Combining forecasts? Keep it simple

Lis, Szymon; Chlebus, Marcin

Combining forecasts? Keep it simple

Central European Economic Journal

Volume 10 (2023): Issue 57 (January 2023)

By: Szymon Lis

and Marcin Chlebus

Open Access

|Oct 2023

Figures & Tables

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

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

Correlations between VaR forecasts for p-value = 0.025

Correlations between VaR forecasts for p-value = 0.01

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

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)
Model	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)
Model	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)
Model	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)
Model	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)
Model	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)
Model	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)
Model	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)
Model	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)
Model	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)
Model	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

DOI: https://doi.org/10.2478/ceej-2023-0020 | Journal eISSN: 2543-6821 | Journal ISSN: 2544-9001

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Language: English

Page range: 343 - 370

Published on: Oct 31, 2023

Published by: Faculty of Economic Sciences, University of Warsaw

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

Related subjects:

Business and economics,

Political economics,

Economic theory, systems and structures,

Microeconomics,

Macroeconomics,

Economic policy

© 2023 Szymon Lis, Marcin Chlebus, published by Faculty of Economic Sciences, University of Warsaw
This work is licensed under the Creative Commons Attribution 4.0 License.

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Combining forecasts? Keep it simple

Figures & Tables

Figure 1.

Figure 2.

Figure 3.

Figure 4.

Figure 5.

Figure 6.

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

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_

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_

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

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_

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_

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

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_

Statistics of prices’ log-returns

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

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_

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

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