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 or b s
13.1 GMM on th pricing rrors Givs a Cross-sctional rgrssion h asst pricing modl is : m = b' f, E ( p) = E ( mx) or simply E( p) = E( xf ') b (13.1) in vctor form p 1 x 1 f 1 b 1 p =, x =, f =, b = pn xn f K b K
Apply th pricing rrors as GMM momnts g () b = E ( xf ' b p) h GMM stimat is formd from: with first-ordr condition dwg ' ( b) = dwe ' ( xf' b p) = 0 whr min b g ( b) ' Wg ( b) g ( b) ' d ' = = E ( fx ') b
W h first stag has,th scond stag has W = S 1.h GMM stimat is 1 fisrstag : b1 = ( d ' d) d ' E ( p) = 1 1 sc ondstag : b2 = d ' S d d ' S E p h first-stag stimat is an OLS crosssctional rgrssion of avrag prics on th scond momnt of payoff with factors, and th scond-stag is a GLS cross-sctional rgrssion Easily shown in (13.1) I 1 ( ) ( )
h standard rror (from 11.2,11.8): 1 fisrtstag :cov b d ' d d ' Sd d ' d ( ) 1 = ( ) ( ) ( ) 1 1 1 2 = ( ) sc ondstag : cov b d ' S d 1 1 h covarianc matrix of th pricing rrors is (from 11.5,11.9,11.10) ( ) 1 = ( ) ( ) ( ) 1 1 2 = ( ) 1 1 fisrtstag : cov[ g b ] ( I d d ' d d ') S( I d d ' d d ') sc ondstag : cov[ g b ] S d d ' S d d '
h modl tst: ( ) ( ) ( ) g b g b g b momnts paramtrs 1 2 'cov( ) χ (# # ) which spcializs for th scond-stag stimat is : ( ) ( ) g b S g b momnts paramtrs 1 2 ' χ (# # ) 2 it turns out that th χ tst has th sam valu for first and scond stag
13.2 h Cas of Excss Rturns If all assts ar xcss rturns, th modl is lack of idntification. Writ th modl as m = a b' f normaliz a=1, thn ( ) = ( ) = ( ) ( ') w hav: g b E mr E R E R f b g ( b) d ' = = E ( fr ') b
th first-ordr condition is ( dw ' db E ( )) R = 0 th GMM stimats of b ar 1 fisrstag : b1 = ( d ' d) d ' E ( R ) 1 ( ) ( ) 1 1 sc ondstag : b2 = d ' S d d ' S E R h GMM stimat is a cross-sctional rgrssion of man xcss rturns on th scond momnts of rturns with factors
Man rturns on covarianc W can obtain a cross-sctional rgrssion of man rturns on covarianc Normaliz a = 1 + b' E( f) rathr than a=1 thn, th modl is : m = 1 b' f E( f) = 1 b' f ( ) with,th pricing rrors ar: E m = ( ) 1 ( ) ( ) ( ) ( = = ') g b E mr E R E R f b g ( b) d ' = = E fr ' = cov R, f b ' ( ) ( )
th first-ordr condition is: d ' W db E ( R ) = 0 h GMM stimats of b ar: 1 fisrstag : b1 = ( d ' d) d ' E ( R ) 1 ( ) ( ) 1 1 sc ondstag : b2 = d ' S d d ' S E R h GMM stimat is a cross-sctional rgrssion of xpctd rturns on th covarianc btwn rturns and factors
h standard rrors and varianc of th pricing rrors ar th sam as in (13.2) (13.3),with diffrnt d matrix. h p = E ( mx) for xcss rturn is quivalnt to ER ( ) = cov ( R, f' ) b thus th covarianc ntr in plac of btas E ( m) E ( R ) ( m R ) ( ) cov (, ) ( ) cov (, ') 0 = + cov, = E R + m R = E R + R f b
hr is on fly in th ointmnt, th man of th factor E( f) is stimatd. h distribution should rcogniz sampling varianc. It is bttr to us som othr non-sampldpndnt normalization for a.
