Panel data measures of price discovery

Figures & data

Table 1. Description of experiments E1–E6.

Experiment	α1	α2	${\Omega }_{ϵ,11}$	${\Omega }_{ϵ,22}$	${\Omega }_{ϵ,12}$	PPT1	PISA1
E1	–0.5	0.5	1	1	0	0.50	0.50
E2	–0.5	0.5	0.82	0.81	0.21	0.50	0.49
E3	–0.8	0.2	1	1	0	0.20	0.05
E4	–0.8	0.2	0.82	0.81	0.21	0.20	0.13
E5	–1.2	0.8	1	1	0	0.40	0.31
E6	–0.5	0.5	0.02	0.02	0	0.50	0.50

Notes: α₁ and α₂ refer to the mean of ${\alpha }_{1,i}$ and ${\alpha }_{2,i},$ respectively, which are in turn the individual error correction parameters. ${\Omega }_{ϵ,11},$ ${\Omega }_{ϵ,12}$ and ${\Omega }_{ϵ,22}$ refer to the elements of the covariance matrix of the VECM error of the two markets. PPT₁ and PISA₁ refer to the true PPT and PISA measures for market 1.

Table 2. Results for E1 when $-{\alpha }_1={\alpha }_2$ and ${\Omega }_{ϵ,12}=0.$

		Bias $\times 100$				RMSE $\times 100$				5% Size
N	T	${\widehat{\mathit{PPT}}}_1$	${\widehat{\mathit{PISA}}}_1$	${\overline{PT}}_1$	${\overline{\mathit{ISA}}}_1$	${\widehat{\mathit{PPT}}}_1$	${\widehat{\mathit{PISA}}}_1$	${\overline{PT}}_1$	${\overline{\mathit{ISA}}}_1$	$t_{PP{T}_1}$	$t_{\mathit{PIS}{A}_1}$
30	30	–0.01	–0.05	–0.03	–0.08	2.66	5.49	2.95	4.53	6.9	8.2
30	50	–0.01	–0.02	–0.02	–0.06	2.06	4.28	2.18	3.80	5.2	7.0
30	100	–0.01	–0.02	–0.01	–0.03	1.56	3.25	1.60	3.02	6.3	7.9
30	200	–0.03	–0.05	–0.02	–0.04	1.20	2.48	1.22	2.38	6.1	7.1
50	30	0.05	0.08	0.05	0.08	2.02	4.20	2.24	3.39	5.5	6.7
50	50	–0.01	0.00	–0.02	0.00	1.62	3.42	1.70	2.99	5.9	7.9
50	100	–0.04	–0.07	–0.03	–0.05	1.19	2.49	1.22	2.32	6.4	7.0
50	200	0.00	–0.02	0.00	–0.02	0.94	1.94	0.95	1.86	5.8	6.4
100	30	0.02	0.00	0.01	–0.01	1.44	3.04	1.61	2.47	5.3	6.9
100	50	0.01	0.00	0.03	0.02	1.13	2.36	1.20	2.08	5.4	6.6
100	100	–0.01	–0.01	–0.01	0.00	0.84	1.76	0.87	1.65	5.4	6.7
100	200	0.00	0.01	0.00	0.01	0.68	1.41	0.69	1.35	6.2	6.6
200	30	–0.01	–0.03	–0.03	–0.06	1.03	2.16	1.15	1.75	5.3	6.6
200	50	0.00	–0.03	0.00	–0.02	0.81	1.71	0.85	1.49	5.3	6.7
200	100	0.02	0.04	0.02	0.03	0.61	1.26	0.62	1.16	5.4	6.4
200	200	–0.02	–0.04	–0.02	–0.04	0.46	0.96	0.47	0.92	5.2	6.3

Notes: $t_{PP{T}_1}$ and $t_{\mathit{PIS}{A}_1}$ are the t-statistics associated with ${\widehat{\mathit{PPT}}}_1$ and ${\widehat{\mathit{PISA}}}_1,$ respectively. ${\overline{PT}}_1$ and ${\overline{\mathit{ISA}}}_1$ refer to the average ${\widehat{PT}}_{i,1}$ and ${\widehat{\mathit{ISA}}}_{i,1},$ respectively. ${\Omega }_{ϵ,12}$ refers to the covariance between the VECM errors of the two markets, and α₁ and α₂ refer to the average error correction parameters of the same model.

