The role of uncertainty on agricultural futures markets momentum trading and volatility

Robert L. Czudaj

doi:10.1515/snde-2018-0054

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The role of uncertainty on agricultural futures markets momentum trading and volatility

Robert L. Czudaj

Veröffentlicht/Copyright: 20. Juli 2019

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Abstract

This paper sheds light on the role of different sources of uncertainty on agricultural futures markets momentum trading and volatility. Momentum trading is proxied by two technical analysis indicators – the moving average convergence divergence and the relative strength index – while we also consider two different concepts of uncertainty – the CBOE volatility index of the S&P500 and daily news about the stance of economic policy in the US. To capture different effects on the transmission mechanism of uncertainty shocks, we implement a Bayesian VAR approach, which accounts for time-variation in the coefficients and the variance covariance structure of the model’s innovations. The results point in favor of a time-dependent uncertainty effect on expectations of daily momentum traders in agricultural futures markets. The corresponding trades in these periods push futures prices upwards and downwards and result in an increased volatility. Direct effects of both uncertainty sources on the volatility of agricultural futures markets confirm this view.

Keywords: agricultural futures markets; momentum trading; time-varying Bayesian VAR; uncertainty; volatility

JEL Classification: C32; G13; Q14

Acknowledgments

Thanks for valuable comments on a previous draft of the paper are due to two anonymous reviewers.

A Appendix

A.1 Agricultural futures prices and trading indicators

Figure 12:

Agricultural futures prices and technical analysis indicators – part I.

The upper panels display the futures prices (in green) for two agricultural commodities (i.e. coffee and corn) and their trading indicators based on technical analysis for a daily sample period from January 2005 to December 2014. The middle panels show the moving average convergence divergence MACD_12,26,t following Eq. (2) (gray solid line), the signal line Signal_9,t according to Eq. (4) (red dotted line) and the MACD histogram Hist_12,26,9,t given in Eq. (5) (gray areas). The bottom panels show the relative strength index RSI_14,t given in Eq. (7) (in blue).

Figure 13:

Agricultural futures prices and technical analysis indicators – part II.

The upper panels display the futures prices (in green) for two agricultural commodities (i.e. cotton and soybean oil) and their trading indicators based on technical analysis for a daily sample period from January 2005 to December 2014. The middle panels show the moving average convergence divergence MACD_12,26,t following Eq. (2) (gray solid line), the signal line Signal_9,t according to Eq. (4) (red dotted line) and the MACD histogram Hist_12,26,9,t given in Eq. (5) (gray areas). The bottom panels show the relative strength index RSI_14,t given in Eq. (7) (in blue).

Figure 14:

Agricultural futures prices and technical analysis indicators – part III.

The upper panels display the futures prices (in green) for three agricultural commodities (i.e. soybeans, sugar and wheat) and their trading indicators based on technical analysis for a daily sample period from January 2005 to December 2014. The middle panels show the moving average convergence divergence MACD_12,26,t following Eq. (2) (gray solid line), the signal line Signal_9,t according to Eq. (4) (red dotted line) and the MACD histogram Hist_12,26,9,t given in Eq. (5) (gray areas). The bottom panels show the relative strength index RSI_14,t given in Eq. (7) (in blue).

A.2 GARCH models

Table 2:

GARCH models for coffee.

	GARCH	EGARCH	AP-G.	GJR-G.	GARCH	EGARCH	AP-G.	GJR-G.	GARCH	EGARCH	AP-G.	GJR-G.	GARCH	EGARCH	AP-G.	GJR-G.	GARCH	EGARCH	AP-G.	GJR-G.
	N(0, 1)	N(0, 1)	N(0, 1)	N(0, 1)	t_ν	t_ν	t_ν	t_ν	sN(0, 1)_κ	sN(0, 1)_κ	sN(0, 1)_κ	sN(0, 1)_κ	st_ν,κ	st_ν,κ	st_ν,κ	st_ν,κ	GED_ν	GED_ν	GED_ν	GED_ν
ω	0.0881	0.0335	0.0561	0.0975	0.0696	0.0245	0.0450	0.0746	0.0881	0.0334	0.0543	0.0975	0.0695	0.0244	0.0444	0.0745	0.0755	0.0257	0.0477	0.0804
se	0.0414	0.0028	0.0358	0.0485	0.0237	0.0017	0.0166	0.0289	0.0414	0.0028	0.0344	0.0483	0.0237	0.0017	0.0162	0.0289	0.0315	0.0018	0.0220	0.0357
p-value	0.0333	0.0000	0.1168	0.0443	0.0033	0.0000	0.0067	0.0099	0.0333	0.0000	0.1142	0.0433	0.0034	0.0000	0.0062	0.0100	0.0164	0.0000	0.0303	0.0242
α	0.0402	0.0462	0.0471	0.0638	0.0361	0.0409	0.0433	0.0549	0.0402	0.0465	0.0474	0.0639	0.0362	0.0410	0.0435	0.0550	0.0382	0.0422	0.0449	0.0577
se	0.0123	0.0137	0.0126	0.0231	0.0073	0.0121	0.0079	0.0152	0.0123	0.0136	0.0125	0.0230	0.0073	0.0121	0.0078	0.0152	0.0097	0.0125	0.0092	0.0179
p-value	0.0011	0.0008	0.0002	0.0057	0.0000	0.0007	0.0000	0.0003	0.0010	0.0006	0.0002	0.0054	0.0000	0.0007	0.0000	0.0003	0.0001	0.0007	0.0000	0.0012
β	0.9374	0.9766	0.9374	0.9333	0.9468	0.9812	0.9457	0.9442	0.9374	0.9767	0.9376	0.9332	0.9467	0.9813	0.9457	0.9441	0.9425	0.9798	0.9428	0.9403
se	0.0198	0.0013	0.0212	0.0233	0.0098	0.0000	0.0085	0.0134	0.0198	0.0013	0.0210	0.0232	0.0099	0.0000	0.0084	0.0134	0.0149	0.0002	0.0127	0.0174
p-value	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
γ		0.0909				0.0828				0.0911				0.0829				0.0866
se		0.0115				0.0082				0.0111				0.0078				0.0039
p-value		0.0000				0.0000				0.0000				0.0000				0.0000
θ			−0.4623				−0.4442				−0.4742				−0.4489				−0.4379
se			0.1978				0.1822				0.2009				0.1828				0.1813
p-value			0.0194				0.0148				0.0183				0.0140				0.0157
δ			1.2291				1.2209				1.1916				1.2040				1.2264
se			0.3674				0.3190				0.3566				0.3124				0.3198
p-value			0.0008				0.0001				0.0008				0.0001				0.0001
ϕ				−0.0443				−0.0356				−0.0443				−0.0355				−0.0377
se				0.0226				0.0162				0.0225				0.0161				0.0178
p-value				0.0500				0.0276				0.0488				0.0278				0.0342
ν					6.7555	7.0265	7.0653	6.9982					6.7667	7.0521	7.0910	7.0075	1.3862	1.4048	1.4054	1.4006
se					0.9180	0.9656	0.9686	0.9635					0.9175	0.9687	0.9723	0.9633	0.0570	0.0571	0.0570	0.0571
p-value					0.0000	0.0000	0.0000	0.0000					0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
κ									1.0131	1.0233	1.0222	1.0119	1.0115	1.0159	1.0153	1.0099
se									0.0300	0.0295	0.0293	0.0290	0.0242	0.0244	0.0243	0.0241
p-value									0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
persis tence	0.9776	0.9766	0.9770	0.9749	0.9829	0.9812	0.9803	0.9813	0.9776	0.9767	0.9772	0.9748	0.9829	0.9813	0.9803	0.9812	0.9807	0.9798	0.9788	0.9792
AIC	4.1649	4.1572	4.1582	4.1596	4.1314	4.1270	4.1282	4.1290	4.1656	4.1576	4.1587	4.1603	4.1321	4.1276	4.1289	4.1297	4.1297	4.1252	4.1263	4.1270
SBC	4.1719	4.1665	4.1698	4.1689	4.1406	4.1386	4.1421	4.1405	4.1749	4.1692	4.1726	4.1719	4.1437	4.1415	4.1451	4.1436	4.1390	4.1368	4.1402	4.1386
WLB	0.2672	0.6837	0.5942	0.5110	0.2748	0.6834	0.5926	0.5015	0.2669	0.6857	0.5988	0.5096	0.2743	0.6827	0.5938	0.4999	0.2662	0.6703	0.5823	0.4956
p-value	0.6052	0.4083	0.4408	0.4747	0.6001	0.4084	0.4414	0.4788	0.6054	0.4076	0.4390	0.4753	0.6004	0.4087	0.4409	0.4795	0.6059	0.4129	0.4454	0.4814
WLB²	2.0105	3.4313	2.6517	1.8427	3.0865	4.5415	3.5096	2.7003	2.0029	3.4063	2.7033	1.8194	3.0709	4.5182	3.5272	2.6813	2.4203	3.9782	3.0942	2.2397
p-value	0.1562	0.0640	0.1034	0.1746	0.0789	0.0331	0.0610	0.1003	0.1570	0.0649	0.1001	0.1774	0.0797	0.0335	0.0604	0.1015	0.1198	0.0461	0.0786	0.1345
SB	1.4362	3.0798	2.8698	2.2361	1.8794	3.0970	2.7551	2.1418	1.4299	3.0750	2.9038	2.2196	1.8664	3.0786	2.7611	2.1250	1.5394	2.9166	2.6376	2.0222
p-value	0.6971	0.3795	0.4121	0.5249	0.5978	0.3769	0.4309	0.5435	0.6985	0.3802	0.4067	0.5281	0.6006	0.3797	0.4299	0.5469	0.6732	0.4047	0.4509	0.5678
APGof	104.2021	84.2180	80.4948	83.1042	47.7820	46.8592	38.7765	54.8465	99.2379	81.6404	85.4590	77.5036	47.0819	41.1949	35.1806	51.5370	53.9554	45.5863	44.3453	47.0183
p-value	0.0000	0.0000	0.0001	0.0000	0.1581	0.1812	0.4800	0.0475	0.0000	0.0001	0.0000	0.0002	0.1754	0.3748	0.6448	0.0862	0.0561	0.2171	0.2564	0.1771

