RDP 1999-04: Value at Risk: On the Stability and Forecasting of the Variance-Covariance Matrix Appendix C: Forecast Performance

James Engel and Marianne Gizycki

May 1999

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An asterisk indicates best performance. The numbers presented are multiplied by 1,000 except for the MAPE and R squared criteria.

Table C1: Foreign Exchange
Daily
125 day window
	Hist	Exp	Dexp	BEKK	GARCHCC
ME	0.004^*	0.004	0.009	−0.005	0.005
MAE	0.049	0.047^*	0.054	0.049	0.049
RMSE	0.098	0.094^*	0.103	0.377	0.108
MAPE	482.510	640.474	431.204	508.773	452.449^*
MMEU	1.939^*	1.975	1.994	2.683	1.968
MMEO	4.122	3.926	4.421	2.778^*	4.069
R²	0.035	0.088^*	0.004	0.001	0.037
250 day window
ME	0.004	0.004^*	0.008	−0.004	0.004
MAE	0.048	0.047	0.053	0.046^*	0.048
RMSE	0.097	0.094^*	0.100	0.111	0.094
MAPE	448.560	640.634	584.847	227.159^*	429.216
MMEU	1.934	1.974	1.917^*	2.415	1.957
MMEO	4.133	3.928	4.427	3.196^*	4.029
R²	0.038	0.088^*	0.012	0.027	0.084
500 day window
ME	0.004	0.004	0.018	−0.003^*	0.003
MAE	0.048	0.047	0.058	0.045^*	0.047
RMSE	0.097	0.094^*	0.101	0.107	0.094
MAPE	530.773	640.634	639.489	461.536^*	521.527
MMEU	1.915	1.974	1.666^*	2.269	1.973
MMEO	4.163	3.928	5.256	3.346^*	3.971
R²	0.039	0.088^*	0.014	0.039	0.083
750 day window
ME	0.005	0.004	0.027	−0.001^*	0.003
MAE	0.049	0.047	0.063	0.046^*	0.047
RMSE	0.097	0.094^*	0.101	0.095	0.094
MAPE	541.055	640.634	1151.385	496.691^*	597.006
MMEU	1.914	1.974	1.453^*	2.197	1.965
MMEO	4.233	3.928	5.915	3.513^*	3.981
R²	0.041	0.088	0.027	0.082	0.095^*
1,250 day window
ME	0.006	0.004	−0.004	0.000^*	0.003
MAE	0.050	0.047	0.045^*	0.046	0.047
RMSE	0.097	0.094^*	0.097	0.095	0.094
MAPE	570.695	640.634	717.589	490.637^*	578.418
MMEU	1.877^*	1.974	2.156	2.133	1.954
MMEO	4.372	3.928	3.655	3.634^*	4.014
R²	0.033	0.088	0.023	0.081	0.092^*

