FINANCIAL ECONOMETRICS

Transcription

1 FINANCIAL ECONOMETRICS SPRING 7 WEEK VII MULTIVARIATE MODELLING OF VOLATILITY Prof. Dr. Burç ÜLENGİN

2 MULTIVARIATE VOLATILITY Tere my e inercions mong e condiionl vrince of e reurn series. Also covrince of e reurn series my cnge over e ime. Terefore e full perspecive of voliliy modelling requires e remen of vrinces nd covrinces ogeer- simulneously. Wen e vrinces nd covrinces re modelled i mens correlions re modelled oo.

3 MOVING CORRELATION OF THE RETURNS OF TWO FINANCIAL ASSETS..9 CORRELATION cor5 cor

4 MULTIVARIATE GARCH In mulivrie GARCH models, y is vecor of e condiionl mens (Nx), e condiionl vrince of y is n mrix H (NxN). Te digonl elemens of H re e vrince erms ii, nd e off-digonl elemens re e covrince erms ij.... N... N H... NN... NN

5 MULTIVARIATE GARCH Tere re numerous differen represenions of e mulivrie GARCH model. Te min represenions re: VECH Digonl BEKK- B, Engle, Krf, Kroner Consn correlion represenion Principle componen represenion

6 VECH REPRESANTATION Full remen of e mrix H In e VECH model, e numer of prmeers cn e exeremely lrge. Esiming lrge numer of prmeers is no in eory prolem s long s ere is lrge enoug smple size. Te prmeers of VECH re esimed y mximum likeliood nd e oining convergence of e ypicl opimizion lgorim employed in prcice e very difficul wen lrge numer of prmeers re involved. Also esimed vrinces mus e posiive nd i requires e ddiionl resricions on prmeers

7 VECH REPRESANTATION Vrile Cse A nd B re {Nx(N+)/, Nx(N+)/} mrices. In e cse of vriles, 3 equions nd prmeers. 5 vriles, equions nd 8 prmeers. vriles, 55 equions nd 45 prmeers.

8 DIAGONAL REPRESENTATION Te digonl represenion is sed on e ssumpion e individul condiionl vrinces nd condiionl covrinces re funcions of only lgged vlues of emselves nd lgged squred residuls. Bollerslev, Engle nd Woodridge (988) proposed In e cse of vriles, is represenion reduces e numer of prmeers o e esimed from o 9. A e expense of losing informion on cerin inerrelionsips, suc s e relionsip eween e individul condiionl vrinces nd e condiionl covrinces. Also esimed vrinces mus e posiive nd i requires e ddiionl resricions on prmeers

9 DIAGONAL REPRESENTATION Vrile Cse ) ( ' BH A H

10 DIAGONAL REPRESENTATION OIL & NATURAL GAS PRICES ROIL r r gs oil 33 RGAZ

11 DIAGONAL REPRESENTATION ESTIMATION OIL & NATURAL GAS PRICES Esimion Meod: ARCH Mximum Likeliood (Mrqurd) Covrince specificion: Digonl VECH Smple: 997M 7M Included oservions: Tol sysem (lnced) oservions 4 Disurnce ssumpion: Suden's disriuion Convergence cieved fer 98 ierions Coefficien Sd. Error z-sisic Pro. C() C() Vrince Equion Coefficiens C(3) C(4) C(5) C(6) C(7) C(8) C(9) C() C() Disriuion (Degree of Freedom) C() Log likeliood Scwrz crierion Avg. log likeliood Hnnn-Quinn crier Akike info crierion r r gs oil 33 33

