. | CA specifications . | EWMA . | DCC . | CA-HAR specifications . | HDRD . | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
. | W . | R . | iR . | F . | FR . | . | . | W . | R . | iR . | F . | FR . | . |
Panel A: daily forecasts | |||||||||||||
Frob | 2.274 | 2.276 | 2.275 | 2.274 | 2.280 | 2.627 | 2.620 | 2.250 | 2.252 | 2.248 | 2.246 | 2.253 | 2.231 |
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) |
QLIK | 4.354 | 4.353 | 4.358 | 4.356 | 4.364 | 4.525 | 4.911 | 4.341 | 4.341 | 4.345 | 4.343 | 4.351 | 4.325 |
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 | |||||||||||||
Frob | 9.369 | 9.314 | 9.271 | 9.303 | 9.295 | 10.799 | 11.230 | 9.259 | 9.240 | 9.229 | 9.239 | 9.288 | 9.507 |
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) |
QLIK | 12.578 | 12.573 | 12.572 | 12.573 | 12.574 | 12.674 | 13.151 | 12.567 | 12.564 | 12.568 | 12.569 | 12.574 | 12.575 |
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 | |||||||||||||
Frob | 18.937 | 18.723 | 18.547 | 18.676 | 18.557 | 20.814 | 21.906 | 19.273 | 19.252 | 18.738 | 18.830 | 18.782 | 18.953 |
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) |
QLIK | 16.175 | 16.165 | 16.160 | 16.165 | 16.162 | 16.218 | 16.718 | 16.162 | 16.158 | 16.150 | 16.153 | 16.155 | 16.173 |
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 . | EWMA . | DCC . | CA-HAR specifications . | HDRD . | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
. | W . | R . | iR . | F . | FR . | . | . | W . | R . | iR . | F . | FR . | . |
Panel A: daily forecasts | |||||||||||||
Frob | 2.274 | 2.276 | 2.275 | 2.274 | 2.280 | 2.627 | 2.620 | 2.250 | 2.252 | 2.248 | 2.246 | 2.253 | 2.231 |
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) |
QLIK | 4.354 | 4.353 | 4.358 | 4.356 | 4.364 | 4.525 | 4.911 | 4.341 | 4.341 | 4.345 | 4.343 | 4.351 | 4.325 |
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 | |||||||||||||
Frob | 9.369 | 9.314 | 9.271 | 9.303 | 9.295 | 10.799 | 11.230 | 9.259 | 9.240 | 9.229 | 9.239 | 9.288 | 9.507 |
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) |
QLIK | 12.578 | 12.573 | 12.572 | 12.573 | 12.574 | 12.674 | 13.151 | 12.567 | 12.564 | 12.568 | 12.569 | 12.574 | 12.575 |
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 | |||||||||||||
Frob | 18.937 | 18.723 | 18.547 | 18.676 | 18.557 | 20.814 | 21.906 | 19.273 | 19.252 | 18.738 | 18.830 | 18.782 | 18.953 |
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) |
QLIK | 16.175 | 16.165 | 16.160 | 16.165 | 16.162 | 16.218 | 16.718 | 16.162 | 16.158 | 16.150 | 16.153 | 16.155 | 16.173 |
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 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.
. | CA specifications . | EWMA . | DCC . | CA-HAR specifications . | HDRD . | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
. | W . | R . | iR . | F . | FR . | . | . | W . | R . | iR . | F . | FR . | . |
Panel A: daily forecasts | |||||||||||||
Frob | 2.274 | 2.276 | 2.275 | 2.274 | 2.280 | 2.627 | 2.620 | 2.250 | 2.252 | 2.248 | 2.246 | 2.253 | 2.231 |
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) |
QLIK | 4.354 | 4.353 | 4.358 | 4.356 | 4.364 | 4.525 | 4.911 | 4.341 | 4.341 | 4.345 | 4.343 | 4.351 | 4.325 |
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 | |||||||||||||
Frob | 9.369 | 9.314 | 9.271 | 9.303 | 9.295 | 10.799 | 11.230 | 9.259 | 9.240 | 9.229 | 9.239 | 9.288 | 9.507 |
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) |
QLIK | 12.578 | 12.573 | 12.572 | 12.573 | 12.574 | 12.674 | 13.151 | 12.567 | 12.564 | 12.568 | 12.569 | 12.574 | 12.575 |
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 | |||||||||||||
Frob | 18.937 | 18.723 | 18.547 | 18.676 | 18.557 | 20.814 | 21.906 | 19.273 | 19.252 | 18.738 | 18.830 | 18.782 | 18.953 |
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) |
QLIK | 16.175 | 16.165 | 16.160 | 16.165 | 16.162 | 16.218 | 16.718 | 16.162 | 16.158 | 16.150 | 16.153 | 16.155 | 16.173 |
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 . | EWMA . | DCC . | CA-HAR specifications . | HDRD . | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
. | W . | R . | iR . | F . | FR . | . | . | W . | R . | iR . | F . | FR . | . |
Panel A: daily forecasts | |||||||||||||
Frob | 2.274 | 2.276 | 2.275 | 2.274 | 2.280 | 2.627 | 2.620 | 2.250 | 2.252 | 2.248 | 2.246 | 2.253 | 2.231 |
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) |
QLIK | 4.354 | 4.353 | 4.358 | 4.356 | 4.364 | 4.525 | 4.911 | 4.341 | 4.341 | 4.345 | 4.343 | 4.351 | 4.325 |
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 | |||||||||||||
Frob | 9.369 | 9.314 | 9.271 | 9.303 | 9.295 | 10.799 | 11.230 | 9.259 | 9.240 | 9.229 | 9.239 | 9.288 | 9.507 |
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) |
QLIK | 12.578 | 12.573 | 12.572 | 12.573 | 12.574 | 12.674 | 13.151 | 12.567 | 12.564 | 12.568 | 12.569 | 12.574 | 12.575 |
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 | |||||||||||||
Frob | 18.937 | 18.723 | 18.547 | 18.676 | 18.557 | 20.814 | 21.906 | 19.273 | 19.252 | 18.738 | 18.830 | 18.782 | 18.953 |
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) |
QLIK | 16.175 | 16.165 | 16.160 | 16.165 | 16.162 | 16.218 | 16.718 | 16.162 | 16.158 | 16.150 | 16.153 | 16.155 | 16.173 |
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 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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