The table provides estimated coefficients and diagnostics for several different GARCH models for coffee futures returns. We apply four different GARCH models: GARCH refers to Eq. (10), EGARCH to Eq. (11), AP-GARCH (AP-G.) to Eq. (12) and GJR-GARCH (GJR-G.) to Eq. (13). For all models we have considered five different distributional assumptions for ε_t: the standard normal (N(0, 1)) and Student’s t with shape ν (t_ν), the skew standard normal with skewness κ (sN(0, 1)_κ), the skew t with shape ν and skewness κ (t_ν,κ), and the generalized error distribution with shape ν (GED_ν). The upper part of the table reports parameter estimates together with robust standard errors (se) following White (1982) and p-values (see Eq. (10) to (13) for the definition of the parameters). The bottom part of the table shows the Akaike information criterion (AIC), the Schwarz Bayesian information criterion (SBC), the weighted Ljung-Box test statistic (WLB) proposed by Fisher and Gallagher (2012) for the null of no serial correlation of order 1 in the standardized residuals and the corresponding p-value, the weighted Ljung-Box test statistic (WLB²) and p-value using squared standardized residuals, the sign bias test statistic (SB) and p-value for the null of no leverage effects in the standardized residuals (i.e. positive and/or negative effects to shocks) according to Engle and Ng (1993), and the adjusted version of Pearson’s χ² goodness-of-fit test statistic (APGof) provided by Palm (1996) and p-value for the null that the empirical distribution of the standardized residuals matches the chosen theoretical density. The lowest values of AIC and SBC are underlined.

Table 3:

GARCH models for corn.

	GARCH	EGARCH	AP-G.	GJR-G.	GARCH	EGARCH	AP-G.	GJR-G.	GARCH	EGARCH	AP-G.	GJR-G.	GARCH	EGARCH	AP-G.	GJR-G.	GARCH	EGARCH	AP-G.	GJR-G.
	N(0, 1)	N(0, 1)	N(0, 1)	N(0, 1)	t_ν	t_ν	t_ν	t_ν	sN(0, 1)_κ	sN(0, 1)_κ	sN(0, 1)_κ	sN(0, 1)_κ	st_ν,κ	st_ν,κ	st_ν,κ	st_ν,κ	GED_ν	GED_ν	GED_ν	GED_ν
ω	0.0656	0.0308	0.0268	0.0769	0.0402	0.0130	0.0195	0.0396	0.0660	0.0309	0.0271	0.0780	0.0400	0.0129	0.0194	0.0395	0.0587	0.0224	0.0278	0.0614
se	0.0227	0.0032	0.0068	0.0246	0.0192	0.0024	0.0109	0.0199	0.0228	0.0032	0.0071	0.0256	0.0190	0.0024	0.0109	0.0197	0.0305	0.0025	0.0218	0.0426
p-value	0.0038	0.0000	0.0001	0.0017	0.0360	0.0000	0.0741	0.0471	0.0037	0.0000	0.0001	0.0023	0.0347	0.0000	0.0759	0.0450	0.0538	0.0000	0.2013	0.1496
α	0.0277	−0.0105	0.0344	0.0244	0.0625	0.0081	0.0729	0.0635	0.0279	−0.0107	0.0347	0.0245	0.0623	0.0078	0.0728	0.0630	0.0477	−0.0019	0.0586	0.0464
se	0.0056	0.0144	0.0057	0.0103	0.0174	0.0110	0.0197	0.0168	0.0055	0.0142	0.0056	0.0102	0.0173	0.0110	0.0198	0.0167	0.0184	0.0133	0.0278	0.0170
p-value	0.0000	0.4641	0.0000	0.0179	0.0003	0.4645	0.0002	0.0002	0.0000	0.4488	0.0000	0.0167	0.0003	0.4796	0.0002	0.0002	0.0093	0.8851	0.0352	0.0064
β	0.9576	0.9816	0.9572	0.9525	0.9317	0.9891	0.9351	0.9324	0.9574	0.9815	0.9569	0.9520	0.9318	0.9891	0.9352	0.9324	0.9398	0.9834	0.9405	0.9380
se	0.0024	0.0001	0.0007	0.0082	0.0181	0.0000	0.0180	0.0204	0.0023	0.0001	0.0017	0.0089	0.0180	0.0000	0.0181	0.0201	0.0232	0.0004	0.0288	0.0328
p-value	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
γ		0.0743				0.1364				0.0744				0.1363				0.1142
se		0.0042				0.0010				0.0042				0.0010				0.0073
p-value		0.0000				0.0000				0.0000				0.0000				0.0000
θ			0.1112				−0.0692				0.1144				−0.0685				0.0013
se			0.2571				0.1005				0.2490				0.1005				0.1568
p-value			0.6654				0.4913				0.6459				0.4954				0.9931
δ			0.6484				0.8859				0.6529				0.8842				0.8162
se			0.0364				0.1744				0.0635				0.1768				0.3048
p-value			0.0000				0.0000				0.0000				0.0000				0.0074
ϕ				0.0125				−0.0032				0.0128				−0.0027				0.0055
se				0.0197				0.0218				0.0203				0.0215				0.0286
p-value				0.5244				0.8823				0.5280				0.9016				0.8472
ν					5.1599	5.3024	5.3675	5.1553					5.1836	5.3262	5.3932	5.1800	1.2506	1.2621	1.2679	1.2512
se					0.5687	0.6002	0.5873	0.5726					0.5801	0.6114	0.5997	0.5843	0.0749	0.0782	0.0733	0.0761
p-value					0.0000	0.0000	0.0000	0.0000					0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
κ									0.9900	0.9912	0.9958	0.9895	1.0365	1.0352	1.0359	1.0363
se									0.0607	0.0618	0.0621	0.0599	0.0355	0.0370	0.0374	0.0357
p-value									0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
persis tence	0.9854	0.9816	0.9848	0.9831	0.9942	0.9891	0.9890	0.9942	0.9853	0.9815	0.9847	0.9829	0.9941	0.9891	0.9890	0.9941	0.9876	0.9834	0.9840	0.9872
AIC	4.2558	4.2445	4.2405	4.2561	4.1195	4.1094	4.1075	4.1203	4.2565	4.2453	4.2413	4.2568	4.1197	4.1096	4.1077	4.1205	4.1572	4.1490	4.1468	4.1580
SBC	4.2627	4.2538	4.2521	4.2654	4.1288	4.1210	4.1214	4.1319	4.2658	4.2568	4.2552	4.2684	4.1312	4.1235	4.1239	4.1343	4.1665	4.1605	4.1607	4.1696
WLB	0.1023	0.1165	0.0680	0.1506	0.0207	0.0101	0.0003	0.0182	0.1026	0.1174	0.0687	0.1521	0.0207	0.0102	0.0003	0.0186	0.0488	0.0442	0.0178	0.0586
p-value	0.7491	0.7329	0.7942	0.6980	0.8855	0.9200	0.9869	0.8928	0.7488	0.7319	0.7932	0.6966	0.8856	0.9196	0.9871	0.8916	0.8251	0.8335	0.8939	0.8087
WLB²	0.5655	0.8783	1.4025	0.4702	0.0024	0.1300	0.2403	0.0033	0.5599	0.8749	1.3912	0.4595	0.0026	0.1296	0.2426	0.0033	0.0578	0.2537	0.4628	0.0473
p-value	0.4520	0.3487	0.2363	0.4929	0.9609	0.7185	0.6240	0.9543	0.4543	0.3496	0.2382	0.4979	0.9596	0.7188	0.6223	0.9543	0.8100	0.6145	0.4963	0.8279
SB	2.2469	2.1765	2.3825	2.2570	0.4035	0.1514	0.1618	0.3792	2.2339	2.1764	2.3737	2.2344	0.4015	0.1540	0.1613	0.3815	0.4771	0.3693	0.3778	0.5076
p-value	0.5228	0.5366	0.4969	0.5208	0.9395	0.9850	0.9835	0.9445	0.5253	0.5366	0.4985	0.5252	0.9399	0.9847	0.9836	0.9440	0.9239	0.9465	0.9448	0.9172
APGof	140.1981	129.5590	130.7975	136.8952	75.5058	87.1612	85.8591	73.7273	125.0810	113.6479	119.7455	120.2219	57.3081	65.9146	62.0719	59.5312	122.5721	134.0052	129.0191	121.6511
p-value	0.0000	0.0000	0.0000	0.0000	0.0004	0.0000	0.0000	0.0006	0.0000	0.0000	0.0000	0.0000	0.0294	0.0045	0.0108	0.0187	0.0000	0.0000	0.0000	0.0000