12 COV_ = *RESID(-)*RESID(-) +.857*COV_(-) DIAGONAL REPRESENTATION ESTIMATION OIL & NATURAL GAS PRICES Covrince specificion: Digonl VECH GARCH = M + A.*RESID(-)*RESID(-)' + B.*GARCH(-) M is n indefinie mrix A is n indefinie mrix B is n indefinie mrix Coefficien C().59 C().438 Coefficien Vrince Eq C(3) M(,) C(4).837 M(,).84 C(5) M(,) 9.48 C(6).4 A(,). C(7) -.56 A(,) -.6 C(8) -. A(,) -. C(9).35 B(,).35 C().858 B(,).86 C().67 B(,).7 Esimed Equions: ===================== RGAZ = C() ROIL = C() Susiued Coefficiens: ===================== RGAZ =.58 ROIL =.438 Vrince-Covrince Represenion: ===================== GARCH = M + A.*RESID(-)*RESID(-)' + B.*GARCH(-) Vrince nd Covrince Equions: ===================== GARCH = M(,) + A(,)*RESID(-)^ + B(,)*GARCH(-) GARCH = M(,) + A(,)*RESID(-)^ + B(,)*GARCH(-) COV_ = M(,) + A(,)*RESID(-)*RESID(-) + B(,)*COV_(-) Susiued Coefficiens: ===================== GARCH = *RESID(-)^ +.35*GARCH(-) GARCH = *RESID(-)^ +.66*GARCH(-) r r gs oil 33 33

13 DIAGONAL REPRESENTATION VOLATILITY FORECAST OF OIL & NATURAL GAS PRICES 8 Vr(RGAZ) Condiionl Covrince..75 Condiionl Correlion Cor(RGAZ,ROIL) Cov(RGAZ,ROIL) Vr(ROIL)

14 DIAGONAL REPRESENTATION REVISED MODEL ESTIMATION OIL & NATURAL GAS PRICES r r r oil oil gs

15 DIAGONAL REPRESENTATION REVISED MODEL ESTIMATION OIL & NATURAL GAS PRICES Esimion Meod: ARCH Mximum Likeliood (Mrqurd) Covrince specificion: Digonl VECH Smple: 997M3 7M Included oservions: 9 Tol sysem (lnced) oservions 38 Disurnce ssumpion: Suden's disriuion Convergence cieved fer 6 ierions Coefficien Sd. Error z-sisic Pro. C() C() C(3) C(4) C(5) C(6) C(7) C(8) Covrince specificion: Digonl VECH GARCH = M + A.*RESID(-)*RESID(-)' + B.*GARCH(-) M is sclr A is rnk one mrix B is rnk one mrix Trnformed Vrince Coefficiens Coefficien M.974 A(,).4 A(,) -.5 A(,). B(,).793 B(,).875 B(,).966 -Disriuion (Degree of Freedom) C(9) Log likeliood Scwrz crierion Avg. log likeliood Hnnn-Quinn crier Akike info crierion

16 DIAGONAL REPRESENTATION REVISED MODEL ESTIMATION OIL & NATURAL GAS PRICES Covrince specificion: Digonl VECH GARCH = M + A.*RESID(-)*RESID(-)' + B.*GARCH(-) M is sclr A is rnk one mrix B is rnk one mrix Coefficien C().548 C().4 C(3).53 Esimed Equions: ===================== RGAZ = C()+C()*ROIL(-) ROIL = C(3) Susiued Coefficiens: ===================== RGAZ = *ROIL(-) ROIL = C(4).974 C(5).474 C(6) -. C(7).89 C(8).983 Trnformed Vrince Coefficiens Coefficien M.974 A(,).4 A(,) -.5 A(,). B(,).793 B(,).875 B(,).966 Vrince-Covrince Represenion: ===================== GARCH = M + A.*RESID(-)*RESID(-)' + B.*GARCH(-) Vrince nd Covrince Equions: ===================== GARCH = M + A(,)*RESID(-)^ + B(,)*GARCH(-) GARCH = M + A(,)*RESID(-)^ + B(,)*GARCH(-) COV_ = M + A(,)*RESID(-)*RESID(-) + B(,)*COV_(-) Susiued Coefficiens: ===================== GARCH = *RESID(-)^ +.79*GARCH(-) GARCH = *RESID(-)^ +.965*GARCH(-) COV_ = *RESID(-)*RESID(-) +.875*COV_(-)

17 DIAGONAL REPRESENTATION VOLATILITY FORECAST OF OIL & NATURAL GAS PRICES Condiionl Covrince Vr(RGAZ) Condiionl Correlion Cor(RGAZ,ROIL) Cov(RGAZ,ROIL) Vr(ROIL)

