	1.8559	1.3839
Skewness	2.9542	2.4951	2.3569	1.6382
Kurtosis	13.1024	10.4010	10.7386	6.7829

Notes: This table presents the descriptive statistics for 16 variables used in this study. More specifically, dlogsp denotes the daily percentage log return of S&P 500; and iv denotes the volatility index (VIX) close as to S&P 500. In addition, ${\mathit{\sigma }}_{dlogsp}^{gf}$ is the forecast volatility of dlogsp, which is derived from the GARCH(1,1) model; ${\mathit{\sigma }}_{dlogsp}^{egf}$ represents the forecast volatility of dlogsp, which is derived from the EGARCH(1,1) model; ${\mathit{\sigma }}_{dlogsp}^{pgf}$ denotes the forecast volatility of dlogsp, which is derived from the PGARCH(1,1) model; and ${\mathit{\sigma }}_{dlogsp}^{tgf}$ is the forecast volatility of dlogsp, which is derived from the TGARCH(1,1) model. Further, def denotes the default spread; dex is the first-difference series of the trade-weighted US dollar index; term represents the term premium; and dff is the first-difference series of the effective federal funds rate. Moreover, ivf ^ARMA(2,1) denotes the forecast VIX, which is derived from the ARMA(2,1) model; ivf ^ARMA(4,4) represents the forecast VIX, which is derived from the ARMA(4,4) model; ${\mathit{\sigma }}_{iv}^{tgf}$ denotes the forecast volatility of the VIX, which is derived from the TGARCH(1,1) model; ${\mathit{\sigma }}_{iv}^{egf}$ is the forecast volatility of the VIX, which is derived from the EGARCH(1,1) model; ${\mathit{\sigma }}_{dlogiv}^{tgf}$ represents the forecast volatility of the first log differences of the VIX, which is derived from the TGARCH(1,1) model; and ${\mathit{\sigma }}_{dlogiv}^{egf}$ denotes the forecast volatility of the first log differences of the VIX, which is derived from the EGARCH(1,1) model. All statistics in this table are those for our out-of-sample period that is from 3 January 2006 to 28 February 2014, and the number of daily observations for this period is 2,038. In this study, we first build and specify forecast models by using the samples of our in-sample period, which is from 2 January 1990 to 30 December 2005, and then empirically compare the predictive power of various variables for downside risk in the US stock market in our out-of-sample period.

Figure 1. Dynamic relations of the S&P 500 and the VIX: daily time-series evolution for the period from 3 January 2006 to 28 February 2014.

Notes: This figure presents the daily time-series evolution of S&P 500 close prices and VIX close values for our out-of-sample period from 3 January 2006 to 28 February 2014. This period includes the date of the US Lehman Brothers bankruptcy in 15 September 2008. S&P 500 close prices are from the Thomson Reuters, while VIX close values are calculated and supplied by the Chicago Board Options Exchange (CBOE). In this study, we first specify forecast models using our in-sample period from 2 January 1990 to 30 December 2005; and then empirically compare the predictive power of various variables for downside risk in the US stock market in the above out-of-sample period.

Table 2. Testing the predictive power for downside risk in the US stock market by univariate logit models: results for the VIX and the forecast volatilities from several GARCH models

Coefficients	Estimates	p-value	Coefficients	Estimates	p-value	Coefficients	Estimates	p-value
Panel A: Downside risk predictive power of the previous day’s VIX
k = 99.0			k = 98.5			k = 98.0
$m_{0,iv(t-1)}^{lg-99\%}$	−7.043***	0.000	$m_{0,iv(t-1)}^{lg-98.5\%}$	−6.309***	0.000	$m_{0,iv(t-1)}^{lg-98\%}$	−5.577***	0.000
$m_{1,iv(t-1)}^{lg-99\%}$	0.083***	0.000	$m_{1,iv(t-1)}^{lg-98.5\%}$	0.076***	0.000	$m_{1,iv(t-1)}^{lg-98\%}$	0.064***	0.000
MF-R^2	0.188166		MF-R^2	0.156169		MF-R^2	0.101679
k = 97.5			k = 97.0			k = 95.0
$m_{0,iv(t-1)}^{lg-97.5\%}$	−5.219***	0.000	$m_{0,iv(t-1)}^{lg-97\%}$	−4.937***	0.000	$m_{0,iv(t-1)}^{lg-95\%}$	−4.372***	0.000
$m_{1,iv(t-1)}^{lg-97.5\%}$	0.060***	0.000	$m_{1,iv(t-1)}^{lg-97\%}$	0.057***	0.000	$m_{1,iv(t-1)}^{lg-95\%}$	0.057***	0.000
MF-R^2	0.087843		MF-R^2	0.078253		MF-R^2	0.078673
Panel B: Downside risk predictive power of the forecast volatility form the GARCH model
k = 99.0			k = 98.5			k = 98.0
$m_{0,gf(t)}^{lg-99\%}$	−6.908***	0.000	$m_{0,gf(t)}^{lg-98.5\%}$	−6.041***	0.000	$m_{0,gf(t)}^{lg-98\%}$	−5.336***	0.000
$m_{1,gf(t)}^{lg-99\%}$	0.081***	0.000	$m_{1,gf(t)}^{lg-98.5\%}$	0.072***	0.000	$m_{1,gf(t)}^{lg-98\%}$	0.060***	0.000
MF-R^2	0.223304		MF-R^2	0.170584		MF-R^2	0.111372
k = 97.5			k = 97.0			k = 95.0
$m_{0,gf(t)}^{lg-97.5\%}$	−4.966***	0.000	$m_{0,gf\left(t\right)}^{lg-97\%}$	−4.661***	0.000	$m_{0,gf(t)}^{lg-95\%}$	−4.054***	0.000
$m_{1,gf(t)}^{lg-97.5\%}$	0.056***	0.000	$m_{1,gf(t)}^{lg-97\%}$	0.052***	0.000	$m_{1,gf(t)}^{lg-95\%}$	0.050***	0.000
MF-R^2	0.093162		MF-R^2	0.078511		MF-R^2	0.073811
Panel C: Downside risk predictive power of the forecast volatility form the EGARCH model
k = 99.0			k = 98.5			k = 98.0
$m_{0,egf(t)}^{lg-99\%}$	−7.095***	0.000	$m_{0,egf(t)}^{lg-98.5\%}$	−6.208***	0.000	$m_{0,egf(t)}^{lg-98\%}$	−5.518***	0.000
$m_{1,egf(t)}^{lg-99\%}$	0.096***	0.000	$m_{1,egf(t)}^{lg-98.5\%}$	0.084***	0.000	$m_{1,egf(t)}^{lg-98\%}$	0.072***	0.000
MF-R^2	0.238500		MF-R^2	0.182498		MF-R^2	0.125834
k = 97.5			k = 97.0			k = 95.0
$m_{0,egf(t)}^{lg-97.5\%}$	−5.128***	0.000	$m_{0,egf(t)}^{lg-97\%}$	−4.838***	0.000	$m_{0,egf(t)}^{lg-95\%}$	−4.194***	0.000
$m_{1,egf(t)}^{lg-97.5\%}$	0.067***	0.000	$m_{1,egf(t)}^{lg-97\%}$	0.064***	0.000	$m_{1,egf(t)}^{lg-95\%}$	0.060***	0.000
MF-R^2	0.104742		MF-R^2	0.092101		MF-R^2	0.082080
Panel D: Downside risk predictive power of the forecast volatility form the PGARCH model
k = 99.0			k = 98.5			k = 98.0
$m_{0,pgf(t)}^{lg-99\%}$	−6.816***	0.000	$m_{0,pgf(t)}^{lg-98.5\%}$	−5.978***	0.000	$m_{0,pgf(t)}^{lg-98\%}$	−5.325***	0.000
$m_{1,pgf(t)}^{lg-99\%}$	0.080***	0.000	$m_{1,pgf(t)}^{lg-98.5\%}$	0.071***	0.000	$m_{1,pgf(t)}^{lg-98\%}$	0.061***	0.000
MF-R^2	0.227879		MF-R^2	0.174846		MF-R^2	0.119585
k = 97.5			k = 97.0			k = 95.0
$m_{0,pgf(t)}^{lg-97.5\%}$	−4.955***	0.000	$m_{0,pgf(t)}^{lg-97\%}$	−4.671***	0.000	$m_{0,pgf(t)}^{lg-95\%}$	−4.045***	0.000
$m_{1,pgf(t)}^{lg-97.5\%}$	0.057***	0.000	$m_{1,pgf(t)}^{lg-97\%}$	0.053***	0.000	$m_{1,pgf(t)}^{lg-95\%}$	0.051***	0.000
MF-R^2	0.099698		MF-R^2	0.086768		MF-R^2	0.078291
Panel E: Downside risk predictive power of the forecast volatility form the TGARCH model
k = 99.0			k = 98.5			k = 98.0
$m_{0,tgf(t)}^{lg-99\%}$	−7.008***	0.000	$m_{0,tgf(t)}^{lg-98.5\%}$	−6.095***	0.000	$m_{0,tgf(t)}^{lg-98\%}$	−5.389***	0.000
$m_{1,tgf(t)}^{lg-99\%}$	0.082***	0.000	$m_{1,tgf(t)}^{lg-98.5\%}$	0.072***	0.000	$m_{1,tgf(t)}^{lg-98\%}$	0.061***	0.000
MF-R^2	0.250857		MF-R^2	0.190221		MF-R^2	0.127788
k = 97.5			k = 97.0			k = 95.0
$m_{0,tgf(t)}^{lg-97.5\%}$	−5.009***	0.000	$m_{0,tgf(t)}^{lg-97\%}$	−4.702***	0.000	$m_{0,tgf(t)}^{lg-95\%}$	−4.069***	0.000
$m_{1,tgf(t)}^{lg-97.5\%}$	0.056***	0.000	$m_{1,tgf(t)}^{lg-97\%}$	0.052***	0.000	$m_{1,tgf(t)}^{lg-95\%}$	0.050***	0.000
MF-R^2	0.106875		MF-R^2	0.090743		MF-R^2	0.081729

Notes: This table exhibits the results of empirical comparisons of the predictive power of the one-day lagged VIX of S&P 500 and the forecast S&P 500 volatilities from various GARCH models. These tests are conducted in our out-of-sample period that is from 3 January 2006 to 28 February 2014, and all investigations are conducted by using univariate logit models. More specifically, we use the one-day lagged VIX close values and the out-of-sample forecast volatilities from GARCH(1,1), EGARCH(1,1), PGARCH(1,1), and TGARCH(1,1) models. All above models used to derive the forecast volatilities are specified in our in-sample period, which is from 2 January 1990 to 30 December 2005. Further, MF-R² in this table denotes the McFadden’s R-squared value.

*Statistical significance of coefficients at the 10% level.

**Statistical significance of coefficients at the 5% level.

***Statistical significance of coefficients at the 1% level.

Table 3. Testing the predictive power for downside risk in the US stock market by multiple logit models: the VIX versus the forecast volatilities from several GARCH models

