A New Fama-French 5-Factor Model Based on SSAEPD Error and GARCH-Type Volatility

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1 Journal of Mahemaical Finance, 06, 6, 7-77 hp:// ISSN Online: 6-44 ISSN Prin: A New Fama-French 5-Facor Model Based on SSAEPD Error and GARCH-Type Volailiy Wenao Zhou, Liuling Li School of Finance, Nankai Universiy, Tianjin, China School of Economics, Nankai Universiy, Tianjin, China How o cie his paper: Zhou, W.T. and Li, L.L. (06) A New Fama-French 5- Facor Model Based on SSAEPD Error and GARCH-Type Volailiy. Journal of Mahemaical Finance, 6, hp://dx.doi.org/0.436/jmf Received: July 30, 06 Acceped: November 3, 06 Published: November 6, 06 Copyrigh 06 by auhors and Scienific Research Publishing Inc. This work is licensed under he Creaive Commons Aribuion Inernaional License (CC BY 4.0). hp://creaivecommons.org/licenses/by/4.0/ Open Access Absrac In his paper, we exend he 5-facor model in Fama and French (05) wih he non-normal errors disribuion of SSAEPD (Sandardized Sandard Asymmeric Exponenial Power Disribuion) in Zhu and Zinde-Walsh (009) and he GARCHype volailiy. The focus is on finding ou wheher our new model can ouperform he original Fama-French 5-facor model. We use Fama-French 5 value-weighed porfolios o conduc our research. The MLE is used o esimae he parameers. The LR es and KS es are used for model diagnosics. Models are compared by AIC. Empirical resuls show ha wih GARCH-ype volailiies and non-normal errors, he Fama-French 5 facors are sill alive. Our new model can successfully capure he skewness, fa-ailness and asymmeric kurosis in he daa and has beer in-sample fi han he 5-facors model in Fama and French (05). Our sudy provides an updae o exising asse pricing lieraure and reference for invesors. Keywords Fama-French 5-Facor Model (FF5), Sandardized Sandard Asymmeric Exponenial Power Disribuion (SSAEPD), GARCH, Asse Pricing. Inroducion The capial asse pricing model of Sharpe and Linner (965) marks he birh of asse pricing heory [], which discovers ha here exiss a posiive linear relaion beween expeced reurns and heir marke beas. Three decades laer, Fama and French (993) proposed a hree-facor model relaing o marke premium, Size, B/M and confirmed ha he 3-facor model ouperformed he single-facor CAPM. [] However, recen sudies have discovered ha many oher imporan paerns in average reurns are lef unexplained by he 3-facor model. Panel A of Table documens DOI: 0.436/jmf November 6, 06