The table provides estimated coefficients and diagnostics for several different GARCH models for corn futures returns. We apply four different GARCH models: GARCH refers to Eq. (10), EGARCH to Eq. (11), AP-GARCH (AP-G.) to Eq. (12) and GJR-GARCH (GJR-G.) to Eq. (13). For all models we have considered five different distributional assumptions for ε_t: the standard normal (N(0, 1)) and Student’s t with shape ν (t_ν), the skew standard normal with skewness κ (sN(0, 1)_κ), the skew t with shape ν and skewness κ (t_ν,κ), and the generalized error distribution with shape ν (GED_ν). The upper part of the table reports parameter estimates together with robust standard errors (se) following White (1982) and p-values (see Eq. (10) to (13) for the definition of the parameters). The bottom part of the table shows the Akaike information criterion (AIC), the Schwarz Bayesian information criterion (SBC), the weighted Ljung-Box test statistic (WLB) proposed by Fisher and Gallagher (2012) for the null of no serial correlation of order 1 in the standardized residuals and the corresponding p-value, the weighted Ljung-Box test statistic (WLB²) and p-value using squared standardized residuals, the sign bias test statistic (SB) and p-value for the null of no leverage effects in the standardized residuals (i.e. positive and/or negative effects to shocks) according to Engle and Ng (1993), and the adjusted version of Pearson’s χ² goodness-of-fit test statistic (APGof) provided by Palm (1996) and p-value for the null that the empirical distribution of the standardized residuals matches the chosen theoretical density. The lowest values of AIC and SBC are underlined.

Table 4:

GARCH models for cotton.

