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Article

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

by
Mehmet Balcilar
1,
Riza Demirer
2,* and
Festus V. Bekun
3
1
Department of Economics, Eastern Mediterranean University, Northern Cyprus, via Mersin 10, Famagusta 99450, Turkey
2
Department of Economics and Finance, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1102, USA
3
Faculty of Economics Administrative and Social Sciences, Istanbul Gelisim University, Istanbul 34310, Turkey
*
Author to whom correspondence should be addressed.
Mathematics 2021, 9(8), 915; https://doi.org/10.3390/math9080915
Submission received: 30 March 2021 / Revised: 13 April 2021 / Accepted: 14 April 2021 / Published: 20 April 2021
(This article belongs to the Special Issue Application of Mathematical Methods in Financial Economics)

Abstract

:
This paper introduces a new methodology to estimate time-varying alphas and betas in conditional factor models, which allows substantial flexibility in a time-varying framework. To circumvent problems associated with the previous approaches, we introduce a Bayesian time-varying parameter model where innovations of the state equation have a spike-and-slab mixture distribution. The mixture distribution specifies two states with a specific probability. In the first state, the innovation variance is set close to zero with a certain probability and parameters stay relatively constant. In the second state, the innovation variance is large and the change in parameters is normally distributed with mean zero and a given variance. The latent state is specified with a threshold that governs the state change. We allow a separate threshold for each parameter; thus, the parameters may shift in an unsynchronized manner such that the model moves from one state to another when the change in the parameter exceeds the threshold and vice versa. This approach offers great flexibility and nests a plethora of other time-varying model specifications, allowing us to assess whether the betas of conditional factor models evolve gradually over time or display infrequent, but large, shifts. We apply the proposed methodology to industry portfolios within a five-factor model setting and show that the threshold Capital Asset Pricing Model (CAPM) provides robust beta estimates coupled with smaller pricing errors compared to the alternative approaches. The results have significant implications for the implementation of smart beta strategies that rely heavily on the accuracy and stability of factor betas and yields.
JEL Classification Code:
C11; C32; G11; G12; G14

Author Contributions

Conceptualization, M.B. and R.D.; methodology, M.B.; software, M.B.; validation, R.D. and M.B.; formal analysis, M.B.; investigation, R.D., M.B. and F.V.B.; resources, M.B.; data curation, M.B.; writing—original draft preparation, R.D., M.B. and F.V.B.; writing—review and editing, R.D.; visualization, M.B.; supervision, R.D.; project administration, R.D. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

Publicly available datasets from Kenneth R. French were analyzed in this study. This data can be found here: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html (access on 24 March 2021).

Conflicts of Interest

The authors declare no conflict of interest.

