# Bayesian Analysis of Intraday Stochastic Volatility Models of High-Frequency Stock Returns with Skew Heavy-Tailed Errors

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Stochastic Volatility Model with Intraday Seasonality

**Step 1**with

**Step 1**. This is the reason we call it the NCP-based ASIS step. After this update, we merely shift the location of ${h}_{1:T+1}$ by ${x}_{t}^{\prime}{\beta}^{(r+0.5)}$ (

**Step 1**) or by $-{x}_{t}^{\prime}{\beta}^{(r+1)}$ (

**Step 1.5**). In ASIS, these shifts are applied with probability 1 even if all elements in ${h}_{1:T+1}$ are not updated at the beginning of

**Step 1**, which is highly probable in practice because we need to use the MH algorithm to generate ${h}_{1:T+1}$. Although we also utilize the MH algorithm to generate $\beta $, as explained later, the acceptance rate of $\beta $ in the MH step is much higher than that of ${h}_{1:T+1}$ in our experience. Thus, we expect that both ${x}_{t}^{\prime}{\beta}^{(r+0.5)}$ and $-{x}_{t}^{\prime}{\beta}^{(r+1)}$ will be updated more often than ${h}_{1:T+1}$ itself. As a result, the above ASIS step may improve mixing of the sample sequence of ${h}_{1:T+1}$. Conversely, we may apply the following CP-based ASIS step:

## 3. Extension: Skew Heavy-Tailed Distributions

#### 3.1. Mean-Variance Mixture of the Normal Distribution

- exponential distribution ($\lambda =1,\xi =0$),
- gamma distribution ($\lambda >0,\xi =0$),
- inverse gamma distribution ($\lambda <0,\psi =0$),
- inverse Gaussian distribution ($\lambda =-\frac{1}{2}$)

- skew variance gamma (VG) distribution($\lambda =\frac{\nu}{2},\psi =\nu ,\xi =0$),
- skew t distribution ($\lambda =-\frac{\nu}{2},\psi =0,\xi =\nu $),

SV-N: | stochastic volatility model with the normal error, |

SV-G: | stochastic volatility model with the VG error, |

SV-SG: | stochastic volatility model with the skew VG error, |

SV-T: | stochastic volatility model with the Student-t error, |

SV-ST: | stochastic volatility model with the skew t error. |

#### 3.2. Conditional Posterior Distributions

#### 3.2.1. Latent Log Volatility ${h}_{1:T+1}$

#### 3.2.2. Regression Coefficients $\beta $

#### 3.2.3. Leverage Parameter $\gamma $

#### 3.2.4. Random Scale ${\delta}_{1:T}$

#### 3.2.5. Asymmetry Parameter $\alpha $

#### 3.2.6. Tail Parameter $\nu $

## 4. Empirical Study

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Conditional Posterior Distributions

#### Appendix A.1. NCP Form

#### Appendix A.1.1. Latent Log Volatility h_{1}:T+1

- 1
- the choice of ${h}_{1:T+1}^{\ast}$ is crucial to make the approximation (A8) workable.
- 2
- the acceptance rate of the MH algorithm tends to be too low when ${h}_{1:T+1}$ is a high-dimensional vector.

**Step****1:**- Initialize ${h}_{1:T+1}^{\ast \left(0\right)}$ and set the counter $r=1$.
**Step****2:**- Update ${h}_{1:T+1}^{\ast \left(r\right)}$ by ${h}_{1:T+1}^{\ast \left(r\right)}={\mu}_{h}\left({h}_{1:T+1}^{\ast (r-1)}\right)$.
**Step****3:**- Let $r=r+1$ and go to
**Step 2**unless ${max}_{t=1,\dots ,T+1}|{h}_{t}^{\ast \left(r\right)}-{h}_{t}^{\ast (r-1)}|$ is less than the preset tolerance level.

#### Appendix A.1.2. Regression Coefficients β

#### Appendix A.1.3. Leverage Parameter γ

#### Appendix A.1.4. Variance τ^{2}

#### Appendix A.1.5. AR(1) Coefficient ϕ

#### Appendix A.2. CP Form

#### Appendix A.2.1. Latent Log Volatility ${\tilde{h}}_{1:T+1}$

#### Appendix A.2.2. Regression Coefficients β

#### Appendix A.2.3. Leverage Parameter γ

#### Appendix A.2.4. Variance τ^{2}

#### Appendix A.2.5. AR(1) Coefficient ϕ

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Date | Skewness | Kurtosis | Min. | Max |
---|---|---|---|---|

