Employing a generalized reduced gradient algorithm method to form combinations of steel price forecasts generated separately by ARIMA-TF and ANN models

Figures & data

Figure 1. Times series observations trends.

Table 1. Unit root test

Unit root test	Level	Steel (predicted variable)			Iron ore (explanatory v)
		t	p	Break	t	p	Break
Augmented Dickey–Fuller	0	−2.06	0.3		−2.4	0.2
Augmented Dickey–Fuller	1	−5.37	0		−8.4	0
Philips–Perron	0	−1.6	0.5		−1.9	0.3
Philips–Perron	1	−6.75	0		−8.3	0
Breakpoint	0	−7.82	.01	2020/10	−3.1	0.6	2020/04
Breakpoint	1				−9.9	<.01	2021/07

Table 2. Data Set

Variable	Explanation	Source
Steel price	Cold Rolled flat steel	Industrial
Iron ore price	IODEX 62 Fe CFR China	Industrial
Coking coal price	HCC 64 Mid Vol FOB Australia	Industrial
Inflation	Consumer Price Index (CPI)	TCMB
Capacity Utilization (CU)	CU of main metal industry	TUIK
GDP	GDP of OECD	FRED
Industrial Production	Industrial Production Index (IPI)	TUIK

FRED: Federal Reserve Economic Data

TCMB: Central Bank of the Republic of Türkiye

TUIK: Turkish Statistical Institute

Table 3. ARIMA-TF (1, 1, 0) forecast of in sample (out of sample) data

ARIMA(1,1,0) Model			Coefficient	SE	t-stat	Sig.
Steel price	AR	Lag 1	0.247	0.105	2.348	0.021
Iron ore	Numerator	Lag 0	1341	0.308	4.360	0.000

Table 4. ANN forecast of in sample (out of sample) data

Explanatory Variable	Weight
Iron ore	0.511
Coal	0.018
Inflation	0.048
CU	0.260
GDP	0.153
IPI	0.010

Table 5. WLS regressions of in sample data: $Y_t={\mathit{B}}_0+\hspace{0.17em}{\mathit{B}}_1\left({\hat{Y}}_{t\left[\mathit{A}\mathit{R}\mathit{I}\mathit{M}\mathit{A}-\mathit{T}\mathit{F}\right]}\right)+\hspace{0.17em}{\mathit{B}}_2\left({\hat{Y}}_{t\left[\mathit{A}\mathit{N}\mathit{N}\right]}\right)\hspace{0.17em}+\hspace{0.17em}{\epsilon }_t$

Model		Coefficient	Std. Error	t-stat	Sig.
Unconstrained	(Constant)	−4839	12,669	− .382	0.703
	ARIMA-TF forecasts	0.365	0.054	6.785	0.000
	ANN forecasts	0.642	0.056	11.543	0.000
Constrained	ARIMA-TF forecasts	0.375	N/A	N/A	N/A
	ANN forecasts	0.625	N/A	N/A	N/A

Table 6. Residual measures of in sample forecasts

Type	Model	MAPE	RMSE	Explanatory Variables
Component Forecasts	ARIMA TF	2.88%	26.20	(1,1,0) Iron ore Lag0
	ANN	2.57%	19.82	Iron ore, Coal, CU, CPI, IPI, GDP

Table 7. Residual measures of out of sample forecasts

Type	Model	MAPE %	RMSE	Weights
				ARIMA	ANN
Component Forecasts	ARIMA TF	6.68	42.68
	ANN	10.67	68.19
Combination Forecasts	GRG unconstrained	6.32	45.38	0.383	0.616
	GRG constrained	6.13	43.66	0.429	0.571
	WLS unconstrained	6.39	46.01	0.370	0.629
	WLS constrained	6.52	46.97	0.357	0.643

Table 8. Wilcoxon signed-rank test for out of sample forecasts

	ARIMA	ANN	GRG	WLS	GRG constrained
ANN	−2.059^b
	0.039
GRG	−0.071^b	−3.574^c
	0.943	0.000
WLS	−0.071^b	−3.574^c	−1.633^b
	0.943	0.000	0.102
GRG constrained	−0.071^c	−3.479^c	−1.349^b	−1.444^b
	0.943	0.001	0.177	0.149
WLS constrained	−0.213^b	−3.574^c	−2.438^b	−3.101^c	−1.728^b
	0.831	0.000	0.015	0.002	0.084

Table 9. Diebold–Mariano (DM) and Harvey–Leybourne–Newbold (HLN) results for out of sample forecasts

	ARIMA		ANN		GRG		WLS		GRG cons		WLS cons
ARIMA			HLN	−1.58	HLN	−0.17	HLN	−0.21	HLN	−0.06	HLN	−0.26
			p	0.13	p	0.87	p	0.84	p	0.95	p	0.80
ANN	DM	−1.39			HLN	4.95	HLN	5.01	HLN	4.78	HLN	5.16
	p	0.17			p	0.00	p	0.00	p	0.00	p	0.00
GRG	DM	−0.15	DM	4.35			HLN	−2.94	HLN	2.87	HLN	−2.45
	p	0.88	p	0.00			p	0.01	p	0.01	p	0.03
WLS	DM	−0.18	DM	4.40	DM	−2.58			HLN	2.89	HLN	−2.19
	p	0.85	p	0.00	p	0.01			p	0.01	p	0.043
GRG cons	DM	−0.06	DM	4.20	DM	2.52	DM	2.54			HLN	−2.67
	p	0.96	p	0.00	p	0.01	p	0.01			p	0.02
WLS cons	DM	−0.23	DM	4.54	DM	−2.15	DM	−1.93	DM	−2.35
	p	0.82	p	0.00	p	0.03	p	0.054	p	0.02

