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Combining forecasts? Keep it simple

Szymon Lis

Szymon Lis

University of Warsaw, Faculty of Economic SciencesWarsaw, Poland
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Lis, Szymon
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Marcin Chlebus

Marcin Chlebus

University of Warsaw, Faculty of Economic SciencesWarsaw, Poland
Profilo Orcid
Cerca questo autore su
Sciendo|Google Scholar
Chlebus, Marcin
  
31 ott 2023
Combining forecasts? Keep it simple's Cover Image
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Pubblicato online: 31 ott 2023
Pagine: 343 - 370
DOI: https://doi.org/10.2478/ceej-2023-0020
Parole chiave
© 2023 Szymon Lis et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Figure 1.

Returns for commodities (light grey for calm period, dark grey for crisis period)
Returns for commodities (light grey for calm period, dark grey for crisis period)

Figure 2.

The diagram of QRNN model with four predictors and two hidden nodes Source: Cannon, 2011
The diagram of QRNN model with four predictors and two hidden nodes Source: Cannon, 2011

Figure 3.

Correlations between VaR forecasts for p-value = 0.025
Correlations between VaR forecasts for p-value = 0.025

Figure 4.

Correlations between VaR forecasts for p-value = 0.01
Correlations between VaR forecasts for p-value = 0.01

Figure 5.

Returns and VaR forecast for confidence levels: 0.975 (on left) and 0.99 (on right)
Returns and VaR forecast for confidence levels: 0.975 (on left) and 0.99 (on right)

Figure 6.

Combing weight for the most promising methods of combining VaR forecasts for all assets for both confidence levels (CL)—0.975 and 0.99
Combing weight for the most promising methods of combining VaR forecasts for all assets for both confidence levels (CL)—0.975 and 0.99

Test results: Excess Ratio (ER), Kupiec (UC), Christoffersen (CC), Dynamic Quantile (DQ) and Traffic Light (TL) divided into the analysed models and periods for silver for confidence level equal to 0_99

Model Period I (Whole period) Period II (All calm periods) Period I (All crisis periods) Period I (COVID period)
ER UC CC DQ TL ER UC CC DQ TL ER UC CC DQ TL ER UC CC DQ TL
GARCH 2.36% 56.00 62.62 114.64 R 2.30% 36.32 39.22 67.05 R 2.50% 19.83 23.85 56.09 R 3.46% 12.30 15.63 76.45 Y
GARCH-t 1.30% 3.43 19.68 58.92 Y 1.06% 0.12 8.50 25.70 G 1.85% 7.28 14.45 46.95 Y 2.23% 11.52 15.63 73.12 Y
GARCH-st 1.03% 0.05 21.68 77.52 G 0.75% 1.94 14.40 42.41 G 1.69% 4.96 13.18 47.30 Y 2.16% 11.30 15.63 75.03 Y
QML-GARCH 2.31% 52.56 62.26 120.78 R 2.20% 31.39 34.78 61.32 R 2.58% 21.72 28.68 71.24 R 3.51% 12.73 15.63 74.50 Y
CaViaR 1.11% 0.46 3.05 19.34 G 0.93% 0.16 1.48 4.38 G 1.53% 3.03 4.14 39.88 Y 2.60% 20.58 20.79 88.35 R
Mean 1.01% 0.00 11.44 36.79 G 0.82% 0.98 12.38 38.30 G 1.45% 2.22 3.49 11.08 Y 3.46% 11.90 12.19 48.59 Y
Highest VaR 2.65% 78.37 84.86 157.60 R 2.44% 43.32 45.64 76.26 R 3.14% 36.67 41.01 101.95 R 3.46% 31.46 32.44 130.63 R
Lowest VaR 0.70% 4.29 10.09 20.68 G 0.62% 4.99 7.69 9.93 G 0.89% 0.17 3.18 21.35 G 2.16% 4.13 6.42 55.81 Y
CQOM 1.64% 14.24 29.29 108.75 R 1.51% 6.61 19.03 76.54 Y 1.93% 8.57 11.55 52.32 Y 2.60% 11.60 12.19 54.94 Y
Elastic Net 1.30% 3.43 19.68 87.01 Y 1.10% 0.27 8.30 44.48 G 1.77% 6.07 13.76 55.34 Y 3.46% 14.19 17.72 82.18 R
LASSO 1.20% 1.62 24.93 104.71 G 1.06% 0.12 13.66 64.24 G 1.53% 3.03 12.44 50.30 Y 3.90% 11.93 15.63 65.06 Y
QRF 2.89% 99.07 99.71 290.41 R 2.64% 54.69 56.27 136.88 R 3.46% 46.41 46.60 187.82 R 4.76% 13.24 12.03 98.63 Y
GBRM 1.56% 11.38 23.52 68.29 Y 1.41% 4.32 13.57 42.35 Y 1.93% 8.57 11.55 40.85 Y 2.60% 2.37 11.96 92.74 Y
QRNN 3.61% 171.00 177.55 776.88 R 3.46% 109.12 115.76 459.11 R 3.95% 62.45 63.00 339.63 R 6.06% 47.94 48.22 305.35 R