Week 1 | −0.1081 | 7.2296 | −7.2569 | 5.4520 |

Week 2 | 0.2494 | 7.7468 | −5.9886 | 5.7911 |

Week 3 | 0.3534 | 7.4500 | −6.4415 | 5.8413 |

Week 4 | −0.0125 | 7.1031 | −6.4212 | 5.6074 |

Week 5 | 0.0346 | 4.9146 | −4.6433 | 5.4437 |

Order 5 | Order 6 | Order 7 | Order 8 | Order 9 | Order 10 | |
---|---|---|---|---|---|---|

SV-N | 3965.1 | 3966.0 | 3962.0 | 3975.1 | 3969.6 | 3971.3 |

SV-G | 3701.0 | 3698.5 | 3702.2 | 3697.4 | 3700.0 | 3701.1 |

SV-SG | 3701.9 | 3701.0 | 3698.9 | 3699.0 | 3702.8 | 3700.8 |

SV-T | 3813.2 | 3813.3 | 3813.8 | 3816.6 | 3813.9 | 3813.0 |

SV-ST | 3819.3 | 3813.8 | 3816.5 | 3816.2 | 3815.7 | 3817.5 |

Order 5 | Order 6 | Order 7 | Order 8 | Order 9 | Order 10 | |
---|---|---|---|---|---|---|

SV-N | 3811.0 | 3813.7 | 3814.5 | 3815.7 | 3818.0 | 3819.6 |

SV-G | 3621.0 | 3622.7 | 3621.0 | 3619.7 | 3621.2 | 3621.8 |

SV-SG | 3618.6 | 3621.3 | 3622.0 | 3623.7 | 3623.4 | 3619.7 |

SV-T | 3685.4 | 3684.1 | 3681.5 | 3859.0 | 3860.4 | 3684.6 |

SV-ST | 3686.9 | 3684.9 | 3683.3 | 3684.1 | 3684.3 | 3686.4 |

Order 5 | Order 6 | Order 7 | Order 8 | Order 9 | Order 10 | |
---|---|---|---|---|---|---|

SV-N | 3976.9 | 3991.1 | 3996.8 | 3975.1 | 3969.6 | 3971.3 |

SV-G | 3750.2 | 3752.4 | 3752.0 | 3755.2 | 3752.9 | 3754.3 |

SV-SG | 3753.6 | 3753.3 | 3755.8 | 3754.1 | 3759.9 | 3752.8 |

SV-T | 3862.0 | 3859.4 | 3861.6 | 3859.0 | 3860.4 | 3861.6 |

SV-ST | 3862.1 | 3861.2 | 3861.3 | 3860.6 | 3861.5 | 3862.1 |

Order 5 | Order 6 | Order 7 | Order 8 | Order 9 | Order 10 | |
---|---|---|---|---|---|---|

SV-N | 3927.5 | 3924.5 | 3924.8 | 3925.8 | 3924.7 | 3927.1 |

SV-G | 3670.3 | 3680.1 | 3679.0 | 3662.8 | 3675.7 | 3670.4 |

SV-SG | 3663.5 | 3668.2 | 3670.3 | 3664.6 | 3672.6 | 3669.1 |

SV-T | 3750.9 | 3751.9 | 3752.3 | 3750.8 | 3753.1 | 3753.9 |

SV-ST | 3753.5 | 3754.3 | 3753.4 | 3755.3 | 3752.8 | 3751.7 |

Order 5 | Order 6 | Order 7 | Order 8 | Order 9 | Order 10 | |
---|---|---|---|---|---|---|

SV-N | 4114.6 | 4114.3 | 4111.3 | 4115.2 | 4113.7 | 4114.5 |

SV-G | 3961.2 | 3959.6 | 3958.3 | 3960.0 | 3961.2 | 3960.0 |

SV-SG | 3953.7 | 3955.6 | 3960.0 | 3963.3 | 3957.7 | 3953.1 |

SV-T | 4033.7 | 4033.8 | 4032.7 | 4031.2 | 4032.8 | 4033.1 |

SV-ST | 4032.5 | 4033.0 | 4033.4 | 4030.4 | 4032.0 | 4031.5 |

$\mathit{\gamma}$ | $\mathit{\varphi}$ | $\mathit{\tau}$ | $\mathit{\alpha}$ | $\mathit{\nu}$ | |
---|---|---|---|---|---|

SV-N (7) ${}^{a}$ | -0.5742 ${}^{b}$ | 0.8909 | 0.1903 | ||

[−1.4270, −0.1241] ${}^{c}$ | [0.8214, 0.9431] | [0.1306, 0.2494] | |||

3.42 ${}^{d}$ | 4.01 | 4.54 | |||

SV-G (8) | −1.3565 | 0.9608 | 0.0798 | 2.5248 | |

[−3.1743, 0.8347] | [0.9320, 0.9815] | [0.0604, 0.1034] | [2.0444, 3.2647] | ||

4.13 | 3.17 | 4.45 | 3.40 | ||

SV-SG (7) | −1.4520 | 0.9598 | 0.0812 | 0.0014 | 2.5489 |

[−3.2233, 0.4995] | [0.9316, 0.9802] | [0.0630, 0.1077] | [−0.0504, 0.0544] | [2.0425, 3.3462] | |