2 Table. Researches abou facor model for sock marke. Auhor (Year) Research Purpose Model Esimaion Mehod Daa Counry Model facors Frequency & Period Panel A: Exension of Facor Model Fama e.al. (993) CAPM Exension FF3 - USA Mk, SMB, HML, WML M963:7-99: Carhar (997) FF3 Exenion CAPM, FF3, C4 OLS USA Mk, SMB, HML, WML M96: - 993: Griffin (00) FF3 Exenion World, Domesic or Inernaional FF3 - Global Mk, SMB, HML M98:995: Bali e.al. (004) FF3 Exenion FF3 wih VAR OLS USA Mk, SMB, HML, VAR M963: - 00: Chan e.al. (005) FF3 Exension FF3 wih IML GMM Ausralia Mk, SMB, HML, IML M990: - 998: Chan e.al. (007) FF3 Exension FF3 wih Defaul facor GMM Ausralia Mk, SMB, HML, DEF M He (008) FF3 Exenion FF3, FF3 wih Sae Swich OLS China Mk, SMB, HML, Sae Swich M995:6-005: Xiao e.al. (007) FF3 Exenion FF3 wih Susainabiliy Facor GMM Global Mk, SMB, HML, SUS M Fama e.al. (03) FF3 Exenion FF4 - USA Mk, SMB, HML, RMW M963:7-0: Yang (03) FF3 Exenion FF3 wih SSAEPD, EGARCH MLE USA Mk, SMB, HML M96-0 Fama e.al. (05) FF4 Exenion FF5 - USA Mk, SMB, HML, RMW, CMA M963:7-03: Mu (05) C Exension C4 wih SSAEPD, EGARCH MLE USA Mk, SMB, HML, WML M97: - 04: Panel B: Fama-French 5-Facor Model comparison Fama e.al. (04) Model Comparison CAPM, FF3, FF4, FF5, FF5 wih WML - USA Mk, SMB, HML, RMW, CMA, WML M963:7-04: Hou e.al. (05) Model Comparison FF5, C, q-facor - USA Harshia e.al. (05) Model Comparison CAPM, FF3, FF5 - India Mk, SMB, HML, RMW, CMA, WML Mk, SMB, HML, RMW, CMA M967: - 03: M999:0-04:9 Chiah e.al. (05) Model Comparison FF3, FF5 HAC-adjused OLS Ausralia Mk, SMB, HML, RMW, CMA M98: - 0: he developmen of he facor model in sock marke. For example, Carhar (997) incorporaes momenum facor ino he Fama-French 3-facor (FF3) model and esablishes a Carhar 4-facor (C4) model which documens ha socks performing he bes in he shor run end o coninue his rend [3]. Chan and Faff (005) consruc a liquidiy-augmened FF3 model [4]. Connor, Hagmann and Linon (0) consider a five-facor exension of he C4 model which suggess an own-volailiy facor [5]. Xiao, Faff, Gharghori and Min (0) incorporae a susainabiliy facor ino 3-facor model which explains he susainabiliy of he world price beer [6]. In 05, Fama and French proposed a 5-facor model direced a capuring he size, value, profiabiliy and invesmen paerns in average sock reurns and found i performed beer han heir 3-facor model [7]. Since hen, many sudies focusing on Fama-French 5-facor (FF5) model have been done. Panel B of Table presens he re- 7

3 searches for he FF5 model. These researches are focused on empirical analysis of FF5 model in differen sock markes and comparison beween he FF5 model and oher models. For example, Hou, Xue and Zhang (05) find ha he 4-facor q-model performs beer han he FF5 model in US marke [8]. Harshia, Singh, S. and Yadav, S. S. (05) discover ha he FF5 model works beer in India han CAPM and FF3 model [9]. Differen from previous researches, our research ries o exend he 5-facor model in Fama and French (05). Many asse pricing models in he exising lieraure jus assume ha financial ime series follow he normal disribuion, bu more and more researches and sudies have observed he unique disribuional properies of financial daa more kurosis and higher peak conradicing he assumpion of normaliy [0]. Thus, insead of adding new facors, we incorporae he GARCH-ype volailiies of Bollerslev (986) ino FF5 model and employ non-normal errors of SSAEPD proposed by Zhu and Zinde-Walsh (009) for he error erm. SSAEPD is capable of capuring many sylized facs in financial ime series such as skewness, fa ails and asymmeric kurosis []. We denoe our new model as FF5-SSAEPD-GARCH. Based on our new model, we ry o figure ou he following wo quesions: ) Wih GARCH-ype volailiies and SSAEPD errors, are he Fama-French 5 facors sill alive? ) Can our new model bea he 5 facor model in Fama and French (05)? To answer hese quesions, we firs run simulaion o es wheher he MaLab program we wrie can be used in our analysis. Then, Fama-French 5 value-weighed porfolios are analyzed. Daa are downloaded from he French s Daa Library, and he sample period is from Jul. 963 o Dec. 03. Mehod of Maximum Likelihood Esimaion (MLE) is used o esimae he parameers. Likelihood Raio es (LR) and Kolmogorov- Smirnov es (KS) are exploied for model diagnosics. Akaike Informaion Crierion (AIC) is employed for model comparison. Simulaion resuls show our MaLab program can be employed for our empirical analysis. According o he empirical resuls, we find ou he 5 facors in Fama and French (05) are sill alive! The new model fis he daa well and has beer in-sample fi han he 5-facor model in Fama and French (05). The paper proceeds as follows. The model and mehodology are discussed in Secion. Simulaion analysis is repored in Secion 3. Empirical resuls and he model comparisons are presened in Secion 4. Secion 5 provides he conclusions and fuure exensions.. Model and Mehodology.. Fama-French 5-Facor Model (FF5-Normal) Fama and French (05) propose a 5-facor model (denoed as FF5) o capure he size, value, profiabiliy, and invesmen paerns in expeced sock reurns, and show his model empirically ouperforms heir 3 facor model. The 5-facor model is: 73

