# Predicting Returns, Volatilities and Correlations of Stock Indices Using Multivariate Conditional Autoregressive Range and Return Models

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

`fmincon`.

## 2. Realised Volatility Measures

## 3. Multivariate Volatility and Return Models

#### 3.1. Two-Stage MCARR-Return Model

#### 3.1.1. Stage-One MCARR Model

#### 3.1.2. Stage-Two Multivariate Return Model

#### 3.2. BEKK-GARCH Model

## 4. Application

#### 4.1. Data and Volatility Measures

#### 4.2. Volatility Modelling and Forecast

`fmincon`in MATLAB. Since MCARR models contain many parameters, the search for starting values in running

`fmincon`is more demanding. In general, we set parameter estimates from the univariate CARR model for each return series as the starting values and set the cross component parameters such as ${b}_{ij}$ in (21) and ${\mathtt{s}}_{ij}$ in (16) to be very small. In the first stage, we model the SRPK measures using MCARR models and consider two different mean functions, denoted by MCARR(1,1,0) and MCARR(1,1,1) models, respectively. When modelling each volatility measure, the errors are assumed to follow a MLN distribution defined in (17). All models have only first order terms ($p=q=1$) to reduce the complexity of the model. The parameter estimates defined in (17), (21) and (24), the LL and AIC of each model are displayed in Table 2. The results show that the parameter estimates are mostly positive and significant. The significance of ${a}_{21}$ and ${b}_{21}$ in (21) and (24), respectively, confirms the short term and long term volatility spillover effects. The fitted volatilities $\mathit{\lambda}{}_{t}$ of the six models in Table 2 are used to estimate $\mathbf{\Sigma}{}_{t}$ in the return model for the two-stage MCARR-return model according to (33). In general, MCARR(1,1,1) model with 27 parameters outperforms MCARR(1,1,0) model with 21 parameters due to larger LL and smaller AIC values.

#### 4.3. Return Model and Forecast

**Table 3.**Robust loss functions at two asymmetric levels (b = $-0.5,-2$) for in-sample estimates and out-of-sample volatility forecasts using CARR (univariate) and MCARR (multivariate) models fitted to SRPK measures for S&P 500, DJIA and DJUSFI indices and compared to SRCO measure as the volatility proxy. The best model with the lowest average RL in each row is highlighted in boldface.

Index | RL${}_{\mathbf{SRCO}}$ | CARR | MCARR | |||||||
---|---|---|---|---|---|---|---|---|---|---|

(1,1,0) | (1,1,1) | (1,1,0)${}_{12}$ | (1,1,1)${}_{12}$ | (1,1,0)${}_{13}$ | (1,1,1)${}_{13}$ | (1,1,0)${}_{23}$ | (1,1,1)${}_{23}$ | |||

In-sample | S&P 500 | ($\mathtt{b}=-0.5$) | 8.0113 | 7.6172 | 8.3019 | 8.1238 | 8.0758 | 8.0154 | - | - |

($i=1$) | ($\mathtt{b}=-2$) | 0.2122 | 0.2036 | 0.2403 | 0.2253 | 0.2163 | 0.2137 | - | - | |

DJIA | ($\mathtt{b}=-0.5$) | 10.556 | 10.127 | 10.780 | 10.590 | - | - | 10.605 | 10.553 | |

($i=2$) | ($\mathtt{b}=-2$) | 0.2201 | 0.2133 | 0.2499 | 0.2330 | - | - | 0.2241 | 0.2227 | |

DJUSFI | ($\mathtt{b}=-0.5$) | 12.863 | 12.361 | - | - | 12.647 | 12.535 | 12.706 | 12.588 | |

($i=3$) | ($\mathtt{b}=-2$) | 0.1968 | 0.1907 | - | - | 0.1949 | 0.1936 | 0.1961 | 0.1945 | |

