A simulation study on the distributions of disturbances in the GARCH model

Figures & data

Table 1. Simulation results: Student’s t-distribution

$\mathit{\alpha }$	$\mathit{\beta }$	T	${\text{Mean}}_{ll}$	${\text{Bias}}_{\mathit{\alpha }}$	${\text{SE}}_{\mathit{\alpha }}$	${\text{RMSE}}_{\mathit{\alpha }}$	${\text{Bias}}_{\mathit{\beta }}$	${\text{SE}}_{\mathit{\beta }}$	${\text{RMSE}}_{\mathit{\beta }}$	${\text{Bias}}_{\mathit{\alpha }+\mathit{\beta }}$	${\text{SE}}_{\mathit{\alpha }+\mathit{\beta }}$	${\text{RMSE}}_{\mathit{\alpha }+\mathit{\beta }}$
Panel A: Normal distribution
0.10	0.65	3,000	$-2,708$	0.0219	0.1211	0.1231	$-0.0481$	0.1998	0.2055	$-0.0262$	0.1605	0.1626
		4,000	$-3,614$	0.0207	0.1125	0.1144	$-0.0448$	0.1719	0.1776	$-0.0241$	0.1352	0.1374
		5,000	$-4,493$	0.0132	0.0882	0.0892	$-0.0451$	0.1599	0.1661	$-0.0318$	0.1258	0.1298
0.15	0.70	3,000	$-3,237$	0.0108	0.1052	0.1058	$-0.0386$	0.1339	0.1394	$-0.0278$	0.1063	0.1098
		4,000	$-4,339$	0.0101	0.0963	0.0968	$-0.0166$	0.1188	0.1199	$-0.0065$	0.0765	0.0768
		5,000	$-5,414$	0.0133	0.0884	0.0894	$-0.0345$	0.1137	0.1188	$-0.0212$	0.0791	0.0819
0.20	0.75	3,000	$-4,112$	0.0138	0.1190	0.1198	$-0.0309$	0.1114	0.1156	$-0.0171$	0.0494	0.0523
		4,000	$-5,474$	$-0.0057$	0.0732	0.0734	$-0.0125$	0.0759	0.0770	$-0.0182$	0.0450	0.0485
		5,000	$-6,843$	0.0068	0.0742	0.0745	$-0.0187$	0.0696	0.0721	$-0.0118$	0.0455	0.0470
Panel B: Student’s t-Distribution
0.10	0.65	3,000	$-2,167$	0.0022	0.0272	0.0273	$-0.0045$	0.0826	0.0828	$-0.0023$	0.0686	0.0686
		4,000	$-2,884$	0.0000	0.0228	0.0228	$-0.0029$	0.0708	0.0708	$-0.0029$	0.0600	0.0600
		5,000	$-3,607$	0.0022	0.0205	0.0206	$-0.0070$	0.0560	0.0564	$-0.0048$	0.0469	0.0471
0.15	0.70	3,000	$-2,714$	0.0032	0.0311	0.0313	$-0.0071$	0.0489	0.0494	$-0.0039$	0.0379	0.0381
		4,000	$-3,627$	$-0.0005$	0.0289	0.0289	$-0.0022$	0.0428	0.0429	$-0.0027$	0.0321	0.0322
		5,000	$-4,530$	0.0013	0.0246	0.0247	$-0.0033$	0.0364	0.0366	$-0.0020$	0.0287	0.0288
0.20	0.75	3,000	$-3,570$	0.0045	0.0380	0.0383	$-0.0051$	0.0332	0.0336	$-0.0006$	0.0281	0.0281
		4,000	$-4,760$	0.0005	0.0292	0.0292	$-0.0017$	0.0250	0.0250	$-0.0012$	0.0231	0.0231
		5,000	$-5,949$	0.0004	0.0264	0.0264	$-0.0009$	0.0226	0.0227	$-0.0005$	0.0217	0.0217
Panel C: GED distribution
0.10	0.65	3,000	$-2,216$	$-0.0150$	0.0250	0.0291	$-0.0077$	0.0957	0.0960	$-0.0227$	0.0812	0.0843
		4,000	$-2,948$	$-0.0148$	0.0210	0.0257	$-0.0079$	0.0764	0.0768	$-0.0227$	0.0656	0.0694
		5,000	$-3,687$	$-0.0139$	0.0194	0.0239	$-0.0126$	0.0641	0.0654	$-0.0265$	0.0534	0.0596
0.15	0.70	3,000	$-2,761$	$-0.0225$	0.0269	0.0350	$-0.0100$	0.0544	0.0553	$-0.0325$	0.0411	0.0524
		4,000	$-3,692$	$-0.0244$	0.0259	0.0356	$-0.0024$	0.0489	0.0489	$-0.0268$	0.0351	0.0441
		5,000	$-4,609$	$-0.0226$	0.0211	0.0309	$-0.0081$	0.0422	0.0430	$-0.0307$	0.0322	0.0445
0.20	0.75	3,000	$-3,619$	$-0.0288$	0.0332	0.0439	-0.0077	0.0360	0.0368	$-0.0364$	0.0237	0.0435
		4,000	-4,824	$-0.0329$	0.0245	0.0410	$-0.0032$	0.0296	0.0297	$-0.0361$	0.0196	0.0411
		5,000	$-6,029$	$-0.0312$	0.0229	0.0387	$-0.0029$	0.0260	0.0262	$-0.0341$	0.0173	0.0383
Panel D: Tempered stable distribution
0.10	0.65	3,000	$-2,162$	0.0099	0.0595	0.0604	$-0.0051$	0.0841	0.0843	0.0048	0.0854	0.0856
		4,000	$-2,879$	$-0.0032$	0.0349	0.0350	$-0.0032$	0.0729	0.0729	$-0.0065$	0.0660	0.0663
		5,000	$-3,602$	$-0.0025$	0.0270	0.0271	$-0.0066$	0.0576	0.0579	$-0.0091$	0.0506	0.0514
0.15	0.70	3,000	$-2709$	0.0021	0.0459	0.0460	$-0.0073$	0.0502	0.0507	$-0.0052$	0.0522	0.0524
		4,000	$-3,622$	$-0.0048$	0.0400	0.0403	$-0.0019$	0.0435	0.0435	$-0.0067$	0.0431	0.0436
		5,000	$-4,525$	$-0.0075$	0.0250	0.0261	$-0.0035$	0.0368	0.0370	$-0.0110$	0.0305	0.0324
0.20	0.75	3,000	$-3,564$	$-0.0006$	0.0421	0.0421	$-0.0054$	0.0338	0.0342	$-0.0060$	0.0332	0.0337
		4,000	$-4,755$	$-0.0075$	0.0324	0.0333	$-0.0015$	0.0257	0.0257	$-0.0090$	0.0282	0.0296
		5,000	$-5,944$	$-0.0123$	0.0249	0.0277	-0.0011	0.0228	0.0229	$-0.0134$	0.0217	0.0255

