In this paper, we empirically test a new model with the data of US services sector, which is an extension of the 5-factor model in Fama and French (2015) [1]. 3 types of 5 factors (Global, North American and US) are compared. Empirical results show the Fama-French 5 factors are still alive! The new model has better in-sample fit than the 5-factor model in Fama and French (2015).
After the Capital Asset Pricing Model (CAPM) was created by Sharpe (1964) [
Author (Year) | Research Purpose | Model | Data | ||
---|---|---|---|---|---|
Country | Factors | Frequency & Period | |||
Panel A: Development of Factor Model | |||||
Fama et al. (1993) [ | CAPM Extension | FF3 | USA | Mkt, SMB, HML, WML | M1963:7-1991:12 |
Carhart (1997) | FF3 Extension | CAPM, FF3, C | USA | Mkt, SMB, HML, WML | M1962:1-1993:12 |
Griffin (2002) [ | FF3 Extension | Domestic or International FF3 | Global | Mkt, SMB, HML | M1981:1995:12 |
Chan et al. (2005) [ | FF3 Extension | FF3 with IML | Australia | Mkt, SMB, HML, IML | M1990:1-1998:12 |
Fama et al. (2012) [ | Model Comparison | Global or Local CAPM, FF3, C | Global | Mkt, SMB, HML, WML | M1990:11-2011:3 |
Connor et al. (2012) | C Extension | C with VOL | USA | Mkt, SMB, HML, WML, VOL | M1970-2007 |
Chai et al. (2013) [ | C Extension | C with IML | Australia | Mkt, SMB, HML, WML, IML | M1982:1-2010:12 |
Fama et al. (2013) [ | FF3 Extension | FF4 | USA | Mkt, SMB, HML, RMW | M1963:7-2012:12 |
Yang (2013) | FF3 Extension | FF3 with SSAEPD, EGARCH | USA | Mkt, SMB, HML | M1926-2011 |
Hou et al. (2014) | Model Comparison | FF5, C, q-factor | USA | Mkt, SMB, RMW, CMA, WML, HML | M1972:1-2011:12 |
Fama et al. (2015a) | FF4 Extension | FF5 | USA | Mkt, SMB, HML, RMW, CMA | M1963:7-2013:12 |
Zhu (2016) | FF5 Extension | FF5 with SSAEPD, EGARCH | USA | Mkt, SMB, HML, RMW, CMA | M1963:7-2013:12 |
Panel B: Researches for Fama-French 5-Factor Model | |||||
Fama et al. (2014) | Model Comparison | CAPM, FF3, FF4, FF5, FF5 with WML | USA | Mkt, SMB, HML, RMW, CMA, WML | M1963:7-2014:12 |
Hou et al. (2015) | Model Comparison | FF5, C, q-factor | USA | Mkt, SMB, HML, RMW, CMA, WML | M1967:1-2013:12 |
Harshita et al. (2015) | Model Comparison | CAPM, FF3, FF5 | India | Mkt, SMB, HML, RMW, CMA | M1999:10-2014:9 |
Fama et al. (2015b) | Empirical Tests | FF5 | Global | Mkt, SMB, HML, RMW, CMA | M1990:7-2014:9 |
Chiah et al. (2016) | Empirical Tests | FF3, C, FF5 | Australia | Mkt, SMB, HML, PMU, LMH | M:1982:1-2013:12 |
Bin Guo et al. (2017) | Empirical Tests | FF5 | China | Mkt, SMB, HML, RMW, CMA,CMAB | M:1995:7-2014:6 |
Rehab et al. (2016) | Empirical Tests | FF5 | Egypt | MKT, SMB, HML, HEMLE, HSMLS, HDMLD, IML and WML | M:2005:7-2013:7 |
Fama et al. (2016) | Empirical Tests | FF5 | Global | Mkt, SMB, HML, RMW, CMA | M:1990:7-2015:12 |
Notes: “-” 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 4-factor model (2013); FF5 = Fama and French (2015) 5-factor model; C = 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; IMV = liquidity; Vol = Own-volatility; RMW = Profitability; CMA = Investment; PMU = Profitable Minus Unprofitable; HML = High Minus Low; HAC-adjusted OLS = Newey-West heteroskedasticity; and autocorrelation-adjusted OLS. WLS = Weighted least squares.
