Table 9 Open in new tab The GMV...

		EWMA	DCC	HDRD	W	R	iR	F	FR
Panel A: MCD/PFE/PG/WMT/XOM

Mean ret		4.414	5.961	5.539	5.432	5.385	5.259	5.297	5.319
${\hat{σ}}_{p}$		10.583	10.892	10.605	10.568	10.569	10.564	10.564	10.563
mcs p-value		(0.01)	(0.00)	(0.01)	(0.01)	(0.01)	(0.40)	(0.40)	(1.00)
StDev ret		10.383	10.600	10.307	10.341	10.348	10.327	10.326	10.317
TO		0.031	0.110	0.152	0.108	0.106	0.089	0.092	0.083
CO		0.500	0.537	0.506	0.501	0.501	0.500	0.500	0.500
SP		−0.005	−0.009	−0.003	−0.005	−0.005	−0.005	−0.005	−0.005

$c = 0 %$	Sharpe	0.425	0.562	0.537	0.525	0.520	0.509	0.513	0.516
	$Δ_{1}$	182	−122	−47	−22	−12	12	5
	$Δ_{10}$	195	−68	−49	−17	−7	14	6
$c = 1 %$	Sharpe	0.424	0.560	0.534	0.523	0.518	0.507	0.511	0.514
	$Δ_{1}$	180	−121	−43	−21	−11	13	5
	$Δ_{10}$	192	−66	−45	−16	−5	15	7
$c = 2 %$	Sharpe	0.424	0.557	0.530	0.520	0.515	0.505	0.509	0.512
	$Δ_{1}$	177	−120	−40	−20	−10	13	5
	$Δ_{10}$	190	−65	−42	−15	−4	15	7

EWMA

DCC

HDRD

Panel A: MCD/PFE/PG/WMT/XOM

Mean ret

4.414

5.961

5.539

5.432

5.385

5.259

5.297

5.319

{\hat{σ}}_{p}

10.583

10.892

10.605

10.568

10.569

10.564

10.563

mcs p-value

(0.01)

(0.00)

(0.01)

(0.40)

(1.00)

StDev ret

10.383

10.600

10.307

10.341

10.348

10.327

10.326

10.317

0.031

0.110

0.152

0.108

0.106

0.089

0.092

0.083

0.500

0.537

0.506

0.501

0.500

−0.005

−0.009

−0.003

−0.005

c = 0 %

Sharpe

0.425

0.562

0.537

0.525

0.520

0.509

0.513

0.516

Δ_{1}

182

−122

−47

−22

−12

Δ_{10}

195

−68

−49

−17

−7

c = 1 %

Sharpe

0.424

0.560

0.534

0.523

0.518

0.507

0.511

0.514

Δ_{1}

180

−121

−43

−21

−11

Δ_{10}

192

−66

−45

−16

−5

c = 2 %

Sharpe

0.424

0.557

0.530

0.520

0.515

0.505

0.509

0.512

Δ_{1}

177

−120

−40

−20

−10

Δ_{10}

190

−65

−42

−15

−4

		EWMA	DCC	HDRD	W	R	iR	F	FR
Panel A: MCD/PFE/PG/WMT/XOM

Mean ret		4.414	5.961	5.539	5.432	5.385	5.259	5.297	5.319
${\hat{σ}}_{p}$		10.583	10.892	10.605	10.568	10.569	10.564	10.564	10.563
mcs p-value		(0.01)	(0.00)	(0.01)	(0.01)	(0.01)	(0.40)	(0.40)	(1.00)
StDev ret		10.383	10.600	10.307	10.341	10.348	10.327	10.326	10.317
TO		0.031	0.110	0.152	0.108	0.106	0.089	0.092	0.083
CO		0.500	0.537	0.506	0.501	0.501	0.500	0.500	0.500
SP		−0.005	−0.009	−0.003	−0.005	−0.005	−0.005	−0.005	−0.005

$c = 0 %$	Sharpe	0.425	0.562	0.537	0.525	0.520	0.509	0.513	0.516
	$Δ_{1}$	182	−122	−47	−22	−12	12	5
	$Δ_{10}$	195	−68	−49	−17	−7	14	6
$c = 1 %$	Sharpe	0.424	0.560	0.534	0.523	0.518	0.507	0.511	0.514
	$Δ_{1}$	180	−121	−43	−21	−11	13	5
	$Δ_{10}$	192	−66	−45	−16	−5	15	7
$c = 2 %$	Sharpe	0.424	0.557	0.530	0.520	0.515	0.505	0.509	0.512
	$Δ_{1}$	177	−120	−40	−20	−10	13	5
	$Δ_{10}$	190	−65	−42	−15	−4	15	7

