Singular Conditional Autoregressive Wishart Model for Realized Covariance Matrices

Figures & data

Table 1 Summary statistics for the realized variance (multiplied by 10⁴) of 50 assets in each of the 12 market sectors considered.

Sector	Nr.assets	Mean	Median	Std.dev.	Skewness	Kurtosis
Basic industries	1	17.23	8.47	24.45	4.34	33.21
Capital goods	4	17.68	7.61	59.31	82.02	9377.41
Consumer durables	1	10.37	2.61	75.15	54.69	3459.49
Consumer nondurables	2	15.90	2.96	130.01	83.19	7769.35
Consumer services	6	7.61	2.98	19.80	18.38	629.25
Energy	1	20.62	12.33	33.13	10.30	206.96
Finance	11	7.02	2.43	39.47	159.67	32358.94
Health care	11	11.75	4.48	36.99	63.56	8048.78
Miscellaneous	2	11.41	3.44	106.37	49.79	2823.32
Public utilities	1	4.11	1.42	8.59	8.30	125.28
Technology	9	10.06	4.23	48.09	146.27	26943.34
Transportation	1	16.24	3.73	63.71	18.54	476.35

Table 2 Summary statistics for the realized variance (multiplied by 10⁴) of 300 assets in each of the 12 market sectors considered.

Sector	Nr.assets	Mean	Median	Std.dev.	Skewness	Kurtosis
Basic industries	8	14.78	5.73	38.07	10.68	184.96
Capital goods	23	11.41	4.91	30.58	17.85	561.30
Consumer durables	8	8.37	3.61	70.77	63.84	4333.05
Consumer nondurables	12	6.98	2.91	88.50	107.42	13240.57
Consumer services	37	8.48	3.22	31.29	57.64	7008.07
Energy	6	21.40	10.64	40.97	8.74	130.46
Finance	63	8.64	3.18	80.44	209.24	51151.97
Health care	67	19.61	7.03	318.11	194.25	43606.03
Miscellaneous	10	11.27	4.96	59.19	79.04	8548.82
Public utilities	6	7.16	3.49	14.59	11.85	256.04
Technology	53	7.46	3.20	80.25	190.35	41194.67
Transportation	7	13.96	4.18	81.76	71.08	6526.02

Table 3 Summary of the rolling window forecasts for each respective model for 20 years of data (mid 1997 to mid 2017) on 50 assets proportionally distributed among the 12 sectors in NASDAQ classification, evaluated on a monthly basis.

