# Selection Criteria in Regime Switching Conditional Volatility Models

## Abstract

**:**

## 1. Introduction

## 2. Theory: Models and Selection Criteria

#### 2.1. Models

#### 2.1.1. Univariate GARCH Model

#### 2.1.2. Asymmetric Volatility Models

#### 2.1.3. MS-GARCH Models

#### 2.2. Selection Criteria: Information Criteria and Loss Functions

## 3. Design of the Experiments

#### 3.1. Common Design: Starting Values and Numerical Method

**Remark 1**From now on, we consider a selection criterion as strongly efficient if and only if it leads to the selection of the true DGP in at least $90\%$ of cases. Moreover, we consider a selection criterion as weakly efficient if and only if it leads to selection of the true DGP regardless of the assumption on the distribution of the error term in at least $90\%$ of cases.

#### 3.2. Experiment 1: Simulation of MS-GARCH-K Processes

#### 3.3. Experiment 2: Simulation of MS-GARCH-H Processes

#### 3.4. Experiment 3: Simulation of LST-GARCH Processes

## 4. Results and Discussion

#### 4.1. Results

#### 4.1.1. Experiment 1

**Table 1.**Summary of the results of experiments 1 and 2. A cross means that the criterion is weakly efficient as defined in remark 1.

DGP | MSE(${h}_{t}$) | QLIKE(${h}_{t}$) | MAE(${h}_{t}$) | MSE(${\u03f5}_{t}^{2}$) | QLIKE(${\u03f5}_{t}^{2}$) | MAE(${\u03f5}_{t}^{2}$) | AIC | BIC | |
---|---|---|---|---|---|---|---|---|---|

Experiment 1 and 2 with Gaussian innovations | |||||||||

${P}_{1}$ | MS-K | ||||||||

MS-H | x | x | x | x | x | ||||

${P}_{2}$ | MS-K | ||||||||

MS-H | x | x | x | x | x | ||||

${P}_{3}$ | MS-K | x | x | x | x | x | x | x | x |

MS-H | x | x | x | x | x | ||||

${P}_{4}$ | MS-K | ||||||||

MS-H | x | x | x | x | x | x | x | x | |

Experiment 2 | |||||||||

$\gamma =0.5$ | LST-G | ||||||||

$\gamma =1.5$ | LST-G | ||||||||

$\gamma =5$ | LST-G |

**Table 2.**Results of experiment 1 with MSG-GARCH-K DGP when many switches occur. Data are generated with the transition matrix ${P}_{1}$. The top row of the table gives the selection criteria. The left column gives the different models. The line in grey indicates the process which should be selected in first position by each criterion. Regime persistence and volatility persistence are respectively $\delta =-0.8$ and $\lambda =0.9228$.

MSE(${h}_{t}$) | QLIKE(${h}_{t}$) | MAE(${h}_{t}$) | MSE(${\u03f5}_{t}^{2}$) | QLIKE(${\u03f5}_{t}^{2}$) | MAE(${\u03f5}_{t}^{2}$) | AIC | BIC | |
---|---|---|---|---|---|---|---|---|