Notes: This table presents the simulation results where the true distribution is Student’s t. $\mathit{\alpha }$ and $\mathit{\beta }$ are the true GARCH parameters T is the sample size. $Mea{n}_{ll}$ is the mean of log-likelihood. Bias is the simulation bias. SE and RMSE stand for standard error and root-mean-square-error, respectively. The simulation study is based on 300 replications.

Table 2. Simulation Results: GED distribution

$\mathit{\alpha }$	$\mathit{\beta }$	T	${\text{Mean}}_{ll}$	${\text{Bias}}_{\mathit{\alpha }}$	${\text{SE}}_{\mathit{\alpha }}$	${\text{RMSE}}_{\mathit{\alpha }}$	${\text{Bias}}_{\mathit{\beta }}$	${\text{SE}}_{\mathit{\beta }}$	${\text{RMSE}}_{\mathit{\beta }}$	${\text{Bias}}_{\mathit{\alpha }+\mathit{\beta }}$	${\text{SE}}_{\mathit{\alpha }+\mathit{\beta }}$	${\text{RMSE}}_{\mathit{\alpha }+\mathit{\beta }}$
Panel A: Normal distribution
0.10	0.65	3,000	$-2,811$	0.0034	0.0295	0.0297	$-0.0206$	0.1021	0.1041	$-0.0172$	0.0875	0.0892
		4,000	$-3,763$	0.0043	0.0233	0.0237	$-0.0264$	0.0985	0.1020	$-0.0221$	0.0862	0.0890
		5,000	$-4,702$	0.0022	0.0217	0.0219	$-0.0116$	0.0834	0.0842	$-0.0094$	0.0704	0.0710
0.15	0.70	3,000	$-3,467$	0.0005	0.0295	0.0295	$-0.0081$	0.0554	0.0560	$-0.0075$	0.0405	0.0412
		4,000	$-4,631$	0.0030	0.0255	0.0257	$-0.0082$	0.0475	0.0482	$-0.0051$	0.0353	0.0357
		5,000	$-5,769$	$-0.0004$	0.0225	0.0225	$-0.0030$	0.0474	0.0475	$-0.0034$	0.0350	0.0352
0.20	0.75	3,000	$-4,585$	0.0041	0.0319	0.0322	$-0.0047$	0.0334	0.0337	$-0.0006$	0.0186	0.0186
		4,000	$-6,117$	0.0017	0.0262	0.0262	$-0.0041$	0.0268	0.0271	$-0.0024$	0.0169	0.0171
		5,000	$-7,637$	$-0.0011$	0.0231	0.0232	$-0.0010$	0.0246	0.0246	$-0.0021$	0.0139	0.0141
Panel B: Student’s t-distribution
0.10	0.65	3,000	$-2,624$	0.0218	0.0317	0.0385	$-0.0137$	0.0902	0.0913	0.0082	0.0769	0.0773
		4,000	$-3,516$	0.0224	0.0266	0.0348	$-0.0178$	0.0760	0.0780	0.0046	0.0649	0.0650
		5,000	$-4,389$	0.0218	0.0244	0.0327	$-0.0125$	0.0754	0.0764	0.0094	0.0621	0.0628
0.15	0.70	3,000	$-3,279$	0.0286	0.0342	0.0446	$-0.0069$	0.0529	0.0534	0.0217	0.0398	0.0453
