# An Experiment on Autoregressive and Threshold Autoregressive Models with Non-Gaussian Error with Application to Realized Volatility

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Model Specification

#### 2.2. Model Estimation

#### 2.3. Empirical Data Analysis Preparation

## 3. Results

#### 3.1. Simulation Study

#### 3.2. Empirical Data Analysis

## 4. Discussion

#### 4.1. Simulation Study

#### 4.2. Empirical Data Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AR | Autoregressive |

AIC | Akaike Information Criterion |

BAR | Buffered Threshold Autoregressive |

BIC | Bayesian Information Criterion |

EU | European Union |

GARCH | Generalized Autoregressive Conditional Heteroscedastic |

MSE | Mean Square Error |

NYSE | New York Stock Exchange |

PACF | Partial Auto-Correlation Function |

RV | Realized Variance |

SV | Stochastic Variance |

TAR | Threshold Autoregressive |

US | United States |

## Appendix A

StockNum | p | d | AIC | Ljung–Box Test | p-Value |
---|---|---|---|---|---|

1 | 2 | 2 | 275.7274 | Significant | 0.0073 |

2 | 2 | 2 | 2.2509 | Insignificant | 0.2034 |

3 | 5 | 3 | 3.8177 | Insignificant | 0.0220 |

4 | 2 | 2 | 246.9670 | Insignificant | 0.3230 |

5 | 5 | 1 | 143.1260 | Significant | 0.0155 |

6 | 4 | 2 | −28.6272 | Significant | 0.0218 |

7 | 4 | 1 | −9.54336 | Insignificant | 0.0716 |

8 | 5 | 1 | −21.6088 | Insignificant | 0.1159 |

9 | 5 | 1 | 5.7717 | Significant | 0.0000 |

10 | 2 | 1 | 144.2935 | Insignificant | 0.6415 |

11 | 2 | 2 | −17.8815 | Somewhat Significant | 0.0093 |

12 | 5 | 2 | 4.7316 | Significant | 0.0012 |

13 | 4 | 1 | −81.0311 | Somewhat Significant | 0.0064 |

14 | 2 | 1 | −241.8272 | Insignificant | 0.4850 |

15 | 2 | 2 | 157.0407 | Significant | 0.0000 |

16 | 3 | 1 | −180.8073 | Significant | 0.0000 |

17 | 2 | 1 | −127.2748 | Somewhat Significant | 0.0194 |

18 | 1 | 1 | −90.0935 | Significant | 0.0000 |

19 | 5 | 1 | −117.8152 | Significant | 0.0000 |

20 | 3 | 1 | 30.9568 | Significant | 0.0136 |

21 | 4 | 1 | −60.7726 | Significant | 0.0005 |

22 | 1 | 3 | −192.5901 | Significant | 0.0000 |

23 | 2 | 1 | −99.0953 | Significant | 0.0000 |

24 | 3 | 2 | −92.6265 | Significant | 0.0000 |

25 | 2 | 2 | 68.0884 | Insignificant | 0.8520 |

26 | 1 | 1 | 16.6098 | Significant | 0.0005 |

27 | 1 | 1 | 44.5472 | Insignificant | 0.2261 |

28 | 2 | 2 | −121.6981 | Significant | 0.0000 |

29 | 2 | 1 | −120.5036 | Significant | 0.0000 |

30 | 1 | 1 | −93.9557 | Significant | 0.0002 |

## Appendix B

StockNum | p | d | BIC | Ljung-Box Test | p-Value |
---|---|---|---|---|---|

1 | 2 | 2 | 582.7805 | Significant | 0.0073 |

2 | 2 | 2 | 35.8275 | Insignificant | 0.2304 |

3 | 1 | 1 | 36.9940 | Significant | 0.0000 |

4 | 2 | 2 | 525.2598 | Insignificant | 0.3203 |

5 | 5 | 0 | 318.3741 | Significant | 0.0000 |

6 | 1 | 1 | −28.8146 | Significant | 0.0000 |

7 | 1 | 3 | 17.7029 | Insignificant | 0.8858 |

