Approximate p-Values of Certain Tests Involving Hypotheses About Multiple Breaks

Alastair Hall; Nikolaos Sakkas

doi:10.1515/jem-2012-0014

Published by De Gruyter March 15, 2013

Approximate p-Values of Certain Tests Involving Hypotheses About Multiple Breaks

Alastair Hall and Nikolaos Sakkas

From the journal Journal of Econometric Methods

https://doi.org/10.1515/jem-2012-0014

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Abstract

We provide formulae for calculating approximate p-values for the non-standard asymptotic null distributions of a variety of tests used for detecting multiple structural change in a wide range of models. Our approximations are based on simulated quantiles obtained from 100,000 replications, and the latter are more accurate than the quantiles reported in the literature by increasing the number of replications by a factor of 10. The p-value response surfaces are approximated using a parametric method proposed by Hansen and their use is illustrated with an example. Using our p-value response surfaces, it is shown that the use of Bai and Perron’s response surfaces for the critical values of these tests can lead to misleading inferences, and thus should be used with extreme caution.

Keywords: least squares; regression models; tests of parameter variation

Corresponding author: Alastair Hall, Economics, School of Social Sciences, University of Manchester, Oxford Road, Manchester M13 9PL, UK

Appendix

Table 1

Absolute Errors in Fitted Distributions.

p-Value	SupF		F(1+l\|l)		UDmax		WDmax
	Median	Maximum	Median	Maximum	Median	Maximum	Median	Maximum
0.00	0.0001	0.0003	0.0002	0.0007	0.0002	0.0007	0.0001	0.0001
0.01	0.0003	0.0012	0.0008	0.0033	0.0011	0.0032	0.0014	0.0033
0.02	0.0004	0.0015	0.0009	0.0037	0.0013	0.0038	0.0016	0.0047
0.03	0.0004	0.0014	0.0009	0.0037	0.0014	0.0034	0.0016	0.0043
0.04	0.0004	0.0015	0.0008	0.0038	0.0014	0.0032	0.0016	0.0041
0.05	0.0004	0.0015	0.0008	0.0034	0.0014	0.0031	0.0015	0.0045
0.06	0.0004	0.0015	0.0007	0.0035	0.0013	0.0028	0.0012	0.0038
0.07	0.0004	0.0014	0.0007	0.0036	0.0011	0.0027	0.0011	0.0047
0.08	0.0004	0.0014	0.0006	0.0030	0.0010	0.0023	0.0012	0.0040
0.09	0.0004	0.0014	0.0007	0.0033	0.0007	0.0020	0.0010	0.0041
0.10	0.0004	0.0013	0.0006	0.0026	0.0005	0.0020	0.0010	0.0041
0.15	0.0003	0.0014	0.0007	0.0033	0.0007	0.0025	0.0013	0.0038
0.20	0.0004	0.0015	0.0007	0.0042	0.0011	0.0027	0.0013	0.0033
0.25	0.0004	0.0017	0.0006	0.0035	0.0013	0.0032	0.0014	0.0043
0.30	0.0004	0.0016	0.0006	0.0034	0.0010	0.0031	0.0012	0.0048
0.40	0.0004	0.0018	0.0007	0.0035	0.0005	0.0014	0.0013	0.0045
0.50	0.0004	0.0017	0.0008	0.0036	0.0012	0.0035	0.0019	0.0044
0.60	0.0005	0.0016	0.0008	0.0036	0.0013	0.0042	0.0016	0.0055
0.70	0.0005	0.0021	0.0009	0.0043	0.0006	0.0019	0.0014	0.0059

Table 2a

SupF Distributions, k=1.

q	ε=0.05			ε=0.10			ε=0.15			ε=0.20			ε=0.25
	θ₀	θ₁	η	θ₀	θ₁	η	θ₀	θ₁	η	θ₀	θ₁	η	θ₀	θ₁	η
1	–1.41	1.08	3.76	–1.17	1.05	3.34	–1.01	1.02	3.02	–0.86	1.00	2.78	–0.74	0.98	2.55
2	–2.34	1.11	5.53	–1.99	1.06	4.94	–1.70	1.03	4.56	–1.44	1.02	4.36	–1.26	0.99	4.01
3	–3.27	1.13	6.94	–2.66	1.10	6.56	–2.22	1.07	6.25	–1.84	1.06	6.05	–1.58	1.04	5.67
4	–4.08	1.16	8.46	–3.29	1.14	8.21	–2.77	1.11	7.81	–2.45	1.07	7.22	–2.08	1.05	6.87
5	–4.86	1.14	9.44	–4.10	1.11	8.88	–3.55	1.07	8.39	–3.00	1.05	8.11	–2.50	1.04	7.91
6	–5.81	1.14	10.25	–4.83	1.11	9.96	–3.89	1.12	10.34	–3.20	1.12	10.25	–2.65	1.10	9.95
7	–6.30	1.15	11.80	–5.26	1.11	11.20	–4.28	1.11	11.47	–3.66	1.08	10.98	–3.15	1.05	10.43
8	–7.23	1.14	12.42	–5.94	1.13	12.38	–4.97	1.11	12.28	–4.26	1.08	11.86	–3.74	1.05	11.28
9	–8.07	1.14	13.19	–6.57	1.13	13.43	–5.67	1.10	12.97	–4.63	1.11	13.42	–4.15	1.06	12.45
10	–8.90	1.11	13.42	–7.51	1.10	13.54	–6.32	1.09	13.78	–5.32	1.08	13.78	–4.24	1.08	14.05
11	–9.36	1.15	15.51	–7.82	1.13	15.34	–6.45	1.13	15.79	–5.57	1.09	15.07	–4.73	1.07	14.82
12	–10.56	1.09	14.42	–8.84	1.08	14.85	–7.73	1.05	14.43	–6.60	1.05	14.69	–5.65	1.03	14.53
13	–10.83	1.14	16.97	–9.07	1.11	16.79	–8.12	1.06	15.61	–6.7	1.06	16.26	–5.51	1.07	16.73
14	–8.36	1.13	17.72	–9.68	1.09	17.30	–8.36	1.06	16.90	–7.36	1.04	16.61	–5.82	1.08	18.08
15	–12.53	1.11	17.47	–10.74	1.07	17.00	–9.23	1.05	17.07	–7.82	1.06	17.84	–6.60	1.06	18.11
16	–13.26	1.08	17.40	–11.61	1.01	15.94	–10.00	1.03	17.20	–8.47	1.05	18.24	–7.42	1.02	17.63
17	–13.90	1.08	18.34	–11.72	1.07	18.92	–10.10	1.06	19.08	–8.46	1.07	20.14	–7.27	1.06	20.17
18	–14.83	1.07	18.44	–12.61	1.03	18.35	–11.06	1.02	18.60	–9.38	1.03	19.61	–7.70	1.05	20.93
19	–15.33	1.06	19.24	–13.13	1.02	18.70	–11.46	1.02	19.50	–9.62	1.04	20.96	–8.55	1.02	20.29
20	–16.16	1.06	19.56	–13.55	1.07	21.37	–11.53	1.08	22.45	–10.02	1.07	22.78	–8.50	1.07	23.09