Coefficients	Estimates	p-value	Coefficients	Estimates	p-value	Coefficients	Estimates	p-value
Panel A: Forecast volatility from the GARCH model versus the previous day’s VIX
k = 99.0			k = 98.5			k = 98.0
$s_{0,gf(t)}^{lg-99\%}$	−6.737***	0.000	$s_{0,gf(t)}^{lg-98.5\%}$	−6.075***	0.000	$s_{0,gf(t)}^{lg-98\%}$	−5.348***	0.000
$s_{1,gf(t)}^{lg-99\%}$	0.104***	0.005	$s_{1,gf(t)}^{lg-98.5\%}$	0.067**	0.031	$s_{1,gf(t)}^{lg-98\%}$	0.058**	0.049
$s_{2,iv(t-1)}^{lg-99\%}$	−0.027	0.523	$s_{2,iv(t-1)}^{lg-98.5\%}$	0.005	0.876	$s_{2,iv(t-1)}^{lg-98\%}$	0.002	0.953
MF-R^2	0.225202		MF-R^2	0.170659		MF-R^2	0.111381
k = 97.5			k = 97.0			k = 95.0
$s_{0,gf(t)}^{lg-97.5\%}$	−5.037***	0.000	$s_{0,gf(t)}^{lg-97\%}$	−4.826***	0.000	$s_{0,gf(t)}^{lg-95\%}$	−4.339***	0.000
$s_{1,gf(t)}^{lg-97.5\%}$	0.045	0.101	$s_{1,gf(t)}^{lg-97\%}$	0.027	0.296	$s_{1,gf(t)}^{lg-95\%}$	0.008	0.718
$s_{2,iv(t-1)}^{lg-97.5\%}$	0.012	0.698	$s_{2,iv(t-1)}^{lg-97\%}$	0.028	0.328	$s_{2,iv(t-1)}^{lg-95\%}$	0.049**	0.043
MF-R^2	0.093475		MF-R^2	0.080229		MF-R^2	0.078833
Panel B: Forecast volatility from the EGARCH model versus the previous day’s VIX
k = 99.0			k = 98.5			k = 98.0
$s_{0,egf(t)}^{lg-99\%}$	−6.571***	0.000	$s_{0,egf(t)}^{lg-98.5\%}$	−5.937***	0.000	$s_{0,egf(t)}^{lg-98\%}$	−5.160***	0.000
$s_{1,egf(t)}^{lg-99\%}$	0.236***	0.000	$s_{1,egf(t)}^{lg-98.5\%}$	0.153***	0.001	$s_{1,egf(t)}^{lg-98\%}$	0.164***	0.000
$s_{2,iv(t-1)}^{lg-99\%}$	−0.141**	0.015	$s_{2,iv(t-1)}^{lg-98.5\%}$	−0.069	0.138	$s_{2,iv(t-1)}^{lg-98\%}$	−0.091**	0.035
MF-R^2	0.267681		MF-R^2	0.189542		MF-R^2	0.137424
k = 97.5			k = 97.0			k = 95.0
$s_{0,egf(t)}^{lg-97.5\%}$	−4.879***	0.000	$s_{0,egf(t)}^{lg-97\%}$	−4.632***	0.000	$s_{0,egf(t)}^{lg-95\%}$	−4.232***	0.000
$s_{1,egf(t)}^{lg-97.5\%}$	0.130***	0.001	$s_{1,egf(t)}^{lg-97\%}$	0.115***	0.002	$s_{1,egf(t)}^{lg-95\%}$	0.051*	0.091
$s_{2,iv(t-1)}^{lg-97.5\%}$	−0.063	0.110	$s_{2,iv(t-1)}^{lg-97\%}$	−0.052	0.157	$s_{2,iv(t-1)}^{lg-95\%}$	0.009	0.752
MF-R^2	0.110232		MF-R^2	0.095827		MF-R^2	0.082204
Panel C: Forecast volatility from the PGARCH model versus the previous day’s VIX
k = 99.0			k = 98.5			k = 98.0
$s_{0,pgf(t)}^{lg-99\%}$	−6.138***	0.000	$s_{0,pgf(t)}^{lg-98.5\%}$	−5.724***	0.000	$s_{0,pgf(t)}^{lg-98\%}$	−4.933***	0.000
$s_{1,pgf(t)}^{lg-99\%}$	0.175***	0.001	$s_{1,pgf(t)}^{lg-98.5\%}$	0.106***	0.010	$s_{1,pgf(t)}^{lg-98\%}$	0.116***	0.003
$s_{2,iv(t-1)}^{lg-99\%}$	−0.111*	0.064	$s_{2,iv(t-1)}^{lg-98.5\%}$	−0.041	0.389	$s_{2,iv(t-1)}^{lg-98\%}$	−0.064	0.147
MF-R^2	0.244708		MF-R^2	0.177212		MF-R^2	0.125031
k = 97.5			k = 97.0			k = 95.0
$s_{0,pgf(t)}^{lg-97.5\%}$	−4.722***	0.000	$s_{0,pgf(t)}^{lg-97\%}$	−4.530***	0.000	$s_{0,pgf(t)}^{lg-95\%}$	−4.239***	0.000
$s_{1,pgf(t)}^{lg-97.5\%}$	0.090**	0.011	$s_{1,pgf(t)}^{lg-97\%}$	0.073**	0.025	$s_{1,pgf(t)}^{lg-95\%}$	0.024	0.369
$s_{2,iv(t-1)}^{lg-97.5\%}$	−0.038	0.341	$s_{2,iv(t-1)}^{lg-97\%}$	−0.023	0.535	$s_{2,iv(t-1)}^{lg-95\%}$	0.031	0.289
MF-R^2	0.101637		MF-R^2	0.087479		MF-R^2	0.079668
Panel D: Forecast volatility from the TGARCH model versus the previous day’s VIX
k = 99.0			k = 98.5			k = 98.0
$s_{0,tgf(t)}^{lg-99\%}$	−6.287***	0.000	$s_{0,tgf(t)}^{lg-98.5\%}$	−5.714***	0.000	$s_{0,tgf(t)}^{lg-98\%}$	−4.964***	0.000
$s_{1,tgf(t)}^{lg-99\%}$	0.172***	0.000	$s_{1,tgf(t)}^{lg-98.5\%}$	0.119***	0.000	$s_{1,tgf(t)}^{lg-98\%}$	0.114***	0.000
$s_{2,iv(t-1)}^{lg-99\%}$	−0.113**	0.022	$s_{2,iv(t-1)}^{lg-98.5\%}$	−0.059	0.136	$s_{2,iv(t-1)}^{lg-98\%}$	−0.066*	0.076
MF-R^2	0.277187		MF-R^2	0.197413		MF-R^2	0.135955
k = 97.5			k = 97.0			k = 95.0
$s_{0,tgf(t)}^{lg-97.5\%}$	−4.697***	0.000	$s_{0,tgf(t)}^{lg-97\%}$	−4.527***	0.000	$s_{0,tgf(t)}^{lg-95\%}$	−4.161***	0.000
$s_{1,tgf(t)}^{lg-97.5\%}$	0.096***	0.001	$s_{1,tgf(t)}^{lg-97\%}$	0.075***	0.005	$s_{1,tgf(t)}^{lg-95\%}$	0.038*	0.092
$s_{2,iv(t-1)}^{lg-97.5\%}$	−0.049	0.154	$s_{2,iv(t-1)}^{lg-97\%}$	−0.027	0.390	$s_{2,iv(t-1)}^{lg-95\%}$	0.014	0.581
MF-R^2	0.111233		MF-R^2	0.092106		MF-R^2	0.082106

Notes: This table exhibits the results of empirical comparisons of the predictive power of the one-day lagged VIX of S&P 500 and the forecast S&P 500 volatilities derived from various GARCH models. These tests are conducted in our out-of-sample period that is from 3 January 2006 to 28 February 2014, and all investigations are conducted using multiple logit models. Each multiple logit model includes a constant term, the one-day lagged VIX, and the forecast volatility derived from one of the four GARCH models. In the tests, we use the one-day lagged VIX close values and the out-of-sample forecast volatilities derived from GARCH(1,1), EGARCH(1,1), PGARCH(1,1), and TGARCH(1,1) models. All above models used to derive the forecast volatilities are specified in our in-sample period, which is from 2 January 1990 to 30 December 2005. Further, MF-R² in this table denotes the McFadden’s R-squared value.

*Statistical significance of coefficients at the 10% level.

**Statistical significance of coefficients at the 5% level.

***Statistical significance of coefficients at the 1% level.

Table 4. Testing the predictive power for downside risk in the US stock market by multiple logit models with control variables: the VIX versus the forecast volatilities from several GARCH models

