Table 5

Out-of-sample point forecasts (dimension 5, set I)

CA specifications
EWMADCCCA-HAR specifications
HDRD
WRiRFFRWRiRFFR
Panel A: daily forecasts

Frob2.2742.2762.2752.2742.2802.6272.6202.2502.2522.2482.2462.2532.231
mcs p-value(0.02)(0.01)(0.01)(0.05)(0.01)(0.01)(0.00)(0.43)(0.37)(0.43)(0.66)(0.34)(1.00)
QLIK4.3544.3534.3584.3564.3644.5254.9114.3414.3414.3454.3434.3514.325
mcs p-value(0.00)(0.00)(0.00)(0.00)(0.00)(0.00)(0.00)(0.07)(0.07)(0.02)(0.06)(0.00)(1.00)

Panel B: weekly forecasts

Frob9.3699.3149.2719.3039.29510.79911.2309.2599.2409.2299.2399.2889.507
mcs p-value(0.25)(0.61)(0.87)(0.61)(0.64)(0.11)(0.02)(0.79)(0.87)(1.00)(0.87)(0.64)(0.61)
QLIK12.57812.57312.57212.57312.57412.67413.15112.56712.56412.56812.56912.57412.575
mcs p-value(0.11)(0.25)(0.33)(0.21)(0.21)(0.01)(0.00)(0.42)(1.00)(0.42)(0.42)(0.21)(0.42)

Panel C: biweekly forecasts

Frob18.93718.72318.54718.67618.55720.81421.90619.27319.25218.73818.83018.78218.953
mcs p-value(0.05)(0.39)(1.00)(0.39)(0.92)(0.01)(0.01)(0.01)(0.04)(0.67)(0.34)(0.39)(0.39)
QLIK16.17516.16516.16016.16516.16216.21816.71816.16216.15816.15016.15316.15516.173
mcs p-value(0.10)(0.30)(0.32)(0.27)(0.32)(0.06)(0.01)(0.32)(0.32)(1.00)(0.32)(0.32)(0.32)
CA specifications
EWMADCCCA-HAR specifications
HDRD
WRiRFFRWRiRFFR
Panel A: daily forecasts

Frob2.2742.2762.2752.2742.2802.6272.6202.2502.2522.2482.2462.2532.231
mcs p-value(0.02)(0.01)(0.01)(0.05)(0.01)(0.01)(0.00)(0.43)(0.37)(0.43)(0.66)(0.34)(1.00)
QLIK4.3544.3534.3584.3564.3644.5254.9114.3414.3414.3454.3434.3514.325
mcs p-value(0.00)(0.00)(0.00)(0.00)(0.00)(0.00)(0.00)(0.07)(0.07)(0.02)(0.06)(0.00)(1.00)

Panel B: weekly forecasts

Frob9.3699.3149.2719.3039.29510.79911.2309.2599.2409.2299.2399.2889.507
mcs p-value(0.25)(0.61)(0.87)(0.61)(0.64)(0.11)(0.02)(0.79)(0.87)(1.00)(0.87)(0.64)(0.61)
QLIK12.57812.57312.57212.57312.57412.67413.15112.56712.56412.56812.56912.57412.575
mcs p-value(0.11)(0.25)(0.33)(0.21)(0.21)(0.01)(0.00)(0.42)(1.00)(0.42)(0.42)(0.21)(0.42)

Panel C: biweekly forecasts

Frob18.93718.72318.54718.67618.55720.81421.90619.27319.25218.73818.83018.78218.953
mcs p-value(0.05)(0.39)(1.00)(0.39)(0.92)(0.01)(0.01)(0.01)(0.04)(0.67)(0.34)(0.39)(0.39)
QLIK16.17516.16516.16016.16516.16216.21816.71816.16216.15816.15016.15316.15516.173
mcs p-value(0.10)(0.30)(0.32)(0.27)(0.32)(0.06)(0.01)(0.32)(0.32)(1.00)(0.32)(0.32)(0.32)

Notes: This table shows the mean of the Frobenius norm and QLIK values based on daily, weekly, and biweekly predictions of the (5×5) covariance matrix of 5 stocks (MCD/PFE/PG/WMT/XOM) according to the Conditional Autoregressive model with and without 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, DCC-GARCH, and the HAR-DRD (HDRD) models as benchmarks. Parameters are estimated with a moving window of 1,000 observations and re-estimated after 250 observations. Panel A shows results of 1-step ahead predictions, and Panels B and C list results of cumulative forecasts for a week (five-step ahead) and two weeks (10 steps ahead). The lowest value of the Frobenius norm and QLIK across the models are marked in bold. In addition, we report the MCS p-values based on a 5% significance level. The p-values of the models within the model confidence set are marked in bold. The out-of-sample period goes from January 2005 until December 2019 and contains 3,696 observations.