Table 2b

SupF Distributions, k=2.

q	ε=0.05			ε=0.10			ε=0.15			ε=0.20			ε=0.25
	θ₀	θ₁	η	θ₀	θ₁	η	θ₀	θ₁	η	θ₀	θ₁	η	θ₀	θ₁	η
1	–5.19	2.32	8.09	–3.06	2.19	7.31	–2.08	2.09	6.34	–1.59	1.95	5.11	–1.19	1.90	4.24
2	–7.92	2.32	10.62	–5.17	2.22	10.06	–3.47	2.18	9.73	–2.46	2.12	8.81	–1.79	2.02	7.46
3	–10.35	2.20	11.49	–6.89	2.17	11.98	–4.60	2.19	12.43	–3.40	2.11	11.30	–2.38	2.07	10.28
4	–12.34	2.19	13.24	–8.50	2.19	14.25	–5.62	2.24	15.43	–3.77	2.20	14.91	–2.78	2.09	12.98
5	–14.08	2.18	14.97	–9.95	2.16	15.86	–7.10	2.14	16.31	–4.77	2.17	16.89	–2.57	2.21	17.17
6	–16.01	2.25	17.50	–11.22	2.25	19.09	–8.15	2.20	19.18	–6.18	2.12	17.95	–4.01	2.12	17.73
7	–17.49	2.26	19.74	–12.51	2.18	20.12	–9.00	2.18	21.15	–7.32	2.03	18.57	–5.28	1.99	17.73
8	–19.36	2.14	19.18	–14.35	2.03	19.00	–10.65	2.06	20.80	–8.13	2.04	20.70	–5.94	1.99	19.74
9	–20.97	2.19	21.63	–15.02	2.19	24.05	–11.02	2.19	25.31	–7.48	2.22	26.81	–4.76	2.19	26.22
10	–22.72	2.13	21.82	–16.63	2.16	24.91	–11.65	2.24	28.59	–8.27	2.20	28.64	–4.31	2.26	30.40
11	–24.32	2.06	21.59	–17.63	2.18	27.35	–13.92	2.04	25.41	–10.73	2.02	25.77	–7.27	2.04	26.71
12	–25.92	2.13	24.56	–19.66	2.02	24.80	–15.56	1.96	24.97	–12.49	1.95	25.16	–8.54	1.99	27.12
13	–27.50	2.16	26.68	–20.87	2.13	28.83	–15.70	2.13	31.02	–11.13	2.16	33.37	–8.48	2.07	31.16
14	–28.85	2.09	26.65	–22.03	2.02	27.74	–17.06	2.04	30.36	–11.72	2.17	35.69	–8.08	2.13	35.10
15	–30.75	2.00	25.10	–22.71	2.18	33.97	–16.26	2.25	38.83	–11.33	2.24	40.42	–7.95	2.16	38.59
16	–31.86	2.02	27.00	–24.20	2.07	32.19	–18.86	2.02	33.34	–14.37	2.03	35.12	–10.29	2.02	35.71
17	–33.45	2.03	28.50	–24.73	2.17	37.13	–19.48	2.08	36.74	–14.05	2.14	40.56	–10.05	2.10	40.26
18	–34.58	1.92	26.70	–26.76	2.03	33.95	–20.39	2.10	39.08	–15.11	2.10	41.06	–12.18	1.99	37.85
19	–35.69	1.89	27.16	–28.22	1.90	31.12	–21.65	2.03	38.46	–15.84	2.08	42.44	–12.17	2.02	40.98
20	–37.70	1.97	30.37	–29.17	2.03	36.76	–20.79	2.20	46.98	–16.72	2.09	44.26	–13.21	2.00	41.81

Table 2c

SupF Distributions, k=3.

q	ε=0.05			ε=0.10			ε=0.15			ε=0.20
	θ₀	θ₁	η	θ₀	θ₁	η	θ₀	θ₁	η	θ₀	θ₁	η
1	–6.01	3.37	11.95	–3.08	3.29	11.04	–2.25	3.04	8.42	–1.69	2.88	6.23
2	–10.04	3.29	14.94	–5.92	3.25	14.81	–3.62	3.20	13.75	–2.82	2.98	10.39
3	–13.07	3.31	18.34	–8.24	3.16	17.51	–5.44	3.10	16.57	–3.28	3.12	15.25
4	–15.85	3.23	20.46	–9.57	3.36	23.32	–5.20	3.43	24.37	–4.26	3.10	18.50
5	–17.63	3.47	26.97	–11.02	3.40	27.55	–8.59	3.05	22.23	–5.80	3.00	20.29
6	–21.00	3.37	27.66	–12.91	3.41	30.90	–9.57	3.16	26.85	–7.42	2.93	21.97
7	–23.07	3.35	30.45	–14.93	3.33	32.65	–10.33	3.20	31.02	–7.02	3.08	27.87
8	–25.42	3.43	34.56	–17.15	3.21	33.40	–11.16	3.22	34.69	–9.46	2.88	26.80
9	–26.61	3.64	42.26	–17.72	3.34	39.40	–12.35	3.19	37.39	–8.89	3.06	33.48
10	–30.66	3.23	35.25	–21.12	3.09	36.01	–13.58	3.21	40.68	–11.55	2.85	31.47
11	–33.11	3.13	35.05	–22.51	3.16	40.16	–14.86	3.16	42.64	–9.98	3.09	40.48
12	–35.16	3.15	37.93	–24.33	3.13	42.05	–17.59	3.06	42.00	–10.76	3.16	44.73
13	–37.66	3.24	41.97	–24.76	3.33	50.20	–18.03	3.20	48.38	–9.26	3.35	53.53
14	–39.57	3.25	44.82	–25.36	3.41	55.92	–18.69	3.21	51.96	–8.59	3.42	59.52
15	–41.97	3.17	44.49	–28.17	3.27	53.94	–18.83	3.28	57.51	–9.06	3.39	62.63
16	–43.78	2.86	38.05	–30.57	3.15	52.74	–19.18	3.31	62.00	–15.34	3.02	52.19
17	–45.72	3.17	49.32	–33.33	2.95	48.46	–25.99	2.82	46.76	–14.63	3.15	59.31
18	–48.01	2.92	43.42	–33.08	3.22	60.34	–26.04	2.95	53.67	–17.45	3.03	57.25
19	–49.73	2.84	42.89	–35.89	3.03	55.61	–23.20	3.20	67.17	–19.52	2.87	54.79
20	–52.17	3.04	50.67	–36.45	3.18	63.72	–23.87	3.24	71.46	–13.67	3.31	75.81