Out-of-sample | S&P 500 | ($\mathtt{b}=-0.5$) | 1.8689 | 1.8207 | 1.9699 | 0.3699 | 1.8334 | 1.8173 | - | - |

($i=1$) | ($\mathtt{b}=-2$) | 0.2000 | 0.2115 | 0.2098 | 0.0601 | 0.2000 | 0.2019 | - | - | |

DJIA | ($\mathtt{b}=-0.5$) | 1.6859 | 1.6237 | 1.7854 | 0.3438 | - | - | 1.6442 | 1.6724 | |

($i=2$) | ($\mathtt{b}=-2$) | 0.1662 | 0.1648 | 0.1665 | 0.0484 | - | - | 0.1648 | 0.1657 | |

DJUSFI | ($\mathtt{b}=-0.5$) | 2.1305 | 4.4748 | - | - | 2.2588 | 2.2254 | 2.2560 | 2.2540 | |

($i=3$) | ($\mathtt{b}=-2$) | 0.1162 | 0.2199 | - | - | 0.1164 | 0.1157 | 0.1193 | 0.1178 |

_{ij}and MCARR(1,1,1)

_{ij}for some i < j and i, j = 1, 2, 3 refer to i and j data pair.

**Table 4.**Summary statistics of daily returns from 1 May 2013 to 31 May 2019 for S&P 500, DJIA and DJUSFI indices.

Summary Statistics | S&P 500 | DJIA | DJUSFI |
---|---|---|---|

mean ${}^{\u2020}$ | 18.702 | 18.963 | 24.545 |

median ${}^{\u2020}$ | 45.714 | 53.483 | 93.577 |

variance ${}^{\u2020}$ | 529,219 | 567,147 | 930,425 |

skewness | −0.4501 | −0.3887 | −0.5750 |

kurtosis | 4.0672 | 4.0951 | 3.6553 |

minimum ${}^{\u2020}$ | −3947.3 | −4273.7 | −6329.2 |

maximum ${}^{\u2020}$ | 4332.8 | 4564.1 | 4773.2 |

LB, Q${}_{6}$ | 4.2167 | 4.7614 | 4.2159 |

LB, Q${}_{12}$ | 7.7828 | 7.0345 | 7.6066 |

^{†}Measures of location and dispersion are rescaled by multiplying 10

^{5}.

**Table 5.**Parameter estimates, standard errors (in parentheses), model performance measures LL and AIC for in-sample estimation of the return model for the two-stage MCARR-return model using volatility estimated from MCARR(1,1,0) and MCARR(1,1,1) volatility models. Models with boldface LL and AIC are the best models used in forecasting.

S&P 500 and DJIA | S&P 500 and DJUSFI | DJIA and DJUSFI | ||||
---|---|---|---|---|---|---|

Return | MCARR | MCARR | MCARR | |||

Model | (1,1,0)${}_{12}$ | (1,1,1)${}_{12}$ | (1,1,0)${}_{13}$ | (1,1,1)${}_{13}$ | (1,1,0)${}_{23}$ | (1,1,1)${}_{23}$ |