Test results: Excess Ratio (ER), Kupiec (UC), Christoffersen (CC), Dynamic Quantile (DQ) and Traffic Light (TL) divided into the analysed models and periods for oil for confidence level equal to 0_975_

Model Period I (Whole period) Period II (All calm periods) Period I (All crisis periods) Period I (COVID period)
ER UC CC DQ TL ER UC CC DQ TL ER UC CC DQ TL ER UC CC DQ TL
GARCH 2.96% 3.39 3.43 10.66 Y 2.37% 0.22 3.56 7.97 G 4.35% 14.30 15.37 19.82 R 5.63% 6.88 8.63 20.99 Y
GARCH-t 3.54% 16.26 37.79 313.36 R 1.75% 7.51 8.57 55.08 G 7.73% 90.37 99.57 311.51 R 4.76% 22.16 24.91 157.82 R
GARCH-st 2.43% 0.09 0.95 5.04 G 1.96% 3.83 6.10 8.43 G 3.54% 4.91 7.86 9.45 Y 4.33% 2.61 6.14 21.69 Y
QML-GARCH 3.04% 3.53 3.43 10.71 Y 2.39% 0.24 3.56 8.09 G 4.35% 14.30 15.37 19.84 R 5.77% 6.97 8.63 20.99 Y
CaViaR 2.57% 0.09 1.66 14.89 G 1.96% 3.83 3.84 7.94 G 4.03% 10.04 11.73 24.06 Y 5.19% 3.85 6.69 21.18 Y
Mean 2.50% 0.00 0.68 18.03 G 1.37% 18.14 19.25 26.72 G 5.15% 27.59 27.74 37.72 R 4.76% 5.28 5.49 11.16 Y
Highest VaR 4.86% 74.74 83.96 264.82 R 3.16% 4.76 4.76 29.49 Y 8.86% 125.64 131.16 328.47 R 6.49% 39.66 41.25 172.61 R
Lowest VaR 1.59% 16.27 18.80 24.20 G 0.99% 34.98 35.56 29.30 G 2.98% 1.10 3.52 7.61 G 3.46% 0.25 1.96 6.54 G
CQOM 3.01% 4.12 22.26 157.69 Y 1.78% 6.80 6.81 45.91 G 5.88% 42.38 56.14 181.95 R 3.90% 27.60 34.10 133.16 R
Elastic Net 2.98% 3.75 5.04 22.36 Y 1.96% 3.83 3.84 9.29 G 5.39% 32.24 32.77 59.60 R 6.49% 14.83 15.27 63.54 R
LASSO 3.03% 4.51 10.03 69.61 Y 1.78% 6.80 7.78 12.84 G 5.96% 44.17 45.66 108.55 R 7.36% 17.15 18.83 85.82 R
QRF 5.82% 137.72 139.48 253.06 R 5.28% 70.54 70.63 217.80 R 7.09% 72.17 74.44 124.92 R 8.66% 19.60 25.83 74.65 R
GBRM 3.39% 12.23 12.24 24.87 Y 2.81% 1.13 2.11 15.93 G 4.75% 20.50 21.00 32.50 R 4.76% 3.85 4.23 13.35 Y
QRNN 5.65% 125.62 126.08 208.16 R 4.73% 47.48 50.07 297.87 R 7.81% 92.75 92.77 156.26 R 9.09% 53.11 53.12 134.19 R

Test results: Excess Ratio (ER), Kupiec (UC), Christoffersen (CC), Dynamic Quantile (DQ) and Traffic Light (TL) divided into the analysed models and periods for gas for confidence level equal to 0_99_