4 R R = β + β * R R + β * SMB + β * HMLO f 0 m f β * RMW + β * CMA + u, u ~ Normal µσ,. where θ = (β 0, β, β, β 3, β 4, β 5, μ, σ) are parameers o be esimaed in his model. R is he reurn on sock porfolio for period. R f is he risk-free reurn. R m is he value-weighed marke reurn. SMB is he reurn on small sock porfolio minus he reurn on big sock porfolio. HMLO is he high book-o-marke raio minus low book-o-marke raio orhogonalized. RMW sands for robus operaing profiabiliy porfolios minus weak operaing profiabiliy porfolios. CMA sands for conservaive invesmen porfolios minus aggressive invesmen porfolios. The error erm u is disribued as he Normal... The FF5-SSAEPD-GARCH Model Based on he GARCH-ype volailiy in Bollerslev (986) and non-normal error disribuion of SSAEPD in Zhu and Zinde-Walsh (009), we exend Fama-French (05) five-facor model in his secion. The new model is denoed as FF5-SSAEPD-GARCH and is mah formula is: R R = β + β R R + β SMB + β HMLO f 0 m f 3 + β4rmw + β5 CMA + u, =,,, T, ( α ) u = σ z, z ~ SSAEPD, p, p, (3) σ r s = a0 + au i i+ biσ i i= i=. (4) ( a b ) max rs, where a 0 > 0, ai 0, bi 0, i= i + i <. (,, 0, { } r s θ = β β a a,{ b i },, i α p, p i= i= ) are he parameer vecors o be esimaed. T is he sample size. The error erm z is disribued as he Sandardized Sandard { 0 = }, { b } i i= 0 r i= Asymmeric Exponenial Power Disribuion (SSAEPD) proposed by Zhu and Zinde-Walsh (009). σ is he condiional sandard deviaion, i.e., volailiy. Wih a = 0 s, α = 0.5, p = p =, he FF5-SSAEPD-GARCH model reduces o he FF5-Normal model. Sandardized Sandard AEPD (SSAEPD) The probabiliy densiy funcion (PDF) of he SSAEPD proposed by Zhu and Zinde- Walsh (009) is where f ( z β ) p α + δ δ * ( ) * α p α w z w K p exp, if z, δ = p α w zδ w δ K * ( p ) exp +, if z. * > α p ( α ) δ α K( p ) + ( α) * α = αk p K p, () () (5) (6) 74

5 K ( p) = p Γ + ( p) p x 0, y Γ x = y e d, y (8) w pγ p pγ p = ( α) α, B Γ p Γ p δ α p Γ 3 p α p Γ 3 p 3 3 = ( ) B Γ p Γ p pγ( p) pγ( p) ( α) α, Γ ( p) Γ ( p) ( α) (7) (9) (0) B = αk p + K p. () > > >. p and p are he parameers conroling he lef ail and righ ail, respecively. Parameer α conrols he skewness of And µ R, σ 0, p 0, p 0, α ( 0,) SSAEPD. The smaller p and p are, he faer he ails are. If α is smaller han 0.5, i indicaes a lef-skewed disribuion. If α is larger han 0.5, i indicaes a righ-skewed disribuion When α = 0.5, p = p =, SSAEPD can be reduced o Normal (0, ). The mean of z is zero and is variance is..3. Maximum Likelihood Esimaion In his paper, we esimae he FF5-SSAEPD-GARCH model wih Maximum Likelihood Esimaion (MLE). The likelihood funcion is where T ({ f, m f,,,, } ; θ = ) L R R R R SMB HMLO RMW CMA T = ( f ) = f R R p ω + δz ( ) p n η α α = p i= ω δz ( ) + η α p δ α ω K p exp, z, δ δ α ω K p exp, z >. ( α ) δ R R f β0 β Rm R f βsmb β3hmlo β4rmw β5cma z =, (3) σ σ r s = a0 + au i i+ biσ i i= i= (). (4) (,, 0, { } r s β β a a,{ b i },, i α p, p i= i= ) θ = is he parameer vecor o be esimaed. 3. Simulaion Analysis In his secion, we firs generae random number series for ( Rm Rf ) Y RMW, CMA and { } T, SMB, HMLO, =. Then we use hem o run he simulaions o es wheher he 75