Coefficients	Estimates	p-value	Coefficients	Estimates	p-value	Coefficients	Estimates	p-value
Panel A: Forecast volatility from the GARCH model versus the previous day’s VIX
k = 99.0			k = 98.5			k = 98.0
${\mathit{\omega }}_{0,gf(t)}^{lg-99\%}$	−6.292***	0.000	${\mathit{\omega }}_{0,gf(t)}^{lg-98.5\%}$	−5.438***	0.000	${\mathit{\omega }}_{0,gf(t)}^{lg-98\%}$	−4.707***	0.000
${\mathit{\omega }}_{1,gf(t)}^{lg-99\%}$	0.188***	0.000	${\mathit{\omega }}_{1,gf(t)}^{lg-98.5\%}$	0.120***	0.002	${\mathit{\omega }}_{1,gf\left(t\right)}^{lg-98\%}$	0.108***	0.002
${\mathit{\omega }}_{2,iv(t-1)}^{lg-99\%}$	−0.051	0.283	${\mathit{\omega }}_{2,iv(t-1)}^{lg-98.5\%}$	−0.001	0.979	${\mathit{\omega }}_{2,iv(t-1)}^{lg-98\%}$	5.6E-05	0.999
${\mathit{\gamma }}_{1,def(t-1)}^{lg-99\%}$	−1.754***	0.003	${\mathit{\gamma }}_{1,def(t-1)}^{lg-98.5\%}$	−1.171**	0.013	${\mathit{\gamma }}_{1,def(t-1)}^{lg-98\%}$	−1.170***	0.006
${\mathit{\gamma }}_{2,dex(t-1)}^{lg-99\%}$	0.075	0.840	${\mathit{\gamma }}_{2,dex(t-1)}^{lg-98.5\%}$	0.021	0.946	${\mathit{\gamma }}_{2,dex(t-1)}^{lg-98\%}$	−0.005	0.987
${\mathit{\gamma }}_{3,term(t-1)}^{lg-99\%}$	0.208	0.535	${\mathit{\gamma }}_{3,term(t-1)}^{lg-98.5\%}$	−0.066	0.772	${\mathit{\gamma }}_{3,term(t-1)}^{lg-98\%}$	−0.084	0.640
${\mathit{\gamma }}_{4,dff(t-1)}^{lg-99\%}$	−2.436**	0.028	${\mathit{\gamma }}_{4,dff(t-1)}^{lg-98.5\%}$	−1.986**	0.048	${\mathit{\gamma }}_{4,dff(t-1)}^{lg-98\%}$	−1.747*	0.065
MF-R^2	0.299330		MF-R^2	0.207457		MF-R^2	0.143596
k = 97.5			k = 97.0			k = 95.0
${\mathit{\omega }}_{0,gf(t)}^{lg-97.5\%}$	−4.411***	0.000	${\mathit{\omega }}_{0,gf(t)}^{lg-97\%}$	−4.226***	0.000	${\mathit{\omega }}_{0,gf(t)}^{lg-95\%}$	−3.960***	0.000
${\mathit{\omega }}_{1,gf(t)}^{lg-97.5\%}$	0.085***	0.008	${\mathit{\omega }}_{1,gf(t)}^{lg-97\%}$	0.066**	0.030	${\mathit{\omega }}_{1,gf(t)}^{lg-95\%}$	0.033	0.186
${\mathit{\omega }}_{2,iv(t-1)}^{lg-97.5\%}$	0.015	0.654	${\mathit{\omega }}_{2,iv(t-1)}^{lg-97\%}$	0.029	0.342	${\mathit{\omega }}_{2,iv(t-1)}^{lg-95\%}$	0.050**	0.046
${\mathit{\gamma }}_{1,def(t-1)}^{lg-97.5\%}$	−0.949**	0.012	${\mathit{\gamma }}_{1,def(t-1)}^{lg-97\%}$	−0.901**	0.011	${\mathit{\gamma }}_{1,def(t-1)}^{lg-95\%}$	−0.594**	0.032
${\mathit{\gamma }}_{2,dex(t-1)}^{lg-97.5\%}$	0.041	0.884	${\mathit{\gamma }}_{2,dex(t-1)}^{lg-97\%}$	0.139	0.606	${\mathit{\gamma }}_{2,dex(t-1)}^{lg-95\%}$	0.110	0.614
${\mathit{\gamma }}_{3,term(t-1)}^{lg-97.5\%}$	−0.156	0.317	${\mathit{\gamma }}_{3,term(t-1)}^{lg-97\%}$	−0.137	0.323	${\mathit{\gamma }}_{3,term(t-1)}^{lg-95\%}$	−0.082	0.447
${\mathit{\gamma }}_{4,dff(t-1)}^{lg-97.5\%}$	−1.299	0.157	${\mathit{\gamma }}_{4,dff(t-1)}^{lg-97\%}$	−1.319	0.134	${\mathit{\gamma }}_{4,dff(t-1)}^{lg-95\%}$	−0.476	0.542
MF-R^2	0.115516		MF-R^2	0.101040		MF-R^2	0.086715
Panel B: Forecast volatility from the EGARCH model versus the previous day’s VIX
k = 99.0			k = 98.5			k = 98.0
${\mathit{\omega }}_{0,egf(t)}^{lg-99\%}$	−6.760***	0.000	${\mathit{\omega }}_{0,egf(t)}^{lg-98.5\%}$	−5.763***	0.000	${\mathit{\omega }}_{0,egf(t)}^{lg-98\%}$	−4.956***	0.000
${\mathit{\omega }}_{1,egf(t)}^{lg-99\%}$	0.241***	0.000	${\mathit{\omega }}_{1,egf(t)}^{lg-98.5\%}$	0.152***	0.002	${\mathit{\omega }}_{1,egf(t)}^{lg-98\%}$	0.162***	0.000
${\mathit{\omega }}_{2,iv(t-1)}^{lg-99\%}$	−0.124**	0.049	${\mathit{\omega }}_{2,iv(t-1)}^{lg-98.5\%}$	−0.046	0.370	${\mathit{\omega }}_{2,iv(t-1)}^{lg-98\%}$	−0.063	0.188
${\mathit{\gamma }}_{1,def(t-1)}^{lg-99\%}$	−0.719	0.200	${\mathit{\gamma }}_{1,def(t-1)}^{lg-98.5\%}$	−0.536	0.218	${\mathit{\gamma }}_{1,def(t-1)}^{lg-98\%}$	−0.638	0.107
${\mathit{\gamma }}_{2,dex(t-1)}^{lg-99\%}$	−0.177	0.633	${\mathit{\gamma }}_{2,dex(t-1)}^{lg-98.5\%}$	−0.132	0.674	${\mathit{\gamma }}_{2,dex(t-1)}^{lg-98\%}$	−0.139	0.642
${\mathit{\gamma }}_{3,term(t-1)}^{lg-99\%}$	0.271	0.416	${\mathit{\gamma }}_{3,term(t-1)}^{lg-98.5\%}$	−0.013	0.956	${\mathit{\gamma }}_{3,term(t-1)}^{lg-98\%}$	−0.026	0.887
${\mathit{\gamma }}_{4,dff(t-1)}^{lg-99\%}$	−1.771*	0.080	${\mathit{\gamma }}_{4,dff(t-1)}^{lg-98.5\%}$	−1.619*	0.093	${\mathit{\gamma }}_{4,dff(t-1)}^{lg-98\%}$	−1.398	0.113
MF-R^2	0.297319		MF-R^2	0.205427		MF-R^2	0.151899
k = 97.5			k = 97.0			k = 95.0
${\mathit{\omega }}_{0,egf(t)}^{lg-97.5\%}$	−4.614***	0.000	${\mathit{\omega }}_{0,egf(t)}^{lg-97\%}$	−4.361***	0.000	${\mathit{\omega }}_{0,egf(t)}^{lg-95\%}$	−4.045***	0.000
${\mathit{\omega }}_{1,egf(t)}^{lg-97.5\%}$	0.124***	0.003	${\mathit{\omega }}_{1,egf(t)}^{lg-97\%}$	0.110***	0.005	${\mathit{\omega }}_{1,egf(t)}^{lg-95\%}$	0.045	0.148
${\mathit{\omega }}_{2,iv(t-1)}^{lg-97.5\%}$	−0.032	0.456	${\mathit{\omega }}_{2,iv(t-1)}^{lg-97\%}$	−0.020	0.629	${\mathit{\omega }}_{2,iv(t-1)}^{lg-95\%}$	0.035	0.292
${\mathit{\gamma }}_{1,def(t-1)}^{lg-97.5\%}$	−0.526	0.135	${\mathit{\gamma }}_{1,def(t-1)}^{lg-97\%}$	−0.585*	0.075	${\mathit{\gamma }}_{1,def(t-1)}^{lg-95\%}$	−0.429*	0.088
${\mathit{\gamma }}_{2,dex(t-1)}^{lg-97.5\%}$	−0.059	0.832	${\mathit{\gamma }}_{2,dex(t-1)}^{lg-97\%}$	0.068	0.802	${\mathit{\gamma }}_{2,dex(t-1)}^{lg-95\%}$	0.078	0.720
${\mathit{\gamma }}_{3,term(t-1)}^{lg-97.5\%}$	−0.108	0.496	${\mathit{\gamma }}_{3,term(t-1)}^{lg-97\%}$	−0.090	0.525	${\mathit{\gamma }}_{3,term(t-1)}^{lg-95\%}$	−0.062	0.575
${\mathit{\gamma }}_{4,dff(t-1)}^{lg-97.5\%}$	−1.072	0.211	${\mathit{\gamma }}_{4,dff(t-1)}^{lg-97\%}$	−1.115	0.179	${\mathit{\gamma }}_{4,dff(t-1)}^{lg-95\%}$	−0.445	0.558
MF-R^2	0.205427		MF-R^2	0.107021		MF-R^2	0.087144
Panel C: Forecast volatility from the PGARCH model versus the previous day’s VIX
k = 99.0			k = 98.5			k = 98.0
${\mathit{\omega }}_{0,pgf(t)}^{lg-99\%}$	−6.039***	0.000	${\mathit{\omega }}_{0,pgf(t)}^{lg-98.5\%}$	−5.397***	0.000	${\mathit{\omega }}_{0,pgf(t)}^{lg-98\%}$	−4.566***	0.000
${\mathit{\omega }}_{1,pgf(t)}^{lg-99\%}$	0.223***	0.000	${\mathit{\omega }}_{1,pgf(t)}^{lg-98.5\%}$	0.126***	0.006	${\mathit{\omega }}_{1,pgf(t)}^{lg-98\%}$	0.138***	0.001
${\mathit{\omega }}_{2,iv(t-1)}^{lg-99\%}$	−0.121*	0.060	${\mathit{\omega }}_{2,iv(t-1)}^{lg-98.5\%}$	−0.031	0.540	${\mathit{\omega }}_{2,iv(t-1)}^{lg-98\%}$	−0.050	0.295
${\mathit{\gamma }}_{1,def(t-1)}^{lg-99\%}$	−1.259**	0.035	${\mathit{\gamma }}_{1,def(t-1)}^{lg-98.5\%}$	−0.788*	0.086	${\mathit{\gamma }}_{1,def(t-1)}^{lg-98\%}$	−0.908**	0.029
${\mathit{\gamma }}_{2,dex(t-1)}^{lg-99\%}$	−0.129	0.728	${\mathit{\gamma }}_{2,dex(t-1)}^{lg-98.5\%}$	−0.106	0.736	${\mathit{\gamma }}_{2,dex(t-1)}^{lg-98\%}$	−0.110	0.712
${\mathit{\gamma }}_{3,term(t-1)}^{lg-99\%}$	0.312	0.338	${\mathit{\gamma }}_{3,term(t-1)}^{lg-98.5\%}$	0.009	0.970	${\mathit{\gamma }}_{3,term(t-1)}^{lg-98\%}$	−0.007	0.971
${\mathit{\gamma }}_{4,dff(t-1)}^{lg-99\%}$	−1.940*	0.069	${\mathit{\gamma }}_{4,dff(t-1)}^{lg-98.5\%}$	−1.762*	0.079	${\mathit{\gamma }}_{4,dff(t-1)}^{lg-98\%}$	−1.511	0.101
MF-R^2	0.293792		MF-R^2	0.200143		MF-R^2	0.146952
k = 97.5			k = 97.0			k = 95.0
${\mathit{\omega }}_{0,pgf(t)}^{lg-97.5\%}$	−4.349***	0.000	${\mathit{\omega }}_{0,pgf(t)}^{lg-97\%}$	−4.152***	0.000	${\mathit{\omega }}_{0,pgf(t)}^{lg-95\%}$	−3.999***	0.000
${\mathit{\omega }}_{1,pgf(t)}^{lg-97.5\%}$	0.100***	0.009	${\mathit{\omega }}_{1,pgf(t)}^{lg-97\%}$	0.084**	0.019	${\mathit{\omega }}_{1,pgf(t)}^{lg-95\%}$	0.027	0.348
${\mathit{\omega }}_{2,iv(t-1)}^{lg-97.5\%}$	−0.017	0.692	${\mathit{\omega }}_{2,iv(t-1)}^{lg-97\%}$	−0.002	0.965	${\mathit{\omega }}_{2,iv(t-1)}^{lg-95\%}$	0.050	0.121
${\mathit{\gamma }}_{1,def(t-1)}^{lg-97.5\%}$	−0.708*	0.052	${\mathit{\gamma }}_{1,def(t-1)}^{lg-97\%}$	−0.729**	0.031	${\mathit{\gamma }}_{1,def(t-1)}^{lg-95\%}$	−0.471*	0.064
${\mathit{\gamma }}_{2,dex(t-1)}^{lg-97.5\%}$	−0.040	0.885	${\mathit{\gamma }}_{2,dex(t-1)}^{lg-97\%}$	0.081	0.763	${\mathit{\gamma }}_{2,dex(t-1)}^{lg-95\%}$	0.080	0.713
${\mathit{\gamma }}_{3,term(t-1)}^{lg-97.5\%}$	−0.097	0.537	${\mathit{\gamma }}_{3,term(t-1)}^{lg-97\%}$	−0.087	0.539	${\mathit{\gamma }}_{3,term(t-1)}^{lg-95\%}$	−0.070	0.524
${\mathit{\gamma }}_{4,dff(t-1)}^{lg-97.5\%}$	−1.154	0.196	${\mathit{\gamma }}_{4,dff(t-1)}^{lg-97\%}$	−1.197	0.165	${\mathit{\gamma }}_{4,dff(t-1)}^{lg-95\%}$	−0.465	0.551
MF-R^2	0.115641		MF-R^2	0.102656		MF-R^2	0.085669
Panel D: Forecast volatility from the TGARCH model versus the previous day’s VIX
k = 99.0			k = 98.5			k = 98.0
${\mathit{\omega }}_{0,tgf(t)}^{lg-99\%}$	−6.524***	0.000	${\mathit{\omega }}_{0,tgf(t)}^{lg-98.5\%}$	−5.498***	0.000	${\mathit{\omega }}_{0,tgf(t)}^{lg-98\%}$	−4.716***	0.000
${\mathit{\omega }}_{1,tgf(t)}^{lg-99\%}$	0.183***	0.000	${\mathit{\omega }}_{1,tgf(t)}^{lg-98.5\%}$	0.125***	0.000	${\mathit{\omega }}_{1,tgf(t)}^{lg-98\%}$	0.118***	0.000
${\mathit{\omega }}_{2,iv(t-1)}^{lg-99\%}$	−0.095*	0.075	${\mathit{\omega }}_{2,iv(t-1)}^{lg-98.5\%}$	−0.037	0.391	${\mathit{\omega }}_{2,iv(t-1)}^{lg-98\%}$	−0.038	0.340
${\mathit{\gamma }}_{1,def(t-1)}^{lg-99\%}$	−0.916*	0.071	${\mathit{\gamma }}_{1,def(t-1)}^{lg-98.5\%}$	−0.703*	0.087	${\mathit{\gamma }}_{1,def(t-1)}^{lg-98\%}$	−0.756**	0.043
${\mathit{\gamma }}_{2,dex(t-1)}^{lg-99\%}$	−0.155	0.676	${\mathit{\gamma }}_{2,dex(t-1)}^{lg-98.5\%}$	−0.103	0.743	${\mathit{\gamma }}_{2,dex(t-1)}^{lg-98\%}$	−0.109	0.713
${\mathit{\gamma }}_{3,term(t-1)}^{lg-99\%}$	0.322	0.349	${\mathit{\gamma }}_{3,term(t-1)}^{lg-98.5\%}$	0.003	0.989	${\mathit{\gamma }}_{3,term(t-1)}^{lg-98\%}$	−0.024	0.896
${\mathit{\gamma }}_{4,dff(t-1)}^{lg-99\%}$	−1.698*	0.093	${\mathit{\gamma }}_{4,dff(t-1)}^{lg-98.5\%}$	−1.541	0.103	${\mathit{\gamma }}_{4,dff(t-1)}^{lg-98\%}$	−1.388	0.117
MF-R^2	0.315876		MF-R^2	0.217889		MF-R^2	0.154566
k = 97.5			k = 97.0			k = 95.0
${\mathit{\omega }}_{0,tgf(t)}^{lg-97.5\%}$	−4.394***	0.000	${\mathit{\omega }}_{0,tgf(t)}^{lg-97\%}$	−4.206***	0.000	${\mathit{\omega }}_{0,tgf(t)}^{lg-95\%}$	−3.944***	0.000
${\mathit{\omega }}_{1,tgf(t)}^{lg-97.5\%}$	0.098***	0.001	${\mathit{\omega }}_{1,tgf(t)}^{lg-97\%}$	0.077***	0.005	${\mathit{\omega }}_{1,tgf(t)}^{lg-95\%}$	0.040*	0.086
${\mathit{\omega }}_{2,iv(t-1)}^{lg-97.5\%}$	−0.021	0.578	${\mathit{\omega }}_{2,iv(t-1)}^{lg-97\%}$	0.001	0.984	${\mathit{\omega }}_{2,iv(t-1)}^{lg-95\%}$	0.035	0.232
${\mathit{\gamma }}_{1,def(t-1)}^{lg-97.5\%}$	−0.637*	0.060	${\mathit{\gamma }}_{1,def(t-1)}^{lg-97\%}$	−0.665**	0.036	${\mathit{\gamma }}_{1,def(t-1)}^{lg-95\%}$	−0.476*	0.057
${\mathit{\gamma }}_{2,dex(t-1)}^{lg-97.5\%}$	−0.033	0.906	${\mathit{\gamma }}_{2,dex(t-1)}^{lg-97\%}$	0.087	0.746	${\mathit{\gamma }}_{2,dex(t-1)}^{lg-95\%}$	0.089	0.683
${\mathit{\gamma }}_{3,term(t-1)}^{lg-97.5\%}$	−0.102	0.513	${\mathit{\gamma }}_{3,term(t-1)}^{lg-97\%}$	−0.095	0.498	${\mathit{\gamma }}_{3,term(t-1)}^{lg-95\%}$	−0.057	0.600
${\mathit{\gamma }}_{4,dff(t-1)}^{lg-97.5\%}$	−1.058	0.212	${\mathit{\gamma }}_{4,dff(t-1)}^{lg-97\%}$	−1.127	0.176	${\mathit{\gamma }}_{4,dff(t-1)}^{lg-95\%}$	−0.447	0.552
MF-R^2	0.123971		MF-R^2	0.106240		MF-R^2	0.088127

Notes: This table exhibits the results of empirical comparisons of the predictive power of the one-day lagged VIX of S&P 500 and the forecast S&P 500 volatilities from various GARCH models. These tests are conducted in our out-of-sample period that is from 3 January 2006 to 28 February 2014, and all investigations are conducted by using multiple logit models. Each multiple logit model includes a constant term, the one-day lagged VIX, the forecast volatility from one of the four GARCH models, and four control variables. Our control variables are the one-day lagged default spread, the one-day lag of the first-difference series of the trade-weighted US dollar index, the one-day lagged term premium, and the one-day lag of the first-difference series of the effective federal funds rate. All GARCH models used to derive the forecast volatilities are specified in our in-sample period, which is from 2 January 1990 to 30 December 2005. Further, MF-R² in this table denotes the McFadden’s R-squared value.

*Statistical significance of coefficients at the 10% level.

