On the Small Sample Distribution

This paper gives an account of the R/S statistic and its known properties. I t assesses the adequacy of the asymptotic distribution of the statistic in the case of samples of small and moderate size, and suggests an improved approximation based on the beta distribution. The results indicate that the proposed beta approximation is superior to the asymptotic distribution for practical purposes. Hence a new table of critical values is presented as an alternative to that of Lo (1991). I I N T R O D U C T I O N T h i s p a p e r i s conce rned w i t h t h e re-scaled a d j u s t e d r a n g e (R/S) s t a t i s t i c as a m e a n s o f i n v e s t i g a t i n g l o n g t e r m s t a t i s t i c a l dependence or pers i s ­ t ence i n t i m e ser ies d a t a . I n t r o d u c e d b y H u r s t ( 1 9 5 1 ) , R/S a n a l y s i s w a s deve loped b y M a n d e l b r o t a n d W a l l i s (1969a , 1969b, 1969c) a n d f i r s t u s e d b y h y d r o l o g i s t s t o s t u d y pers i s tence i n geophys ica l t i m e series. A l t h o u g h i t w a s also u s e d b y M a n d e l b r o t (1971) t o e x a m i n e pers i s tence i n asset r e t u r n s , t h e R/S s t a t i s t i c h a s o n l y r e c e n t l y been u t i l i s e d b y economis t s . H o w e v e r , t h e r e a re a l r e a d y a few n o t a b l e a p p l i c a t i o n s t o f i n a n c i a l t i m e ser ies (e.g., Pe te r s (1989 , 1 9 9 1 , 1992, 1994); L o (1991) ; M o o d y a n d W u (1995)) a n d a t l ea s t one t o p o l i t i c a l o p i n i o n p o l l series (Bye r s et al. ( 1 9 9 6 ) ) . 1 Paper presented at the Eleventh Annual Conference of the Ir ish Economic Association. *An earlier version of the paper appeared as Technical Paper No. 6 (June 1997) in the Trinity Economic Papers series. The authors are grateful to an anonymous referee for helpful comments on the present version. 1. The financial time series used tend to be very long. For example, although Peters used some moderate sample sizes of the order of 300 or 400, Lo used daily price series with between 1,500 and 6,000 observations, while Moody and W u used tick-by-tick data and had samples of hundreds of thousands of observations. T h e case fo r p e r s i s t e n c e t o be p l a c e d a m o n g t h e p r a c t i c a l concerns o f e conomis t s h a s been s t a t e d s t r o n g l y b y H a u b r i c h a n d L o (1989) , H a u b r i c h (1990) a n d L o (1991) . I n d e e d , as L o (1991) p o i n t s ou t , l ongrange dependence w a s a n i m p l i c i t h y p o t h e s i s i n s eve ra l e a r l y t h e o r i e s o f t r a d e a n d bus ines s cycles , a n d i t i s c o n s i s t e n t w i t h t h e G r a n g e r (1966) d i scovery o f t h e t y p i c a l s p e c t r a l shape o f a n e c o n o m i c v a r i a b l e . F u r t h e r m o r e , i t h a s i m p o r t a n t i m p l i c a t i o n s fo r seve ra l t heo r i e s i n m o d e r n f i n a n c i a l economics, such as those conce rned w i t h o p t i m a l c o n s u m p t i o n a n d p o r t f o l i o decis ions a n d t h e p r i c i n g o f d e r i v a t i v e s e c u r i t i e s , as w e l l as fo r s t a n d a r d tes t s o f t h e c a p i t a l asset p r i c i n g m o d e l a n d t h e a r b i t r a g e p r i c i n g t h e o r y . R/S a n a l y s i s offers a r e l a t i v e l y s t r a i g h t f o r w a r d means o f i n v e s t i g a t i n g t h e i s sue o f l o n g t e r m dependency . I t i s ba sed o n s i m p l e g r a p h i c a l p l o t s a n d H u r s t e x p o n e n t s ; i t i s c l a i m e d t h a t i t c an de tec t l o n g r a n g e dependence i n h i g h l y n o n n o r m a l t i m e series w i t h l a r g e skewness a n d k u r t o s i s , a n d n o n p e r i o d i c cycles; a n d t h e R/S s t a t i s t i c i t uses i s c h a r a c t e r i s e d b y a l m o s t sure convergence fo r s tochas t i c processes w i t h i n f i n i t e v a r i a n c e . 