18 DIAGONAL REPRESENTATION TARCH MODEL ESTIMATION OIL & NATURAL GAS PRICES ) )( ( I d I I d I d r r r oil oil gs ) ( )) ( )( ( ' BH D I I D A A H I = if - < = oerwise I = if - < = oerwise

19 DIAGONAL REPRESENTATION TARCH MODEL ESTIMATION OIL & NATURAL GAS PRICES Esimion Meod: ARCH Mximum Likeliood (Mrqurd) Covrince specificion: Digonl VECH TARCH De: 8/5/8 Time: 8:36 Smple: 997M3 7M Included oservions: 9 Tol sysem (lnced) oservions 38 Disurnce ssumpion: Suden's disriuion Presmple covrince: ckcs (prmeer =.5) Convergence cieved fer 69 ierions Coefficien Sd. Error z-sisic Pro. Covrince specificion: Digonl VECH GARCH = M + A.*RESID(-)*RESID(-)' + D.*(RESID(-)*(RESID( -)<))*(RESID(-)*(RESID(-)<))'D.*(RESID(-)*(RESID(-)< *(RESID(-)*(RESID(-)<))' + B.*GARCH(-) M is sclr A is sclr D is rnk one mrix B is sclr C() C() C(3) Vrince Equion Coefficiens C(4) C(5) C(6) C(7) C(8) Trnformed Vrince Coefficiens Coefficien M.465 A -.3 D(,).5 D(,) -.5 D(,).55 B.97 -Disriuion (Degree of Freedom) C(9) Log likeliood Scwrz crierion Avg. log likeli Hnnn-Quinn crier Akike info cr 5.485

20 DIAGONAL REPRESENTATION TARCH MODEL ESTIMATION OIL & NATURAL GAS PRICES Coefficien Esimed Equions: ===================== RGAZ = C()+C()*ROIL(-) ROIL = C(3) Susiued Coefficiens: ===================== RGAZ = *ROIL(-) ROIL = C().44 C().3 C(3).543 Vrince Equion Coefficiens C(4).465 C(5) -.3 C(6).4 C(7) -.34 C(8).97 Trnformed Vrince Coefficiens Coefficien M.465 A -.3 D(,).5 D(,) -.5 D(,).55 B.97 Vrince-Covrince Represenion: ===================== GARCH = M + A.*RESID(-)*RESID(-)' + D.*(RESID(-)*(RESID(-)<))*(RESID(-)*(RESID(-)<))'D.*(RESID(-)*(RESID(-)<))*(RESID(-)*(RESID(-)<))' + B.*GARCH Vrince nd Covrince Equions: ===================== GARCH = M + A*RESID(-)^ + D(,)*RESID(-)^*(RESID(-)<) + B*GARCH(-) GARCH = M + A*RESID(-)^ + D(,)*RESID(-)^*(RESID(-)<) + B*GARCH(-) COV_ = M + A*RESID(-)*RESID(-) + D(,)*RESID(-)*(RESID(-)<)*RESID(-)*(RESID(-)<) + B*COV_(-) Susiued Coefficiens: ===================== GARCH = *RESID(-)^ +.5*RESID(-)^*(RESID(-)<) +.97*GARCH(-) GARCH = *RESID(-)^ +.54*RESID(-)^*(RESID(-)<) +.97*GARCH(-) COV_ = *RESID(-)*RESID(-) -.5*RESID(-)*(RESID(-)<)*RESID(-)*(RESID(-)<) +.97*COV_(-)

21 DIAGONAL REPRESENTATION TARCH MODEL FORECAST OIL & NATURAL GAS PRICES VOLATILITY Condiionl Covrince Condiionl Correlion 6 Vr(RGAZ).8 Cor(RGAZ,ROIL) Cov(RGAZ,ROIL) Vr(ROIL)