[
In 2015, Fama and French proposed 5 factor model(FF5), it adds profitability and investment factors into their 3-factor model proposed in 1993. Since then, many studies about Fama-French 5-factor (FF5) model have been done. Panel B of
For example, Hou, Xue and Zhang (2015) [
Although FF5 model has better performance in many case, it’s not adapted to every situation. Fama and French (2017) [
In 2017, Li et al. [
1) With EGARCH-type volatilities and SSAEPD errors, are Fama-French 5 factors still alive?
2) Can this new model explain services industry better than the 5 factor model in Fama and French (2015)?
To answer these questions, we run simulation to test the validity of MatLab program used in this paper. Then, the industry of services in US are analyzed. Data are downloaded from the French’s Data Library, and the sample period is from Jul. 1990 to Feb. 2017. 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.
We find out the Fama-French 5 factors are still alive. The new model has better in-sample fit than the 5-factor model in Fama and French (2015). The industry of services can earn extra Alpha returns since the constant term in the new model is statistically significant. The Beta ( β 1 ) coefficient (for US, North American) is very close to 1. We also find out models with GARCH-typed volatility fit data better than those with EGARCH-typed volatility. To capture fat-tailedness, GARCH equation is better than non-normal error terms of SSAEPD.
The organization of this paper is as follows: The model and methodology are discussed in Section 2; Empirical results and the model comparisons will be presented in Section 3; Section 4 is the conclusions and future extensions.
Fama and French(2015) propose a 5-factor model (denoted as FF5) to explain market, size, value, profitability, and investment patterns in expected stock returns, and show this model empirically outperforms their 3 factor model. The 5-factor model is:
R t − R f t = β 0 + β 1 ∗ ( R m t − R f t ) + β 2 ∗ S M B t + β 3 ∗ H M L O t (1)
+ β 4 ∗ R M W t + β 5 ∗ C M A t + u t , u t ~ Normal ( μ , σ 2 ) . (2)
where θ = ( β 0 , β 1 , β 2 , β 3 , β 4 , β 5 , μ , σ ) are parameters to be estimated in this model. R t is the rate of return for stock portfolio. R f t is the rate of return for the risk-free asset. R m t is the rate of return for the market. S M B t stands for small market capitalization minus big market capitalization. H M L O t is the high book-to-market ratio minus low book-to-market ratio orthogonalized1. RMWt stands for robust operating profitability portfolios minus weak operating profitability portfolios. C M A t stands for conservative investment portfolios minus aggressive investment portfolios. The error term u t is distributed as the Normal. t = 1 , 2 , ⋯ , T .
Li et al. (2016) extend Fama-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 FF5-SSAEPD-EGARCH model):
R t − R f t = β 0 + β 1 ( R m t − R f t ) + β 2 S M B t + β 3 H M L O t (5)
+ β 4 R M W t + β 5 C M A t + u t ,
u t = σ t z t , z t ~ S S A E P D ( α , p 1 , p 2 ) , (6)
ln ( σ t 2 ) = a + ∑ i = 1 s g ( z t − i ) + ∑ j = 1 m b j ln ( σ t − j 2 ) , (7)
g ( z t − i ) = c i z t − i + d i [ | z t − i | − E ( | z t − i | ) ] = ( ( c i + d i ) z t − i − d i E ( | z t − i | ) , if z t − i ≥ 0, ( c i − d i ) z t − i − d i E ( | z t − i | ) , else . (8)
where θ = ( β 0 , β 1 , β 2 , β 3 , β 4 , β 5 , α , p 1 , p 2 , a , { b j } j = 1 m , { c i } i = 1 s , { d i } i = 1 s ) are the parameters to be estimated. Definitions of variables are the same as before. σ t is the conditional standard deviation, i.e., volatility. The error term z t is distributed as the Standardized Standard Asymmetric Exponential Power Distribution (SSAEPD) proposed in Zhu and Zinde-Walsh (2009).