		4,000	$-4,380$	0.0293	0.0302	0.0421	$-0.0046$	0.0454	0.0456	0.0247	0.0346	0.0425
		5,000	$-5,457$	0.0281	0.0262	0.0384	$-0.0052$	0.0423	0.0426	0.0229	0.0306	0.0382
0.20	0.75	3,000	$-4,398$	0.0353	0.0325	0.0479	$-0.0027$	0.0299	0.0300	0.0325	0.0179	0.0371
		4,000	$-5,867$	0.0352	0.0258	0.0437	$-0.0032$	0.0241	0.0243	0.0320	0.0173	0.0364
		5,000	$-7,325$	0.0344	0.0245	0.0423	$-0.0007$	0.0242	0.0242	0.0337	0.0159	0.0372
Panel C: GED distribution
0.10	0.65	3,000	$-2,596$	0.0027	0.0255	0.0256	$-0.0166$	0.0898	0.0913	$-0.0139$	0.0780	0.0792
		4,000	$-3,478$	0.0035	0.0205	0.0208	$-0.0191$	0.0774	0.0797	$-0.0156$	0.0674	0.0692
		5,000	$-4,342$	0.0025	0.0194	0.0196	$-0.0117$	0.0731	0.0741	$-0.0092$	0.0619	0.0626
0.15	0.70	3,000	$-3,252$	0.0003	0.0271	0.0271	$-0.0069$	0.0504	0.0509	$-0.0066$	0.0367	0.0373
		4,000	$-4,344$	0.0016	0.0229	0.0229	$-0.0054$	0.0434	0.0437	$-0.0038$	0.0322	0.0324
		5,000	$-5,410$	$-0.0001$	0.0200	0.0200	$-0.0040$	0.0412	0.0414	$-0.0041$	0.0301	0.0304
0.20	0.75	3,000	$-4,370$	0.0025	0.0282	0.0284	$-0.0034$	0.0296	0.0298	$-0.0009$	0.0167	0.0167
		4,000	$-5,830$	0.0012	0.0225	0.0225	$-0.0035$	0.0234	0.0237	$-0.0023$	0.0161	0.0163
		5,000	$-7,278$	$-0.0012$	0.0200	0.0201	$-0.0008$	0.0229	0.0229	$-0.0019$	0.0138	0.0139
Panel D: Tempered stable distribution
0.10	0.65	3,000	$-2,590$	0.0030	0.0258	0.0259	$-0.0173$	0.0878	0.0895	$-0.0143$	0.0759	0.0772
		4,000	$-3,473$	0.0032	0.0213	0.0215	$-0.0190$	0.0802	0.0824	$-0.0157$	0.0709	0.0727
		5,000	$-4,337$	0.0026	0.0202	0.0204	$-0.0124$	0.0753	0.0763	$-0.0098$	0.0641	0.0648
0.15	0.70	3,000	$-3246$	0.0007	0.0297	0.0298	$-0.0065$	0.0540	0.0544	$-0.0058$	0.0387	0.0391
		4,000	$-4,338$	0.0022	0.0236	0.0237	$-0.0077$	0.0453	0.0460	$-0.0055$	0.0340	0.0344
		5,000	$-5,405$	$-0.0004$	0.0207	0.0207	$-0.0034$	0.0414	0.0415	$-0.0037$	0.0302	0.0304
0.20	0.75	3,000	$-4,364$	0.0031	0.0295	0.0297	$-0.0037$	0.0311	0.0313	$-0.0006$	0.0182	0.0183
		4,000	$-5,824$	0.0023	0.0239	0.0240	$-0.0047$	0.0241	0.0245	$-0.0024$	0.0173	0.0175
		5,000	$-7,272$	$-0.0015$	0.0208	0.0208	$-0.0007$	0.0239	0.0239	$-0.0021$	0.0145	0.0146