**Statistical significance of coefficients at the 5% level.

***Statistical significance of coefficients at the 1% level.

Table 5. Testing the predictive power for downside risk in the US stock market by univariate quantile regressions: results for the VIX and the forecast volatilities from several GARCH models

Coefficients	Estimates	p-value	Coefficients	Estimates	p-value	Coefficients	Estimates	p-value
Panel A: Downside risk predictive power of the previous day’s VIX
j = 1.0			j = 1.5			j = 2.0
$l_{0,iv(t-1)}^{Q-1\%}$	−9.621	0.178	$l_{0,iv(t-1)}^{Q-1.5\%}$	−7.923**	0.033	$l_{0,iv(t-1)}^{Q-2\%}$	−7.852**	0.029
$l_{1,iv(t-1)}^{Q-1\%}$	−1.381***	0.000	$l_{1,iv(t-1)}^{Q-1.5\%}$	−1.304***	0.000	$l_{1,iv(t-1)}^{Q-2\%}$	−1.231***	0.000
Adj. R^2	0.168087		Adj. R^2	0.157998		Adj. R^2	0.148048
j = 2.5			j = 3.0			j = 5.0
$l_{0,iv(t-1)}^{Q-2.5\%}$	−6.160*	0.086	$l_{0,iv(t-1)}^{Q-3\%}$	−8.153***	0.004	$l_{0,iv(t-1)}^{Q-5\%}$	−3.168	0.268
$l_{1,iv(t-1)}^{Q-2.5\%}$	−1.197***	0.000	$l_{1,iv(t-1)}^{Q-3\%}$	−1.009***	0.000	$l_{1,iv(t-1)}^{Q-5\%}$	−1.031***	0.000
Adj. R^2	0.139410		Adj. R^2	0.133981		Adj. R^2	0.107816
Panel B: Downside risk predictive power of the forecast volatility form the GARCH model
j = 1.0			j = 1.5			j = 2.0
$l_{0,gf(t)}^{Q-1\%}$	−20.751***	0.000	$l_{0,gf(t)}^{Q-1.5\%}$	−17.039***	0.000	$l_{0,gf(t)}^{Q-2\%}$	−15.407***	0.000
$l_{1,gf(t)}^{Q-1\%}$	−1.045***	0.000	$l_{1,gf(t)}^{Q-1.5\%}$	−1.058***	0.000	$l_{1,gf(t)}^{Q-2\%}$	−0.985***	0.000
Adj. R^2	0.165250		Adj. R^2	0.157120		Adj. R^2	0.145393
j = 2.5			j = 3.0			j = 5.0
$l_{0,gf(t)}^{Q-2.5\%}$	−13.295***	0.000	$l_{0,gf(t)}^{Q-3\%}$	−11.773***	0.000	$l_{0,gf(t)}^{Q-5\%}$	−11.902***	0.000
$l_{1,gf(t)}^{Q-2.5\%}$	−1.017***	0.000	$l_{1,gf(t)}^{Q-3\%}$	−0.977***	0.000	$l_{1,gf(t)}^{Q-5\%}$	−0.716***	0.000
Adj. R^2	0.138662		Adj. R^2	0.131503		Adj. R^2	0.106557
Panel C: Downside risk predictive power of the forecast volatility form the EGARCH model
j = 1.0			j = 1.5			j = 2.0
$l_{0,egf(t)}^{Q-1\%}$	−16.446***	0.000	$l_{0,egf(t)}^{Q-1.5\%}$	−13.566***	0.000	$l_{0,egf(t)}^{Q-2\%}$	−11.693***	0.000
$l_{1,egf(t)}^{Q-1\%}$	−1.223***	0.000	$l_{1,egf(t)}^{Q-1.5\%}$	−1.254***	0.000	$l_{1,egf(t)}^{Q-2\%}$	−1.244***	0.000
Adj. R^2	0.194512		Adj. R^2	0.186060		Adj. R^2	0.175194
j = 2.5			j = 3.0			j = 5.0
$l_{0,egf(t)}^{Q-2.5\%}$	−12.063***	0.000	$l_{0,egf(t)}^{Q-3\%}$	−10.509***	0.000	$l_{0,egf(t)}^{Q-5\%}$	−7.745***	0.004
$l_{1,egf(t)}^{Q-2.5\%}$	−1.128***	0.000	$l_{1,egf(t)}^{Q-3\%}$	−1.128***	0.000	$l_{1,egf(t)}^{Q-5\%}$	−1.038***	0.000
Adj. R^2	0.165310		Adj. R^2	0.155252		Adj. R^2	0.124073
Panel D: Downside risk predictive power of the forecast volatility form the PGARCH model
j = 1.0			j = 1.5			j = 2.0
$l_{0,pgf(t)}^{Q-1\%}$	−17.997**	0.022	$l_{0,pgf(t)}^{Q-1.5\%}$	−16.334***	0.000	$l_{0,pgf(t)}^{Q-2\%}$	−14.557***	0.000
$l_{1,pgf(t)}^{Q-1\%}$	−1.128**	0.019	$l_{1,pgf(t)}^{Q-1.5\%}$	−1.082***	0.000	$l_{1,pgf(t)}^{Q-2\%}$	−1.054***	0.000
Adj. R^2	0.189403		Adj. R^2	0.180956		Adj. R^2	0.171419
j = 2.5			j = 3.0			j = 5.0
$l_{0,pgf(t)}^{Q-2.5\%}$	−13.310***	0.000	$l_{0,pgf(t)}^{Q-3\%}$	−11.015***	0.000	$l_{0,pgf(t)}^{Q-5\%}$	−8.737***	0.001
$l_{1,pgf(t)}^{Q-2.5\%}$	−1.036***	0.000	$l_{1,pgf(t)}^{Q-3\%}$	−1.074***	0.000	$l_{1,pgf(t)}^{Q-5\%}$	−0.958***	0.000
Adj. R^2	0.162030		Adj. R^2	0.151920		Adj. R^2	0.121389
Panel E: Downside risk predictive power of the forecast volatility form the TGARCH model
j = 1.0			j = 1.5			j = 2.0
$l_{0,tgf(t)}^{Q-1\%}$	−16.818**	0.012	$l_{0,tgf(t)}^{Q-1.5\%}$	−15.898**	0.022	$l_{0,tgf(t)}^{Q-2\%}$	−14.515***	0.000
$l_{1,tgf(t)}^{Q-1\%}$	−1.148***	0.005	$l_{1,tgf(t)}^{Q-1.5\%}$	−1.089**	0.015	$l_{1,tgf(t)}^{Q-2\%}$	−1.019***	0.000
Adj. R^2	0.196455		Adj. R^2	0.184850		Adj. R^2	0.172468
j = 2.5			j = 3.0			j = 5.0
$l_{0,tgf(t)}^{Q-2.5\%}$	−13.714***	0.000	$l_{0,tgf(t)}^{Q-3\%}$	−11.896***	0.000	$l_{0,tgf(t)}^{Q-5\%}$	−9.496***	0.000
$l_{1,tgf(t)}^{Q-2.5\%}$	−0.969***	0.000	$l_{1,tgf(t)}^{Q-3\%}$	−0.964***	0.000	$l_{1,tgf(t)}^{Q-5\%}$	−0.876***	0.000
Adj. R^2	0.162715		Adj. R^2	0.154355		Adj. R^2	0.123703

Notes: This table exhibits the results of empirical comparisons of the predictive power of the one-day lagged VIX of S&P 500 and the forecast S&P 500 volatilities from various GARCH models. These tests are conducted in our out-of-sample period that is from 3 January 2006 to 28 February 2014, and all examinations are conducted using univariate quantile regression models. In the tests, we use the one-day lagged VIX close values and the out-of-sample forecast volatilities from GARCH(1,1), EGARCH(1,1), PGARCH(1,1), and TGARCH(1,1) models. All above models used to derive the forecast volatilities are specified in our in-sample period, which is from 2 January 1990 to 30 December 2005. Further, Adj. R² in this table denotes the adjusted R-squared value.

*Statistical significance of coefficients at the 10% level.

**Statistical significance of coefficients at the 5% level.

***Statistical significance of coefficients at the 1% level.

Table 6. Testing the predictive power for downside risk in the US stock market by multiple quantile regressions: the VIX versus the forecast volatilities from several GARCH models

Coefficients	Estimates	p-value	Coefficients	Estimates	p-value	Coefficients	Estimates	p-value
Panel A: Forecast volatility from the GARCH model versus the previous day’s VIX
j = 1.0			j = 1.5			j = 2.0
${\mathit{\theta }}_{0,gf(t)}^{Q-1\%}$	−16.436**	0.023	${\mathit{\theta }}_{0,gf(t)}^{Q-1.5\%}$	−13.191***	0.001	${\mathit{\theta }}_{0,gf(t)}^{Q-2\%}$	−10.820***	0.003
${\mathit{\theta }}_{1,gf(t)}^{Q-1\%}$	−0.263	0.652	${\mathit{\theta }}_{1,gf(t)}^{Q-1.5\%}$	−0.616**	0.034	${\mathit{\theta }}_{1,gf(t)}^{Q-2\%}$	−0.478	0.116
${\mathit{\theta }}_{2,iv(t-1)}^{Q-1\%}$	−0.845	0.291	${\mathit{\theta }}_{2,iv(t-1)}^{Q-1.5\%}$	−0.519	0.166	${\mathit{\theta }}_{2,iv(t-1)}^{Q-2\%}$	−0.685*	0.080
Adj. R^2	0.168743		Adj. R^2	0.163054		Adj. R^2	0.152365
j = 2.5			j = 3.0			j = 5.0
${\mathit{\theta }}_{0,gf(t)}^{Q-2.5\%}$	−7.769**	0.018	${\mathit{\theta }}_{0,gf(t)}^{Q-3\%}$	−8.465**	0.027	${\mathit{\theta }}_{0,gf(t)}^{Q-5\%}$	−8.116**	0.017
${\mathit{\theta }}_{1,gf(t)}^{Q-2.5\%}$	−0.485	0.118	${\mathit{\theta }}_{1,gf(t)}^{Q-3\%}$	−0.111	0.762	${\mathit{\theta }}_{1,gf(t)}^{Q-5\%}$	−0.413*	0.055
${\mathit{\theta }}_{2,iv(t-1)}^{Q-2.5\%}$	−0.731*	0.057	${\mathit{\theta }}_{2,iv(t-1)}^{Q-3\%}$	−0.899**	0.028	${\mathit{\theta }}_{2,iv(t-1)}^{Q-5\%}$	−0.441	0.156
Adj. R^2	0.142256		Adj. R^2	0.133822		Adj. R^2	0.110916
Panel B: Forecast volatility from the EGARCH model versus the previous day’s VIX
j = 1.0			j = 1.5			j = 2.0
${\mathit{\theta }}_{0,egf(t)}^{Q-1\%}$	−22.024***	0.000	${\mathit{\theta }}_{0,egf(t)}^{Q-1.5\%}$	−17.575***	0.000	${\mathit{\theta }}_{0,egf(t)}^{Q-2\%}$	−12.939***	0.000
${\mathit{\theta }}_{1,egf(t)}^{Q-1\%}$	−2.030***	0.000	${\mathit{\theta }}_{1,egf(t)}^{Q-1.5\%}$	−1.858***	0.000	${\mathit{\theta }}_{1,egf(t)}^{Q-2\%}$	−1.725***	0.000
${\mathit{\theta }}_{2,iv(t-1)}^{Q-1\%}$	0.893***	0.008	${\mathit{\theta }}_{2,iv(t-1)}^{Q-1.5\%}$	0.651*	0.072	${\mathit{\theta }}_{2,iv(t-1)}^{Q-2\%}$	0.439	0.229
Adj. R^2	0.200778		Adj. R^2	0.189994		Adj. R^2	0.176695
j = 2.5			j = 3.0			j = 5.0
${\mathit{\theta }}_{0,egf(t)}^{Q-2.5\%}$	−14.036***	0.000	${\mathit{\theta }}_{0,egf(t)}^{Q-3\%}$	−12.598***	0.000	${\mathit{\theta }}_{0,egf(t)}^{Q-5\%}$	−8.128**	0.018
${\mathit{\theta }}_{1,egf(t)}^{Q-2.5\%}$	−1.503***	0.000	${\mathit{\theta }}_{1,egf(t)}^{Q-3\%}$	−1.550***	0.000	${\mathit{\theta }}_{1,egf(t)}^{Q-5\%}$	−1.079***	0.008
${\mathit{\theta }}_{2,iv(t-1)}^{Q-2.5\%}$	0.386	0.192	${\mathit{\theta }}_{2,iv(t-1)}^{Q-3\%}$	0.420	0.158	${\mathit{\theta }}_{2,iv(t-1)}^{Q-5\%}$	0.052	0.892
Adj. R^2	0.167572		Adj. R^2	0.156119		Adj. R^2	0.123669
Panel C: Forecast volatility from the PGARCH model versus the previous day’s VIX
j = 1.0			j = 1.5			j = 2.0
${\mathit{\theta }}_{0,pgf(t)}^{Q-1\%}$	−23.259*	0.096	${\mathit{\theta }}_{0,pgf(t)}^{Q-1.5\%}$	−19.873***	0.000	${\mathit{\theta }}_{0,pgf(t)}^{Q-2\%}$	−16.265***	0.000
${\mathit{\theta }}_{1,pgf(t)}^{Q-1\%}$	−1.532**	0.013	${\mathit{\theta }}_{1,pgf(t)}^{Q-1.5\%}$	−1.479***	0.000	${\mathit{\theta }}_{1,pgf(t)}^{Q-2\%}$	−1.242***	0.000
${\mathit{\theta }}_{2,iv(t-1)}^{Q-1\%}$	0.567	0.412	${\mathit{\theta }}_{2,iv(t-1)}^{Q-1.5\%}$	0.512	0.275	${\mathit{\theta }}_{2,iv(t-1)}^{Q-2\%}$	0.243	0.591
Adj. R^2	0.192705		Adj. R^2	0.182644		Adj. R^2	0.171725
j = 2.5			j = 3.0			j = 5.0
${\mathit{\theta }}_{0,pgf(t)}^{Q-2.5\%}$	−15.943***	0.000	${\mathit{\theta }}_{0,pgf(t)}^{Q-3\%}$	−14.359***	0.000	${\mathit{\theta }}_{0,pgf(t)}^{Q-5\%}$	−8.575***	0.005
${\mathit{\theta }}_{1,pgf(t)}^{Q-2.5\%}$	−1.447***	0.000	${\mathit{\theta }}_{1,pgf(t)}^{Q-3\%}$	−1.436***	0.000	${\mathit{\theta }}_{1,pgf(t)}^{Q-5\%}$	−0.940**	0.021
${\mathit{\theta }}_{2,iv(t-1)}^{Q-2.5\%}$	0.472	0.169	${\mathit{\theta }}_{2,iv(t-1)}^{Q-3\%}$	0.434	0.209	${\mathit{\theta }}_{2,iv(t-1)}^{Q-5\%}$	−0.022	0.952
Adj. R^2	0.164183		Adj. R^2	0.153320		Adj. R^2	0.120977
Panel D: Forecast volatility from the TGARCH model versus the previous day’s VIX
j = 1.0			j = 1.5			j = 2.0
${\mathit{\theta }}_{0,tgf(t)}^{Q-1\%}$	−21.119*	0.054	${\mathit{\theta }}_{0,tgf(t)}^{Q-1.5\%}$	−20.075	0.104	${\mathit{\theta }}_{0,tgf(t)}^{Q-2\%}$	−15.780***	0.010
${\mathit{\theta }}_{1,tgf(t)}^{Q-1\%}$	−1.508***	0.000	${\mathit{\theta }}_{1,tgf(t)}^{Q-1.5\%}$	−1.514***	0.001	${\mathit{\theta }}_{1,tgf(t)}^{Q-2\%}$	−1.138**	0.018
${\mathit{\theta }}_{2,iv(t-1)}^{Q-1\%}$	0.525	0.402	${\mathit{\theta }}_{2,iv(t-1)}^{Q-1.5\%}$	0.530	0.482	${\mathit{\theta }}_{2,iv(t-1)}^{Q-2\%}$	0.161	0.808
Adj. R^2	0.200419		Adj. R^2	0.186502		Adj. R^2	0.172670
j = 2.5			j = 3.0			j = 5.0
${\mathit{\theta }}_{0,tgf(t)}^{Q-2.5\%}$	−16.563***	0.001	${\mathit{\theta }}_{0,tgf(t)}^{Q-3\%}$	−12.963***	0.003	${\mathit{\theta }}_{0,tgf(t)}^{Q-5\%}$	−9.740***	0.003
${\mathit{\theta }}_{1,tgf(t)}^{Q-2.5\%}$	−1.308***	0.002	${\mathit{\theta }}_{1,tgf(t)}^{Q-3\%}$	−0.992***	0.000	${\mathit{\theta }}_{1,tgf(t)}^{Q-5\%}$	−0.936***	0.000
${\mathit{\theta }}_{2,iv(t-1)}^{Q-2.5\%}$	0.422	0.435	${\mathit{\theta }}_{2,iv(t-1)}^{Q-3\%}$	0.072	0.837	${\mathit{\theta }}_{2,iv(t-1)}^{Q-5\%}$	0.059	0.848
Adj. R^2	0.163315		Adj. R^2	0.154202		Adj. R^2	0.123355