Model Period I (Whole period) Period II (All calm periods) Period I (All crisis periods) Period I (COVID period)
ER UC CC DQ TL ER UC CC DQ TL ER UC CC DQ TL ER UC CC DQ TL
GARCH 1.03% 0.05 0.95 3.12 G 0.79% 1.41 1.78 5.43 G 1.61% 3.94 4.60 9.29 Y 3.46% 2.37 2.60 26.03 Y
GARCH-t 0.67% 5.05 5.43 8.05 G 0.48% 9.84 9.98 10.05 G 1.13% 0.20 0.51 2.29 G 2.16% 0.19 0.27 2.44 G
GARCH-st 0.99% 0.01 0.82 9.81 G 0.75% 1.94 2.27 6.23 G 1.53% 3.03 3.62 13.25 Y 2.09% 1.02 1.17 27.33 G
QML-GARCH 1.08% 0.28 1.26 3.39 G 0.81% 1.48 1.78 5.38 G 1.77% 6.07 6.87 12.93 Y 3.54% 2.45 2.60 26.19 Y
CaViaR 1.05% 0.07 0.95 5.60 G 0.82% 0.98 1.38 5.06 G 1.53% 3.03 3.62 10.03 Y 2.60% 1.02 1.17 9.73 G
Mean 0.84% 1.11 1.70 4.82 G 0.65% 4.07 4.32 8.78 G 1.29% 0.96 1.37 4.27 G 3.46% 0.29 0.27 2.80 G
Highest VaR 1.25% 2.45 3.77 10.26 Y 0.99% 0.00 0.58 2.86 G 1.85% 7.28 8.14 19.35 Y 3.46% 2.37 2.60 26.07 Y
Lowest VaR 0.63% 6.80 7.12 9.37 G 0.41% 13.10 13.20 12.22 G 1.13% 0.20 0.51 2.41 G 2.16% 0.19 0.27 2.70 G
CQOM 4.40% 264.51 268.51 748.76 R 4.67% 209.24 212.38 615.08 R 3.78% 56.92 57.68 156.26 R 2.60% 11.30 12.03 29.88 Y
Elastic Net 3.49% 158.07 167.67 609.97 R 3.43% 106.60 113.48 430.37 R 3.62% 51.57 54.28 217.59 R 3.46% 61.53 61.96 212.30 R
LASSO 1.35% 4.56 11.81 67.51 Y 1.10% 0.27 8.30 4.77 G 1.93% 8.57 9.06 39.17 Y 3.90% 6.24 6.68 26.86 Y
QRF 3.32% 140.58 140.62 422.57 R 2.78% 62.80 62.83 203.90 R 4.59% 86.18 86.24 262.53 R 4.76% 17.29 18.39 63.33 R
GBRM 1.54% 10.50 10.50 39.50 Y 1.23% 1.51 2.04 16.13 G 2.25% 14.56 15.85 44.46 R 2.60% 2.37 2.60 25.28 Y
QRNN 3.78% Inf Inf 1181.52 R 4.12% 160.80 160.80 1024.17 R 2.98% 32.11 34.39 194.18 R 6.06% 4.13 4.46 71.51 Y

Test results: Excess Ratio (ER), Kupiec (UC), Christoffersen (CC), Dynamic Quantile (DQ) and Traffic Light (TL) divided into the analysed models and periods for gold for confidence level equal to 0_99

Model Period I (Whole period) Period II (All calm periods) Period I (All crisis periods) Period I (COVID period)
ER UC CC DQ TL ER UC CC DQ TL ER UC CC DQ TL ER UC CC DQ TL
GARCH 1.85% 24.38 24.59 47.78 R 1.65% 10.30 10.36 20.23 Y 2.33% 16.25 16.39 40.37 R 3.46% 8.64 9.89 31.80 Y
GARCH-t 1.13% 0.69 6.40 18.19 G 1.06% 0.12 1.02 3.17 G 1.29% 0.96 6.88 34.25 G 2.19% 2.39 5.38 30.03 Y
GARCH-st 0.91% 0.32 4.16 9.17 G 0.89% 0.36 1.79 3.81 G 0.97% 0.01 2.69 9.69 G 2.16% 2.37 5.38 30.07 Y
QML-GARCH 1.90% 24.66 24.59 47.38 R 1.67% 10.35 10.36 20.02 Y 2.37% 16.43 16.39 40.04 R 3.55% 8.87 9.89 31.55 Y
CaViaR 1.03% 0.05 3.07 15.54 G 1.10% 0.27 1.09 6.80 G 0.89% 0.17 3.18 13.09 G 2.60% 4.13 6.42 26.36 Y
Mean 1.30% 3.43 5.09 18.31 Y 1.30% 2.48 2.89 8.34 G 1.29% 0.96 2.61 18.09 G 3.46% 8.64 9.89 31.23 Y
Highest VaR 1.88% 25.64 26.93 52.17 R 1.65% 10.30 10.36 19.95 Y 2.42% 18.01 19.63 49.68 R 3.46% 8.64 9.89 31.27 Y
Lowest VaR 0.84% 1.11 5.53 12.07 G 0.86% 0.63 2.19 3.78 G 0.81% 0.51 3.89 13.52 G 2.16% 2.37 5.38 30.17 Y
CQOM 1.54% 10.50 13.28 51.94 Y 1.44% 5.04 7.12 34.06 Y 1.77% 6.07 6.77 35.15 Y 2.60% 4.13 6.42 52.00 Y
Elastic Net 1.32% 3.98 11.47 62.74 Y 1.10% 0.27 8.30 39.08 G 1.85% 7.28 7.86 40.96 Y 3.46% 8.64 9.89 43.82 Y
LASSO 1.27% 2.92 4.68 34.99 Y 1.06% 0.12 1.02 7.58 G 1.77% 6.07 6.77 42.25 Y 3.90% 11.30 12.19 48.62 Y
QRF 3.61% 171.00 171.03 448.16 R 3.64% 122.05 122.39 323.10 R 3.54% 48.97 52.20 148.36 R 4.76% 17.29 18.39 73.79 R
GBRM 1.35% 4.56 11.81 80.37 Y 0.96% 0.05 4.83 17.66 G 2.25% 14.56 16.57 99.99 R 2.60% 4.13 6.42 47.14 Y
QRNN 3.42% 150.49 160.82 541.10 R 3.29% 96.70 102.40 349.83 R 3.70% 54.22 58.90 206.00 R 6.06% 27.68 27.71 80.57 R

