A New 5-factor Model Based on the EGARCH-typed Volatilities and the SSAEPD Errors
Liuling Li, Qingyu Zhu and Yanjia Yang [A] [A] Liuling Li is an Assistant Professor from the Institute of Statistics and Econometrics, School of Economics, Nankai University, Tianjin, P.R. China, 300071 (liliuling@nankai.edu.cn). Qingyu Zhu is a student from the College of Software, Nankai University, P.R.China (zhuqingyu_cs@outlook.com). Yanjia Yang is a Ph.D. student at Business School, Nanyang Technological University (yanjia_yang@yahoo.com). Corresponding author: Qingyu Zhu, zhuqingyu_cs@outlook.com. Funding number: 63140013.
In this paper, we extend the five-factor model in Fama and French(2015) with the EGARCH-typed volatilities in Nelson(1991) and the SSAEPD distribution in Zhu and Zinde-Walsh(2009). MLE is used for parameter estimation and AIC for model comparion. Simulation results show our MatLab program is valid. Empirical results find out 1) Fama-French five factors are still alive! 2) Our new model has better in-sample fit than the one in Fama and French(2015).
Keywords: Fama-French Five-Factor Model(FF5F), Standardized Standard Asymmetric Exponential Power Distribution(SSAEPD), EGARCH
Table of Contents
1 Introduction
Sharpe(1964) and Lintner(1965) make a fundamental contribution to understand the relationship between expected returns of stock market by proposing the Capital Asset Pricing Model(CAPM). Based on that, Fama and French(1993) add two more factors relating to size and book-to-market into CAPM and show that their 3-factor model performs better.
After that, many researchers suggest a lot of new factor models to extend Fama-French 3 factor model (see Table 1↓ Panel A). For example, Carhart(1997) introduces a 4-factor model by augumenting the Fama-French 3-factor model with momentum factor which can explain the short-term persistence in expected returns. He(2008) analyzes state switch information. Chan and Faff(2005) advocate a liquidity factor. Connor, Hagmann and Linton(2012) consider an own-volatility factor. Fama and French(2015) suggest a 5 factor model and show it is better than their 3 factor model proposed in 1993.
Since year 2015, many empirical studies about Fama-French 5-factor model have been done (see Table 1↓ Panel B). And these researches can be divided into two groups. One group applies the Fama-French 5-factor model to different countries and compares it with others. For example, Hou, Xue and Zhang(2015) find that the 4-factor q-model performs better than Fama-French 5-factor model in US market. Harshita, Singh, S., Yadav, and Surendra(2015) point out that the Fama-French 5-factor model works better in India.
The other group is to extend Fama-French(2015)’s 5-factor model by adding new factors. Harvey and Liu(2015) create a 14-factor model combining Betting againist beta (bab), Gross protability (gp) and other 9 factors. In their paper, they document that the market factor is the most important factor for describing expected returns.
Our research falls into the 2nd group and tries to extend the 5-factor model in Fama and French(2015). But different from previous researches, instead of adding different factors, we consider the EGARCH-typed volatilities suggested in Nelson(1991) and non-normal errors of SSAEPD proposed by Zhu and Zinde-Walsh(2009). We denote our new model as FF5-SSAEPD-EGARCH. Based on our new 5-factor model, we try to test following two hypotheses:
- With EGARCH-type volatilities and SSAEPD errors, are Fama-French 5 factors still alive?
- Can our new model beat the 5 factor model in Fama and French(2015)?
Table 1 Researches about Fama-French Factor Model
| Author(Year) | Research Purpose | Model | Estimation | Data | ||
| Method | Country | Factor | Frequency | |||
| Panel A: Model Extension (propose a new model) | ||||||
| Fama et.al.(1993) | Model Extension | FF3 | - | USA | Mkt, SMB, HML | M1963:7-1991:12 |
| Carhart(1997) | Model Extension | C4 | OLS | USA | Mkt, SMB, HML, WML | M1962:1-1993:12 |
| Bali et.al.(2004) | Model Extension | FF3+VaR | OLS | USA | Mkt, SMB, HML, VaR | M1963:1-2001:12 |
| Chan et.al.(2005) | Model Extension | FF3+IML | GMM | Australia | Mkt, SMB, HML, IML | M1990:1-1998:12 |
| He(2008) | Model Extension | FF3+State Switch | OLS | China | Mkt, SMB, HML, State Switch | M1995:6-2005:12 |
| Connor et.al.(2012) | Model Extension | C4+VOL | Kernel | USA | Mkt, SMB, HML, WML, VOL | M1970-2007 |
| Chai et.al.(2013) | Model Extension | C4+IML | OLS | Australia | Mkt, SMB, HML, WML, IML | M1982:1-2010:12 |
| Fama et.al.(2013) | Model Extension | FF4 | - | USA | Mkt, SMB, HML, RMW | M1963:7-2012:12 |
| Fama et.al.(2015) | Model Extension | FF5 | - | USA | Mkt, SMB, HMLO, RMW, CMA | M1963:7-2013:12 |
| Panel B: Empirical Application | ||||||
| Griffin(2002) | Robustness Exam | FF3 | - | Global | Mkt, SMB, HML | M1981:1995:12 |
| Fama et.al.(2012) | Robustness Exam | CAPM, FF3, C4 | - | Global | Mkt, SMB, HML, WML | M1990:11-2011:3 |
| Fama et.al.(2014) | Model Comparison | FF5, FF5+WML | - | USA | Mkt, SMB, HMLO, RMW, CMA, WML | M1963:7-2014:12 |
| Hou et.al.(2014) | Model Comparison | FF5, C4, q-factor | - | USA | Mkt, SMB, HMLO, RMW, CMA, WML | M1972:1-2011:12 |
| Hou et.al.(2015) | Model Comparison | FF5, C4, q-factor | - | USA | Mkt, SMB, HMLO, RMW, CMA, WML | M1967:1-2013:12 |
| Harshita et.al.(2015) | Robustness Exam | CAPM, FF3, FF5 | - | India | Mkt, SMB, HMLO, RMW, CMA | M1999:10-2014:9 |
| Chiah et.al.(2015) | Robustness Exam | FF3, FF5 | H.OLS | Australia | Mkt, SMB, HMLO, RMW, CMA | M1982:12-2012:12 |
| Fama et.al.(2015) | Robustness Exam | FF5 | - | Global | Mkt, SMB, HMLO, RMW, CMA | M1990:7-2014:9 |
Notes: Model extension means a new model is proposed in this paper. Robustness exam means a old model is checked with other dataset such as global data or data from other countries instead of USA, with which Fama and French 3-factor model was first proposed. Model comparison means different old models are compared with USA dataset. “-” means that no information is available in this paper. CAPM= Capital Asset Pricing Model. FF3= Fama and French(1993) 3-factor model. FF4= Fama and French(2013) 4-factor model. FF5 = Fama and French(2015) 5-factor model. C4= Carhart(1997) 4-factor. q-factor= Hou, Xue, and Zhang(2012) q-factor model. 14-factor= Harvay and Liu(2015) 14-factor model. Mkt= Market. SMB= Size. HML= Book-to-market. WML= Momentum. VaR= Value-at-Risk. IMV= liquidity. Vol= Own-volatility. RMW= Profitability. CMA= Investment. H.OLS= Newey-West heteroskedasticity and autocorrelation-adjusted OLS. WLS= Weighted least squares.