${{\mu}_{01}}^{\u2020}$ | 0.3067 | 0.3022 | 0.2891 | 0.2817 | 0.3431 | 0.3504 |

(0.1170) | (0.1267) | (0.1281) | (0.1194) | (0.1287) | (0.1235) | |

${{\mu}_{02}}^{\u2020}$ | 0.3084 | 0.3167 | 0.3758 | 0.4055 | 0.4806 | 0.4912 |

(0.1196) | (0.1314) | (0.1892) | (0.1705) | (0.1804) | (0.1708) | |

${\omega}_{1}$ | 0.9965 | 0.9921 | 0.9270 | 0.9423 | 0.9614 | 0.9604 |

(0.0168) | (0.0191) | (0.0184) | (0.0172) | (0.0176) | (0.0201) | |

${\omega}_{2}$ | 1.0120 | 1.0012 | 0.9624 | 0.9667 | 0.9691 | 0.9669 |

(0.0189) | (0.0194) | (0.0238) | (0.0184) | (0.0193) | (0.0222) | |

$\nu $ | 5.6716 | 6.0415 | 5.3052 | 5.8563 | 6.4431 | 6.2247 |

(0.4133) | (0.6267) | (0.3357) | (0.4734) | (0.4716) | (0.6569) | |

LL | 12,937 | 12,936 | 11834 | 11,853 | 11,733 | 11,728 |

AIC | −25,863 | −25,862 | −23,658 | −23,695 | −23,456 | −23,445 |

^{†}Estimates and standard errors are rescaled by multiplying 10

^{3}.

#### 4.4. VaR and CVaR Forecasts and Risk Forecast Performance Measures

**Table 6.**Parameter estimates and model performance measures LL and AIC for in-sample estimation of the DBEKK-GARCH(1,1) and FBEKK-GARCH(1,1) models. Models with boldface LL and AIC are the best for each index pair.

Error Distribution | MVN | MVT | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Model Type | DBEKK-GARCH | FBEKK-GARCH | DBEKK-GARCH | FBEKK-GARCH | ||||||||

(1,1)${}_{\mathbf{12}}$ | (1,1)${}_{\mathbf{13}}$ | (1,1)${}_{\mathbf{23}}$ | (1,1)${}_{\mathbf{12}}$ | (1,1)${}_{\mathbf{13}}$ | (1,1)${}_{\mathbf{23}}$ | (1,1)${}_{\mathbf{12}}$ | (1,1)${}_{\mathbf{13}}$ | (1,1)${}_{\mathbf{23}}$ | (1,1)${}_{\mathbf{12}}$ | (1,1)${}_{\mathbf{13}}$ | (1,1)${}_{\mathbf{23}}$ | |

${{c}_{11}}^{\u2020}$ | 0.1828 | 0.1692 | 0.1558 | 0.1701 | 0.1978 | 0.1938 | 0.1661 | 0.1215 | 0.1210 | 0.1649 | 0.1896 | 0.1877 |

${{c}_{21}}^{\u2020}$ | 0.1722 | 0.2123 | 0.1981 | 0.1539 | 0.2401 | 0.2248 | 0.1556 | 0.1593 | 0.1493 | 0.1544 | 0.2343 | 0.2201 |

${{c}_{22}}^{\u2020}$ | 0.0694 | 0.1290 | 0.1340 | 0.0680 | 0.1566 | 0.1707 | 0.0670 | 0.1117 | 0.1138 | 0.0671 | 0.1559 | 0.1644 |

${a}_{11}$ | 0.4623 | 0.3480 | 0.3456 | 0.4623 | 0.4554 | 0.4575 | 0.4605 | 0.3124 | 0.3210 | 0.4605 | 0.4540 | 0.4582 |

${{a}_{12}}^{\star}$ | - | - | - | 0.0629 | 0.0029 | 0.0104 | - | - | - | 0.0220 | −0.0083 | −0.0230 |

${{a}_{21}}^{\star}$ | - | - | - | 0.0683 | 0.3540 | 0.0229 | - | - | - | 0.0089 | 0.1370 | 0.3910 |

${a}_{22}$ | 0.4623 | 0.3200 | 0.3167 | 0.4623 | 0.4554 | 0.4575 | 0.4605 | 0.2980 | 0.2961 | 0.4605 | 0.4540 | 0.4582 |

${b}_{11}$ | 0.8607 | 0.9008 | 0.9084 | 0.8607 | 0.8505 | 0.8517 | 0.8606 | 0.9301 | 0.9289 | 0.8606 | 0.8514 | 0.8512 |

${{b}_{12}}^{\star}$ | - | - | - | 0.1420 | 0.0620 | 0.0210 | - | - | - | 0.0357 | −0.0180 | −0.1300 |