GARCH | 7.2 | 3.8 | 0.5 | 0 | 0 | 0 | 16.6 | 67.8 |

GARCH-T | 4.2 | 2.1 | 0.5 | 0 | 0 | 0 | 32.2 | 31.1 |

MSG-GARCH-H | 8.0 | 4.9 | 13.2 | 33.5 | 52.2 | 59.2 | 6.6 | 0 |

MST-GARCH-H | 0.4 | 0.6 | 0.6 | 9 | 5.3 | 2.3 | 0.3 | 0 |

MSG-GARCH-K | 30.7 | 46.1 | 54.3 | 49.1 | 34.2 | 34.0 | 27.1 | 0 |

MST-GARCH-K | 40.6 | 40.6 | 29.4 | 8.4 | 8.3 | 4.5 | 1.1 | 0 |

LST-GARCH | 1.1 | 0.9 | 0.1 | 0 | 0 | 0 | 0.4 | 0 |

LST-GARCH-T | 1.3 | 0.2 | 0 | 0 | 0 | 0 | 1.3 | 0 |

GJR | 0.7 | 0.4 | 0.1 | 0 | 0 | 0 | 3.8 | 0.8 |

GJR-T | 0.7 | 0.3 | 0.1 | 0 | 0 | 0 | 5.5 | 0.1 |

EGARCH | 3.1 | 0.1 | 0.4 | 0 | 0 | 0 | 2.0 | 0.1 |

EGARCH-T | 2.0 | 0 | 0.8 | 0 | 0 | 0 | 2.1 | 0 |

**Table 3.**Results of experiment 1 with MS-GARCH-K DGP when probabilities of remaining are equal to probabilities of switching. Data are generated with the transition matrix ${P}_{2}$. The top row of the table gives the selection criteria. The left column gives the different models. The line in grey indicates the process which should be selected in first position, by each criterion. Regime persistence and volatility persistence are respectively $\delta =0$ and $\lambda \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.9165$.

MSE(${h}_{t}$) | QLIKE(${h}_{t}$) | MAE(${h}_{t}$) | MSE(${\u03f5}_{t}^{2}$) | QLIKE(${\u03f5}_{t}^{2}$) | MAE(${\u03f5}_{t}^{2}$) | AIC | BIC | |
---|---|---|---|---|---|---|---|---|

GARCH | 0.5 | 0.1 | 0 | 0 | 0 | 0 | 0 | 1.8 |

GARCH-T | 0.2 | 0 | 0.1 | 0 | 0 | 0 | 8.2 | 82.3 |

MSG-GARCH-H | 24.0 | 17.3 | 23.8 | 13.3 | 21.6 | 19.3 | 21.4 | 5.2 |

MST-GARCH-H | 9.3 | 7.0 | 5.4 | 5.6 | 1.7 | 1.5 | 8.4 | 0.2 |

MSG-GARCH-K | 36.1 | 46.1 | 49.5 | 74.6 | 72.7 | 74.4 | 42.1 | 5.4 |

MST-GARCH-K | 21.9 | 29.4 | 19.6 | 6.5 | 4.0 | 4.8 | 12.7 | 0.1 |

LST-GARCH | 0 | 0.1 | 0 | 0 | 0 | 0 | 0 | 0 |

LST-GARCH-T | 0.4 | 0 | 0 | 0 | 0 | 0 | 0.2 | 0 |

GJR | 0.2 | 0 | 0 | 0 | 0 | 0 | 0 | 0.1 |

GJR-T | 0 | 0 | 0 | 0 | 0 | 0 | 3.3 | 0.5 |

EGARCH | 6.1 | 0 | 1.6 | 0 | 0 | 0 | 0 | 0.1 |

EGARCH-T | 1.3 | 0 | 0 | 0 | 0 | 0 | 3.7 | 4.3 |

**Table 4.**Results of experiment 1 with MSG-GARCH-K DGP when regime-specific variances are persistent. Data are generated with the transition matrix ${P}_{3}$. The top row of the table gives the selection criteria. The left column gives the different models. The line in grey indicates the process which should be selected in first position, by each criterion. Regime persistence and volatility persistence are respectively $\delta =0.8$ and $\lambda =0.9147$.

MSE(${h}_{t}$) | QLIKE(${h}_{t}$) | MAE(${h}_{t}$) | MSE(${\u03f5}_{t}^{2}$) | QLIKE(${\u03f5}_{t}^{2}$) | MAE(${\u03f5}_{t}^{2}$) | AIC | BIC | |
---|---|---|---|---|---|---|---|---|