	GARCH	EGARCH	AP-G.	GJR-G.	GARCH	EGARCH	AP-G.	GJR-G.	GARCH	EGARCH	AP-G.	GJR-G.	GARCH	EGARCH	AP-G.	GJR-G.	GARCH	EGARCH	AP-G.	GJR-G.
	N(0, 1)	N(0, 1)	N(0, 1)	N(0, 1)	t_ν	t_ν	t_ν	t_ν	sN(0, 1)_κ	sN(0, 1)_κ	sN(0, 1)_κ	sN(0, 1)_κ	st_ν,κ	st_ν,κ	st_ν,κ	st_ν,κ	GED_ν	GED_ν	GED_ν	GED_ν
ω	0.1001	0.0359	0.0354	0.1021	0.0742	0.0165	0.0310	0.0628	0.1052	0.0367	0.0364	0.1069	0.0742	0.0165	0.0310	0.0629	0.0960	0.0209	0.0351	0.0897
se	0.0749	0.0048	0.0199	0.0705	0.0788	0.0028	0.0193	0.1246	0.0698	0.0048	0.0188	0.0665	0.0788	0.0028	0.0193	0.1248	0.0673	0.0030	0.0151	0.0716
p-value	0.1810	0.0000	0.0752	0.1478	0.3466	0.0000	0.1075	0.6140	0.1320	0.0000	0.0530	0.1078	0.3467	0.0000	0.1075	0.6144	0.1538	0.0000	0.0197	0.2103
α	0.1004	0.0159	0.1019	0.1156	0.0681	0.0117	0.0880	0.0721	0.1018	0.0158	0.1023	0.1170	0.0681	0.0117	0.0880	0.0721	0.0821	0.0139	0.0951	0.0902
se	0.0335	0.0185	0.0189	0.0349	0.0491	0.0151	0.0299	0.0762	0.0324	0.0184	0.0187	0.0350	0.0491	0.0151	0.0299	0.0763	0.0372	0.0155	0.0192	0.0430
p-value	0.0027	0.3909	0.0000	0.0009	0.1657	0.4381	0.0032	0.3441	0.0017	0.3903	0.0000	0.0008	0.1658	0.4381	0.0032	0.3446	0.0271	0.3693	0.0000	0.0361
β	0.8814	0.9796	0.8967	0.8827	0.9150	0.9859	0.9133	0.9245	0.8788	0.9789	0.8956	0.8801	0.9150	0.9859	0.9133	0.9245	0.8946	0.9820	0.9035	0.9005
se	0.0499	0.0008	0.0243	0.0478	0.0654	0.0015	0.0330	0.1068	0.0472	0.0007	0.0234	0.0458	0.0654	0.0015	0.0330	0.1070	0.0517	0.0006	0.0227	0.0565
p-value	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
γ		0.1880				0.1441				0.1889				0.1441				0.1619
se		0.0263				0.0188				0.0262				0.0187				0.0270
p-value		0.0000				0.0000				0.0000				0.0000				0.0000
θ			−0.0852				−0.0631				−0.0846				−0.0631				−0.0783
se			0.1058				0.1073				0.1055				0.1073				0.1006
p-value			0.4206				0.5565				0.4225				0.5564				0.4361
δ			0.5471				0.6596				0.5457				0.6596				0.6149
se			0.1535				0.1278				0.1536				0.1278				0.1201
p-value			0.0004				0.0000				0.0004				0.0000				0.0000
ϕ				−0.0355				−0.0223				−0.0355				−0.0223				−0.0256
se				0.0384				0.0290				0.0389				0.0290				0.0314
p-value				0.3556				0.4423				0.3609				0.4432				0.4142
ν					4.9113	5.0338	5.1571	4.8972					4.9114	5.0333	5.1565	4.8969	1.2319	1.2565	1.2696	1.2327
se					0.5854	0.5069	0.5541	0.6791					0.5857	0.5069	0.5540	0.6793	0.0705	0.0648	0.0627	0.0702
p-value					0.0000	0.0000	0.0000	0.0000					0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
κ									0.9771	0.9836	0.9827	0.9780	0.9999	0.9993	0.9998	1.0003
se									0.0405	0.0425	0.0359	0.0400	0.0246	0.0251	0.0251	0.0247
p-value									0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
persis tence	0.9818	0.9796	0.9797	0.9806	0.9831	0.9859	0.9794	0.9855	0.9806	0.9789	0.9789	0.9795	0.9831	0.9859	0.9794	0.9855	0.9767	0.9820	0.9756	0.9778
AIC	4.0844	4.0603	4.0513	4.0839	3.9816	3.9678	3.9649	3.9818	4.0848	4.0609	4.0518	4.0843	3.9823	3.9686	3.9657	3.9826	3.9956	3.9817	3.9777	3.9958
SBC	4.0914	4.0696	4.0629	4.0932	3.9908	3.9794	3.9789	3.9934	4.0940	4.0725	4.0658	4.0959	3.9939	3.9825	3.9820	3.9965	4.0049	3.9933	3.9916	4.0074
WLB	2.0429	2.1436	2.6084	1.9456	1.7883	1.9394	2.4412	1.6822	2.0610	2.1544	2.6179	1.9631	1.7883	1.9394	2.4412	1.6824	1.9498	2.0488	2.5290	1.8519
p-value	0.1529	0.1432	0.1063	0.1631	0.1811	0.1637	0.1182	0.1946	0.1511	0.1422	0.1057	0.1612	0.1811	0.1637	0.1182	0.1946	0.1626	0.1523	0.1118	0.1736
WLB²	0.1594	0.0704	0.0984	0.1865	0.0008	0.0361	0.0002	0.0086	0.1745	0.0763	0.1059	0.2012	0.0008	0.0362	0.0002	0.0086	0.0499	0.0021	0.0372	0.0455
p-value	0.6897	0.7908	0.7538	0.6659	0.9781	0.8492	0.9886	0.9261	0.6762	0.7824	0.7449	0.6538	0.9781	0.8491	0.9886	0.9263	0.8233	0.9638	0.8471	0.8311
SB	0.5804	0.6992	0.7922	0.3606	1.5568	1.5821	0.9982	1.5439	0.5610	0.6826	0.7723	0.3404	1.5569	1.5826	0.9982	1.5435	0.8984	1.0537	0.7904	0.7252
p-value	0.9009	0.8734	0.8513	0.9483	0.6692	0.6634	0.8017	0.6722	0.9053	0.8773	0.8561	0.9523	0.6692	0.6633	0.8017	0.6723	0.8258	0.7883	0.8518	0.8673
APGof	143.0565	121.6404	110.0891	134.7510	55.5784	45.7772	45.1408	52.8417	134.1464	123.8998	115.0851	135.8648	56.4694	44.9181	45.3317	52.2371	73.8759	64.4566	66.4614	68.3707
p-value	0.0000	0.0000	0.0000	0.0000	0.0413	0.2114	0.2307	0.0686	0.0000	0.0000	0.0000	0.0000	0.0348	0.2377	0.2248	0.0763	0.0006	0.0063	0.0040	0.0025

The table provides estimated coefficients and diagnostics for several different GARCH models for cotton futures returns. We apply four different GARCH models: GARCH refers to Eq. (10), EGARCH to Eq. (11), AP-GARCH (AP-G.) to Eq. (12) and GJR-GARCH (GJR-G.) to Eq. (13). For all models we have considered five different distributional assumptions for ε_t: the standard normal (N(0, 1)) and Student’s t with shape ν (t_ν), the skew standard normal with skewness κ (sN(0, 1)_κ), the skew t with shape ν and skewness κ (t_ν,κ), and the generalized error distribution with shape ν (GED_ν). The upper part of the table reports parameter estimates together with robust standard errors (se) following White (1982) and p-values (see Eq. (10) to (13) for the definition of the parameters). The bottom part of the table shows the Akaike information criterion (AIC), the Schwarz Bayesian information criterion (SBC), the weighted Ljung-Box test statistic (WLB) proposed by Fisher and Gallagher (2012) for the null of no serial correlation of order 1 in the standardized residuals and the corresponding p-value, the weighted Ljung-Box test statistic (WLB²) and p-value using squared standardized residuals, the sign bias test statistic (SB) and p-value for the null of no leverage effects in the standardized residuals (i.e. positive and/or negative effects to shocks) according to Engle and Ng (1993), and the adjusted version of Pearson’s χ² goodness-of-fit test statistic (APGof) provided by Palm (1996) and p-value for the null that the empirical distribution of the standardized residuals matches the chosen theoretical density. The lowest values of AIC and SBC are underlined.

Table 5:

GARCH models for soybean oil.

