A Comparative Study of VaR Estimation for Structured Products

Table 3

Estimation of conditional heteroscedastic models for alternative notes during various periods.

RUD notes
Time series models	Parameter
Panel A: The whole period
	𝜇	𝛼0	𝛼1	𝛼2	𝛼3	𝛼4	𝛼5		RMSE	MAE
AR(1)-ARCH(5)	0.000075	0.000133	0.224618	0.155303	−0.123168	0.091281	0.070586		0.007834	0.005241
	𝜇	𝛼0	𝛼1	𝛽1
AR(1)-GARCH(1,1)	0.001243	0.000123	0.141169	0.814768					0.007834	0.005246
	𝜇	𝛼0	𝛼1	𝛽1	𝛽2	𝛽3	𝛽4
AR(1)-GARCH(1,4)	0.001139	0.000134	0.152539	0.664518	−0.343514	0.887827	−0.41614		0.007834	0.005236
	𝜇	𝛼0	𝛼1	𝛾	𝛽1
AR(1)-EGARCH(1,1)	0.001513	-1.465525	0.432311	0.211102	0.7875371				0.007848	0.005224
Panel B: The low oil price period
	𝜇	𝛼0	𝛼1	𝛼2	𝛼3	𝛼4	𝛼5		RMSE	MAE
AR(1)-ARCH(5)	−0.001177	0.001250	0.123340	0.087360	−0.033873	0.123157	0.062676		0.007835	0.005246
	𝜇	𝛼0	𝛼1	𝛽1
AR(1)-GARCH(1,1)	−0.001377	0.000118	0.112652	0.811888					0.007840	0.005291
	𝜇	𝛼0	𝛼1	𝛽1	𝛽2	𝛽3	𝛽4
AR(1)-GARCH(1,4)	−0.0002145	0.000113	0.166107	0.566635	−0.416205	0.718833	−0.344506		0.007837	0.005272
	𝜇	𝛼0	𝛼1	𝛾	𝛽1
AR(1)-EGARCH(1,1)	0.001235	−10.5564833	0.223816	0.212933	−0.088939				0.007302	0.005282
Panel C: The high oil price period
	𝜇	𝛼0	𝛼1	𝛼2	𝛼3	𝛼4	𝛼5		RMSE	MAE
AR(1)-ARCH(5)	0.001123	0.001137	0.245690	0.237896	−0.013347	−0.035673	0.071133		0.007295	0.004973
	𝜇	𝛼0	𝛼1	𝛽1
AR(1)-GARCH(1,1)	0.000334	0.000102	0.177786	0.799431					0.007294	0.004967
	𝜇	𝛼0	𝛼1	𝛽1	𝛽2	𝛽3	𝛽4
AR(1)-GARCH(1,4)	0.001132	0.000103	0.245556	0.789879	−0.539841	0.245112	0.217447		0.007294	0.004968
	𝜇	𝛼0	𝛼1	𝛾	𝛽1
AR(1)-EGARCH(1,1)	0.000551	−1.126772	0.343221	0.132112	0.912626				0.007302	0.005011
SPO notes
Time series models	Parameter
Panel A: The whole period
	𝜇	𝛼0	𝛼1	𝛼2	𝛼3	𝛼4	𝛼5		RMSE	MAE
AR(1)-ARCH(5)	0.000778	0.000102	0.124557	0.166728	0.098039	0.234964	0.567833		0.008156	0.005566
	𝜇	𝛼0	𝛼1	𝛽1
AR(1)-GARCH(1,1)	0.000665	0.000112	0.099794	0.991196					0.008553	0.005556
	𝜇	𝛼0	𝛼1	𝛼2	𝛽1	𝛽2	𝛽3	𝛽4	𝛽5
AR(1)-GARCH(2,5)	0.001520	0.000112	0.088600	0.075643	0.823604	−0.4127508	0.665673	−0.888641	0.882347
									0.008155	0.005565
	𝜇	𝛼0	𝛼1	𝛾	𝛽1
AR(1)-EGARCH(1,1)	0.000558	−0.366564	0.213535	0.005563	0.988207				0.008157	0.005570
Panel B: The low oil price period
	𝜇	𝛼0	𝛼1	𝛼2	𝛼3	𝛼4	𝛼5		RMSE	MAE
AR(1)-ARCH(5)	0.000112	0.000138	0.096679	−0.002138	−0.041125	0.089648	0.134568		0.008152	0.005520
	𝜇	𝛼0	𝛼1	𝛽1
AR(1)-GARCH(1,1)	0.000114	0.000211	0.048385	0.933170					0.008152	0.005547
	𝜇	𝛼0	𝛼1	𝛼2	𝛽1	𝛽2	𝛽3	𝛽4	𝛽5
AR(1)-GARCH(2,5)	0.000719	0.0001122	0.111361	−0.088981	0.397539	0.0311331	0.385764	0.299079	−0.21435
									0.008152	0.005525
	𝜇	𝛼0	𝛼1	𝛾	𝛽1
AR(1)-EGARCH(1,1)	0.001122	−5.998389	0.064714	0.299825	0.455299				0.008163	0.005585
