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A preliminary examination of the no arbitrage property in the Canadian security market Hung, Reynold 1984

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A PRELIMINARY EXAMINATION ARBITRAGE  P R O P E R T Y I N THE  OF THE  CANADIAN  NO  SECURITY  MARKET  By  R E Y N O L D HUNG B.Comm.(Hon.),  Queen's U n i v e r s i t y ,  A T H E S I S SUBMITTED THE  1982  IN PARTIAL FULFILLMENT  R E Q U I R E M E N T S FOR  THE  DEGREE  OF  OF  MASTER OF SCIENCE IN BUSINESS ADMINISTRATION  in THE Faculty  FACULTY  o f Commerce  We  accept to  THE  this  OF GRADUATE  STUDIES  and Business  thesis  the required  as  © Reynold  conforming  standard  U N I V E R S I T Y OF B R I T I S H March  Administration  COLUMBIA  1984 Hung,  1984  In presenting/ requirements  this thesis f o r an  of  British  it  freely available  Library  shall  for reference  and  study.  I  for extensive  p u r p o s e s may  department or  by  h i s or  her  copying or  f i n a n c i a l gain  be  shall  copying of  g r a n t e d by  the  not  be  of  further this  this  a l l o w e d w i t h o u t my  Date  March  27,  1984  Columbia  my  thesis  COMMERCE AND BUSINESS ADMINISTRATION  The U n i v e r s i t y o f B r i t i s h 1956 Main Mall Vancouver, Canada V6T 1Y3  thesis  It is  permission.  Department o f  make  head o f  representatives. publication  the  University  the  for scholarly  for  the  I agree that  permission  understood that  f u l f i l m e n t of  advanced degree a t  Columbia,  agree t h a t  in partial  written  ABSTRACT This no  arbitrage  market. the  paper examines t h e e m p i r i c a l property  using  In addition,  market The  "beta"  kernels  i s introduced  and t e s t e d .  that  there  s e t o f complex  empirical  security prices.  a difference and  Canada.  strongly  i n market c o n d i t i o n  t h e no a r b i t r a g e  A s e t o f market  property  property  paper.  there i s  between t h e U.S. and Canada,  s h o u l d n o t be r e j e c t e d i n  Furthermore, the usefulness  of this  d a t a on  i s based.  suggests that  of the a l t e r n a t i v e  s e c u r i t y r i s k measure c a n n o t be e s t a b l i s h e d results  with the  f r o m t h e C a n a d i a n common s t o c k  evidence  prefer-  exists a set of positive  w h i c h t h e t e s t i n g o f t h e no a r b i t r a g e The  from t h e s t a t e  s e c u r i t y p r i c e s which i s c o n s i s t e n t  i s estimated  security  an a l t e r n a t i v e s e c u r i t y r i s k measure t o  ence model which s t a t e s  observed  of the  t h e C a n a d i a n common s t o c k  model f o r t e s t i n g i s d e r i v e d  primitive  implications  b a s e d on t h e  i i i  Table o f  Contents Page  Abstract List  i  o f Tables  i v  Acknowledgement 1.  Introduction  2.  The  3.  Methodology  State  v i i 1  Preference  Model  3 6  a)  The  Analysis  b)  The  Estimation  c)  The T e s t i n g i n Canada  o f t h e No  Alternative  Estimates  d)  i  6 of the Market Kernel Arbitrage  8  Property 12  of Security  Risk  Premium  16  4.  Data  20  5.  Results  21  a)  The  b)  The A p p l i c a b i l i t y as  No  Arbitrage  a Risk  Measure  Property of Using  21 cov(X,Z) •  26  6.  Conclusion  30  7.  