8 | 2 | 0 | −11.8445 | Significant | 0.0060 |

9 | 1 | 1 | 38.3802 | Significant | 0.0000 |

10 | 2 | 1 | 319.9127 | Insignificant | 0.6415 |

11 | 2 | 2 | −4.4373 | Somewhat Significant | 0.0093 |

12 | 1 | 1 | 43.3797 | Significant | 0.0000 |

13 | 1 | 1 | −134.143 | Significant | 0.0000 |

14 | 2 | 1 | −452.3287 | Insignificant | 0.4850 |

15 | 2 | 2 | 345.4071 | Significant | 0.0000 |

16 | 1 | 1 | −331.1404 | Significant | 0.0000 |

17 | 2 | 1 | −223.2238 | Somewhat Significant | 0.0194 |

18 | 1 | 1 | −155.8226 | Significant | 0.0000 |

19 | 1 | 3 | −192.3672 | Significant | 0.0000 |

20 | 1 | 1 | 93.1204 | Significant | 0.0000 |

21 | 1 | 1 | −86.9778 | Significant | 0.0183 |

22 | 1 | 3 | −360.8156 | Significant | 0.0000 |

23 | 2 | 1 | −166.8648 | Significant | 0.0000 |

24 | 1 | 2 | −159.1344 | Significant | 0.0000 |

25 | 2 | 2 | 167.5026 | Insignificant | 0.8520 |

26 | 1 | 1 | 57.5842 | Significant | 0.0002 |

27 | 1 | 1 | 113.4588 | Insignificant | 0.2261 |

28 | 2 | 2 | −212.0704 | Significant | 0.0000 |

29 | 2 | 1 | −209.6814 | Significant | 0.0000 |

30 | 1 | 1 | −163.5470 | Significant | 0.0002 |

## Appendix C

StockNum | p | d | Info Cri * | Ljung-Box Test | p-Value |
---|---|---|---|---|---|

1 | 2 | 2 | BIC | Significant | 0.0073 |

2 | 2 | 2 | AIC | Insignificant | 0.2304 |

3 | 5 | 3 | AIC | Insignificant | 0.0220 |

4 | 2 | 2 | AIC | Insignificant | 0.3203 |

5 | 5 | 0 | BIC | Significant | 0.0000 |

6 | 1 | 1 | BIC | Significant | 0.0000 |

7 | 1 | 3 | BIC | Insignificant | 0.8858 |

8 | 5 | 1 | AIC | Insignificant | 0.1159 |

9 | 1 | 1 | BIC | Significant | 0.0000 |

10 | 2 | 1 | AIC/BIC | Insignificant | 0.6415 |

11 | 2 | 2 | AIC/BIC | Somewhat Significant | 0.0093 |

12 | 1 | 1 | BIC | Significant | 0.0000 |

13 | 4 | 1 | AIC | Somewhat Significant | 0.0064 |

14 | 2 | 1 | AIC/BIC | Insignificant | 0.4850 |

15 | 2 | 2 | AIC/BIC | Significant | 0.0000 |

16 | 1 | 1 | BIC | Significant | 0.0000 |

17 | 2 | 1 | AIC/BIC | Somewhat Significant | 0.0194 |

18 | 1 | 1 | AIC/BIC | Significant | 0.0000 |

19 | 1 | 3 | BIC | Significant | 0.0000 |

20 | 1 | 1 | BIC | Significant | 0.0000 |

21 | 1 | 1 | BIC | Significant | 0.0000 |

22 | 1 | 3 | AIC/BIC | Significant | 0.0000 |

23 | 2 | 1 | AIC/BIC | Significant | 0.0000 |

24 | 1 | 2 | BIC | Significant | 0.0000 |

25 | 2 | 2 | AIC/BIC | Insignificant | 0.8520 |

26 | 1 | 1 | AIC/BIC | Significant | 0.0005 |

27 | 1 | 1 | AIC/BIC | Insignificant | 0.2261 |

28 | 2 | 2 | AIC/BIC | Significant | 0.0000 |

29 | 2 | 1 | AIC/BIC | Significant | 0.0000 |

30 | 1 | 1 | AIC/BIC | Significant | 0.0002 |

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True Model | ${\mathit{\alpha}}_{1}$ | ${\mathit{\beta}}_{1}$ | ${\mathit{\alpha}}_{2}$ | ${\mathit{\beta}}_{2}$ | ${\mathit{\phi}}_{1,1}$ | ${\mathit{\phi}}_{1,2}$ | ${\mathit{\phi}}_{2,1}$ | ${\mathit{\phi}}_{2,2}$ | T |
---|---|---|---|---|---|---|---|---|---|