Panel C: The high oil price period
	𝜇	𝛼0	𝛼1	𝛼2	𝛼3	𝛼4	𝛼5		RMSE	MAE
AR(1)-ARCH(5)	0.000565	0.000078	0.114599	0.211361	0.121363	0.097749	0.056788		0.008874	0.005949
	𝜇	𝛼0	𝛼1	𝛽1
AR(1)-GARCH(1,1)	0.000394	0.000113	0.126451	0.866762					0.008874	0.005949
	𝜇	𝛼0	𝛼1	𝛼2	𝛽1	𝛽2	𝛽3	𝛽4	𝛽5
AR(1)-GARCH(2,5)	0.001233	0.0001124	0.126651	0.088795	0.611582	-0.113913	0.311316	−0.711249	0.698910
									0.008874	0.005949
	𝜇	𝛼0	𝛼1	𝛾	𝛽1
AR(1)-EGARCH(1,1)	0.000778	−0.512671	0.256784	0.044575	0.958854				0.008884	0.005969
DJQ notes
Time series models	Parameter
Panel A: The whole period
	𝜇	𝛼0	𝛼1	𝛼2	𝛼3	𝛼4	𝛼5		RMSE	MAE
AR(1)-ARCH(5)	0.000443	0.000113	0.211583	0.13348	0.114672	0.11126	0.066886		0.007630	0.005398
	𝜇	𝛼0	𝛼1	𝛼2	𝛽1
AR(1)-GARCH(2,1)	0.001130	0.000121	0.215542	-0.131214	0.889394				0.007628	0.005391
	𝜇	𝛼0	𝛼1	𝛼2	𝛽1	𝛽2	𝛽3	𝛽4	𝛽5
AR(1)-GARCH(2,5)	0.001137	0.000111	0.188956	-0.177846	1.184288	0.418862	-0.340559	-0.811254	0.526681
									0.007629	0.005395
	𝜇	𝛼0	𝛼1	𝛼2	𝛾	𝛽1
AR(1)-EGARCH(2,1)	0.000313	-0.281558	0.378891	-0.221350	-0.011877	0.898797			0.007628	0.005389
	𝜇	𝛼0	𝛼1	𝛼2	𝛾	𝛽1	𝛽2
AR(1)-EGARCH(2,2)	0.001134	-0.055808	0.411404	-0.311069	-0.021166	1.566290	-0.778983		0.007628	0.005392
Panel B: The low oil price period
	𝜇	𝛼0	𝛼1	𝛼2	𝛼3	𝛼4	𝛼5		RMSE	MAE
AR(1)-ARCH(5)	-0.000113	0.000167	0.322399	0.099110	0.026004	-0.021233	-0.045188		0.007632	0.005377
	𝜇	𝛼0	𝛼1	𝛼2	𝛽1
AR(1)-GARCH(2,1)	0.0001083	0.000112	0.466754	-0.055689	0.244596				0.007631	0.003701
	𝜇	𝛼0	𝛼1	𝛼2	𝛽1	𝛽2	𝛽3	𝛽4	𝛽5
AR(1)-GARCH(2,5)	-0.000116	0.000111	0.356711	-0.211299	0.611261	0.088741	-0.091258	0.021123	-0.092397
									0.007633	0.005382
	𝜇	𝛼0	𝛼1	𝛼2	𝛾	𝛽1
AR(1)-EGARCH(2,1)	0.000441	-5.522416	0.622357	-0.055101	0.211319	0.511383			0.007628	0.005390
	𝜇	𝛼0	𝛼1	𝛼2	𝛾	𝛽1	𝛽2
AR(1)-EGARCH(2,2)	0.000668	-13.512605	0.694437	0.711222	0.112577	- 0.391220	0.181336		0.007628	0.005386
Panel C: The high oil price period
	𝜇	𝛼0	𝛼1	𝛼2	𝛼3	𝛼4	𝛼5		RMSE	MAE
AR(1)-ARCH(5)	0.000771	0.000123	0.144517	0.133165	0.171292	0.213814	0.113118		0.008166	0.005746
	𝜇	𝛼0	𝛼1	𝛼2	𝛽1
AR(1)-GARCH(2,1)	0.001124	0.000111	0.161129	-0.033211	0.877278				0.008160	0.005744
	𝜇	𝛼0	𝛼1	𝛼2	𝛽1	𝛽2	𝛽3	𝛽4	𝛽5
AR(1)-GARCH(2,5)	0.001125	0.000111	0.111213	-0.088396	1.112524	0.733184	-0.491514	-0.988785	0.772727
									0.008160	0.0057435
	𝜇	𝛼0	𝛼1	𝛼2	𝛾	𝛽1
AR(1)-EGARCH(2,1)	0.001145	-0.413491	0.222399	-0.011715	-0.001136	0.982226			0.008160	0.005744
	𝜇	𝛼0	𝛼1	𝛼2	𝛾	𝛽1	𝛽2
AR(1)-EGARCH(2,2)	0.000661	-0.534123	0.277669	0.099854	0.015515	0.0414396	0.581893		0.008161	0.005744

Note that RMSE and MAE demonstrate root mean squared errors and mean absolute errors, respectively.