. Author manuscript; available in PMC: 2015 Dec 1.

Published in final edited form as: Econ Theory (N Y). 2014 Jun 6;30(6):1272–1314. doi: 10.1017/S0266466614000176

Table 4.

(Location-scale model) Empirical relative efficiency of OWQAE with estimated weights compared to four methods: OLS, LAD, QAO, OWQAE using unweighted quantile regression in (27). Mixture 1: 0.5N(0,1)+0.5N(0,0.5⁶); Mixture 2: 0.5N(−2,1)+0.5N(2,1). For Student-t₁, t₂, due to infinite variance, the OLS is not stable and varies significantly in simulations. [Numbers ≥ 1 indicate better performance of OWQAE].

(a): Model 5 in (60)

	location parameter				scale parameter
ε_t	β = 0.6				γ₀ = 0.5				γ₁ = 1.0
Distribution	OLS	LAD	QAO	QAEU	OLS	LAD	QAO	QAEU	OLS	LAD	QAO	QAEU
Student-t₁	NA	0.96	0.85	1.17	NA	1.17	0.88	1.20	NA	1.22	0.92	1.25
Student-t₂	NA	1.05	0.91	1.13	NA	1.66	0.94	1.28	NA	1.70	0.96	1.27
N(0,1)	0.89	1.43	0.93	1.12	0.68	2.29	0.92	1.28	0.68	2.37	0.94	1.26
Mixture 1	7.05	1.21	0.91	1.18	0.55	1.88	0.85	1.23	0.55	2.05	0.89	1.27
Mixture 2	2.57	23.03	1.03	1.23	0.66	2.07	0.86	1.28	0.69	2.58	0.86	1.20
Laplace	1.53	0.80	0.80	1.11	0.93	1.89	0.96	1.25	0.96	1.81	0.96	1.21
Gamma(1)	7.38	6.40	0.79	1.17	7.31	3.72	0.50	1.13	6.38	3.18	0.44	1.14
Gamma(2)	2.32	2.89	0.92	1.13	2.70	2.61	0.79	1.23	2.60	2.76	0.81	1.21
Gamma(3)	1.67	2.37	0.92	1.13	1.79	2.63	0.83	1.15	1.76	2.84	0.88	1.20
Beta(1,1)	1.48	4.39	0.95	1.09	0.66	3.87	0.83	1.19	0.65	3.56	0.85	1.17
Beta(1,2)	1.95	4.48	0.87	1.12	1.04	3.41	0.71	1.10	1.07	3.75	0.73	1.21
Beta(1,3)	2.75	4.97	0.84	1.11	1.71	3.64	0.69	1.14	1.63	3.30	0.65	1.10

(b): Model 6 in (61)

	location parameter				scale parameter
ε_t	β = 0.4				γ₀ = 0.5				γ₁ = 0.5
Distribution	OLS	LAD	QAO	QAEU	OLS	LAD	QAO	QAEU	OLS	LAD	QAO	QAEU
Student-t₁	NA	0.65	0.55	NA	NA	16.84	0.37	NA	NA	3.85	0.30	NA
Student-t₂	NA	1.03	0.89	6.76	NA	3.69	0.88	3.38	NA	4.01	0.96	6.02
N(0,1)	0.89	1.45	0.97	1.12	0.62	1.94	0.92	1.21	0.67	2.10	0.97	1.19
Mixture 1	6.47	1.12	0.85	1.22	0.51	1.76	0.77	1.07	0.60	1.92	0.99	1.10
Mixture 2	2.40	22.77	1.00	6.59	0.70	20.72	0.91	34.50	0.64	9.51	0.84	5.82
Laplace	1.48	0.80	0.81	1.53	0.92	2.31	0.92	1.50	0.91	2.35	0.97	1.44
Gamma(1)	5.71	6.16	0.78	1.10	5.13	3.65	0.40	1.12	7.79	5.61	0.62	1.21
Gamma(2)	2.13	2.86	0.93	1.22	2.63	4.37	0.69	1.58	2.73	4.64	0.80	1.53
Gamma(3)	1.44	2.04	0.95	1.57	1.89	4.55	0.84	2.12	1.77	3.85	0.84	1.76
Beta(1,1)	1.59	4.30	0.92	1.01	0.67	3.14	0.71	1.00	0.83	3.79	0.98	1.01
Beta(1,2)	1.73	4.27	0.86	0.98	1.00	2.44	0.65	0.99	1.15	3.21	0.89	0.99
Beta(1,3)	2.48	5.17	0.81	1.01	1.50	2.45	0.54	0.98	1.48	3.79	0.87	1.00