GARCH | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

GARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

MSG-GARCH-H | 5.1 | 5.5 | 4.4 | 1.9 | 0.1 | 6.4 | 7.9 | |

MST-GARCH-H | 3.2 | 4.3 | 1.3 | 0.1 | 0.1 | 0 | 1.7 | 0.2 |

MSG-GARCH-K | 48.8 | 43.1 | 64.7 | 85.7 | 93.6 | 90.1 | 72.2 | 88.8 |

MST-GARCH-K | 42.9 | 47.1 | 29.6 | 12.2 | 5.9 | 9.8 | 19.7 | 3.1 |

LST-GARCH | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

LST-GARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

GJR | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

GJR-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

EGARCH | 0 | 0 | 0 | 0.1 | 0 | 0 | 0 | 0 |

EGARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

**Table 5.**Results of experiment 1 with MS-GARCH-K DGP when the first regime has a high variance which occurs very few times. Data are generated with the transition matrix ${P}_{4}$. The top row of the table gives the selection criteria. The left column gives the different models. The line in grey indicates the process which should be selected in first position, by each criterion. Regime persistence and volatility persistence are respectively $\delta =0$ and $\lambda =0.8888$.

MSE(${h}_{t}$) | QLIKE(${h}_{t}$) | MAE(${h}_{t}$) | MSE(${\u03f5}_{t}^{2}$) | QLIKE(${\u03f5}_{t}^{2}$) | MAE(${\u03f5}_{t}^{2}$) | AIC | BIC | |
---|---|---|---|---|---|---|---|---|

GARCH | 8.4 | 5.7 | 5.7 | 0 | 0 | 0 | 21.9 | 69.4 |

GARCH-T | 7.3 | 5.5 | 5.9 | 0.1 | 0 | 0.1 | 43.8 | 28.8 |

MSG-GARCH-H | 13.8 | 15.5 | 14.5 | 34.7 | 25.8 | 27.2 | 5.3 | 0 |

MST-GARCH-H | 10.0 | 10.3 | 6.0 | 27.6 | 16.8 | 20.2 | 0.2 | 0 |

MSG-GARCH-K | 22.2 | 34.2 | 41.7 | 30.0 | 51.9 | 45.3 | 5.0 | 0 |

MST-GARCH-K | 14.9 | 19.7 | 16.7 | 7.4 | 5.2 | 7.0 | 0.5 | 0 |

LST-GARCH | 3.8 | 1.7 | 1.2 | 0 | 0 | 0 | 0.2 | 0 |

LST-GARCH-T | 4.8 | 3.3 | 1.8 | 0.2 | 0.2 | 0 | 1.0 | 0 |

GJR | 2.3 | 1.1 | 1.5 | 0 | 0.1 | 0 | 6.9 | 1.4 |

GJR-T | 2.2 | 2.3 | 2.1 | 0 | 0 | 0 | 9.9 | 0.1 |

EGARCH | 5.1 | 0.5 | 1.1 | 0 | 0 | 0 | 1.9 | 0.2 |

EGARCH-T | 5.2 | 0.2 | 1.8 | 0 | 0 | 0.1 | 3.4 | 0.1 |

#### 4.1.2. Experiment 2

**Table 6.**Results of experiment 2 with MS-GARCH-H DGP when many switches occur. Data are generated with the transition matrix ${P}_{1}$. The top row of the table gives the selection criteria. The left column gives the different models. The line in grey indicates the process which should be selected in first position, by each criterion. Regime persistence and volatility persistence are respectively $\delta =-0.8$ and $\lambda =0.9228$.

MSE(${h}_{t}$) | QLIKE(${h}_{t}$) | MAE(${h}_{t}$) | MSE(${\u03f5}_{t}^{2}$) | QLIKE(${\u03f5}_{t}^{2}$) | MAE(${\u03f5}_{t}^{2}$) | AIC | BIC | |
---|---|---|---|---|---|---|---|---|