Table 2d

SupF Distributions, k=4.

q	ε=0.05			ε=0.15			ε=0.15
	θ₀	θ₁	η	θ₀	θ₁	η	θ₀	θ₁	η
1	–6.31	4.60	17.10	–2.95	4.38	14.24	–2.47	3.96	9.47
2	–11.33	4.46	21.22	–6.68	4.16	18.05	–4.11	4.09	15.44
3	–15.32	4.36	24.62	–9.13	4.09	22.23	–6.01	4.01	19.41
4	–18.80	4.28	27.77	–10.69	4.34	29.45	–6.62	4.21	26.15
5	–20.52	4.63	37.24	–11.44	4.51	37.17	–8.66	4.10	29.04
6	–23.68	4.71	42.45	–13.32	4.53	42.05	–10.88	3.99	31.20
7	–27.90	4.40	40.53	–16.21	4.42	44.12	–10.93	4.18	38.75
8	–29.74	4.65	49.46	–18.19	4.39	47.86	–10.75	4.30	45.93
9	–29.93	4.98	62.12	–18.78	4.44	54.12	–12.20	4.24	49.06
10	–36.15	4.44	52.23	–23.30	4.14	50.30	–16.36	4.00	46.50
11	–39.96	4.15	48.54	–24.71	4.26	56.73	–14.81	4.20	56.58
12	–41.45	4.49	60.50	–25.94	4.33	62.50	–16.70	4.23	60.30
13	–43.96	4.62	67.11	–27.97	4.38	67.17	–17.26	4.30	66.46
14	–47.61	4.33	62.45	–27.01	4.59	78.56	–19.52	4.16	66.58
15	–51.32	4.06	57.25	–33.35	4.12	66.77	–19.18	4.29	74.64
16	–53.71	4.01	58.80	–32.38	4.41	80.22	–17.28	4.48	86.24
17	–56.40	3.96	59.85	–36.95	4.15	74.35	–26.75	3.92	68.48
18	–58.88	4.01	64.08	–39.13	4.11	76.36	–27.05	3.99	74.65
19	–59.11	4.42	82.05	–37.29	4.31	88.96	–23.23	4.23	89.69
20	–63.31	4.25	78.24	–37.90	4.48	99.04	–27.58	4.08	86.01

Table 2e

SupF Distributions, k=5.

q	ε=0.05			ε=0.10			ε=0.15
	θ₀	θ₁	η	θ₀	θ₁	η	θ₀	θ₁	η
1	–6.95	5.49	19.54	–3.30	5.26	15.53	–2.20	4.78	8.86
2	–11.84	5.58	27.13	–7.24	5.06	20.65	–3.72	4.83	14.82
3	–15.49	5.78	35.52	–8.97	5.18	28.19	–5.26	4.79	19.76
4	–21.31	5.19	33.28	–11.35	5.31	35.00	–5.16	5.13	28.41
5	–21.94	5.87	49.15	–11.50	5.61	45.64	–6.29	5.16	34.00
6	–25.60	5.88	54.75	–14.33	5.52	49.78	–9.83	4.72	33.02
7	–31.25	5.44	51.16	–16.61	5.50	54.99	–11.17	4.79	38.12
8	–32.69	5.79	63.43	–20.66	5.23	54.53	–9.92	5.06	48.27
9	–27.66	6.70	93.08	–18.94	5.56	68.47	–10.30	5.16	55.24
10	–39.50	5.67	70.78	–23.40	5.31	67.01	–16.06	4.59	47.78
11	–45.00	5.29	65.11	–24.98	5.37	73.57	–14.84	4.90	59.03
12	–46.84	5.49	75.01	–30.09	5.05	69.06	–13.15	5.17	70.88
13	–45.99	6.06	96.55	–29.42	5.42	83.89	–12.58	5.32	80.00
14	–51.46	5.75	91.70	–25.36	5.83	103.75	–14.71	5.16	80.98
15	–57.68	5.21	78.73	–27.35	5.72	106.59	–16.66	5.10	83.84
16	–58.94	5.49	92.09	–34.38	5.40	98.93	–19.91	4.97	83.72
17	–64.70	5.09	81.92	–36.91	5.40	102.89	–22.89	4.86	84.01
18	–68.18	4.84	77.74	–42.06	5.15	97.81	–25.46	4.80	85.73
19	–66.71	5.56	107.76	–38.65	5.39	114.30	–21.03	5.02	100.86
20	–70.06	5.56	111.62	–37.12	5.68	131.05	–26.94	4.80	94.89

Table 2f

SupF Distributions, k=6.

q	ε=0.05			ε=0.10
	θ₀	θ₁	η	θ₀	θ₁	η
1	–7.24	6.47	22.42	–3.99	6.00	15.46
2	–12.22	6.69	32.68	–7.24	6.08	23.63
3	–15.57	7.01	44.17	–9.09	6.14	31.90
4	–22.75	6.31	41.55	–11.70	6.28	39.65
5	–22.38	7.08	60.94	–10.24	6.81	55.14
6	–28.25	6.75	61.49	–17.16	6.13	50.21
7	–32.94	6.67	65.53	–17.11	6.49	62.97
8	–34.16	6.99	79.02	–21.06	6.22	64.02
9	–32.09	7.60	101.85	–18.41	6.69	81.74
10	–42.45	6.83	87.30	–24.25	6.33	78.78
11	–49.15	6.37	80.66	–26.28	6.33	84.79
12	–47.08	7.05	106.03	–32.02	5.98	80.48
13	–44.98	7.53	128.72	–28.86	6.54	102.30
14	–50.66	7.33	127.99	–25.11	6.85	120.57
15	–62.21	6.33	100.46	–30.00	6.59	118.34
16	–61.31	6.94	126.03	–36.85	6.29	112.43
17	–69.62	6.35	110.21	–38.42	6.36	120.20
18	–73.51	6.20	110.01	–43.53	6.15	117.34
19	–67.09	7.13	152.80	–35.19	6.73	149.05
20	–69.97	7.19	160.97	–31.19	7.04	170.03