•Standardized Standard AEPD (SSAEPD)
According to Zhu and Zinde-Walsh (2009), the AEPD density has following form2:
f AEPD ( x ) = { ( α α * ) 1 σ K ( p 1 ) exp ( − 1 p 1 | x − μ 2 α * σ | p 1 ) , if x ≤ μ , ( 1 − α 1 − α * ) 1 σ K ( p 2 ) exp ( − 1 p 2 | x − μ 2 ( 1 − α * ) σ | p 2 ) , if x > μ . (10)
f AEPD ( x ) = { 1 σ exp ( − 1 p 1 | x − μ 2 α σ K ( p 1 ) | p 1 ) , if x ≤ μ , 1 σ exp ( − 1 p 2 | x − μ 2 ( 1 − α ) σ K ( p 2 ) | p 2 ) , if x > μ . (9)
θ = ( α , p 1 , p 2 , μ , σ )
where θ = ( α , p 1 , p 2 , μ , σ ) is the parameter vector. μ ∈ R and σ > 0 represent location and scale, respectively3. α ∈ ( 0,1 ) is the skewness parameter. p 1 > 0 and p 2 > 0 are the left and the right tail parameters, respectively. K ( p ) and α * are defined as
K ( p ) = 1 2 p 1 / p Γ ( 1 + 1 / p ) , (11)
α * = α K ( p 1 ) α K ( p 1 ) + ( 1 − α ) K ( p 2 ) . (12)
If we set the location parameter μ = 0 and the scale parameter σ = 1 , then we say X is a random variable distributed as Standard AEPD, denote it as X ~ SAEPD ( α , p 1 , p 2 ,0,1 ) . Its PDF4, mean and variance are
f SAEPD ( x ) = { ( α α * ) K ( p 1 ) exp ( − 1 p 1 | x 2 α * | p 1 ) , if x ≤ 0 , ( 1 − α 1 − α * ) K ( p 2 ) exp ( − 1 p 2 | x 2 ( 1 − α * ) | p 2 ) , if x > 0 , (14)
E ( X ) = 1 B [ ( 1 − α ) 2 p 2 Γ ( 2 / p 2 ) Γ 2 ( 1 / p 2 ) − α 2 p 1 Γ ( 2 / p 1 ) Γ 2 ( 1 / p 1 ) ] , (15)
V a r ( X ) = 1 B 2 { ( 1 − α ) 3 p 2 2 Γ ( 3 / p 2 ) Γ 3 ( 1 / p 2 ) + α 2 p 1 2 Γ ( 3 / p 1 ) Γ 3 ( 1 / p 1 ) − [ ( 1 − α ) p 2 Γ ( 2 / p 2 ) Γ 2 ( 1 / p 2 ) − α 2 p 1 Γ ( 2 / p 1 ) Γ 2 ( 1 / p 1 ) ] 2 } . (16)
Then, if we standardize X with its mean and standard deviation, we can get
θ = ( α , p 1 , p 2 , 0 , 1 )
Z = X − E ( X ) V a r ( X ) , which we call Standardized Standard AEPD (SSAEPD). The
PDF of Z can be got by transformation.
f SSAEPD ( Z ) = | J | f SAEPD ( E ( X ) + Z V a r ( X ) ) (17)
= δ f SAEPD ( ω + Z δ ) (18)
where ω = E ( X ) , | J | = δ and δ = V a r ( X ) , we can get the probability density function (PDF) of the SSAEPD
f SSAEPD ( z ) = { δ ( α α * ) K ( p 1 ) exp ( − 1 p 1 | ω + z δ 2 α * | p 1 ) , if z ≤ − ω δ , δ ( 1 − α 1 − α * ) K ( p 2 ) exp ( − 1 p 2 | ω + z δ 2 ( 1 − α * ) | p 2 ) , if z > − ω δ . (19)
E ( z ) = 0 , V a r ( z ) = 1. With α = 0.5 , p 1 = p 2 = 2 , SSAEPD reduces to Normal (0,1).