To answer these questions, we run simulation to test the MatLab program used in this paper. Then, Fama-French 25 value-weighted portfolios are analyzed. Data are downloaded from the French’s Data Library, and the sample period is from Jul. 1963 to Dec. 2013. Method of Maximum Likelihood Estimation(MLE) is used to estimate the parameters. Likelihood Ratio test(LR) and Kolmogorov-Smirnov test(KS) are used for model diagnostics. Akaike Information Criterion(AIC) is used for model comparsion.
Simulation results show our MatLab program is valid. According to the empirical results, we find out the 5 factors in Fama and French(2015) are still alive! The EGARCH-type volatility can capture the excess kurtosis. The new model fits the data well and has better in-sample fit than the one in Fama and French(2015).
The organization of this paper is as follows. The model and methodology are discussed in section 2. Simulation result is in section 3. Empirical results and the model comparisons are presented in section 4. Section 5 is the conclusions and future extensions.
2 Model and Methodology
2.1 Fama-French 5-Factor Model
Fama and French(2015) propose a 5-factor model (denoted as FF5F-Normal) to explain the size, value, profitability, and investment patterns in expected stock returns, and show this model empirically outperforms their 3 factor model. The 5-factor model [B] [B] See equation (6) on page 12 of Fama and French(2015). is:
(1)
Rt − Rft
=
β0 + β1*(Rmt − Rft) + β2*SMBt + β3*HMLOt
+ β4*RMWt + β5*CMAt + zt, zt ~ Normal(μ, σ2).
Where θ = (β0, β1, β2, β3, β4, β5, μ, σ) are parameters to be estimated in this model. Rt is the rate of return for stock portfolio. Rft is the rate of return for the risk-free asset. Rmt is the rate of return for the market. SMBt stands for small minus big market capitalization. HMLOt is the high minus low book-to-market ratio orthogonalized [C] [C] Fama and French(2015) first try following 5-factor model.
(2)
Rt − Rft
=
β0 + β1*(Rmt − Rft) + β2*SMBt + β3*HMLt
+ β4*RMWt + β5*CMAt + ut,
ut
~
N(μ, σ2), t = 1, ..., T.
And they find out HMLt (the high minus low book-to-market ratio) is redudant. Then, they run the regression of HMLt on other 4 factors as follows.
. RMWt stands for robust minus weak operating profitability portfolios. CMAt stands for conservative minus aggressive investment portfolios. The error term zt is distributed as the Normal. t = 1, 2, ..., T.
(3)
HMLt
=
β0 + β1*(Rmt − Rft) + β2*SMBt + β3*RMWt + β4*CMAt + ϵt,
ϵt
~
N(μ, σ2), t = 1, ..., T.
The sum of the intercept β̂0 and the residual ϵ̂t from equation (3↑) is defined as HMLOt̂.
2.2 FF5F-SSAEPD-EGARCH Model
We extend Fama and French(2015) five-factor model by introducing a Standardized Standard Asymmetric Exponential Power Distribution (SSAEPD) errors and the EGARCH -type volatilites. The new model we proposed is (denoted as the FF5F-SSAEPD-EGARCH model):
(4)
Rt − Rft
=
β0 + β1(Rmt − Rft) + β2SMBt + β3HMLOt
+ β4RMWt + β5CMAt + ut,
(5) ut = σtzt, zt ~ SSAEPD(α, p1, p2),
(7)
g(zt − i)
=
cizt − i + di[∣zt − i∣ − E(∣zt − i∣)],
=
⎧⎨⎩
(ci + di)zt − i − diE(∣zt − i∣),
if zt − i ≥ 0,
(ci − di)zt − i − diE(∣zt − i∣),
else.
where θ = (β0, β1, β2, β3, β4, β5, α, p1, p2, a, {bj}mj = 1, {ci}si = 1, {di}si = 1) are the parameters to be estimated. Definitions of variables are the same as before. σt is the conditional standard deviation, i.e., volatility. The error term zt is distributed as the Standardized Standard Asymmetric Exponential Power Distribution(SSAEPD) proposed in Zhu and Zinde-Walsh(2009).
- Standardized Standard AEPD (SSAEPD)
If a random variable Y is distributed as the Standardized Standard AEPD (denoted as Y ~ SSAEPD(α, p1, p2)), then its probability density function (PDF) is
(8) f(y∣β) = ⎧⎨⎩
δ⎛⎝(α)/(α∗)⎞⎠K(p1)exp⎛⎝ − (1)/(p1)||(ω + δy)/(2α∗)||p1⎞⎠,
if y≤ − (ω)/(δ),
δ⎛⎝(1 − α)/(1 − α∗)⎞⎠K(p2)exp⎛⎝ − (1)/(p2)||(ω + δy)/(2(1 − α∗))||p2⎞⎠,
if y > − (ω)/(δ).
Where
(9) Y = (X − ω)/(δ),
(11)
δ2 = Var(X)
=
(1)/(B2)⎡⎣(1 − α)3(p22Γ(3 ⁄ p2))/(Γ3(1 ⁄ p2)) + α3(p21Γ(3 ⁄ p1))/(Γ3(1 ⁄ p1))⎤⎦
− (1)/(B2)⎡⎣(1 − α)2(p2Γ(2 ⁄ p2))/(Γ2(1 ⁄ p2)) − α2(p1Γ(2 ⁄ p1))/(Γ2(1 ⁄ p1))⎤⎦2,
(13) B = αK(p1) + (1 − α)K(p2).
And X is a random variable distributed as standard AEPD(α, p1, p2). [D] [D] The PDF of SSAEPD(α, p1, p2) is derived from SAEPD(α, p1, p2) by changing variable techniques. If X is distributed as the standard AEPD, then its probability density function is
f(x∣β) = ⎧⎨⎩
⎛⎝(α)/(α∗)⎞⎠K(p1)exp⎛⎝ − (1)/(p1)||(x)/(2α∗)||p1⎞⎠,
if x≤0,
⎛⎝(1 − α)/(1 − α∗)⎞⎠K(p2)exp⎛⎝ − (1)/(p2)||(x)/(2(1 − α∗))||p2⎞⎠,
if x > 0.
