Oracle Estimation of a Change Point in High-Dimensional Quantile Regression

Figures & data

Table 1. Estimation results from quantile regression.

	No. of reg.	No. of selected reg. in step 3b	$\widehat{\tau }$	CI for τ0	$\widehat{\delta }$
γ = 0.25
6 control variables
No interaction	26	17	5.65	[4.75, 6.17]	−4.07
Two-way interaction	41	20	2.35	[1.00, 4.44]	−1.82
Three-way interaction	61	24	2.35	[1.00, 4.15]	−2.19
Four-way interaction	76	21	5.65	[4.69, 6.08]	−5.50
Five-way interaction	82	22	2.45	[1.00, 4.93]	−1.55
Six-way interaction	83	22	2.45	[1.00, 4.75]	−1.55
12 control variables
No interaction	32	17	5.65	[4.75, 6.17]	−4.07
Two-way interaction	98	18	5.25	[3.55, 6.09]	−3.40
Three-way interaction	318	22	5.25	[3.63, 5.94]	−3.61
Four-way interaction	813	26	5.25	[3.79, 5.97]	−3.53
Five-way interaction	1605	27	5.25	[4.57, 5.65]	−5.37
Six-way interaction	2529	28	5.65	[4.96, 6.06]	−5.50
γ = 0.50
6 control variables
No interaction	26	15	5.65	[1.67, 11.85]	−2.24
Two-way interaction	41	17	5.05	[2.25, 7.46]	−2.63
Three-way interaction	61	20	5.25	[4.22, 6.38]	−4.15
Four-way interaction	76	19	5.05	[3.60, 7.00]	−3.14
Five-way interaction	82	20	5.05	[1.23, 9.16]	−1.90
Six-way interaction	83	20	5.05	[1.33, 9.39]	−1.90
12 control variables
No interaction	32	16	1.95	[0.77, 4.61]	−3.69
Two-way interaction	98	21	6.75	[1.00, 45.57]	0.48
Three-way interaction	318	25	4.05	[1.00, 13.15]	−0.97
Four-way interaction	813	27	3.65	[1.00, 15.91]	−0.56
Five-way interaction	1605	29	3.25	[1.00, 13.16]	−0.68
Six-way interaction	2529	28	3.25	[1.00, 11.67]	−0.74
γ = 0.75
6 control variables
No interaction	26	15	10.05	[9.37, 11.29]	−10.62
Two-way interaction	41	14	NA	NA	0.00
Three-way interaction	61	21	NA	NA	0.00
Four-way interaction	76	18	NA	NA	0.00
Five-way interaction	82	18	NA	NA	0.00
Six-way interaction	83	18	NA	NA	0.00
12 control variables
No interaction	32	14	10.05	[8.44, 11.94]	−7.14
Two-way interaction	98	20	NA	NA	0.00
Three-way interaction	318	21	NA	NA	0.00
Four-way interaction	813	25	NA	NA	0.00
Five-way interaction	1605	28	NA	NA	0.00
Six-way interaction	2529	24	NA	NA	0.00

NOTES: The sample size is n = 1813. The parameter τ₀ is estimated by the grid search on the 591 equi-spaced points over [1, 60]. Both $\widehat{\tau }$ and the 95% confidence interval are based on reestimation after Step 3b: that is, τ is estimated again using $({\tilde{U}}_i,\tilde{\alpha })$ from Step 3b.

Figure 1. Estimation results: 12 control variables and six-way interaction.