Model	Parameters	${\text{FN}}_{\mathcal{l}}$	${\text{SD}}_{\text{EW},\mathcal{l}}$	${\text{SD}}_{\text{GMV},\mathcal{l}}$	${\text{EGMV}}_{\text{var},\mathcal{l}}$	${\text{EGMV}}_{\text{tu},\mathcal{l}}$	${\text{EGMV}}_{\text{lev},\mathcal{l}}$	Sharpe Ratio
SCAWCT (0, 1)	50	${0.0189}^{(\text{MSN})}$	$7.44e^{-3^{(\mathrm{M})}}$	${0.105}^{(\text{MSN})}$	$1.87e^{-2^{(\text{SN})}}$	${0.31}^{(\text{SN})}$	$\mathbf{1}.\mathbf{5}{\mathbf{8}}^{(\text{MSN})}$	0.014
SCAWCT (1, 1)	100	${0.0189}^{(\text{MSN})}$	$5.53e^{-3^{(\text{MSN})}}$	$1.92e^{-2^{(\text{MSN})}}$	$1.21e^{-3^{(\text{MSN})}}$	${0.70}^{(\mathrm{M})}$	${1.72}^{(\text{MS})}$	0.021
SCAW-SSCT (0, 1)	12	${0.0187}^{(\text{MSN})}$	$6.99e^{-3^{(\mathrm{M})}}$	${0.105}^{(\text{MSN})}$	$1.85e^{-2^{(\text{SN})}}$	${0.34}^{(\text{SN})}$	${1.59}^{(\text{MSN})}$	0.016
SCAW-SSCT (1, 1)	24	${0.0185}^{(\text{MSN})}$	$4.95e^{-3^{(\text{MSN})}}$	${0.107}^{(\text{MSN})}$	$1.76e^{-2^{(\text{MSN})}}$	0.65	${1.71}^{(\text{MS})}$	$\mathbf{0}.\mathbf{0}\mathbf{3}\mathbf{5}$
SCAW-SSCT (1, 2)	36	$\mathbf{0}.\mathbf{0}\mathbf{1}\mathbf{8}{\mathbf{3}}^{(\text{MSN})}$	$4.84e^{-3^{(\text{MSN})}}$	${0.106}^{(\text{MSN})}$	$1.83e^{-2^{(\text{MSN})}}$	0.54	${1.69}^{(\text{MS})}$	0.026
SCAW-SSCT (2, 2)	48	${0.0184}^{(\text{MSN})}$	$4.61e^{-3^{(\text{MSN})}}$	${0.105}^{(\text{MSN})}$	$1.81e^{-2^{(\text{MSN})}}$	${0.55}^{(\mathrm{M})}$	${1.69}^{(\text{MS})}$	0.026
SCAW-SS-HARCT (1, 5)	24	${0.0197}^{(\text{MSN})}$	$8.73e^{-3^{(\mathrm{M})}}$	${0.107}^{(\text{MSN})}$	$1.96e^{-2^{(\text{SN})}}$	${0.40}^{(\text{SN})}$	${1.61}^{(\text{MSN})}$	0.019
SCAW-SS-HARCT (1, 20)	24	${0.0201}^{(\text{MSN})}$	$1.00e^{-2^{(\mathrm{M})}}$	${0.109}^{(\text{MSN})}$	$1.98e^{-2^{(\text{SN})}}$	${0.33}^{(\text{SN})}$	${1.58}^{(\text{MSN})}$	0.020
SCAW-SCALAR(0, 1)	1	${0.0187}^{(\text{MSN})}$	$6.88e^{-3^{(\mathrm{M})}}$	${0.104}^{(\text{MSN})}$	$1.85e^{-2^{(\text{SN})}}$	${0.36}^{(\text{SN})}$	${1.59}^{(\text{MSN})}$	0.018
SCAW-SCALAR(1, 1)	2	${0.0184}^{(\text{MSN})}$	$4.71e^{-3^{(\text{MSN})}}$	${0.103}^{(\text{MSN})}$	$1.79e^{-2^{(\text{SN})}}$	${0.31}^{(\text{SN})}$	${1.65}^{(\text{MSN})}$	$\mathbf{0}.\mathbf{0}\mathbf{3}\mathbf{5}$
SCAW-SCALAR(1, 2)	3	${0.0184}^{(\text{MSN})}$	$4.69e^{-3^{(\text{MSN})}}$	${0.103}^{(\text{MSN})}$	$1.78e^{-2^{(\text{SN})}}$	${0.29}^{(\text{SN})}$	${1.65}^{(\text{MSN})}$	$\mathbf{0}.\mathbf{0}\mathbf{3}\mathbf{5}$
SCAW-SCALAR(2, 2)	4	$\mathbf{0}.\mathbf{0}\mathbf{1}\mathbf{8}{\mathbf{3}}^{(\text{MSN})}$	$\mathbf{4}.\mathbf{5}\mathbf{3}{\mathbf{e}}^{-{\mathbf{3}}^{(\text{MSN})}}$	$\mathbf{0}.\mathbf{1}\mathbf{0}{\mathbf{2}}^{(\text{MSN})}$	$\mathbf{1}.\mathbf{7}\mathbf{1}{\mathbf{e}}^{-{\mathbf{2}}^{(\text{SN})}}$	${0.32}^{(\text{SN})}$	${1.66}^{(\text{MSN})}$	0.031
SCAW-SCALAR-HAR(1, 5)	2	${0.0196}^{(\text{MSN})}$	$8.51e^{-3^{(\mathrm{M})}}$	${0.106}^{(\text{MSN})}$	$1.88e^{-2^{(\text{SN})}}$	${0.40}^{(\text{SN})}$	${1.61}^{(\text{MSN})}$	0.020
SCAW-SCALAR-HAR(1, 20)	2	${0.0199}^{(\text{MSN})}$	$9.86e^{-3^{(\mathrm{M})}}$	${1.07}^{(\text{MSN})}$	$1.95e^{-2^{(\text{SN})}}$	${0.35}^{(\text{SN})}$	${1.59}^{(\text{MSN})}$	0.020
MGARCHCT (0, 1)	50	0.0207	$1.18e^{-2}$	0.121	$1.79e^{-2}$	0.19	2.07	0.017
MGARCHCT (1, 1)	100	0.0219	$1.11e^{-2}$	0.148	$3.66e^{-2}$	0.84	2.34	0.011
MGARCH-SSCT (0, 1)	12	0.0207	$1.20e^{-2}$	0.121	$1.77e^{-2}$	0.13	2.06	0.017
MGARCH-SSCT (1, 1)	24	0.0215	$1.13e^{-2}$	0.131	$2.19e^{-2}$	0.36	2.15	0.007
MGARCH-SSCT (1, 2)	36	0.0215	$1.05e^{-2}$	0.128	$2.05e^{-2}$	0.42	2.14	0.010
MGARCH-SSCT (2, 2)	48	0.0212	$1.00e^{-2}$	0.136	$2.34e^{-2}$	0.59	2.20	0.019
MGARCH-SS-HARCT (1, 5)	24	0.0207	$1.20e^{-2}$	0.121	$1.79e^{-2}$	0.14	2.01	0.017
MGARCH-SS-HARCT (1, 20)	24	0.0207	$1.20e^{-2}$	0.121	$1.79e^{-3}$	0.14	2.06	0.017
MGARCH-SCALAR(0, 1)	1	0.0207	$1.19e^{-2}$	0.121	$1.79e^{-2}$	0.14	2.06	0.017
MGARCH-SCALAR(1, 1)	2	0.0207	$1.15e^{-2}$	0.121	$1.79e^{-2}$	0.14	2.06	0.015
MGARCH-SCALAR(1, 2)	3	0.0206	$1.19e^{-2}$	0.121	$1.78e^{-2}$	$\mathbf{0}.\mathbf{1}\mathbf{2}$	2.06	0.017
MGARCH-SCALAR(2, 2)	4	0.0206	$1.19e^{-2}$	0.121	$1.79e^{-2}$	$\mathbf{0}.\mathbf{1}\mathbf{2}$	2.06	0.017
MGARCH-SCALAR-HAR(1, 5)	2	0.0207	$1.20e^{-2}$	0.121	$1.79e^{-2}$	0.14	2.06	0.017
MGARCH-SCALAR-HAR(1, 20)	2	0.0207	$1.20e^{-2}$	0.121	$1.79e^{-2}$	0.14	2.06	0.017
DCC-S	153	0.0234	$6.04e^{-3}$	0.148	$3.03e^{-2}$	0.52	1.79	0.019
DCC-NL	153	0.0229	$5.79e^{-3}$	0.144	$2.94e^{-2}$	0.47	1.67	0.023