3.99 | 3.01 | 4.45 | 1.23 | 3.48 | |

SV-T (10) | −1.4209 | 0.9864 | 0.0766 | 4.2950 | |

[−4.3298, 1.5726] | [0.9745, 0.9955] | [0.0594, 0.1010] | [3.3294, 5.5097] | ||

3.85 | 2.90 | 4.42 | 3.44 | ||

SV-ST (6) | −1.6137 | 0.9870 | 0.0753 | 0.0006 | 4.2338 |

[−5.0449, 1.6958] | [0.9751, 0.9957] | [0.0586, 0.0962] | [−0.0400, 0.0407] | [3.3247, 5.5254] | |

3.94 | 2.78 | 4.38 | 1.62 | 3.54 |

$\mathit{\gamma}$ | $\mathit{\varphi}$ | $\mathit{\tau}$ | $\mathit{\alpha}$ | $\mathit{\nu}$ | |
---|---|---|---|---|---|

SV-N (5) ${}^{a}$ | −0.2127 ${}^{b}$ | 0.7149 | 0.3435 | ||

[−0.5550, 0.0923] ${}^{c}$ | [0.5967, 0.8106] | [0.2750, 0.4187] | |||

3.04 ${}^{d}$ | 3.87 | 4.39 | |||

SV-G (8) | 0.0397 | 0.9191 | 0.0917 | 2.2475 | |

[−1.7042, 2.0695] | [0.8122, 0.9694] | [0.0604, 0.1442] | [2.0089, 2.7055] | ||

4.27 | 4.05 | 4.58 | 2.6736 | ||

SV-SG (5) | 0.0380 | 0.8965 | 0.0991 | −0.0001 | 2.2622 |

[−1.5878, 1.9299] | [0.6761, 0.9664] | [0.0647, 0.1610] | [−0.0530, 0.0531] | [2.0102, 2.7308] | |

4.24 | 4.46 | 4.61 | 1.24 | 2.85 | |

SV-T (7) | −0.7862 | 0.914 | 0.0673 | 3.30 | |

[−4.7784, 2.8361] | [0.9825, 0.9979] | [0.0511, 0.0886] | [2.7141, 4.0299] | ||

3.91 | 2.83 | 4.43 | 3.17 | ||

SV-ST (7) | −0.6862 | 0.9909 | 0.0690 | −0.0027 | 3.3647 |

[−4.4268, 2.9433] | [0.9816, 0.9976] | [0.0535, 0.0902] | [−0.0376, 0.0323] | [2.7611, 4.0958] | |

3.93 | 2.77 | 4.42 | 1.50 | 3.14 |

$\mathit{\gamma}$ | $\mathit{\varphi}$ | $\mathit{\tau}$ | $\mathit{\alpha}$ | $\mathit{\nu}$ | |
---|---|---|---|---|---|

SV-N (9) ${}^{a}$ | 0.0514 ${}^{b}$ | 0.6249 | 0.2950 | ||

[−0.3345, 0.4436] ${}^{c}$ | [0.3506, 0.8190] | [0.2042, 0.3872] | |||

2.71 ${}^{d}$ | 4.39 | 4.54 | |||

SV-G (5) | 0.0639 | 0.4533 | 0.0919 | 2.2888 | |

[−1.8134, 1.9573] | [0.1344, 0.7393] | [0.0632, 0.1413] | [2.0155, 2.7501] | ||

4.20 | 4.33 | 4.57 | 2.65 | ||

SV-SG (10) | −0.2023 | 0.7992 | 0.0851 | −0.0031 | 2.3419 |

[−2.1985, 1.7331] | [0.2317, 0.9511] | [0.0596, 0.1250] | [−0.0546, 0.0485] | [2.0232, 2.9008] | |

4.20 | 4.59 | 4.55 | 1.17 | 2.99 | |

SV-T (8) | −0.2369 | 0.9871 | 0.0661 | 4.0539 | |

[−3.6943, 3.4453] | [0.9755, 0.9960] | [0.0514, 0.0837] | [3.2156, 5.1327] | ||

3.86 | 2.83 | 4.39 | 3.39 | ||

SV-ST (8) | −0.3313 | 0.9866 | 0.0670 | −0.0034 | 4.1237 |

[−4.2240, 3.6835] | [0.9730, 0.9957] | [0.0521, 0.0900] | [−0.0426, 0.0361] | [3.2114, 5.2538] | |

