Detection of Multiple Structural Breaks in Large Covariance Matrices

Figures & data

Table 1 Comparison of break detection results.

		Common components						Idiosyncratic error components
	Methods	$\#$ break(%)				${\text{ACU}}_1$(%)	${\text{ACU}}_2$(%)	$\#$ break(%)			${\text{ACU}}_1$(%)	${\text{ACU}}_2$(%)	${\text{ACU}}_3$(%)
		0	1	2	$>2$	${\eta }_1^c=133$	${\eta }_2^c=267$	$<3$	3	$>3$	${\eta }_1^e=100$	${\eta }_2^e=200$	${\eta }_3^e=300$
${\mathrm{\varrho }}_1^e=1$	W(S)BS-Cov	0	2	98	0	65	93	0	100	0	100	100	100
	BCF	12	86	2	0	0	78	0	100	0	100	100	100
${\mathrm{\varrho }}_1^e=0.5$	W(S)BS-Cov	0	2	98	0	64	95	5	95	0	100	96	99
	BCF	12	87	1	0	0	81	26	74	0	86	95	83
${\mathrm{\varrho }}_1^e=0.1$	W(S)BS-Cov	0	3	97	0	64	95	67	33	0	73	58	71
	BCF	11	88	1	0	0	81	100	0	0	3	14	1

Table 2 Detection results with breaks on weak factors.

		Common components						Idiosyncratic error components
	$\widehat{q}$	$\#$ break(%)				${\text{ACU}}_1$(%)	${\text{ACU}}_2$(%)	$\#$ break(%)				${\text{ACU}}_1$(%)	${\text{ACU}}_2$(%)
		0	1	2	$>2$	${\eta }_1^c=100$	${\eta }_2^c=300$	0	1	2	$>2$	${\eta }_1^e=150$	${\eta }_2^e=250$
Case 1	8.94	0	0	100	0	85	85	0	1	98	1	98	100
Case 2	8.92	0	0	100	0	87	93	0	0	100	0	100	100
Case 3	8.93	0	0	100	0	81	87	0	1	99	0	100	99
Case 4	8.72	3	2	95	0	84	84	0	0	95	5	89	98
Case 5	8.65	1	4	95	0	83	83	0	1	93	6	88	96
Case 6	8.02	21	10	69	0	73	73	0	1	73	26	78	74
Case 7	6.60	100	0	0	0	0	0	0	2	86	12	70	73
Case 8	6.84	100	0	0	0	0	0	0	0	92	8	91	89
Case 9	6.49	100	0	0	0	0	0	0	0	82	18	67	75

Fig. 1 The detected break times in the covariance structure of the 375 daily returns of the S&P 500 constituents from January 01, 2000 to February 28, 2021. The estimated break dates for common components are April 24, 2000, November 25, 2002, September 12, 2008, May 08, 2009 and March 06, 2020 (in red lines); and the estimated break dates for idiosyncratic components are August 06, 2002, September 11, 2008 and May 11, 2009 (in blue dotted lines).

Table 3 Detection results on calibrated data.

	W(S)BS-Cov 5.01 3.31		W(S)BS-Cov (fix $q_0=12$) 5.02 3.08
Mean ${\widehat{K}}_1$
Mean ${\widehat{K}}_2$
	$\text{ACU}(\%)$	$\text{RMSE}$	$\text{ACU}(\%)$	$\text{RMSE}$
Breaks in common components
24/04/2000	79	8.721	79	8.731
25/11/2002	88	5.725	85	6.813
12/09/2008	99	1.643	100	1.300
08/05/2009	99	2.502	99	2.240
06/03/2020	97	3.838	97	3.929
Breaks in idiosyncratic error components
06/08/2002	97	3.425	96	2.466
11/09/2008	100	1.503	100	1.562
11/05/2009	99	2.995	99	1.912

Fig. 2 Variation of eight commonly used risk factors from January 01, 2000 to February 28, 2021. The estimated break dates are April 19, 2001, July 01, 2008, September 12, 2008, May 13, 2009 and March 06, 2020 (in red lines).

Table 4 Canonical correlation between estimated factors and observed factors.

	Period 1	Period 2	Period 3	Period 4	Period 5	Period 6	Total
From	01/01/2000	25/04/2000	26/11/2002	13/09/2008	09/05/2009	07/03/2020	01/01/2000
To	24/04/2000	25/11/2002	12/09/2008	08/05/2009	06/03/2020	28/02/2021	28/02/2021
First	0.988	0.984	0.993	0.998	0.994	0.996	0.990
p-value 1:8	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
Second	0.964	0.898	0.873	0.947	0.861	0.949	0.850
p-value 2:8	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
Third	0.742	0.588	0.817	0.877	0.651	0.784	0.726
p-value 3:8	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
Fourth	0.605	0.508	0.627	0.760	0.553	0.577	0.554
p-value 4:8	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
Fifth	0.517	0.359	0.444	0.615	0.384	0.383	0.292
p-value 5:8	(0.007)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
Sixth	0.418	0.280	0.239	0.431	0.256	0.308	0.207
p-value 6:8	(0.129)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
Seventh	0.259	0.107	0.134	0.398	0.155	0.219	0.119
p-value 7:8	(0.605)	(0.255)	(0.000)	(0.000)	(0.000)	(0.007)	(0.000)
Eighth	0.051	0.027	0.087	0.256	0.022	0.160	0.039
p-value 8:8	(0.916)	(0.795)	(0.004)	(0.005)	(0.510)	(0.047)	(0.019)

NOTE: p-values are reported in the brackets using the F-approximations of Rao’s test statistics related to Wilks’ Lambda.