	GARCH	EGARCH	AP-G.	GJR-G.	GARCH	EGARCH	AP-G.	GJR-G.	GARCH	EGARCH	AP-G.	GJR-G.	GARCH	EGARCH	AP-G.	GJR-G.	GARCH	EGARCH	AP-G.	GJR-G.
	N(0, 1)	N(0, 1)	N(0, 1)	N(0, 1)	t_ν	t_ν	t_ν	t_ν	sN(0, 1)_κ	sN(0, 1)_κ	sN(0, 1)_κ	sN(0, 1)_κ	st_ν,κ	st_ν,κ	st_ν,κ	st_ν,κ	GED_ν	GED_ν	GED_ν	GED_ν
ω	0.0199	0.0062	0.0219	0.0200	0.0175	0.0039	0.0177	0.0176	0.0186	0.0055	0.0204	0.0188	0.0167	0.0785	0.0170	0.0170	0.0189	0.0043	0.0202	0.0190
se	0.0089	0.0017	0.0107	0.0089	0.0055	0.0015	0.0060	0.0056	0.0080	0.0017	0.0096	0.0080	0.0052	0.0268	0.0058	0.0053	0.0073	0.0015	0.0081	0.0073
p-value	0.0253	0.0003	0.0410	0.0242	0.0016	0.0067	0.0032	0.0016	0.0198	0.0009	0.0335	0.0187	0.0013	0.0033	0.0033	0.0013	0.0095	0.0044	0.0125	0.0089
α	0.0473	0.0027	0.0431	0.0462	0.0421	0.0018	0.0420	0.0405	0.0469	0.0013	0.0436	0.0448	0.0418	−0.0006	0.0418	0.0397	0.0447	0.0021	0.0420	0.0434
se	0.0129	0.0091	0.0190	0.0144	0.0076	0.0088	0.0166	0.0091	0.0123	0.0090	0.0178	0.0135	0.0072	0.0194	0.0167	0.0087	0.0108	0.0089	0.0177	0.0120
p-value	0.0003	0.7674	0.0233	0.0014	0.0000	0.8385	0.0112	0.0000	0.0001	0.8845	0.0141	0.0009	0.0000	0.9739	0.0122	0.0000	0.0000	0.8168	0.0176	0.0003
β	0.9439	0.9931	0.9438	0.9439	0.9501	0.9937	0.9500	0.9500	0.9448	0.9938	0.9446	0.9448	0.9506	0.8954	0.9504	0.9504	0.9468	0.9933	0.9468	0.9468
se	0.0145	0.0000	0.0156	0.0144	0.0068	0.0001	0.0084	0.0069	0.0134	0.0000	0.0144	0.0134	0.0061	0.0286	0.0080	0.0063	0.0114	0.0000	0.0125	0.0114
p-value	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
γ		0.1049				0.0972				0.1046				0.2474				0.1010
se		0.0017				0.0035				0.0028				0.0456				0.0030
p-value		0.0000				0.0000				0.0000				0.0000				0.0000
θ			0.0177				0.0204				0.0283				0.0274				0.0200
se			0.0604				0.0696				0.0605				0.0690				0.0627
p-value			0.7693				0.7700				0.6397				0.6915				0.7495
δ			2.2228				2.0075				2.1861				2.0107				2.1530
se			0.6510				0.6116				0.6054				0.6192				0.6026
p-value			0.0006				0.0010				0.0003				0.0012				0.0004
ϕ				0.0022				0.0034				0.0045				0.0046				0.0028
se				0.0114				0.0107				0.0112				0.0107				0.0109
p-value				0.8483				0.7520				0.6907				0.6697				0.7959
ν					13.1015	12.9534	13.0992	13.1032					13.5441	9.4481	13.5708	13.5665	1.6955	1.6894	1.6962	1.6954
se					3.1330	3.1880	3.1044	3.1389					3.2422	1.7944	3.2391	3.2596	0.0766	0.0796	0.0753	0.0766
p-value					0.0000	0.0000	0.0000	0.0000					0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
κ									1.0624	1.0677	1.0633	1.0638	1.0455	1.0208	1.0466	1.0466
se									0.0382	0.0394	0.0372	0.0379	0.0372	0.0358	0.0371	0.0370
p-value									0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
persis tence	0.9912	0.9931	0.9909	0.9912	0.9922	0.9937	0.9922	0.9922	0.9918	0.9938	0.9914	0.9918	0.9925	0.8954	0.9924	0.9925	0.9915	0.9933	0.9916	0.9915
AIC	3.5584	3.5615	3.5598	3.5592	3.5489	3.5513	3.5505	3.5497	3.5572	3.5600	3.5587	3.5579	3.5488	3.5829	3.5503	3.5495	3.5522	3.5549	3.5537	3.5530
SBC	3.5653	3.5708	3.5714	3.5684	3.5582	3.5629	3.5644	3.5613	3.5665	3.5716	3.5726	3.5696	3.5604	3.5968	3.5666	3.5635	3.5615	3.5665	3.5676	3.5646
WLB	0.6443	0.5165	0.6442	0.6419	0.6026	0.4873	0.5997	0.5995	0.6501	0.5246	0.6480	0.6457	0.6056	0.5804	0.6021	0.6019	0.6226	0.5004	0.6221	0.6196
p-value	0.4222	0.4723	0.4222	0.4230	0.4376	0.4851	0.4387	0.4388	0.4201	0.4689	0.4208	0.4216	0.4364	0.4461	0.4378	0.4379	0.4301	0.4793	0.4303	0.4312
WLB²	0.0217	0.2766	0.0308	0.0259	0.1239	0.4429	0.1404	0.1401	0.0266	0.3018	0.0394	0.0362	0.1313	1.1462	0.1537	0.1532	0.0591	0.3523	0.0741	0.0684
p-value	0.8829	0.5989	0.8608	0.8720	0.7248	0.5057	0.7078	0.7082	0.8706	0.5827	0.8426	0.8491	0.7171	0.2844	0.6950	0.6955	0.8079	0.5528	0.7855	0.7936
SB	4.5530	4.3227	4.7416	4.6513	4.6831	4.5172	4.8311	4.8278	4.5081	4.3641	4.7782	4.7126	4.6496	6.7322	4.8489	4.8442	4.6091	4.4398	4.8006	4.7344
p-value	0.2076	0.2287	0.1917	0.1992	0.1965	0.2108	0.1846	0.1848	0.2116	0.2247	0.1888	0.1941	0.1993	0.0809	0.1832	0.1836	0.2028	0.2177	0.1870	0.1923
APGof	64.1812	67.9088	68.2911	66.3477	62.5564	62.5882	61.3775	61.6324	48.7292	55.2923	47.5185	51.5010	49.6213	57.2039	48.0601	47.6460	68.8646	58.7969	74.9498	68.0362
p-value	0.0067	0.0028	0.0026	0.0041	0.0097	0.0097	0.0126	0.0119	0.1366	0.0436	0.1644	0.0867	0.1186	0.0301	0.1515	0.1613	0.0022	0.0218	0.0005	0.0027

The table provides estimated coefficients and diagnostics for several different GARCH models for soybean oil futures returns. We apply four different GARCH models: GARCH refers to Eq. (10), EGARCH to Eq. (11), AP-GARCH (AP-G.) to Eq. (12) and GJR-GARCH (GJR-G.) to Eq. (13). For all models we have considered five different distributional assumptions for ε_t: the standard normal (N(0, 1)) and Student’s t with shape ν (t_ν), the skew standard normal with skewness κ (sN(0, 1)_κ), the skew t with shape ν and skewness κ (t_ν,κ), and the generalized error distribution with shape ν (GED_ν). The upper part of the table reports parameter estimates together with robust standard errors (se) following White (1982) and p-values (see Eq. (10) to (13) for the definition of the parameters). The bottom part of the table shows the Akaike information criterion (AIC), the Schwarz Bayesian information criterion (SBC), the weighted Ljung-Box test statistic (WLB) proposed by Fisher and Gallagher (2012) for the null of no serial correlation of order 1 in the standardized residuals and the corresponding p-value, the weighted Ljung-Box test statistic (WLB²) and p-value using squared standardized residuals, the sign bias test statistic (SB) and p-value for the null of no leverage effects in the standardized residuals (i.e. positive and/or negative effects to shocks) according to Engle and Ng (1993), and the adjusted version of Pearson’s χ² goodness-of-fit test statistic (APGof) provided by Palm (1996) and p-value for the null that the empirical distribution of the standardized residuals matches the chosen theoretical density. The lowest values of AIC and SBC are underlined.

Table 6:

GARCH models for soybeans.