Table 3. Results for E2 when $-{\alpha }_1={\alpha }_2$ and ${\Omega }_{ϵ,12}\ne 0.$

		Bias $\times 100$				RMSE $\times 100$				5% Size
N	T	${\widehat{\mathit{PPT}}}_1$	${\widehat{\mathit{PISA}}}_1$	${\overline{PT}}_1$	${\overline{\mathit{ISA}}}_1$	${\widehat{\mathit{PPT}}}_1$	${\widehat{\mathit{PISA}}}_1$	${\overline{PT}}_1$	${\overline{\mathit{ISA}}}_1$	$t_{PP{T}_1}$	$t_{\mathit{PIS}{A}_1}$
30	30	0.12	0.24	0.03	0.34	3.28	5.25	3.56	4.33	6.5	7.7
30	50	0.07	0.07	0.02	0.16	2.51	4.05	2.67	3.63	6.2	7.4
30	100	0.04	0.07	0.01	0.12	1.81	2.92	1.85	2.74	5.9	6.6
30	200	–0.01	–0.01	–0.02	0.04	1.41	2.23	1.43	2.18	7.6	8.0
50	30	0.11	0.18	0.00	0.33	2.47	3.98	2.72	3.29	5.2	6.4
50	50	0.08	0.11	0.01	0.19	1.95	3.11	2.03	2.76	6.5	7.6
50	100	0.00	–0.01	–0.03	0.04	1.39	2.23	1.44	2.10	5.1	5.8
50	200	0.02	0.04	0.00	0.07	1.06	1.70	1.07	1.63	5.5	6.5
100	30	0.07	0.12	–0.12	0.21	1.78	2.88	2.09	2.35	5.6	6.6
100	50	0.03	0.05	–0.04	0.15	1.38	2.23	1.46	1.99	5.7	6.8
100	100	0.02	0.01	–0.02	0.07	0.98	1.59	1.00	1.49	4.5	5.5
100	200	0.02	0.03	0.01	0.07	0.76	1.21	0.77	1.17	5.3	5.8
200	30	0.06	0.08	–0.07	0.22	1.24	2.01	1.37	1.66	4.7	5.6
200	50	0.04	0.06	–0.02	0.16	0.95	1.53	1.00	1.36	4.7	5.5
200	100	0.04	0.06	0.00	0.11	0.70	1.13	0.72	1.07	5.4	5.9
200	200	0.00	–0.01	–0.02	0.04	0.53	0.85	0.54	0.83	4.9	5.2

Note: See Table 2 for an explanation.

Table 4. Results for E3 when $-{\alpha }_1>{\alpha }_2$ and ${\Omega }_{ϵ,12}=0.$

		Bias $\times 100$				RMSE $\times 100$				5% Size
N	T	${\widehat{\mathit{PPT}}}_1$	${\widehat{\mathit{PISA}}}_1$	${\overline{PT}}_1$	${\overline{\mathit{ISA}}}_1$	${\widehat{\mathit{PPT}}}_1$	${\widehat{\mathit{PISA}}}_1$	${\overline{PT}}_1$	${\overline{\mathit{ISA}}}_1$	$t_{PP{T}_1}$	$t_{\mathit{PIS}{A}_1}$
30	30	1.16	1.15	–2.28	7.75	3.42	2.86	5.24	8.20	8.5	8.0
30	50	0.81	0.75	–0.97	4.91	2.66	2.11	3.21	5.32	8.2	7.9
30	100	0.38	0.37	–0.56	2.67	1.94	1.48	2.09	3.05	7.6	7.7
30	200	0.20	0.20	–0.36	1.58	1.47	1.07	1.54	1.91	7.0	7.4
50	30	1.37	1.18	–1.83	7.86	2.81	2.32	6.16	8.12	9.4	8.3
50	50	0.80	0.67	–1.03	4.89	2.13	1.66	2.44	5.14	8.4	7.9
50	100	0.34	0.29	–0.62	2.63	1.48	1.13	1.64	2.86	6.4	7.7
50	200	0.20	0.19	–0.37	1.58	1.19	0.88	1.26	1.81	7.4	8.2
100	30	1.35	1.05	–2.11	7.84	2.23	1.80	3.60	7.97	12.5	11.8
100	50	0.82	0.64	–1.00	4.91	1.59	1.24	1.83	5.03	9.8	9.1
100	100	0.42	0.32	–0.54	2.69	1.10	0.83	1.19	2.81	7.3	7.2
100	200	0.23	0.18	–0.35	1.58	0.85	0.62	0.91	1.69	7.4	7.9
200	30	1.34	1.00	–2.16	7.83	1.83	1.42	3.44	7.90	20.2	16.7
200	50	0.82	0.61	–1.04	4.89	1.27	0.97	1.51	4.95	14.6	12.9
200	100	0.41	0.30	–0.55	2.67	0.83	0.62	0.94	2.73	9.5	9.1
200	200	0.21	0.15	–0.37	1.57	0.61	0.44	0.69	1.63	7.5	7.6