22 BEKK REPRESENTATION Engle nd Kroner(995) developed e B(99) pproc. BEKK represenion of mulivrie GARCH improves on o e VECH nd digonl represenion, since H is lmos gurneed o e posiive definie. BEKK represenion require more prmeers n Digonl rep. u less prmeers n VECH. I is more generl n digonl rep. s i llows for inercion effecs digonl rep. does no. H ' A A BH ( ) B

23 BEKK REPRESENTATION OF OIL & NATURAL GAS PRICES r r oil gs

24 Log likelioo Scwrz crierion Avg. log like Hnnn-Quinn crier Akike info c BEKK ESTIMATION OF OIL & NATURAL GAS PRICES Esimion Meod: ARCH Mximum Likeliood (Mrqurd) Covrince specificion: BEKK Smple: 997M 7M Included oservions: Tol sysem (lnced) oservions 4 Disurnce ssumpion: Suden's disriuion Presmple covrince: ckcs (prmeer =.5) Convergence cieved fer 5 ierions Covrince specificion: BEKK GARCH = M + A*RESID(-)*RESID(-)'*A + B*GARCH(-)*B M is sclr A is digonl mrix B is digonl mrix Coefficien Sd. Error z-sisic Pro. C() C() Vrince Equion Coefficiens C(3) C(4) C(5) C(6) C(7) Trnformed Vrince Coefficiens Coefficien M.34 A(,).47 A(,) -.79 B(,).98 B(,).987 -Disriuion (Degree of Freedom) C(8)

25 BEKK ESTIMATION OF OIL & NATURAL GAS PRICES Esimed Equions: ===================== RGAZ = C() Coefficien ROIL = C() Susiued Coefficiens: ===================== RGAZ = ROIL = Vrince-Covrince Represenion: ===================== GARCH = M + A*RESID(-)*RESID(-)'*A + B*GARCH(-)*B Vrince nd Covrince Equions: ===================== GARCH = M + A(,)^*RESID(-)^ + B(,)^*GARCH(-) GARCH = M + A(,)^*RESID(-)^ + B(,)^*GARCH(-) COV_ = M + A(,)*A(,)*RESID(-)*RESID(-) + B(,)*B(,)*COV_(-) Susiued Coefficiens: ===================== GARCH =.34+.8*RESID(-)^+.84*GARCH(-) C().89 C().346 Vrince Equion Coefficiens C(3).34 C(4).47 C(5) -.79 C(6).98 C(7).987 Trnformed Vrince Coefficien Coefficien M.34 A(,).47 A(,) -.79 B(,).98 B(,).987 GARCH = *RESID(-)^+.974*GARCH(-) COV_ = *RESID(-)*RESID(-) +.896*COV_(-)

26 REVISED BEKK REPRESENTATION OF OIL & NATURAL GAS PRICES r r r oil oil gs

27 Log likelioo Scwrz crierion Avg. log likel Hnnn-Quinn crier Akike info c REVISED BEKK REPRESENTATION OF OIL & NATURAL GAS PRICES Esimion Meod: ARCH Mximum Likeliood (Mrqurd Covrince specificion: BEKK Smple: 997M3 7M Included oservions: 9 Tol sysem (lnced) oservions 38 Disurnce ssumpion: Suden's disriuion Presmple covrince: ckcs (prmeer =.5) Convergence cieved fer 6 ierions Covrince specificion: BEKK GARCH = M + A*RESID(-)*RESID(-)'*A + B*GARCH(-)*B M is sclr A is digonl mrix B is digonl mrix Trnformed Vrince Coefficiens Coefficien Sd. Error z-sisic Pro. Coefficien Sd. Error z-sisic Pro. C() C() C(3) Vrince Equion Coefficiens M A(,) A(,) B(,) B(,) C(4) C(5) C(6) C(7) C(8) Disriuion (Degree of Freedom) C(9)