Notes: This table exhibits the results of empirical comparisons of the predictive power of the one-day lagged VIX of S&P 500 and the forecast S&P 500 volatilities from various GARCH models. These tests are conducted in our out-of-sample period that is from 3 January 2006 to 28 February 2014. All examinations are conducted by using the multiple quantile regression models. Each multiple quantile regression model includes a constant term, the one-day lagged VIX, and the forecast volatility from one of the four GARCH models. In the tests, we use the one-day lagged VIX close values and the out-of-sample forecast volatilities derived from GARCH(1,1), EGARCH(1,1), PGARCH(1,1), and TGARCH(1,1) models. All above models used to derive the forecast volatilities are specified in our in-sample period, which is from 2 January 1990 to 30 December 2005. In this table, Adj. R² denotes the adjusted R-squared value.

*Statistical significance of coefficients at the 10% level.

**Statistical significance of coefficients at the 5% level.

***Statistical significance of coefficients at the 1% level.

Table 7. Testing the predictive power for downside risk in the US stock market by multiple quantile regression models with control variables: the VIX versus the forecast volatilities from several GARCH models

Coefficients	Estimates	p-value	Coefficients	Estimates	p-value	Coefficients	Estimates	p-value
Panel A: Forecast volatility from the GARCH model versus the previous day’s VIX
j = 1.0			j = 1.5			j = 2.0
${\mathit{\chi }}_{0,gf(t)}^{Q-1\%}$	−21.689*	0.055	${\mathit{\chi }}_{0,gf(t)}^{Q-1.5\%}$	−17.317*	0.090	${\mathit{\chi }}_{0,gf(t)}^{Q-2\%}$	−19.288***	0.000
${\mathit{\chi }}_{1,gf(t)}^{Q-1\%}$	−1.060	0.207	${\mathit{\chi }}_{1,gf(t)}^{Q-1.5\%}$	−0.815	0.296	${\mathit{\chi }}_{1,gf(t)}^{Q-2\%}$	−0.677	0.303
${\mathit{\chi }}_{2,iv(t-1)}^{Q-1\%}$	−0.620	0.530	${\mathit{\chi }}_{2,iv(t-1)}^{Q-1.5\%}$	−0.953	0.290	${\mathit{\chi }}_{2,iv(t-1)}^{Q-2\%}$	−1.144	0.116
${\mathit{\gamma }}_{1,def(t-1)}^{Q-1\%}$	11.809*	0.059	${\mathit{\gamma }}_{1,def(t-1)}^{Q-1.5\%}$	12.262*	0.062	${\mathit{\gamma }}_{1,def(t-1)}^{Q-2\%}$	16.514**	0.016
${\mathit{\gamma }}_{2,dex(t-1)}^{Q-1\%}$	3.941	0.656	${\mathit{\gamma }}_{2,dex(t-1)}^{Q-1.5\%}$	4.282	0.625	${\mathit{\gamma }}_{2,dex(t-1)}^{Q-2\%}$	−0.547	0.904
${\mathit{\gamma }}_{3,term(t-1)}^{Q-1\%}$	1.346	0.243	${\mathit{\gamma }}_{3,term(t-1)}^{Q-1.5\%}$	1.095	0.330	${\mathit{\gamma }}_{3,term(t-1)}^{Q-2\%}$	1.596	0.104
${\mathit{\gamma }}_{4,dff(t-1)}^{Q-1\%}$	15.486***	0.002	${\mathit{\gamma }}_{4,dff(t-1)}^{Q-1.5\%}$	17.720***	0.001	${\mathit{\gamma }}_{4,dff(t-1)}^{Q-2\%}$	16.845*	0.053
Adj. R^2	0.199998		Adj. R^2	0.190151		Adj. R^2	0.175999
j = 2.5			j = 3.0			j = 5.0
${\mathit{\chi }}_{0,gf(t)}^{Q-2.5\%}$	−17.518***	0.000	${\mathit{\chi }}_{0,gf(t)}^{Q-3\%}$	−15.791***	0.000	${\mathit{\chi }}_{0,gf(t)}^{Q-5\%}$	−10.663***	0.001
${\mathit{\chi }}_{1,gf(t)}^{Q-2.5\%}$	−0.588	0.218	${\mathit{\chi }}_{1,gf(t)}^{Q-3\%}$	−0.357*	0.076	${\mathit{\chi }}_{1,gf(t)}^{Q-5\%}$	−0.475**	0.026
${\mathit{\chi }}_{2,iv(t-1)}^{Q-2.5\%}$	−0.992*	0.093	${\mathit{\chi }}_{2,iv(t-1)}^{Q-3\%}$	−1.255***	0.001	${\mathit{\chi }}_{2,iv(t-1)}^{Q-5\%}$	−1.043***	0.002
${\mathit{\gamma }}_{1,def(t-1)}^{Q-2.5\%}$	12.165**	0.039	${\mathit{\gamma }}_{1,def(t-1)}^{Q-3\%}$	13.042***	0.003	${\mathit{\gamma }}_{1,def(t-1)}^{Q-5\%}$	12.272***	0.000
${\mathit{\gamma }}_{2,dex(t-1)}^{Q-2.5\%}$	−0.934	0.836	${\mathit{\gamma }}_{2,dex(t-1)}^{Q-3\%}$	0.200	0.953	${\mathit{\gamma }}_{2,dex(t-1)}^{Q-5\%}$	1.664	0.464
${\mathit{\gamma }}_{3,term(t-1)}^{Q-2.5\%}$	1.941*	0.067	${\mathit{\gamma }}_{3,term(t-1)}^{Q-3\%}$	1.895*	0.084	${\mathit{\gamma }}_{3,term(t-1)}^{Q-5\%}$	0.662	0.534
${\mathit{\gamma }}_{4,dff(t-1)}^{Q-2.5\%}$	9.812	0.372	${\mathit{\gamma }}_{4,dff(t-1)}^{Q-3\%}$	7.279*	0.065	${\mathit{\gamma }}_{4,dff(t-1)}^{Q-5\%}$	10.787	0.135
Adj. R^2	0.165755		Adj. R^2	0.156493		Adj. R^2	0.125050
Panel B: Forecast volatility from the EGARCH model versus the previous day’s VIX
j = 1.0			j = 1.5			j = 2.0
${\mathit{\chi }}_{0,egf(t)}^{Q-1\%}$	−21.145***	0.000	${\mathit{\chi }}_{0,egf(t)}^{Q-1.5\%}$	−18.377***	0.000	${\mathit{\chi }}_{0,egf(t)}^{Q-2\%}$	−17.142***	0.000
${\mathit{\chi }}_{1,egf(t)}^{Q-1\%}$	−2.167***	0.001	${\mathit{\chi }}_{1,egf(t)}^{Q-1.5\%}$	−1.813***	0.001	${\mathit{\chi }}_{1,egf(t)}^{Q-2\%}$	−1.346***	0.007
${\mathit{\chi }}_{2,iv(t-1)}^{Q-1\%}$	0.669	0.347	${\mathit{\chi }}_{2,iv(t-1)}^{Q-1.5\%}$	0.152	0.758	${\mathit{\chi }}_{2,iv(t-1)}^{Q-2\%}$	−0.203	0.744
${\mathit{\gamma }}_{1,def(t-1)}^{Q-1\%}$	5.593	0.210	${\mathit{\gamma }}_{1,def(t-1)}^{Q-1.5\%}$	8.315*	0.059	${\mathit{\gamma }}_{1,def(t-1)}^{Q-2\%}$	8.434	0.245
${\mathit{\gamma }}_{2,dex(t-1)}^{Q-1\%}$	6.965	0.325	${\mathit{\gamma }}_{2,dex(t-1)}^{Q-1.5\%}$	2.005	0.607	${\mathit{\gamma }}_{2,dex(t-1)}^{Q-2\%}$	2.516	0.617
${\mathit{\gamma }}_{3,term(t-1)}^{Q-1\%}$	−0.728	0.639	${\mathit{\gamma }}_{3,term(t-1)}^{Q-1.5\%}$	0.505	0.687	${\mathit{\gamma }}_{3,term(t-1)}^{Q-2\%}$	0.803	0.511
${\mathit{\gamma }}_{4,dff(t-1)}^{Q-1\%}$	24.679*	0.064	${\mathit{\gamma }}_{4,dff(t-1)}^{Q-1.5\%}$	8.585	0.383	${\mathit{\gamma }}_{4,dff(t-1)}^{Q-2\%}$	16.638***	0.001
Adj. R^2	0.225475		Adj. R^2	0.203841		Adj. R^2	0.188400
j = 2.5			j = 3.0			j = 5.0
${\mathit{\chi }}_{0,egf(t)}^{Q-2.5\%}$	−14.212***	0.000	${\mathit{\chi }}_{0,egf(t)}^{Q-3\%}$	−15.569***	0.000	${\mathit{\chi }}_{0,egf(t)}^{Q-5\%}$	−9.955***	0.000
${\mathit{\chi }}_{1,egf(t)}^{Q-2.5\%}$	−1.170**	0.019	${\mathit{\chi }}_{1,egf(t)}^{Q-3\%}$	−0.978***	0.006	${\mathit{\chi }}_{1,egf(t)}^{Q-5\%}$	−1.041***	0.002
${\mathit{\chi }}_{2,iv(t-1)}^{Q-2.5\%}$	−0.372	0.545	${\mathit{\chi }}_{2,iv(t-1)}^{Q-3\%}$	−0.484	0.345	${\mathit{\chi }}_{2,iv(t-1)}^{Q-5\%}$	−0.374	0.432
${\mathit{\gamma }}_{1,def(t-1)}^{Q-2.5\%}$	8.163	0.242	${\mathit{\gamma }}_{1,def(t-1)}^{Q-3\%}$	10.639***	0.001	${\mathit{\gamma }}_{1,def(t-1)}^{Q-5\%}$	9.447***	0.000
${\mathit{\gamma }}_{2,dex(t-1)}^{Q-2.5\%}$	1.180	0.806	${\mathit{\gamma }}_{2,dex(t-1)}^{Q-3\%}$	0.964	0.762	${\mathit{\gamma }}_{2,dex(t-1)}^{Q-5\%}$	2.349	0.343
${\mathit{\gamma }}_{3,term(t-1)}^{Q-2.5\%}$	0.398	0.746	${\mathit{\gamma }}_{3,term(t-1)}^{Q-3\%}$	0.320	0.801	${\mathit{\gamma }}_{3,term(t-1)}^{Q-5\%}$	−0.464	0.681
${\mathit{\gamma }}_{4,dff(t-1)}^{Q-2.5\%}$	16.400***	0.002	${\mathit{\gamma }}_{4,dff(t-1)}^{Q-3\%}$	7.997	0.380	${\mathit{\gamma }}_{4,dff(t-1)}^{Q-5\%}$	5.993	0.144
Adj. R^2	0.176804		Adj. R^2	0.164761		Adj. R^2	0.133495
Panel C: Forecast volatility from the PGARCH model versus the previous day’s VIX
j = 1.0			j = 1.5			j = 2.0
${\mathit{\chi }}_{0,pgf(t)}^{Q-1\%}$	−28.532***	0.000	${\mathit{\chi }}_{0,pgf(t)}^{Q-1.5\%}$	−23.275***	0.000	${\mathit{\chi }}_{0,pgf(t)}^{Q-2\%}$	−19.839***	0.000
${\mathit{\chi }}_{1,pgf(t)}^{Q-1\%}$	−1.992***	0.000	${\mathit{\chi }}_{1,pgf(t)}^{Q-1.5\%}$	−1.687***	0.000	${\mathit{\chi }}_{1,pgf(t)}^{Q-2\%}$	−1.304***	0.008
${\mathit{\chi }}_{2,iv(t-1)}^{Q-1\%}$	0.572	0.229	${\mathit{\chi }}_{2,iv(t-1)}^{Q-1.5\%}$	0.296	0.535	${\mathit{\chi }}_{2,iv(t-1)}^{Q-2\%}$	−0.208	0.724
${\mathit{\gamma }}_{1,def(t-1)}^{Q-1\%}$	11.675**	0.030	${\mathit{\gamma }}_{1,def(t-1)}^{Q-1.5\%}$	8.986**	0.035	${\mathit{\gamma }}_{1,def(t-1)}^{Q-2\%}$	10.736*	0.099