Bibliography  58  iv  List  o f Tables  Table 1 2  3 4  5 6 7 8 9 10 11  12  13  14  Page F - t e s t and C h i - s q u a r e t e s t s by u s i n g f o r J a n u a r y 1956 - December 1960  32  F - t e s t and Chi-square t e s t s f o r J a n u a r y 1956 - December  33  by u s i n g R 1960  F - t e s t and C h i - s q u a r e t e s t s by u s i n g f o r J a n u a r y 1961 - June 1968  R^ .  34  F - t e s t and C h i - s q u a r e t e s t s by u s i n g R f o r J a n u a r y 1961 - J u n e 1968 ™  35  30 s e c u r i t i e s s u b s a m p l e s s e l e c t e d f r o m t h e B r e n n a n & Thompson p a p e r f o r c o m p a r i s o n p u r p o s e A l t e r n a t i v e estimates o f market J a n u a r y 1956 - December 1960  kernel,  A l t e r n a t i v e estimates o f market J a n u a r y 1961 - June 1968  kernel,  .  36 37 38  Correlation matrix of kernels, D e c e m b e r 196 0  January  Correlation June 1968  matrix of kernels,  January  Correlation J u n e 196 8  matrix of kernels,  1956 39 1961 40  January  1956 41  C o v a r i a n c e s b e t w e e n t h e 118 s e c u r i t i e s r i s k p r e m i u m a n d t h e CAPM k e r n e l Zm f o r t h e p e r i o d J a n u a r y 1956 - December 196 0  42  C o v a r i a n c e s b e t w e e n t h e 118 s e c u r i t i e s r i s k p r e m i u m a n d t h e CAPM k e r n e l Zm f o r t h e p e r i o d J a n u a r y 1956 - J u n e 1958  43  C o v a r i a n c e s b e t w e e n t h e 118 s e c u r i t i e s r i s k p r e m i u m a n d t h e CAPM k e r n e l Zm f o r t h e p e r i o d J u l y 1958 - December 1960  44  C o v a r i a n c e s b e t w e e n t h e 118 s e c u r i t i e s r i s k p r e m i u m a n d t h e CAPM k e r n e l Zm f o r t h e p e r i o d J a n u a r y 1 9 6 1 - J u n e 196 8  .  45  V  List  of Tables  (continued)  Table 15  16  Page C o v a r i a n c e s b e t w e e n t h e 118 s e c u r i t i e s r i s k p r e m i u m a n d t h e CAPM k e r n e l Zm f o r t h e p e r i o d J a n u a r y 1961 - September 1964  46  C o v a r i a n c e between t h e 118 s e c u r i t i e s r i s k p r e m i u m a n d t h e CAPM k e r n e l Zm f o r t h e p e r i o d O c t o b e r 1964 - J u n e 1968  47  2 17  18  19  20  21  22  23  24  25  Regression coefficient, R and t - s t a t i s t i c s of t h e r e g r e s s i o n between c o v a r i a n c e s o f t h e 1 1 8 s e c u r i t i e s a n d Zm o v e r d i f f e r e n t p e r i o d s of time 2 Regression coefficient, R and t - s t a t i s t i c s of t h e r e g r e s s i o n between c o v a r i a n c e s o f t h e 1 1 8 s e c u r i t i e s a n d Zm o v e r d i f f e r e n t p e r i o d s of time  48  49  C o v a r i a n c e s b e t w e e n t h e 118 s e c u r i t i e s r i s k premium and t h e u n c o n s t r a i n e d k e r n e l Zu f o r t h e p e r i o d J a n u a r y 1956 - December 1960  50  C o v a r i a n c e s b e t w e e n t h e 118 s e c u r i t i e s r i s k premium and t h e u n c o n s t r a i n e d k e r n e l Zu f o r t h e p e r i o d J a n u a r y 1956 - J u n e 195 8  51  C o v a r i a n c e s b e t w e e n t h e 118 s e c u r i t i e s r i s k premium a n d t h e u n c o n s t r a i n e d k e r n e l Zu f o r t h e p e r i o d J u l y 1958 - December 1960  52  C o v a r i a n c e s b e t w e e n t h e 118 s e c u r i t i e s r i s k p r e m i u m a n d t h e u n c o n s t r a i n e d k e r n e l Zu f o r the p e r i o d J a n u a r y 1961 - June 1968  53  C o v a r i a n c e s b e t w e e n t h e 118 s e c u r i t i e s r i s k premium a n d t h e u n c o n s t r a i n e d k e r n e l Zu f o r t h e p e r i o d J a n u a r y 1961 - S e p t e m b e r 1964 . . . .  