# Flexible Time-Varying Betas in a Novel Mixture Innovation Factor Model with Latent Threshold

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Data and Methodology

#### 3.1. Data

#### 3.2. Methodology

## 4. Empirical Results

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Monthly industry portfolio excess returns: (

**a**) Consumer durables; (

**b**) Manufacturing; (

**c**) Energy; (

**d**) High tech. Note: The figure presents the plots for monthly excess portfolio returns over the sample period of July 1963–December 2020.

**Figure 2.**Threshold time-varying parameter beta estimates—Consumer Durables. Note: The figure presents the plots for the posterior medians (solid line) with 68% confidence bands (gray shaded region) of the time varying beta estimates from the TTVP model with stochastic volatility. Gibbs sampling is used with 10,000 posterior and 10,000 burn-in draws. MKT, SMB, HML, RMW and CMA represent the Market, Size, Book-to-Market, Profitability and Investment factors based on the five-factor specification of Fama and French [57].

**Figure 3.**Threshold time-varying parameter beta estimates—Manufacturing. Note: The figure presents the plots for the posterior medians (solid line) with 68% confidence bands (gray shaded region) of the time varying beta estimates from the TTVP model with stochastic volatility. Gibbs sampling is used with 10,000 posterior and 10,000 burn-in draws. MKT, SMB, HML, RMW and CMA represent the Market, Size, Book-to-Market, Profitability and Investment factors based on the five-factor specification of Fama and French [57].

**Figure 4.**Threshold time-varying parameter beta estimates—Energy. Note: The figure presents the plots for the posterior medians (solid line) with 68% confidence bands (gray shaded region) of the time varying beta estimates from the TTVP model with stochastic volatility. Gibbs sampling is used with 10,000 posterior and 10,000 burn-in draws. MKT, SMB, HML, RMW and CMA represent the Market, Size, Book-to-Market, Profitability and Investment factors based on the five-factor specification of Fama and French [57].

**Figure 5.**Threshold time-varying parameter beta estimates—High Tech. Note: The figure presents the plots for the posterior medians (solid line) with 68% confidence bands (gray shaded region) of the time varying beta estimates from the TTVP model with stochastic volatility. Gibbs sampling is used with 10,000 posterior and 10,000 burn-in draws. MKT, SMB, HML, RMW and CMA represent the Market, Size, Book-to-Market, Profitability and Investment factors based on the five-factor specification of Fama and French [57].

Statistic | DURBL | MANUF | ENRGY | HITEC | MKT | SMB | HML | RMW | CMA |
---|---|---|---|---|---|---|---|---|---|

Mean | 0.624 | 0.610 | 0.537 | 0.711 | 0.568 | 0.230 | 0.251 | 0.249 | 0.256 |

S.D. | 6.726 | 4.955 | 5.901 | 6.394 | 4.468 | 3.026 | 2.867 | 2.167 | 1.988 |

Min | −32.710 | −27.930 | −34.730 | −26.530 | −23.240 | −14.890 | −13.960 | −18.480 | −6.860 |

Max | 42.620 | 17.500 | 32.330 | 20.320 | 16.100 | 18.080 | 12.580 | 13.380 | 9.560 |

Skewness | 0.602 | −0.504 | 0.005 | −0.241 | −0.507 | 0.334 | 0.013 | −0.327 | 0.316 |

Kurtosis | 5.975 | 2.506 | 4.032 | 1.290 | 1.878 | 2.947 | 2.358 | 12.238 | 1.603 |

Jarque–Bera (JB) | 1077.237 * | 212.099 * | 472.145 * | 55.471 * | 132.675 * | 265.584 * | 161.942 * | 4349.274 * | 86.602 * |

Q(1) | 8.152 * | 1.676 | 0.717 | 2.408 | 2.652 | 2.947 * | 21.859 * | 15.273 * | 9.901 * |

Q(6) | 23.302 * | 7.253 | 4.158 | 4.806 | 7.417 | 10.664 * | 30.461 * | 20.396 * | 19.610 * |

ARCH(1) | 14.192 * | 10.178 * | 83.967 * | 55.253 * | 18.336 * | 58.167 * | 40.465 * | 122.214 * | 69.473 * |

Variable | Mean | Median | S.D. | Min | Max | 5th Percentile | 95th Percentile |
---|---|---|---|---|---|---|---|