Table 2g

SupF Distributions. k=7.

q	ε=0.05			ε=0.10
	θ₀	θ₁	η	θ₀	θ₁	η
1	–8.08	7.17	23.23	–4.08	6.98	16.07
2	–12.31	7.78	37.87	–7.49	6.98	24.68
3	–17.16	7.80	47.07	–9.56	7.05	33.69
4	–24.42	7.16	46.28	–11.28	7.38	44.50
5	–22.19	8.33	73.08	–9.58	7.97	61.55
6	–28.40	8.01	75.22	–16.42	7.32	58.77
7	–30.84	8.22	87.47	–19.00	7.23	64.75
8	–34.86	8.16	94.27	–18.41	7.53	78.38
9	–32.95	8.74	117.45	–18.69	7.67	89.63
10	–45.23	7.83	100.34	–26.61	7.05	81.84
11	–52.67	7.35	93.96	–28.98	7.08	88.74
12	–43.70	8.66	140.63	–36.27	6.57	81.03
13	–45.38	8.74	152.33	–30.74	7.40	110.29
14	–53.94	8.33	145.31	–22.89	8.00	139.26
15	–69.56	6.85	103.40	–32.72	7.35	125.13
16	–65.72	7.95	145.04	–34.56	7.47	134.78
17	–71.20	7.73	143.63	–19.46	8.49	182.98
18	–78.58	7.21	131.01	–38.01	7.55	151.04
19	–51.71	9.31	230.80	–33.83	7.79	170.45
20	–71.62	8.46	195.76	–28.67	8.17	195.04

Table 2h

SupF Distributions. k=8.

q	ε=0.05			ε=0.10
	θ₀	θ₁	η	θ₀	θ₁	η
1	–8.74	7.95	24.30	–3.59	8.09	15.84
2	–12.36	8.85	42.54	–7.11	7.82	23.81
3	–17.03	8.94	54.17	–8.90	8.03	34.12
4	–24.55	8.36	55.55	–11.17	8.18	43.54
5	–22.79	9.42	81.86	–9.77	8.73	59.77
6	–29.47	9.07	84.82	–18.10	7.72	53.23
7	–33.21	9.03	93.61	–16.50	8.33	70.78
8	–37.74	9.00	101.32	–21.74	7.84	69.96
9	–38.06	9.39	120.01	–19.72	8.31	87.41
10	–50.94	8.31	100.42	–23.51	8.16	91.53
11	–53.52	8.62	115.01	–31.34	7.50	83.27
12	–43.60	9.83	161.54	–29.50	8.09	103.61
13	–36.25	10.55	197.66	–24.07	8.73	129.72
14	–56.51	9.36	163.33	–27.56	8.42	130.58
15	–71.50	8.16	130.66	–43.05	7.12	98.98
16	–69.34	8.94	163.69	–40.80	7.68	121.29
17	–66.95	9.38	189.36	–37.24	8.16	144.16
18	–73.45	9.17	188.75	–22.73	9.12	190.43
19	–59.52	10.14	243.14	–34.95	8.46	172.41
20	–66.70	10.01	244.45	12.34	10.57	283.52

Table 2i

SupF Distributions, k=9.

q	ε=0.05
	θ₀	θ₁	η
1	–9.32	8.69	25.10
2	–12.20	9.93	47.07
3	–14.57	10.50	66.24
4	–24.91	9.42	62.67
5	–22.70	10.56	91.44
6	–29.64	10.16	95.11
7	–30.04	10.53	114.25
8	–38.89	10.00	112.09
9	–37.56	10.56	136.33
10	–53.14	9.23	111.33
11	–54.89	9.61	129.09
12	–34.69	11.59	202.38
13	–45.52	11.06	195.16
14	–55.04	10.61	189.53
15	–71.78	9.44	157.89
16	–74.81	9.69	173.25
17	–60.45	10.94	233.31
18	–67.03	10.79	236.23
19	–47.35	11.87	301.11
20	–43.71	12.18	328.79

Table 3a

F_T(ℓ+1|ℓ) Distributions, ℓ=1.

q	ε=0.05			ε=0.10			ε=0.15			ε=0.20			ε=0.25
	θ₀	θ₁	η	θ₀	θ₁	η	θ₀	θ₁	η	θ₀	θ₁	η	θ₀	θ₁	η
1	–2.4	1.17	4.8	–2.00	1.14	4.44	–1.75	1.11	4.06	–1.48	1.11	3.93	–1.28	1.10	3.72
2	–3.82	1.25	7.07	–3.24	1.20	6.54	–2.78	1.19	6.35	–2.44	1.18	6.10	–2.16	1.15	5.67
3	–5.43	1.20	7.63	–4.61	1.20	7.61	–3.92	1.21	7.78	–3.37	1.22	7.83	–2.94	1.20	7.54
4	–6.72	1.25	9.21	–5.82	1.22	8.92	–5.13	1.19	8.60	–4.47	1.18	8.54	–3.91	1.18	8.46
5	–7.82	1.23	10.20	–6.85	1.18	9.54	–6.11	1.16	9.27	–5.29	1.19	9.77	–4.56	1.18	9.81
6	–9.15	1.17	10.02	–8.06	1.14	9.69	–7.16	1.17	10.33	–6.46	1.16	10.19	–5.76	1.14	9.85
7	–10.10	1.23	12.06	–8.76	1.21	11.98	–7.81	1.20	12.01	–6.92	1.19	12.04	–6.16	1.17	11.79
8	–11.33	1.15	11.18	–10.06	1.14	11.38	–8.93	1.14	11.74	–8.01	1.14	11.79	–7.13	1.15	12.10
9	–12.32	1.21	13.25	–10.91	1.16	12.55	–9.80	1.16	13.00	–8.96	1.14	12.63	–7.97	1.14	12.74
10	–13.29	1.16	13.07	–11.92	1.11	12.30	–10.81	1.12	12.82	–9.68	1.14	13.57	–8.51	1.17	14.53
11	–14.55	1.17	13.76	–12.83	1.16	14.18	–11.55	1.16	14.60	–10.18	1.20	15.93	–9.29	1.15	15.04
12	–15.34	1.10	13.07	–13.65	1.10	13.70	–12.43	1.09	13.77	–11.29	1.09	13.99	–10.40	1.04	13.10
13	–16.44	1.16	15.04	–14.56	1.15	15.55	–13.20	1.12	15.14	–11.92	1.14	16.16	–10.87	1.12	15.95
14	–17.29	1.15	15.70	–15.42	1.12	15.69	–13.85	1.15	17.12	–12.62	1.14	17.10	–11.67	1.11	16.41
15	–18.35	1.12	15.45	–16.36	1.11	15.77	–14.82	1.12	16.75	–13.62	1.10	16.50	–12.53	1.06	15.73
16	–19.02	1.09	15.46	–16.99	1.12	17.21	–15.64	1.02	14.69	–14.52	1.02	14.86	–13.30	1.01	14.92
17	–20.00	1.11	16.50	–17.92	1.07	16.31	–16.40	1.05	16.08	–15.14	1.04	16.07	–13.92	1.06	17.11
18	–20.95	1.09	16.36	–18.76	1.13	18.63	–17.04	1.13	19.41	–15.67	1.09	18.60	–14.44	1.06	17.96
19	–21.59	1.06	16.18	–19.32	1.12	19.30	–17.84	1.05	17.42	–16.48	1.02	16.92	–15.29	1.00	16.64
20	–22.57	1.05	16.40	–20.45	1.02	16.39	–18.89	1.02	16.92	–17.53	1.01	17.10	–16.22	1.03	17.91