6 program we wrie in MaLab can be applied o our empirical analysis. The FF5- SSAEPD-GARCH (, ) is simulaed as follows: R R = β + β R R + β SMB + β HMLO f 0 m f 3 + β4rmw + β5 CMA + u, =,,, T, ( α ) (5) u = σ z, z ~ SSAEPD, p, p, (6) σ = + + (7). a0 au bσ We choose β0 = 0., β =, β = 0.5, β3 = 0.5, β4 = 0.5, β5 = 0.5, α = 0.5, p = p =, a = 0.3, b = 0.5, c = 0.4 as he rue values of he parameers. The daa is generaed as follows: ) Given α = 0.5, p = p =, we can generae SSAEPD random number series ) Se he iniial value σ =, ε 0 = 0, and given a = 0.3, b = 0.5, c = 0.4, we can T = 5 generae { σ } u = zσ. u 0 T and { } =, wih he following formula: σ σ σ = a0 + a, 0z + b 0 3) Generae random number series ( Rm Rf ) Uniform (0, )., SMB, HMLO, RMW, CMA from 4) Se β 0 = 0., β =, β = 0.5, β 3 = 0.5, β 4 = 0.5, β 5 = 0.5 and we can ge { } he following formula: Afer geing he simulaed daa ( Rm Rf ) T { Y } Y T =, wih, SMB, HMLO, RMW, CMA and =, we can use hem o esimae he parameers in he FF5-SSAEPD-GARCH model. Using above-menioned rue values of he parameers, we do he simulaions. The simulaion resuls are repored in Table, all he esimaes are close o he rue values of he parameers. For robusness exam, we hen aler he rue values of he parameers and re-run he simulaion. All he simulaion resuls show ha mos of esimaes are very close o he rue values of he parameers, since mos of errors are equal o or less han 0%. Also, we ge he same esimaes as hose in Fama and French (05) (see. Appendix ). Hence, we can draw he conclusion ha his MaLab program can be applied o esimae and analyze empirical daa for FF5-SSAEPD-GARCH. 4. Empirical Analysis 4.. Daa The daa we analyze are he monhly reurns from he Fama-French 5 value-weighed porfolios for US sock marke, which are he same as daa used in Fama and French (05). The descripive saisics of sample daa are calculaed by MaLab and lised in Table 3. For each observaion, he skewness esimaes (excep one case) is no 0 and all kurosis esimaes are more han 3, which suggess ha he daa follows a lepokuric disribuion wih he high peak and fa ail. The P-value of Jarque-Bera es for each porfolio is 0, which is smaller han 5% significance level. Hence, we can rejec he null hypohesis and conclude ha he asse reurns do no follow Normal disribuion. Thus, non-normal error of SSAEPD migh be able o fi he daa beer. 76

7 Table. Simulaion resuls. β 0 β β β 3 β 4 β 5 α p p a b c T E P 8.84%.60%.9% 3.97% 6.%.93% 0.97%.5% 0.89% 0.07% 0.57% 0.70% T E P.85% 3.03% 4.69% 0.85% 7.6% 7.98%.09% 0.49% 0.90%.78%.7%.43% T E P 4.0%.08% 5.4% 5.4% 4.5% 0.89% 0.9% 0.0% 0.03% 0.65%.4% 0.4% T E P 4.43% 4.85% 5.5% 4.5% 6.40% 7.% 0.00% 0.00% 0.00%.73% 3.7%.05% T E P 6.43% 0.70% 0.48% 0.38% 3.87%.69% 0.00% 0.0% 0.00% 9.38% 5.78% 9.5% T E P 5.78%.49% 3.% 0.33% 3.7% 0.073% 0.00% 0.00% 0.00% 8.5%.64% 5.3% T E P 3.90% 3.8% 4.05% 4.49%.8% 0.0% 0.07% 0.% 0.6%.95% 6.5% 4.34% T E P.8% 4.9%.5% 7.50% 0.4% 4.% 0.00% 0.00% 0.00%.94%.09%.% T E P.0% 0.0%.05% 0.5% 3.3% 7.59% 0.0% 0.0% 0.03% 3.7%.30% 0.5% Noes: T means he rue value of parameers. E means he esimaes. P means he error in percenage. 77

