The DCCX model advanced in this paper provides a structural form model for conditional
correlations while the standard DCC model is a reduced-form model. That is, the DCCX
model is more general to apply to the financial or economic issues. This article represents a
modified DCC model for the empirical analysis to discuss whether a day’s change in stock
33
market uncertainty is associated with differences in the stock-bond returns relation. This
investigation further evaluates the empirical relevance of cross-market hedging and addresses
the notion of flight-to-quality versus flight-from-quality with increased versus decreased
stock uncertainty. It is discovered several striking results in our empirical investigation. First,
it is found a negative stock-bond returns relation with the two measures of market uncertainty,
i.e. the VIX and the stock turnover. More accurately, by means of the modified DCC model,
it is explored that stock-bond returns correlation tends to be negative (positive), during
periods when VIX increases (decreases) and during periods when unexpected stock turnover
is high (low).
In the future, we can extend the sample period including the 1980-1990 to search if the
similar phenomenon exists. Besides, to consider more other exogenous variables is also
necessary for finding out other explanation about the correlation. And the empirical study in
other countries needs more investigation to support our consequence.
34
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38
TABLE 1 Descriptive Statistics
This table reports the descriptive statistics for the data used in this article. S&P500 and T-bond refer to the stock and 10-year Treasury bond return series, respectively. The returns are in daily percentage units. VIX is the CBOE’s Volatility Index. TV is taken as the natural log of turnover volume of S&P500 at the period t−1 in daily and in thousand units. Std. Dev. denotes standard deviation and ρi refers to the i th autocorrelation. Panel A reports the sample moments of the data from 1990 to 2007. Panel B reports the sample moments of the data from 1990 to 1997. Panel C reports the sample moments of the data from 1998 to 2007. Panel D reports the correlation matrix over the 1990-2007 sample period. Panel E presents the subsample correlation matrix. The correlation coefficients of the 1990-1997 sample period are shown in brackets on the upper triangle and the correlation coefficients of the 1998-2007 sample period are on the lower triangle.
Panel A: Sample Moments, 1990-2007
Jarque-Bera 2705.956 1312.158 845.985 357.399
ρ1 -0.009 0.053 0.982 0.980
ρ2 -0.023 -0.026 0.965 0.973
ρ3 -0.027 -0.035 0.952 0.970
ρ10 0.010 0.012 0.893 0.962
39
Jarque-Bera 2485.692 986.044 885.337 367.412
ρ1 0.040 0.066 0.974 0.871
Jarque-Bera 737.305 355.743 171.077 333.892
ρ1 -0.028 0.048 0.982 0.918
ρ2 -0.023 -0.036 0.966 0.885
ρ3 -0.019 -0.039 0.953 0.872
ρ10 0.002 0.003 0.894 0.842
40
TABLE 1 (Continued)
Panel D:Correlation Matrix, 1990-2007
Panel E:Correlation Matrix, 1990-1997；1998-2007
S&P500 T-bond VIX TV
S&P500 1.000 -0.049 -0.111 -0.012
T-bond -0.049 1.000 0.022 -0.006
VIX -0.111 0.022 1.000 0.184
TV -0.012 -0.006 0.184 1.000
S&P500 T-bond VIX TV
S&P500 1.000 [0.390] [-0.104] [ 0.024]
T-bond -0.224 1.000 [-0.056] [-0.022]
VIX -0.114 0.055 1.000 [-0.015]
TV -0.010 -0.001 -0.318 1.000
41 TABLE 2
Estimation of Bivariate Return-based DCC Model Using Daily S&P 500 Index and T-bond, 1990-2007.
Stage 1 of DCC estimation: 2
This table provides the estimation for the bivariate return-based DCC model using daily S&P 500 index and T-bond. The two formulas above two steps esimation are GARCH and the conditional correlation equation respectively of the standard DCC model with mean reversion. In the first stage, we use the GARCH model to estimate their volatilities (
ht
∧ ) for each assets and computes their standardized residuals ( Zt).
Then, in the second stage, the conditional correlation process can be obtained by using their standardized residuals andq1 2 = E z z[ 1 2].The conditional correlation matrix is given by
1 2 ,t/ 1 1,t 2 2 ,t
q q q .The conditional covariance can then be expressed using the product of conditional correlation between these two variables and their individual conditional standard deviations. The table shows estimations of the two models using the MLE method. It is presented the estimation of the DCCX with lagged VIX (
1,t 1
x − ) and the lagged stock turnover (
2 ,t 1
x − ) respectively. LLF is the log likelihood function and numbers in parentheses are T-values. We allow a free intercept parameter
Panel A: Step 1 of DCC estimation
S&P500 T-bond
42 TABLE 3
Estimation of Bivariate Return-based DCC Model Using Daily S&P 500 Index and T-bond, 1990-1997.