Panel A: TTVP-SV Model | |||||||

Intercept | −0.338 | −0.381 | 0.093 | −0.436 | −0.037 | −0.431 | −0.153 |

MKT | 1.165 | 1.162 | 0.010 | 1.153 | 1.180 | 1.154 | 1.180 |

SMB | 0.224 | 0.378 | 0.192 | 0.000 | 0.415 | 0.002 | 0.414 |

HML | 0.287 | 0.267 | 0.073 | 0.162 | 0.421 | 0.171 | 0.416 |

RMW | 0.157 | 0.097 | 0.110 | 0.047 | 0.356 | 0.050 | 0.350 |

CMA | 0.094 | 0.057 | 0.098 | −0.023 | 0.269 | −0.018 | 0.266 |

Panel B: Rolling Model | |||||||

Intercept | −0.443 | −0.506 | 0.288 | −0.978 | 0.208 | −0.827 | 0.105 |

MKT | 1.221 | 1.181 | 0.182 | 0.954 | 1.623 | 0.991 | 1.583 |

SMB | 0.303 | 0.256 | 0.246 | −0.149 | 0.838 | −0.106 | 0.762 |

HML | 0.376 | 0.406 | 0.333 | −0.408 | 1.030 | −0.277 | 0.887 |

RMW | 0.264 | 0.198 | 0.424 | −0.382 | 1.150 | −0.287 | 1.034 |

CMA | 0.117 | 0.071 | 0.300 | −0.575 | 0.632 | −0.340 | 0.583 |

Panel C: Static Model | |||||||

Intercept | −0.340 | 0.150 | −0.586 | −0.094 | |||

MKT | 1.267 | 0.037 | 1.207 | 1.327 | |||

SMB | 0.214 | 0.052 | 0.128 | 0.300 | |||

HML | 0.377 | 0.069 | 0.263 | 0.491 | |||

RMW | 0.254 | 0.072 | 0.136 | 0.373 | |||

CMA | 0.147 | 0.106 | −0.027 | 0.320 |

Variable | Mean | Median | S.D. | Min | Max | 5th Percentile | 95th Percentile |
---|---|---|---|---|---|---|---|

Panel A: TTVP-SV Model | |||||||

Intercept | −0.086 | −0.083 | 0.033 | −0.135 | −0.001 | −0.133 | −0.025 |

MKT | 1.067 | 1.067 | 0.000 | 1.066 | 1.067 | 1.066 | 1.067 |

SMB | 0.102 | 0.102 | 0.001 | 0.101 | 0.104 | 0.101 | 0.104 |

HML | 0.085 | 0.085 | 0.001 | 0.084 | 0.086 | 0.085 | 0.086 |

RMW | 0.240 | 0.240 | 0.003 | 0.235 | 0.245 | 0.236 | 0.245 |

CMA | 0.053 | 0.075 | 0.036 | 0.002 | 0.108 | 0.006 | 0.107 |

Panel B: Rolling Model | |||||||

Intercept | −0.118 | −0.095 | 0.148 | −0.471 | 0.120 | −0.369 | 0.078 |

MKT | 1.105 | 1.091 | 0.058 | 0.980 | 1.265 | 1.022 | 1.200 |

SMB | 0.103 | 0.115 | 0.062 | −0.024 | 0.258 | 0.001 | 0.198 |

HML | 0.014 | 0.018 | 0.106 | −0.253 | 0.263 | −0.204 | 0.161 |

RMW | 0.222 | 0.281 | 0.230 | −0.337 | 0.606 | −0.225 | 0.478 |

CMA | 0.116 | 0.127 | 0.134 | −0.175 | 0.406 | −0.105 | 0.336 |

Panel C: Static Model | |||||||

Intercept | −0.164 | 0.061 | −0.265 | −0.063 | |||

MKT | 1.084 | 0.015 | 1.059 | 1.108 | |||

SMB | 0.110 | 0.021 | 0.074 | 0.145 | |||

HML | 0.088 | 0.028 | 0.041 | 0.134 | |||

RMW | 0.320 | 0.030 | 0.271 | 0.369 | |||

CMA | 0.126 | 0.043 | 0.055 | 0.197 |

Variable | Mean | Median | S.D. | Min | Max | 5th Percentile | 95th Percentile |
---|---|---|---|---|---|---|---|

