Sequential Variable Selection as Bayesian Pragmatism in Linear Factor Models

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Journal of Mathmatical Financ, 3, 3, 3-36 http://dxdoiorg/436/jmf33a Publishd Onlin March 3 (http://wwwscirporg/journal/jmf) Squntial Variabl Slction as Baysian Pragmatism in Linar Factor Modls John

1 Journal of Mathmatical Financ, 3, 3, Publishd Onlin March 3 ( Squntial Variabl Slction as Baysian Pragmatism in Linar Factor Modls John Knight, Stphn Satchll, Jssica Qi Zhang 3 Dpartmnt of Economics, Univrsity of Wstrn Ontario, Ontario, Canada Dpartmnt of Financ, Univrsity of Sydny, Sydny, Australia 3 Dpartmnt of Accounting Financ, Univrsity of Grnwich, London, UK jknight@uwoca, ss999gb@yahoocouk, zq3@gracuk Rcivd January, 3; rvisd Fbruary 9, 3; accptd Fbruary, 3 ABSTRACT W xamin a popular practitionr mthodology usd in th construction of linar factor modls whrby particular factors ar incrasd or dcrasd in rlativ importanc within th modl This allows modl buildrs to customis modls, as such, rflct thos factors that th clint modllr may think important W call this procss Pragmatic Baysianism (or prag-bays for short) w provid analysis which shows whn such a procdur is likly to b succssful Kywords: Linar Factor Modls; Baysian Statistics; Squntial Rgrssion Introduction Th purpos of this papr is to invstigat statistical procdurs frquntly usd by practitionrs to build factor modls In particular, w ar intrstd in th variabl slction mthodologis that ar usd to giv a particular rturns modl a particular styl natur For xampl, in th contxt of global modls, on may wish th modl to dpnd mor or lss upon domstic factors such as country s indics rathr than, say, global factors such as currncy or world quity bond markts Likwis at th domstic lvl, on may want on s modl to b built around styls (valu, growth tc) rathr than industris or sctors altrnativly, th opposit may b prfrrd Th litratur on this topic is vry spars W prsnt a brif survy of altrnativ approachs Th problm can b viwd as a practical altrnativ to wll-known Baysian procdurs, such as Jorion s (986) [] Bays Stin adjustmnt Black-Littrman s BL modl (99, 99) [,3] Ths modls ar both xampls of Baysian adjustmnt which ffctivly updats currntly hld opinions with data to form nw opinions Satchll Scowcroft () [4] also prsnt dtails of Baysian portfolio construction procdurs basd on Black-Littrman modls Th ssntial ida in this procss is to hav a prior distribution ovr xpctd rturns or ovr th rgrssion Btas In ithr cas, on nds to spcify hyprparamtrs which ar, in practic, vry troublsom Th procdur w advocat, which is usd by practitionrs, is to convrt blifs about th magnitud of btas into procdurs of squntial rgrssion In Sction w shall dscrib how this is don in practic how it could b analysd in thory In Sction 3 w shall prsnt conditions undr which ths mthodologis should work Sction 4 prsnts som mpirical rsults Conclusions furthr discussion ar prsntd in Sction 5 Thorm Thr ar a numbr of procdurs that can b usd to facilitat on factor bing prfrrd to anothr Hr w shall assum that our rturn sris is dnotd by th n vctor y, th two factors ovr which w may hav prfrncs ar dnotd by X X rspctivly, both n vctors Ltting X X, X, w will facilitat calculations n latr by making th following assumption: XX Our tru modl is y X X u () whr y u ar n vctors, β β ar scalars u ~ N, IN This is obviously a simplification of th gnral cas, but littl is lost in so doing it allows us to focus on th ssntial faturs of th problm W now dfin th squntial variabl slction mthod (SVSM), which is th ssntial componnt of th prag-bays approach Dfinition: Th SVSM is dfind by th following Copyright 3 SciRs

J KNIGHT ET AL 3 procdur If you want variabl to xplain mor of y asst rturns than variabl, you rgrss variabl first in a univariat rgrssion Th cofficint for variabl is thn calculatd by rgrssing th