28 REVISED BEKK REPRESENTATION OF OIL & NATURAL GAS PRICES Esimed Equions: ===================== RGAZ = C()+C()*ROIL(-) ROIL = C(3) Susiued Coefficiens: ===================== RGAZ = *ROIL(-) ROIL = Vrince-Covrince Represenion: ===================== GARCH = M + A*RESID(-)*RESID(-)'*A + B*GARCH(-)*B Vrince nd Covrince Equions: ===================== GARCH = M + A(,)^*RESID(-)^ + B(,)^*GARCH(-) GARCH = M + A(,)^*RESID(-)^ + B(,)^*GARCH(-) COV_ = M + A(,)*A(,)*RESID(-)*RESID(-) + B(,)*B(,)*COV_(-) Susiued Coefficiens: ===================== GARCH = *RESID(-)^+.79*GARCH(-) GARCH =.974+.*RESID(-)^+.965*GARCH(-) COV_ = *RESID(-)*RESID(-) +.875*COV_(-)

29 REVISED BEKK FORECASTING OF OIL & NATURAL Condiionl Correlion GAS PRICES VOLATILITY Cor(RGAZ,ROIL) Condiionl Covrince.6 Vr(RGAZ) Cov(RGAZ,ROIL) Vr(ROIL)

30 CONSTANT CORRELATION REPRESENTATION Bollerslev(99) employes e condiionl corelion mrix R o derive represenion of e mulivrie GARCH model. In is R mrix, Bollerslev resrics e condiionl correlions o e equl o e correlion coefficiens eween vriles, wic re simply consns. Tus R is consn over ime. Tis represenion s e dvnge H will e posiive definie.

31 CONSTANT CORRELATION REPRESENTATION N N N N R NN N N N N NN H Te individul vrince erms ii re ken o e individul GARCH processes

32 Log likeliood Scwrz crierion Avg. log likeli Hnnn-Quinn crier Akike info cri CONSTANT CORRELATION REPRESENTATION OF OIL & NATURAL GAS PRICES Esimion Meod: ARCH Mximum Likeliood (Mrqurd) Covrince specificion: Consn Condiionl Correlion Smple: 997M 7M Included oservions: Tol sysem (lnced) oservions 4 Disurnce ssumpion: Suden's disriuion Presmple covrince: ckcs (prmeer =.5) Convergence cieved fer ierions Coefficien Sd. Error z-sisic Pro. C() C() Vrince Equion Coefficiens C(3) C(4) C(5) C(6) C(7) C(8) C(9) Disriuion (Degree of Freedom) C() Covrince specificion: Consn Condiionl Correlion GARCH(i) = M(i) + A(i)*RESID(i)(-)^ + B(i)*GARCH(i)(- COV(i,j) = R(i,j)*@SQRT(GARCH(i)*GARCH(j)) Trnformed Vrince Coefficiens Coefficien Sd. Error z-sisic Pro. M() A() B() M() A() B() R(,)

33 CONSTANT CORRELATION REPRESENTATION OF OIL & NATURAL GAS PRICES Susiued Coefficiens: ===================== RGAZ = ROIL = Vrince nd Covrince Represenions: ===================== GARCH(i) = M(i) + A(i)*RESID(i)(-)^ + B(i)*GARCH(i)(-) COV(i,j) = R(i,j)*@SQRT(GARCH(i)*GARCH(j)) Vrince Equion Coefficiens C(3) C(4).5 C(5).449 C(6) C(7) -.45 C(8).33 C(9).54 Vrince nd Covrince Equions: ===================== GARCH = C(3) + C(4)*RESID(-)^ + C(5)*GARCH(-) GARCH = C(6) + C(7)*RESID(-)^ + C(8)*GARCH(-) COV_ = C(9)*@SQRT(GARCH*GARCH) Susiued Coefficiens: ===================== GARCH = *RESID(-)^ +.448*GARCH(-) GARCH = *RESID(-)^ +.33*GARCH(-) Trnformed Vrince Coefficiens Coefficien M() A().5 B().449 M() A() -.45 B().33 R(,).54 COV_ =.54*@SQRT(GARCH*GARCH)

34 CONSTANT CORRELATION FORECAST OF OIL & NATURAL GAS PRICES VOLATILITY Condiionl Covrince Condiionl Correlion Vr(RGAZ) Cor(RGAZ,ROIL) Cov(RGAZ,ROIL) Vr(ROIL)