Table C2: Foreign Exchange
Quarter-average
125 day window
	Hist	Exp	Dexp	BEKK	GARCHCC
ME	0.000^*	−0.001	0.004	−0.010	0.002
MAE	0.019^*	0.021	0.027	0.043	0.022
RMSE	0.030^*	0.033	0.042	0.441	0.125
MAPE	1.370	1.693	5.663	1.209^*	1.312
MMEU	1.802^*	2.099	2.041	4.368	1.847
MMEO	2.039	1.930	2.554	0.984^*	2.099
R²	0.335^*	0.327	0.104	0.002	0.025
250 day window
ME	0.000^*	−0.001	0.004	0.018	0.001
MAE	0.017^*	0.021	0.024	0.066	0.019
RMSE	0.027^*	0.033	0.035	3.388	0.062
MAPE	2.671	1.693^*	1.873	2.188	2.072
MMEU	1.698^*	2.097	1.809	3.960	1.714
MMEO	1.960	1.932	2.539	1.054^*	2.008
R²	0.389^*	0.327	0.176	0.000	0.096
500 day window
ME	0.000^*	−0.001	0.014	−0.015	0.000
MAE	0.017^*	0.021	0.026	0.029	0.017
RMSE	0.025^*	0.033	0.035	0.106	0.027
MAPE	2.887	1.693	4.096	0.997^*	2.688
MMEU	1.637	2.097	1.147^*	3.713	1.663
MMEO	1.966	1.932	3.498	0.952^*	1.960
R²	0.442^*	0.327	0.237	0.023	0.410
750 day window
ME	0.000^*	−0.001	0.023	−0.011	0.000
MAE	0.017^*	0.021	0.030	0.028	0.017
RMSE	0.025^*	0.033	0.036	0.108	0.027
MAPE	3.360	1.693	7.563	1.379^*	3.344
MMEU	1.643	2.097	0.637^*	3.392	1.685
MMEO	2.044	1.932	4.542	1.133^*	1.970
R²	0.448^*	0.327	0.357	0.026	0.401
1,250 day window
ME	0.002	−0.001	−0.009	0.668	0.001^*
MAE	0.018^*	0.021	0.019	0.703	0.018
RMSE	0.026^*	0.033	0.030	96.804	0.029
MAPE	3.473	1.693^*	3.527	21.811	3.319
MMEU	1.522^*	2.097	2.499	3.766	1.614
MMEO	2.221	1.932	1.390^*	1.523	2.147
R²	0.401^*	0.327	0.284	0.000	0.309

Table C3: Interest Rates
Daily
125 day window
	Hist	Exp	Dexp	BEKK	GARCHCC
ME	0.000^*	−0.001	0.168	−0.006	−0.017
MAE	0.068	0.067^*	0.253	0.093	0.089
RMSE	0.283^*	0.284	2.161	0.570	0.415
MAPE	90.571	1388.708	464.170	469.233	5.818^*
MMEU	1.824^*	1.905	2.484	3.055	2.361
MMEO	3.932	3.654	7.205	3.274^*	4.498
R²	0.041^*	0.039	0.017	0.004	0.036
250 day window
ME	0.001^*	−0.001	0.122	−0.001	−0.012
MAE	0.069	0.067^*	0.204	0.102	0.087
RMSE	0.287	0.284^*	1.367	0.600	0.399
MAPE	90.406	1388.757	436.538	113.641	5.566^*
MMEU	1.779^*	1.904	2.346	3.180	2.254
MMEO	4.078	3.656	7.054	3.419^*	4.693
R²	0.029	0.039^*	0.013	0.000	0.020
500 day window
ME	0.008	−0.001^*	0.106	0.007	0.001
MAE	0.076	0.067^*	0.187	0.111	0.093
RMSE	0.296	0.284^*	0.951	0.553	0.361
MAPE	243.407	1,388.757	398.508	1,147.206	6.641^*
MMEU	1.746^*	1.904	2.277	3.225	2.201
MMEO	4.488	3.656^*	7.180	3.722	5.215
R²	0.015	0.039	0.028	0.001	0.072^*
750 day window
ME	0.013	−0.001^*	0.061	0.020	0.013
MAE	0.080	0.067^*	0.141	0.124	0.104
RMSE	0.301	0.284^*	0.429	0.785	0.367
MAPE	231.027	1,388.757	326.254	1,778.201	9.810^*
MMEU	1.715^*	1.904	2.048	3.256	2.228
MMEO	4.756	3.656^*	7.401	3.824	5.750
R²	0.013	0.039^*	0.012	0.001	0.030
1,250 day window
ME	0.032	−0.001^*	0.056	0.096	0.041
MAE	0.097	0.067^*	0.138	0.200	0.119
RMSE	0.424	0.284^*	0.428	2.140	0.469
MAPE	346.281^*	1,388.757	3019.276	2,321.331	806.829
MMEU	1.684^*	1.904	2.156	3.422	2.081
MMEO	5.177	3.656^*	6.957	4.248	6.279
R²	0.020	0.039^*	0.010	0.002	0.028

