Valueat-Risk

Table 9: S&P 500 Futures Return
QS(τ)
0.01 0.05 0.10 0.20 0.30 0.40 0.60 0.70 0.80 0.90 0.95 0.99
GJR-EDF vs: h = 1
rv t, rv w t , rv m t 0.44 -1.01 -1.07 -0.96 -1.27 0.53 -0.28 -1.47 -1.98 -2.40 -2.23 -0.93
e t, |e t|, rv t, rv w t , rv m t 0.32 -1.19 -1.57 -2.02 -2.74 -1.31 -1.21 -2.21 -2.69 -2.86 -2.56 -1.19
|r t|, |r t|I(r t < 0), rv t, rv w t , rvm t -0.06 -1.26 -1.39 -2.21 -2.69 -1.53 -1.36 -2.79 -3.03 -2.71 -2.32 -0.82
rvn t, rvn w t , rvn m t -1.80 0.16 -0.27 -0.69 -1.14 0.54 0.69 0.78 0.34 -0.83 -0.88 0.24
rv(2) t, rv(2) w t , rv(2)m t
0.17 -1.13 -1.48 -0.79 -1.61 0.21 0.06 -0.79 -1.64 -1.92 -2.22 -1.39
c t, c w , c m 0.03 -0.77 -0.85 -0.60 -1.15 0.34 -0.13 -1.80 -2.36 -2.55 -2.19 -1.03
c t, c w , c m , j t, j w t , jm t 0.48 -0.45 -0.74 -0.35 -0.35 1.29 -0.17 -1.65 -1.82 -2.24 -2.08 -0.52
|r t| ewma 1.15 2.48 2.71 2.25 0.78 -0.19 2.68 4.92 5.05 3.92 3.83 2.14
|r t|, |r t| w , |r t| m 1.90 3.44 2.67 0.78 -0.03 -0.84 2.06 4.55 4.78 3.94 3.63 2.63
V IX t, V IX w t , V IX m t -0.54 -0.03 0.65 0.52 0.50 0.79 -1.09 -2.34 -2.40 -1.82 -1.71 -0.76
rv t, rv w t , rv m t -2.16 -1.60 -0.93 -0.71 1.38 2.73 -0.21 -1.44 -1.44 -1.34 -1.67 -1.68
e t, |e t|,rv t, rv w t , rv m t -2.04 -1.50 -0.74 -0.64 1.23 2.80 -0.98 -2.50 -2.35 -2.33 -2.60 -2.34
|r t|, |r t|I(r t < 0), rv t, rv w t , rv m t -1.10 -1.23 -0.50 -0.56 0.84 2.37 -1.35 -2.53 -2.53 -2.35 -2.68 -2.75
rvn t, rvn w t , rvn m t -1.72 -1.26 -0.69 -0.37 2.01 2.69 1.53 0.37 0.74 0.38 -0.36 -0.75
rv(2) t, rv(2) w t , rv(2)m t
-2.29 -1.50 -0.99 -0.59 1.25 2.49 0.33 -1.03 -1.09 -0.98 -1.57 -1.36
c t, c w , c m -1.68 -1.58 -0.91 0.02 1.92 2.63 -0.52 -1.57 -1.39 -1.04 -1.50 -1.81
c t, c w , c m , j t, j w t , jm t -1.35 -1.03 -0.69 -0.03 1.67 2.74 0.12 -0.97 -1.02 -0.78 -0.59 -0.18
|r t| ewma 0.95 2.05 2.16 1.74 1.50 2.44 3.00 4.64 4.18 4.96 3.78 1.16
|r t|, |r t| w , |r t| m 1.03 2.28 3.12 2.74 1.40 2.17 3.13 3.19 4.12 4.07 2.72 0.88
V IX t, V IX w t , V IX m t -0.61 -0.13 0.91 0.23 0.76 2.19 -1.14 -2.68 -2.10 -1.83 -1.78 0.52
rv t, rv w t , rv m t 0.55 0.66 0.88 1.73 2.75 3.58 1.71 1.08 0.76 0.06 -1.34 -1.14
e t, |e t|,rv t, rv w t , rv m t 0.50 0.68 0.82 1.73 2.94 3.69 1.30 0.42 0.28 -0.42 -1.44 -1.03
|r t|, |r t|I(r t < 0), rv t, rv w t , rv m t 0.61 0.70 0.88 1.79 2.71 3.77 1.40 0.09 0.00 -0.31 -1.46 -0.88
rvn t, rvn w t , rvn m t 0.64 0.92 1.16 1.92 2.80 3.95 2.48 1.56 1.32 0.68 -0.48 -0.62
rv(2) t, rv(2) w t , rv(2)m t
0.57 0.71 1.00 1.73 2.53 3.40 1.86 1.18 1.12 -0.05 -1.16 -1.24
c t, c w , c m 0.65 0.70 1.01 1.88 2.67 3.27 1.56 0.66 0.46 0.10 -1.11 -0.36
c t, c w , c m , j t, j w t , jm t 0.74 1.16 1.18 2.01 2.59 2.84 1.31 0.81 0.81 0.51 -0.60 0.41
|r t| ewma 1.29 1.09 0.90 1.83 2.58 2.98 2.83 2.15 2.17 1.70 0.98 1.25
|r t|, |r t| w , |r t| m 1.31 1.53 1.63 2.27 2.84 3.50 2.87 1.99 2.12 1.68 0.88 1.57
V IX t, V IX w t , V IX m t 0.91 0.81 0.98 1.72 2.30 2.84 0.53 -1.02 -1.56 -1.43 -1.94 -0.77
h = 2
h = 5
QS(τ) refers to the difference in Quantile Score test at level τ (ranging from 0.01 to 0.99) discussed in Equation (13).
A negative value indicates that the alternative model outperforms the GJR-EDF benchmark, and the opposite for a
positive value. The value reported is the t-statistic as described in Equation (16) which is standard normal distributed.
In bold are denoted the test statistics that are significantly negative at 5% (one-sided), and in italics the ones that are
significantly positive at the same level. h denotes the forecasting horizon.
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