${{b}_{21}}^{\star}$ | - | - | - | 0.0899 | 0.4580 | 0.3120 | - | - | - | 0.0112 | 0.1810 | 0.5760 |

${b}_{22}$ | 0.8607 | 0.9071 | 0.9113 | 0.8607 | 0.8505 | 0.8517 | 0.8606 | 0.9291 | 0.9316 | 0.8606 | 0.8514 | 0.8512 |

$\nu $ | - | - | - | - | - | - | 11.100 | 6.0926 | 6.1433 | 11.100 | 11.100 | 11.100 |

LL | 12,804 | 11,732 | 11,597 | 12,801 | 11,716 | 11,579 | 12,921 | 11,838 | 11,708 | 12,920 | 11,805 | 11,674 |

AIC | −25,594 | −23,452 | −23,180 | −25,580 | −23,409 | −23,136 | −25,826 | −23,661 | −23,400 | −25,818 | −23,587 | −23,324 |

^{†}Estimates are rescaled by multiplying 10

^{2}.

^{⋆}Estimates are rescaled by multiplying 10

^{5}.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Histogram, ACF and time series plots for SRPK measures from 1 May 2013 to 31 May 2019 with S&P 500, DJIA, DJUSFI and their pairwise sum indices.

**Figure 2.**Plots of in-sample fits of variance and correlation using MCARR(1,1,0) and MCARR(1,1,1) (in grey background) models with S&P 500, DJIA and DJUSFI indices.

**Figure 3.**Histogram, ACF and time series plots for daily returns of S&P 500, DJIA and DJUSFI indices from 1 May 2013 to 31 May 2019.

**Figure 4.**One-day-ahead VaR forecasts of returns (left 2 panels) and closing prices (right 2 panels) at varying levels for return series using two-stage MCARR-return model with S&P 500, DJIA and DJUSFI indices from 3 June 2019 to 30 August 2019.

**Table 1.**Summary statistics of SRPK measures from 1 May 2013 to 31 May 2019 for S&P 500, DJIA, DJUSFI and their pairwise combined indices.

Summary Statistics | SRPK_{1} | SRPK_{2} | SRPK_{3} | SRPK_{12} | SRPK_{13} | SRPK_{23} |
---|---|---|---|---|---|---|

mean | 4.5189 | 4.7778 | 7.6544 | 18.205 | 22.664 | 22.963 |

median | 2.3250 | 2.4656 | 4.5843 | 9.2849 | 12.579 | 12.732 |

variance | 70.652 | 95.635 | 127.02 | 1278.6 | 1348.4 | 1514.6 |

skewness | 11.594 | 13.901 | 9.2989 | 12.955 | 9.5787 | 10.812 |

kurtosis | 231.03 | 301.58 | 155.21 | 274.10 | 164.31 | 199.75 |

minimum | 0.1769 | 0.2183 | 0.5553 | 0.7354 | 1.1851 | 1.3046 |

maximum | 205.32 | 253.12 | 248.48 | 908.41 | 820.49 | 909.92 |

LB, Q${}_{6}$ | 997.80 | 820.63 | 1010.6 | 897.51 | 1124.9 | 1005.1 |

LB, Q${}_{12}$ | 1277.0 | 1031.4 | 1310.3 | 1139.9 | 1455.6 | 1288.8 |

**Table 2.**Parameter estimates, standard errors (in parentheses), model performance measures LL and AIC for in-sample estimation of the MCARR(1,1,0) and MCARR(1,1,1) models with the pairwise combined indices. Models with boldface LL and AIC are the best models.