13.3 Hors Racs How to tst whthr on st of factors drivs anothr. h gnral modl is : m = b f + b f ' ' 1 1 2 2 wo mthods to tst 1. Wald tst. 1 2 2 2 2 # b 2 = 0? b 'var( b ) b χ b 2
χ 2 2. Diffrnc tst rstrict condition: b 2 = 0 J rstrictd J unrstrictd ofrstrictions 2 ( ) ( ) χ (# ) his is vry much lik a liklihood ratio tst
13.4 sting for Charactristics In a good asst pricing modl, th alphas or pricing rrors should driv out th charactristic. Dnot th charactristic of portfolio y i by i i. Lt y t dnot th tim sris whos man E( y i t ) dtrmins th charactristic
writ th momnt condition for th assts as: g = E m () b x p γy ( ) t+ 1 t+ 1 t t h GMM stimat of is : γ = ( E ( y) ' WE ( y) ) E ( y) ' Wg o tst whthr γ = 0 is statistically significant. γ
13.5 sting for pricd Factors:Lambdas or b s? b asks whthr factor j hlps to pric j assts givn othr factors. m= b f = b f + b f + + b f + + b f ' 1 1 2 2 j j K K bj is th multipl rgrssion cofficint of m on f j givn all th othr factors. λ j asks whthr factor j is pricd, or whthr its factor-mimicking portfolio carris a positiv risk prmium
to xplain ths, us man-zro factors, xcss rturns, normaliz E( m ) = 1 s Sction 6.3 : m = 1 f ' b = E( mr ) = E R ( f b) 1 ( ) ( ' ) cov (, ') cov (, ') ( ') ( ') 0 1 ' E R = E R f b = R f b= R f E ff E ff b= λ = ( ') writ E ff b ( ' ) ( 1 ) ( ) λ = E f f b = E f m = E mf λ j So capturs whthr factor f is pricd j βλ
λ j is proportional to th singl rgrssion cofficint of m on ( m f ) j ( f ) j cov, λ = E( mf ) = cov ( m, f ) = var f var λ ask is th factor corrlatd j = 0 f j with th tru discount factor? Whn th factors ar orthogonal, f ( ) j j j j λ = 0 b = 0 λ = E ( ff ') b j j
If you want to know whthr factor i is pricd, look at λ ( ore ( mf i )) If you want to know whthr factor i hlps to pric assts, look at b i For xampl, suppos CAPM is tru, m m= a br x considr anothr R (positivly corrlatd m with R )as spurious factor
h answr is : m x m= a br bxr b x = 0 indicating that this factor dos not hlp to pric assts. howvr ( i ) E R = β λ + β λ im m ix x
Man-Varianc Frontir and Prformanc Evalution Stock rturns from many countris ar not prfctly corrlatd, so it looks lik on can rduc portfolio varianc by holding an intrnationally divrsifid portfolio. Is it ral or just sampling rrors? Whn valuating fund managr, w want to know whthr th managr is truly abl to form a portfolio that bats man-varianc fficint portfolios, or just du to luck. A factor modl is tru if only if a linar combination of th factors is man-varianc fficint p A CAPM, p = E ( mx), m = a br tst analogously tsts p whthr R is on th man-varianc frontir of th tst assts.
d o tst whthr th R assts span th d man-varianc frontir of R and R i
DSantis (1992) and Chn and Knz (1992,1993): m = a b ' R for intrsction, d will pric d f both R and only for on valu of a. R thus, w can tst for coincidnt frontirs d by tsting whthr m = a bd ' R prics d f both R and R for two prspcifid valus of a simulanously d
d1 d 2 If R and R on th frontir, thr must b discount factors: m = a b R 1 1 1 d1 m = a b R 2 2 2 d 2 st for spanning with a J tst, fixd E a b R R 1 1 d 1 d [( ) ] = 0 E a b R R = 1 1 d 1 i [( ) ] 0 E a b R R = 2 2 d 2 d [( ) ] 0 E a b R R = 2 2 d 2 i [( ) ] 0 1 2 a, a