We estimate the FF5-SSAEPD-EGARCH model with the method of Maximum Likelihood Estimation (MLE). The likelihood function is
L ( { R t − R f t , R m t − R f t , S M B t , H M L O t , R M W t , C M A t } t = 1 T ; θ ) (20)
= ∏ t = 1 T f ( R t − R f t )
= ∏ i = 1 n { δ η ( α α ∗ ) K ( p 1 ) exp ( − 1 p 1 | ω + δ z t 2 α ∗ | p 1 ) , z t ≤ − ω δ , δ η ( 1 − α 1 − α ∗ ) K ( p 2 ) exp ( − 1 p 2 | ω + δ z t 2 ( 1 − α ∗ ) | p 2 ) , z t > − ω δ . (21)
where
z t = R t − R f t − β 0 − β 1 ( R m t − R f t ) − β 2 S M B t − β 3 H M L O t − β 4 R M W t − β 5 C M A t σ t , (22)
ln ( σ t 2 ) = a + ∑ i = 1 s g ( z t − i ) + ∑ j = 1 m b j ln ( σ t − j 2 ) , (23)
g ( z t − i ) = c z t − i + d i [ | z t − i | − E ( | z t − i | ) ] , = ( ( c i + d i ) z t − i − d i E ( | z t − i | ) , if z t − i ≥ 0 , ( c i − d i ) z t − i − d i E ( | z t − i | ) , else . (24)
In this paper, the sector of services in US is analyzed. Monthly return and 3 types of 5 factors (US 5 factors, North American 5 factors and Global 5 factors)5 are downloaded from French’s Data Library6. Sample period is from July 1990 to Feb. 2017. 3 types of 5 factors (US, north American, global) are compared.
North American 5 factors are construted by Canada and United States.
US 5 factors are constructed by United States.
Mean | Med. | Max. | Min. | St.De. | Ske. | Kur. | P | |
---|---|---|---|---|---|---|---|---|
Panel A: Excess Returns of US Sector of Services | ||||||||
US | 1.00 | 1.74 | 22.22 | −19.16 | 5.98 | −0.30 | 3.97 | 0.00 |
Panel B: US 5 factors | ||||||||
ME | 0.65 | 1.19 | 11.35 | −17.23 | 4.27 | −0.68 | 4.26 | 0.00 |
SMB | 0.19 | 0.07 | 18.73 | −15.28 | 3.11 | 0.47 | 8.05 | 0.00 |
HML | 0.27 | −0.05 | 12.91 | −11.25 | 3.03 | 0.15 | 5.57 | 0.00 |
RMW | 0.33 | 0.37 | 13.52 | −19.11 | 2.75 | −0.45 | 13.86 | 0.00 |
CMA | 0.26 | 0.09 | 9.55 | −6.88 | 2.10 | 0.56 | 5.22 | 0.00 |
Panel C: North American 5 factors | ||||||||
ME | 0.65 | 1.14 | 11.54 | −18.42 | 4.25 | −0.72 | 4.55 | 0.00 |
SMB | 0.18 | 0.12 | 16.48 | −13.54 | 2.79 | 0.33 | 7.59 | 0.00 |
HML | 0.24 | 0.19 | 16.75 | −13.36 | 3.24 | 0.56 | 7.69 | 0.00 |
RMW | 0.32 | 0.28 | 13.13 | −15.32 | 2.43 | 0.14 | 12.25 | 0.00 |
CMA | 0.30 | 0.02 | 14.23 | −10.03 | 2.67 | 0.93 | 7.63 | 0.00 |
Panel D: Global 5 factors | ||||||||
ME | 0.45 | 0.86 | 11.45 | −19.54 | 4.33 | −0.70 | 4.62 | 0.00 |
SMB | 0.12 | 0.08 | 8.00 | −8.43 | 1.98 | −0.34 | 5.20 | 0.00 |
HML | 0.33 | 0.22 | 11.65 | −9.54 | 2.29 | 0.54 | 8.17 | 0.00 |
RMW | 0.34 | 0.34 | 6.10 | −5.44 | 1.46 | −0.04 | 5.06 | 0.00 |
CMA | 0.26 | 0.06 | 9.60 | −6.55 | 1.89 | 0.66 | 6.92 | 0.00 |
Notes: Med. = Median; Max. = Maximum; Min. = Minimum; St.De. = Standard Devistion; Ske. = Skewness; Kur. = Kurtosis; P = englishP-value of Jarque-Bera Test; ME = Market Excess Return; SMB = Small minus Big; HML = High minus Low; RMW = Robust minus Weak; CMA = Conservation minus Aggressive; The null hypothesis of JB test is H0: Data are distributed as Normal(0,1).