- Method of Maximum Likelihood Estimation (MLE)
We estimate the FF5F-SSAEPD-EGARCH model with the method of Maximum Likelihood Estimation (MLE). The likelihood function is
(14) L({Rt − Rft, Rmt − Rft, SMBt, HMLOt, RMWt, CMAt}Tt = 1;θ)
= ∏Tt = 1f(Rt − Rft)
(15) = n∏i = 1⎧⎨⎩
(δ)/(η)⎛⎝(α)/(α∗)⎞⎠K(p1)exp⎛⎝ − (1)/(p1)||(ω + δzt)/(2α∗)||p1⎞⎠,
zt≤ − (ω)/(δ),
(δ)/(η)⎛⎝(1 − α)/(1 − α∗)⎞⎠K(p2)exp⎛⎝ − (1)/(p2)||(ω + δzt)/(2(1 − α∗))||p2⎞⎠,
zt > − (ω)/(δ).
Where
(16)
zt
=
[Rt − Rft − β0 − β1(Rmt − Rft) − β2SMBt − β3HMLOt
− β4RMWt − β5CMAt] ⁄ σt,
(18)
g(zt − i)
=
czt − i + di[∣zt − i∣ − E(∣zt − i∣)],
=
⎧⎨⎩
(ci + di)zt − i − diE(∣zt − i∣),
if zt − i ≥ 0,
(ci − di)zt − i − diE(∣zt − i∣),
else.
3 Simulation Results
In this section, we simulate the data and analyze the results to confirm that the program in MatLab is valid. The 5-factor model is simulated as follows.
(19) Yt = β0 + β1X1t + β2X2t + β3X3t + β4X4t + β5X5t + ut,
(20) ut = σtzt, zt ~ SSAEPD(α, p1, p2), t = 1, 2, ..., T,
(22)
g(zt − 1)
= czt − 1 + d[∣zt − 1∣ − E(∣zt − 1∣)]
= ⎧⎨⎩
(c + d)zt − 1 − dE(∣zt − 1∣),
if zt − 1 ≥ 0
(c − d)zt − 1 − dE(∣zt − 1∣),
else.
The data generation process(DGP) has following steps.
- Given α = 0.5, p1 = p2 = 2, we can generate SSAEPD random number series {zt}Tt = 1 [E] [E] For the method to generate SSAEPD random variable, one can refer to Li, Tian and Zhen(2011)..
-
Set the initial value σ20 = 1, z0 = 1, and given α = 0.5, p1 = p2 = 2,a = 0.7, b = 0.3, c = 0.4, d = 0.6, we can generate {σ2t}Tt = 1 and {ut}Tt = 1 with the following formula:(23) g(zt − 1) = czt − 1 + d[∣zt − 1∣ − E(∣zt − 1∣)] = ⎧⎨⎩ (c + d)zt − 1 − dE(∣zt − 1∣), if zt − 1 ≥ 0 (c − d)zt − 1 − dE(∣zt − 1∣), else.
- Generate random number series {X1t}Tt = 1, {X2t}Tt = 1, {X3t}Tt = 1, {X4t}Tt = 1, {X5t}Tt = 1from Uniform(0,1).
- Set β0 = 0.2, β1 = 0.5, β2 = 0.5, β3 = 0.5, β4 = 0.5, β5 = 0.5, and we can get {Yt}Tt = 1 with equation (19↑).
After getting the simulated data {X1t, X2t, X3t,X4t, X5t,Yt}Tt = 1, we estimate the parameters in the model. The simulation results are reported in Table 2↓. The estimates from MatLab program are β0 = 0.2237, β1 = 0.4962, β2 = 0.4915, β3 = 0.5345, β4 = 0.4467, β5 = 0.5021, α = 0.5000, p1 = 1.9999, p2 = 1.9995, a = 0.7050, b = 0.3066, c = 0.4129, d = 0.6448, which are very close to the true values of the parameters (β0 = 0.2, β1 = 0.5, β2 = 0.5, β3 = 0.5, β4 = 0.5, β5 = 0.5, α = 0.5, p1 = p2 = 2, a = 0.7, b = 0.3, c = 0.4, d = 0.6).
For robustness check, we also change the true values of the parameters and redo the simulation. All estimates are very closed to the true values of the parameters, since most of errors are equal to or less than 20%. Hence, we conclude the MatLab program can be applied to analyze empirical data.
Table 2 Simulation Results
| β0 | β1 | β2 | β3 | β4 | β5 | α | p1 | p2 | a | b | c | d | |
| T | 0.2 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 2 | 2 | 0.7 | 0.3 | 0.4 | 0.6 |
| E | 0.2237 | 0.4962 | 0.4915 | 0.5345 | 0.4467 | 0.5021 | 0.5000 | 1.9999 | 1.9995 | 0.7050 | 0.3066 | 0.4129 | 0.6448 |
| P | -11.87% | 0.76% | 1.70% | -6.89% | 10.66% | -0.41% | 0.00% | 0.00% | 0.02% | -0.71% | -2.20% | -3.22% | -7.46% |
| T | 0.2 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 1.5 | 1.5 | 0.7 | 0.3 | 0.4 | 0.6 |
| E | 0.2206 | 0.5048 | 0.5074 | 0.4894 | 0.5083 | 0.4523 | 0.5000 | 1.4999 | 1.5001 | 0.7159 | 0.2924 | 0.4029 | 0.5557 |
| P | -10.29% | -0.96% | -1.48% | 2.12% | -1.65% | 9.55% | 0.00% | 0.00% | 0.00% | -2.27% | 2.52% | -0.72% | 7.39% |
| T | 0.2 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.4 | 2 | 2 | 0.7 | 0.3 | 0.4 | 0.6 |
| E | 0.1844 | 0.5326 | 0.5438 | 0.4617 | 0.5300 | 0.4872 | 0.3997 | 2.0001 | 2.0002 | 0.6759 | 0.3238 | 0.4025 | 0.5591 |
| P | 7.82% | -6.52% | -8.75% | 7.66% | -5.99% | 2.55% | 0.07% | -0.01% | -0.01% | 3.44% | -7.92% | -0.62% | 6.82% |
| T | 0.2 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 0.5 | 2 | 2 | 0.7 | 0.3 | 0.4 | 0.6 |
| E | 0.2120 | 1.4128 | 1.5438 | 1.5350 | 1.5038 | 1.4756 | 0.5000 | 2.0007 | 1.9986 | 0.7012 | 0.3015 | 0.3918 | 0.5914 |
| P | -6.00% | 5.81% | -2.92% | -2.33% | -0.25% | 1.62% | -0.01% | -0.04% | 0.07% | -0.16% | -0.50% | 2.05% | 1.44% |
| T | 0.8 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 2 | 2 | 0.7 | 0.3 | 0.4 | 0.6 |
| E | 0.7941 | 0.4986 | 0.5491 | 0.4812 | 0.5200 | 0.4857 | 0.5000 | 2.0000 | 2.0000 | 0.6755 | 0.3196 | 0.3911 | 0.5926 |
| P | 0.73% | 0.28% | -9.82% | 3.75% | -4.01% | 2.86% | 0.00% | 0.00% | 0.00% | 3.50% | -6.52% | 2.22% | 1.23% |
| T | 0.2 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 2 | 2 | 0.6 | 0.3 | 0.4 | 0.6 |