Panel A: TTVP-SV Model | |||||||

Intercept | −0.087 | −0.117 | 0.129 | −0.302 | 0.099 | −0.292 | 0.094 |

MKT | 0.963 | 0.954 | 0.017 | 0.951 | 1.007 | 0.952 | 1.003 |

SMB | −0.118 | −0.181 | 0.143 | −0.255 | 0.335 | −0.250 | 0.261 |

HML | 0.096 | 0.026 | 0.172 | −0.018 | 0.547 | −0.011 | 0.527 |

RMW | −0.086 | −0.103 | 0.184 | −0.404 | 0.195 | −0.400 | 0.192 |

CMA | 0.415 | 0.168 | 0.403 | −0.084 | 1.261 | −0.047 | 1.256 |

Panel B: Rolling Model | |||||||

Intercept | 0.022 | 0.131 | 0.529 | −1.181 | 1.010 | −0.823 | 0.815 |

MKT | 0.967 | 0.973 | 0.135 | 0.680 | 1.348 | 0.739 | 1.153 |

SMB | −0.161 | −0.196 | 0.218 | −0.577 | 0.518 | −0.487 | 0.172 |

HML | 0.031 | −0.063 | 0.261 | −0.362 | 0.733 | −0.279 | 0.563 |

RMW | 0.042 | 0.052 | 0.545 | −0.991 | 0.921 | −0.910 | 0.773 |

CMA | 0.304 | 0.145 | 0.605 | −0.689 | 1.748 | −0.510 | 1.675 |

Panel C: Static Model | |||||||

Intercept | −0.224 | 0.168 | −0.499 | 0.052 | |||

MKT | 1.008 | 0.041 | 0.941 | 1.075 | |||

SMB | −0.063 | 0.058 | −0.159 | 0.033 | |||

HML | 0.230 | 0.078 | 0.102 | 0.357 | |||

RMW | 0.209 | 0.081 | 0.076 | 0.342 | |||

CMA | 0.365 | 0.118 | 0.170 | 0.559 |

Variable | Mean | Median | S.D. | Min | Max | 5th Percentile | 95th Percentile |
---|---|---|---|---|---|---|---|

Panel A: TTVP-SV Model | |||||||

Intercept | 0.227 | 0.247 | 0.057 | 0.105 | 0.318 | 0.151 | 0.315 |

MKT | 1.054 | 1.053 | 0.002 | 1.051 | 1.056 | 1.051 | 1.056 |

SMB | 0.075 | 0.084 | 0.021 | 0.025 | 0.088 | 0.026 | 0.087 |

HML | −0.234 | −0.278 | 0.119 | −0.362 | −0.010 | −0.359 | −0.028 |

RMW | −0.149 | −0.163 | 0.219 | −0.646 | 0.181 | −0.619 | 0.179 |

CMA | −0.466 | −0.462 | 0.054 | −0.561 | −0.387 | −0.556 | −0.392 |

Panel B: Rolling Model | |||||||

Intercept | 0.326 | 0.178 | 0.390 | −0.150 | 1.452 | −0.104 | 1.181 |

MKT | 1.043 | 1.032 | 0.124 | 0.835 | 1.376 | 0.873 | 1.266 |

SMB | 0.125 | 0.107 | 0.127 | −0.222 | 0.398 | −0.035 | 0.379 |

HML | −0.290 | −0.263 | 0.306 | −0.860 | 0.318 | −0.768 | 0.202 |

RMW | −0.131 | −0.246 | 0.320 | −0.718 | 0.573 | −0.508 | 0.493 |

CMA | −0.398 | −0.419 | 0.407 | −1.357 | 0.365 | −1.088 | 0.242 |

Panel C: Static Model | |||||||

Intercept | −0.224 | 0.168 | −0.499 | 0.052 | |||

MKT | 1.008 | 0.041 | 0.941 | 1.075 | |||

SMB | −0.063 | 0.058 | −0.159 | 0.033 | |||

HML | 0.230 | 0.078 | 0.102 | 0.357 | |||

RMW | 0.209 | 0.081 | 0.076 | 0.342 | |||

CMA | 0.365 | 0.118 | 0.170 | 0.559 |

Model | DURBL | MANUF | ENRGY | HITEC |
---|---|---|---|---|

TTVP-SV | 3.670 | 1.531 | 3.796 | 2.403 |

TTVP | 3.525 | 1.520 | 3.696 | 2.372 |

Rolling | 3.708 | 1.544 | 4.163 | 2.484 |

Static | 3.747 | 1.533 | 4.195 | 2.575 |

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**MDPI and ACS Style**

Balcilar, M.; Demirer, R.; Bekun, F.V.
Flexible Time-Varying Betas in a Novel Mixture Innovation Factor Model with Latent Threshold. *Mathematics* **2021**, *9*, 915.
https://doi.org/10.3390/math9080915

**AMA Style**

Balcilar M, Demirer R, Bekun FV.
Flexible Time-Varying Betas in a Novel Mixture Innovation Factor Model with Latent Threshold. *Mathematics*. 2021; 9(8):915.
https://doi.org/10.3390/math9080915

**Chicago/Turabian Style**

Balcilar, Mehmet, Riza Demirer, and Festus V. Bekun.
2021. "Flexible Time-Varying Betas in a Novel Mixture Innovation Factor Model with Latent Threshold" *Mathematics* 9, no. 8: 915.
https://doi.org/10.3390/math9080915