Table 5

Out-of-sample point forecasts (dimension 5, set I)

CA specifications
EWMADCCCA-HAR specifications
HDRD
WRiRFFRWRiRFFR
Panel A: daily forecasts

Frob2.2742.2762.2752.2742.2802.6272.6202.2502.2522.2482.2462.2532.231
mcs p-value(0.02)(0.01)(0.01)(0.05)(0.01)(0.01)(0.00)(0.43)(0.37)(0.43)(0.66)(0.34)(1.00)
QLIK4.3544.3534.3584.3564.3644.5254.9114.3414.3414.3454.3434.3514.325
mcs p-value(0.00)(0.00)(0.00)(0.00)(0.00)(0.00)(0.00)(0.07)(0.07)(0.02)(0.06)(0.00)(1.00)

Panel B: weekly forecasts

Frob9.3699.3149.2719.3039.29510.79911.2309.2599.2409.2299.2399.2889.507
mcs p-value(0.25)(0.61)(0.87)(0.61)(0.64)(0.11)(0.02)(0.79)(0.87)(1.00)(0.87)(0.64)(0.61)
QLIK12.57812.57312.57212.57312.57412.67413.15112.56712.56412.56812.56912.57412.575
mcs p-value(0.11)(0.25)(0.33)(0.21)(0.21)(0.01)(0.00)(0.42)(1.00)(0.42)(0.42)(0.21)(0.42)

Panel C: biweekly forecasts

Frob18.93718.72318.54718.67618.55720.81421.90619.27319.25218.73818.83018.78218.953
mcs p-value(0.05)(0.39)(1.00)(0.39)(0.92)(0.01)(0.01)(0.01)(0.04)(0.67)(0.34)(0.39)(0.39)
QLIK16.17516.16516.16016.16516.16216.21816.71816.16216.15816.15016.15316.15516.173
mcs p-value(0.10)(0.30)(0.32)(0.27)(0.32)(0.06)(0.01)(0.32)(0.32)(1.00)(0.32)(0.32)(0.32)
CA specifications
EWMADCCCA-HAR specifications
HDRD
WRiRFFRWRiRFFR
Panel A: daily forecasts

Frob2.2742.2762.2752.2742.2802.6272.6202.2502.2522.2482.2462.2532.231
mcs p-value(0.02)(0.01)(0.01)(0.05)(0.01)(0.01)(0.00)(0.43)(0.37)(0.43)(0.66)(0.34)(1.00)
QLIK4.3544.3534.3584.3564.3644.5254.9114.3414.3414.3454.3434.3514.325
mcs p-value(0.00)(0.00)(0.00)(0.00)(0.00)(0.00)(0.00)(0.07)(0.07)(0.02)(0.06)(0.00)(1.00)

Panel B: weekly forecasts

Frob9.3699.3149.2719.3039.29510.79911.2309.2599.2409.2299.2399.2889.507
mcs p-value(0.25)(0.61)(0.87)(0.61)(0.64)(0.11)(0.02)(0.79)(0.87)(1.00)(0.87)(0.64)(0.61)
QLIK12.57812.57312.57212.57312.57412.67413.15112.56712.56412.56812.56912.57412.575
mcs p-value(0.11)(0.25)(0.33)(0.21)(0.21)(0.01)(0.00)(0.42)(1.00)(0.42)(0.42)(0.21)(0.42)

Panel C: biweekly forecasts

Frob18.93718.72318.54718.67618.55720.81421.90619.27319.25218.73818.83018.78218.953
mcs p-value(0.05)(0.39)(1.00)(0.39)(0.92)(0.01)(0.01)(0.01)(0.04)(0.67)(0.34)(0.39)(0.39)
QLIK16.17516.16516.16016.16516.16216.21816.71816.16216.15816.15016.15316.15516.173
mcs p-value(0.10)(0.30)(0.32)(0.27)(0.32)(0.06)(0.01)(0.32)(0.32)(1.00)(0.32)(0.32)(0.32)

Notes: This table shows the mean of the Frobenius norm and QLIK values based on daily, weekly, and biweekly predictions of the (5×5) covariance matrix of 5 stocks (MCD/PFE/PG/WMT/XOM) according to the Conditional Autoregressive model with and without 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, DCC-GARCH, and the HAR-DRD (HDRD) models as benchmarks. Parameters are estimated with a moving window of 1,000 observations and re-estimated after 250 observations. Panel A shows results of 1-step ahead predictions, and Panels B and C list results of cumulative forecasts for a week (five-step ahead) and two weeks (10 steps ahead). The lowest value of the Frobenius norm and QLIK across the models are marked in bold. In addition, we report the MCS p-values based on a 5% significance level. The p-values of the models within the model confidence set are marked in bold. The out-of-sample period goes from January 2005 until December 2019 and contains 3,696 observations.

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