Notes: This table presents the simulation results where the true distribution is GED. For explanation of variables, please see Table 1.

Table 3. Simulation results: Tempered stable distribution results

$\mathit{\alpha }$	$\mathit{\beta }$	${\mathit{\alpha }}^{+}$	${\mathit{\alpha }}^{-}$	${\mathit{\lambda }}^{+}$	${\mathit{\lambda }}^{-}$	${\text{Mean}}_{ll}$	${\text{Bias}}_{\mathit{\alpha }}$	${\text{SE}}_{\mathit{\alpha }}$	${\text{RMSE}}_{\mathit{\alpha }}$	${\text{Bias}}_{\mathit{\beta }}$	${\text{SE}}_{\mathit{\beta }}$	${\text{RMSE}}_{\mathit{\beta }}$	${\text{Bias}}_{\mathit{\alpha }+\mathit{\beta }}$	${\text{SE}}_{\mathit{\alpha }+\mathit{\beta }}$	${\text{RMSE}}_{\mathit{\alpha }+\mathit{\beta }}$
Panel A: Normal distribution
0.10	0.65	0.5	0.5	1.0	1.0	$-4,689$	$-0.0009$	0.0228	0.0228	$-0.0052$	0.0807	0.0809	$-0.0061$	0.0677	0.0680
		0.2	1.2	1.2	0.2	$-4,546$	0.0035	0.0349	0.0350	$-0.0178$	0.1138	0.1152	$-0.0143$	0.0954	0.0965
		1.2	0.2	0.2	1.2	$-4,531$	$-0.0019$	0.0312	0.0312	$-0.0108$	0.1048	0.1054	$-0.0128$	0.0885	0.0894
0.15	0.70	0.5	0.5	1.0	1.0	$-5,778$	0.0002	0.0240	0.0240	$-0.0020$	0.0439	0.0440	$-0.0018$	0.0313	0.0314
		0.2	1.2	1.2	0.2	$-5,515$	$-0.0035$	0.0351	0.0353	$-0.0100$	0.0621	0.0629	$-0.0135$	0.0435	0.0455
		1.2	0.2	0.2	1.2	$-5,531$	$-0.0030$	0.0359	0.0360	$-0.0071$	0.0696	0.0700	$-0.0102$	0.0473	0.0484
0.20	0.75	0.5	0.5	1.0	1.0	$-7,599$	0.0031	0.0261	0.0263	$-0.0051$	0.0279	0.0284	$-0.0020$	0.0173	0.0174
		0.2	1.2	1.2	0.2	$-7,051$	$-0.0019$	0.0360	0.0361	$-0.0069$	0.0386	0.0392	$-0.0088$	0.0241	0.0257
		1.2	0.2	0.2	1.2	$-7,029$	$-0.0040$	0.0349	0.0351	$-0.0066$	0.0372	0.0377	$-0.0107$	0.0262	0.0283
Panel B: Student’s t-distribution
0.10	0.65	0.5	0.5	1.0	1.0	$-4,331$	0.0141	0.0228	0.0268	0.0009	0.0660	0.0660	0.0151	0.0548	0.0569
		0.2	1.2	1.2	0.2	$-3,793$	0.0482	0.0350	0.0595	0.0014	0.0612	0.0612	0.0496	0.0538	0.0732
		1.2	0.2	0.2	1.2	$-3,782$	0.0475	0.0355	0.0593	$-0.0047$	0.0644	0.0646	0.0427	0.0547	0.0694
0.15	0.70	0.5	0.5	1.0	1.0	$-5,419$	0.0244	0.0282	0.0373	$-0.0010$	0.0394	0.0394	0.0234	0.0272	0.0359
		0.2	1.2	1.2	0.2	$-4,757$	0.0729	0.0441	0.0852	$-0.0043$	0.0376	0.0378	0.0685	0.0399	0.0793