| E | 0.2119 | 0.5008 | 0.5148 | 0.4609 | 0.4764 | 0.5135 | 0.4999 | 2.0000 | 2.0000 | 0.6135 | 0.2826 | 0.4021 | 0.6105 |
| P | -5.96% | -0.15% | -2.96% | 7.82% | 4.73% | -2.70% | 0.00% | 0.00% | 0.00% | -2.25% | 5.79% | -0.52% | -1.75% |
| T | 0.2 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 2 | 2 | 0.7 | 0.4 | 0.4 | 0.6 |
| E | 0.1895 | 0.5498 | 0.5192 | 0.5152 | 0.5148 | 0.4637 | 0.5000 | 2.0000 | 2.0000 | 0.7073 | 0.4076 | 0.4234 | 0.6313 |
| P | 5.23% | -9.97% | -3.84% | -3.05% | -2.96% | 7.25% | 0.00% | 0.00% | 0.00% | -1.04% | -1.91% | -5.85% | -5.22% |
| T | 0.2 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 2 | 2 | 0.7 | 0.3 | 0.8 | 0.6 |
| E | 0.1835 | 0.5196 | 0.4895 | 0.4774 | 0.5454 | 0.5100 | 0.5001 | 2.0003 | 2.0003 | 0.7167 | 0.2781 | 0.7818 | 0.5861 |
| P | 8.23% | -3.93% | 2.10% | 4.52% | -9.08% | -2.00% | -0.01% | -0.02% | -0.02% | -2.38% | 7.31% | 2.27% | 2.32% |
| T | 0.42 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 2 | 2 | 0.7 | 0.3 | 0.4 | 1 |
| E | 0.1825 | 0.5292 | 0.5318 | 0.4573 | 0.4907 | 0.4900 | 0.5000 | 2.0005 | 1.9995 | 0.6945 | 0.3062 | 0.4087 | 1.0404 |
| P | 8.74% | -5.83% | -6.36% | 8.53% | 1.86% | 2.00% | 0.00% | 0.00% | 0.00% | 0.78% | -2.05% | -2.17% | -4.04% |
Notes: T means the true value of parameters. E means the estimates. P means the error in percentage.
4 Empirical Analysis
4.1 Data
Fama and French (2015) analyze the monthly US stocks and show that their 5-factor model is better than their 3-factor model proposed in 1993. For comparison, we use the same data as Fama and French (2015) to empirically test our new model. [F] [F] The data are downloaded from the French’s Data Library website. There are 606 observations from 1963:07 to 2013:12. The descriptive statistics of sample data are calculated by MatLab and listed in Table 3↓. For each observation, the skewness (except one portfolio) is not 0 and the kurtosis is more than 3. The P-value of Jarque-Bera test for each portfolio is 0, which is smaller than 5% significance level. Hence, we can reject the null hypothesis and conclude that the asset returns do not follow Normal distribution and non-Normal error of SSAEPD may be better.
Table 3 Descriptive Statistics(1963:7-2013:12)
| Size | Book-to-market quintiles | ||||||||||
| quintile | Low | 2 | 3 | 4 | High | Low | 2 | 3 | 4 | High | |
| Mean | Median | ||||||||||
| Small | 0.68 | 1.22 | 1.26 | 1.42 | 1.56 | 1.06 | 1.43 | 1.36 | 1.54 | 1.57 | |
| 2 | 0.89 | 1.14 | 1.35 | 1.36 | 1.44 | 1.37 | 1.56 | 1.60 | 1.55 | 1.85 | |
| 3 | 0.91 | 1.19 | 1.21 | 1.29 | 1.49 | 1.51 | 1.47 | 1.57 | 1.50 | 1.61 | |
| 4 | 1.01 | 0.99 | 1.12 | 1.26 | 1.27 | 1.09 | 1.24 | 1.45 | 1.54 | 1.60 | |
| Big | 0.87 | 0.93 | 0.89 | 0.97 | 1.03 | 0.97 | 1.04 | 1.15 | 1.10 | 1.19 | |
| Maximum | Minimum | ||||||||||
| Small | 39.85 | 38.56 | 28.13 | 27.78 | 33.27 | -34.24 | -30.94 | -28.69 | -28.88 | -28.88 | |
| 2 | 27.45 | 26.12 | 26.34 | 27.34 | 30.04 | -32.71 | -31.56 | -27.80 | -26.04 | -28.84 | |
| 3 | 24.69 | 25.03 | 21.94 | 23.40 | 29.20 | -29.79 | -28.99 | -24.26 | -23.03 | -26.17 | |
| 4 | 25.91 | 20.44 | 24.01 | 24.32 | 27.90 | -25.94 | -28.83 | -26.33 | -21.02 | -23.84 | |
| Big | 22.35 | 16.53 | 19.08 | 19.76 | 17.57 | -21.64 | -22.36 | -21.74 | -19.32 | -19.13 | |
| Standard Deviation | Skewness | ||||||||||
| Small | 8.01 | 6.90 | 6.02 | 5.68 | 6.11 | -0.02 | 0.00 | -0.17 | -0.22 | -0.26 | |
| 2 | 7.23 | 6.00 | 5.46 | 5.32 | 6.03 | -0.34 | -0.46 | -0.48 | -0.50 | -0.46 | |
| 3 | 6.67 | 5.51 | 5.03 | 4.94 | 5.52 | -0.36 | -0.52 | -0.53 | -0.33 | -0.41 | |
| 4 | 5.94 | 5.21 | 5.10 | 4.84 | 5.53 | -0.22 | -0.60 | -0.51 | -0.28 | -0.30 | |
| Big | 4.70 | 4.47 | 4.38 | 4.39 | 5.05 | -0.22 | -0.38 | -0.31 | -0.25 | -0.30 | |
| Kurtosis | P-value of Jarque-Bera Test | ||||||||||
| Small | 5.26 | 6.12 | 5.63 | 5.90 | 6.34 | 0 | 0 | 0 | 0 | 0 | |
| 2 | 4.45 | 5.33 | 5.91 | 6.11 | 6.16 | 0 | 0 | 0 | 0 | 0 | |
| 3 | 4.44 | 5.64 | 5.26 | 5.43 | 6.26 | 0 | 0 | 0 | 0 | 0 | |
| 4 | 4.72 | 5.90 | 6.35 | 5.06 | 5.52 | 0 | 0 | 0 | 0 | 0 | |
| Big | 4.67 | 4.76 | 5.24 | 4.96 | 4.07 | 0 | 0 | 0 | 0 | 0 | |
4.2 Estimation Results
The estimation results for our new model are shown in Table 4↓. For comparison, we also estimate the 5-factor model of Fama and French(2015). Table 8↓ lists the estimates [G] [G] In Table 9↓ of Appendix 1, we present the results estimated in Fama and French (2015). . These results are similar to those in Fama and French (2015), which indicates that the MatLab program in our paper is reasonable.