8 Table 3. Descripive saisics (963:7-03:). Size Book-o-marke quiniles Quinile Low 3 4 High Low 3 4 High Mean Median Small Big Maximum Minimum Small Big Sandard Deviaion Skewness Small Big Kurosis P-value of Jarque-Bera Tes Small Big Esimaion Resuls Esimaes for he FF5-SSAEPD-GARCH Model The esimaes for our new model are displayed in Table 4. We find ou ha our model can successfully capure he skewness, fa-ailness and asymmeric kurosis of he daa. To be specific, he skewness parameers α are all no equal o 0.5, which capures he skewness in he daa. 44 ou of 50 esimaes for he ail parameers pi (i =, ) are smaller han, which suggess ha porfolio reurns are fa-ailed disribued. Besides, all he ail parameers p and p are no equal o each oher, which documen he asymmeric kurosis. And 5 ou of 5 porfolios have bigger esimaes for he lef ail parameer p, which means ha hese reurns have hinner lef ails. 78

9 Table 4. Esimaes on FF5-SSAEPD-GARCH (Monhly, 963:07-03:). Size Book-o-marke quiniles Quinile Low 3 4 High Low 3 4 High β 0 β Small Big β β 3 Small Big β 4 β 5 Small Big α p Small Big p a Small Big b c Small Big

10 4.3. Model Diagnosics To es he significance of coefficiens in our new model, Likelihood Raio es (LR) is applied, LR formula is from Neyman and Pearson (993), which is Equaion (8). LR = ln ( likelihood for null) + ln ( likelihood for alernaive) (8) Tess for Parameer Resricions Tess for Parameers in he Mean Equaion The P-values of LR are lised in Table 5. The null hypohesis of he join significance es is H 0 : β = β = β 3 = β 4 = β 5 = 0. The P-values of he join significance es for all he 5 porfolios are 0, which means β, β, β 3, β 4 and β 5 are saisically joinly significan under 5% significance level. The individual significance ess show ha under 5% significance level he coefficien β in all 5 porfolios are saisically significan; 4/5 porfolios have a saisically coefficien β and β 5 ; 3/5 and 9/5 porfolios have a saisically coefficien β 3 and β 4, respecively. As for coefficien β 0, 6 ou of he 5 porfolios don have a saisically significan coefficien β 0 under 5% significance level. Thus, we can conclude ha wih non-normal errors such as SSAEPD and GARCH- ype volailiies, he Fama-French 5-facor model is sill alive. Tess for Parameers in he GARCH Equaion In his par, some resricions on he parameers in he GARCH equaion are esed wih Likelihood Raio es (LR). And he resuls are lised in Table 5. Resuls show he GARCH-ype volailiy should be included in Fama-French 5 facor model. For insance, we do he join significance es for hypohesis H 0 : b = c = 0. The P-values of he LR are all smaller han he significance level 5%, which means our GARCH-ype volailiy is quie necessary. As for individual hypoheses, we discover ha mos P-values of LR are smaller han he significance level 5%. And o be specific, ARCH erm (H 0 : b = 0) is significan in 0 ou of 5 porfolios and GARCH erm (H0: c = 0) is significan in 8 ou of 5 porfolios. Tess for Parameers in SSAEPD We also run significance ess for he parameers in he SSAEPD and he resuls of parameer resricions show srong non-normaliy. And he resuls are lised in Table 5. For example, for he Hypohesis H 0 : α = 0.5, p = p =, ou of 5 p-values are smaller han he significance level 5%, which means ha Normal error assumpion is no suppored by mos of our daa. Besides, Asymmery is documened (H 0 : α = 0.5 is rejeced by 7 ou of 5 porfolios). And non-normaliy is found (H 0 : p = is rejeced by 8 ou of 5 porfolios and ou of 5 porfolios rejec he null H 0 : p =.) Residual Check In his subsecion, he residuals for previous models are checked wih boh Kolmogorov-Smirnov es and graphs. Our resuls show ha 0 ou of he 5 porfolios have residuals which do follow SSAEPD. Tha means our new model is adequae for he Fama- French 5 porfolios. Bu he FF5-Normal model is no adequae for he daa, since porfolios have residuals which do no follow he Normal error disribuion. 70