2 J KNIGHT ET AL 3 procdur If you want variabl to xplain mor of y asst rturns than variabl, you rgrss variabl first in a univariat rgrssion Th cofficint for variabl is thn calculatd by rgrssing th rsidual of y on variabl upon th rsidual of variabl on variabl Th qustion w wish to ask is: undr what circumstancs will this procdur lad to a largr stimatd x- posur ˆ of variabl vrsus that of variabl, ˆ A closly rlatd qustion is th conditions undr which th nw slop stimats will b biggr or smallr than thos calculatd from convntional ordinary last squars (OLS) It is worth discussing a variant on ths procdurs which concrns tsting Rathr than just focusing on th magnitud of ˆ : w could also altrnativly mak inclusion xclusion dcisions basd on t-statistics Our rsults can b tiltd in th dsird dirction by moving th critical valus of our tsts In trms of th Equation (), w do not wish to impos β > β for all stocks This is bcaus w rcognis that particular stocks may not b modld subjct to such a constraint To illustrat, in th cas of factor bing a global factor factor bing a domstic factor, w can imagin cass of multinationals whr β > β but thr will also b Japans railway stocks, for xampl, whr th opposit is tru Accordingly, a Baysian approach whr β β ar variabl allows us to approach this qustion in a thortically appaling way W may hav a prior, that P(β β ) d whr d is som thrshold probability, P() dnots th probability of th vnt in brackts This can b asily imposd by an adroit choic of hypr-paramtrs in th prior joint distribution of β β Thn w can comput th liklihood in th usual way, finally, th postrior distribution of β β whr th postrior probability of β β can b computd in a straightforward mannr Howvr, implmntation of hirarchical Bays modls rquird a numbr of ancillary assumptions that ar not particularly transparnt, s Glman (4) [5] for xampl W shall not dtail how a Baysian might procd, but rturn to our SVSM mthod to s if it can achiv similar rsults now addrss th scond qustion as to whthr th SVSM mthod will incras th magnitud, rlativ to OLS, of stimatd β With th abov modl w now considr th two sti- mators of β ) ˆ from y X whr X u ˆ XX Xy X X X X u ) from y X X u i X Px X x X P y, whr x P I X X X X With th assumption on X X w hav immdiatly that ˆ X y X yx y sinc u N, I This implis y N X, I ˆ X y X y And Nu, Whr W now calculat th following probability illustratd in th following diagram Figur, whr th horizontal axis givs valus of ˆ whil th vrtical givs valus of P ˆ P ˆ, ˆ, ˆ P ˆ ˆ P ˆ ˆ P ˆ ˆ,,,,,, Th rsult is statd in th following Thorm Thorm Undr th SVSM stimation procdur w hav th following probability: Whn ρ > Figur Ara dfining th probability Copyright 3 SciRs

3 3 J KNIGHT ET AL For ρ < whr gr r Proof: S Appndix ˆ d d P g r f r r g r f r r g r f r dr P ˆ ˆ h r f r dr h r f r dr h r f d, f r r π r r r xp f r xp r h r 3 Statistical Analysis Th rsults show that if th rgrssion was a high R if th two variabls ar positivly corrlatd, thn To illustrat our calculations, w carrid out som nuthis procdur lads to a high probability that ˆ xmrical calculations; w calculatd th probability that cds not just whn xcds, but vn whn ˆ xcds for diffrnt valus of ρ; w also ˆ is lss than (s Tabls 3-5) In th cas whn computd th R of th rgrssion Th valus of wr R is low or whn th rturns ar ngativly corrlatd,,, 5,, whilst th valus of ρ wr 8, th mthodology is lss succssful 5,,, 5 8 Diffrnt combinations of wr usd, namly (8, 4), (5, 4), (4, 4), 4 Empirical Exampls (3, 4), (, 4) Th output constituts Tabls -5 For illustrativ purposs, w us six Fama-Frnch styl Tabl Probability R-squard for ˆβ = 8 basd portfolios formd on siz book-to-markt Ths ar: Small Growth (SG), Small Nutral (SN), Small Valu (SV), Big Growth (BG), Big Nutral (BN), ˆβ = 8 ; Probability for ˆβ = 8 ˆβ =4; Big Valu (BV) Thr ar two rturn factors, th first, R-squard for ˆβ = 8 ˆβ =4 SMB (Small Minus Big) is th rturn diffrnc btwn th avrag of thr small portfolios th avrag of thr larg portfolios, Likwis, th scond factor, HML (High Minus Low) is th rturn diffrnc btwn th avrag of two valu portfolios th avrag of two growth portfolios W choos two diffrnt sampl priods, whr SMB HML ar ithr positivly or ngativly corrlatd Tabl 6 lists th rgrssion rsults for th priod from Jan to 954 Dc, whr SMB HML ar posi tivly corrlatd with ρ = 59; Tabl 7 lists th rgrssion rsults for th priod from 99 Jan to Dc with ρ = 348 In our squntial variabl slction modl, SMB is variabl HML is variabl is th stimatd cofficint from th univariat rgrssion π Th stocks ar rankd basd on two indpndnt critria: siz (markt capitalization) book-to-pric (th ratio of book valu to markt valu) Th mdian NYSE markt quity is chosn to divid th stocks into two groups: big small; th 3th 7th prcntils of bookto-pric ratio ar usd to split th stocks into thr groups: growth, nutral valu Six portfolios ar formd from th intrsction of ths indpndnt sorts Six portfolios ar formd from th intrsction of ths indpndnt sorts Copyright 3 SciRs

4 J KNIGHT ET AL 33 Tabl Probability R-squard for ˆβ =5 β = 4 ; Probability for β =4 ˆβ = 5 R-squard for ˆβ = 5 β =4 Tabl 4 Probability R-squard for β = 4 ; Probability for β =4 β = 3 ; R-squard for β = 3 β =4 ˆ ˆ β ˆ = Tabl 3 Probability R-squard for β = 4 β = 4 ; Probability for β =4 ; R-squard for β = 4 β = Tabl 5 Probability R-squard for β = 4 ; Probability for β =4 β = ; R-squard for β = β =4 ˆ ˆ β ˆ = Copyright 3 SciRs