Table 4 Estimates on FF5F-SSAEPD-EGARCH (Monthly, 1963:07-2013:12)
| Size | Book-to-market quintiles | ||||||||||||||||||||||
| quintile | Low | 2 | 3 | 4 | High | Low | 2 | 3 | 4 | High | Low | 2 | 3 | 4 | High | Low | 2 | 3 | 4 | High | |||
| β0 | β1 | β2 | β3 | ||||||||||||||||||||
| Small | -0.31 | 0.02 | 0.01 | 0.07 | 0.11 | 1.04 | 0.97 | 0.94 | 0.89 | 0.97 | 1.23 | 1.18 | 1.05 | 1.05 | 1.10 | -0.41 | -0.08 | 0.11 | 0.27 | 0.49 | |||
| 2 | -0.08 | -0.05 | 0.11 | 0.03 | -0.05 | 1.07 | 1.02 | 0.97 | 0.98 | 1.09 | 0.96 | 0.89 | 0.83 | 0.76 | 0.85 | -0.51 | -0.04 | 0.27 | 0.42 | 0.68 | |||
| 3 | 0.05 | 0.08 | 0.00 | 0.04 | -0.02 | 1.06 | 1.02 | 0.97 | 0.99 | 1.08 | 0.67 | 0.63 | 0.53 | 0.48 | 0.60 | -0.42 | 0.03 | 0.29 | 0.46 | 0.62 | |||
| 4 | 0.16 | -0.09 | -0.01 | 0.08 | -0.11 | 1.04 | 1.07 | 1.04 | 1.02 | 1.11 | 0.34 | 0.27 | 0.23 | 0.24 | 0.27 | -0.41 | 0.00 | 0.26 | 0.52 | 0.77 | |||
| Big | 0.09 | -0.08 | -0.02 | -0.09 | -0.12 | 0.99 | 1.02 | 0.99 | 0.98 | 1.06 | -0.20 | -0.15 | -0.22 | -0.15 | -0.08 | -0.32 | 0.01 | 0.23 | 0.53 | 0.82 | |||
| β4 | β5 | ||||||||||||||||||||||
| Small | -0.57 | -0.26 | -0.03 | 0.10 | 0.14 | -0.52 | -0.11 | 0.17 | 0.36 | 0.64 | |||||||||||||
| 2 | -0.25 | 0.02 | 0.21 | 0.22 | 0.19 | -0.64 | -0.02 | 0.27 | 0.51 | 0.74 | |||||||||||||
| 3 | -0.19 | -0.06 | 0.16 | 0.12 | 0.23 | -0.66 | 0.00 | 0.31 | 0.51 | 0.75 | |||||||||||||
| 4 | -0.17 | 0.06 | 0.00 | 0.08 | 0.22 | -0.53 | 0.11 | 0.29 | 0.55 | 0.71 | |||||||||||||
| Big | 0.12 | 0.17 | -0.03 | 0.11 | 0.00 | -0.36 | 0.22 | 0.34 | 0.54 | 0.68 | |||||||||||||
| α | p1 | p2 | |||||||||||||||||||||
| Small | 0.35 | 0.49 | 0.48 | 0.43 | 0.43 | 1.20 | 2.20 | 2.00 | 1.90 | 1.80 | 1.89 | 2.00 | 2.30 | 1.70 | 1.70 | ||||||||
| 2 | 0.48 | 0.54 | 0.43 | 0.45 | 0.45 | 1.70 | 2.00 | 1.80 | 1.70 | 1.80 | 1.90 | 1.50 | 1.90 | 1.90 | 1.78 | ||||||||
| 3 | 0.40 | 0.56 | 0.55 | 0.60 | 0.51 | 1.40 | 1.80 | 1.70 | 2.00 | 1.90 | 1.80 | 1.65 | 1.70 | 1.50 | 1.50 | ||||||||
| 4 | 0.52 | 0.45 | 0.60 | 0.47 | 0.46 | 1.70 | 1.80 | 1.60 | 1.40 | 1.75 | 1.60 | 1.70 | 1.70 | 1.70 | 1.90 | ||||||||
| Big | 0.47 | 0.54 | 0.55 | 0.45 | 0.47 | 1.70 | 2.20 | 1.60 | 1.80 | 1.60 | 1.90 | 1.90 | 1.50 | 1.70 | 1.80 | ||||||||
| a | b | c | d | ||||||||||||||||||||
| Small | 0.10 | 0.06 | 0.03 | 0.03 | 0.02 | 0.93 | 0.93 | 0.95 | 0.96 | 0.97 | 0.04 | 0.02 | 0.01 | -0.02 | -0.03 | 0.24 | 0.26 | 0.23 | 0.20 | 0.18 | |||
| 2 | 0.04 | 0.05 | 0.03 | 0.08 | 0.10 | 0.95 | 0.90 | 0.94 | 0.82 | 0.89 | -0.05 | 0.04 | -0.01 | -0.05 | 0.05 | 0.19 | 0.34 | 0.19 | 0.22 | 0.26 | |||
| 3 | 0.03 | 0.03 | 0.05 | 0.07 | 0.08 | 0.96 | 0.96 | 0.94 | 0.89 | 0.94 | -0.03 | -0.02 | -0.06 | -0.05 | -0.02 | 0.17 | 0.23 | 0.26 | 0.44 | 0.25 | |||
| 4 | 0.03 | 0.05 | 0.06 | 0.10 | 0.04 | 0.96 | 0.94 | 0.95 | 0.90 | 0.97 | 0.00 | -0.07 | -0.13 | 0.06 | -0.07 | 0.22 | 0.28 | 0.30 | 0.17 | 0.13 | |||
| Big | 0.00 | 0.02 | 0.07 | 0.04 | 0.10 | 0.97 | 0.96 | 0.92 | 0.95 | 0.94 | -0.03 | 0.02 | 0.05 | 0.03 | -0.04 | 0.17 | 0.27 | 0.30 | 0.21 | 0.32 | |||
To test the significance of coefficients in our new model, Likelihood Ratio test (LR) is applied [H] [H] LR formula is from Neyman and Pearson(1993), which is calculated using Equation (26↓).