NOTE: The one-step ahead forecasts ($\mathcal{l}=1$) are then evaluated toward the observed realized covariance with the measures: Frobenius norm (${\text{FN}}_{\mathcal{l}}$), discrepancy in the standard deviation of the equally weighted portfolio (${\text{SD}}_{\text{EW},\mathcal{l}}$), forecasted standard deviation of the GMV portfolio (${\text{SD}}_{\text{GMV},\mathcal{l}}$), variance of the empirical return computed from the forecasted weights of the GMV portfolio (${\text{EGMV}}_{\text{var},\mathcal{l}}$), the turnover (${\text{EGMV}}_{\text{tu},\mathcal{l}}$), and the leverage (${\text{EGMV}}_{\text{lev},\mathcal{l}}$). Each SCAW model’s forecast is compared with the forecast of the equivalent MGARCH model, the DCC-S model and the DCC-NL model. The performance of the SCAW models is also compared with equivalent MGARCH model, the DCC-S model and the DCC-NL model through the test based on prewhitened ${\text{HAC}}_{\text{PW}}$ method described in Ledoit and Wolf (2011) in the case of ${\text{EGMV}}_{\text{var},\mathcal{l}}$ criteria and the pairwise forecast test for other measures. Letters M, S, and N indicate statistical significance at 5% or less in favor of the SCAW model when compared to MGARCH, DCC-S, and DCC-NL models, respectively. In the last column, the reward to risk through the Sharpe ratio is presented. The strategy with the best performance with respect to each model is highlighted bold in each row.