Table C4: Interest Rates
Quarter-average
125 day window
	Hist	Exp	Dexp	BEKK	GARCHCC
ME	0.000^*	0.000	0.169	0.028	0.175
MAE	0.033^*	0.037	0.219	0.092	0.225
RMSE	0.081^*	0.082	2.166	0.685	2.348
MAPE	4.626	6.410	17.833	2.361^*	39.055
MMEU	2.114^*	2.374	2.971	4.113	3.060
MMEO	2.225	2.161^*	5.573	2.433	5.959
R²	0.191	0.209^*	0.036	0.017	0.000
250 day window
ME	0.002	0.000^*	0.123	0.036	0.040
MAE	0.034^*	0.037	0.170	0.103	0.068
RMSE	0.090	0.082^*	1.351	0.756	1.083
MAPE	5.597	6.412	14.340	3.256^*	12.369
MMEU	2.002	2.373	2.766	4.217	1.991^*
MMEO	2.343	2.162^*	5.352	2.559	3.417
R²	0.167	0.209^*	0.049	0.017	0.006
500 day window
ME	0.009	0.000^*	0.107	0.053	0.062
MAE	0.041	0.037^*	0.154	0.123	0.098
RMSE	0.110	0.082^*	0.935	0.884	0.880
MAPE	7.082	6.412	15.057	3.701^*	11.379
MMEU	2.010	2.373	2.700	4.298	2.282^*
MMEO	2.774	2.162^*	5.524	2.869	4.365
R²	0.126	0.209^*	0.054	0.032	0.013
750 day window
ME	0.014	0.000^*	0.062	0.069	0.062
MAE	0.044	0.037^*	0.098	0.139	0.098
RMSE	0.122	0.082^*	0.313	1.058	0.726
MAPE	8.467	6.412^*	14.021	14.118	12.450
MMEU	1.888^*	2.373	2.120	4.321	2.187
MMEO	3.057	2.162^*	5.631	3.047	4.983
R²	0.107	0.209^*	0.052	0.009	0.013
1,250 day window
ME	0.033	0.000^*	0.056	0.453	0.073
MAE	0.063	0.037^*	0.096	0.527	0.106
RMSE	0.345	0.082^*	0.308	10.772	0.565
MAPE	10.097	6.412^*	17.106	8.469	12.484
MMEU	1.814^*	2.373	2.284	4.826	2.050
MMEO	3.572	2.162^*	5.099	3.876	5.394
R²	0.053	0.209^*	0.054	0.008	0.051

Table C5: Foreign Exchange Variances
Variance – daily
125 day window
	Hist	Exp	Dexp	BEKK	GARCHCC
ME	0.00001	0.00001	0.00000	0.00002	0.00001
MMEU	0.00210	0.00210	0.00210	0.00243	0.00208
MMEO	0.00470	0.00451	0.00530	0.00421	0.00473
250 day window
ME	0.00001	0.00001	0.00000	0.00001	0.00001
MMEU	0.00212	0.00210	0.00203	0.00224	0.00207
MMEO	0.00473	0.00451	0.00532	0.00435	0.00464
500 day window
ME	0.00001	0.00001	0.00003	0.00006	0.00001
MMEU	0.00201	0.00210	0.00184	0.00217	0.00209
MMEO	0.00482	0.00451	0.00631	0.00437	0.00459
750 day window
ME	0.00001	0.00001	0.00004	0.00000	0.00001
MMEU	0.00203	0.00210	0.00163	0.00213	0.00206
MMEO	0.00494	0.00451	0.00682	0.00444	0.00461
1,250 day window
ME	0.00001	0.00001	0.00000	0.00001	0.00001
MMEU	0.00201	0.00210	0.00232	0.00211	0.00204
MMEO	0.00513	0.00451	0.00444	0.00458	0.00472