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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 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 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 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 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].
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].
Table 1. Descriptive statistics.
Table 1. Descriptive statistics.
StatisticDURBLMANUFENRGYHITECMKTSMBHMLRMWCMA
Mean0.6240.6100.5370.7110.5680.2300.2510.2490.256
S.D.6.7264.9555.9016.3944.4683.0262.8672.1671.988
Min−32.710−27.930−34.730−26.530−23.240−14.890−13.960−18.480−6.860
Max42.62017.50032.33020.32016.10018.08012.58013.3809.560
Skewness0.602−0.5040.005−0.241−0.5070.3340.013−0.3270.316
Kurtosis5.9752.5064.0321.2901.8782.9472.35812.2381.603
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 *1.6760.7172.4082.6522.947 *21.859 *15.273 *9.901 *
Q(6)23.302 *7.2534.1584.8067.41710.664 *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 *
Note: This table presents the descriptive statistics for monthly industry portfolio excess returns for Consumer Durables (DURBL), Manufacturing (MANUF), Energy (ENRGY) and Technology (HITEC) firms over July 1963–December 2020. The last five columns are the Fama–French factors including Market (MKT), Size (SMB), Book-to-Market (HML), Profitability (RMW) and Investment (CMA) factors based on the five-factor specification of [57]. The Jarque–Bera normality test (JB), the first- [Q(1)] and sixth-order [Q(6)] serial correlation test and first- [ARCH(1)] and sixth-order [ARCH(6)] autoregressive conditional heteroskedasticity test. * denotes significance at 1% level.
Table 2. Time-varying, rolling and static factor betas—Consumer Durables.
Table 2. Time-varying, rolling and static factor betas—Consumer Durables.
VariableMeanMedianS.D.MinMax5th Percentile95th Percentile
Panel A: TTVP-SV Model
Intercept−0.338−0.3810.093−0.436−0.037−0.431−0.153
MKT1.1651.1620.0101.1531.1801.1541.180
SMB0.2240.3780.1920.0000.4150.0020.414
HML0.2870.2670.0730.1620.4210.1710.416
RMW0.1570.0970.1100.0470.3560.0500.350
CMA0.0940.0570.098−0.0230.269−0.0180.266
Panel B: Rolling Model
Intercept−0.443−0.5060.288−0.9780.208−0.8270.105
MKT1.2211.1810.1820.9541.6230.9911.583
SMB0.3030.2560.246−0.1490.838−0.1060.762
HML0.3760.4060.333−0.4081.030−0.2770.887
RMW0.2640.1980.424−0.3821.150−0.2871.034
CMA0.1170.0710.300−0.5750.632−0.3400.583
Panel C: Static Model
Intercept−0.340 0.150 −0.586−0.094
MKT1.267 0.037 1.2071.327
SMB0.214 0.052 0.1280.300
HML0.377 0.069 0.2630.491
RMW0.254 0.072 0.1360.373
CMA0.147 0.106 −0.0270.320
Note: Panels A and B present the descriptive statistics for the time-varying beta estimates from the threshold TVP with stochastic volatility (TTVP-SV) and rolling regression models, respectively. MKT, SMB, HML, RMW and CMA represent the Market, Size, Book-to-Market, Profitability and Investment factors based on the five-factor specification of [57]. Static beta estimates are also reported in Panel C for comparison purposes. The window size for the rolling beta estimates is 120 months. Estimation is carried out over the July 1963–December 2020 period with 690 months (covering July 1963–December 2020 period) and 570 months (covering July 1973–December 2020 period) for the TTVP-SV and rolling methods, respectively. The static model is estimated using ordinary least squares for which the Median, Min and Max estimates are not available.
Table 3. Time varying, rolling and static factor betas—Manufacturing.
Table 3. Time varying, rolling and static factor betas—Manufacturing.
VariableMeanMedianS.D.MinMax5th Percentile95th Percentile
Panel A: TTVP-SV Model
Intercept−0.086−0.0830.033−0.135−0.001−0.133−0.025
MKT1.0671.0670.0001.0661.0671.0661.067
SMB0.1020.1020.0010.1010.1040.1010.104
HML0.0850.0850.0010.0840.0860.0850.086
RMW0.2400.2400.0030.2350.2450.2360.245
CMA0.0530.0750.0360.0020.1080.0060.107
Panel B: Rolling Model
Intercept−0.118−0.0950.148−0.4710.120−0.3690.078
MKT1.1051.0910.0580.9801.2651.0221.200
SMB0.1030.1150.062−0.0240.2580.0010.198
HML0.0140.0180.106−0.2530.263−0.2040.161
RMW0.2220.2810.230−0.3370.606−0.2250.478
CMA0.1160.1270.134−0.1750.406−0.1050.336
Panel C: Static Model
Intercept−0.164 0.061 −0.265−0.063
MKT1.084 0.015 1.0591.108
SMB0.110 0.021 0.0740.145
HML0.088 0.028 0.0410.134
RMW0.320 0.030 0.2710.369
CMA0.126 0.043 0.0550.197
Note: Panels A and B present the descriptive statistics for the time-varying beta estimates from the threshold TVP with stochastic volatility (TTVP-SV) and rolling regression models, respectively. MKT, SMB, HML, RMW and CMA represent the Market, Size, Book-to-Market, Profitability and Investment factors based on the five-factor specification of [57]. Static beta estimates are also reported in Panel C for comparison purposes. The window size for the rolling beta estimates is 120 months. Estimation is carried out over the July 1963–December 2020 period with 690 months (covering July 1963–December 2020 period) and 570 months (covering July 1973–December 2020 period) for the TTVP-SV and rolling methods, respectively. The static model is estimated using ordinary least squares for which the Median, Min and Max estimates are not available.