GARCH | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

GARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

MSG-GARCH-H | 52.3 | 46.9 | 67.8 | 12.2 | 49.8 | 7.8 | 93.6 | 99.4 |

MST-GARCH-H | 45.7 | 45.9 | 30.2 | 7.6 | 7.6 | 2.3 | 6.3 | 5.0 |

MSG-GARCH-K | 0.9 | 3.9 | 1.1 | 65.8 | 30.6 | 70.8 | 0.1 | 0.1 |

MST-GARCH-K | 1 | 3.3 | 0.9 | 12.0 | 12.0 | 19.1 | 0 | 0 |

LST-GARCH | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

LST-GARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

GJR | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

GJR-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

EGARCH | 0.1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

EGARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

**Table 7.**Results of experiment 2 with MS-GARCH-H DGP when probabilities of remaining are equal to probabilities of switching. Data are generated with the transition matrix ${P}_{2}$. The top row of the table gives the selection criteria. The left column gives the different models. The line in grey indicates the process which should be selected in first position, by each criterion. Regime persistence and volatility persistence are respectively $\delta =0$ and $\lambda =0.9165$.

MSE(${h}_{t}$) | QLIKE(${h}_{t}$) | MAE(${h}_{t}$) | MSE(${\u03f5}_{t}^{2}$) | QLIKE(${\u03f5}_{t}^{2}$) | MAE(${\u03f5}_{t}^{2}$) | AIC | BIC | |
---|---|---|---|---|---|---|---|---|

GARCH | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

GARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

MSG-GARCH-H | 57.9 | 54.6 | 76.8 | 34.2 | 57.2 | 39.0 | 94.9 | 99.2 |

MST-GARCH-H | 41.1 | 39.5 | 19.1 | 5.1 | 3.2 | 2.8 | 5.0 | 0.6 |

MSG-GARCH-K | 0.7 | 3.2 | 3.7 | 53.4 | 36.8 | 54.1 | 0.1 | 0.2 |

MST-GARCH-K | 0.2 | 2.7 | 0.4 | 7.3 | 3.3 | 4.1 | 0 | 0 |

LST-GARCH | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

LST-GARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

GJR | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

GJR-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

EGARCH | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

EGARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

**Table 8.**Results of experiment 2 with MSG-GARCH-H DGP when regime specific variances are persistent. Data are generated with the transition matrix ${P}_{3}$. The top row of the table gives the selection criteria. The left column gives the different models. The line in grey indicates the process which should be selected in first position, by each criterion. Regime persistence and volatility persistence are respectively $\delta =0.8$ and $\lambda =0.9147$.

MSE(${h}_{t}$) | QLIKE(${h}_{t}$) | MAE(${h}_{t}$) | MSE(${\u03f5}_{t}^{2}$) | QLIKE(${\u03f5}_{t}^{2}$) | MAE(${\u03f5}_{t}^{2}$) | AIC | BIC | |
---|---|---|---|---|---|---|---|---|

GARCH | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

GARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

MSG-GARCH-H | 59.9 | 64.6 | 79.2 | 48.7 | 19.1 | 35.9 | 97.9 | 99.8 |

MST-GARCH-H | 39.5 | 32.8 | 19.1 | 13.1 | 2.8 | 7.5 | 2.0 | 0 |

MSG-GARCH-K | 0.6 | 1.8 | 1.7 | 31.2 | 73.0 | 50.0 | 0.1 | 0.2 |

MST-GARCH-K | 0 | 0.1 | 0 | 6.9 | 5.1 | 6.6 | 0 | 0 |

LST-GARCH | 0 | 0 | 0 | 0.1 | 0 | 0 | 0 | 0 |

LST-GARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

GJR | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

GJR-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

EGARCH | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

EGARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

**Table 9.**Results of experiment 2 with MSG-GARCH-H DGP when the first regime has a high variance which occurs very few times. Data are generated with the transition matrix ${P}_{4}$. The top row of the table gives the selection criteria. The left column gives the different models. The line in grey indicates the process which should be selected in first position by each criterion. Regime persistence and volatility persistence are respectively $\delta =0$ and $\lambda =0.8888$.