5 | 2 | 5 | 2 | 0.5 | 0.3 | 0.3 | 0.2 | 30 | |

AIC | Proportion of correct estimation of Autoregressive (AR) order and Lag: 44/50 | ||||||||

AIC Bias | 0.032 | 0.023 | 0.341 | −0.022 | 0.013 | −0.007 | 0.003 | −0.004 | 0.001 |

AIC MSE | 1.245 | 0.083 | 2.759 | 0.123 | 0.002 | 0.003 | 0.002 | 0.002 | 0.001 |

BIC | Proportion of correct estimation of AR order and Lag: 50/50 | ||||||||

BIC Bias | 0.015 | 0.022 | 0.384 | −0.015 | 0.012 | −0.006 | 0.004 | −0.005 | 0.001 |

BIC MSE | 1.198 | 0.08 | 3.887 | 0.143 | 0.002 | 0.003 | 0.002 | 0.002 | 0.001 |

True Model | ${\mathit{\alpha}}_{1}$ | ${\mathit{\beta}}_{1}$ | ${\mathit{\alpha}}_{2}$ | ${\mathit{\beta}}_{2}$ | ${\mathit{\phi}}_{1,1}$ | ${\mathit{\phi}}_{2,1}$ | T |
---|---|---|---|---|---|---|---|

4 | 2 | 4 | 2 | 0.7 | 0.3 | 15 | |

AIC | Proportion of correct estimation of AR order and Lag: 36/50 | ||||||

AIC Bias | 0.34 | −0.074 | 0.068 | −0.006 | −0.008 | 0.007 | 0.019 |

AIC MSE | 0.929 | 0.086 | 0.722 | 0.078 | 0.003 | 0.001 | 0.002 |

BIC | Proportion of correct estimation of AR order and Lag: 50/50 | ||||||

BIC Bias | 0.199 | −0.035 | 0.016 | 0.024 | −0.002 | 0.007 | 0.019 |

BIC MSE | 0.825 | 0.079 | 0.711 | 0.082 | 0.003 | 0.001 | 0.002 |

True Model | $\mathit{\alpha}$ | $\mathit{\beta}$ | ${\mathit{\phi}}_{1}$ | ${\mathit{\phi}}_{2}$ |
---|---|---|---|---|

5 | 2 | 0.6 | 0.2 | |

AIC | Proportion of correct estimation of AR order and Lag: 7/50 | |||

AIC Bias | 0.627275 | −0.08103 | 0.020833 | −0.02968 |

AIC MSE | 2.004022 | 0.087312 | 0.001747 | 0.002992 |

BIC | Proportion of correct estimation of AR order and Lag: 46/50 | |||

BIC Bias | 0.137581 | −0.02197 | 0.006732 | −0.00646 |

BIC MSE | 0.605966 | 0.037434 | 0.001058 | 0.001104 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, Z.; Li, W.K.
An Experiment on Autoregressive and Threshold Autoregressive Models with Non-Gaussian Error with Application to Realized Volatility. *Economies* **2019**, *7*, 58.
https://doi.org/10.3390/economies7020058

**AMA Style**

Zhang Z, Li WK.
An Experiment on Autoregressive and Threshold Autoregressive Models with Non-Gaussian Error with Application to Realized Volatility. *Economies*. 2019; 7(2):58.
https://doi.org/10.3390/economies7020058

**Chicago/Turabian Style**

Zhang, Ziyi, and Wai Keung Li.
2019. "An Experiment on Autoregressive and Threshold Autoregressive Models with Non-Gaussian Error with Application to Realized Volatility" *Economies* 7, no. 2: 58.
https://doi.org/10.3390/economies7020058