2 H o w e v e r , i t i s k n o w n t o be a d v e r s e l y af fec ted w h e n t h e r e i s s h o r t t e r m dependence i n t h e ser ies (see M c L e o d a n d H i p e l (1978) , a n d H i p e l a n d M c L e o d (1978) ) , a n d i t w a s t h i s de f i c i ency t h a t l e d L o (1991) , a n d t h e n M o o d y a n d W u (1995) , t o propose m o d i f i c a t i o n s o f t h e R/S s t a t i s t i c . U n f o r t u n a t e l y , c e r t a i n s h o r t c o m i n g s r e m a i n i n t h e c a l c u l a t i o n a n d p lo t s o f t h e R/S s t a t i s t i c u s e d i n s t a n d a r d R/S a n a l y s i s . M o r e o v e r , a l t h o u g h t h e a s y m p t o t i c d i s t r i b u t i o n o f t h e s t a t i s t i c i s k n o w n , t h e r e i s v e r y l i t t l e i n f o r ­ m a t i o n a v a i l a b l e o n i t s d i s t r i b u t i o n fo r s m a l l s amples a n d t h e a d e q u a c y o f t h e a s y m p t o t i c d i s t r i b u t i o n as a n a p p r o x i m a t i o n i n t h e s m a l l s a m p l e s i t u a t i o n . T h e r e f o r e w e i n v e s t i g a t e t h e s m a l l s a m p l e b e h a v i o u r o f t h e R/S s t a t i s t i c , h a v i n g p a r t i c u l a r r e g a r d to t h e q u a l i t y o f t h e a s y m p t o t i c a p p r o x i ­ m a t i o n t o i t s f i n i t e s a m p l e d i s t r i b u t i o n a n d o f a n a l t e r n a t i v e a p p r o x i m a t i o n b a s e d o n a b e t a d i s t r i b u t i o n . W e conc lude t h a t o u r b e t a p rocedure g ives a b e t t e r a p p r o x i m a t i o n t o t h e f i n i t e s a m p l e d i s t r i b u t i o n o f t h e R/S s t a t i s t i c t h a n t h e k n o w n a s y m p t o t i c d i s t r i b u t i o n does. Hence w e p r o v i d e a n e w t a b l e o f c r i t i c a l va lue s as a n a l t e r n a t i v e to t h a t g i v e n b y L o (1991). T h e p a p e r i s o r g a n i s e d as f o l l o w s . Sec t ion I I describes t h e s t a n d a r d a n d m o d i f i e d R/S s t a t i s t i c s a n d o u t l i n e s t h e i r m a i n p roper t i e s . Sec t ion I I I focuses o n issues r e l a t i n g to t h e s m a l l s ample d i s t r i b u t i o n o f t h e R/S s t a t i s t i c a n d i t s a p p r o x i m a t i o n b y m e a n s o f a b e t a d i s t r i b u t i o n . Sec t ion I V r e p o r t s o n o u r e x p e r i m e n t a l w o r k a n d f i n d i n g s , a n d Sec t ion V presen ts o u r t a b l e o f c r i t i c a l 2. An introduction to long-range dependence is given in the new textbook by Campbell et al. (1997, C h . 2). In particular, their footnote 21, pages 62-63, provides a straightforward account of the behaviour of the R/S statistic and a simple example that shows why it takes on large values in the presence of persistence. v a l u e s fo r t h e R/S s t a t i s t i c . T h e p a p e r ends w i t h a s h o r t s u m m a r y a n d conc lus ion i n Sec t ion V I . I I T H E R/S S T A T I S T I C H u r s t w a s a c i v i l e n g i n e e r i n t e r e s t e d i n r e s e r v o i r s t o r a g e a n d , i n p a r t i c u l a r , t h e i d e a l r e s e r v o i r size. T h u s he w a s conce rned w i t h i n f l o w s a n d o u t f l o w s o f w a t e r . F o r economic app l i ca t i ons , t h e i dea o f f l ows has i m m e d i a t e r e l e v a n c e t o m a n y v a r i a b l e s a n d p r o b l e m s , s u c h as those i n v o l v i n g s t o c k r e t u r n s or f l o w s i n t o a n d o u t o f u n e m p l o y m e n t p e r p e r i o d , a n d i t m a y be u s e f u l t o b e a r s u c h e x a m p l e s i n m i n d . H o w e v e r , w e p roceed a l o n g m o r e g e n e r a l l i n e s , d e n o t i n g a ser ies o f one p e r i o d changes i n a n y v a r i a b l e b y { x t } t _ 1 , a n d i t s s ample m e a n a n d s t a n d a r d d e v i a t i o n over t h e p e r i o d t = l , . . . , T b y m a n d s, r e spec t ive ly . T h e p a r t i a l or c u m u l a t i v e s u m s o f d e v i a t i o n s o f t h e Xt f r o m t h e i r m e a n are de f ined as p ( k ) = I ( x t m ) , k = l , . . . , T . t=i C l e a r l y , w h e n x k ( k > 1) exceeds m , t h e k t h p a r t i a l s u