${\mathit{\gamma }}_{2,dex(t-1)}^{Q-1\%}$	5.089	0.239	${\mathit{\gamma }}_{2,dex(t-1)}^{Q-1.5\%}$	1.903	0.646	${\mathit{\gamma }}_{2,dex(t-1)}^{Q-2\%}$	2.209	0.641
${\mathit{\gamma }}_{3,term(t-1)}^{Q-1\%}$	−0.272	0.848	${\mathit{\gamma }}_{3,term(t-1)}^{Q-1.5\%}$	0.157	0.900	${\mathit{\gamma }}_{3,term(t-1)}^{Q-2\%}$	0.661	0.585
${\mathit{\gamma }}_{4,dff(t-1)}^{Q-1\%}$	21.705*	0.061	${\mathit{\gamma }}_{4,dff(t-1)}^{Q-1.5\%}$	16.511***	0.008	${\mathit{\gamma }}_{4,dff(t-1)}^{Q-2\%}$	17.502***	0.001
Adj. R^2	0.222179		Adj. R^2	0.203816		Adj. R^2	0.188705
j = 2.5			j = 3.0			j = 5.0
${\mathit{\chi }}_{0,pgf(t)}^{Q-2.5\%}$	−17.228***	0.000	${\mathit{\chi }}_{0,pgf(t)}^{Q-3\%}$	−16.114***	0.000	${\mathit{\chi }}_{0,pgf(t)}^{Q-5\%}$	−11.659***	0.000
${\mathit{\chi }}_{1,pgf(t)}^{Q-2.5\%}$	−1.281**	0.012	${\mathit{\chi }}_{1,pgf(t)}^{Q-3\%}$	−1.017*	0.053	${\mathit{\chi }}_{1,pgf(t)}^{Q-5\%}$	−0.963**	0.012
${\mathit{\chi }}_{2,iv(t-1)}^{Q-2.5\%}$	−0.166	0.771	${\mathit{\chi }}_{2,iv(t-1)}^{Q-3\%}$	−0.447	0.389	${\mathit{\chi }}_{2,iv(t-1)}^{Q-5\%}$	−0.362	0.463
${\mathit{\gamma }}_{1,def(t-1)}^{Q-2.5\%}$	9.465	0.143	${\mathit{\gamma }}_{1,def(t-1)}^{Q-3\%}$	11.780***	0.002	${\mathit{\gamma }}_{1,def(t-1)}^{Q-5\%}$	10.035***	0.000
${\mathit{\gamma }}_{2,dex(t-1)}^{Q-2.5\%}$	1.353	0.760	${\mathit{\gamma }}_{2,dex(t-1)}^{Q-3\%}$	1.564	0.684	${\mathit{\gamma }}_{2,dex(t-1)}^{Q-5\%}$	2.698	0.413
${\mathit{\gamma }}_{3,term(t-1)}^{Q-2.5\%}$	0.163	0.896	${\mathit{\gamma }}_{3,term(t-1)}^{Q-3\%}$	−0.024	0.986	${\mathit{\gamma }}_{3,term(t-1)}^{Q-5\%}$	−0.565	0.635
${\mathit{\gamma }}_{4,dff(t-1)}^{Q-2.5\%}$	16.065***	0.005	${\mathit{\gamma }}_{4,dff(t-1)}^{Q-3\%}$	10.109	0.530	${\mathit{\gamma }}_{4,dff(t-1)}^{Q-5\%}$	6.983	0.480
Adj. R^2	0.176892		Adj. R^2	0.164559		Adj. R^2	0.132807
Panel D: Forecast volatility from the TGARCH model versus the previous day’s VIX
j = 1.0			j = 1.5			j = 2.0
${\mathit{\chi }}_{0,tgf(t)}^{Q-1\%}$	−26.113***	0.000	${\mathit{\chi }}_{0,tgf(t)}^{Q-1.5\%}$	−23.142***	0.000	${\mathit{\chi }}_{0,tgf(t)}^{Q-2\%}$	−21.326***	0.000
${\mathit{\chi }}_{1,tgf(t)}^{Q-1\%}$	−2.118**	0.024	${\mathit{\chi }}_{1,tgf\left(t\right)}^{Q-1.5\%}$	−1.635***	0.000	${\mathit{\chi }}_{1,tgf(t)}^{Q-2\%}$	−1.252*	0.090
${\mathit{\chi }}_{2,iv(t-1)}^{Q-1\%}$	0.680	0.310	${\mathit{\chi }}_{2,iv(t-1)}^{Q-1.5\%}$	0.253	0.623	${\mathit{\chi }}_{2,iv(t-1)}^{Q-2\%}$	−0.266	0.665
${\mathit{\gamma }}_{1,def(t-1)}^{Q-1\%}$	10.459**	0.030	${\mathit{\gamma }}_{1,def(t-1)}^{Q-1.5\%}$	9.882**	0.033	${\mathit{\gamma }}_{1,def(t-1)}^{Q-2\%}$	12.322**	0.025
${\mathit{\gamma }}_{2,dex(t-1)}^{Q-1\%}$	2.484	0.563	${\mathit{\gamma }}_{2,dex(t-1)}^{Q-1.5\%}$	0.224	0.953	${\mathit{\gamma }}_{2,dex(t-1)}^{Q-2\%}$	1.186	0.774
${\mathit{\gamma }}_{3,term(t-1)}^{Q-1\%}$	−0.436	0.803	${\mathit{\gamma }}_{3,term(t-1)}^{Q-1.5\%}$	0.143	0.908	${\mathit{\gamma }}_{3,term(t-1)}^{Q-2\%}$	0.881	0.502
${\mathit{\gamma }}_{4,dff(t-1)}^{Q-1\%}$	19.590	0.549	${\mathit{\gamma }}_{4,dff(t-1)}^{Q-1.5\%}$	17.669	0.468	${\mathit{\gamma }}_{4,dff(t-1)}^{Q-2\%}$	13.781	0.290
Adj. R^2	0.224285		Adj. R^2	0.206828		Adj. R^2	0.190458
j = 2.5			j = 3.0			j = 5.0
${\mathit{\chi }}_{0,tgf(t)}^{Q-2.5\%}$	−19.456***	0.000	${\mathit{\chi }}_{0,tgf(t)}^{Q-3\%}$	−15.004***	0.000	${\mathit{\chi }}_{0,tgf(t)}^{Q-5\%}$	−11.161***	0.000
${\mathit{\chi }}_{1,tgf(t)}^{Q-2.5\%}$	−1.033**	0.012	${\mathit{\chi }}_{1,tgf(t)}^{Q-3\%}$	−1.023**	0.018	${\mathit{\chi }}_{1,tgf(t)}^{Q-5\%}$	−0.775***	0.002
${\mathit{\chi }}_{2,iv(t-1)}^{Q-2.5\%}$	−0.418	0.410	${\mathit{\chi }}_{2,iv(t-1)}^{Q-3\%}$	−0.447	0.408	${\mathit{\chi }}_{2,iv(t-1)}^{Q-5\%}$	−0.564	0.195
${\mathit{\gamma }}_{1,def(t-1)}^{Q-2.5\%}$	11.382**	0.031	${\mathit{\gamma }}_{1,def(t-1)}^{Q-3\%}$	10.702*	0.053	${\mathit{\gamma }}_{1,def(t-1)}^{Q-5\%}$	11.245***	0.000
${\mathit{\gamma }}_{2,dex(t-1)}^{Q-2.5\%}$	1.728	0.668	${\mathit{\gamma }}_{2,dex(t-1)}^{Q-3\%}$	1.190	0.765	${\mathit{\gamma }}_{2,dex(t-1)}^{Q-5\%}$	1.819	0.426
${\mathit{\gamma }}_{3,term(t-1)}^{Q-2.5\%}$	0.932	0.434	${\mathit{\gamma }}_{3,term(t-1)}^{Q-3\%}$	0.124	0.928	${\mathit{\gamma }}_{3,term(t-1)}^{Q-5\%}$	−0.727	0.523
${\mathit{\gamma }}_{4,dff(t-1)}^{Q-2.5\%}$	17.043***	0.001	${\mathit{\gamma }}_{4,dff(t-1)}^{Q-3\%}$	17.865***	0.002	${\mathit{\gamma }}_{4,dff(t-1)}^{Q-5\%}$	5.363*	0.076
Adj. R^2	0.177116		Adj. R^2	0.165545		Adj. R^2	0.134190

Notes: This table exhibits the results of empirical comparisons of the predictive power of the one-day lagged VIX of S&P 500 and the forecast S&P 500 volatilities from various GARCH models. These tests are conducted in our out-of-sample period that is from 3 January 2006 to 28 February 2014. All examinations are conducted using multiple quantile regression models. Each multiple quantile regression model includes a constant term, the one-day lagged VIX, the forecast volatility from one of the four GARCH models, and four control variables. Our control variables are the one-day lagged default spread, the one-day lag of the first-difference series of the trade-weighted US dollar index, the one-day lagged term premium, and the one-day lag of the first-difference series of the effective federal funds rate. All GARCH models used to derive the forecast volatilities are specified in our in-sample period, which is from 2 January 1990 to 30 December 2005. Further, Adj. R² in this table denotes the adjusted R-squared value.

*Statistical significance of coefficients at the 10% level.

**Statistical significance of coefficients at the 5% level.

***Statistical significance of coefficients at the 1% level.

Figure 2. Dynamic relations of the VIX and the forecast S&P 500 volatilities from the EGARCH and TGARCH models: daily time-series evolution for the period from 3 January 2006 to 28 February 2014.

Notes: This figure presents the daily time-series evolution of the VIX close and the forecast volatilities of the percentage log return of the S&P 500, which is derived from the EGARCH (1,1) model with the generalized error distribution (GED) errors and the TGARCH (1,1) model with the GED errors. These series are exhibited for our out-of-sample period, which spans 3 January 2006 to 28 February 2014, and this period includes the date of the Lehman Brothers bankruptcy in the US. Two kinds of GARCH models used to derive the forecast volatilities are specified in our in-sample period, which is from 2 January 1990 to 30 December 2005.

Figure 3. Dynamic relations of the S&P 500 and the forecast VIX from the ARMA models: daily time-series evolution for the period from 3 January 2006 to 28 February 2014.

Notes: This figure presents the daily time-series evolution of the close prices of the S&P 500 and the forecast values of the VIX close, which are derived from the ARMA(2,1) model (Panel A) and from the ARMA(4,4) model (Panel B). These series are exhibited for our out-of-sample period from 3 January 2006 to 28 February 2014, and this period includes the date of the Lehman Brothers bankruptcy in the US. Two kinds of ARMA models for deriving the forecast values of the VIX are specified in our in-sample period, which is from 2 January 1990 to 30 December 2005.