Table 4 Ranking among models computed by using the MCS test (see Hansen, Lunde, and Nason 2011) with three loss functions (${\text{FN}}_{\mathcal{l}},{\text{SD}}_{\text{EW},\mathcal{l}}$, and ${\text{SD}}_{\text{GMV},\mathcal{l}}$, respectively) computed for rolling window forecasting in the case of data consisting of 50 stocks.

Model	$\text{MCS}\mathrm{ }-\mathrm{ }\text{Ranking}$
	${\text{FN}}_{\mathcal{l}}$	${\text{SD}}_{\text{EW},\mathcal{l}}$	${\text{SD}}_{\text{GMV},\mathcal{l}}$
SCAWCT (0, 1)	10	–	–
SCAWCT (1, 1)	9	–	–
SCAW-SSCT (0, 1)	8	–	–
SCAW-SSCT (1, 1)	6	–
SCAW-SSCT (1, 2)	1	5	–
SCAW-SSCT (2, 2)	5	2	–
SCAW-SS-HARCT (1, 5)	–	–	–
SCAW-SS-HARCT (1, 20)	–	–	–
SCAW-SCALAR(0, 1)	7	–	–
SCAW-SCALAR(1, 1)	3	4	–
SCAW-SCALAR(1, 2)	2	3	–
SCAW-SCALAR(2, 2)	4	1	1
SCAW-SCALAR-HAR(1, 5)	–	–	–
SCAW-SCALAR-HAR(1, 20)	–	–	–
MGARCHCT (0, 1)	–	–	–
MGARCHCT (1, 1)	–	–	–
MGARCH-SSCT (0, 1)	–	–	–
MGARCH-SSCT (1, 1)	–	–	–
MGARCH-SSCT (1, 2)	–	–	–
MGARCH-SSCT (2, 2)	–	–	–
MGARCH-SS-HARCT (1, 5)	–	–	–
MGARCH-SS-HARCT (1, 20)	–	–	–
MGARCH-SCALAR(0, 1)	–	–	–
MGARCH-SCALAR(1, 1)	–	–	–
MGARCH-SCALAR(1, 2)	–	–	–
MGARCH-SCALAR(2, 2)	–	–	–
MGARCH-SCALAR-HAR(1, 5)	–	–	–
MGARCH-SCALAR-HAR(1, 20)	–	–	–
DCC-S	–	–	–
DCC-NL	–	–	–

NOTE: The symbol ‘-’ indicates the absence of the model in the confidence set with respect to the considered loss function. The numbers are the ranking of the models, which are included in the 90% confidence set.