35 Log likeliood Scwrz crierion Avg. log likeli Hnnn-Quinn crier Akike info cri REVISED CONSTANT CORRELATION REPRESENTATION OF OIL & NATURAL GAS PRICES Esimion Meod: ARCH Mximum Likeliood (Mrqurd) Covrince specificion: Consn Condiionl Correlion Smple: 997M3 7M Included oservions: 9 Tol sysem (lnced) oservions 38 Disurnce ssumpion: Suden's disriuion Presmple covrince: ckcs (prmeer =.5) Convergence cieved fer 6 ierions Covrince specificion: Consn Condiionl Correlion GARCH(i) = M(i) + A(i)*RESID(i)(-)^ + B(i)*GARCH(i)(-) COV(i,j) = R(i,j)*@SQRT(GARCH(i)*GARCH(j)) Coefficien Sd. Error z-sisic Pro. C() C() C(3) Vrince Equion Coefficiens C(4) C(5) C(6) C(7) C(8) C(9) C() Trnformed Vrince Coefficiens Coefficien Sd. Error z-sisic Pro. M() A() B() M() A() B() R(,) Disriuion (Degree of Freedom) C()

36 COV_ REVISED CONSTANT CORRELATION REPRESENTATION OF OIL & NATURAL GAS PRICES Esimed Equions: ===================== RGAZ = C()+C()*ROIL(-) ROIL = C(3) Susiued Coefficiens: ===================== RGAZ = *ROIL(-) ROIL = Vrince nd Covrince Represenions: ===================== GARCH(i) = M(i) + A(i)*RESID(i)(-)^ + B(i)*GARCH(i)(-) Vrince Equion Coefficiens C(4) C(5).98 C(6).34 C(7) 7.93 C(8) -.35 C(9).53 C(). COV(i,j) = R(i,j)*@SQRT(GARCH(i)*GARCH(j)) Vrince nd Covrince Equions: ===================== GARCH = C(4) + C(5)*RESID(-)^ + C(6)*GARCH(-) GARCH = C(7) + C(8)*RESID(-)^ + C(9)*GARCH(-) COV_ = C()*@SQRT(GARCH*GARCH) Susiued Coefficiens: ===================== GARCH = *RESID(-)^ +.34*GARCH(-) Trnformed Vrince Coefficiens Coefficien M() A().98 B().34 M() 7.93 A() -.35 B().53 R(,). GARCH = *RESID(-)^ +.5*GARCH(-)

37 REVISED CONSTANT CORRELATION FORECAST OF OIL & NATURAL GAS PRICES VOLATILITY Condiionl Correlion Condiionl Covrince Cor(RGAZ,ROIL) Vr(RGAZ) Cov(RGAZ,ROIL) Vr(ROIL)

38 FACTOR VOLATILITY MODELS

39 Principl Componen Anlysis

40 Principl Componen Anlysis We collec finncil rios in order o sses finncil el of e firms. How cn we reduce ese rios few indices? Te producion conrol deprmen collec severl mesures in order o conrol process. Cn we develop some indices in order o summrize e process oucomes? In order o crry ou efficien regression nlysis we ve o reduce mulicollineriy mong e explnory vriles if i exiss. Cn we genere some new indices in order o ge orogonl explnory series lso conin mos of e informion of e originl vriles?

41 Principl Componen Anlysis in Finnce To reduce numer of risk fcors o mngele dimension. For exmple, insed of 6 yields of differen muriies s risk fcors, we mig use jus 3 principl componen. To ideny e key sources of risk. Typiclly e mos imporn risk fcors re prllel sifs, cnges in slope nd cnges in convexiy of e curves. To fcilie e mesuremen of porfolio risk, for insnce y inroducing scenrios on e movemens in e mjor risk fcors.

