f a c to r is s p e c if ie d in a f o u r fa c to r m o d e l th e m e a n e s tim a te o f a b n o rm a l p e rfo r m a n c e is lo w e r (0 .2 0 % p e r m o n th ) th o u g h r e m a in s s t a t is t ic a lly s ig n if ic a n t f r o m z e ro . E v id e n c e is a ls o p re s e n te d o n p e rs is te n c e in c o n v e r t ib le a rb itra g e h e d g e fu n d p e rfo r m a n c e . 5 .6 L im it a t io n s a n d a v e n u e s f o r fu r th e r re s e a rc h 5.6.1 S u it a b ilit y o f th e la g o f th e h e d g e fu n d as a n e x p la n a to r y v a r ia b le E v id e n c e p re s e n te d h e re su g g e sts th a t i f a n ill iq u id it y fa c to r is s p e c if ie d in a m u lt i- f a c t o r m o d e l c o n t a in in g la g g e d a n d c o n te m p o ra n e o u s r is k fa c to r s , th e n e s tim a te s o f a b n o rm a l p e rfo r m a n c e w i l l b e r e d u c e d to 2 .5 % p e r a n n u m . O m it t in g th e i ll iq u id it y fa c to r f r o m th e m o d e l le a d s to th e c o n c lu s io n th a t c o n v e r t ib le a rb itra g e h e d g e fu n d s g e n e ra te a b n o rm a l re tu rn s o f 4 .1 % p e r a n n u m . T h e s e re s u lts are s e n s itiv e to th e s p e c if ic a t io n o f an i l l iq u id it y f a c to r a n d m a y b e s e n s itiv e to th e ill iq u id it y fa c t o r s p e c ifie d .
G e t m a n s k y , L o a n d M a k a r o v (2 0 0 4 ) d is c u s s th e p o s s ib ilit y th a t th e s e r ia l c o r r e la tio n in h e d g e fu n d re tu rn s is p a r t ia lly c a u se d b y d e lib e ra te p e rfo r m a n c e s m o o t h in g in a d d it io n to th e ill iq u id it y in th e s e c u ritie s h e ld b y th e fu n d s . In t h is ca se , as in c lu d in g th e la g o f th e d e p e n d e n t v a r ia b le as an e x p la n a to r y v a r ia b le re d u c e s th e e s tim a te d a lp h a fo r c o n v e r t ib le a rb itra g e fu n d s , p e rfo r m a n c e m a y be u n d e rs ta te d . H o w e v e r , i f th e s e ria l c o r r e la t io n is c a u s e d b y a n o m itte d ill iq u id it y v a r ia b le , a n d th e e ffe c t is s y m m e tric , it m u s t b e a c c o u n te d f o r in th e r is k fa c to r m o d e l. I f a n i ll iq u id it y fa c to r is n o t s p e c ifie d th e n e s tim a te s o f p e rfo r m a n c e w i l l b e o v e r state d. T h e s p e c if ic a t io n o f th e la g o f th e h e d g e fu n d re tu rn as a d e p e n d e n t v a r ia b le s h o u ld le a d to a n e s tim a te o f p e rfo rm a n c e c lo s e r to th e tru e v a lu e th a n a r is k fa c to r m o d e l o m it t in g a n i ll iq u id it y v a ria b le .
I f a ll o f th e a u t o c o r r e la tio n is ca u se d b y s m o o th in g th e e s tim a te d m e a n h e d g e fu n d r e tu rn s h o u ld b e u n c h a n g e d , b u t th e sta n d a rd d e v ia t io n s h o u ld b e la rg e r, h a v in g a r e s u lt in g n e g a tiv e e ffe c t o n p e rfo r m a n c e e v a lu a tio n , t h ro u g h th e la rg e r v a ria n c e u s in g m e a n v a r ia n c e a n a ly s is . I n th e p re s e n t a n a ly s is , s t r ip p in g o u t p a s t re tu rn s a n d f a ilin g to in c lu d e th e p o r tio n o f c u rre n t re a l re tu rn s w h ic h w i l l n o t b e re p o rte d u n t il fu tu re m o n th s , w i l l lo w e r th e m e a n re tu rn le a d in g to a n u n d e rs ta te m e n t o f p e rfo r m a n c e . H o w e v e r , it w o u ld b e d if f ic u lt to r e b u f f a n a rg u m e n t th a t d e lib e ra te p e rfo r m a n c e s m o o th in g is n o t an a d d it io n a l r is k f o r a n in v e s to r, a n d in v e s to r s w o u ld f a v o u r fu n d s th a t d o n o t p e rfo rm a n c e s m o o th o v e r fu n d s th a t d o , in th e sa m e w a y th a t in v e s t o r s p re fe r f u n d s th a t h o ld s e c u ritie s w it h g re a te r liq u id it y th a n fu n d s w h o h o ld i ll iq u id s e c u ritie s
ceteris paribus.
