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. RMW_t 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).

• 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 Table 4. We find out with non-Normal errors such as SSAEPD and EGARCH-type volatilities, the Fama-French 5 factors are still alive for the sector of services.

Panel A of Table 4 lists the test results for the FF5-SSAEPD-EGARCH model. For example, the P-values of the joint significance test (see column T1)⁹ for all 3 types of 5 factors (US, North Ameirican and Global) are approximately equal to 0, which means the coefficient of β 1 , β 2 , β 3 , β 4 and β 5 are statistically joint significant under 5% significance level.

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 Table 4 lists the test results for the FF5-Normal model. For this model, this sector doesn’t have a statistically significant coefficient β 0 under 5% significance level (see column T2) which means they can not earn statistically signicant Alpha returns. But with FF5-SSAEPD-EGARCH model, this sector in both US and Global market have a statistically significant coefficient β 0 under 5% significance level, expecially we can earn more Alpha return from Global market because β 0 in this market is 0.78 (see Table 3, column 2), the highest among 3 types of 5 factors. Furthermore, the size factor seems statistically significant for 2/3 kinds of 5 factors and is not significant in Global Market¹⁰.

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 Table 4. Results show the EGARCH-type volatility should be included in Fama-French 5 factor model. For instance, we do the joint significance test for hypothesis H 0 : b = c = d = 0 . The P-value of the LR are all smaller than the significance level 5%, which means our EGARCH-type volatilities is necessary, english ARCH and GARCH terms should be added into Fama-French 5-factor model since they are all statistically signicant (see column T13).

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 Table 5, the P-value of the Global five factors is 0.07, greater than 5%, which means under 5% significance level, the null hypothesis is not rejected and the residuals from FF5-SSAEPD-EGARCH do follow the SSAEPD. 2/3 markets support this result. For the FF5-Normal model¹¹, the P-values of the KS test are also listed in Table 5. All of them have smaller P-values than 0.05, which means reject the nulls. Hence, the residuals of the FF5-Normal model don’t follow Normal distribution. And the FF5-Normal model is not adequate for the data.

We compare models with AIC. Results in Table 6 show FF5-SSAEPD-EGARCH model has smaller AIC value than FF5-Normal model. That is, this new model is better than the one in Fama and French(2015). However, FF5-GARCH model or FF5-SSAEPD-GARCH model seem to be the best because it has the smallest AIC values. That means, models with GARCH is better than the ones with EGARCH. Also, since AIC values are the same for both FF5-GARCH and FF5-SSAEPD-GARCH, we conclude that, to capture fat-tailedness, GARCH equation is better than SSAEPD, which is consistent with what we found out in our previous researches.