MSE(${h}_{t}$) | QLIKE(${h}_{t}$) | MAE(${h}_{t}$) | MSE(${\u03f5}_{t}^{2}$) | QLIKE(${\u03f5}_{t}^{2}$) | MAE(${\u03f5}_{t}^{2}$) | AIC | BIC | |
---|---|---|---|---|---|---|---|---|

GARCH | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

GARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 1.3 | 21.8 |

MSG-GARCH-H | 88.2 | 82.0 | 87.2 | 73.0 | 83.6 | 64.2 | 89.9 | 74.9 |

MST-GARCH-H | 8.8 | 8.7 | 5.3 | 13.5 | 3.8 | 5.9 | 1.8 | 0.1 |

MSG-GARCH-K | 3.0 | 9.3 | 7.5 | 13.2 | 12.5 | 29.7 | 6.3 | 3.0 |

MST-GARCH-K | 0.1 | 0 | 0 | 0.2 | 0.1 | 0.2 | 0 | 0 |

LST-GARCH | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

LST-GARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

GJR | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

GJR-T | 0 | 0 | 0 | 0 | 0 | 0 | 0.6 | 0.1 |

EGARCH | 0 | 0 | 0 | 0.1 | 0 | 0 | 0 | 0 |

EGARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0.1 | 0.1 |

#### 4.1.3. Experiment 3

**Table 10.**Selection results of experiment 3 with a smooth logistic function. Data are generated with a LST-GARCH model with $\gamma =0.5$. The top row of the table gives the selection criteria. The left column gives the different models. The line in grey indicates the process which should be selected in first position, by each criterion. The volatility persistence is $\lambda =0.825$.

MSE(${h}_{t}$) | QLIKE(${h}_{t}$) | MAE(${h}_{t}$) | MSE(${\u03f5}_{t}^{2}$) | QLIKE(${\u03f5}_{t}^{2}$) | MAE(${\u03f5}_{t}^{2}$) | AIC | BIC | |
---|---|---|---|---|---|---|---|---|

GARCH | 4.3 | 9.9 | 6.9 | 0.1 | 0 | 0 | 56.2 | 96.3 |

GARCH-T | 4.0 | 8.8 | 6.4 | 0.3 | 0 | 0.2 | 2.6 | 0.2 |

MSG-GARCH-H | 1.8 | 2.2 | 4.0 | 29.6 | 21.9 | 13.7 | 1.3 | 0 |

MST-GARCH-H | 2.8 | 2.5 | 5.2 | 1.5 | 3.5 | 25.0 | 0 | 0 |

MSG-GARCH-K | 1.1 | 1.9 | 2.3 | 41.8 | 61.3 | 45.2 | 4 | 0 |

MST-GARCH-K | 4.3 | 7.5 | 6.5 | 3.9 | 3.4 | 5.1 | 0 | 0 |

LST-GARCH | 30.9 | 32.3 | 27.3 | 7.3 | 2.5 | 1.9 | 3.5 | 0 |

LST-GARCH-T | 16.7 | 11.1 | 18.8 | 6.0 | 4.6 | 1.9 | 0.1 | 0 |

GJR | 14.5 | 10.0 | 9.9 | 1.4 | 1.9 | 0.9 | 26.5 | 3.3 |

GJR-T | 14.8 | 13.5 | 11.4 | 1.1 | 0 | 0.3 | 1.0 | 0 |

EGARCH | 2.2 | 0.2 | 0.8 | 3.9 | 0.8 | 3.0 | 8.0 | 0.2 |

EGARCH-T | 2.6 | 0.1 | 0.5 | 3.1 | 0.1 | 2.8 | 0.4 | 0 |

**Table 11.**Selection results of experiment 3 with a smooth logistic function. Data are generated with a LST-GARCH model with $\gamma =1.5$. The top row of the table gives the selection criteria. The left column gives the different models. The line in grey indicates the process which should be selected in first position, by each criterion. Volatility persistence is $\lambda =0.825$.