	GARCH	EGARCH	AP-G.	GJR-G.	GARCH	EGARCH	AP-G.	GJR-G.	GARCH	EGARCH	AP-G.	GJR-G.	GARCH	EGARCH	AP-G.	GJR-G.	GARCH	EGARCH	AP-G.	GJR-G.
	N(0, 1)	N(0, 1)	N(0, 1)	N(0, 1)	t_ν	t_ν	t_ν	t_ν	sN(0, 1)_κ	sN(0, 1)_κ	sN(0, 1)_κ	sN(0, 1)_κ	st_ν,κ	st_ν,κ	st_ν,κ	st_ν,κ	GED_ν	GED_ν	GED_ν	GED_ν
ω	0.0637	0.0285	0.0565	0.0639	0.0292	0.0125	0.0206	0.0250	0.0686	0.0297	0.0550	0.0684	0.0327	0.0103	0.0229	0.0286	0.0448	0.0147	0.0349	0.0435
se	0.0256	0.0034	0.0247	0.0254	0.0103	0.0022	0.0080	0.0085	0.0275	0.0033	0.0241	0.0274	0.0123	0.0020	0.0097	0.0104	0.0208	0.0023	0.0187	0.0205
p-value	0.0130	0.0000	0.0222	0.0117	0.0044	0.0000	0.0106	0.0032	0.0126	0.0000	0.0223	0.0125	0.0076	0.0000	0.0183	0.0058	0.0317	0.0000	0.0615	0.0342
α	0.0506	0.0009	0.0562	0.0489	0.0422	−0.0107	0.0545	0.0493	0.0519	0.0035	0.0626	0.0531	0.0430	0.0102	0.0569	0.0509	0.0464	0.0051	0.0569	0.0499
se	0.0147	0.0170	0.0186	0.0219	0.0095	0.0118	0.0111	0.0116	0.0155	0.0165	0.0170	0.0217	0.0108	0.0109	0.0118	0.0128	0.0150	0.0126	0.0153	0.0185
p-value	0.0006	0.9586	0.0026	0.0255	0.0000	0.3633	0.0000	0.0000	0.0008	0.8330	0.0002	0.0145	0.0001	0.3486	0.0000	0.0001	0.0020	0.6890	0.0002	0.0070
β	0.9280	0.9774	0.9278	0.9281	0.9484	0.9871	0.9467	0.9502	0.9247	0.9761	0.9234	0.9246	0.9468	0.9894	0.9442	0.9485	0.9381	0.9847	0.9371	0.9384
se	0.0202	0.0006	0.0202	0.0207	0.0102	0.0001	0.0091	0.0072	0.0220	0.0004	0.0208	0.0222	0.0123	0.0000	0.0112	0.0098	0.0200	0.0015	0.0189	0.0198
p-value	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
γ		0.1218				0.1171				0.1250				0.1129				0.1162
se		0.0271				0.0090				0.0240				0.0047				0.0158
p-value		0.0000				0.0000				0.0000				0.0000				0.0000
θ			0.0094				−0.0785				−0.0204				−0.0841				−0.0353
se			0.1159				0.0915				0.1175				0.0927				0.0968
p-value			0.9353				0.3910				0.8623				0.3644				0.7154
δ			1.7177				1.3957				1.4962				1.3366				1.4940
se			0.4352				0.2560				0.3475				0.2455				0.2983
p-value			0.0001				0.0000				0.0000				0.0000				0.0000
ϕ				0.0028				−0.0142				−0.0019				−0.0152				−0.0062
se				0.0205				0.0141				0.0208				0.0145				0.0163
p-value				0.8928				0.3160				0.9267				0.2956				0.7033
ν					5.9575	5.7868	5.9905	5.9465					5.8731	5.8458	5.9257	5.8679	1.3467	1.3381	1.3454	1.3460
se					0.6991	0.6043	0.6872	0.6927					0.6772	0.6431	0.6689	0.6737	0.0666	0.0644	0.0664	0.0667
p-value					0.0000	0.0000	0.0000	0.0000					0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
κ									0.8748	0.8672	0.8703	0.8746	0.8822	0.8788	0.8793	0.8817
se									0.0297	0.0307	0.0308	0.0299	0.0246	0.0248	0.0248	0.0248
p-value									0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
persis tence	0.9786	0.9774	0.9790	0.9784	0.9905	0.9871	0.9901	0.9924	0.9766	0.9761	0.9772	0.9768	0.9899	0.9894	0.9888	0.9922	0.9845	0.9847	0.9841	0.9852
AIC	3.8360	3.8401	3.8373	3.8368	3.7655	3.7665	3.7651	3.7658	3.8236	3.8258	3.8241	3.8244	3.7579	3.7570	3.7570	3.7581	3.7820	3.7831	3.7826	3.7827
SBC	3.8430	3.8494	3.8489	3.8461	3.7748	3.7780	3.7790	3.7774	3.8329	3.8374	3.8380	3.8360	3.7694	3.7709	3.7732	3.7720	3.7912	3.7947	3.7965	3.7943
WLB	0.1388	0.0338	0.0921	0.1449	0.2109	0.0881	0.0677	0.1717	0.1318	0.0299	0.0485	0.1279	0.1994	0.0521	0.0533	0.1594	0.1755	0.0471	0.0711	0.1606
p-value	0.7095	0.8542	0.7615	0.7035	0.6461	0.7667	0.7948	0.6786	0.7166	0.8627	0.8257	0.7206	0.6552	0.8195	0.8174	0.6897	0.6752	0.8281	0.7897	0.6886
WLB²	0.5329	0.9558	0.5828	0.5415	0.9067	0.9735	0.9827	0.9657	0.4784	0.8828	0.5604	0.4747	0.8894	1.1858	0.9545	0.9411	0.6629	1.0425	0.7385	0.6620
p-value	0.4654	0.3282	0.4452	0.4618	0.3410	0.3238	0.3215	0.3258	0.4891	0.3474	0.4541	0.4908	0.3456	0.2762	0.3286	0.3320	0.4156	0.3072	0.3901	0.4159
SB	4.4285	4.9017	4.3225	4.2761	4.2765	3.1976	4.7457	5.1792	4.3698	5.0220	4.5785	4.4780	4.3835	4.9182	4.8513	5.3413	4.2943	4.7819	4.5469	4.6671
p-value	0.2188	0.1791	0.2287	0.2332	0.2331	0.3621	0.1914	0.1591	0.2242	0.1702	0.2054	0.2143	0.2229	0.1779	0.1830	0.1484	0.2314	0.1885	0.2081	0.1979
APGof	110.3977	124.0330	113.6397	110.0799	62.3723	72.6067	80.9023	71.1128	70.8903	89.4521	74.8315	70.1911	51.0890	41.5221	50.8665	48.7688	98.9555	85.7334	88.4350	97.0167
p-value	0.0000	0.0000	0.0000	0.0000	0.0101	0.0009	0.0001	0.0013	0.0013	0.0000	0.0005	0.0016	0.0930	0.3614	0.0966	0.1358	0.0000	0.0000	0.0000	0.0000

The table provides estimated coefficients and diagnostics for several different GARCH models for soybeans futures returns. We apply four different GARCH models: GARCH refers to Eq. (10), EGARCH to Eq. (11), AP-GARCH (AP-G.) to Eq. (12) and GJR-GARCH (GJR-G.) to Eq. (13). For all models we have considered five different distributional assumptions for ε_t: the standard normal (N(0, 1)) and Student’s t with shape ν (t_ν), the skew standard normal with skewness κ (sN(0, 1)_κ), the skew t with shape ν and skewness κ (t_ν,κ), and the generalized error distribution with shape ν (GED_ν). The upper part of the table reports parameter estimates together with robust standard errors (se) following White (1982) and p-values (see Eq. (10) to (13) for the definition of the parameters). The bottom part of the table shows the Akaike information criterion (AIC), the Schwarz Bayesian information criterion (SBC), the weighted Ljung-Box test statistic (WLB) proposed by Fisher and Gallagher (2012) for the null of no serial correlation of order 1 in the standardized residuals and the corresponding p-value, the weighted Ljung-Box test statistic (WLB²) and p-value using squared standardized residuals, the sign bias test statistic (SB) and p-value for the null of no leverage effects in the standardized residuals (i.e. positive and/or negative effects to shocks) according to Engle and Ng (1993), and the adjusted version of Pearson’s χ² goodness-of-fit test statistic (APGof) provided by Palm (1996) and p-value for the null that the empirical distribution of the standardized residuals matches the chosen theoretical density. The lowest values of AIC and SBC are underlined.

Table 7:

GARCH models for sugar.