. The P-values of LR are listed in Table 5↓. We find out with non-Normal errors such as SSAEPD and EGARCH-type volatilities, the Fama-French 5 factors are still alive [I] [I] The null hypothesis of the joint significance test is H0:β1 = β2 = β3 = β4 = β5 = 0. The P-values of the joint significance test for all the 25 portfolios are 0, which means the coefficient of β0, β1, β2, β3, β4 and β5 are statistically joint significance under 5% significance level. Under 5% significance level, the coefficient β1 in all 25 portfolios are statistically significant; 24/25 (24/25, 19/25 and 23/25 ) portfolios have a statistically coefficient β2 (β3, β4 and β5, respectively.).. And most portfolios in high book-to-market quintiles can not earn the Alpha returns [J] [J] For example, 17 out of the 25 portfolios don’t have a statistically significant coefficient β0 under 5% significance level. And most of these 17 portfolios belong to high book-to-market quintiles. . Results show the EGARCH-type volatility should be included in Fama-French 5 factor model [K] [K] For instance, we do the joint significance test (H0:b = c = d = 0) for parameters in the EGARCH equation. The P-value of the LR are all smaller than the significance level 5%, which means our EGARCH-type volatilities is necessary. And for other individual tests, we found that most P-values of LR are smaller than the significance level 5%, which means the parameters are individually significant. Actually, a is significant in 20 out of 25 portfolios. b is significant in all 25 portfolios. c is significant in 6 portfolios. And d is significant in 22 out of 25 portfolios. . Test results show SSAEPD is weakly necessary [L] [L] We find out that all α do not equal to 0.5, which may document the skewness. 43 out of 50 values of pi (i = 1, 2) are smaller than 2, which may show portfolio returns are fat-tailed distributed. 24 out of 25 estimates of p1 do not equal to those of p2, which may capture the asymmetry. For example, asymmetry is documented (H0:α = 0.5 is rejected by 9 out of 25 portfolios). And non-normality is found (H0:p1 = 2 is rejected by 12 out of 25 portfolios and 8 out of 25 portfolios reject the null H0:p1 = 2.).. We think the reason may be EGARCH has more power to absort non-Normality.
(26) LR = − 2ln(likelihood\kern1pt\kern1ptfor\kern1ptnull) + 2ln(likelihood\kern1pt\kern1ptfor\kern1ptalternative).
Table 5 P-values of Likelihood Ratio Test (LR)
| Size | Book-to-market quintiles | ||||||||||||||||||||||
| quintile | Low | 2 | 3 | 4 | High | Low | 2 | 3 | 4 | High | Low | 2 | 3 | 4 | High | Low | 2 | 3 | 4 | High | |||
| H0:β1 = β2 = β3 = β4 = β5 = 0 | H0:β0 = 0 | H0:β1 = 0 | H0:β2 = 0 | ||||||||||||||||||||
| Small | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0.99 | 0.65 | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | |||
| 2 | 0* | 0* | 0* | 0* | 0* | 0.56 | 0.97 | 0* | 0.82 | 1 | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | |||
| 3 | 0* | 0* | 0* | 0* | 0* | 0* | 0.78 | 1 | 0.09 | 1 | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | |||
| 4 | 0* | 0* | 0* | 0* | 0* | 0.03* | 0.62 | 0.99 | 1 | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | |||
| Big | 0* | 0* | 0* | 0* | 0* | 0.22 | 0* | 0.99 | 0.83 | 0.59 | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0.23 | |||
| H0:β3 = 0 | H0:β4 = 0 | H0:β5 = 0 | H0: b = c = d = 0 | ||||||||||||||||||||
| Small | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | |||
| 2 | 0* | 0* | 0* | 0* | 0* | 0* | 0.95 | 0* | 0* | 0* | 0* | 0.98 | 0* | 0* | 0* | 0* | 0* | 0* | 0.01* | 0.04* | |||
| 3 | 0* | 0* | 0* | 0* | 0* | 0* | 0.15 | 0* | 0* | 0* | 0* | 1 | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | |||
| 4 | 0* | 0* | 0* | 0* | 0* | 0* | 1 | 0* | 0.50 | 0* | 0* | 0.01* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | |||
| Big | 0* | 1 | 0* | 0* | 0* | 0* | 0* | 0.36 | 0* | 0.92 | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | |||
| H0:a = 0 | H0:b = 0 | H0:c = 0 | H0:d = 0 | ||||||||||||||||||||
| Small | 0* | 0* | 0.86 | 0.06 | 0* | 0* | 0* | 0* | 0* | 0* | 0.57 | 0.99 | 1 | 0* | 0.64 | 0* | 0* | 0* | 0.56 | 0* | |||
| 2 | 0* | 0.72 | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0.37 | 0.98 | 1 | 0.82 | 1 | 0* | 0* | 0* | 0.14 | 0.05 | |||
| 3 | 0.15 | 0* | 0.03* | 0.04* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0.96 | 0.40 | 0.75 | 1 | 0.01* | 0* | 0* | 0* | 0* | |||
| 4 | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0.99 | 0* | 0.01* | 0.56 | 0.06 | 0* | 0* | 0* | 0* | 0* | |||
| Big | 0* | 0.15 | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0.63 | 0* | 0* | 0.93 | 0.84 | 0* | 0* | 0* | 0* | 0* | |||
| H0:α = 0.5 | H0:p1 = p2 = 2 | H0:p1 = 2 | H0:p2 = 2 | ||||||||||||||||||||
| Small | 0* | 0.96 | 0.56 | 0* | 0.96 | 0* | 0* | 1 | 0.17 | 0* | 0* | 0.85 | 0* | 0.64 | 0* | 0.90 | 1 | 1 | 0.27 | 0.72 | |||
| 2 | 0.81 | 0.28 | 0* | 0.94 | 1 | 0.27 | 0.02* | 0.40 | 0.12 | 0.22 | 0* | 1 | 0.35 | 0.04* | 0.18 | 0.74 | 0.01* | 0* | 0.99 | 0.38 | |||
| 3 | 0.29 | 0* | 0* | 0* | 1 | 0* | 0.15 | 0.18 | 0* | 0* | 0* | 0.96 | 0.97 | 1 | 1 | 0* | 0.06 | 0.04* | 0* | 0* | |||
| 4 | 0.89 | 1 | 1 | 0.45 | 0.77 | 0.16 | 0.76 | 0..22 | 0.03* | 0.38 | 0.32 | 0.13 | 0.02* | 0.01* | 0* | 0.10 | 1 | 0* | 0.11 | 0.97 | |||
| Big | 0* | 0* | 0.59 | 0* | 0.71 | 0* | 0.87 | 0.05 | 0.57 | 0.02* | 0.82 | 0* | 0* | 0.16 | 0.02* | 0.92 | 0.97 | 0* | 1 | 0.41 | |||
Note: * means the null hypothesis is rejected under 5% significance level.
The residuals for previous models are checked with both Kolmogorov-Smirnov test [M] [M] The null hypothesis of KS test is H0 : Data follows a sepcified distribution. If the P-value of KS test is bigger than 5% significance level, the null hypothesis is not rejected. Otherwise, the null hypothesis is rejected. and graphs. Our results show our new model (FF5-SSAEPD-EGARCH) is adequate for the 25 portfolios while the FF5-Normal model is not. Since all 25 portfolios have residuals which do follow SSAEPD. But for the FF5F-Normal model, 21 out of 25 portfolios have residuals do not follow the Normal distribution.