Parameter | S&P 500 and DJIA | S&P 500 and DJUSFI | DJIA and DJUSFI | |||
---|---|---|---|---|---|---|

MCARR | MCARR | MCARR | ||||

(1,1,0) | (1,1,1) | (1,1,0) | (1,1,1) | (1,1,0) | (1,1,1) | |

${c}_{1}$ | 0.0614 | 0.1013 | 0.1263 | 0.1347 | 0.1132 | 0.1180 |

(0.0117) | (0.0088) | (0.0215) | (0.0219) | (0.0164) | (0.0133) | |

${c}_{2}$ | 0.0593 | 0.0997 | 0.2760 | 0.2797 | 0.2622 | 0.2561 |

(0.0117) | (0.0097) | (0.0403) | (0.0433) | (0.0346) | (0.0294) | |

${c}_{3}$ | 0.2181 | 0.3857 | 0.6942 | 0.7266 | 0.6554 | 0.6668 |

(0.0445) | (0.0353) | (0.1129) | (0.1166) | (0.0901) | (0.0737) | |

${a}_{11}$ | 0.3510 | 0.3141 | 0.4010 | 0.3917 | 0.3591 | 0.3510 |

(0.0110) | (0.0100) | (0.0121) | (0.0244) | (0.0118) | (0.0189) | |

${a}_{21}$ | 0.0831 | 0.0202 | 0.0046 | 0.0061 | −0.0205 | −0.0180 |

(0.0104) | (0.0082) | (0.0089) | (0.0101) | (0.0059) | (0.0074) | |

${a}_{22}$ | 0.3333 | 0.3023 | 0.3966 | 0.3843 | 0.3970 | 0.3805 |

(0.0100) | (0.0102) | (0.0145) | (0.0204) | (0.0143) | (0.0167) | |

${a}_{31}$ | 0.0099 | 0.0390 | 0.0319 | 0.0285 | 0.0413 | 0.0380 |

(0.0050) | (0.0043) | (0.0055) | (0.0067) | (0.0039) | (0.0052) | |

${a}_{32}$ | 0.0133 | 0.0406 | 0.0597 | 0.0551 | 0.0576 | 0.0527 |

(0.0045) | (0.0043) | (0.0063) | (0.0067) | (0.0051) | (0.0052) | |

${a}_{33}$ | 0.4664 | 0.4654 | 0.5499 | 0.5290 | 0.5168 | 0.4947 |

(0.0077) | (0.0114) | (0.0214) | (0.0247) | (0.0190) | (0.0193) | |

${b}_{11}$ | 0.5334 | 0.6217 | 0.5328 | 0.5365 | 0.6021 | 0.6072 |

(0.0077) | (0.0092) | (0.0062) | (0.0275) | (0.0138) | (0.0198) | |

${b}_{21}$ | −0.0958 | −0.0028 | −0.0149 | −0.0174 | 0.0197 | 0.0177 |

(0.0095) | (0.0078) | (0.0096) | (0.0123) | (0.0063) | (0.0085) | |

${b}_{22}$ | 0.5655 | 0.6409 | 0.5336 | 0.5400 | 0.5573 | 0.5678 |

(0.0125) | (0.0098) | (0.0138) | (0.0226) | (0.0163) | (0.0164) | |

${b}_{31}$ | 0.0127 | −0.0374 | −0.0185 | −0.0157 | −0.0362 | −0.0343 |

(0.0039) | (0.0046) | (0.0050) | (0.0080) | (0.0034) | (0.0060) | |

${b}_{32}$ | 0.0063 | −0.0395 | −0.0415 | −0.0388 | −0.0494 | −0.0469 |

(0.0054) | (0.0047) | (0.0052) | (0.0066) | (0.0042) | (0.0053) | |

${b}_{33}$ | 0.4855 | 0.4966 | 0.4232 | 0.4339 | 0.4610 | 0.4734 |

(0.0087) | (0.0109) | (0.0197) | (0.0230) | (0.0184) | (0.0192) | |