Table 3b

F_T(ℓ+1|ℓ) Distributions, ℓ=2.

q	ε=0.05			ε=0.10			ε=0.15			ε=0.20
	θ₀	θ₁	η	θ₀	θ₁	η	θ₀	θ₁	η	θ₀	θ₁	η
1	–3.15	1.22	5.34	–2.66	1.20	5.08	–2.31	1.18	4.82	–2.00	1.18	4.69
2	–5.05	1.30	7.52	–4.30	1.27	7.25	–3.76	1.27	7.18	–3.36	1.25	6.86
3	–6.90	1.22	7.66	–6.04	1.22	7.69	–5.37	1.22	7.75	–4.79	1.23	7.88
4	–8.48	1.24	8.86	–7.54	1.22	8.55	–6.74	1.22	8.75	–6.08	1.22	8.72
5	–9.73	1.23	9.78	–8.64	1.22	9.70	–7.83	1.20	9.55	–7.06	1.23	10.11
6	–11.11	1.22	10.32	–9.95	1.16	9.61	–9.16	1.18	10.01	–8.45	1.14	9.42
7	–12.44	1.30	12.76	–11.10	1.25	12.01	–10.18	1.22	11.56	–9.30	1.19	11.25
8	–13.50	1.20	11.70	–12.16	1.12	10.44	–11.23	1.14	10.93	–10.34	1.12	10.77
9	–14.99	1.23	12.91	–13.43	1.17	12.05	–12.37	1.19	12.57	–11.38	1.14	11.83
10	–15.93	1.19	12.77	–14.40	1.15	12.33	–13.30	1.12	11.90	–12.34	1.15	12.89
11	–17.25	1.14	12.12	–15.66	1.17	13.34	–14.47	1.17	13.63	–13.32	1.20	14.78
12	–18.09	1.14	13.07	–16.39	1.11	12.93	–15.23	1.14	13.90	13.99	1.18	15.27
13	–19.30	1.15	13.83	–17.59	1.13	13.80	–16.22	1.12	14.19	–15.16	1.13	14.55
14	–20.27	1.12	13.76	–18.51	1.12	14.29	–17.24	1.15	15.56	–16.03	1.14	15.68
15	–21.45	1.12	14.21	–19.50	1.11	14.77	–18.15	1.13	15.69	–16.93	1.12	15.79
16	–22.29	1.12	15.08	–20.47	1.14	16.20	–18.70	1.08	15.05	–17.47	1.07	15.14
17	–23.40	1.12	15.42	–21.36	1.12	16.25	–19.69	1.08	15.81	–18.33	1.08	16.04
18	–24.34	1.10	15.29	–22.56	1.13	17.16	–20.96	1.13	17.70	–19.42	1.13	18.29
19	–24.91	1.07	15.11	–23.26	1.12	17.50	–21.40	1.06	16.22	–19.96	1.08	17.53
20	–25.70	1.04	14.72	–23.61	1.01	14.70	–22.08	1.00	14.81	–20.98	1.02	15.70

Table 3c

F_T(ℓ+1|ℓ) Distributions, ℓ=3.

q	ε=0.05			ε=0.10			ε=0.15
	θ₀	θ₁	η	θ₀	θ₁	η	θ₀	θ₁	η
1	–3.77	1.26	5.81	–3.22	1.25	5.60	–2.72	1.27	5.67
2	–6.05	1.31	7.60	–5.21	1.30	7.49	–4.64	1.30	7.50
3	–7.95	1.23	7.68	–7.12	1.21	7.39	–6.45	1.20	7.40
4	–9.63	1.23	8.47	–8.68	1.19	8.03	–7.94	1.23	8.58
5	–11.11	1.28	10.17	–9.98	1.22	9.42	–9.11	1.21	9.41
6	–12.59	1.27	10.81	–11.29	1.19	9.77	–10.56	1.20	9.96
7	–14.28	1.34	12.76	–12.79	1.26	11.74	–11.82	1.23	11.30
8	–15.30	1.31	13.34	–13.73	1.20	11.58	–12.75	1.15	10.82
9	–16.82	1.22	11.98	–15.18	1.19	11.81	–14.17	1.19	12.01
10	–17.73	1.19	12.20	–16.18	1.17	12.38	–15.07	1.16	12.40
11	–18.79	1.11	10.91	–17.58	1.16	12.46	–16.41	1.16	12.90
12	–19.83	1.13	12.12	–18.32	1.14	12.89	–17.24	1.16	13.56
13	–21.37	1.16	13.40	–19.63	1.13	13.19	–18.18	1.11	13.12
14	–22.09	1.10	12.50	–20.40	1.10	13.13	–19.37	1.12	13.89
15	–23.33	1.11	13.19	–21.63	1.12	13.97	–20.38	1.13	14.70
16	–24.51	1.13	14.50	–22.82	1.15	15.71	–21.04	1.12	15.36
17	–25.57	1.11	14.18	–23.78	1.13	15.48	–22.11	1.12	15.84
18	–26.50	1.09	14.36	–25.00	1.12	15.81	–23.55	1.14	16.78
19	–27.03	1.06	14.03	–25.70	1.11	16.12	–23.76	1.06	15.33
20	–28.00	1.05	14.26	–25.85	1.03	14.35	–24.20	1.00	13.94