Panel B: AA/AXP/BA/CAT/GE/HD/HON/IBM/JPM/KO/MCD/PFE/PG/WMT/XOM

Mean ret

6.755

8.943

7.583

7.167

7.179

7.038

7.061

7.055

{\hat{σ}}_{p}

9.651

10.343

9.737

9.669

9.672

9.661

9.659

9.660

mcs p-value

(1.00)

(0.00)

(0.17)

(0.52)

StDev ret

10.149

10.681

10.317

10.144

10.137

10.113

10.128

10.108

0.063

0.213

0.331

0.166

0.154

0.132

0.145

0.123

0.416

0.492

0.427

0.426

0.425

0.424

−0.104

−0.181

−0.110

−0.109

−0.108

−0.106

−0.107

−0.106

c = 0 %

Sharpe

0.666

0.837

0.735

0.707

0.708

0.696

0.697

0.698

Δ_{1}

−366

−103

−22

−24

−1

Δ_{10}

−254

−63

−15

−19

c = 1 %

Sharpe

0.664

0.832

0.727

0.702

0.704

0.693

0.694

0.695

Δ_{1}

−361

−93

−20

−23

Δ_{10}

−250

−53

−13

−17

c = 2 %

Sharpe

0.663

0.827

0.719

0.698

0.701

0.689

0.690

0.692

Δ_{1}

−356

−82

−17

−21

Δ_{10}

−245

−42

−11

−16

Table 9

Open in new tab

The GMV portfolio (biweekly)

		EWMA	DCC	HDRD	W	R	iR	F	FR
Panel A: MCD/PFE/PG/WMT/XOM

Mean ret		4.414	5.961	5.539	5.432	5.385	5.259	5.297	5.319
${\hat{σ}}_{p}$		10.583	10.892	10.605	10.568	10.569	10.564	10.564	10.563
mcs p-value		(0.01)	(0.00)	(0.01)	(0.01)	(0.01)	(0.40)	(0.40)	(1.00)
StDev ret		10.383	10.600	10.307	10.341	10.348	10.327	10.326	10.317
TO		0.031	0.110	0.152	0.108	0.106	0.089	0.092	0.083
CO		0.500	0.537	0.506	0.501	0.501	0.500	0.500	0.500
SP		−0.005	−0.009	−0.003	−0.005	−0.005	−0.005	−0.005	−0.005

$c = 0 %$	Sharpe	0.425	0.562	0.537	0.525	0.520	0.509	0.513	0.516
	$Δ_{1}$	182	−122	−47	−22	−12	12	5
	$Δ_{10}$	195	−68	−49	−17	−7	14	6
$c = 1 %$	Sharpe	0.424	0.560	0.534	0.523	0.518	0.507	0.511	0.514
	$Δ_{1}$	180	−121	−43	−21	−11	13	5
	$Δ_{10}$	192	−66	−45	−16	−5	15	7
$c = 2 %$	Sharpe	0.424	0.557	0.530	0.520	0.515	0.505	0.509	0.512
	$Δ_{1}$	177	−120	−40	−20	−10	13	5
	$Δ_{10}$	190	−65	−42	−15	−4	15	7

		EWMA	DCC	HDRD	W	R	iR	F	FR
Panel A: MCD/PFE/PG/WMT/XOM

Mean ret		4.414	5.961	5.539	5.432	5.385	5.259	5.297	5.319
${\hat{σ}}_{p}$		10.583	10.892	10.605	10.568	10.569	10.564	10.564	10.563
mcs p-value		(0.01)	(0.00)	(0.01)	(0.01)	(0.01)	(0.40)	(0.40)	(1.00)
StDev ret		10.383	10.600	10.307	10.341	10.348	10.327	10.326	10.317
TO		0.031	0.110	0.152	0.108	0.106	0.089	0.092	0.083
CO		0.500	0.537	0.506	0.501	0.501	0.500	0.500	0.500
SP		−0.005	−0.009	−0.003	−0.005	−0.005	−0.005	−0.005	−0.005

$c = 0 %$	Sharpe	0.425	0.562	0.537	0.525	0.520	0.509	0.513	0.516
	$Δ_{1}$	182	−122	−47	−22	−12	12	5
	$Δ_{10}$	195	−68	−49	−17	−7	14	6
$c = 1 %$	Sharpe	0.424	0.560	0.534	0.523	0.518	0.507	0.511	0.514
	$Δ_{1}$	180	−121	−43	−21	−11	13	5
	$Δ_{10}$	192	−66	−45	−16	−5	15	7
$c = 2 %$	Sharpe	0.424	0.557	0.530	0.520	0.515	0.505	0.509	0.512
	$Δ_{1}$	177	−120	−40	−20	−10	13	5
	$Δ_{10}$	190	−65	−42	−15	−4	15	7