Note: See Table 2 for an explanation.

Table 5. Results for E4 when $-{\alpha }_1>{\alpha }_2$ and ${\Omega }_{ϵ,12}\ne 0.$

		Bias $\times 100$				RMSE $\times 100$				5% Size
N	T	${\widehat{\mathit{PPT}}}_1$	${\widehat{\mathit{PISA}}}_1$	${\overline{PT}}_1$	${\overline{\mathit{ISA}}}_1$	${\widehat{\mathit{PPT}}}_1$	${\widehat{\mathit{PISA}}}_1$	${\overline{PT}}_1$	${\overline{\mathit{ISA}}}_1$	$t_{PP{T}_1}$	$t_{\mathit{PIS}{A}_1}$
30	30	1.33	1.32	–1.81	6.78	3.96	3.58	12.07	7.43	9.5	9.2
30	50	0.80	0.76	–0.98	4.17	3.04	2.59	3.38	4.81	7.6	9.6
30	100	0.39	0.39	–0.55	2.22	2.17	1.84	2.30	2.85	7.1	9.2
30	200	0.25	0.26	–0.30	1.30	1.59	1.32	1.64	1.82	6.0	7.3
50	30	1.45	1.28	–1.77	6.78	3.26	2.90	4.71	7.17	9.4	9.3
50	50	0.87	0.78	–0.95	4.23	2.44	2.09	2.70	4.61	8.2	9.0
50	100	0.39	0.35	–0.57	2.20	1.67	1.40	1.79	2.57	5.9	7.3
50	200	0.23	0.22	–0.34	1.27	1.32	1.08	1.36	1.64	7.5	8.1
100	30	1.46	1.22	–1.76	6.79	2.53	2.19	5.39	7.00	12.2	11.9
100	50	0.95	0.79	–0.81	4.25	1.85	1.57	2.69	4.45	10.0	10.0
100	100	0.48	0.39	–0.48	2.27	1.25	1.05	1.30	2.46	7.8	8.6
100	200	0.23	0.19	–0.34	1.26	0.95	0.78	1.01	1.46	7.4	8.4
200	30	1.41	1.11	–2.05	6.76	2.03	1.70	6.49	6.86	17.0	15.8
200	50	0.93	0.76	–0.90	4.25	1.45	1.22	1.53	4.35	13.9	13.9
200	100	0.46	0.36	–0.51	2.24	0.93	0.77	0.98	2.34	8.4	9.6
200	200	0.23	0.18	–0.34	1.25	0.67	0.55	0.73	1.36	6.8	8.1

Note: See Table 2 for an explanation.