${{d}_{1}}^{\u2020}$ | - | 0.0328 | - | 0.0514 | - | 0.0396 |

- | (0.0135) | - | (0.0227) | - | (0.0180) | |

${{d}_{2}}^{\u2020}$ | - | 0.0233 | - | 0.1055 | - | 0.1118 |

- | (0.0125) | - | (0.0340) | - | (0.0303) | |

${{d}_{3}}^{\u2020}$ | - | 0.0537 | - | 0.1407 | - | 0.1351 |

- | (0.0224) | - | (0.0470) | - | (0.0384) | |

${{e}_{1}}^{\u2020}$ | - | −0.1418 | - | −0.1080 | - | −0.0792 |

- | (0.0219) | - | (0.0264) | - | (0.0211) | |

${{e}_{2}}^{\u2020}$ | - | −0.1313 | - | −0.1551 | - | −0.1621 |

- | (0.0213) | - | (0.0397) | - | (0.0331) | |

${{e}_{3}}^{\u2020}$ | - | −0.2644 | - | −0.2391 | - | −0.2184 |

- | (0.0412) | - | (0.0600) | - | (0.0481) | |

${\mathtt{s}}_{11}$ | 0.2087 | 0.1915 | 0.2924 | 0.2895 | 0.2690 | 0.2661 |

(0.0079) | (0.0050) | (0.0110) | (0.0080) | (0.0073) | (0.0094) | |

${\mathtt{s}}_{21}$ | 0.1809 | 0.1678 | 0.2308 | 0.2287 | 0.2184 | 0.2154 |

(0.0066) | (0.0051) | (0.0093) | (0.0066) | (0.0066) | (0.0084) | |

${\mathtt{s}}_{22}$ | 0.1794 | 0.1737 | 0.2329 | 0.2311 | 0.2366 | 0.2330 |

(0.0056) | (0.0056) | (0.0088) | (0.0064) | (0.0069) | (0.0084) | |

${\mathtt{s}}_{31}$ | 0.1955 | 0.1805 | 0.2604 | 0.2580 | 0.2444 | 0.2414 |

(0.0073) | (0.0051) | (0.0100) | (0.0072) | (0.0068) | (0.0088) | |

${\mathtt{s}}_{32}$ | 0.1812 | 0.1720 | 0.2338 | 0.2319 | 0.2307 | 0.2274 |

(0.0061) | (0.0054) | (0.0089) | (0.0065) | (0.0067) | (0.0083) | |

${\mathtt{s}}_{33}$ | 0.1893 | 0.1773 | 0.2481 | 0.2460 | 0.2402 | 0.2370 |

(0.0067) | (0.0053) | (0.0093) | (0.0068) | (0.0067) | (0.0085) | |

LL | −2411.7 | −2279.9 | −5315.0 | −5302.6 | −5571.4 | −5555.4 |

AIC | 4865.5 | 4613.7 | 10671 | 10,659 | 11,185 | 11,164 |

^{†}Estimates and standard errors are rescaled by multiplying 10

^{3}.

**Table 7.**Forecast performance measures of the test statistic and p-value of KUC test for one-day-ahead VaR return forecasts at the lower and upper risk levels 0.005, 0.025 and 0.05 using the best two-stage MCARR-return model with volatility forecasts from MCARR(1,1,0)${}_{12}$, MCARR(1,1,1)${}_{13}$ and MCARR(1,1,0)${}_{23}$ models for the S&P 500, DJIA and DJUSFI indices. The subscripts $ij$ in MCARR(1,1,0)${}_{ij}$ and MCARR(1,1,1)${}_{ij}$ for some $i<j$ and $i,j=1,2,3$ refer to i and j data pair.