Table 5 Ranking among models computed by using the MCS test (see Hansen, Lunde, and Nason 2011) with three loss functions (${\text{FN}}_{\mathcal{l}}$/${\text{SD}}_{\text{EW},\mathcal{l}}$/${\text{SD}}_{\text{GMV},\mathcal{l}}$) computed for fixed window estimation with forecasting horizon $\mathcal{l}=1,5,10$ in the case of data consisting of 300 stocks.

Model	$\text{MCS}\mathrm{ }-\mathrm{ }\text{Ranking}$
	$\mathcal{l}=1$	$\mathcal{l}=5$	$\mathcal{l}=10$
SCAW-SSCT (0, 1)	2∕.-∕.-	5∕.-∕.-	6∕.-∕.-
SCAW-SSCT (1, 1)	3∕.-∕.-	1∕.-∕.-	1∕.-∕.-
SCAW-SSCT (1, 2)	8∕.-∕.-	11∕.-∕.-	10∕.-∕.-
SCAW-SSCT (2, 2)	10∕.-∕.-	14∕.-∕.	12∕.-∕.-
SCAW-SS-HARCT (1, 5)	19∕.-∕.-	3∕.-∕.-	4∕.-∕.-
SCAW-SS-HARCT (1, 20)	-∕.-∕.-	2∕.-∕.3	2∕.3∕.-
SCAW-SCALAR(0, 1)	1∕.-∕.-	6∕.-∕.-	7∕.-∕.-
SCAW-SCALAR(1, 1)	6∕.-∕.-	7∕.-∕.-	3∕.-∕.-
SCAW-SCALAR(1, 2)	9∕.-∕.-	9∕.-∕.2	9∕.1∕.2
SCAW-SCALAR(2, 2)	7∕.1∕.1	8∕.1∕.1	8∕.1∕.2
SCAW-SCALAR-HAR(1, 5)	11∕.-∕.-	4∕.-∕.-	5∕.-∕.-
SCAW-SCALAR-HAR(1, 20)	-∕.-∕.-	10∕.-∕.-	11∕.-∕.-
MGARCH-SSCT (0, 1)	20∕.-∕.-	-∕.-∕.-	-∕.-∕.-
MGARCH-SSCT (1, 1)	17∕.-∕.-	-∕.-∕.-	-∕.-∕.-
MGARCH-SSCT (1, 2)	12∕.-∕.-	-∕.-∕.-	-∕.-∕.-
MGARCH-SSCT (2, 2)	13∕.-∕.-	-∕.-∕.-	-∕.-∕.-
MGARCH-SS-HARCT (1, 5)	23∕.-∕.-	-∕.-∕.-	-∕.-∕.-
MGARCH-SS-HARCT (1, 20)	-∕.-∕.-	-∕.-∕.-	-∕.-∕.-
MGARCH-SCALAR(0, 1)	18∕.-∕.-	-∕.-∕.-	-∕.-∕.-
MGARCH-SCALAR(1, 1)	15∕.-∕.-	-∕.-∕.-	-∕.-∕.-
MGARCH-SCALAR(1, 2)	16∕.-∕.-	-∕.-∕.-	-∕.-∕.-
MGARCH-SCALAR(2, 2)	14∕.-∕.-	-∕.-∕.-	-∕.-∕.-
MGARCH-SCALAR-HAR(1, 5)	21∕.-∕.-	-∕.-∕.-	-∕.-∕.-
MGARCH-SCALAR-HAR(1, 20)	22∕.-∕.-	-∕.-∕.-	-∕.-∕.-
DCC-S	4∕.-∕.-	12∕.-∕.-	13∕.-∕.-
DCC-NL	5∕.-∕.-	13∕.-∕.-	14∕.-∕.-

NOTE: The symbol ‘-’ indicates the absence of the model in the confidence set with respect to the considered loss function. The numbers are the ranking of the models, which are included in the 90% confidence set.

Ledoit, O., and Wolf, M. (2011), Robust Performances Hypothesis Testing With the Variance, Wilmott, 2011, 86–89.

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