0.05. That means, under 5% significance level, we can reject the null hypothesis and conclude that data do not follow Normal distribution. Hence, non-Normal error of SSAEPD may be proper. And from
The estimates are listed in
β 0 | β 1 | β 2 | β 3 | β 4 | β 5 | α | p 1 | p 2 | μ | σ | a | b | c | d | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Panel A: FF5-SSAEPD-EGARCH | |||||||||||||||
US 5 factors | 0.35 | 1.00 | 0.11 | −0.26 | −0.31 | −0.70 | 0.43 | 1.68 | 1.74 | −0.03 | 1.93 | 0.12 | 0.91 | 0.10 | 0.09 |
North American 5 factors | 0.27 | 0.97 | 0.04 | −0.29 | −0.25 | −0.65 | 0.43 | 1.70 | 1.79 | 0.03 | 2.01 | 0.10 | 0.94 | −0.01 | 0.16 |
Global 5 factors | 0.78 | 0.81 | −0.26 | −0.53 | −0.81 | −1.02 | 0.43 | 1.70 | 1.80 | 0.54 | 3.02 | 0.06 | 0.97 | −0.02 | 0.21 |
Panel B: FF5-Normal | |||||||||||||||
US 5 factors | 0.33 | 1.02 | 0.12 | −0.26 | −0.27 | −0.77 | - | - | - | 0 | 1.92 | - | - | - | - |
North American 5 factors | 0.31 | 1.00 | 0.10 | −0.29 | −0.16 | −0.72 | - | - | - | 0 | 1.99 | - | - | - | - |
Global 5 factors | 0.94 | 0.79 | −0.13 | −0.27 | −0.59 | −1.21 | - | - | - | 0 | 2.95 | - | - | - | - |
Notes: FF5-Normal is the model used in Fama-French (2015); FF5-SSAEPD-EGARCH is the new 5-factor model suggested by Zhu and Li (2016) supposing the error term meet the EGARCH-type volatilities and SSAEPD errors.
• Parameter Restriction Tests
Likelihood Ratio test (LR)8 is used to test the significance of regressors in these models. The P-values for Likelihood Ratio tests are listed in
Panel A of
For 3 kinds of five factors, the individual significance tests show β 1 is statistically significant (see column T3). That is, market returns have significant effect on this sector returns. Same is true for β 3 , β 4 and β 5 . For 2/3 types of 5 factors, β 0 and β 2 are statistically significant (see column T2 and T4, respectively).
Panel B of
T1 | T2 | T3 | T4 | T5 | T6 | T7 | T8 | T9 | T10 | T11 | T12 | T13 | T14 | T15 | T16 | T17 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Panel A: FF5-SSAEPD-EGARCH | |||||||||||||||||
US 5 factors | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0.04* | 0.06 | 0.03* | 0.01* | 0.11 | 0* | 0* | 0* | 0.01* | 0.99 |
North Ame. 5 factors | 0* | 0.13 | 0* | 0.99 | 0* | 0* | 0* | 0.07 | 0.99 | 0.03* | 0* | 0* | 0* | 0* | 0* | 0.99 | 0.99 |
Global 5 factors | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0.01* | 0.99 | 0* | 0* | 0.07 | 0* | 0* | 0* | 0.70 | 0.99 |
Panel B:FF5-Normal | |||||||||||||||||
US 5 factors | 0* | 1 | 0* | 0* | 0* | 0* | 0* | - | - | - | - | - | - | - | - | - | - |
North Ame. 5 factors | 0* | 1 | 0* | 0.03* | 0* | 0* | 0* | - | - | - | - | - | - | - | - | - | - |
Global 5 factors | 0* | 1 | 0* | 0.13 | 0.01* | 0* | 0* | - | - | - | - | - | - | - | - | - | - |
Notes: | T1 means H 0 : β 1 = β 2 = β 3 = β 4 = β 5 = 0 | T2 means H 0 : β 0 = 0 . |
---|---|---|
T3 means H 0 : β 1 = 0 . | T4 means H 0 : β 2 = 0 . | |
T5 means H 0 : β 3 = 0 . | T6 means H 0 : β 4 = 0 . | |
T7 means H 0 : β 5 = 0 . | T8 means H 0 : α = 0.5 , p 1 = p 2 = 2 . | |
T9 means H 0 : α = 0.5 . | T10 means H 0 : p 1 = p 2 = 2 . | |
T11 means H 0 : p 1 = 2 . | T12 means H 0 : p 2 = 2 . | |
T13 means H 0 : b = c = d = 0 . | T14 means H 0 : a = 0 . | |
T15 means H 0 : b = 0 . | T16 means H 0 : c = 0 . | |
T17 means H 0 : d = 0 . | ||
*means the null hypothesis is rejected under 5% significance level; North Ame. = North American |
For the FF5-SSAEPD-EGARCH model, among both US 5 factors and Global 5 factors, all individual coefficients in the mean equation are statistically significant. Hence, we conclude that Fama-French 5 factors are alive even if EGARCH and SSAEPD considered.