Test results: Excess Ratio (ER), Kupiec (UC), Christoffersen (CC), Dynamic Quantile (DQ) and Traffic Light (TL) divided into the analysed models and periods for gas for confidence level equal to 0_975_

Model Period I (Whole period) Period II (All calm periods) Period I (All crisis periods) Period I (COVID period)
ER UC CC DQ TL ER UC CC DQ TL ER UC CC DQ TL ER UC CC DQ TL
GARCH 2.19% 1.72 1.72 5.62 G 1.78% 6.80 6.81 8.64 G 3.14% 1.93 1.98 7.75 G 5.63% 0.79 1.36 10.87 G
GARCH-t 2.09% 2.99 3.01 6.79 G 1.75% 7.51 7.52 8.66 G 2.90% 0.77 0.77 5.75 G 4.76% 1.58 2.32 10.97 G
GARCH-st 2.45% 0.04 0.13 2.93 G 2.09% 2.10 2.47 2.96 G 3.30% 2.98 3.08 10.15 Y 4.33% 1.49 2.32 11.11 G
QML-GARCH 2.24% 1.22 1.22 4.27 G 1.85% 5.50 5.50 6.19 G 3.14% 1.93 1.98 7.59 G 5.63% 0.79 1.36 11.01 G
CaViaR 2.53% 0.01 0.06 12.09 G 1.99% 3.34 3.88 9.26 G 3.78% 7.28 7.71 20.93 Y 5.19% 1.57 2.32 13.96 G
Mean 2.24% 1.22 1.60 9.07 G 1.82% 6.13 7.02 11.42 G 3.22% 2.43 2.50 10.27 Y 4.76% 0.79 1.36 10.88 G
Highest VaR 2.86% 2.14 2.20 8.51 G 2.37% 0.22 0.30 1.96 G 4.03% 10.04 10.71 24.08 Y 6.49% 2.61 3.52 14.67 Y
Lowest VaR 1.83% 8.48 8.72 12.70 G 1.41% 16.94 17.20 17.60 G 2.82% 0.50 0.50 5.85 G 3.46% 0.79 1.36 10.63 G
CQOM 4.52% 56.49 67.33 232.88 R 4.22% 29.41 43.33 169.76 R 5.23% 29.10 29.21 76.81 R 3.90% 0.63 0.77 10.55 G
Elastic Net 2.45% 0.04 0.84 8.38 G 2.02% 2.90 7.23 11.76 G 3.46% 4.22 7.31 17.85 Y 6.49% 1.79 2.32 10.66 G
LASSO 2.45% 0.04 2.09 13.69 G 2.09% 2.10 6.04 14.24 G 3.30% 2.98 3.08 17.62 Y 7.36% 1.86 2.32 9.07 G
QRF 5.53% 117.24 117.29 357.06 R 5.25% 68.99 69.14 221.09 R 6.20% 49.73 49.74 141.98 R 8.66% 6.88 8.44 18.56 Y
GBRM 3.30% 9.83 12.20 37.10 Y 2.85% 1.38 2.39 12.48 G 4.35% 14.30 15.37 39.06 R 4.76% 2.61 3.22 12.55 Y
QRNN 5.27% 99.63 105.52 453.83 R 5.21% 67.45 70.62 349.00 R 5.39% 32.24 35.09 119.62 R 9.09% 6.88 6.98 64.95 Y

Test results: Excess Ratio (ER), Kupiec (UC), Christoffersen (CC), Dynamic Quantile (DQ) and Traffic Light (TL) divided into the analysed models and periods for copper for confidence level equal to 0_99_