Table 10. Hypotheses of testing summary

H0i	Null Hypothesis	DM	HLN	Wilcoxon
H01	There is no statistically significant difference between ARIMA-TF and ANN	accept	accept	reject
H02	There is no statistically significant difference between ARIMA-TF and GRG (unrestricted)	accept	accept	accept
H03	There is no statistically significant difference between ARIMA-TF and WLS (unrestricted)	accept	accept	accept
H04	There is no statistically significant difference between ARIMA-TF and GRG constrained	accept	accept	accept
H05	There is no statistically significant difference between ARIMA-TF and WLS constrained	accept	accept	accept
H06	There is no statistically significant difference between ANN and GRG (unrestricted)	reject	reject	reject
H07	There is no statistically significant difference between ANN and WLS (unrestricted)	reject	reject	reject
H08	There is no statistically significant difference between ANN and GRG constrained	reject	reject	reject
H09	There is no statistically significant difference between ANN and WLS constrained	reject	reject	reject
H010	There is no statistically significant difference between GRG (unrestricted) and WLS (unrestricted)	reject	reject	accept
H011	There is no statistically significant difference between GRG (unrestricted) and GRG constrained	reject	reject	accept
H012	There is no statistically significant difference between GRG (unrestricted) and WLS constrained	reject	reject	reject
H013	There is no statistically significant difference between WLS (unrestricted) and GRG constrained	reject	reject	accept
H014	There is no statistically significant difference between WLS (unrestricted) and WLS constrained	accept	reject	reject
H015	There is no statistically significant difference between GRG constrained and WLS constrained	reject	reject	reject

Table 11. WLS regression results for in sample data of sub segments $Y_t={\mathit{B}}_0+\hspace{0.17em}{\mathit{B}}_1\left({\hat{Y}}_{t\left[\mathit{A}\mathit{R}\mathit{I}\mathit{M}\mathit{A}-\mathit{T}\mathit{F}\right]}\right)+\hspace{0.17em}{\mathit{B}}_2\left({\hat{Y}}_{t\left[\mathit{A}\mathit{N}\mathit{N}\right]}\right)\hspace{0.17em}+\hspace{0.17em}{\epsilon }_t$

Data	Model	Variable	Coefficient	Std. Error	t-stat	Sig.
First half of data	Unconstrained	(Constant)	−34,101	13,771	−2.476	0.018
		ARIMA-TF forecasts	0.265	0.096	2.750	0.009
		ANN forecasts	0.785	0.093	8.464	0.000
	Constrained	ARIMA-TF forecasts	0.072	N/A	N/A	N/A
		ANN forecasts	0.928	N/A	N/A	N/A
Second half of data	Unconstrained	(Constant)	−3.169	55.685	−.057	0.955
		ARIMA-TF forecasts	0.704	0.098	7.170	0.000
		ANN forecasts	0.304	0.133	2.289	0.028
	Constrained	ARIMA-TF forecasts	0.434	N/A	N/A	N/A
		ANN forecasts	0.566	N/A	N/A	N/A

Table 12. Residual measures of in sample data for sub segments

Data	Model	MAPE	RMSE	Explanatory Variables
First half of the data	ARIMA-TF	2.15%	20.24	(0,1,0) Iron ore
	ANN	1.63%	13.88	Iron ore, Coal, CU, CPI, IPI, GDP
Second half of the data	ARIMA TF	3.70%	30.32	(0,1,0) Iron ore
	ANN	4.54%	36.08	Iron ore, Coal, CU, CPI, IPI, GDP

Table 13. Residual measures of out of sample data for sub-segments

Data	Model	MAPE %	RMSE	Weights
				ARIMA	ANN
First half of data	ARIMA TF	11.51%	56.61
	ANN	11.79%	55.99
	GRG unconstrained	11.19%	54.00	0.388	0.610
	GRG constrained	11.46%	54.96	0.345	0.655
	WLS unconstrained	17.13%	79.17	0.265	0.785
	WLS constrained	11.72%	55.67	0.072	0.928
Second half of data	ARIMA TF	4.57%	26.68
	ANN	5.08%	30.74
	GRG unconstrained	4.55%	27.22	0.631	0.363
	GRG constrained	4.35%	25.84	0.659	0.341
	WLS unconstrained	4.85%	30.40	0.704	0.304
	WLS constrained	4.54%	26.62	0.434	0.566

Note that for the out of sample period of the second sub segment data set, Table 13 indicates significantly lower forecast error measures for all respective models, compared to those reported for the first sub segment; with the unrestricted GRG model performing best in the first sub segment; and the constrained GRG model performing best in the second sub segment. Note that the weight of the ARIMA-TF forecast is higher than the weight of the ANN forecast in the first three out of four cases in the second sub set; whereas in the first sub segment the ANN model forecast has higher weights in all four cases.

Figure 2. Vertex line indicating lowest RMSE with forecast weights constrained to be nonnegative and sum to one.