42 Bsics & Bckground Ax x x x x 3 x x x 3 A is squre mrix X is column vecor is sclr quniyeigenvlue u normlized eigenvecor Bsic properies A u ' u n i i u ' Tr (A) u n i i

43 Bsics & Bckground IF mrix A composes of some oserved x vlues Pricipl Componen Scores y i in u ' ' ' n (x i i x) y i u (xi x) y u (x x)

44 MATHEMATICAL BACKGROUND n ' n n ' ' nxn i ' i n i i nxn u u... u u u u A u u A A nxn squre mrix n ' n n ' ' i ' i n i i u u... u u u u A u u A

45 A BASIC EXAMPLE OF EIGENVALUES AND EIGENVECTORS , * * ) ( x x x x x Ax Normlizion

46 MATHEMATICAL EXAMPLE A 3 4 A I 3 ( ) *(4 ) 4 3* 88l <, 8l 5<< <, 8, 3<< 88-, U U

47 Bsics & Bckground Eigenvlue nd Eigenvecor: Eigen origines in e Germn lnguge nd cn e loosely rnsled s of iself Tus n Eigenvlue of mrix could e concepulized s vlue of iself Eigenvlues nd Eigenvecors re uilized in wide rnge of pplicions (PCA, clculing power of mrix, finding soluions for sysem of differenil equions, nd grow models)

48 GEOMETRICAL APPROACH TO PRINCIPAL COMPONENT ANALYSIS x x Men correced d

49 AXIS ROTATION x x

50 AXIS ROTATION DIMEMSION REDUCTION x x x x

51 AXIS ROTATION x A x x x X = x *cos + x *sin X = -x *sin + x *cos

52 AXIS ROTATION = Oservion x x xcor xcor x' x' Men Vrince Tol Vrince= 44.8 Tol Vrince= 44.8 Covrinc Sre of x'= 5% Correlio Correlion.746 X = x *cos + x *sin X = -x *sin + x *cos

53 AXIS ROTATION = Oservion x x xcor xcor. x' x' Men Vrince Tol Vrince= 44.8 Tol Vrince= 44.8 Covrinc Sre of x'= 65% Correlio Correlion.77 X = x *cos + x *sin X = -x *sin + x *cos

54 AXIS ROTATION = 3 Oservion x x xcor xcor 3 x' x' Men Vrince Tol Vrince= 44.8 Tol Vrince= 44.8 Covrin Sre of x'= 83% Correlio Correlion.448 X = x *cos3 + x *sin3 X = -x *sin3 + x *cos3

55 AXIS ROTATION = 4 Oservion x x xcor xcor 4 x' x' Men Vrince Tol Vrince= 44.8 Tol Vrince= 44.8 Covrin Sre of x'= 87% Correli Correlion.7 X = x *cos4 + x *sin4 X = -x *sin4 + x *cos4

56 AXIS ROTATION = 44 Oservio x x xcor xcor x' x' Men Vrince Tol Vrince= 44.8 Tol Vrince= 44.8 Covrince Sre of x'= 87% Correlion Correlion. X = x *cos44 + x *sin44 X = -x *sin44 + x *cos44

57 AXIS ROTATION = 7 Oservion x x xcor xcor 7 x' x' Men Vrince Tol Vrince= 44.8 Tol Vrince= 44.8 Covrin Sre of x'= 7% Correli Correlion X = x *cos7 + x *sin7 X = -x *sin7 + x *cos7

58 FINDING OPTIMAL Porion of x over ol vrince Wile x dimension explins 5% of e ol vrince, Wen = 4, new x dimension explins 87% of ol vrince 87% *

59 ANALYTICAL APPROACH TO PRINCIPAL COMPONENT ANALYSIS Assume ere re p vriles = w *x +w *x +...+w p *x p = w *x +w *x +...+w p *x p... p = w p *x +w p *x +...+w pp *x p s re principl componens nd w ij is e weig of e j vriles on e i principl componen Vr( ) > Vr( )>... >Vr( p ) w i + w i +...+w ip = i=,,...p w i *w j +w i *w j +...+w ip *w jp = for ll ij

60 ANALYTICAL APPROACH TO PRINCIPAL COMPONENT ANALYSIS Assume ere re p vriles = w *x +w *x +...+w p *x p = w *x +w *x +...+w p *x p... p = w p *x +w p *x +...+w pp *x p s re principl componens nd w ij is e weig of e j vriles on e i principl componen X = cos * x + sin * x X = -sin * x + cos * x = w *x + w *x = w *x + w *x