In this part, some restrictions on the parameters in the EGARCH equation are also tested with Likelihood Ratio test (LR). And the results are also listed in
Next, we test the parameters in the SSAEPD with same method and the results show englishparameter α is statistically significant equal to 0.5, so skewnessenglish is not documentedenglish. Non-Normality is confirmed (see column T8, T10, T11, T12). The left and the right tail parametersenglish ( p 1 and p 2 ) are jointly statistically different from 2 (see column T10). And leftenglish tail is statistically different from 2 in all markets (see column T11) but right tail is only statistically
different from 2 in North American market (see column T12). i.e., strong left fat-tailedness is documented. Therefore. this new 5-factor model can capture the fat-tailedness better than FF5-Normal model.
• Kolmogorov-Smirnov Test for Residuals
We check the residuals for models with Kolmogorov-Smirnov test (KS). The P-values of KS test are listed in
We compare models with AIC. Results in
In this paper, US sector of services is studied. A new Fama-French 5-factor model (denoted as FF5-SSAEPD-EGARCH) is empirically tested. This new model uses the non-normal error term of SSAEPD of Zhu and Zinde-Walsh
Model | FF5-SSAEPD-EGARCH | FF5-Normal |
---|---|---|
US 5factors | 0.04 | 0.00 |
North American 5 factors | 0.05 | 0.00 |
Global 5 factors | 0.07 | 0.00 |
Note: The null hypothesis of KS test is H 0 : Data follow a specified distribution. We set the significance level of all tests at 5%. If the P-value of KS test is bigger than 5%, then we do not reject the null hypothesis. Otherwise, we reject the null hypothesis. For example, We apply KS test for the FF5-SSAEPD-EGARCH model residuals with the null hypothesis of H 0 : FF5-SSEAPD-EGARCH model residuals are distributed as SSAEPD ( α ^ , p ^ 1 , p ^ 2 ) . For Global five factors, its P-value is 0.07, which is bigger than 0.05. That means, under 5% significance level, we cannot reject the null hypothesis and conclude that the residuals from FF5-SSEAPD-EGARCH model follow SSAEPD.
Model | FF5-SSAEPD-EGARCH | FF5-Normal | FF5-SSAEPD | FF5-GARCH | FF5-SSAEPD-GARCH | FF5-EGARCH |
---|---|---|---|---|---|---|
US 5 factors | 4.16 | 4.19 | 4.90 | 4.14 | 4.14 | 4.18 |
North Ame. 5 factors | 4.30 | 4.27 | 5.06 | 4.22 | 4.22 | 4.31 |
Global 5 factors | 5.09 | 5.05 | 6.59 | 4.91 | 4.91 | 5.12 |
Note: North Ame.=North American.
(2009) and EGARCH-type volatility of Nelson (1991) to extend the 5 factor model of Fama and French (2015). The return of services industry and 3 types of 5 factors (US five factors, North American five factors, Global five factors) from French’s Data Library are analyzed and compared. Sample period is from Jul. 1990 to Feb. 2017. Likelihood Ratio test (LR) is used for parameter restriction test, Kolmogorov-Smirnov test (KS) for residual check and AIC for model comparison. Maximum Likelihood Estimation method (MLE) is used to estimate models via MatLab.
Empirical results show: 1) With EGARCH-typed volatilities and non-normal errors, the Fama-French 5 factors are still alive; 2) The new model fits the data better than Fama-French (2015)’s 5-factor model; 3) Models with GARCH-typed volatility are a little bit better than the ones with EGARCH-typed volatility; 4) To capture fat-tailedness, GARCH equation is better than SSAEPD; 5) Using SSAEPD, model can capture stronger left fat-tailedness.
Future extensions will include but not limited to the following: First, we can construct a new index for services industry; Second, other sectors can be analyzed; Last, different factors can be considered.
Yang, Q., Li, L.L., Zhu, Q.Y. and Mizrach, B. (2017) Analysis of US Sector of Services with a New Fama-French 5-Factor Model. Applied Mathematics, 8, 1307-1319. https://doi.org/10.4236/am.2017.89096