		1.2	0.2	0.2	1.2	$-4,774$	0.0724	0.0422	0.0838	$-0.0038$	0.0404	0.0406	0.0686	0.0406	0.0798
0.20	0.75	0.5	0.5	1.0	1.0	$-7,239$	0.0330	0.0262	0.0421	$-0.0029$	0.0237	0.0239	0.0301	0.0168	0.0344
		0.2	1.2	1.2	0.2	$-6,303$	0.0468	0.0234	0.0524	0.0003	0.0226	0.0226	0.0471	0.0088	0.0479
		1.2	0.2	0.2	1.2	$-6,287$	0.0462	0.0255	0.0528	$-0.0003$	0.0236	0.0236	0.0459	0.0102	0.0470
Panel C: GED distribution
0.10	0.65	0.5	0.5	1.0	1.0	$-4,317$	$-0.0032$	0.0195	0.0198	$-0.0002$	0.0685	0.0685	$-0.0034$	0.0579	0.0580
		0.2	1.2	1.2	0.2	$-3,816$	$-0.0086$	0.0283	0.0296	$-0.0017$	0.0663	0.0663	$-0.0103$	0.0564	0.0573
		1.2	0.2	0.2	1.2	$-3,805$	$-0.0100$	0.0294	0.0311	$-0.0071$	0.0639	0.0643	$-0.0133$	0.0557	0.0573
0.15	0.7	0.5	0.5	1.0	1.0	$-5,405$	$-0.0025$	0.0234	0.0235	$-0.0012$	0.0403	0.0403	$-0.0036$	0.0273	0.0275
		0.2	1.2	1.2	0.2	$-4,781$	$-0.0145$	0.0290	0.0324	$-0.0058$	0.0398	0.0402	$-0.0190$	0.0312	0.0365
		1.2	0.2	0.2	1.2	$-4,798$	$-0.0143$	0.0303	0.0335	$-0.0040$	0.0434	0.0436	$-0.0175$	0.0354	0.0395
0.20	0.75	0.5	0.5	1.0	1.0	$-7,224$	0.0026	0.0227	0.0228	$-0.0039$	0.0241	0.0244	$-0.0053$	0.0149	0.0158
		0.2	1.2	1.2	0.2	$-6,326$	$-0.0205$	0.0268	0.0337	$-0.0015$	0.0236	0.0236	$-0.0220$	0.0209	0.0303
		1.2	0.2	0.2	1.2	$-6,310$	$-0.0213$	0.0289	0.0359	$-0.0031$	0.0236	0.0238	$-0.0232$	0.0244	0.0337
Panel D: Tempered stable distribution
0.10	0.65	0.5	0.5	1.0	1.0	$-4,305$	$-0.0013$	0.0187	0.0187	$-0.0001$	0.0662	0.0662	$-0.0014$	0.0559	0.0559
		0.2	1.2	1.2	0.2	$-3,727$	0.0067	0.0196	0.0207	$-0.0006$	0.0622	0.0622	0.0061	0.0557	0.0560
		1.2	0.2	0.2	1.2	$-3,716$	0.0072	0.0186	0.0199	$-0.0032$	0.0606	0.0607	0.0001	0.0537	0.0537
0.15	0.7	0.5	0.5	1.0	1.0	$-5,392$	0.0004	0.0216	0.0216	$-0.0007$	0.0382	0.0382	$-0.0003$	0.0266	0.0266
		0.2	1.2	1.2	0.2	$-4,691$	0.0088	0.0216	0.0233	$-0.0044$	0.0356	0.0359	0.0030	0.0288	0.0290
		1.2	0.2	0.2	1.2	$-4,710$	0.0093	0.0214	0.0233	$-0.0032$	0.0404	0.0405	0.0054	0.0307	0.0312
0.20	0.75	0.5	0.5	1.0	1.0	$-7,212$	$-0.0014$	0.0217	0.0217	$-0.0035$	0.0236	0.0239	$-0.0009$	0.0145	0.0145
		0.2	1.2	1.2	0.2	$-6,236$	0.0062	0.0218	0.0227	$-0.0014$	0.0223	0.0223	0.0048	0.0154	0.0161
		1.2	0.2	0.2	1.2	$-6,220$	0.0075	0.0209	0.0222	$-0.0019$	0.0222	0.0223	0.0043	0.0170	0.0175