Data Pair | Risk Level | ${\mathit{\alpha}}_{1}$ | ${\mathit{\alpha}}_{2}$ | ||||
---|---|---|---|---|---|---|---|

0.005 | 0.025 | 0.05 | 0.05 | 0.025 | 0.005 | ||

S&P 500 | ${\mathrm{LR}}_{\mathrm{KUC}}$ | 0.9262 | 1.0033 | 3.6012 | 2.1524 | 3.2407 | 0.6416 |

($i=1$) | p-value | 0.3359 | 0.3165 | 0.0577 | 0.1423 | 0.0718 | 0.4231 |

DJIA | ${\mathrm{LR}}_{\mathrm{KUC}}$ | 0.6416 | 0.0951 | 0.1957 | 6.5655 | 3.2407 | 0.6416 |

($j=2$) | p-value | 0.4231 | 0.7578 | 0.6582 | 0.1040 | 0.0718 | 0.4231 |

S&P 500 | ${\mathrm{LR}}_{\mathrm{KUC}}$ | 0.9262 | 2.6238 | 3.6012 | 2.1524 | 3.2407 | 0.6416 |

($i=1$) | p-value | 0.3359 | 0.1053 | 0.0577 | 0.1423 | 0.0718 | 0.4231 |

DJUSFI | ${\mathrm{LR}}_{\mathrm{KUC}}$ | 0.6416 | 0.2657 | 0.1957 | 6.5655 | 3.2407 | 0.6416 |

($j=3$) | p-value | 0.4231 | 0.6062 | 0.6582 | 0.1040 | 0.0718 | 0.4231 |

DJIA | ${\mathrm{LR}}_{\mathrm{KUC}}$ | 0.9262 | 2.6238 | 0.9167 | 2.1524 | 3.2407 | 0.6416 |

($i=2$) | p-value | 0.3359 | 0.1053 | 0.3383 | 0.1423 | 0.0718 | 0.4231 |

DJUSFI | ${\mathrm{LR}}_{\mathrm{KUC}}$ | 0.6416 | 0.2657 | 0.1957 | 2.1524 | 3.2407 | 0.6416 |

($j=3$) | p-value | 0.4231 | 0.6062 | 0.6582 | 0.1423 | 0.0718 | 0.4231 |

**Table 8.**Forecast performance measures of the test statistic and p-value of CI test for one-day-ahead VaR return forecasts at the lower and upper risk levels 0.005, 0.025 and 0.05 using the best two-stage MCARR-return model with volatility forecasts from MCARR(1,1,0)${}_{12}$, MCARR(1,1,1)${}_{13}$ and MCARR(1,1,0)${}_{23}$ models for the S&P 500, DJIA and DJUSFI indices. The subscripts $ij$ in MCARR(1,1,0)${}_{ij}$ and MCARR(1,1,1)${}_{ij}$ for some $i<j$ and $i,j=1,2,3$ refer to i and j data pair.

Data Pair | Risk Level | ${\mathit{\alpha}}_{1}$ | ${\mathit{\alpha}}_{2}$ | ||||
---|---|---|---|---|---|---|---|

0.005 | 0.025 | 0.05 | 0.05 | 0.025 | 0.005 | ||

S&P 500 | ${\mathrm{LR}}_{\mathrm{CI}}$ | 0.0323 | 0.3001 | 1.7546 | 0.0323 | 0.0000 | 0.0000 |

($i=1$) | p-value | 0.8575 | 0.5838 | 0.1853 | 0.8575 | 1.0000 | 1.0000 |

DJIA | ${\mathrm{LR}}_{\mathrm{CI}}$ | 0.0000 | 0.1312 | 0.5428 | 0.0000 | 0.0000 | 0.0000 |

($j=2$) | p-value | 1.0000 | 0.7172 | 0.4613 | 1.0000 | 1.0000 | 1.0000 |

S&P 500 | ${\mathrm{LR}}_{\mathrm{CI}}$ | 0.0323 | 0.5428 | 1.7546 | 0.0323 | 0.0000 | 0.0000 |

($i=1$) | p-value | 0.8575 | 0.4613 | 0.1853 | 0.8575 | 1.0000 | 1.0000 |

DJUSFI | ${\mathrm{LR}}_{\mathrm{CI}}$ | 0.0000 | 0.0323 | 0.5428 | 0.0000 | 0.0000 | 0.0000 |