Table C6: Foreign Exchange Variances
Variance – quarter-average
125 day window
	Hist	Exp	Dexp	BEKK	GARCHCC
ME	0.00000	0.00000	0.00000	0.00004	0.00001
MMEU	0.00201	0.00235	0.00212	0.00388	0.00205
MMEO	0.00232	0.00216	0.00322	0.00293	0.00251
250 day window
ME	0.00000	0.00000	0.00000	0.00017	0.00000
MMEU	0.00202	0.00235	0.00201	0.00435	0.00196
MMEO	0.00233	0.00216	0.00332	0.00260	0.00242
500 day window
ME	0.00000	0.00000	0.00002	−0.00001	0.00000
MMEU	0.00192	0.00235	0.00133	0.00436	0.00188
MMEO	0.00234	0.00216	0.00434	0.00172	0.00232
750 day window
ME	0.00000	0.00000	0.00003	−0.00001	0.00000
MMEU	0.00182	0.00235	0.00082	0.00415	0.00187
MMEO	0.00243	0.00216	0.00521	0.00188	0.00234
1,250 day window
ME	0.00000	0.00000	−0.00001	0.00338	0.00000
MMEU	0.00172	0.00235	0.00303	0.00735	0.00177
MMEO	0.00273	0.00216	0.00202	0.00320	0.00267

Table C7: Interest Rate Variances
Variance – daily
125 day window
	Hist	Exp	Dexp	BEKK	GARCHCC
ME	0.00000	0.00000	0.00025	0.00006	−0.00002
MMEU	0.00274	0.00276	0.00353	0.00278	0.00285
MMEO	0.00651	0.00595	0.01202	0.00652	0.00599
250 day window
ME	0.00001	0.00000	0.00026	0.00008	−0.00001
MMEU	0.00272	0.00276	0.00311	0.00305	0.00276
MMEO	0.00693	0.00595	0.01145	0.00694	0.00636
500 day window
ME	0.00003	0.00000	0.00026	0.00011	0.00002
MMEU	0.00262	0.00276	0.00307	0.00313	0.00271
MMEO	0.00774	0.00595	0.01155	0.00783	0.00715
750 day window
ME	0.00005	0.00000	0.00010	0.00016	0.00004
MMEU	0.00262	0.00276	0.00268	0.00308	0.00275
MMEO	0.00833	0.00595	0.01105	0.00851	0.00804
1,250 day window
ME	0.00014	0.00000	0.00018	0.00051	0.00014
MMEU	0.00262	0.00276	0.00282	0.00351	0.00253
MMEO	0.00951	0.00595	0.01054	0.01049	0.00927

Table C8: Interest Rate Variances
Variance – quarter-average
125 day window
	Hist	Exp	Dexp	BEKK	GARCHCC
ME	0.00001	0.00000	0.00026	0.00020	0.00066
MMEU	0.00342	0.00382	0.00436	0.00354	0.00390
MMEO	0.00378	0.00351	0.01016	0.00720	0.01240
250 day window
ME	0.00001	0.00000	0.00026	0.00024	0.00015
MMEU	0.00325	0.00382	0.00396	0.00381	0.00272
MMEO	0.00422	0.00351	0.00954	0.00773	0.00546
500 day window
ME	0.00003	0.00000	0.00030	0.00032	0.00022
MMEU	0.00325	0.00382	0.00382	0.00402	0.00319
MMEO	0.00516	0.00351	0.00975	0.00894	0.00715
750 day window
ME	0.00005	0.00000	0.00015	0.00038	0.00020
MMEU	0.00305	0.00382	0.00296	0.00391	0.00301
MMEO	0.00562	0.00351	0.00894	0.00984	0.00815
1,250 day window
ME	0.00014	0.00000	0.00019	0.00213	0.00024
MMEU	0.00305	0.00382	0.00324	0.00594	0.00271
MMEO	0.00704	0.00351	0.00832	0.01391	0.00910