Table 4. Time varying, rolling and static factor betas—Energy.
Table 4. Time varying, rolling and static factor betas—Energy.
VariableMeanMedianS.D.MinMax5th Percentile95th Percentile
Panel A: TTVP-SV Model
Intercept−0.087−0.1170.129−0.3020.099−0.2920.094
MKT0.9630.9540.0170.9511.0070.9521.003
SMB−0.118−0.1810.143−0.2550.335−0.2500.261
HML0.0960.0260.172−0.0180.547−0.0110.527
RMW−0.086−0.1030.184−0.4040.195−0.4000.192
CMA0.4150.1680.403−0.0841.261−0.0471.256
Panel B: Rolling Model
Intercept0.0220.1310.529−1.1811.010−0.8230.815
MKT0.9670.9730.1350.6801.3480.7391.153
SMB−0.161−0.1960.218−0.5770.518−0.4870.172
HML0.031−0.0630.261−0.3620.733−0.2790.563
RMW0.0420.0520.545−0.9910.921−0.9100.773
CMA0.3040.1450.605−0.6891.748−0.5101.675
Panel C: Static Model
Intercept−0.224 0.168 −0.4990.052
MKT1.008 0.041 0.9411.075
SMB−0.063 0.058 −0.1590.033
HML0.230 0.078 0.1020.357
RMW0.209 0.081 0.0760.342
CMA0.365 0.118 0.1700.559
Note: Panels A and B present the descriptive statistics for the time-varying beta estimates from the threshold TVP with stochastic volatility (TTVP-SV) and rolling regression models, respectively. MKT, SMB, HML, RMW and CMA represent the Market, Size, Book-to-Market, Profitability and Investment factors based on the five-factor specification of [57]. Static beta estimates are also reported in Panel C for comparison purposes. The window size for the rolling beta estimates is 120 months. Estimation is carried out over the July 1963–December 2020 period with 690 months (covering July 1963–December 2020 period) and 570 months (covering July 1973–December 2020 period) for the TTVP-SV and rolling methods, respectively. The static model is estimated using ordinary least squares for which the Median, Min and Max estimates are not available.
Table 5. Time varying, rolling and static factor betas—High Tech.
Table 5. Time varying, rolling and static factor betas—High Tech.
VariableMeanMedianS.D.MinMax5th Percentile95th Percentile
Panel A: TTVP-SV Model
Intercept0.2270.2470.0570.1050.3180.1510.315
MKT1.0541.0530.0021.0511.0561.0511.056
SMB0.0750.0840.0210.0250.0880.0260.087
HML−0.234−0.2780.119−0.362−0.010−0.359−0.028
RMW−0.149−0.1630.219−0.6460.181−0.6190.179
CMA−0.466−0.4620.054−0.561−0.387−0.556−0.392
Panel B: Rolling Model
Intercept0.3260.1780.390−0.1501.452−0.1041.181
MKT1.0431.0320.1240.8351.3760.8731.266
SMB0.1250.1070.127−0.2220.398−0.0350.379
HML−0.290−0.2630.306−0.8600.318−0.7680.202
RMW−0.131−0.2460.320−0.7180.573−0.5080.493
CMA−0.398−0.4190.407−1.3570.365−1.0880.242
Panel C: Static Model
Intercept−0.224 0.168 −0.4990.052
MKT1.008 0.041 0.9411.075
SMB−0.063 0.058 −0.1590.033
HML0.230 0.078 0.1020.357
RMW0.209 0.081 0.0760.342
CMA0.365 0.118 0.1700.559
Note: Panels A and B present the descriptive statistics for the time-varying beta estimates from the threshold TVP with stochastic volatility (TTVP-SV) and rolling regression models, respectively. MKT, SMB, HML, RMW and CMA represent the Market, Size, Book-to-Market, Profitability and Investment factors based on the five-factor specification of [57]. Static beta estimates are also reported in Panel C for comparison purposes. The window size for the rolling beta estimates is 120 months. Estimation is carried out over the July 1963–December 2020 period with 690 months (covering July 1963–December 2020 period) and 570 months (covering July 1973–December 2020 period) for the TTVP-SV and rolling methods, respectively. The static model is estimated using ordinary least squares for which the Median, Min and Max estimates are not available.
Table 6. In-sample Pricing Errors.
Table 6. In-sample Pricing Errors.
ModelDURBLMANUFENRGYHITEC
TTVP-SV3.6701.5313.7962.403
TTVP3.5251.5203.6962.372
Rolling3.7081.5444.1632.484
Static3.7471.5334.1952.575
Note: The table reports in-sample root mean square error (pricing error) estimates from the threshold TVP with stochastic volatility (TTVP-SV), the threshold TVP without stochastic volatility (TTVP), rolling regression and static regression models. The window size for the rolling beta estimates is 120 months. Estimation is carried out over the July 1963–December 2020 period with 690 months (covering July 1963–December 2020 period) and 570 months (covering July 1973–December 2020 period) for the TVP and rolling methods, respectively. The static model is estimated using ordinary least squares.
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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

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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

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