In r e la t iv e p e rfo r m a n c e e v a lu a tio n , th e in c lu s io n o f a la g g e d c o n v e r t ib le a rb itra g e re tu rn as a r is k fa c to r, e v e n i f it s lig h t ly re d u c e s e s tim a te s o f o v e r a ll p e rfo r m a n c e is s u p e r io r to a fa c to r m o d e l w h ic h d o e s n o t d iffe r e n tia t e b e tw e e n f u n d s w h o e n g a g e in d e lib e ra te p e r fo r m a n c e s m o o th in g a n d th o s e th a t d o n o t. N o n e t h e le s s , w h e n d a ta b e c o m e s a v a ila b le o n t u r n o v e r in th e c o n v e r t ib le b o n d m a rk e t, a n i l l iq u id it y r is k fa c to r d e riv e d f r o m t h is d a ta is lik e l y to b e a m o r e d ir e c t m o d e l o f ill iq u id it y r is k w h ile a v o id in g th e p o t e n t ia l b ia s e s f r o m in c lu d in g a la g o f a h e d g e fu n d b e n c h m a r k in d e x as a r is k fa c to r.Chapter 6: A review of non-linear time series models
6
.1 I n t r o d u c t io nT h is c h a p te r p r o v id e s a r e v ie w o f n o n - lin e a r m o d e ls f o c u s in g in d e ta il o n th e s m o o th t r a n s it io n a u to re g re s s iv e ( S T A R ) a n d s m o o th tr a n s it io n r e g r e s s iv e ( S T R ) f a m ily o f m o d e ls f ir s t p r o p o s e d b y C h a n a n d T o n g ( 1 9 8 6 ) a n d e x te n d e d b y T e r a s v ir ta a n d A n d e r s o n (1 9 9 2 ) f o r m o d e llin g n o n - lin e a r it ie s in th e b u s in e s s c y c le . S e v e r a l stu d ie s o f h e d g e fu n d s h a v e n o te d th e n o n - lin e a r it y in h e re n t in th e re tu rn s o f d y n a m ic t r a d in g s tra te g ie s
.53
G iv e n th e se c h a r a c te r is tic s , a lin e a r m o d e l m a y b e f u n c t io n a lly m is s p e c if ie d w h e n e x a m in in g th e d a ta g e n e ra tin g p ro c e s s o f a d y n a m ic t r a d in g stra te g y. M it c h e ll a n d P u lv in o ( 2 0 0 1 ) a n d A g a r w a l a n d N a i k ( 2 0 0 4 ) n o te th a t th e p a y o f f to d y n a m ic t r a d in g s tra te g ie s sh a re c h a r a c te r is tic s w it h s h o rt p o s it io n s in e q u it y p u t o p tio n s im p ly in g th a t th e re a re t w o re g im e s ; o n e r e g im e w it h lit t le e q u it y e x p o s u re a n d o n e w it h a h ig h c o r r e la t io n w it h e q u itie s . I n C h a p te r 3 o f t h is s tu d y , e v id e n c e w a s p re s e n te d w h ic h in d ic a te d th a t th e r e la t io n s h ip b e tw e e n c o n v e r t ib le b o n d a rb itra g e a n d e q u it ie s has th re e r e g im e s . W h e n e q u it y re tu rn s w e re e x t r e m e ly n e g a tiv e th e re is a s tro n g p o s it iv e c o r r e la t io n w it h c o n v e r t ib le b o n d a r b itra g e re tu rn s ; w h e n e q u it y re tu rn s w e re w it h in o n e s ta n d a rd d e v ia tio n o f th e m e a n th e re is a w e a k p o s it iv e c o r r e la tio n w it h e q u itie s ; and w h e n e q u it y re tu rn s w e re e x t r e m e ly p o s it iv e th e re is a n e g a tiv e r e la t io n s h ip w it h e q u itie s .53 A g a r w a l and N a i k (2004), F u n g a n d Hsieh (2001), F u n g and Hsieh (2002) an d Mitchell and Pulvino (2001) a m o n g s t others d o c u m e n t this feature of h e dg e fund returns.
A n in n o v a t io n o f th e S T A R m o d e ls d is c u s s e d la te r in t h is c h a p te r is th a t th e y a llo w f o r a s m o o th t r a n s it io n f r o m o n e r e g im e to a n o th e r, ra th e r th a n a ju m p , a c h a r a c te r is tic b e tte r s u it e d to e x a m in in g d y n a m ic t r a d in g stra te g ie s w h e r e p o r t f o lio s a re re b a la n c e d , b y v a r io u s m a r k e t p a r t ic ip a n t s a t d if f e r e n t in te r v a ls a n d b y v a r y in g d e g re e s , in r e a c tio n to e v o lv in g m a r k e t c o n d itio n s .