Table 3d

F_T(ℓ+1|ℓ) Distributions, ℓ=4.

q	ε=0.05			ε=0.10			ε=0.15
	θ₀	θ₁	η	θ₀	θ₁	η	θ₀	θ₁	η
1	–4.33	1.29	6.07	–3.73	1.29	5.94	–3.24	1.30	5.99
2	–6.85	1.33	7.64	–6.00	1.31	7.57	–5.41	1.31	7.46
3	–8.75	1.22	7.48	–7.89	1.20	7.30	–7.26	1.21	7.34
4	–10.56	1.29	9.20	–9.47	1.17	7.67	–8.85	1.21	8.24
5	–12.24	1.30	10.32	–11.02	1.24	9.47	–10.11	1.22	9.40
6	–13.82	1.33	11.70	–12.38	1.22	10.07	–11.64	1.22	9.99
7	–15.71	1.35	12.60	–14.08	1.27	11.52	–13.07	1.24	11.23
8	–16.80	1.32	13.04	–15.12	1.26	12.46	–13.99	1.22	11.84
9	–18.03	1.20	11.07	–16.49	1.19	11.41	–15.48	1.19	11.56
10	–19.06	1.18	11.53	–17.51	1.18	12.13	–16.54	1.21	12.87
11	–19.80	1.09	10.26	–18.88	1.14	11.63	–17.84	1.15	12.04
12	–21.20	1.13	11.90	–19.79	1.14	12.46	–18.75	1.16	13.11
13	–22.89	1.17	12.98	–21.13	1.14	12.80	–19.55	1.09	12.25
14	–23.41	1.09	11.88	–21.85	1.10	12.63	–20.70	1.09	12.49
15	–24.65	1.10	12.58	–23.25	1.13	13.66	–21.99	1.12	13.76
16	–26.19	1.14	13.98	–24.63	1.16	15.20	–22.93	1.15	15.54
17	–27.02	1.09	13.20	–25.48	1.12	14.49	–23.97	1.12	15.16
18	–28.10	1.09	13.72	–26.58	1.10	14.46	–25.38	1.13	15.59
19	–28.42	1.04	13.07	–27.36	1.10	15.07	–25.43	1.05	14.35
20	–29.77	1.06	13.99	–27.76	1.05	14.45	–25.79	1.00	13.32

Table 3e

F_T(ℓ+1|ℓ) Distributions, ℓ=5.

q	ε=0.05			ε=0.10
	θ₀	θ₁	η	θ₀	θ₁	η
1	–4.82	1.31	6.22	–4.17	1.31	6.18
2	–7.50	1.33	7.61	–6.64	1.32	7.58
3	–9.36	1.21	7.31	–8.52	1.20	7.15
4	–11.39	1.34	9.82	–10.18	1.22	8.24
5	–13.19	1.32	10.39	–11.86	1.25	9.49
6	–14.92	1.40	12.62	–13.34	1.28	10.79
7	–16.85	1.35	12.17	–15.12	1.28	11.35
8	–18.00	1.32	12.74	–16.26	1.28	12.42
9	–18.91	1.18	10.45	–17.46	1.17	10.73
10	–20.05	1.17	11.05	–18.66	1.19	12.01
11	–20.40	1.06	9.60	–19.86	1.14	11.25
12	–22.38	1.14	11.78	–20.85	1.13	11.78
13	–23.94	1.15	12.30	–22.26	1.13	12.23
14	–24.46	1.09	11.54	–23.10	1.11	12.53
15	–25.89	1.11	12.43	–24.54	1.13	13.37
16	–27.52	1.14	13.59	–25.97	1.15	14.37
17	–28.09	1.08	12.54	–26.73	1.11	13.67
18	–29.18	1.08	12.85	–27.62	1.08	13.34
19	–29.53	1.04	12.37	–28.52	1.08	14.04
20	–31.17	1.07	13.71	–29.22	1.06	14.13

Table 3f

F_T(ℓ+1|ℓ) Distributions, ℓ=6.

q	ε=0.05			ε=0.10
	θ₀	θ₁	η	θ₀	θ₁	η
1	–5.24	1.32	6.31	–4.57	1.33	6.36
2	–8.06	1.32	7.47	–7.20	1.32	7.56
3	–9.82	1.19	7.08	–9.02	1.19	7.05
4	–12.16	1.36	9.88	–10.84	1.26	8.70
5	–13.99	1.33	10.29	–12.56	1.26	9.50
6	–15.93	1.42	12.70	–14.20	1.33	11.50
7	–17.79	1.34	11.67	–16.02	1.29	11.39
8	–18.93	1.31	12.21	–17.24	1.29	12.32
9	–19.63	1.16	9.92	–18.17	1.15	10.12
10	–20.89	1.17	10.86	–19.60	1.19	11.77
11	–21.16	1.07	9.58	–20.60	1.13	10.82
12	–23.28	1.14	11.40	–21.76	1.13	11.52
13	–24.70	1.14	11.65	–23.10	1.12	11.68
14	–25.43	1.10	11.46	–24.26	1.13	12.71
15	–26.92	1.12	12.26	–25.52	1.12	12.84
16	–28.68	1.15	13.42	–27.14	1.16	14.07
17	–28.78	1.06	11.78	–27.68	1.10	13.05
18	–29.89	1.06	11.98	–28.48	1.07	12.64
19	–30.35	1.02	11.75	–29.23	1.05	12.89
20	–32.21	1.07	13.19	–30.42	1.07	13.89

Table 3g

F_T(ℓ+1|ℓ) Distributions, ℓ=7.

