Table 1.
GARCH model specification.
Table 2.
Summary statistics.
Fig 1.
Notes: The upper graph shows the autocorrelation of portfolio logarithm realized standard deviation over the full sample period from February 01, 2001 to December 31, 2009 (2,242 observations). The lower graph shows the portfolio realized standard deviation against forecasts from HAR-RV and DCC models over the out-of-sample period from January 27, 2005 to December 31, 2009 (1,242 observations).
Table 3.
GARCH models estimation.
Table 4.
ARFIMA and HAR-RV models estimation.
Table 5.
Volatility forecast evaluation.
Fig 2.
Notes: The graphs show the fluctuation test statistics with the 95% critical values for the MSE differences between competing conditional volatility models. The MSE statistics are calculated using rolling windows of 124 observations over the out-of-sample evaluation period from January 27, 2005 to December 31, 2009 (1,242 observations).
Fig 3.
Notes: The graphs show the fluctuation test statistics with the 95% critical values for the MSE differences between competing conditional volatility models. The MSE statistics are calculated using rolling windows of 124 observations over the out-of-sample evaluation period from January 27, 2005 to December 31, 2009 (1,242 observations).
Table 6.
Unconditional coverage test.
Table 7.
Conditional coverage test.
Table 8.
Berkowitz test.
Table 9.
Tick loss.
Table 10.
Quadratic loss.
Table 11.
VaR efficiency.
Fig 4.
Notes: The pie chart shows the covariance risk contribution of each stock in an equally-weighted portfolio. The HighRisk portfolio is made up of the top 5 risky stocks, which are equally weighted. The LowRisk portfolio consists of the remaining 5 stocks, which are also equally weighted. Their return densities are shown in the upper panel. The bottom right graph (from top to bottom) shows the return distribution of the HighRisk portfolio mixed with 0%, 10%, 30%, 70%, 90% and 100% of LowRisk portfolio.
Fig 5.
Portfolio allocation: Univariate models.
Notes: The graphs show the p-values of Berkowitz, unconditional and conditional coverage tests for the VaR models (HAR-ST and GARCH-ST) under four confidence levels, i.e. 95%, 97.5%, 99%, and 99.5%. The portfolio includes long positions in 10 DJIA stocks with allocations to the HighRisk portfolio ranging from 10%, 30%, 50%, 70%, and 90% with the remainder allocated to the LowRisk portfolio. The out-of-sample evaluation period is from January 27, 2005 to December 31, 2009 (1,242 observations).
Fig 6.
Portfolio allocation: Multivariate models.
Notes: The graphs show the p-values of Berkowitz, unconditional and conditional coverage tests for the VaR models (EWMA and DCC-T) under four confidence levels, i.e. 95%, 97.5%, 99%, and 99.5%. The portfolio includes long positions in 10 DJIA stocks with allocations to the HighRisk portfolio ranging from 10%, 30%, 50%, 70%, and 90% with the remainder allocated to the LowRisk portfolio. The out-of-sample evaluation period is from January 27, 2005 to December 31, 2009 (1,242 observations).
Fig 7.
Notes: The graphs show the number of exceptions, mean absolute deviation (MeanAD), and maximum absolute deviation (MaxAD) over different sample periods. The full out-of-sample evaluation period is from January 27, 2005 to December 31, 2009. The Pre global financial crisis (GFC) is from January 27, 2005 to August 29, 2008. The over GFC period is from September 02, 2008 to December 31, 2009. Under each model, the upper bar is for VaR99 while the lower bar is for VaR95. The expected number of exceptions is shown at the top of each graph on the left.