Stage 1 of DCC estimation: 2
This table provides the estimation for the bivariate return-based DCC model using daily S&P 500 index and T-bond. The two formulas above two steps esimation are GARCH and the conditional correlation equation respectively of the standard DCC model with mean reversion. In the first stage, we use the GARCH model to estimate their volatilities (
ht
∧ ) for each assets and computes their standardized residuals ( Z ). t
Then, in the second stage, the conditional correlation process can be obtained by using their standardized residuals andq1 2 = E z z[ 1 2].The conditional correlation matrix is given by
1 2 ,t/ 1 1,t 2 2 ,t
q q q .The conditional covariance can then be expressed using the product of conditional correlation between these two variables and their individual conditional standard deviations. The table shows estimations of the two models using the MLE method. It is presented the estimation of the DCCX with lagged VIX (
1,t 1
x − ) and the lagged stock turnover (x2, 1t− ) respectively. LLF is the log likelihood function and numbers in parentheses are T-values. We allow a free intercept parameter
Panel A: Step 1 of DCC estimation
S&P500 T-bond
43 TABLE 4
Estimation of Bivariate Return-based DCC Model Using Daily S&P 500 Index and T-bond, 1998-2007.
Stage 1 of DCC estimation: 2
This table provides the estimation for the bivariate return-based DCC model using daily S&P 500 index and T-bond. The two formulas above two steps esimation are GARCH and the conditional correlation equation respectively of the standard DCC model with mean reversion. In the first stage, we use the GARCH model to estimate their volatilities (
ht
∧ ) for each assets and computes their standardized residuals (
Zt).
Then, in the second stage, the conditional correlation process can be obtained by using their standardized residuals andq1 2 = E z z[ 1 2].The conditional correlation matrix is given by
1 2 ,t/ 1 1,t 2 2 ,t
q q q .The conditional covariance can then be expressed using the product of conditional correlation between these two variables and their individual conditional standard deviations. The table shows estimations of the two models using the MLE method. It is presented the estimation of the DCCX with lagged VIX (
1,t 1
x − ) and the lagged stock turnover (x2, 1t− ) respectively. LLF is the log likelihood function and numbers in parentheses are T-values. We allow a free intercept parameterc0in the estimation, where 12 1 1 2 2
0
Panel A: Step 1 of DCC estimation
S&P500 T-bond
44 TABLE 5
The Daily Stock-Bond Returns Correlation with VIX and Stock Turnover
This table reports results from estimating the following regression:
0 1 1+ 2ln( 1) 3ln(TV )t-1 correlation at the period t-1 and νtis the residual. ai are estimated coefficients. The overall sample period is 1990 to 2007. The sub-period of 1990-1997 and the 1998-2007 are also reported. The regression is estimated by OLS and T-statistics are in parentheses, calculated with White Heteroskedasticity-Consistent Standard Errors.
Coefficient 1990-2007 1990-1997 1998-2007
a
0 2.258 0.021 0.39645
S&P 500 Index and T-Bond Yield Daily Closing Prices and Returns, 1990-2007. This figure shows the daily close prices and returns of S&P 500 index and 10-year treasury bond (T-bond) over the sample period.
1990 1992 1994 1996 1998 2000 2002 2004 2006
1990 1992 1994 1996 1998 2000 2002 2004 2006
1990 1992 19941996 1998 2000 20022004 2006
1990 1992 1994 1996 1998 2000 2002 2004 2006
46
Figure 2
Panel A. Stock-Bond Returns Correlation with DCCX, 1990-2007
Panel B. CBOE’s Volatility Index (VIX), 1990-2007
Panel C. Stock Turnover by Volume (TV), 1990-2007
This figure displays the time-series of dynamic conditional correlations estimated by the DCCX model between U.S S&P500 stock and 10-year Treasury bond returns in Panel A. The CBOE’s Volatility Index (VIX) at day t-1 (Panel B), and the Stock Turnover by volume (Panel C). The sample spans 1990 to 2007.
7
1990 1992 1994 1996 1998 2000 2002 2004 2006
0
1990 1992 1994 1996 1998 2000 2002 2004 2006
1990 1992 1994 1996 1998 2000 2002 2004 2006
47
Figure 3
Panel A. Stock-Bond Returns Correlation, 1990-1997
Panel B. CBOE’s Volatility Index (VIX), 1990-1997
Panel C. Stock Turnover by Volume (TV), 1990-1997
This figure displays the time-series of dynamic conditional correlations estimated by the DCCX model between U.S S&P500 stock and 10-year Treasury bond returns in Panel A. The CBOE’s Volatility Index (VIX) at day t-1 (Panel B), and the Stock Turnover by volume (Panel C). The sample spans 1990 to 1997.
7
1990 1991 1992 1993 1994 1995 1996 1997
8
1990 1991 1992 1993 1994 1995 1996 1997
1990 1991 1992 1993 1994 1995 1996 1997
48
Figure 4
Panel A. Stock-Bond Returns Correlation, 1998-2007
Panel B. CBOE’s Volatility Index (VIX), 1998-2007
Panel C. Stock Turnover by Volume (TV), 1998-2007
This figure displays the time-series of dynamic conditional correlations estimated by the DCCX model between U.S S&P500 stock and 10-year Treasury bond returns in Panel A. The CBOE’s Volatility Index (VIX) at day t-1 (Panel B), and the Stock Turnover by volume (Panel C). The sample spans 1998 to 2007.
11
49
Figure 5
Panel A. Stock-Bond Returns Correlation with DCCX model, 1990-2007
Panel B. Stock-Bond Returns Correlation with DCC model, 1990-2007
This figure displays the time-series of dynamic conditional correlations estimated by the DCCX model between U.S S&P500 stock and 10-year Treasury bond returns in Panel A. The Panel B reports the time-series of dynamic conditional correlations estimated by the standard DCC model. The sample spans 1990 to 2007.