MSE(${h}_{t}$) | QLIKE(${h}_{t}$) | MAE(${h}_{t}$) | MSE(${\u03f5}_{t}^{2}$) | QLIKE(${\u03f5}_{t}^{2}$) | MAE(${\u03f5}_{t}^{2}$) | AIC | BIC | |
---|---|---|---|---|---|---|---|---|

GARCH | 0 | 0 | 0 | 0.1 | 0 | 0 | 7.4 | 34.8 |

GARCH-T | 0.1 | 0.4 | 0.1 | 0.2 | 0 | 0 | 0.6 | 0 |

MSG-GARCH-H | 0 | 0.2 | 0.1 | 28.0 | 25.1 | 14.1 | 0.1 | 0 |

MST-GARCH-H | 0 | 0.1 | 0.1 | 0.6 | 1.9 | 23.9 | 0 | 0 |

MSG-GARCH-K | 0.1 | 0.2 | 0 | 31.9 | 58.9 | 41.8 | 0.1 | 0 |

MST-GARCH-K | 0.1 | 0.4 | 0 | 3.2 | 4.0 | 4.2 | 0 | 0 |

LST-GARCH | 43.4 | 63.5 | 43.1 | 10.4 | 5.3 | 3.2 | 23.4 | 1.3 |

LST-GARCH-T | 33.0 | 19.8 | 34.5 | 8.0 | 2.3 | 1.8 | 2.0 | 0 |

GJR | 9.7 | 6.0 | 9.2 | 1.5 | 1.2 | 1.2 | 46 | 41.3 |

GJR-T | 11.1 | 9.3 | 10.8 | 2.7 | 0 | 0.9 | 3.0 | 0.1 |

EGARCH | 1.4 | 0.7 | 1.3 | 5.9 | 1.2 | 4.7 | 16.3 | 22.5 |

EGARCH-T | 1.1 | 0.4 | 0.8 | 7.5 | 2 | 4.2 | 1.1 | 0 |

**Table 12.**Selection results of experiment 3 with a smooth logistic function. Data are generated with a LST-GARCH model with $\gamma =5$. The top row of the table gives the selection criteria. The left column gives the different models. The line in grey indicates the process which should be selected in first position, by each criterion. The volatility persistence is $\lambda =0.825$.

MSE(${h}_{t}$) | QLIKE(${h}_{t}$) | MAE(${h}_{t}$) | MSE(${\u03f5}_{t}^{2}$) | QLIKE(${\u03f5}_{t}^{2}$) | MAE(${\u03f5}_{t}^{2}$) | AIC | BIC | |
---|---|---|---|---|---|---|---|---|