3.98 | 3.07 | 4.42 | 1.59 | 3.44 |

$\mathit{\gamma}$ | $\mathit{\varphi}$ | $\mathit{\tau}$ | $\mathit{\alpha}$ | $\mathit{\nu}$ | |
---|---|---|---|---|---|

SV-N (6) ${}^{a}$ | −0.6502 ${}^{b}$ | 0.9336 | 0.1526 | ||

[−1.8098, 0.2658] ${}^{c}$ | [0.8831, 0.9704] | [0.1038, 0.2117] | |||

3.53 ${}^{d}$ | 4.01 | 4.57 | |||

SV-G (8) | −1.4435 | 0.9689 | 0.0853 | 2.9487 | |

[−3.6134, 0.6862] | [0.9454, 0.9859] | [0.0657, 0.1098] | [2.1879, 3.9161] | ||

4.11 | 3.10 | 4.43 | 3.63 | ||

SV-SG (5) | −1.7146 | 0.9694 | 0.0846 | −0.0068 | 2.93 |

[−4.0432, −0.4361] | [0.9458, 0.9868] | [0.0650, 0.1116] | [−0.0599, 0.0471] | [2.14, 3.94] | |

4.18 | 3.11 | 4.46 | 1.54 | 3.64 | |

SV-T (8) | −1.4043 | 0.9869 | 0.0824 | 4.5805 | |

[−4.4147, −1.6809] | [0.9747, 0.9960] | [0.0634, 0.1060] | [3.5659, 5.8472] | ||

3.96 | 2.89 | 4.41 | 3.43 | ||

SV-ST (10) | −1.6482 | 0.9882 | 0.0788 | −0.0045 | 4.4738 |

[−5.1451, 1.6390] | [0.9777, 0.9964] | [0.0623, 0.0982] | [−0.0460, 0.0362] | [3.4944, 5.7573] | |

4.05 | 2.64 | 4.36 | 1.70 | 3.39 |

$\mathit{\gamma}$ | $\mathit{\varphi}$ | $\mathit{\tau}$ | $\mathit{\alpha}$ | $\mathit{\nu}$ | |
---|---|---|---|---|---|

SV-N (7) ${}^{a}$ | −0.2602 ${}^{b}$ | 0.8529 | 0.1428 | ||

[−1.5977, 0.8891] ${}^{c}$ | [0.6617, 0.9395] | [0.0858, 0.2348] | |||

3.47 ${}^{d}$ | 4.38 | 4.62 | |||

SV-G (7) | −1.1175 | 0.8638 | 0.0873 | 3.6578 | |

[−3.7543, 1.4340] | [0.6609, 0.9457] | [0.0642, 0.1144] | [2.4812, 5.1414] | ||

4.21 | 4.24 | 4.45 | 3.86 | ||

SV-SG (10) | −1.0562 | 0.8226 | 0.0828 | −0.0042 | 3.5728 |

[−4.0335, 2.0096] | [0.1974, 0.9485] | [0.0587, 0.1170] | [−0.0565, 0.0490] | [2.4627, 4.8738] | |

4.27 | 4.60 | 4.52 | 1.19 | 3.73 | |

SV-T (8) | −1.2853 | 0.9727 | 0.0730 | 6.2435 | |

[−4.7632, 1.8139] | [0.9448, 0.9898] | [0.0547, 0.0998] | [4.5496, 8.6440] | ||

3.92 | 3.59 | 4.50 | 3.71 | ||

SV-ST (8) | −1.5744 | 0.9726 | 0.0735 | −0.0084 | 6.2388 |

[−5.5186, 1.9908] | [0.9459, 0.9898] | [0.0566, 0.0992] | [−0.0536, 0.0378] | [4.6016, 8.6696] | |

4.04 | 3.56 | 4.46 | 1.84 | 3.74 |

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**MDPI and ACS Style**

Nakakita, M.; Nakatsuma, T.
Bayesian Analysis of Intraday Stochastic Volatility Models of High-Frequency Stock Returns with Skew Heavy-Tailed Errors. *J. Risk Financial Manag.* **2021**, *14*, 145.
https://doi.org/10.3390/jrfm14040145

**AMA Style**

Nakakita M, Nakatsuma T.
Bayesian Analysis of Intraday Stochastic Volatility Models of High-Frequency Stock Returns with Skew Heavy-Tailed Errors. *Journal of Risk and Financial Management*. 2021; 14(4):145.
https://doi.org/10.3390/jrfm14040145

**Chicago/Turabian Style**

Nakakita, Makoto, and Teruo Nakatsuma.
2021. "Bayesian Analysis of Intraday Stochastic Volatility Models of High-Frequency Stock Returns with Skew Heavy-Tailed Errors" *Journal of Risk and Financial Management* 14, no. 4: 145.
https://doi.org/10.3390/jrfm14040145