5 34 J KNIGHT ET AL Tabl 6 Rgrssion rsults for six portfolios whn ρ = 59 SG SN SV BG BN BV ˆ Rindi ˆ Rmulti Tabl 7 Rgrssion rsults for six portfolios whn ρ = 348 SG SN SV BG BN BV ˆ Rindi ˆ Rmulti of y on SMB; is th cofficint on SMB from th multipl rgrssion of y on SMB HML; ˆ is th cofficint on HML calculatd by rgrssing th rsidual of y on SMB upon th rsidual of HML on SMB W ar intrstd in th following qustion Undr what circumstancs will thr b a largr stimatd xposur ˆ than? Th rsults show that whn th two variabls ar positivly corrlatd as in Tabl 6, this procdur always gnrats highr ˆ than Whn th two variabls ar ngativly corrlatd as in Tabl 7, w idntify highr ˆ than only for two portfolios SG BG; for th othr four portfolios, ˆ is lowr than Thrfor comparing th two diffrnt cass, w find out that th mthodology is mor succssful whn ρ is positivly corrlatd This confirms our finding in sction 3 5 Conclusions Baysian mthods ar notoriously difficult to implmnt practitionrs oftn us tricks to allow thir modls to rflct thir blifs W discuss such a procdur, show analytically conditions whn it will work Th particular procdur w discuss is usd by practitionrs to build factor modls W ar intrstd in th variabl slction mthodologis that ar usd to giv a particular rturns modl a particular styl natur For xampl, in th contxt of global modls on may wish th modl to dpnd mor/lss upon domstic factors such as country indics rathr than, say, global factors such as currncy or world quity/bond markts Th mthod w discuss allows for favorabl slction of a variabl by spcifying th ordr in which variabls ntr a rgrssion W strip th problm down to its bar ssntials by considring bivariat situations W valuat ths conditions using numrical intgration furthr confirm thir rlvanc by looking at an mpirical xampl Th xampls usd US quity data ovr yars priod Ths illustrat th fficacy of th procdur REFERENCES [] P Jorion, Bays-Stin Estimation for Portfolio Analysis, Journal of Financial Quantitativ Analysis, Vol, No 3, 986, pp 79-9 doi:37/334 [] F Black R Littrman, Global Asst Allocation with Equitis, Bonds Currncis, Goldman Sachs Co, Nw York, 99 _6/Black_Littrman_GAA_99pdf [3] F Black, R Littrman, Global Portfolio Optimization, Financial Analysts Journal, Vol 48, No 5, 99, pp 8-43 doi:469/fajv48n58 [4] S Satchll A Scowcroft, A Dmystification of th Black-Littrman Modl: Managing Quantitativ Traditional Portfolio Construction, Journal of Asst Managmnt, Vol, No,, pp 38-5 doi:57/palgravjam4 [5] A Glman, J Carlin, H Strn D Rubin, Baysian Data Analysis, nd Edition, Chaptr 5, Chapman & Hall/ CRC, London, 4 Copyright 3 SciRs

6 J KNIGHT ET AL 35 Appndix: Proof of Thorm Diagrammatically w nd to calculat th two aras in Figur on ithr sid of th origin Now whr r RHS diagram f r, s dsdr r LHS diagram f r, s dsdr r r, f rs f sr f r sr~ Nr, r N, First w shall calculat RHS r RHS f s rds f rdr r Transforming from s to ω, s sr d w hav d s, giving RHS d f r d r π r For r r π π r r RHS d f r dr d f r dr Ltting g r d r π r whr x is th cumulativ distribution function of th stard normal distribution w hav: d d RHS g r f r r g r f r r Having compltd th calculation of RHS, w now turn to LHS r LHS f r, s dsdr g r f r dr r W can mak furthr simplifications dpnding upon th sign of ρ For ρ > ˆ P g r f r dr whr d g r f r r g r f r r d f r π r xp Whn ρ< rwrit using ρ thn lt ρ > Thus w now hav: Now, r ~ N, s sr~ Nr, r ~ N, r r RHS f s r f r dd s r again transforming from s to ω, s sr with ds d Copyright 3 SciRs

7 36 J KNIGHT ET AL Ltting h r r RHS d f r d r π r d r π r For RHS h r f r dr r r LHS h r f r dr d h r f r r h r f r dr Thus for ρ < P ˆ ˆ h r f r dr whr d h r f r r h r f r dr f r π r xp Copyright 3 SciRs

Chapter 13 GMM for Linear Factor Models in Discount Factor form. GMM on the pricing errors gives a crosssectional

Chapter 13 GMM for Linear Factor Models in Discount Factor form. GMM on the pricing errors gives a crosssectional Chaptr 13 GMM for Linar Factor Modls in Discount Factor form GMM on th pricing rrors givs a crosssctional rgrssion h cas of xcss rturns Hors rac sting for charactristic sting for pricd factors: lambdas