q	ε=0.05			ε=0.10
	θ₀	θ₁	η	θ₀	θ₁	η
1	–5.63	1.31	6.26	–4.96	1.33	6.36
2	–8.52	1.32	7.41	–7.68	1.32	7.44
3	–10.25	1.20	7.14	–9.41	1.17	6.84
4	–12.83	1.37	9.93	–11.43	1.27	8.77
5	–14.66	1.32	10.00	–13.15	1.26	9.39
6	–16.81	1.43	12.51	–15.02	1.38	12.31
7	–18.55	1.33	11.17	–16.80	1.30	11.35
8	–19.70	1.30	11.66	–18.07	1.29	12.10
9	–20.27	1.16	9.62	–18.79	1.14	9.83
10	–21.58	1.17	10.63	–20.39	1.20	11.59
11	–22.14	1.10	10.11	–21.15	1.11	10.40
12	–24.11	1.15	11.28	–22.51	1.12	11.13
13	–25.26	1.12	11.11	–23.62	1.09	10.92
14	–26.20	1.10	11.24	–25.26	1.15	12.81
15	–27.86	1.13	12.23	–26.35	1.12	12.45
16	–29.60	1.15	13.09	–28.04	1.15	13.61
17	–29.18	1.04	10.98	–28.38	1.08	12.41
18	–30.66	1.06	11.66	–29.19	1.06	12.10
19	–31.09	1.02	11.40	–29.76	1.03	11.96
20	–32.98	1.06	12.70	–31.41	1.07	13.62

Table 3h

F_T(ℓ+1|ℓ) Distributions, ℓ=8.

q	ε=0.05
	θ₀	θ₁	η
1	–5.99	1.29	6.08
2	–8.93	1.32	7.35
3	–10.66	1.20	7.13
4	–13.44	1.39	9.99
5	–15.24	1.32	9.79
6	–17.58	1.44	12.41
7	–19.17	1.31	10.74
8	–20.39	1.30	11.37
9	–20.81	1.15	9.40
10	–22.00	1.15	10.11
11	–23.09	1.14	10.67
12	–24.83	1.15	11.18
13	–25.67	1.10	10.55
14	–27.12	1.12	11.48
15	–28.73	1.14	12.25
16	–30.33	1.15	12.67
17	–29.48	1.02	10.37
18	–31.18	1.05	11.20
19	–31.68	1.02	11.10
20	–33.62	1.05	12.29

Table 4a

UDmax Distributions.

q	ε=0.05			ε=0.10			ε=0.15			ε=0.20			ε=0.25
	θ₀	θ₁	η	θ₀	θ₁	η	θ₀	θ₁	η	θ₀	θ₁	η	θ₀	θ₁	η
1	–4.73	1.45	4.36	–3.04	1.34	4.06	–2.12	1.26	3.76	–1.49	1.18	3.39	–1.06	1.11	2.97
2	–6.71	1.34	4.86	–4.68	1.24	4.73	–3.47	1.18	4.62	–2.56	1.14	4.52	–1.85	1.09	4.28
3	–8.19	1.23	4.99	–5.96	1.18	5.27	–4.55	1.15	5.51	–3.44	1.12	5.59	–2.52	1.08	5.47
4	–9.56	1.20	5.42	–7.20	1.15	5.85	–5.59	1.13	6.24	–4.32	1.10	6.36	–3.22	1.08	6.45
5	–10.68	1.15	5.72	–8.21	1.11	6.18	–6.48	1.07	6.45	–5.10	1.06	6.87	–3.85	1.07	7.30
6	–11.89	1.13	6.03	–9.28	1.09	6.66	–7.48	1.08	7.22	–5.94	1.07	7.76	–4.52	1.06	8.17
7	–12.88	1.10	6.32	–10.05	1.05	6.89	–8.18	1.05	7.64	–6.55	1.03	8.10	–5.04	1.02	8.60
8	–13.80	1.05	6.22	–11.01	1.03	7.10	–9.01	1.01	7.79	–7.34	1.01	8.49	–5.74	1.02	9.27
9	–14.98	1.06	6.81	–11.94	1.03	7.69	–9.87	1.02	8.58	–8.12	1.03	9.46	–6.36	1.02	10.16
10	–15.69	1.01	6.59	–12.80	1.00	7.72	–10.69	1.01	8.93	–8.77	1.01	9.87	–6.80	1.03	11.18
11	–16.64	1.00	6.91	–13.60	0.99	8.27	–11.41	1.01	9.73	–9.37	1.01	10.69	–7.36	1.02	11.76
12	–17.36	0.96	6.75	–14.32	0.96	8.08	–12.11	0.95	9.10	–10.23	0.96	10.03	–8.22	0.98	11.43
13	–18.44	0.97	7.26	–15.16	0.95	8.32	–12.83	0.94	9.27	–10.78	0.96	10.85	–8.69	0.98	12.25
14	–18.92	0.93	7.14	–15.63	0.91	8.29	–13.44	0.93	9.87	–11.39	0.94	10.90	–9.14	1.00	13.50
15	–19.89	0.93	7.36	–16.62	0.92	8.71	–14.33	0.95	10.55	–12.10	0.95	11.79	–9.94	0.96	13.18
16	–20.30	0.90	7.22	–17.20	0.90	8.85	–14.87	0.91	10.16	–12.71	0.94	12.03	–10.48	0.94	13.24
17	–21.44	0.91	7.67	–18.01	0.90	9.34	–15.43	0.89	10.27	–13.26	0.92	12.07	–11.00	0.95	14.11
18	–22.11	0.89	7.70	–18.64	0.88	9.28	–16.19	0.89	10.64	–13.93	0.91	12.45	–11.49	0.95	14.95
19	–22.16	0.85	7.25	–19.02	0.85	8.84	–16.83	0.89	11.09	–14.49	0.92	13.21	–12.13	0.92	14.59
20	–23.56	0.87	7.96	–20.00	0.86	9.50	–17.61	0.90	11.89	–15.26	0.93	14.00	–12.64	0.96	16.38

Table 4b

WDmax Distributions.