Model Period I (Whole period) Period II (All calm periods) Period I (All crisis periods) Period I (COVID period)
ER UC CC DQ TL ER UC CC DQ TL ER UC CC DQ TL ER UC CC DQ TL
GARCH 1.71% 17.36 21.56 53.94 R 1.44% 5.04 6.26 9.79 G 2.33% 16.25 24.63 98.67 R 3.46% 11.30 15.63 110.23 Y
GARCH-t 1.18% 1.27 10.40 38.94 G 0.93% 0.16 0.67 8.74 G 1.77% 6.07 18.76 79.58 Y 2.16% 4.13 11.96 99.07 Y
GARCH-st 1.08% 0.28 10.67 40.15 G 0.89% 0.36 0.82 8.16 G 1.53% 3.03 18.17 92.27 Y 2.12% 4.11 11.96 99.36 Y
QML-GARCH 1.78% 20.75 24.47 55.48 R 1.54% 7.47 8.88 12.95 G 2.33% 16.25 24.63 98.80 R 3.49% 11.41 15.63 131.62 Y
CaViaR 1.20% 1.62 3.71 27.18 G 1.13% 0.49 1.25 16.74 G 1.37% 1.53 6.99 30.44 G 2.60% 2.37 5.38 82.34 Y
Mean 1.01% 0.00 11.44 45.52 G 0.89% 0.36 0.82 6.08 G 1.29% 0.96 19.09 118.12 G 3.46% 4.13 11.96 107.81 Y
Highest VaR 1.95% 29.58 32.33 73.24 R 1.72% 12.41 14.15 27.03 G 2.50% 19.83 27.24 95.92 R 3.46% 11.30 15.63 110.69 Y
Lowest VaR 0.87% 0.79 5.01 21.97 G 0.75% 1.94 2.27 8.58 G 1.13% 0.20 7.18 38.09 G 2.16% 2.37 5.38 59.22 Y
CQOM 1.85% 24.38 33.04 376.94 R 1.34% 3.04 3.40 84.60 G 3.06% 34.37 42.11 389.48 R 2.60% 31.46 32.44 159.44 R
Elastic Net 1.27% 2.92 7.43 78.15 Y 1.13% 0.49 1.25 53.05 G 1.61% 3.94 12.74 66.78 Y 3.46% 2.37 5.38 26.06 Y
LASSO 1.23% 2.02 2.21 136.46 G 1.06% 0.12 0.78 55.47 G 1.61% 3.94 4.90 112.99 Y 3.90% 2.42 5.38 27.19 Y
QRF 3.63% 173.63 178.33 542.73 R 3.36% 101.61 101.76 317.23 Y 4.27% 74.00 81.26 275.12 R 4.76% 13.24 12.19 81.67 Y
GBRM 1.73% 18.46 20.31 64.03 R 1.48% 5.80 7.09 29.37 G 2.33% 16.25 20.93 56.39 R 2.60% 9.54 12.19 49.56 Y
QRNN 3.56% 165.79 168.16 374.11 R 4.01% 152.20 153.27 351.61 R 2.50% 19.83 21.28 76.59 R 6.06% 17.29 17.67 104.84 R

The best model for each commodity (rows) and for all periods (columns) for confidence level of 0_975 (upper part), and 0_99 (lower part) achieved using MCS procedure

Model Period I (Whole period) Period II (All calm periods) Period III (All crisis periods) Period IV (COVID period)
Confidence level = 0.025
Gold GARCH-t GARCH-t GARCH-t CQOM
Silver Mean GARCH-st Mean Mean
Oil GARCH GARCH GARCH-st GARCH-st
Gas GARCH-st Highest VaR GARCH-st LASSO
Copper Mean GARCH Elastic Net Elastic Net
Confidence level = 0.01
Gold Lowest VaR Lowest VaR GARCH-st GARCH-st
Silver Lowest VaR Lowest VaR Lowest VaR Lowest VaR
Oil Lowest VaR Mean Lowest VaR Lowest VaR
Gas Lowest VaR GARCH-t Lowest VaR CQOM
Copper Lowest VaR Lowest VaR Lowest VaR GARCH-st

Test results: Excess Ratio (ER), Kupiec (UC), Christoffersen (CC), Dynamic Quantile (DQ) and Traffic Light (TL) divided into the analysed models and periods for oil for confidence level equal to 0_99_