m w i l l i nc rease , a n d w h e n x t i s less t h a n m , t h e k t h p a r t i a l s u m w i l l decrease. T h e r a n g e o f t h e p a r t i a l sums is r = m a x [ p ( k ) ] m i n [ p ( k ) ] , k k a n d t h e re -sca led ad jus t ed r a n g e , i.e. t h e R/S s t a t i s t i c , f o r t h e t i m e p e r i o d t = l , . . . , T i s j u s t t h e r a t i o o f r to s. Speci f ica l ly , t h e R/S s t a t i s t i c i s c a l c u l a t e d as m a x [ p ( k ) ] m i n [ p ( k ) ] R / S = = k . k . T h e a d j u s t m e n t o f t h e r a n g e re la tes to s u b t r a c t i o n o f t h e m e a n , a n d t h e r e s c a l i n g t o d i v i s i o n b y t h e m a x i m u m l i k e l i h o o d e s t i m a t e o f t h e s t a n d a r d d e v i a t i o n . T h e R/S s t a t i s t i c i s o f a r a t h e r complex f o r m w h i c h m a k e s t h e a n a l y s i s o f i t s s m a l l s ample b e h a v i o u r d i f f i c u l t , even i n t h e n u l l s i t u a t i o n i n w h i c h t h e x t a re i n d e p e n d e n t l y a n d i d e n t i c a l l y d i s t r i b u t e d ( i . i . d . ) . I n d e e d , i t s s a m p l i n g d i s t r i b u t i o n fo r f i n i t e samples i n t h i s case r e m a i n s u n k n o w n except fo r t h e e x p e c t a t i o n . T h e a s y m p t o t i c p r o p e r t i e s o f R/S a re eas ier t o h a n d l e , t h o u g h t h e y a re b y no means u n c o m p l i c a t e d . U s i n g s i m p l e e x p e r i m e n t a t i o n , H u r s t (1951) d e r i v e d a n a p p r o x i m a t i o n for t h e e x p e c t e d v a l u e o f R/S f o r l a r g e s a m p l e s i n t h e i . i . d . case, n a m e l y E [ R / S ] = ^ | T . T h i s r e s u l t w a s c e n t r a l t o H u r s t ' s e a r l y a p p l i c a t i o n s o f R/S a n a l y s i s . R o u n d a b o u t t h e same t i m e , F e l l e r (1951) d e r i v e d t h e a s y m p t o t i c d i s t r i b u t i o n o f R/S u n d e r t h e i . i . d . n u l l hypo thes i s , u s i n g a r e s u l t due t o Doob (1949) , a n d t h e r e b y c o n f i r m e d t h e accuracy o f H u r s t ' s a p p r o x i m a t i o n . I m p l i c i t i n t h e w o r k o f F e l l e r i s t h a t ( T ~ * ) R / S > V , w h e r e t h e d i s t r i b u t i o n f u n c t i o n o f V , as g i v e n e x p l i c i t l y b y L o (1991) , i s F v ( v ) = 1 + 2 I ( 1 4 k 2 v 2 ) e " 2 ( k v > k = l T h e p r o b a b i l i t y d e n s i t y , a n d hence t h e m o m e n t s o f V , m a y t h e r e f o r e be d e r i v e d ; a n d ( n o n t r i v i a l ) c a l c u l a t i o n shows t h a t E [ V ] = ^ a n d E J V 2 ] = • — . I t f o l l o w s t h a t for l a r g e T , E [ R / S ] = J | T = L 2 5 3 3 V T a n d V [ R / S ] = n 6 T = 0 . 0 7 4 I T . These r e s u l t s a re w e l l k n o w n . H o w e v e r , to o u r k n o w l e d g e , h i g h e r m o m e n t s h a v e n o t b e e n r e p o r t e d i n t h e l i t e r a t u r e . O u r c a l c u l a t i o n s show t h a t t h e s t a n d a r d skewness a n d k u r t o s i s coefficients are, r e spec t i ve ly , 3 0.6132, Pi [ R / S ] = w h e r e q(.) denotes t h e R i e m a n n Z e t a f u n c t i o n , a n d P 2 [ R / S ] = 7t 6 V V 7t*_ , ' 30 2 — 3jcq(3) =3.4178. 3. The standard skewness coefficient is the ratio of the third moment about the :mean to the cube of the standard deviation; the standard kurtosis coefficient is the ratio of the fourth moment about the mean to the square of the variance. The symbols used here, i.e., Vp x and p2> a r e the conventional ones for the skewness and kurtosis coefficients, respectively! These f ea tu res o f s l i g h t p o s i t i v e skewness a n d l e p t o k u r t o s i s a re e v i d e n t i n p lo t s o f t h e d e n s i t y f u n c t i o n o f V ; see F i g u r e 1 w h i c h also i n c l u d e s a n o r m a l d i s t r i b u t i o n (da shed ) w i t h t h e same m e a n a n d v a r i a n c e as V (a s i m i l a r d i a g r a m , b u t also c o n t a i n i n g t h e d i s t r i b u t i o n f u n c t i o n , i s g i v e n b y L o ( 1 9 9 1 , p . 1292)).