($j=3$) | p-value | 1.0000 | 0.8575 | 0.4613 | 1.0000 | 1.0000 | 1.0000 |

DJIA | ${\mathrm{LR}}_{\mathrm{CI}}$ | 0.0323 | 0.5428 | 0.8144 | 0.0323 | 0.0000 | 0.0000 |

(i = 2) | p-value | 0.8575 | 0.4613 | 0.3668 | 0.8575 | 1.0000 | 1.0000 |

DJUSFI | ${\mathrm{LR}}_{\mathrm{CI}}$ | 0.0000 | 0.0323 | 0.5428 | 0.0323 | 0.0000 | 0.0000 |

(j = 3) | p-value | 1.0000 | 0.8575 | 0.4613 | 0.8575 | 1.0000 | 1.0000 |

**Table 9.**RMSFE and MAFE loss functions for out-of-sample closing price forecasts using the two-stage CARR-return and MCARR-return models. Volatilities are estimated from both CARR and MCARR models for each of the S&P 500, DJIA and DJUSFI index series. The stage-two return models in the two-stage CARR-return and MCARR-return models use constant mean function and Student-t and MVT error distributions, respectively.

Index | Loss | CARR | MCARR | ||||||
---|---|---|---|---|---|---|---|---|---|

Function | (1,1,0) | (1,1,1) | (1,1,0)${}_{12}$ | (1,1,1)${}_{12}$ | (1,1,0)${}_{13}$ | (1,1,1)${}_{13}$ | (1,1,0)${}_{23}$ | (1,1,1)${}_{23}$ | |

S&P 500 | RMSFE ${}^{\u2020}$ | 0.7516 | 0.7499 | 0.7560 | 0.7587 | 0.7553 | 0.7536 | - | - |

($i=1$) | MAFE ${}^{\u2020}$ | 0.5404 | 0.5390 | 0.5451 | 0.5464 | 0.5431 | 0.5416 | - | - |

DJIA | RMSFE ${}^{\u2020}$ | 0.7932 | 0.7970 | 0.7701 | 0.7754 | - | - | 0.7874 | 0.7874 |

($i=2$) | MAFE ${}^{\u2020}$ | 0.5698 | 0.5740 | 0.5460 | 0.5517 | - | - | 0.5637 | 0.5637 |

DJUSFI | RMSFE ${}^{\u2020}$ | 0.8771 | 0.8791 | - | - | 0.8494 | 0.8546 | 0.8810 | 0.8690 |

($i=3$) | MAFE ${}^{\u2020}$ | 0.6975 | 0.6981 | - | - | 0.6848 | 0.6872 | 0.7069 | 0.6934 |

^{†}Estimates are rescaled by multiplying 10

^{2}. The best out-of-sample closing price forecast performances for each index series are highlighted in boldface.

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## Share and Cite

**MDPI and ACS Style**

Tan, S.K.; Ng, K.H.; Chan, J.S.-K.
Predicting Returns, Volatilities and Correlations of Stock Indices Using Multivariate Conditional Autoregressive Range and Return Models. *Mathematics* **2023**, *11*, 13.
https://doi.org/10.3390/math11010013

**AMA Style**

Tan SK, Ng KH, Chan JS-K.
Predicting Returns, Volatilities and Correlations of Stock Indices Using Multivariate Conditional Autoregressive Range and Return Models. *Mathematics*. 2023; 11(1):13.
https://doi.org/10.3390/math11010013

**Chicago/Turabian Style**

Tan, Shay Kee, Kok Haur Ng, and Jennifer So-Kuen Chan.
2023. "Predicting Returns, Volatilities and Correlations of Stock Indices Using Multivariate Conditional Autoregressive Range and Return Models" *Mathematics* 11, no. 1: 13.
https://doi.org/10.3390/math11010013