11 Table 5. P-values of likelihood raio es (LR). Size Book-o-marke quiniles Quinile Low 3 4 High Low 3 4 High H 0 :β = β = β 3 = β 4 = β 5 = 0 H 0 :β 0 = 0 Small 0* 0* 0* 0* 0* 0* * 0* 0* 0* 0* * * 3 0* 0* 0* 0* 0* * * 0* 0* 0* 0* 0.0* * Big 0* 0* 0* 0* 0* 0* 0* * H 0 :β = 0 H 0 :β = 0 Small 0* 0* 0* 0* 0* 0* 0* 0* 0* 0* 0* 0* 0* 0* 0* 0* 0* 0* 0* 0* 3 0* 0* 0* 0* 0* 0* 0* 0* 0* 0* 4 0* 0* 0* 0* 0* 0* 0* 0* 0* 0* Big 0* 0* 0* 0* 0* 0* 0* 0* 0* 0.0 H 0 :β 3 = 0 H 0 :β 4 = 0 Small 0* 0.03* 0* 0* 0* 0* 0*.00 0* 0* 0* * 0* 0* 0* * 0* 0* 3 0* 0* 0* 0* 0* 0* 0.0* 0* 0.0* 0* 4 0* 0* 0* 0* 0* 0* 0.9 0* * Big 0*.00 0* 0* 0* 0* 0* *.00 H 0 :β 5 = 0 H 0 :b = c = 0 Small 0* 0* 0* 0* 0* 0* 0* 0* 0* 0* 0* * 0* 0* 0* 0* 0* 0.04* 0.0* 3 0* 0* 0* 0* 0* 0* 0* 0* 0* 0* 4 0* 0* 0* 0* 0* 0* 0* 0* 0* 0* Big 0* 0* 0* 0* 0* 0* 0* 0* 0* 0* H 0 :a = 0 H 0 :b = 0 Small 0* 0* 0* 0* 0* 0* 0* 0*.00 0* 0* 0* 0* 0* 0* 0.0* 0* 0* * 3 0* 0* 0* 0* 0* 0* 0* 0* 0* * 0* 0* 0* 0* 0* 0* 0* 0* 0* Big 0* 0* 0* 0* 0* * 0* 0* H 0 :c = 0 H 0 :α = 0.5,p = p = Small 0*.00 0* 0* 0* 0* 0* 0* 0* 0* 0*.00 0*.00 0* 0* 0* * 0* 3 0* 0* 0*.00 0* 0.0* 0* 0* 0* 0* 4 0* * 0* 0* 0* 0* 0.04* Big 0* 0* 0* 0* 0* * 0.3 0* H 0 :α = 0.5 H 0 :p = p = Small * * 0* 0.0* 0* 0* * 0* 0* 0.0* 0* 0* * 0* 0* * 0* 0* 0* 0* 4 0.0* * * 0* 0* 0* 0.04* Big * 0.3 0* H 0 :p = H 0 :p = Small * * * 0.0* 0* * * 0* 0* * * 0* 0* 0* * * 0.4 0* 0. 0* Big * * Noe: * means he null is rejeced under 5% significan level. 7