61 MATRIX ALGEBRA APPROACH TO PRINCIPAL COMPONENT ANALYSIS Assume ere re p vriles = w *x +w *x +...+w p *x p = w *x +w *x +...+w p *x p... p = w p *x +w p *x +...+w pp *x p MATRIX REPRESANTATION = W X = W X W *W = W *W =

62 MATRIX ALGEBRA APPROACH TO PRINCIPAL COMPONENT ANALYSIS Vr( ) = Vr(W X) = W S W S = Te Vrince-Covrince mrix of originl vriles Mx. Vr( ) = W S W s. W *W = If,,..., p re e eigenvlues of S Sw = w Vr( )= W S W = W w = W w = Vrince explined y e firs principl componen = /Trce(S) Trce(S) = Sum of ll i s Vrince explined y e firs k principl componens = ( k )/Trce(S)

63 VARIANCE EXTRACTION METHODS VARIANCE-COVARIANCE MATRIX THE SIZE EFFECT INCLUDED THE ANALYSIS CORRELATION MATRIX THE VARIABLES ARE STANDIZED FIRST, SO THAT MEAN= & VARIANCE= OF ALL VARIABLES THE VARIANCE COVARIANCE MATRIX OF THE STANDARDIZED VARIABLES IS THE CORRELATION MATRIX THE SIZE EFFECT EXCLUDED FROM THE ANALYSIS

64 VARIANCE EXTRACTION METHODS A principl componen represnion sed on e vrincecovrince mrix s e dvnge of providing liner fcor model for e reurns, nd no liner fcor model for e sndrdized reurns, s is e cse wen correlion mrix is used. Sndrdizion mkes ec vrile s common men nd vrince, nd respecively. A PCA on e covrince mrix cpures ll e movemens in e vriles, wic my e domined y e differing voliliies of individul vriles. A PCA on e correlion mrix only cpures e comovemens in reurns nd ignores eir individul voliliies. I is only wen ll vriles ve similir voliliies e PCA will ve similr crcerisics.

65 HOW MANY PRINCIPAL COMPONENTS? IF ALL THE INFORMATION IS EXTRACTED, P COMPONENTS SHOULD BE SELECTED RESEARCHERS CAN WANT TO ELIMINATE MARGINAL INFORMATION SO THAT ONLY MAIN INFO. IS UNDERLINED EXPLAIN RELATIVELY LARGE PERCENTAGE OF THE TOTAL VARIATION. 7-9% ARE USUALLY SUGGESTED FIGURES. EXCLUDE THOSE PRINCIPLE COMPONENTS WHOSE EIGENVALUES ARE LESS THAN AVERAGE EIGENVALUE. IF CORRELATION MATRIX IS USED, EXCLUDE PC WHOSE EIGENVALUES ARE LESS THAN. IF SAMPLE SIZE SMALL, THE CUT OFF POINT CAN BE LOWER,.7. USE SCREE PLOT TO CATCH THE ELBOW AND THE ELBOW POINTS THE NUMBER OF EIGENVALUES SHOULD BE EXCLUDED.

66 Inerpreion of PCs Te firs principle componen cpures common rend in sses or ineres res. If e firs PC cnges ime wen e oer componens re fixed, en e reurns(or cnges in ineres res) ll move y rougly e sme moun. Te second nd iger order PC ve no inuiive inerpreion. Bu wen we use fcor roion, some PC my represen sugroups of e vriles.

67 ORTHOGONAL GARCH X X F X X F F F X F F X ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ )] )( [( ], [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ F VAR F VAR F F COV F F COV F F COV F F COV F F COV F F COV F F COV F F COV F F COV F F COV F F F F COV X X COV F Vr F Vr F F VAR X VAR F Vr F Vr F F VAR X VAR COV[F,F ]=

68 FACTOR GARCH. Clcule eigenvlues nd eigenvecors. Deermine e numer of eigenvecors 3. Clcule e fcor scores nd keep e equions 4. Esime GARCH models for e ec fcor scores. 5. Using fcor score equions esime Vrinces nd Covrinces