54  C o v a r i a n c e s b e t w e e n t h e 118 s e c u r i t i e s r i s k p r e m i u m a n d t h e u n c o n s t r a i n e d k e r n e l Zu f o r t h e p e r i o d O c t o b e r 1964 - J u n e 1968  55  2 Regression c o e f f i c i e n t , R and t - s t a t i s t i c s of t h e r e g r e s s i o n between c o v a r i a n c e s o f t h e 118 s e c u r i t i e s a n d Zu o v e r d i f f e r e n t p e r i o d s of time  56  vi  List  of Tables  (continued)  Table  Page 2  26  Regression coefficient, R and t - s t a t i s t i c s of t h e r e g r e s s i o n between c o v a r i a n c e s o f the 118 s e c u r i t i e s a n d Zu o v e r d i f f e r e n t p e r i o d s of time  57  vii  Acknowledgement  I w i s h t o t h a n k my Philippe my  Jorion  committee  suggestions.  M.Sc.  Committee  and Robert Jones  chairman,  - Rex  Thompson,  - e s p e c i a l l y Rex  f o r t h e i r h e l p f u l comments  Naturally,  a l lerrors  Thompson, and  a r e my r e s p o n s i b i l i t y .  1  1.  INTRODUCTION During  the  the past  determination  of risky  have been proposed, capital model  asset  asset  a number o f m o d e l s f o r  prices i n security  markets  t h e most p r o m i n e n t o f w h i c h a r e t h e  pricing  (APM),[7].  assumption  twenty years,  model  (CAPM)  [8] a n d t h e a r b i t r a g e  Amongst a l l t h e s e  that the security  models, there  market  pricing  i s t h e common  i s i n equilibrium.  When e q u i l i b r i u m i s c o m b i n e d w i t h t h e a s s u m p t i o n o f n o n satiation is  greater than  markets  a  this  return with  zero  probability  which w i l l  and a p o s i t i v e  I t i s therefore a very by a l l f i n a n c e model  i sultimately  a l l o f the present  purpose o f t h i s  proven  have return  basic  builders.  t o be f a l s e ,  then  asset v a l u a t i o n models w i l l  Brennan  testing  paper  i s t o perform  have t o  a preliminary  o f t h e no a r b i t r a g e p r o p e r t y w i t h C a n a d i a n  market data.  ing  of security  revised.  testing  an  function  states that i t i snot possible  zero n e t investment  t h a t has been used property  The  in  with  non-zero p r o b a b i l i t y .  almost be  no a r b i t r a g e p r o p e r t y  negative  property If  of the u t i l i t y  z e r o ) , t h e no a r b i t r a g e p r o p e r t y  form a p o r t f o l i o  with  derivative  i s obtained.  The to  (i.e.the first  This  test  a n d Thompson  procedure  alternative  i s based  o n U.S. k e r n e l s  risk  i t s explanatory  estimated  [1983] a n d i s an e x t e n s i o n  t o t h e Canadian c a p i t a l measure  security  market.  of their Moreover,  i s introduced and a t e s t  and p r e d i c t i v e  power over  time  concern-  i s per-  2  formed. In  s e c t i o n two,  empirical part the  testing  the theoretical  model  i s based i s discussed.  on w h i c h In section  o f t h e B r e n n a n and Thompson m e t h o d o l o g y extension  used  i n performing  discussed sions  i s described. this  the results  and recommendations  Lastly,  results  with  the  data  are  i n section s i x , conclu-  obtained  f o r future  and  three,  together  four describes  empirical test,  i n section five.  concerning  Section  this  i n previous  research  are  sections  presented.  