12 Kolmogorov-Smirnov Tes for Residuals To check he residuals, he Kolmogorov-Smirov es (KS) is employed. The P-value of KS es is displayed in Table 6. The P-values of KS es show he residuals from he new model do follow SSAEPD. For example, he P-value of he porfolio wih Small Size and Low Book-o-marke is 0.7, greaer han 5%, which means under 5% significance level, he null hypohesis is no rejeced and he residuals from our model do follow he SSAEPD. Similarly, he null hypohesis canno be rejeced for oher 9 porfolios. Then, we apply he KS es for he residuals from he FF5-Normal model. The P-values of he KS es are also lised in Table 6. ou of 5 porfolios have smaller P-values han 0.05, which means hese porfolios rejec he nulls. Hence, he error erms of he porfolios do no follow Normal disribuion. And he FF-Normal model is no adequae for he daa. PDFs of Residuals By mehod of eye-rolling, he PDF of residuals is compared wih heoreical PDFs. Taking he porfolio wih Small Size and Low Book-o-marke for example, in Figure, he probabiliy densiy funcion (PDF) for he esimaed residuals z ˆ in FF5-SSAEPD- SSAEPD ˆ α, pˆ, pˆ are ploed. These curves are very close o GARCH and ha of each oher, indicaing ha he residuals are disribued as SSAEPD. Hence, he FF5- SSAEPD-GARCH model fis he daa well. Similarly, he probabiliy densiy funcion (PDF) for he esimaed residuals u ˆ in Normal ˆ µσ, ˆ are shown in Figure. And here is a big FF5-Normal and ha of difference beween hese wo curves, indicaing he residuals do no follow Normal disribuion Model Diagnosics In his subsecion, we compare our new model wih he 5-facor model of Fama and French (05). The Akaike Informaion Crierion (AIC) is used as he model selecion crierion. Table 7 displays he AIC values. We find ha 3 ou of 5 AIC values of he Table 6. P-values of KS es for residuals. Size Book-o-marke quiniles Quinile Low 3 4 High Low 3 4 High FF5-SSAEPD-GARCH a FF5-Normal b Small * 0.8 0* 0* 0* 0.0* 0* * * * * 0* 0* 0* 0* 0* * 0* 0* 0* 0* Big 0* 0* * 0* 0* 0*. b. The null hy- a. The null hypohesis H 0 is: FF5-SSAEPD-GARCH residuals are disribued as SSAEPD ( ˆ α, pˆ, pˆ ) pohesis H 0 is: FF5-Normal residuals are disribued as Normal ( ˆ, ˆ ) µσ.* means he null is rejeced under 5% significan level. 7

13 Figure. PDFs of he residuals (FF5-SSAEPD-GARCH) and SSAEPD ( ˆ, pˆ, pˆ ) α. Figure. PDFs of he residuals (FF5-Normal) and Normal ( ˆ, ˆ ) µσ. 73

14 Table 7. AIC values (Monhly, 963:07-03:). Size Book-o-marke quiniles Quinile Low 3 4 High Low 3 4 High FF5-SSAEPD-GARCH FF5-Normal Small 4.9* 3.60* 3.9* * * 3.6* 3.4* 3.9* 3.44* * 3.6* 3.57* 3.54* 3.96* * 3.76* 3.80* 3.77* 4.6* Big.98* * 3.5* 4.7* Noe: Numbers wih * are smaller AIC values. FF5-SSAEPD-GARCH model are smaller han hose of he FF5-Normal model. Hence, we can conclude ha our new model (FF5-SSAEPD-GARCH) performs beer han he 5-facor model in Fama and French (05). 5. Conclusions In his paper, we exend he 5-facor model in Fama and French (05) by inroducing a non-normal error erm and ime-varying volailiies. The non-normal error assumpion we used is he SSAEPD in Zhu and Zinde-Walsh (009). And he ime-varying volailiies are he GARCH model in Bollerslev (986). For comparison, monhly US sock reurns in Fama and French (05) (963:07-03:) are analyzed. Mehod of Maximum Likelihood is used. Likelihood Raio Tes (LR) is used o es he hypoheses of parameer resricions. Kolmogorov-Smirnov es (KS) is used o check residuals. Akaike Informaion Crierion (AIC) is used o compare models. Simulaion resuls show our MaLab program for he new model is valid. And empirical resuls show: ) his new model can capure he skewness, fa ails and asymmeric kurosis in he daa; ) Wih GARCH-ype volailiies and non-normal errors, he Fama-French 5 facors are sill alive, since he esimaes are all significan; and 3) our new model can fi he daa much beer han 5-facor model in Fama and French (05). Our sudy provides an updae o exising asse pricing lieraure and reference for invesors. Fuure exensions will include bu no limied o he following. Firs, we can exam our resuls wih differen daa. Second, we can compare our resuls wih hose from oher models such as ARIMA model. Las bu no he leas, oher facors can be inroduced ino his model. Acknowledgemens We also wan o hank paricipans in he 6h World Business Research Conference a San Francisco (8-9 July, 06) and he seminars organized by Insiue of Saisics and Economerics, Nankai Universiy. The suppor of Jiayi Zhu and Qingyu Zhu is graefully acknowledged. The auhors are responsible for all errors. 74