Model Period I (Whole period) Period II (All calm periods) Period I (All crisis periods) Period I (COVID period)
ER UC CC DQ TL ER UC CC DQ TL ER UC CC DQ TL ER UC CC DQ
GARCH 1.54% 10.50 13.28 25.45 Y 1.27% 1.97 2.92 11.38 G 2.17% 12.95 18.36 24.49 R 3.43% 8.41 13.93 72.02 Y
GARCH-t 2.41% 59.53 80.88 490.49 R 0.96% 0.05 1.25 106.88 G 5.80% 136.83 145.95 536.93 R 2.16% 39.45 44.28 346.76 R
GARCH-st 1.01% 0.00 0.58 3.13 G 0.89% 0.36 0.82 10.06 G 1.29% 0.96 2.61 6.28 G 2.16% 4.13 6.42 24.85 Y
QML-GARCH 1.52% 9.64 12.55 24.38 Y 1.29% 2.01 2.92 11.47 G 2.09% 11.41 17.23 23.12 Y 3.45% 8.55 13.93 72.03 Y
CaViaR 1.49% 8.81 8.82 14.83 Y 1.10% 0.27 0.98 5.46 G 2.42% 18.01 18.10 27.26 R 2.60% 6.24 7.95 36.61 Y
Mean 1.44% 7.26 7.28 20.93 Y 0.96% 0.05 0.59 21.48 G 2.58% 21.72 21.76 30.45 R 3.41% 8.24 7.95 37.61 Y
Highest VaR 3.49% 158.07 169.88 460.74 R 1.99% 22.40 22.42 79.80 R 7.00% 194.17 201.30 574.54 R 3.46% 56.90 61.42 306.55 R
Lowest VaR 0.46% 15.51 18.66 17.14 G 0.27% 21.77 21.81 15.72 G 0.89% 0.17 3.18 8.15 G 2.16% 1.02 4.98 40.55 G
CQOM 2.36% 56.00 94.71 707.99 R 1.51% 6.61 11.24 151.64 Y 4.35% 76.99 104.99 671.01 R 2.60% 39.45 44.28 297.04 R
Elastic Net 1.64% 14.24 14.83 27.95 R 0.99% 0.00 0.58 15.67 G 3.14% 36.67 37.11 69.57 R 3.46% 14.19 14.79 46.58 R
LASSO 1.59% 12.31 14.83 120.13 Y 0.79% 1.41 1.78 41.45 G 3.46% 46.41 47.70 137.00 R 3.90% 17.29 17.67 112.70 R
QRF 3.68% 178.14 179.09 390.91 R 3.26% 94.28 94.76 279.50 R 4.67% 89.32 92.69 240.24 R 4.76% 20.58 26.33 125.31 R
GBRM 1.80% 21.93 22.21 34.37 R 1.48% 5.80 7.09 38.64 Y 2.58% 21.72 23.01 40.31 R 2.60% 6.24 7.95 25.84 Y
QRNN 4.43% 267.55 269.31 1197.81 R 3.40% 104.09 104.84 533.39 R 6.84% 186.18 186.44 816.55 R 6.06% 66.27 69.41 542.78 R

Statistics of prices’ log-returns

Commodity Min. 1st Qu. Median Mean 3rd Qu. Max J-B test Skewness Ex. Kurtosis
Gold −0.0982 −0.0049 0.0005 0.0004 0.006 0.0864 6398 (<0.001) –0.2658 8.7160
Silver −0.1955 −0.0080 0.0011 0.0003 0.0090 0.1220 13942 (<0.001) –0.9263 10.8079
Oil −0.2799 −0.0128 0.0008 0.0001 0.0130 0.3196 52559 (<0.001) –1.9164 52.6291
Gas −0.1990 −0.1911 −0.0007 −0.0001 0.0173 0.3238 6833 (<0.001) 0.5643 8.7537
Copper −0.1169 −0.0082 0.0002 0.0003 0.0089 0.1177 4279 (<0.001) –0.1731 7.6239

Test results: Excess Ratio (ER), Kupiec (UC), Christoffersen (CC), Dynamic Quantile (DQ) and Traffic Light (TL) divided into the analysed models and periods for copper for confidence level equal to 0_975