3  2.  THE STATE PREFERENCE MODEL Securities  have a t i m e d i m e n s i o n .  vestment decisions  of individuals are decisions  the  timing  o f consumption over  The  passage o f time  and  hence t h e f u t u r e  uncertainty payoffs of  ferent  value  a t some f u t u r e  uncertainty  date,  i s a matrix  and an i n d i v i d u a l ' s p o r t f o l i o  of probable payoffs  preference,  uncertainty  t h e s e c u r i t y i s a s s u m e d t o be known w i t h  to  the investor,  the state  i n nature that  assigns  plete  uncertain  a cashflow  certainty.  s e c u r i t i e s as there  exclusive  there  payoff Thus,  payoffs,  model l i s t s a l l  state.  that  be  o f the world.  preference  f o revery  market which requires  independent  states  are mutually  will  the state, the  a security i s a set of certain  More f o r m a l l y ,  takes  what t h e s t a t e o f t h e world  of  and  on t h e d i f -  the p o r t f o l i o . ([5], Chapter 5 ) .  However, g i v e n  with  This  of probable  some f u t u r e  states  interval.  about t h e f u t u r e ,  at  each associated  date.  time  as a v e c t o r  the language o f state  form o f not knowing  regarding  o f a s e c u r i t y investment.  s e c u r i t i e s comprising  In the  involves  some f u t u r e  c a n be r e p r e s e n t e d  investments  The s e c u r i t i e s i n -  and  exhaustive,  I t requires a r e a s many  are states  of  a comlinearly  nature.  Given a complete market, an i n d i v i d u a l can t h e o r e t i c a l l y reduce  the uncertainty  to  zero.  A l l that  of  the world  will  about t h e value  remains  of h i s future  i s uncertainty  a c t u a l l y occur.  about which  wealth state  That i s , by d i v i d i n g h i s  4  wealth if  among a l l t h e a v a i l a b l e  he c h o o s e s ,  same p a y o f f individual The  tive of  $1  analysis  This  even though t h e p a y o f f s o f over  states.  of the security  by t h e concept  markets w i t h i t s model i s  of a primitive  i s d e f i n e d as a s e c u r i t y  i f a given  occurs.  state,  securities vary  security  state  occurs  and nothing  allows the logical  every  security  primitive as  follows  securities.  be c o n s i d e r e d Mathematically,  ([5] C h a p t e r  P  ±  =  I  P  A  primi-  return  i f any other  decomposition  into portfolios of primitive may  security.  which pays a  securities real  could,  c o n s t r u c t a p o r t f o l i o w h i c h would have t h e  i n every  facilitated  securities, the investor  state  of market  securities.  a combination i t c a n be  Thus, of various  represented  5).  (w) Y  ±  (1)  (w) i s I  where (w)  p  (w)  =  the return  =  the i n i t i a l  =  a s e t o f non-negative state prices  A direct observation state  w , the payoff  P^ = £p(w)  Y^  from  — - —  IP(W)  =  price  (1)  i s always  denotes t h e payoff  l/£p(w) i s a l w a y s o n e p l u s  on s e c u r i t y  i  i n state  of the security  i s that  primitive  i f i n every  t h e same a m o u n t ,  of a riskless security  the r i s k l e s s rate  1 + r f  w  of  then and  return:  5  where  r f i s the riskless Note a l s o  owing  £p(w)  t o the time value It  of great  to  empirical  derived prices  v a l u e when testing.  a procedure  payoffs.  property  i s equivalent  smaller  unity  money.  