Table 8. Testing the predictive power for downside risk in the US stock market in terms of the forecast VIX from ARMA models: results from logit models and quantile regressions

Coefficients	Estimates	p-value	Coefficients	Estimates	p-value	Coefficients	Estimates	p-value
Panel A: Downside risk predictive power of the forecast VIX from the ARMA(2,1) model: results from logit models
k = 99.0			k = 98.5			k = 98.0
$u_{\begin{array}{c}0,ivf\left(2,1\right)\left(t\right)\\ \\ \\ \\ \end{array}}^{lg-99\%}$	−7.086***	0.000	$u_{0,ivf\left(2,1\right)\left(t\right)}^{lg-98.5\%}$	−6.342***	0.000	$u_{0,ivf\left(2,1\right)\left(t\right)}^{lg-98\%}$	−5.600***	0.000
$u_{1,ivf(2,1)(t)}^{lg-99\%}$	0.085***	0.000	$u_{1,ivf(2,1)(t)}^{lg-98.5\%}$	0.078***	0.000	$u_{1,ivf(2,1)(t)}^{lg-98\%}$	0.065***	0.000
MF-R^2	0.187138		MF-R^2	0.155051		MF-R^2	0.100719
k = 97.5			k = 97.0			k = 95.0
$u_{0,ivf(2,1)(t)}^{lg-97.5\%}$	−5.241 ***	0.000	$u_{0,ivf(2,1)(t)}^{lg-97\%}$	−4.957***	0.000	$u_{0,ivf(2,1)(t)}^{lg-95\%}$	−4.395***	0.000
$u_{1,ivf(2,1)(t)}^{lg-97.5\%}$	0.061***	0.000	$u_{1,ivf(2,1)(t)}^{lg-97\%}$	0.058***	0.000	$u_{1,ivf(2,1)(t)}^{lg-95\%}$	0.058***	0.000
MF-R^2	0.087131		MF-R^2	0.077690		MF-R^2	0.078521
Panel B: Downside risk predictive power of the forecast VIX from the ARMA(4,4) model: results from logit models
k = 99.0			k = 98.5			k = 98.0
$u_{0,ivf(4,4)(t)}^{lg-99\%}$	−7.075***	0.000	$u_{0,ivf(4,4)(t)}^{lg-98.5\%}$	−6.341***	0.000	$u_{0,ivf(4,4)(t)}^{lg-98\%}$	−5.599***	0.000
$u_{1,ivf(4,4)(t)}^{lg-99\%}$	0.084***	0.000	$u_{1,ivf(4,4)(t)}^{lg-98.5\%}$	0.078***	0.000	$u_{1,ivf(4,4)(t)}^{lg-98\%}$	0.065***	0.000
MF-R^2	0.186474		MF-R^2	0.155450		MF-R^2	0.100886
k = 97.5			k = 97.0			k = 95.0
$u_{0,ivf(4,4)(t)}^{lg-97.5\%}$	−5.240***	0.000	$u_{0,ivf(4,4)(t)}^{lg-97\%}$	−4.956***	0.000	$u_{0,ivf(4,4)(t)}^{lg-95\%}$	−4.397	0.268
$u_{1,ivf(4,4)(t)}^{lg-97.5\%}$	0.061***	0.000	$u_{1,ivf(4,4)(t)}^{lg-97\%}$	0.058***	0.000	$u_{1,ivf(4,4)(t)}^{lg-95\%}$	0.058***	0.000
MF-R^2	0.087363		MF-R^2	0.077818		MF-R^2	0.078891
Panel C: Downside risk predictive power of the forecast VIX from the ARMA(2,1) model: results from quantile regressions
j = 1.0			j = 1.5			j = 2.0
$u_{0,ivf(2,1)(t)}^{Q-1\%}$	−11.817**	0.017	$u_{0,ivf(2,1)(t)}^{Q-1.5\%}$	−8.018**	0.014	$u_{0,ivf(2,1)(t)}^{Q-2\%}$	−8.134**	0.014
$u_{1,ivf(2,1)(t)}^{Q-1\%}$	−1.264***	0.000	$u_{1,ivf(2,1)(t)}^{Q-1.5\%}$	−1.293***	0.000	$u_{1,ivf(2,1)(t)}^{Q-2\%}$	−1.214***	0.000
Adj. R^2	0.165542		Adj. R^2	0.156664		Adj. R^2	0.146785
j = 2.5			j = 3.0			j = 5.0
$u_{0,ivf(2,1)(t)}^{Q-2.5\%}$	−5.239	0.149	$u_{0,ivf(2,1)(t)}^{Q-3\%}$	−7.506**	0.032	$u_{0,ivf(2,1)(t)}^{Q-5\%}$	−3.379	0.269
$u_{1,ivf(2,1)(t)}^{Q-2.5\%}$	−1.252***	0.000	$u_{1,ivf(2,1)(t)}^{Q-3\%}$	−1.042***	0.000	$u_{1,ivf(2,1)(t)}^{Q-5\%}$	−1.019***	0.000
Adj. R^2	0.137579		Adj. R^2	0.132064		Adj. R^2	0.106947
Panel D: Downside risk predictive power of the forecast VIX from the ARMA(4,4) model: results from quantile regressions
j = 1.0			j = 1.5			j = 2.0
$u_{0,ivf\left(4,4\right)\left(t\right)}^{Q-1\%}$	−11.868**	0.034	$u_{0,ivf(4,4)(t)}^{Q-1.5\%}$	−10.000***	0.001	$u_{0,ivf\left(4,4\right)\left(t\right)}^{Q-2\%}$	−7.971***	0.006
$u_{1,ivf\left(4,4\right)\left(t\right)}^{Q-1\%}$	−1.268***	0.000	$u_{1,ivf\left(4,4\right)\left(t\right)}^{Q-1.5\%}$	−1.226***	0.000	$u_{1,ivf\left(4,4\right)\left(t\right)}^{Q-2\%}$	−1.218***	0.000
Adj. R^2	0.165017		Adj. R^2	0.156591		Adj. R^2	0.146850
j = 2.5			j = 3.0			j = 5.0
$u_{0,ivf(4,4)(t)}^{Q-2.5\%}$	−5.112*	0.075	$u_{0,ivf(4,4)(t)}^{Q-3\%}$	−8.039**	0.011	$u_{0,ivf(4,4)(t)}^{Q-5\%}$	−3.511	0.254
$u_{1,ivf(4,4)(t)}^{Q-2.5\%}$	−1.270***	0.000	$u_{1,ivf(4,4)(t)}^{Q-3\%}$	−1.014***	0.000	$u_{1,ivf(4,4)(t)}^{Q-5\%}$	−1.007***	0.000
Adj. R^2	0.137631		Adj. R^2	0.131963		Adj. R^2	0.106753

Notes: This table exhibits the predictive power of the forecast VIX from ARMA(2,1) and ARMA(4,4) models. These tests are conducted in our out-of-sample period from 3 January 2006 to 28 February 2014 using univariate logit models (Panels A and B) and univariate quantile regressions (Panels C and D). Two kinds of ARMA models for deriving the forecast values of the VIX are specified in our in-sample period, which is from 2 January 1990 to 30 December 2005. In this table, MF-R² denotes the McFadden’s R-squared value and Adj. R² denotes the adjusted R-squared value.

*Statistical significance of coefficients at the 10% level.

**Statistical significance of coefficients at the 5% level.

***Statistical significance of coefficients at the 1% level.

Figure 4. Dynamic relations of the VIX and the forecast volatilities of the VIX from the EGARCH and TGARCH models: daily time-series evolution for the period from 3 January 2006 to 28 February 2014.

Notes: This figure presents the daily time-series evolution of the VIX and the forecast volatilities of the VIX from the EGARCH(1,1) model with GED errors (Panel A) and the TGARCH(1,1) model with GED errors (Panel B). These series are exhibited for our out-of-sample period from 3 January 2006 to 28 February 2014, and this period includes the date of the Lehman Brothers bankruptcy in the US. Two kinds of GARCH models used to derive the forecast volatilities of the VIX are specified in our in-sample period, which is from 2 January 1990 to 30 December 2005.

Table 9. Testing the predictive power for downside risk in the US stock market in terms of the forecast volatilities of the VIX from EGARCH and TGARCH models: results from logit models and quantile regressions

Coefficients	Estimates	p-value	Coefficients	Estimates	p-value	Coefficients	Estimates	p-value
Panel A: Predictive power of the forecast volatility of the VIX from the EGARCH model: results from logit models
k = 99.0			k = 98.5			k = 98.0
${\mathit{\beta }}_{0,viv-egf(t)}^{lg-99\%}$	−6.180***	0.000	${\mathit{\beta }}_{0,viv-egf(t)}^{lg-98.5\%}$	−5.464***	0.000	${\mathit{\beta }}_{0,viv-egf(t)}^{lg-98\%}$	−4.838***	0.000
${\mathit{\beta }}_{1,viv-egf(t)}^{lg-99\%}$	0.165***	0.000	${\mathit{\beta }}_{1,viv-egf(t)}^{lg-98.5\%}$	0.149***	0.000	${\mathit{\beta }}_{1,viv-egf(t)}^{lg-98\%}$	0.122***	0.000
MF-R^2	0.198383		MF-R^2	0.155018		MF-R^2	0.093926
k = 97.5			k = 97.0			k = 95.0
${\mathit{\beta }}_{0,viv-egf(t)}^{lg-97.5\%}$	−4.523***	0.000	${\mathit{\beta }}_{0,viv-egf(t)}^{lg-97\%}$	−4.261***	0.000	${\mathit{\beta }}_{0,viv-egf(t)}^{lg-95\%}$	−3.687***	0.000
${\mathit{\beta }}_{1,viv-egf(t)}^{lg-97.5\%}$	0.114***	0.000	${\mathit{\beta }}_{1,viv-egf(t)}^{lg-97\%}$	0.107***	0.000	${\mathit{\beta }}_{1,viv-egf(t)}^{lg-95\%}$	0.105***	0.000
MF-R^2	0.080073		MF-R^2	0.068501		MF-R^2	0.066565
Panel B: Predictive power of the forecast volatility of the VIX from the TGARCH model: results from logit models
k = 99.0			k = 98.5			k = 98.0
${\mathit{\beta }}_{0,viv-tgf(t)}^{lg-99\%}$	−5.829***	0.000	${\mathit{\beta }}_{0,viv-tgf(t)}^{lg-98.5\%}$	−5.177***	0.000	${\mathit{\beta }}_{0,viv-tgf(t)}^{lg-98\%}$	−4.628***	0.000
${\mathit{\beta }}_{1,viv-tgf(t)}^{lg-99\%}$	0.102***	0.000	${\mathit{\beta }}_{1,viv-tgf(t)}^{lg-98.5\%}$	0.093***	0.000	${\mathit{\beta }}_{1,viv-tgf(t)}^{lg-98\%}$	0.077***	0.000
MF-R^2	0.183171		MF-R^2	0.145232		MF-R^2	0.089643
k = 97.5			k = 97.0			k = 95.0
${\mathit{\beta }}_{0,viv-tgf(t)}^{lg-97.5\%}$	−4.325***	0.000	${\mathit{\beta }}_{0,viv-tgf(t)}^{lg-97\%}$	−4.075***	0.000	${\mathit{\beta }}_{0,viv-tgf(t)}^{lg-95\%}$	−3.516***	0.000
${\mathit{\beta }}_{1,viv-tgf(t)}^{lg-97.5\%}$	0.072***	0.000	${\mathit{\beta }}_{1,viv-tgf(t)}^{lg-97\%}$	0.067***	0.000	${\mathit{\beta }}_{1,viv-tgf(t)}^{lg-95\%}$	0.067***	0.000
MF-R^2	0.075311		MF-R^2	0.063524		MF-R^2	0.063225
Panel C: Predictive power of the forecast volatility of the VIX from the EGARCH model: results from quantile regressions
j = 1.0			j = 1.5			j = 2.0
${\mathit{\beta }}_{0,viv-egf(t)}^{Q-1\%}$	−30.322***	0.000	${\mathit{\beta }}_{0,viv-egf(t)}^{Q-1.5\%}$	−26.640***	0.000	${\mathit{\beta }}_{0,viv-egf(t)}^{Q-2\%}$	−23.060***	0.000
${\mathit{\beta }}_{1,viv-egf(t)}^{Q-1\%}$	−2.212***	0.000	${\mathit{\beta }}_{1,viv-egf(t)}^{Q-1.5\%}$	−2.017***	0.000	${\mathit{\beta }}_{1,viv-egf(t)}^{Q-2\%}$	−2.099***	0.000
Adj. R^2	0.139215		Adj. R^2	0.124371		Adj. R^2	0.113298
j = 2.5			j = 3.0			j = 5.0
${\mathit{\beta }}_{0,viv-egf(t)}^{Q-2.5\%}$	−22.340***	0.000	${\mathit{\beta }}_{0,viv-egf(t)}^{Q-3\%}$	−20.254***	0.000	${\mathit{\beta }}_{0,viv-egf(t)}^{Q-5\%}$	−17.245***	0.000
${\mathit{\beta }}_{1,viv-egf(t)}^{Q-2.5\%}$	−1.742***	0.000	${\mathit{\beta }}_{1,viv-egf(t)}^{Q-3\%}$	−1.836***	0.000	${\mathit{\beta }}_{1,viv-egf(t)}^{Q-5\%}$	−1.535***	0.000
Adj. R^2	0.111053		Adj. R^2	0.107057		Adj. R^2	0.082926
Panel D: Predictive power of the forecast volatility of the VIX from the TGARCH model: results from quantile regressions
j = 1.0			j = 1.5			j = 2.0
${\mathit{\beta }}_{0,viv-tgf(t)}^{Q-1\%}$	−33.503***	0.000	${\mathit{\beta }}_{0,viv-tgf(t)}^{Q-1.5\%}$	−28.535***	0.000	${\mathit{\beta }}_{0,viv-tgf(t)}^{Q-2\%}$	−25.115***	0.000
${\mathit{\beta }}_{1,viv-tgf(t)}^{Q-1\%}$	−1.409***	0.000	${\mathit{\beta }}_{1,viv-tgf(t)}^{Q-1.5\%}$	−1.495***	0.000	${\mathit{\beta }}_{1,viv-tgf(t)}^{Q-2\%}$	−1.470***	0.000
Adj. R^2	0.141416		Adj. R^2	0.125359		Adj. R^2	0.114556
j = 2.5			j = 3.0			j = 5.0
${\mathit{\beta }}_{0,viv-tgf(t)}^{Q-2.5\%}$	−23.654***	0.000	${\mathit{\beta }}_{0,viv-tgf(t)}^{Q-3\%}$	−21.361***	0.000	${\mathit{\beta }}_{0,viv-tgf(t)}^{Q-5\%}$	−18.787***	0.000
${\mathit{\beta }}_{1,viv-tgf(t)}^{Q-2.5\%}$	−1.319***	0.000	${\mathit{\beta }}_{1,viv-tgf(t)}^{Q-3\%}$	−1.365***	0.000	${\mathit{\beta }}_{1,viv-tgf(t)}^{Q-5\%}$	−1.115***	0.000
Adj. R^2	0.111407		Adj. R^2	0.106773		Adj. R^2	0.084665

Notes: This table exhibits the predictive power of the forecast volatilities of the VIX from the EGARCH(1,1) model with GED errors and the TGARCH(1,1) model with GED errors. These tests are conducted in our out-of-sample period from 3 January 2006 to 28 February 2014 using univariate logit models (Panels A and B) and univariate quantile regressions (Panels C and D). Two kinds of GARCH models used to derive the forecast volatilities of the VIX are specified in our in-sample period, which is from 2 January 1990 to 30 December 2005. In this table, MF-R² denotes the McFadden’s R-squared value and Adj. R² is the adjusted R-squared value.