Model Period I (Whole period) Period II (All calm periods) Period I (All crisis periods) Period I (COVID period)
ER UC CC DQ TL ER UC CC DQ TL ER UC CC DQ TL ER UC CC DQ TL
GARCH 2.96% 3.39 4.77 19.79 Y 2.61% 0.14 0.14 4.71 G 3.78% 7.28 9.54 30.64 Y 5.63% 3.85 6.69 29.71 Y
GARCH-t 2.67% 0.48 3.05 18.85 G 2.33% 0.34 0.45 2.93 G 3.46% 4.22 7.42 36.25 Y 4.76% 1.58 5.92 35.76 G
GARCH-st 2.65% 0.36 3.04 20.04 G 2.37% 0.22 0.30 3.05 G 3.30% 2.98 6.71 39.01 Y 4.33% 2.61 6.14 31.14 Y
QML-GARCH 2.99% 3.44 4.77 19.46 Y 2.65% 0.16 0.14 4.21 G 3.78% 7.28 9.54 30.72 Y 5.63% 3.85 6.69 42.44 Y
CaViaR 2.57% 0.09 0.62 10.85 G 2.26% 0.69 3.75 9.95 G 3.30% 2.98 6.71 16.07 Y 5.19% 2.71 6.14 51.69 Y
Mean 2.43% 0.09 0.95 10.86 G 2.23% 0.91 3.87 4.65 G 2.90% 0.77 6.12 22.07 G 4.65% 2.55 6.14 40.23 Y
Highest VaR 3.27% 9.27 10.61 29.07 Y 2.88% 1.66 1.74 12.79 G 4.19% 12.09 15.09 45.05 Y 6.49% 3.85 6.69 30.17 Y
Lowest VaR 2.12% 2.64 4.58 13.18 G 1.92% 4.35 6.54 7.42 G 2.58% 0.03 6.99 23.06 G 3.46% 1.58 5.92 40.33 G
CQOM 4.43% 51.67 Inf 960.75 R 2.98% 2.65 13.39 230.53 G 7.81% 92.75 121.91 806.55 R 3.90% 218.89 219.00 703.95 R
Elastic Net 2.62% 0.25 1.65 17.45 G 2.37% 0.22 0.51 9.27 G 3.22% 2.43 6.46 38.25 Y 6.49% 3.85 6.69 25.60 Y
LASSO 2.62% 0.25 1.65 43.06 G 2.37% 0.22 0.51 36.09 G 3.22% 2.43 6.46 30.80 Y 7.36% 5.28 7.53 31.78 Y
QRF 4.91% 77.52 78.43 231.96 R 4.39% 35.03 35.06 121.60 R 6.12% 47.85 49.03 128.29 R 8.66% 10.57 13.88 49.29 Y
GBRM 3.08% 5.33 7.41 33.89 Y 2.95% 2.29 3.56 21.25 G 3.38% 3.57 12.79 40.09 Y 4.76% 3.85 6.69 23.83 Y
QRNN 4.52% 56.49 61.01 421.32 R 4.49% 38.60 39.35 325.86 R 4.59% 17.91 23.66 136.29 R 9.09% 10.57 11.54 78.96 Y

Test results: Excess Ratio (ER), Kupiec (UC), Christoffersen (CC), Dynamic Quantile (DQ) and Traffic Light (TL) divided into the analysed models and periods for silver for confidence level equal to 0_975_

Model Period I (Whole period) Period II (All calm periods) Period I (All crisis periods) Period I (COVID period)
ER UC CC DQ TL ER UC CC DQ TL ER UC CC DQ TL ER UC CC DQ TL
GARCH 3.58% 17.72 21.25 41.94 R 3.46% 9.96 11.56 22.90 Y 3.86% 8.16 10.22 22.05 Y 5.63% 3.85 6.69 28.62 Y
GARCH-t 3.42% 12.86 21.07 38.19 Y 3.16% 4.76 9.27 19.21 Y 4.03% 10.04 13.55 24.26 Y 4.76% 2.85 6.14 22.35 Y
GARCH-st 2.91% 2.74 13.82 34.46 Y 2.68% 0.36 8.13 24.97 G 3.46% 4.22 7.42 16.69 Y 4.33% 2.74 6.14 23.15 Y
QML-GARCH 3.44% 13.51 21.50 43.07 R 3.40% 8.65 11.93 24.52 Y 3.54% 4.91 10.27 22.85 Y 5.69% 2.90 6.14 23.09 Y
CaViaR 2.84% 1.87 3.74 25.47 G 2.64% 0.24 0.24 9.71 G 3.30% 2.98 6.86 29.91 Y 5.19% 10.57 11.72 38.85 Y
Mean 2.77% 1.17 4.91 16.11 G 2.47% 0.01 0.75 5.15 G 3.46% 4.22 7.55 18.32 Y 4.76% 6.88 8.88 31.12 Y
Highest VaR 4.14% 38.32 44.18 74.44 R 3.81% 17.68 20.61 38.17 R 4.91% 23.23 25.99 42.50 R 6.49% 12.63 13.45 34.59 R
Lowest VaR 2.21% 1.46 6.86 19.48 G 2.09% 2.10 2.47 10.64 G 2.50% 0.00 7.41 22.49 G 3.46% 2.61 6.14 30.82 Y
CQOM 2.38% 0.24 18.45 51.46 G 2.16% 1.44 14.15 34.88 G 2.90% 0.77 6.12 20.39 G 3.90% 2.69 6.14 27.15 Y
Elastic Net 2.48% 0.01 8.00 30.60 G 2.30% 0.50 3.40 15.17 G 2.91% 0.78 6.12 22.45 G 6.49% 3.85 6.69 34.39 Y
LASSO 2.72% 0.79 17.15 72.79 G 2.26% 0.69 12.27 39.12 G 3.78% 7.28 11.65 55.77 Y 7.36% 17.15 17.42 73.08 R
QRF 4.81% 72.00 73.95 217.41 R 4.39% 35.03 35.37 107.96 R 5.80% 40.61 42.44 126.89 R 8.66% 10.57 10.57 55.56 Y
GBRM 3.32% 10.41 19.53 59.65 Y 3.05% 3.42 6.62 22.87 Y 3.95% 9.08 15.27 47.33 Y 4.76% 6.88 8.63 29.33 Y
QRNN 5.00% 83.19 102.20 395.17 R 4.70% 46.17 55.58 253.17 R 5.72% 38.88 48.45 157.84 R 9.09% 39.66 40.11 147.69 R

