Forecasting cryptocurrencies’ price with the financial stress index: a graph neural network prediction strategy

Figures & data

Figure 1. Framework of the model.

Note: the formulations of mathematical terms for GNN are given in the appendix.

Table 1. Parameter settings of the deep learning methods.

	GNN	Univariate LSTM	Multivariate LSTM
Activation	relu	tanh	tanh
Dropout	0	0	0
History step	10	1	10
Batch_size	50	1	16
Epochs	50	50	50
LR	0.001	0.001	0.001
Optimizer	RMSprop	adam	adam

Table 2. Comparing prediction errors of GNN using FSI with other models.

		GNN using FSI	GNN using sub-FSI	Univariate LSTM	Multivariate LSTM
+1 day	MAE	0.275	0.230	1.235	4.924
	MAPE	0.065	0.124	0.085	0.090
	RMSE	0.676	0.675	1.510	4.590
+2 days	MAE	0.378	0.296	1.786	3.881
	MAPE	0.096	0.186	0.090	0.073
	RMSE	0.817	0.783	1.990	4.820
+3 days	MAE	0.404	0.320	1.854	4.015
	MAPE	0.105	0.175	0.092	0.075
	RMSE	0.906	0.821	2.169	5.564
+4 days	MAE	1.409	0.862	1.644	5.430
	MAPE	0.155	0.297	0.095	0.099
	RMSE	3.236	2.306	2.282	6.255
+5 days	MAE	1.218	2.168	1.692	4.883
	MAPE	0.153	0.314	0.101	0.090
	RMSE	2.717	5.845	2.053	5.661
+6 days	MAE	0.601	1.046	1.731	5.129
	MAPE	0.104	0.214	0.098	0.095
	RMSE	1.506	2.949	1.904	5.710
+7 days	MAE	2.119	1.952	1.818	5.275
	MAPE	0.275	0.260	0.096	0.098
	RMSE	4.887	5.137	2.342	6.732

Note: The definitions of MAE, MAPE, and RMSE are given in Table A4 in the appendix.

Figure 2. Training loss of the forecasting steps.

Table A1. Variable definition and summary statistics.

Variable	Definition	Mean	Max	Min	Unit
BTC	Bitcoin price	12.241	67.566	2.105	1,000 USD
LTC	Litecoin price	0.073	0.377	0.003	1,000 USD
ETH	Ethereum price	0.625	4.812	0.000	1,000 USD
DASH	Dash Coin price	0.150	1.551	0.002	1,000 USD
USDT	Tether price/USD	1.001	1.078	0.914	-
FSI	Financial Stability Index	−1.886	10.266	−4.364	-
UFSI	Financial stability index of USA	−0.866	4.988	−2.008	-
AFSI	Financial stability index of other advanced economies	−1.003	4.255	−2.480	-
EFSI	Financial stability index of emerging markets	−0.016	1.232	−0.471	-

Table A2. Spillover index matrix with FSI.

	BTC	LTC	ETH	DASH	USDT	FSI	Contribution from others
BTC	98.68	0.14	0.59	0.03	0.43	0.13	1.32
LTC	45.78	53.23	0.39	0.16	0.30	0.14	46.77
ETH	31.71	5.36	62.15	0.28	0.44	0.06	37.85
XRP	33.33	6.77	4.30	55.14	0.16	0.28	44.84
USDT	0.64	1.61	0.08	0.11	94.86	2.71	5.15
FSI	3.78	0.59	0.15	0.25	2.69	92.56	7.46
Contribution to others	115.24	14.47	5.51	0.83	4.02	3.32	143.39

Table A3. Spillover index matrix with sub-FSI.

	BTC	LTC	ETH	DASH	USDT	UFSI	AFSI	EFSI	Contribution from others
BTC	45.04	42.73	6.76	1.59	0.15	3.24	0.05	0.44	1.82
LTC	33.60	20.48	21.87	15.9	0.85	5.74	0.81	0.75	46.98
ETH	50.39	28.93	10.93	8.63	0.15	0.75	0.06	0.16	37.99
DASH	7.28	6.68	14.08	63.38	0.98	4.85	2.00	0.75	44.94
USDT	0.08	0.69	1.05	0.84	93.1	1.75	1.39	1.09	6.35
UFSI	3.44	1.29	2.56	2.23	0.41	83.41	1.72	4.94	9.79
AFSI	1.97	0.62	4.12	2.63	0.58	76.05	13.74	0.30	58.33
EFSI	1.08	0.59	1.90	2.00	0.15	84.07	2.64	7.58	42.36
Contribution to others	119.39	15.24	5.90	1.49	7.06	91.62	4.62	3.24	248.56

Figure A1. Structural of prediction graph with FSI and with sub-FSI.