15 References [] Sharpe, W.F. (964) Capial Asse Prices: A Theory of Marke Equilibrium under Condiions of Risk. Journal of Finance, 9, [] Fama, E. and French, K. (993) Common Risk Facors in he Reurns on Socks and Bonds. Journal of Financial Economics, 33, hp://dx.doi.org/0.06/ x(93) [3] Carhar, M.M. (997) On Persisence in Muual Fund Performance. Journal of Finance, 5, hp://dx.doi.org/0./j b03808.x [4] Chan, H.W. and Faff, R.W. (005) Asse Pricing and he Illiquidiy Premium. Financial Review, 40, hp://dx.doi.org/0./j x [5] Connor, G., Hagmann, M. and Linon, O. (007) Efficien Semiparameric Esimaion of he Fama-French Model and Exensions. Economerica, 80, [6] Xiao, Y., Faff, R., Gharghori, P. and Min, B.K. (03) Pricing Innovaions in Consumpion Growh: A Re-Evaluaion of he Recursive Uiliy Model. Journal of Banking & Finance, 37, hp://dx.doi.org/0.06/j.jbankfin [7] Fama, E.F. and French, K.R. (05) A Five-Facor Asse Pricing Model. Journal of Financial Economics, 6, -. hp://dx.doi.org/0.06/j.jfineco [8] Hou, K., Xue, C. and Zhang, L. (05) Edior s Choice Digesing Anomalies: An Invesmen Approach. Review of Financial Sudies, 8, hp://dx.doi.org/0.093/rfs/hhu068 [9] Harshia, Singh, S. and Yadav, S.S. (05) Indian Sock Marke and he Asse Pricing Models. Procedia Economics & Finance, 30, hp://dx.doi.org/0.06/s-567(5)097-6 [0] Malmsen, H. and Svira, T. (0) Sylized Facs of Financial Time Series and Three Popular Models of Volailiy. European Journal of Pure & Applied Mahemaics, 3, [] Zhu, D.M. and Zinde-Walsh, V. (009) Properies and Esimaion of Asymmeric Exponenial Power Disribuion. Journal of Economerics, 48, hp://dx.doi.org/0.06/j.jeconom

16 Appendix. Esimaes for he FF5-Normal Model To es our MaLab program, we also esimae he FF5-Normal model using he program by seing { a = i }, { } i= b i i= 0 r =, α = 0.5, p = p =. The esimaes are lised in 0 s Table 8. The esimaes and heir - saisics are very close o hose in Table 9, respecively. Thus, he MaLab program we wrie is valid. Table 8. Esimaes for FF5-normal model by our MaLab program (Monhly, 963:07-03:). Size Book-o-marke quiniles Quinile Low 3 4 High Low 3 4 High a (a) Small Big h (h) Small Big r (r) Small Big c (c) Small Big

17 Table 9. Esimaes in Fama and French (05) (Monhly, 963:07-03:). Size Book-o-marke quiniles Quinile Low 3 4 High Low 3 4 High a (a) Small Big h (h) Small Big r (r) Small Big c (c) Small Big Noe: This able is quoed from he resuls in Table 7 on page 3 of Fama and French (05). 77

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