	GARCH	EGARCH	AP-G.	GJR-G.	GARCH	EGARCH	AP-G.	GJR-G.	GARCH	EGARCH	AP-G.	GJR-G.	GARCH	EGARCH	AP-G.	GJR-G.	GARCH	EGARCH	AP-G.	GJR-G.
	N(0, 1)	N(0, 1)	N(0, 1)	N(0, 1)	t_ν	t_ν	t_ν	t_ν	sN(0, 1)_κ	sN(0, 1)_κ	sN(0, 1)_κ	sN(0, 1)_κ	st_ν,κ	st_ν,κ	st_ν,κ	st_ν,κ	GED_ν	GED_ν	GED_ν	GED_ν
ω	0.0258	0.0132	0.0142	0.0253	0.0176	0.0055	0.0098	0.0171	0.0216	0.0117	0.0125	0.0214	0.0171	0.0053	0.0098	0.0168	0.0214	0.0069	0.0118	0.0208
se	0.0132	0.0022	0.0057	0.0131	0.0068	0.0013	0.0041	0.0067	0.0104	0.0021	0.0053	0.0103	0.0066	0.0013	0.0041	0.0066	0.0082	0.0014	0.0045	0.0081
p-value	0.0496	0.0000	0.0128	0.0537	0.0091	0.0000	0.0159	0.0105	0.0375	0.0000	0.0183	0.0382	0.0095	0.0000	0.0171	0.0106	0.0095	0.0000	0.0085	0.0100
α	0.0489	0.0141	0.0532	0.0518	0.0387	0.0120	0.0423	0.0404	0.0480	0.0134	0.0525	0.0504	0.0383	0.0107	0.0423	0.0392	0.0433	0.0129	0.0472	0.0455
se	0.0099	0.0129	0.0099	0.0127	0.0029	0.0089	0.0033	0.0064	0.0087	0.0123	0.0113	0.0115	0.0029	0.0089	0.0034	0.0064	0.0048	0.0102	0.0054	0.0080
p-value	0.0000	0.2741	0.0000	0.0000	0.0000	0.1762	0.0000	0.0000	0.0000	0.2728	0.0000	0.0000	0.0000	0.2312	0.0000	0.0000	0.0000	0.2072	0.0000	0.0000
β	0.9465	0.9928	0.9525	0.9474	0.9577	0.9954	0.9622	0.9583	0.9484	0.9938	0.9536	0.9490	0.9582	0.9955	0.9621	0.9586	0.9524	0.9943	0.9577	0.9531
se	0.0099	0.0001	0.0065	0.0097	0.0014	0.0001	0.0013	0.0013	0.0081	0.0001	0.0067	0.0078	0.0013	0.0001	0.0014	0.0012	0.0038	0.0001	0.0030	0.0032
p-value	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
γ		0.1055				0.0861				0.1044				0.0862				0.0949
se		0.0048				0.0113				0.0045				0.0108				0.0036
p-value		0.0000				0.0000				0.0000				0.0000				0.0000
θ			−0.1248				−0.1111				−0.1114				−0.0923				−0.1160
se			0.1047				0.0920				0.1021				0.0909				0.1042
p-value			0.2334				0.2272				0.2753				0.3100				0.2658
δ			1.2375				1.3526				1.2919				1.3805				1.3084
se			0.4232				0.1002				0.4757				0.1083				0.2198
p-value			0.0035				0.0000				0.0066				0.0000				0.0000
ϕ				−0.0076				−0.0046				−0.0063				−0.0026				−0.0060
se				0.0162				0.0106				0.0153				0.0107				0.0125
p-value				0.6368				0.6636				0.6809				0.8041				0.6307
ν					6.9706	7.0754	7.1212	6.9950					6.9716	7.0489	7.1048	6.9861	1.4081	1.4162	1.4171	1.4091
se					0.9723	0.9738	0.9937	0.9783					0.9651	0.9588	0.9833	0.9693	0.0720	0.0705	0.0724	0.0728
p-value					0.0000	0.0000	0.0000	0.0000					0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
κ									1.0833	1.0745	1.0776	1.0825	1.0626	1.0579	1.0584	1.0619
se									0.0403	0.0411	0.0389	0.0412	0.0303	0.0307	0.0308	0.0307
p-value									0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
persis tence	0.9954	0.9928	0.9961	0.9953	0.9964	0.9954	0.9961	0.9964	0.9964	0.9938	0.9971	0.9962	0.9965	0.9955	0.9962	0.9965	0.9957	0.9943	0.9953	0.9957
AIC	4.1779	4.1752	4.1752	4.1784	4.1285	4.1282	4.1277	4.1292	4.1743	4.1725	4.1721	4.1749	4.1273	4.1273	4.1268	4.1280	4.1388	4.1380	4.1378	4.1394
SBC	4.1849	4.1845	4.1868	4.1877	4.1377	4.1398	4.1416	4.1408	4.1836	4.1841	4.1861	4.1865	4.1389	4.1412	4.1430	4.1420	4.1481	4.1496	4.1517	4.1510
WLB	0.0424	0.0180	0.0176	0.0402	0.0148	0.0014	0.0021	0.0138	0.0353	0.0146	0.0155	0.0337	0.0138	0.0013	0.0023	0.0133	0.0265	0.0069	0.0076	0.0247
p-value	0.8369	0.8932	0.8945	0.8411	0.9031	0.9699	0.9638	0.9065	0.8510	0.9037	0.9010	0.8543	0.9064	0.9712	0.9617	0.9083	0.8708	0.9339	0.9307	0.8751
WLB²	0.7428	2.1066	1.6424	0.5753	1.8053	3.5609	2.6871	1.5464	0.7681	2.1452	1.5610	0.6156	1.8600	3.7810	2.7966	1.7002	1.2001	2.8002	2.1012	0.9880
p-value	0.3888	0.1467	0.2000	0.4482	0.1791	0.0592	0.1012	0.2137	0.3808	0.1430	0.2115	0.4327	0.1726	0.0518	0.0945	0.1923	0.2733	0.0943	0.1472	0.3202
SB	0.1094	0.7206	0.5757	0.1320	0.4617	1.2696	1.0007	0.4738	0.0916	0.6281	0.4565	0.1010	0.4766	1.2791	0.9730	0.4801	0.2440	0.9732	0.7546	0.2653
p-value	0.9907	0.8683	0.9020	0.9877	0.9272	0.7364	0.8011	0.9246	0.9928	0.8900	0.9283	0.9917	0.9240	0.7341	0.8078	0.9233	0.9702	0.8077	0.8603	0.9664
APGof	103.3947	88.5281	89.1011	105.9733	71.9105	66.7851	58.7628	65.3526	67.4536	53.9240	60.4819	63.5698	31.4489	46.3792	31.5125	29.4751	70.4461	66.9125	67.6765	69.3637
p-value	0.0000	0.0000	0.0000	0.0000	0.0010	0.0037	0.0219	0.0051	0.0031	0.0564	0.0153	0.0077	0.7997	0.1942	0.7974	0.8652	0.0015	0.0036	0.0030	0.0020

The table provides estimated coefficients and diagnostics for several different GARCH models for sugar futures returns. We apply four different GARCH models: GARCH refers to Eq. (10), EGARCH to Eq. (11), AP-GARCH (AP-G.) to Eq. (12) and GJR-GARCH (GJR-G.) to Eq. (13). For all models we have considered five different distributional assumptions for ε_t: the standard normal (N(0, 1)) and Student’s t with shape ν (t_ν), the skew standard normal with skewness κ (sN(0, 1)_κ), the skew t with shape ν and skewness κ (t_ν,κ), and the generalized error distribution with shape ν (GED_ν). The upper part of the table reports parameter estimates together with robust standard errors (se) following White (1982) and p-values (see Eq. (10) to (13) for the definition of the parameters). The bottom part of the table shows the Akaike information criterion (AIC), the Schwarz Bayesian information criterion (SBC), the weighted Ljung-Box test statistic (WLB) proposed by Fisher and Gallagher (2012) for the null of no serial correlation of order 1 in the standardized residuals and the corresponding p-value, the weighted Ljung-Box test statistic (WLB²) and p-value using squared standardized residuals, the sign bias test statistic (SB) and p-value for the null of no leverage effects in the standardized residuals (i.e. positive and/or negative effects to shocks) according to Engle and Ng (1993), and the adjusted version of Pearson’s χ² goodness-of-fit test statistic (APGof) provided by Palm (1996) and p-value for the null that the empirical distribution of the standardized residuals matches the chosen theoretical density. The lowest values of AIC and SBC are underlined.

Table 8:

GARCH models for wheat.