*Statistical significance of coefficients at the 10% level.

**Statistical significance of coefficients at the 5% level.

***Statistical significance of coefficients at the 1% level.

Figure 5. Dynamic relations of the first log differences of the VIX and the forecast volatilities from the EGARCH and TGARCH models: daily time-series evolution for the period from 3 January 2006 to 28 February 2014.

Notes: This figure presents the daily time-series evolution of the first log differences of the VIX and the forecast volatilities from the EGARCH(1,1) model with GED errors (Panel A) and the TGARCH(1,1) model with GED errors (Panel B). These series are exhibited for our out-of-sample period from 3 January 2006 to 28 February 2014, and this period includes the date of the US Lehman Brothers bankruptcy. Two kinds of GARCH models used to derive the forecast volatilities of the first log differences of the VIX are specified in our in-sample period, which is from 2 January 1990 to 30 December 2005.

Table 10. Testing the predictive power for downside risk in the US stock market in terms of the forecast volatilities of the first log differences of the VIX: results from logit models and quantile regressions

Coefficients	Estimates	p-value	Coefficients	Estimates	p-value	Coefficients	Estimates	p-value
Panel A: Predictive power of the forecast volatility of the first log differences of VIX from the EGARCH model: results from logit models
k = 99.0			k = 98.5			k = 98.0
${\mathit{\rho }}_{0,vdliv-egf(t)}^{lg-99\%}$	−8.758***	0.000	${\mathit{\rho }}_{0,vdliv-egf\left(t\right)}^{lg-98.5\%}$	−7.842***	0.000	${\mathit{\rho }}_{0,vdliv-egf(t)}^{lg-98\%}$	−7.016***	0.000
${\mathit{\rho }}_{1,vdliv-egf(t)}^{lg-99\%}$	0.607***	0.000	${\mathit{\rho }}_{1,vdliv-egf(t)}^{lg-98.5\%}$	0.550***	0.000	${\mathit{\rho }}_{1,vdliv-egf(t)}^{lg-98\%}$	0.479***	0.000
MF-R^2	0.149994		MF-R^2	0.119788		MF-R^2	0.086614
k = 97.5			k = 97.0			k = 95.0
${\mathit{\rho }}_{0,vdliv-egf(t)}^{lg-97.5\%}$	−6.381***	0.000	${\mathit{\rho }}_{0,vdliv-egf(t)}^{lg-97\%}$	−6.071***	0.000	${\mathit{\rho }}_{0,vdliv-egf\left(t\right)}^{lg-95\%}$	−5.026***	0.000
${\mathit{\rho }}_{1,vdliv-egf(t)}^{lg-97.5\%}$	0.423***	0.000	${\mathit{\rho }}_{1,vdliv-egf(t)}^{lg-97\%}$	0.405***	0.000	${\mathit{\rho }}_{1,vdliv-egf(t)}^{lg-95\%}$	0.332***	0.000
MF-R^2	0.064664		MF-R^2	0.059250		MF-R^2	0.038593
Panel B: Predictive power of the forecast volatility of the first log differences of VIX from the TGARCH model: results from logit models
k = 99.0			k = 98.5			k = 98.0
${\mathit{\rho }}_{0,vdliv-tgf(t)}^{lg-99\%}$	−7.174***	0. 000	${\mathit{\rho }}_{0,vdliv-tgf(t)}^{lg-98.5\%}$	−6.418***	0.000	${\mathit{\rho }}_{0,vdliv-tgf(t)}^{lg-98\%}$	−5.781***	0.000
${\mathit{\rho }}_{1,vdliv-tgf(t)}^{lg-99\%}$	0.356***	0.000	${\mathit{\rho }}_{1,vdliv-tgf(t)}^{lg-98.5\%}$	0.321***	0.000	${\mathit{\rho }}_{1,vdliv-tgf(t)}^{lg-98\%}$	0.278***	0.000
MF-R^2	0.113873		MF-R^2	0.087000		MF-R^2	0.059770
k = 97.5			k = 97.0			k = 95.0
${\mathit{\rho }}_{0,vdliv-tgf(t)}^{lg-97.5\%}$	−5.284***	0.000	${\mathit{\rho }}_{0,vdliv-tgf(t)}^{lg-97\%}$	−5.003***	0.000	${\mathit{\rho }}_{0,vdliv-tgf(t)}^{lg-95\%}$	−4.195***	0.000
${\mathit{\rho }}_{1,vdliv-tgf(t)}^{lg-97.5\%}$	0.242***	0.000	${\mathit{\rho }}_{1,vdliv-tgf(t)}^{lg-97\%}$	0.229***	0.000	${\mathit{\rho }}_{1,vdliv-tgf(t)}^{lg-95\%}$	0.192***	0.000
MF-R^2	0.042306		MF-R^2	0.037245		MF-R^2	0.024952
Panel C: Predictive power of the forecast volatility of the first log differences of VIX from the EGARCH model: results from quantile regressions
j = 1.0			j = 1.5			j = 2.0
${\mathit{\rho }}_{0,vdliv-egf(t)}^{Q-1\%}$	13.538*	0.067	${\mathit{\rho }}_{0,vdliv-egf\left(t\right)}^{Q-1.5\%}$	10.206	0.382	${\mathit{\rho }}_{0,vdliv-egf(t)}^{Q-2\%}$	7.397	0.203
${\mathit{\rho }}_{1,vdliv-egf(t)}^{Q-1\%}$	−9.389***	0.000	${\mathit{\rho }}_{1,vdliv-egf(t)}^{Q-1.5\%}$	−8.298***	0.000	${\mathit{\rho }}_{1,vdliv-egf(t)}^{Q-2\%}$	−7.205***	0.000
Adj. R^2	0.152211		Adj. R^2	0.130403		Adj. R^2	0.115972
j = 2.5			j = 3.0			j = 5.0
${\mathit{\rho }}_{0,vdliv-egf\left(t\right)}^{Q-2.5\%}$	7.691	0.133	${\mathit{\rho }}_{0,vdliv-egf(t)}^{Q-3\%}$	10.253*	0.051	${\mathit{\rho }}_{0,vdliv-egf(t)}^{Q-5\%}$	7.705	0.231
${\mathit{\rho }}_{1,vdliv-egf(t)}^{Q-2.5\%}$	−6.982***	0.000	${\mathit{\rho }}_{1,vdliv-egf(t)}^{Q-3\%}$	−7.172***	0.000	${\mathit{\rho }}_{1,vdliv-egf(t)}^{Q-5\%}$	−5.771***	0.000
Adj. R^2	0.106554		Adj. R^2	0.095151		Adj. R^2	0.064069
Panel D: Predictive power of the forecast volatility of the first log differences of VIX from the TGARCH model: results from quantile regressions
j = 1.0			j = 1.5			j = 2.0
${\mathit{\rho }}_{0,vdliv-tgf(t)}^{Q-1\%}$	−3.972	0.559	${\mathit{\rho }}_{0,vdliv-tgf(t)}^{Q-1.5\%}$	−1.409	0.892	${\mathit{\rho }}_{0,vdliv-tgf(t)}^{Q-2\%}$	−2.852	0.682
${\mathit{\rho }}_{1,vdliv-tgf(t)}^{Q-1\%}$	−6.326***	0.000	${\mathit{\rho }}_{1,vdliv-tgf(t)}^{Q-1.5\%}$	−6.180***	0.001	${\mathit{\rho }}_{1,vdliv-tgf(t)}^{Q-2\%}$	−5.334***	0.000
Adj. R^2	0.124626		Adj. R^2	0.103989		Adj. R^2	0.090095
j = 2.5			j = 3.0			j = 5.0
${\mathit{\rho }}_{0,vdliv-tgf(t)}^{Q-2.5\%}$	−1.207	0.844	${\mathit{\rho }}_{0,vdliv-tgf(t)}^{Q-3\%}$	−1.373	0.828	${\mathit{\rho }}_{0,vdliv-tgf(t)}^{Q-5\%}$	0.712	0.883
${\mathit{\rho }}_{1,vdliv-tgf(t)}^{Q-2.5\%}$	−5.237***	0.000	${\mathit{\rho }}_{1,vdliv-tgf(t)}^{Q-3\%}$	−4.978***	0.000	${\mathit{\rho }}_{1,vdliv-tgf(t)}^{Q-5\%}$	−4.468***	0.000
Adj. R^2	0.084158		Adj. R^2	0.076957		Adj. R^2	0.048767

Notes: This table exhibits the predictive power of the forecast volatilities of the first log differences of the VIX, which are from the EGARCH(1,1) model with GED errors and the TGARCH(1,1) model with GED errors. These tests are conducted in our out-of-sample period from 3 January 2006 to 28 February 2014 using univariate logit models (Panels A and B) and univariate quantile regressions (Panels C and D). Two kinds of GARCH models used to derive the forecast volatilities of the first log differences of the VIX are specified in our in-sample period, which is from 2 January 1990 to 30 December 2005. In this table, MF-R² denotes the McFadden’s R-squared value and Adj. R² denotes the adjusted R-squared value.

*Statistical significance of coefficients at the 10% level.

**Statistical significance of coefficients at the 5% level.

***Statistical significance of coefficients at the 1% level.

Figure 6. Dynamic relations of the time-varying correlation coefficients from the VECH-MGARCH models: daily time-series evolution for the period from 3 January 2006 to 28 February 2014.

Notes: This figure exhibits the daily time-series evolution of the correlation coefficients among the one-day lagged VIX and the forecast volatilities from EGARCH(1,1) and TGARCH(1,1) models with GED errors. All correlation coefficients are derived from the VECH-MGARCH models, and these correlations are presented for our out-of-sample period from 3 January 2006 to 28 February 2014, and this period includes the date of the Lehman shock in the US. More specifically, the correlation coefficients between the one-day lagged VIX and the forecast S&P 500 volatility from the EGARCH(1,1) model are exhibited in Panel A; the correlation coefficients between the one-day lagged VIX and the forecast S&P 500 volatility from the TGARCH(1,1) model are displayed in Panel B; and those between the forecast volatility from the EGARCH(1,1) model and that from the TGARCH(1,1) model are presented in Panel C.

Figure 7. Dynamic relations of the time-varying correlation coefficients from the BEKK-MGARCH models: daily time-series evolution for the period from 3 January 2006 to 28 February 2014.

Notes: This figure presents the daily time-series evolution of the correlation coefficients among the one-day lagged VIX and the forecast volatilities from EGARCH(1,1) and TGARCH(1,1) models with GED errors. All correlation coefficients are obtained from the BEKK-MGARCH models, and these correlations are exhibited for our out-of-sample period from 3 January 2006 to 28 February 2014, and this period includes the date of the Lehman shock in the US. More concretely, the correlation coefficients between the one-day lagged VIX and the forecast S&P 500 volatility from the EGARCH(1,1) model are shown in Panel A; the correlation coefficients between the one-day lagged VIX and the forecast S&P 500 volatility from the TGARCH(1,1) model are presented in Panel B; and those between the forecast volatility from the EGARCH(1,1) model and that from the TGARCH(1,1) model are shown in Panel C.

Figure 8. Dynamic relations of the time-varying correlation coefficients from the DCC-MGARCH models: daily time-series evolution for the period from 3 January 2006 to 28 February 2014.

Notes: This figure displays the daily time-series evolution of the correlation coefficients among the one-day lagged VIX and the forecast volatilities from EGARCH(1,1) and TGARCH(1,1) models with GED errors. All correlation coefficients are derived from the DCC-MGARCH models, and these correlations are shown for our out-of-sample period from 3 January 2006 to 28 February 2014, and this period includes the date of the Lehman shock in the US. Specifically, the correlation coefficients between the one-day lagged VIX and the forecast S&P 500 volatility from the EGARCH(1,1) model are displayed in Panel A; the correlation coefficients between the one-day lagged VIX and the forecast S&P 500 volatility from the TGARCH(1,1) model are shown in Panel B; and those between the forecast volatility from the EGARCH(1,1) model and that from the TGARCH(1,1) model are exhibited in Panel C.

Figure 9. Dynamic relations of the time-varying correlation coefficients from the ADCC-MGARCH models: daily time-series evolution for the period from 3 January 2006 to 28 February 2014.

Notes: This figure shows the daily time-series evolution of the correlation coefficients among the one-day lagged VIX and the forecast volatilities from EGARCH(1,1) and TGARCH(1,1) models with GED errors. All correlation coefficients are provided by the ADCC-MGARCH models, and these correlations are displayed for our out-of-sample period from 3 January 2006 to 28 February 2014, and this period includes the date of the Lehman shock in the US. More concretely, the correlation coefficients between the one-day lagged VIX and the forecast S&P 500 volatility from the EGARCH(1,1) model are presented in Panel A; the correlation coefficients between the one-day lagged VIX and the forecast S&P 500 volatility from the TGARCH(1,1) model are exhibited in Panel B; and those between the forecast volatility from the EGARCH(1,1) model and that from the TGARCH(1,1) model are displayed in Panel C.