Panel B: AA/AXP/BA/CAT/GE/HD/HON/IBM/JPM/KO/MCD/PFE/PG/WMT/XOM
Mean ret		6.755	8.943	7.583	7.167	7.179	7.038	7.061	7.055
${\hat{σ}}_{p}$		9.651	10.343	9.737	9.669	9.672	9.661	9.659	9.660
mcs p-value		(1.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.17)	(0.52)	(0.52)
StDev ret		10.149	10.681	10.317	10.144	10.137	10.113	10.128	10.108
TO		0.063	0.213	0.331	0.166	0.154	0.132	0.145	0.123
CO		0.416	0.492	0.427	0.426	0.425	0.424	0.424	0.424
SP		−0.104	−0.181	−0.110	−0.109	−0.108	−0.106	−0.107	−0.106
$c = 0 %$	Sharpe	0.666	0.837	0.735	0.707	0.708	0.696	0.697	0.698
	$Δ_{1}$	61	−366	−103	−22	−24	3	−1
	$Δ_{10}$	68	−254	−63	−15	−19	4	3
$c = 1 %$	Sharpe	0.664	0.832	0.727	0.702	0.704	0.693	0.694	0.695
	$Δ_{1}$	58	−361	−93	−20	−23	4	0
	$Δ_{10}$	65	−250	−53	−13	−17	5	4
$c = 2 %$	Sharpe	0.663	0.827	0.719	0.698	0.701	0.689	0.690	0.692
	$Δ_{1}$	55	−356	−82	−17	−21	4	1
	$Δ_{10}$	62	−245	−42	−11	−16	5	5

Panel B: AA/AXP/BA/CAT/GE/HD/HON/IBM/JPM/KO/MCD/PFE/PG/WMT/XOM
Mean ret		6.755	8.943	7.583	7.167	7.179	7.038	7.061	7.055
${\hat{σ}}_{p}$		9.651	10.343	9.737	9.669	9.672	9.661	9.659	9.660
mcs p-value		(1.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.17)	(0.52)	(0.52)
StDev ret		10.149	10.681	10.317	10.144	10.137	10.113	10.128	10.108
TO		0.063	0.213	0.331	0.166	0.154	0.132	0.145	0.123
CO		0.416	0.492	0.427	0.426	0.425	0.424	0.424	0.424
SP		−0.104	−0.181	−0.110	−0.109	−0.108	−0.106	−0.107	−0.106
$c = 0 %$	Sharpe	0.666	0.837	0.735	0.707	0.708	0.696	0.697	0.698
	$Δ_{1}$	61	−366	−103	−22	−24	3	−1
	$Δ_{10}$	68	−254	−63	−15	−19	4	3
$c = 1 %$	Sharpe	0.664	0.832	0.727	0.702	0.704	0.693	0.694	0.695
	$Δ_{1}$	58	−361	−93	−20	−23	4	0
	$Δ_{10}$	65	−250	−53	−13	−17	5	4
$c = 2 %$	Sharpe	0.663	0.827	0.719	0.698	0.701	0.689	0.690	0.692
	$Δ_{1}$	55	−356	−82	−17	−21	4	1
	$Δ_{10}$	62	−245	−42	−11	−16	5	5

Note: This table shows portfolio statistics of the Global Minimum Variance portfolio, based on biweekly predictions of the covariance matrix, according to the Conditional Autoregressive model with HAR dynamics, assuming a Wishart (W), Riesz (R), Inverse Riesz (iR), matrix-F (F), or a F-Riesz (FR) distribution. In addition, we include the EWMA, the DCC-GARCH, and HAR-DRD models as benchmarks. We report the average annualized return and standard deviation (both using the true realized covariance matrix and daily returns), as well as the turnover (TO), portfolio concentration (CO), and short positions (SP). Lowest portfolio volatilities are marked in bold. We also list the p-value that belongs to the model confidence set approach (MCS) of lowest ex-post daily volatility using the true realized covariance matrix based on a 5% significance level. Bold p-values correspond to models that belong to the model confidence set. Finally, the table reports the economic gains of switching from each model listed in the column to the CAFr HAR model in annual basis points, $Δ_{γ}$ ⁠, for various transaction cost levels c and risk aversion coefficients γ. A bold $Δ_{γ}$ means significantly different from zero at the 5% level. Panel A shows results of dimension 5, and Panel B lists results of dimension 15. The out-of-sample period goes from January 2005 until December 2019 and contains 3696 observations.

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