Notes: This table presents the simulation results where the true distribution is tempered stable. For explanation of variables, please see Table 1.

Table 4. Empirical results: S&P 500 index

	Norm	Std-t	GED	T.Stb
$\mathit{\mu }$	0.0512 (0.0007)	0.0641 (0.0000)	0.0689 (0.0000)	0.0443 (0.0000)
$\mathit{\omega }$	0.0159 (0.0000)	0.0113 (0.0010)	0.0133 (0.0003)	0.0103 (0.0000)
$\mathit{\alpha }$	0.0866 (0.0000)	0.0848 (0.0000)	0.0863 (0.0000)	0.0879 (0.0000)
$\mathit{\beta }$	0.9013 (0.0000)	0.9091 (0.0000)	0.9049 (0.0000)	0.9064 (0.0000)
$\mathit{\nu }$		7.6493 (0.0000)	1.3833 (0.0000)
$a_1$				$-0.1377$ (0.0251)
$a_2$				$-1.2381$ (0.0178)
${\mathit{\lambda }}_1$				2.7546 (0.0000)
${\mathit{\lambda }}_2$				3.0146 (0.0000)
p				0.4954 (0.0000)
$\mathit{\alpha }+\mathit{\beta }$	0.9879	0.9939	0.9912	0.9943
log.lik	$-4,707$	$-4,663$	$-4,653$	$-4,639$
AIC	9,422	9,334	9,314	9,296

Notes: This table presents the estimates of daily SP500 data fitted by the GARCH model with Normal, Student’s t, GED and tempered stable distributions. The data are ranging from 1 January 2001 to 31 December 2013. Norm, Std-t and T .Stb stand for Normal, Student’s t and tempered stable distributions, respectively. log.lik is the log- likelihood. AIC is the Akaike Information Criterion. Values in the parentheses are the corresponding p-values.

Table 5. Forecasting results: S&P 500 index

	Mean	SD	Median	$Q_1$	$Q_3$
Panel A: In-sample forecasts
Norm	0.0732	0.2182	0.0720	0.0215	0.1647
Std-t	0.0681	0.2103	0.0711	0.0193	0.1513
GED	0.0637	0.2201	0.0693	0.0187	0.1481
T-Stb	0.0529	0.2142	0.0631	0.0131	0.1392
Panel B: Out-of-sample forecasts
Norm	0.1831	0.1321	0.1181	0.0981	0.2771
Std-t	0.1713	0.1331	0.1132	0.0870	0.2693
GED	0.1641	0.1362	0.1056	0.0873	0.2631
T-Stb	0.1362	0.1314	0.0973	0.0802	0.2045

Notes: This table presents the in- and out-of-sample forecasting performance of the GARCH model with Normal, Student’s t, GED and tempered stable distributions with the SP500 data. Norm, Std -t and T .Stb stand for Normal, Student’s t and tempered stable distributions, respectively. SD, Q1 and Q3 stand for the standard deviation, first quartile and third quartile, respectively.

Figure 1. S&P 500 daily index and return.

Figure 2. Density plots of S&P 500 return and choices of distributions.

Figure 3. Estimated daily volatility.