- Kolmogorov-Smirnov Test for Residual Check
The P-value of KS test is displayed in Table 6↓. The P-values of KS test [N] [N] The null hypothesis is H0 : FF5F-SSAEPD-EGARCH residuals are distributed as SSAEPD(α̂, p̂1, p̂2). show the residuals from the new model do follow SSAEPD. For example, the P-value of the portfolio with Small Size and Low Book-to-market is 0.61, greater than 5%, which means under 5% significance level, the null hypothesis is not rejected and the residuals from FF5F-SSAEPD-EGARCH do follow the SSAEPD. Similarly, the null hypothesis can’t be rejected for all other 24 portfolios. Hence, we conclude the new model is adequate for all 25 portfolios.
Then, we apply the KS test for the residuals from the FF5F-Normal model [O] [O] The null hypothesis H0 : FF5F-Normal residuals are distributed as Normal(μ̂, σ̂2).. The P-values of the KS test are also listed in Table 6↓. 21 out of 25 portfolios have smaller P-values than 0.05, which means these 21 portfolios reject the nulls. Hence, the residuals of the FF5F-Normal model don’t follow Normal distribution. And the FF5F-Normal model is not adequate for the data.
Table 6 P-values of KS Test for Residuals
| Size | Book-to-market quintile | ||||||||||
| qunintiles | Low | 2 | 3 | 4 | High | Low | 2 | 3 | 4 | High | |
| FF5F-SSAEPD-EGARCH | FF5F-Normal | ||||||||||
| Small | 0.61 | 0.75 | 0.53 | 0.15 | 0.67 | 0* | 0* | 0* | 0.01* | 0* | |
| 2 | 0.77 | 0.60 | 0.24 | 0.94 | 0.07 | 0* | 0.07 | 0.13 | 0.07 | 0.01* | |
| 3 | 0.56 | 0.22 | 0.93 | 0.60 | 0.50 | 0* | 0* | 0* | 0* | 0* | |
| 4 | 0.93 | 0.83 | 0.14 | 0.77 | 0.85 | 0* | 0* | 0* | 0* | 0* | |
| Big | 0.91 | 0.88 | 0.44 | 0.79 | 0.93 | 0.81 | 0* | 0* | 0* | 0* | |
Note: * means the data doesn’t follow the sepcified distribution under 5% significance level. The null hypothesis for FF5F-SSAEPD-EGARCH model is H0: Its residuals are distributed as SSAEPD(α̂, p̂1, p̂2). The null hypothes for FF5F-Normal is H0: Its residuals are distributed as Normal(μ̂, σ̂2).
- PDFs for Residual Check
By method of “eye-rolling”, the PDF of residuals is compared with theoretical PDFs. Taking the portfolio with Small Size and High Book-to-market for example, in Figure a↓, the probability density function (PDF) for the estimated residuals zt̂ in our new model and that of SSAEPD(α̂, p̂1, p̂2) are plotted. These curves are very close to each other, which means the residuals are distributed as SSAEPD. Hence, our new model fits the data well.
Similarly, the probability density function (PDF) for the estimated residuals ut̂ in FF5F-Normal and that of Normal(μ̂, σ̂2) are shown in Figure b↓. And there are big differences between these two curves, which means the residuals are not distributed as Normal.
4.3 Model Comparision
Here, we compare our new model with the 5-factor model of Fama and French(2015). The Akaike Information Criterion (AIC) is used as the model selection criterion. Table 7↓ lists the AIC values. We find that all AIC values of our new model are smaller. Hence, we conclude that the new model we proposed is better than the 5-factor model in Fama and French(2015).
Table 7 AIC Values (Monthly, 1963:07-2013:12)
| Size | Book-to-market quintiles | ||||||||||
| quintiles | Low | 2 | 3 | 4 | High | Low | 2 | 3 | 4 | High | |
| FF5F-Normal | FF5F-SSAEPD-EGARCH | ||||||||||
| Small | 4.31 | 3.70 | 3.35 | 3.41 | 3.59 | 4.18* | 3.59* | 3.27* | 3.30* | 3.49* | |
| 2 | 3.62 | 3.37 | 3.26 | 3.31 | 3.52 | 3.53* | 3.23* | 3.22* | 3.28* | 3.47* | |
| 3 | 3.58 | 3.73 | 3.67 | 3.67 | 3.99 | 3.53* | 3.55* | 3.56* | 3.49* | 3.85* | |
| 4 | 3.66 | 3.84 | 3.92 | 3.81 | 4.17 | 3.54* | 3.69* | 3.69* | 3.76* | 4.13* | |
| Big | 2.98 | 3.47 | 3.88 | 3.61 | 4.46 | 2.91* | 3.38* | 3.75* | 3.52* | 4.26* | |
Note: Numbers with * are smaller AIC values.
5 Conclusions
In this paper, we extend the 5-factor model in Fama and French(2015) by introducing a non-normal error term and time-varying volitilities. The non-normal error assumption we used is the SSAEPD in Zhu and Zinde-Walsh (2009). And the time-varying volatilities is the EGARCH model in Nelson(1991). For comparision, monthly US stock returns in Fama and French (2015) (1963:07-2013:12) are analyzed. Method of Maximum Likelihood is used.
Simulation results show our MatLab program for the new model is valid. And empirical results show 1) with EGARCH-type volatilities and non-normal errors, the Fama-French 5 factors are still alive. 2) EGARCH-type volatility enables the model to respond asymmetrically to positive and negative shocks. 3) The new model has better in-sample fit than the 5-factor model in Fama and French(2015).
In the future, we can compare our results with those from other models such as ARIMA model. Or, we can analyze different data with our new model. Last but not the least, other factors can be introduced into this model.
Appendix 1. Estimates for the FF5F-Normal Model
To test our MatLab program, we also estimate the FF5F-Normal model using the program written by us. The estimates are listed in Table 8↓. The estimates and their t − statistics are close to those [P] [P] The estimation results in Fama and French (2015) are listed in Table 9↓. in Table 9↓, respectively. Thus, the MatLab program we wrote is valid.