	GARCH	EGARCH	AP-G.	GJR-G.	GARCH	EGARCH	AP-G.	GJR-G.	GARCH	EGARCH	AP-G.	GJR-G.	GARCH	EGARCH	AP-G.	GJR-G.	GARCH	EGARCH	AP-G.	GJR-G.
	N(0, 1)	N(0, 1)	N(0, 1)	N(0, 1)	t_ν	t_ν	t_ν	t_ν	sN(0, 1)_κ	sN(0, 1)_κ	sN(0, 1)_κ	sN(0, 1)_κ	st_ν,κ	st_ν,κ	st_ν,κ	st_ν,κ	GED_ν	GED_ν	GED_ν	GED_ν
ω	0.0350	0.0184	0.0240	0.0356	0.0238	0.0129	0.0193	0.0260	0.0332	0.0181	0.0230	0.0345	0.0229	0.0122	0.0182	0.0250	0.0296	0.0149	0.0218	0.0313
se	0.0162	0.0019	0.0072	0.0140	0.0115	0.0016	0.0059	0.0110	0.0146	0.0018	0.0069	0.0133	0.0107	0.0016	0.0055	0.0103	0.0129	0.0016	0.0062	0.0118
p-value	0.0304	0.0000	0.0009	0.0111	0.0378	0.0000	0.0011	0.0175	0.0234	0.0000	0.0009	0.0098	0.0319	0.0000	0.0010	0.0153	0.0219	0.0000	0.0004	0.0079
α	0.0453	0.0338	0.0481	0.0558	0.0460	0.0367	0.0539	0.0594	0.0442	0.0333	0.0492	0.0548	0.0453	0.0354	0.0530	0.0581	0.0452	0.0354	0.0503	0.0571
se	0.0097	0.0125	0.0097	0.0119	0.0066	0.0107	0.0069	0.0097	0.0085	0.0120	0.0086	0.0109	0.0059	0.0105	0.0069	0.0088	0.0077	0.0115	0.0081	0.0103
p-value	0.0000	0.0070	0.0000	0.0000	0.0000	0.0006	0.0000	0.0000	0.0000	0.0055	0.0000	0.0000	0.0000	0.0008	0.0000	0.0000	0.0000	0.0020	0.0000	0.0000
β	0.9481	0.9882	0.9511	0.9512	0.9501	0.9902	0.9486	0.9514	0.9493	0.9883	0.9505	0.9518	0.9509	0.9907	0.9498	0.9523	0.9494	0.9890	0.9503	0.9516
se	0.0101	0.0000	0.0085	0.0066	0.0060	0.0000	0.0065	0.0048	0.0087	0.0000	0.0082	0.0058	0.0051	0.0000	0.0064	0.0039	0.0075	0.0000	0.0074	0.0051
p-value	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
γ		0.0969				0.1064				0.0985				0.1051				0.1005
se		0.0032				0.0037				0.0033				0.0034				0.0033
p-value		0.0000				0.0000				0.0000				0.0000				0.0000
θ			−0.3721				−0.3994				−0.3824				−0.3928				−0.3925
se			0.1612				0.1260				0.1570				0.1279				0.1461
p-value			0.0209				0.0015				0.0149				0.0021				0.0072
δ			1.1022				0.9438				1.0196				0.9395				1.0262
se			0.4524				0.2709				0.3858				0.2899				0.3820
p-value			0.0148				0.0005				0.0082				0.0012				0.0072
ϕ				−0.0299				−0.0326				−0.0290				−0.0316				−0.0313
se				0.0150				0.0132				0.0143				0.0127				0.0138
p-value				0.0465				0.0133				0.0429				0.0129				0.0237
ν					10.0756	10.1465	10.1947	10.1246					10.2514	10.4111	10.4928	10.2963	1.6041	1.6091	1.6122	1.6098
se					2.0462	2.0401	2.0381	2.0157					2.0981	2.1170	2.1215	2.0736	0.0806	0.0818	0.0798	0.0790
p-value					0.0000	0.0000	0.0000	0.0000					0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
κ									1.1084	1.1150	1.1178	1.1070	1.1127	1.1132	1.1161	1.1118
se									0.0413	0.0420	0.0408	0.0409	0.0359	0.0361	0.0364	0.0357
p-value									0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
persis tence	0.9933	0.9882	0.9902	0.9920	0.9961	0.9902	0.9900	0.9945	0.9935	0.9883	0.9899	0.9917	0.9962	0.9907	0.9905	0.9941	0.9946	0.9890	0.9895	0.9930
AIC	4.2904	4.2849	4.2838	4.2876	4.2686	4.2622	4.2614	4.2663	4.2852	4.2790	4.2777	4.2825	4.2642	4.2579	4.2569	4.2620	4.2769	4.2715	4.2707	4.2745
SBC	4.2974	4.2942	4.2954	4.2968	4.2778	4.2738	4.2753	4.2779	4.2944	4.2906	4.2916	4.2941	4.2758	4.2718	4.2731	4.2759	4.2861	4.2831	4.2846	4.2861
WLB	0.0801	0.6149	0.5660	0.2453	0.0418	0.5184	0.5650	0.1891	0.0765	0.5951	0.5933	0.2379	0.0401	0.4919	0.5388	0.1821	0.0611	0.5792	0.5774	0.2228
p-value	0.7771	0.4330	0.4519	0.6204	0.8380	0.4715	0.4522	0.6637	0.7821	0.4405	0.4412	0.6257	0.8412	0.4831	0.4629	0.6696	0.8047	0.4466	0.4473	0.6369
WLB²	0.2802	0.2409	0.2593	0.3126	0.2445	0.0926	0.1298	0.2102	0.3251	0.2071	0.2467	0.3347	0.2756	0.1103	0.1563	0.2419	0.2781	0.1696	0.2024	0.2764
p-value	0.5965	0.6236	0.6106	0.5761	0.6210	0.7609	0.7186	0.6466	0.5685	0.6490	0.6194	0.5629	0.5996	0.7398	0.6926	0.6228	0.5979	0.6804	0.6528	0.5991
SB	9.7989	6.0838	6.1214	7.9721	9.6788	5.6860	5.3348	7.5649	9.8660	6.0570	5.8884	8.0423	9.7150	5.8021	5.4557	7.6495	9.7603	5.8332	5.6887	7.7738
p-value	0.0204	0.1076	0.1059	0.0466	0.0215	0.1279	0.1489	0.0559	0.0197	0.1089	0.1172	0.0451	0.0212	0.1216	0.1413	0.0538	0.0207	0.1200	0.1278	0.0509
APGof	59.5803	64.2544	58.7854	56.2734	55.2560	61.3609	59.4213	58.1494	35.7647	38.4992	44.8903	42.1876	25.8442	44.2862	45.1765	31.2814	64.9857	63.8092	61.9968	67.0207
p-value	0.0185	0.0066	0.0218	0.0361	0.0439	0.0126	0.0191	0.0249	0.6182	0.4925	0.2386	0.3348	0.9477	0.2584	0.2296	0.8058	0.0056	0.0073	0.0110	0.0035

The table provides estimated coefficients and diagnostics for several different GARCH models for wheat futures returns. We apply four different GARCH models: GARCH refers to Eq. (10), EGARCH to Eq. (11), AP-GARCH (AP-G.) to Eq. (12) and GJR-GARCH (GJR-G.) to Eq. (13). For all models we have considered five different distributional assumptions for ε_t: the standard normal (N(0, 1)) and Student’s t with shape ν (t_ν), the skew standard normal with skewness κ (sN(0, 1)_κ), the skew t with shape ν and skewness κ (t_ν,κ), and the generalized error distribution with shape ν (GED_ν). The upper part of the table reports parameter estimates together with robust standard errors (se) following White (1982) and p-values (see Eq. (10) to (13) for the definition of the parameters). The bottom part of the table shows the Akaike information criterion (AIC), the Schwarz Bayesian information criterion (SBC), the weighted Ljung-Box test statistic (WLB) proposed by Fisher and Gallagher (2012) for the null of no serial correlation of order 1 in the standardized residuals and the corresponding p-value, the weighted Ljung-Box test statistic (WLB²) and p-value using squared standardized residuals, the sign bias test statistic (SB) and p-value for the null of no leverage effects in the standardized residuals (i.e. positive and/or negative effects to shocks) according to Engle and Ng (1993), and the adjusted version of Pearson’s χ² goodness-of-fit test statistic (APGof) provided by Palm (1996) and p-value for the null that the empirical distribution of the standardized residuals matches the chosen theoretical density. The lowest values of AIC and SBC are underlined.

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Schlagwörter für diesen Artikel

agricultural futures markets; momentum trading; time-varying Bayesian VAR; uncertainty; volatility

The role of uncertainty on agricultural futures markets momentum trading and volatility

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Abstract

Acknowledgments

A Appendix

A.1 Agricultural futures prices and trading indicators

A.2 GARCH models

References

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