q	ε=0.05			ε=0.10			ε=0.15			ε=0.20			ε=0.25
	θ₀	θ₁	η	θ₀	θ₁	η	θ₀	θ₁	η	θ₀	θ₁	η	θ₀	θ₁	η
1	–3.15	1.45	5.06	–2.47	1.38	4.59	–2.11	1.31	4.12	–1.53	1.24	3.71	–1.14	1.19	3.29
2	–5.05	1.34	6.03	–4.04	1.37	6.36	–3.33	1.33	6.15	–2.51	1.30	5.86	–1.84	1.17	5.02
3	–6.70	1.28	6.57	–5.49	1.23	6.39	–4.80	1.24	6.48	–3.55	1.19	6.42	–2.07	1.26	7.56
4	–8.35	1.43	9.06	–6.87	1.38	8.89	–5.74	1.44	9.97	–4.76	1.27	8.00	–3.01	1.26	8.70
5	–9.55	1.37	9.72	–8.13	1.21	7.92	–7.32	1.25	8.52	–5.18	1.30	10.05	–3.72	1.25	9.77
6	–10.83	1.38	10.91	–8.95	1.34	11.06	–7.86	1.28	10.38	–6.44	1.15	8.93	–4.63	1.12	9.26
7	–12.14	1.50	13.93	–10.14	1.40	12.99	–9.25	1.31	11.56	–6.25	1.37	14.13	–3.72	1.35	14.93
8	–13.19	1.36	12.65	–11.30	1.37	13.34	–9.58	1.46	15.75	–7.45	1.33	14.19	–3.30	1.43	18.48
9	–14.66	1.41	14.21	–12.51	1.41	14.98	–10.75	1.46	16.70	–6.99	1.43	18.27	–5.54	1.30	16.13
10	–15.12	1.13	9.96	–13.52	1.19	11.55	–11.98	1.30	14.56	–9.08	1.26	15.01	–4.75	1.37	19.82
11	–17.20	1.31	13.71	–14.25	1.46	18.58	–12.78	1.44	18.51	–9.10	1.37	18.97	–5.37	1.35	20.70
12	–16.90	1.09	10.47	–15.04	1.13	12.03	–13.48	1.20	14.31	–10.71	1.13	14.22	–8.72	1.06	13.41
13	–19.45	1.29	14.70	–17.23	1.25	14.64	–15.32	1.21	14.52	–12.29	1.08	12.87	–8.48	1.27	19.48
14	–19.70	1.26	15.46	–16.81	1.21	15.71	–15.13	1.20	16.14	–12.68	1.07	13.90	–9.59	1.10	16.29
15	–21.15	1.28	16.23	–17.79	1.23	16.90	–16.53	1.20	16.24	–12.37	1.26	20.32	–9.85	1.15	18.54
16	–22.33	1.30	17.45	–19.82	1.23	16.50	–17.98	1.35	20.84	–14.64	1.19	18.14	–11.07	1.16	19.25
17	–23.55	1.25	16.65	–20.90	1.27	18.31	–19.06	1.29	19.78	–15.09	1.24	20.41	–11.72	1.16	20.01
18	–25.01	1.26	17.13	–21.75	1.18	16.44	–20.21	1.26	19.49	–14.20	1.42	28.49	–12.53	1.17	20.98
19	–24.56	1.23	18.38	–21.76	1.10	15.48	–20.60	1.14	17.03	–16.77	1.05	16.31	–13.65	0.97	15.47
20	–26.56	1.20	17.01	–23.30	1.12	15.96	–21.75	1.18	18.40	–17.87	1.08	17.49	–14.14	1.11	20.23

Table 5

p-Value accuracy comparison of Hall and Sakkas (2011) (HS) and Bai and Perron (2003b) (BP).

k	q=1		q=5		q=10		q=15		q=20
	BP	HS	BP	HS	BP	HS	BP	HS	BP	HS
1	0.0383	0.0485	0.0540	0.0494	0.0526	0.0497	0.0768	0.0496	0.1620	0.0492
2	0.0524	0.0490	0.0653	0.0499	0.0622	0.0494	0.1096	0.0487	0.2993	0.0490
3	0.0487	0.0494	0.0608	0.0495	0.0562	0.0491	0.1130	0.0494	0.3588	0.0497
4	0.0577	0.0497	0.0624	0.0499	0.0532	0.0492	0.1156	0.0496	0.4059	0.0495
5	0.0767	0.0494	0.0684	0.0496	0.0527	0.0495	0.1198	0.0503	0.4460	0.0497
6	0.1161	0.0496	0.0815	0.0497	0.0550	0.0493	0.1271	0.0508	0.4876	0.0495
7	0.1912	0.0498	0.1034	0.0497	0.0610	0.0501	0.1388	0.0507	0.5305	0.0502
8	0.3269	0.0504	0.1383	0.0499	0.0714	0.0503	0.1543	0.0501	0.5767	0.0498
9	0.5419	0.0501	0.1913	0.0501	0.0861	0.0505	0.1751	0.0497	0.6242	0.0494

The authors acknowledge the support of the ESRC grant RES-062-23-1351. We would like to thank Jonathan Boyle and Robin Pinning of the University of Manchester Research Computing Services for their assistance in using Condor. We are also grateful to Denise Osborn for useful comments on various aspects of the work. This paper was presented at the 4th CSDA international conference on computational and financial econometrics (CFE ’10), 10-12 December 2010, London, UK.

¹
The error must also satisfy certain other conditions detailed in the sources mentioned below.
²
If p=0 then all parameters are regime specific and the model is said to exhibit pure structural change.
³
It should be noted that these limiting distributions only apply for the statistics based on F_T(λ₁,…,λ_k;q) if the regression error u_t is homoscedastic and serially uncorrelated. If the errors are heteroscedastic and/or serially correlated then the same limiting distributions hold for the analogous functions of Wald statistics that satisfactorily account for the error structure; see Bai and Perron (1998) for further details.
⁴
Such large values of ε are reported because Andrews (1993) allows for asymmetric trimming.
⁵
This methodology has also been employed by Sen and Hall (1999) for the distributions of their overidentifying restrictions test (O_T) and by Sen (1999) for the Ghysels, Guay, and Hall’s (1997) predictive test.
⁶
From the case of q=1, it can be seen that this case occurs for k=9. Further investigation reveals the main source of the distortion is Bai and Perron’s (2003b) response surface: the appropriate critical value reported in Bai and Perron (1998) is 5.20 which translates, using our response surfaces, to a p-value of 0.0758; the critical value predicted by Bai and Perron’s (2003b) response surface is 3.8023. In contrast, our simulations (based on an enhanced design) yield a critical value of 5.4210 for which our response surface gives a p-value of 0.05006.
⁷
In the original version of this paper, we restricted attention to q=1,5,10 but, given the nature of the results for those cases, a referee conjectured Bai and Perron’s (2003b) response surfaces would work well for cases q>10. While natural given results for q=1, 5, 10, this conjecture is erroneous as our results demonstrate. We therefore included these cases to avoid the reader making the same mistake. We attribute the inaccuracy of Bai and Perron’s (2003b) response surfaces to these cases to the fact that the surfaces were calculated using cases for which q≤10 only.
⁸
For completeness, we note that we performed similar calculations for ε=0.10,0.15,0.20,0.25 for q=1,5,10. The distortions are smaller than those reported for ε=0.05 but are nevertheless non-trivial in a number of circumstances. These results are available from the authors upon request.

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Published Online: 2013-03-15

Published in Print: 2013-07-01

Approximate p-Values of Certain Tests Involving Hypotheses About Multiple Breaks

Abstract

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References

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