Table 6. Results for E5 when ${\alpha }_1<-1$ and ${\Omega }_{ϵ,12}=0.$

		Bias $\times 100$				RMSE $\times 100$				5% Size
N	T	${\widehat{\mathit{PPT}}}_1$	${\widehat{\mathit{PISA}}}_1$	${\overline{PT}}_1$	${\overline{\mathit{ISA}}}_1$	${\widehat{\mathit{PPT}}}_1$	${\widehat{\mathit{PISA}}}_1$	${\overline{PT}}_1$	${\overline{\mathit{ISA}}}_1$	$t_{PP{T}_1}$	$t_{\mathit{PIS}{A}_1}$
30	30	0.28	0.55	–0.16	2.51	1.89	3.73	2.05	4.14	6.8	9.5
30	50	0.17	0.37	–0.07	1.53	1.42	2.76	1.46	2.99	6.0	8.9
30	100	0.00	0.04	–0.12	0.66	1.02	2.00	1.04	2.05	6.5	8.9
30	200	–0.01	0.00	–0.07	0.32	0.74	1.41	0.75	1.43	5.1	7.5
50	30	0.30	0.57	–0.15	2.51	1.46	2.90	1.56	3.57	5.8	8.8
50	50	0.15	0.30	–0.09	1.47	1.07	2.13	1.10	2.45	5.2	8.7
50	100	0.02	0.06	–0.09	0.67	0.77	1.50	0.78	1.60	4.8	8.1
50	200	–0.03	–0.05	–0.10	0.28	0.59	1.12	0.60	1.14	6.5	8.1
100	30	0.31	0.57	–0.13	2.55	1.07	2.08	1.11	3.13	6.6	8.9
100	50	0.16	0.29	–0.08	1.48	0.76	1.50	0.78	2.03	5.2	7.6
100	100	0.03	0.05	–0.09	0.67	0.54	1.08	0.55	1.24	4.5	7.6
100	200	–0.03	–0.06	–0.10	0.27	0.42	0.81	0.43	0.84	5.2	7.3
200	30	0.28	0.45	–0.18	2.45	0.76	1.48	0.78	2.76	6.3	9.1
200	50	0.14	0.26	–0.10	1.46	0.57	1.10	0.59	1.77	6.8	8.4
200	100	0.03	0.05	–0.10	0.66	0.39	0.77	0.41	1.00	5.5	7.9
200	200	–0.04	–0.07	–0.11	0.26	0.29	0.57	0.31	0.61	5.2	7.5

Note: See Table 2 for an explanation.

Table 7. Results for E6 when $-{\alpha }_1={\alpha }_2$ and ${\Omega }_{ϵ,12}=0.$

		Bias $\times 100$				RMSE $\times 100$				5% Size
N	T	${\widehat{\mathit{PPT}}}_1$	${\widehat{\mathit{PISA}}}_1$	${\overline{PT}}_1$	${\overline{\mathit{ISA}}}_1$	${\widehat{\mathit{PPT}}}_1$	${\widehat{\mathit{PISA}}}_1$	${\overline{PT}}_1$	${\overline{\mathit{ISA}}}_1$	$t_{PP{T}_1}$	$t_{\mathit{PIS}{A}_1}$
30	30	0.01	0.04	–0.02	0.05	2.68	5.65	2.95	4.54	5.5	7.8
30	50	0.03	0.09	0.03	0.08	2.08	4.35	2.18	3.78	6.3	8.0
30	100	–0.05	–0.12	–0.06	–0.12	1.57	3.31	1.62	3.09	5.4	8.1
30	200	–0.01	–0.02	0.00	–0.01	1.19	2.50	1.21	2.38	5.7	7.0
50	30	0.07	0.12	0.10	0.12	2.05	4.33	2.59	3.51	5.4	7.2
50	50	–0.03	–0.03	–0.03	–0.01	1.64	3.48	1.74	3.05	5.6	7.8
50	100	–0.03	–0.03	–0.04	–0.05	1.22	2.60	1.26	2.40	5.8	8.0
50	200	0.02	0.05	0.02	0.05	0.96	2.00	0.97	1.92	5.8	7.3
100	30	0.09	0.20	0.10	0.14	1.45	3.06	1.59	2.44	5.5	7.3
100	50	–0.02	–0.05	–0.02	–0.05	1.14	2.43	1.22	2.12	4.9	7.4
100	100	0.01	0.04	0.01	0.03	0.87	1.85	0.89	1.71	5.3	7.0
100	200	0.00	–0.01	0.00	–0.01	0.68	1.42	0.69	1.35	5.3	7.1
200	30	–0.01	–0.01	–0.01	0.01	1.02	2.19	1.14	1.75	4.8	6.6
200	50	–0.02	–0.05	–0.03	–0.04	0.79	1.69	0.83	1.46	5.0	6.0
200	100	0.02	0.03	0.01	0.02	0.61	1.28	0.62	1.18	4.9	6.9
200	200	0.01	0.02	0.01	0.02	0.47	0.99	0.48	0.95	5.2	5.9

Note: See Table 2 for an explanation.

Table 8. Cross-correlation, unit root and cointegration results.