Table 8 Estimates for FF5F-Normal Model by our MatLab Program (Monthly, 1963:07-2013:12)
| Size | Book-to-market quintiles | ||||||||||
| quintiles | Low | 2 | 3 | 4 | High | Low | 2 | 3 | 4 | High | |
| a | t(a) | ||||||||||
| Small | -0.29 | 0.11 | 0.01 | 0.12 | 0.12 | -3.28 | 1.60 | 0.14 | 2.03 | 1.95 | |
| 2 | -0.12 | -0.11 | 0.04 | 0.00 | -0.04 | -1.84 | -1.94 | 0.81 | 0.06 | -0.66 | |
| 3 | 0.02 | -0.01 | -0.06 | -0.03 | 0.05 | 0.38 | -0.20 | -0.99 | -0.40 | 0.59 | |
| 4 | 0.17 | -0.23 | -0.15 | 0.05 | -0.10 | 2.59 | -3.22 | -1.98 | 0.65 | -1.18 | |
| Big | 0.11 | -0.10 | -0.11 | -0.15 | -0.10 | 2.49 | -1.66 | -1.55 | -2.42 | -0.99 | |
| h | t(h) | ||||||||||
| Small | -0.42 | -0.14 | 0.11 | 0.28 | 0.52 | -9.92 | -4.39 | 4.07 | 10.41 | 17.90 | |
| 2 | -0.46 | -0.01 | 0.29 | 0.42 | 0.70 | -15.41 | -0.42 | 11.67 | 16.54 | 24.60 | |
| 3 | -0.43 | 0.12 | 0.37 | 0.52 | 0.67 | -14.63 | 3.81 | 12.25 | 17.15 | 18.83 | |
| 4 | -0.46 | 0.09 | 0.38 | 0.52 | 0.80 | -15.22 | 2.63 | 11.16 | 15.78 | 20.57 | |
| Big | -0.31 | 0.03 | 0.26 | 0.62 | 0.84 | -14.29 | 1.13 | 7.68 | 20.86 | 18.56 | |
| r | t(r) | ||||||||||
| Small | -0.56 | -0.33 | 0.01 | 0.11 | 0.11 | -13.00 | -10.32 | 0.34 | 3.90 | 3.74 | |
| 2 | -0.20 | 0.14 | 0.27 | 0.25 | 0.20 | -6.65 | 5.14 | 10.36 | 9.36 | 6.79 | |
| 3 | -0.20 | 0.22 | 0.32 | 0.28 | 0.32 | -6.77 | 6.77 | 10.30 | 8.83 | 8.58 | |
| 4 | -0.18 | 0.26 | 0.27 | 0.14 | 0.25 | -5.88 | 7.67 | 7.69 | 4.00 | 6.10 | |
| Big | 0.13 | 0.24 | 0.08 | 0.22 | 0.01 | 5.86 | 8.45 | 2.29 | 7.31 | 0.23 | |
| c | t(c) | ||||||||||
| Small | -0.57 | -0.11 | 0.19 | 0.39 | 0.61 | -12.36 | -3.23 | 6.70 | 13.17 | 18.98 | |
| 2 | -0.58 | 0.06 | 0.31 | 0.54 | 0.71 | -17.65 | 2.21 | 11.49 | 19.50 | 22.86 | |
| 3 | -0.66 | 0.13 | 0.42 | 0.64 | 0.77 | -20.53 | 3.83 | 12.67 | 19.13 | 19.58 | |
| 4 | -0.49 | 0.31 | 0.51 | 0.60 | 0.78 | -14.82 | 8.56 | 13.50 | 16.73 | 18.19 | |
| Big | -0.38 | 0.25 | 0.41 | 0.65 | 0.72 | -16.04 | 8.31 | 11.15 | 20.17 | 14.50 | |
Table 9 Estimates in Fama and French (2015) (Monthly, 1963:07-2013:12)
| Size | Book-to-market quintiles | ||||||||||
| quintiles | Low | 2 | 3 | 4 | High | Low | 2 | 3 | 4 | High | |
| a | t(a) | ||||||||||
| Small | -0.29 | 0.11 | 0.01 | 0.12 | 0.12 | -3.31 | 1.61 | 0.17 | 2.12 | 1.99 | |
| 2 | -0.11 | -0.10 | 0.05 | -0.00 | -0.04 | -1.73 | -1.88 | 0.95 | -0.04 | -0.64 | |
| 3 | 0.02 | -0.01 | -0.07 | -0.02 | 0.05 | 0.40 | -0.10 | -1.06 | -0.25 | 0.60 | |
| 4 | 0.18 | -0.23 | -0.13 | 0.05 | -0.09 | 2.73 | -3.29 | -1.81 | 0.73 | -1.09 | |
| Big | 0.12 | -0.11 | -0.10 | -0.15 | -0.09 | 2.50 | -1.82 | -1.39 | -2.33 | -0.93 | |
| h | t(h) | ||||||||||
| Small | -0.43 | -0.14 | 0.10 | 0.27 | 0.52 | -10.11 | -4.38 | 3.90 | 10.12 | 17.55 | |
| 2 | -0.46 | -0.01 | 0.29 | 0.43 | 0.69 | -15.22 | -0.45 | 11.77 | 16.78 | 24.44 | |
| 3 | -0.43 | 0.12 | 0.37 | 0.52 | 0.67 | -14.70 | 3.71 | 12.28 | 17.07 | 18.75 | |
| 4 | -0.46 | 0.09 | 0.38 | 0.52 | 0.80 | -15.18 | 2.76 | 11.03 | 15.88 | 20.26 | |
| Big | -0.31 | 0.03 | 0.26 | 0.62 | 0.85 | -14.12 | 1.09 | 7.54 | 21.05 | 18.74 | |
| r | t(r) | ||||||||||
| Small | -0.58 | -0.34 | 0.01 | 0.11 | 0.12 | -13.26 | -10.56 | 0.31 | 3.89 | 3.95 | |
| 2 | -0.21 | 0.13 | 0.27 | 0.26 | 0.21 | -6.75 | 4.89 | 10.35 | 9.86 | 7.04 | |
| 3 | -0.21 | 0.22 | 0.33 | 0.28 | 0.33 | -6.99 | 6.77 | 10.36 | 8.98 | 8.88 | |
| 4 | -0.19 | 0.27 | 0.28 | 0.14 | 0.25 | -6.06 | 7.75 | 7.99 | 4.16 | 6.14 | |
| Big | 0.13 | 0.25 | 0.07 | 0.23 | 0.02 | 5.64 | 8.79 | 2.07 | 7.62 | 0.49 | |
| c | s(c) | ||||||||||
| Small | -0.57 | -0.12 | 0.19 | 0.39 | 0.62 | -12.27 | -3.46 | 6.59 | 13.15 | 19.10 | |
| 2 | -0.59 | 0.06 | 0.31 | 0.55 | 0.72 | -17.76 | 1.94 | 11.27 | 19.39 | 22.92 | |
| 3 | -0.67 | 0.13 | 0.42 | 0.64 | 0.78 | -20.59 | 3.64 | 12.52 | 18.97 | 19.62 | |
| 4 | -0.51 | 0.31 | 0.51 | 0.60 | 0.79 | -15.11 | 8.33 | 13.35 | 16.41 | 18.03 | |
| Big | -0.39 | 0.26 | 0.41 | 0.66 | 0.73 | -16.08 | 8.38 | 10.80 | 19.88 | 14.54 | |
Note:This table is quoted from the results in Table 7 on page 13 of Fama and French (2015).
The comparison results between the parameter values estimated from our new model (FF5F-SSAEPD-EGARCH) and those from the FF5-Normal model are listed in Table 10↓. In the FF5F-SSAEPD-EGARCH model, for example, the absolute values of β0, for 16 portfolios, are greater than or equal to those in FF5F-Normal.
Table 10 Comparison Between Estimates
| FF5F-SSAEPD-EGARCH vs. FF5F-Normal | β0 | β1 | β2 | β3 | β4 | β5 |
| ≥ | 16 | 13 | 13 | 7 | 5 | 5 |
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