GARCH | 0 | 0 | 0 | 0.1 | 0 | 0 | 0 | 0.1 |

GARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

MSG-GARCH-H | 0 | 0 | 0 | 33.4 | 33.2 | 22.8 | 0 | 0 |

MST-GARCH-H | 0 | 0 | 0 | 0.8 | 1.0 | 19.7 | 0 | 0 |

MSG-GARCH-K | 0 | 0 | 0 | 30.0 | 47.3 | 37.0 | 0 | 0 |

MST-GARCH-K | 0 | 0 | 0 | 4.4 | 5.4 | 3.9 | 0 | 0 |

LST-GARCH | 48.0 | 62.8 | 49.5 | 7.2 | 3.8 | 2.4 | 28.9 | 1.8 |

LST-GARCH-T | 19.6 | 26.0 | 26.3 | 8.8 | 8.4 | 6.3 | 1.3 | 0 |

GJR | 18.1 | 5.3 | 13.5 | 6.3 | 0.8 | 3.2 | 55.8 | 92.8 |

GJR-T | 14.3 | 5.9 | 10.7 | 8.4 | 0.1 | 4.5 | 3.1 | 0.3 |

EGARCH | 0 | 0 | 0 | 0.4 | 0 | 5.7 | 10.9 | 5.0 |

EGARCH-T | 0 | 0 | 0 | 0.2 | 0 | 0.2 | 0 | 0 |

#### 4.2. Discussion

**Figure 2.**Box plots of MSG-GARCH parameter estimations. (

**a**) MS-GARCH-K with ${P}_{1}$; (

**b**) MS-GARCH-H with ${P}_{1}$; (

**c**) MS-GARCH-K with ${P}_{2}$; (

**d**) MS-GARCH-H with ${P}_{2}$; (

**e**) MS-GARCH-K with ${P}_{3}$; (

**f**) MS-GARCH-H with ${P}_{3}$; (

**g**) MS-GARCH-K with ${P}_{4}$; (

**h**) MS-GARCH-H with ${P}_{4}$.

**Figure 3.**Non-parametric density estimation of the simulated and estimated conditional volatility for one replication in Experiment 1 and 2. (

**a**) Transition matrix ${P}_{1}$; (

**b**) Transition matrix ${P}_{3}$.

**Figure 4.**Non-parametric density estimation of ${\alpha}_{2}$ and γ for the three sets of parameters. (

**a**,

**b**) correspond to the first set $\gamma =0.5$; (

**c**,

**d**) correspond to the second set $\gamma =1.5$; and (

**e**,

**f**) correspond to the third set $\gamma =5$.

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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^{3.}See [11] for example.^{4.}There are a number of expansions of these two MS-GARCH processes. For example, Gallo and Otrento [27] introduce asymmetric effects in each regime variance.^{5.}Hu and Shin [32] introduced a test procedure which tests the null hypothesis of a GARCH process against an MS-GARCH process.^{6.}For each experiment, we estimate these models: GARCH, GARCH-T, LST-GARCH, LST-GARCH-T, GJR-GARCH, GJR-GARCH-T, EGARCH, AEGARCH-T, MSG(2)-GARCH-H, MSG(2)-GARCH-K, MST(2)-GARCH-H and MST(2)-GARCH-K.^{7.}Results for $T=1000$ are available on demand, results remain the same. We do not consider smaller sample size since in financial application, we used to study daily data.^{9.}We set ${\u03f5}_{0}=0$, ${h}_{0}=1$ and ${\Delta}_{0}=1$. We generate 2000 more observations than required to minimize any starting bias.^{10.}The probabilities of being in regime $i=1,2$. The long-run probability of the first regime: ${\pi}_{1}$ is equal to ${\pi}_{1}=\frac{1-{p}_{22}}{2-{p}_{11}-{p}_{22}}$.^{11.}δ is computed as follows: $\delta ={p}_{11}+{p}_{22}-1$.^{12.}We set ${\u03f5}_{0}=0$, $h{\left({\Delta}_{0}\right)}_{0}=1$ and ${\Delta}_{0}=1$. We generate 2000 more observations than required, to minimize any starting bias.^{13.}We generate 2000 more observations than required to minimize any starting bias.^{14.}Estimation computed with Gaussian kernel and Silverman’s rule of thumb.^{15.}Figure 3(a) is related to the 40th replication of the first and the second experiments with matrix ${P}_{1}$, BIC selects the right specification when data are simulated with MSG-GARCH-H but it selects the GARCH model for data simulated with MS-GARCHG-K. Figure 3(b) is related to the 66th replication of the first and second experiments with ${P}_{3}$ where there is no selection problem.^{16.}Estimation computed with Gaussian kernel and Silverman’s rule of thumb.

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**MDPI and ACS Style**

Chuffart, T.
Selection Criteria in Regime Switching Conditional Volatility Models. *Econometrics* **2015**, *3*, 289-316.
https://doi.org/10.3390/econometrics3020289

**AMA Style**

Chuffart T.
Selection Criteria in Regime Switching Conditional Volatility Models. *Econometrics*. 2015; 3(2):289-316.
https://doi.org/10.3390/econometrics3020289

**Chicago/Turabian Style**

Chuffart, Thomas.
2015. "Selection Criteria in Regime Switching Conditional Volatility Models" *Econometrics* 3, no. 2: 289-316.
https://doi.org/10.3390/econometrics3020289