Note: The one-way arrow shows time series effect and two-way arrow represents bilateral relationship between two variables.

Table A4. The definitions of MAE, MAPE, and RMSE.

	Definition
MAE	$\mathit{MAE}\left(\mathit{x},\hat{x}\right)=\frac{1}{\left|\mathrm{\Omega }\right|}\sum _{\mathit{i}\in \mathrm{\Omega }}\left|x_i-{\hat{x}}_{\mathit{i}}\right|$
RMSE	$\mathit{RMSE}\left(\mathit{x},\hat{x}\right)=\sqrt{\frac{1}{\left|\mathrm{\Omega }\right|}\sum _{\mathit{i}\in \mathrm{\Omega }}{\left|x_i-{\hat{x}}_{\mathit{i}}\right|}^2}$
MAPE	$\mathit{MAPE}\left(\mathit{x},\hat{x}\right)=\frac{1}{\left|\mathrm{\Omega }\right|}\sum _{\mathit{i}\in \mathrm{\Omega }}\left|\frac{x_i-{\hat{x}}_{\mathit{i}}}{x_i}\right|$

Note: where x = x₁…, x_n denotes the ground truth, $\hat{x}={\hat{x}}_1,,{\hat{x}}_{\mathit{n}}$ represents the predicted values, and Ω denotes the indices of observed samples. In our experiments, the Ω is 12.

Table A5. Robustness check of using the matrix obtained by Granger causality and Bayesian graphic.

		Granger causality matrix		Bayesian graphic matrix
		GNN using FSI	GNN using sub-FSI	GNN using FSI	GNN using sub-FSI
+1 day	MAE	0.357	0.481	0.325	0.262
	MAPE	0.068	0.097	0.027	0.112
	RMSE	0.768	1.431	0.717	0.683
+2 days	MAE	0.845	0.913	0.458	0.307
	MAPE	0.073	0.126	0.013	0.163
	RMSE	1.940	2.431	0.987	0.741
+3 days	MAE	1.570	0.480	0.427	0.431
	MAPE	0.175	0.134	0.021	0.173
	RMSE	3.606	0.692	0.894	1.072
+4 days	MAE	1.408	0.514	0.593	0.505
	MAPE	0.147	0.125	0.038	0.201
	RMSE	3.270	1.360	1.134	0.983
+5 days	MAE	1.597	1.280	1.459	1.283
	MAPE	0.239	0.530	0.207	0.387
	RMSE	3.538	1.347	1.831	1.617
+6 days	MAE	1.626	1.389	1.557	1.540
	MAPE	0.223	0.564	0.270	0.280
	RMSE	3.533	1.490	2.170	2.299
+7 days	MAE	1.334	1.370	1.906	2.334
	MAPE	0.149	0.248	0.257	0.370
	RMSE	2.787	1.449	2.897	3.230

Figure A2: Comparison of the predicted prices of the cryptocurrencies between with FSI and sub-FSI.

Table A6. Sensitivity check of excluding the FSI and USDT.

		GNN without FSI/sub-FSI	GNN without USDT		GNN without FSI/sub-FSI and USDT
			with FSI	with sub-FSI
+1 day	MAE	0.338	0.375	0.693	1.342
	MAPE	0.064	0.097	0.160	0.123
	RMSE	0.863	0.756	0.998	1.850
+2 days	MAE	0.509	0.495	1.574	1.778
	MAPE	0.101	0.102	0.212	0.123
	RMSE	1.313	1.025	4.086	3.426
+3 days	MAE	0.625	0.485	1.230	1.999
	MAPE	0.093	0.112	0.174	0.155
	RMSE	1.678	1.001	3.207	3.936
+4 days	MAE	0.904	0.563	0.497	3.592
	MAPE	0.138	0.132	0.189	0.201
	RMSE	2.170	1.189	1.326	6.748
+5 days	MAE	1.037	0.645	2.062	3.408
	MAPE	0.124	0.115	0.281	0.190
	RMSE	2.391	1.603	5.217	6.533
+6 days	MAE	0.844	0.627	2.253	2.962
	MAPE	0.112	0.135	0.291	0.193
	RMSE	2.116	1.546	5.605	5.641
+7 days	MAE	1.040	0.653	2.199	2.480
	MAPE	0.099	0.152	0.334	0.173
	RMSE	2.378	1.539	5.518	4.675