Statistic	Futures prices	Spot prices
Cross-correlation
Raw	0.7727	0.7694
Demeaned	0.0143	0.0041
Unit root
AR	0.9981	0.9978
$t_{\widehat{\Xi }}$	–0.4491 (0.3267)	0.3202 (0.6256)
Cointegration
$t_{\widehat{\Xi }}$	–10.1289 (0.0000)

Notes: The reported cross-correlations are the average pairwise correlation coefficients. “Raw” refers to the raw data, while “Demeaned” refers to the cross-sectionally demeaned data. $t_{\widehat{\Xi }}$ is the unit root test of Demetrescu and Hanck (2012) with p-values within parentheses. “AR” refers to the pooled OLS estimator of the largest AR root. The test for cointegration is implemented by applying the $t_{\widehat{\Xi }}$ test to the difference between the futures and spot prices. Hence, in this case the null hypothesis is that the two variables not cointegrated with cointegrating vector $[1,-1]\prime .$ The data have been cross-sectionally demeaned prior to the application of the unit root and cointegration tests.

Table 9. Error heteroskedasticity and covariance results.

Commodity	Futures prices	Spot prices
	Heteroskedasticity
Corn	4.38 (0.885)	3.88 (0.919)
Cotton	22.23 (0.008)	1.28 (0.998)
Coffee	29.80 (0.000)	0.75 (1.000)
Cocoa	347.72 (0.000)	108.53 (0.000)
Lean hogs	25.65 (0.002)	3.32 (0.950)
Canola	16.26 (0.062)	6.19 (0.721)
Soybean	1.25 (0.999)	0.97 (1.000)
Soybean meal	9.14 (0.424)	734.02 (0.000)
Soybean oil	14.71 (0.099)	202.10 (0.000)
Wheat	41.80 (0.000)	26.78 (0.002)
Crude oil	76.89 (0.000)	291.42 (0.000)
Natural gas	49.52 (0.000)	11.60 (0.237)
Gasoline	54.59 (0.000)	62.85 (0.000)
Copper	50.04 (0.000)	293.56 (0.000)
Tin	2.08 (0.990)	18.99 (0.025)
Gold	149.46 (0.000)	8.09 (0.525)
Silver	100.47 (0.000)	57.02 (0.000)
Error covariance structure
Variance	0.0037	0.0041
Correlation	0.07

Notes: The numbers reported in the upper part of the table are the test values obtained by subjecting the estimated VECM residuals to White’s test for heteroskedasticity. The numbers within parentheses are the associated p-values. The null hypothesis is that the errors are homoskedastic in time. “Variance” refers to the estimated variance of the VECM residuals, and “Correlation” refers to the correlation between the residuals of the two equations.

Table 10. Error correction results.

Commodity	Futures prices	Spot prices
Commodity-specific results
Corn	–0.0026	–0.0045
Cotton	0.0005	–0.0004
Coffee	0.0086	0.0008
Cocoa	0.0015	–0.0014
Lean hogs	0.0004	–0.0005
Canola	0.0037	–0.0025
Soybean	0.0306	0.0218
Soybean meal	–0.0019	–0.0784
Soybean oil	0.0118	–0.0377
Wheat	–0.0006	–0.0022
Crude oil	0.0043	–0.0721
Natural gas	0.0003	–0.0042
Gasoline	0.0002	–0.0023
Copper	–0.0023	–0.0413
Tin	0.0037	–0.0094
Gold	0.0045	–0.0202
Silver	0.0209	–0.0192
Summary results
Average ($\widehat{\alpha }$)	0.0049	–0.0160
STD	0.0088	0.0267

Notes: The commodity-specific results are the estimates ${\widehat{\alpha }}_i$ of the individual error correction parameters, α_i, for the two markets, which are taken from the estimated VECMs. “Average” and “STD” refer to the average and standard deviation of the commodity-specific error correction parameters, respectively, where the former is identically $\widehat{\alpha }.$

Table 11. Price discovery results.

Statistic	Futures prices	Spot prices
$\widehat{\mathit{PPT}}$	0.7662	0.2338
tPPT	6.5997 (0.0000)	2.0137 (0.0440)
$\widehat{\mathit{PISA}}$	0.8448	0.1552
tPISA	15.3248 (0.0000)	1.3717 (0.1702)

Notes: t_PPT and t_PISA test the null hypothesis of no price discovery. The numbers within parentheses are the associated p-values.

Demetrescu, M., Hanck, C. (2012). Unit root testing in heteroscedastic panels using the Cauchy estimator. Journal of Business & Economic Statistics 30(2):256–264. doi:https://doi.org/10.1080/07350015.2011.638839

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