Combining forecasts? Keep it simple

Szymon Lis; Marcin Chlebus

doi:10.2478/ceej-2023-0020

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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

Table 1.

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

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)

Table 2.

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

[i] Note: Gray fields indicate p-values greater than 5%. GARCH stands for GARCH(1,1), GARCH-t - AR-GARCH-t(1,1), GARCH-st - AR-GARCH(1,1), QML-GARCH - QML-GARCH(1,1), CaViaR - Indirect GARCH(1,1), Mean stands for simple average from GARCH and CaViaR, Highest VaR means the maximum from GARCH, GARCH-t, GARCH-st, QML-GARCH, and CaViaR, Lowest VaR stands for the minimum from individual models, CQOM stands for Conditional Quantile Optimisation Method applied for GARCH and CaViaR (described in section 2.3.4), Elastic Net stands for forecast combined using quantile regression with elastic net regularisation (described in section 2.3.6), LASSO stands for forecast combined using quantile regression with LASSO regularisation (described in section 2.3.5), QRF stands for forecast combined using Quantile Regression Forests (described in section 2.3.7), GBRM stands for forecast combined using Gradient Boosting Regression Model (described in section 2.3.8), QRNN stands for forecast combined using Quantile Regression Neural Network (described in section 2.3.9). In TL: 1) G stands for green, 2) Y stands for yellow, 3) R stands for red.

Table 3.

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

[i] Note: The same as for the pervious table.

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

Table 4.

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

[i] Note: The same as for the table 2, but here GARCH stands for GARCH(1,2), QML-GARCH stands for AR-QML-GARCH(1,1), Mean stands for simple average from GARCH-st and CaViaR.

Table 5.

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

[i] Note: The same as for the previous table.

Table 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 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

[i] Note: The same as for the table 2, but here GARCH-t stands for GARCH-t(1,1), GARCH-st stands for GARCH-st(1,1), Mean stands for simple average from GARCH-t and CaViaR.

Table 7.

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

[i] Note: The same as for the previous table.

Table 8.

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

[i] Note: The same as for the table 2, but here GARCH-t stands for GARCH-t(1,1), GARCH-st stands for GARCH-st(1,1), Mean stands for simple average from GARCH-t and CaViaR.

Table 9.

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

[i] Note: The same as for the previous table.

Table 10.

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

[i] Note: The same as for the table 2, but here GARCH-t stands for GARCH-t(1,2), GARCH-st stands for GARCH-st(1,1), Mean stands for simple average from GARCH-st and CaViaR.

Table 11.

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

[i] Note: The same as for the previous table.

Table 12.

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

References

Authors

Metrics

Articles in this issue

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

Keywords:

JEL:

C53,

G32,

Q01

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.

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

Combining forecasts? Keep it simple

Figures & Tables

Figure 1.

Figure 2.

Table 1.

Figure 3.

Figure 4.

Figure 5.

Table 2.

Table 3.

Figure 6.

Table 4.

Table 5.

Table 6.

Table 7.

Table 8.

Table 9.

Table 10.

Table 11.

Table 12.

Paradigm

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