Test results: Excess Ratio (ER), Kupiec (UC), Christoffersen (CC), Dynamic Quantile (DQ) and Traffic Light (TL) divided into the analysed models and periods for gold for confidence level equal to 0_975

Model Period I (Whole period) Period II (All calm periods) Period I (All crisis periods) Period I (COVID period)
ER UC CC DQ TL ER UC CC DQ TL ER UC CC DQ TL ER UC CC DQ TL
GARCH 3.10% 5.77 6.02 21.07 Y 2.85% 1.38 1.55 9.22 G 3.70% 6.45 6.50 16.92 Y 5.63% 6.88 6.98 26.48 Y
GARCH-t 2.81% 1.62 1.77 18.08 G 2.57% 0.06 0.07 2.48 G 3.38% 3.57 3.80 28.92 Y 4.76% 3.85 4.23 21.16 Y
GARCH-st 2.38% 0.24 0.41 16.80 G 2.20% 1.16 1.29 5.88 G 2.82% 0.50 1.36 21.53 G 4.33% 2.61 3.22 22.78 Y
QML-GARCH 3.17% 5.91 6.02 22.59 Y 2.88% 1.66 1.80 9.14 G 3.62% 5.66 5.74 19.26 Y 5.59% 6.80 6.98 26.36 Y
CaViaR 2.65% 0.36 0.74 20.85 G 2.47% 0.01 0.04 15.69 G 3.06% 1.49 2.02 14.36 G 5.19% 5.28 7.53 31.81 Y
Mean 2.81% 1.62 1.77 14.83 G 2.64% 0.24 0.24 9.74 G 3.22% 2.43 2.79 11.04 Y 4.76% 3.85 4.23 15.45 Y
Highest VaR 3.46% 14.18 14.38 41.56 R 3.19% 5.25 5.25 15.70 Y 4.11% 11.05 11.42 33.81 Y 6.49% 10.57 11.54 41.72 Y
Lowest VaR 2.07% 3.37 3.39 10.49 G 1.92% 4.35 4.36 8.59 G 2.42% 0.04 0.14 7.87 G 3.46% 0.79 2.04 7.83 G
CQOM 2.69% 0.63 9.01 56.89 G 2.47% 0.01 4.21 39.29 G 3.25% 2.48 6.46 44.63 Y 3.90% 1.58 2.48 11.99 G
Elastic Net 2.50% 0.00 1.85 24.77 G 2.20% 1.16 1.39 7.99 G 3.27% 2.49 4.23 30.53 Y 6.49% 10.57 11.54 81.74 Y
LASSO 2.62% 0.25 6.95 30.04 G 2.26% 0.69 6.14 14.90 G 3.46% 4.22 5.51 44.51 Y 7.36% 14.83 15.27 91.46 R
QRF 5.15% 92.01 92.12 246.98 R 4.87% 52.90 52.90 158.42 R 5.80% 40.61 41.02 93.76 R 8.66% 22.16 22.59 57.90 R
GBRM 2.89% 2.43 2.51 16.97 G 2.57% 0.06 0.07 5.19 G 3.62% 5.66 5.74 19.85 Y 4.76% 3.85 4.23 17.65 Y
QRNN 4.71% 66.66 70.04 223.39 R 4.49% 38.60 39.35 105.05 R 5.23% 29.10 32.44 212.53 R 9.09% 24.82 29.35 176.38 R
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Economia e business, Economia politica, Teoria economica, sistemi e strutture, Microeconomia, Macroeconomia, Politica economica
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