analyzing  that  this  problems,  state  c a n be c o m b i n e d this  I t has been  t h e no  to the existence  with  In light  the observed of this,  they  complex  arbitrage  i s the subject  o f the next  which  security  f o r the existence  such a s e t o f p r i m i t i v e s e c u r i t y p r i c e s . methodology  shown  of a not necessarily  s e t o f complex tested  have  primitive  to duplicate  result,  subject  however,  the state form.  preference  but i s not  B r e n n a n and Thompson,  By u s i n g  than  s e t of non-negative p r i m i t i v e security prices  consistent  prices.  i s always  i n t o an e m p i r i c a l l y t e s t a b l e  security  is  return.  f o r transforming  primitive payoffs  unique  of  of  has been w e l l - r e c o g n i s e d  is  that  that  rate  Part  section.  of  their  of  6  3.  METHODOLOGY  a)  The A n a l y s i s In  implies state both  section  2, i t i s shown  the existence  prices sides  o f (1) b y p(w)  t h e no a r b i t r a g e  property  o f a set o f non-negative p r i m i t i v e  p(w) such t h a t  I  that  equation  (1) h o l d s .  Dividing  yields:  R (w) = 1  iG I  ±  (2)  where R.(w)  = Y.(w)/P.  o r one plus on  Z (w) = p ( w ) / r r ( w )  Define that  state  w  occurs.  security where  Then e q u a t i o n  > 0. To  test  assumptions of  (Since  about  the r i s k l e s s  stationarity  part  o f the returns  We k n o w t h a t  I p(w) =  interest rate  B r e n n a n a n d Thompson c o n s i d e r of  t o make  Z(w)  some  and hence cannot be  because E [ Z (w) ] =  and  operation, and  a r e b o t h > 0.)  the s t a t i o n a r i t y Z(w).  (3)  ±  (3) e m p i r i c a l l y i t i s n e c e s s a r y  t h e random v a r i a b l e s  stationary  R (w)] = 1  the expectation  p (w) a n d TT (w)  i s the p r o b a b i l i t y  (2) becomes:  ±  Z(«)  of return  i.  rr (w)  I TT(W) Z (w) R ( w ) = E [ Z ( w ) where E [ ] denotes  the rate  o f (3).  a n d u s e them  One o f t h e s e  1  /1 + r f varies  (4) over  time.  Therefore,  two a l t e r n a t i v e s p e c i f i c a t i o n s t o derive  the empirical  specifications will  be  counter-  described  7  next. From tunities the  ( 3 ) we k n o w t h a t i n period  existence  t  (since  (3) i s d e r i v e d  o f a random v a r i a b l e  E[Z (w)  R  t  E[Z (w)]  ± t  from  (1)) i m p l i e s  Z (w) > 0 such  that  t  (w) ] = 1  oppor-  16 I  (5)  = V l + r f t  t  Define  t h e absence o f a r b i t r a g e  (6)  Z(w) b y  Z(w) = (1 + r f t ) Z ( w )  (7)  t  T h e n s u b s t i t u t i n g f o r Z ^ ( w ) i n (5) y i e l d s E[Z(w) Let  R  From  then  (1 + r f t ) (6) a n d  (8) (7) i m p l i e s (9)  ( 8 ) a n d ( 9 ) , we g e t R  ± t  (w)]  = E[z(w)  R  p t  ]  (10)  equivalently:  E[Z Equation  (R  ± t  - R  p t  )] = 0  (11)  (11) i s known a s t h e r i s k  assumption which of return  states (  R  random v a r i a b l e  that  i t ~ Ft^  t h e n t h e no a r b i t r a g e  has  =  = 1  E[Z(w)  rate  (w)]  R„. = 1 + r f t , E[Z(w)]  or  ± t  R  premium  stationarity  i fthe distribution """ s t a t i o n a r y s  property  Z > 0 such that  a stationary distribution.  implies  f°  r  o f excess  t=l,...,T,  the existence  equation  (11) h o l d s  Moreover, equation  of a and Z (11) i s  8  conveniently  w r i t t e n as:  E[Z x..] it  =  i e i ,  0  t e  T  (12)  where X  i t  "  (  Furthermore, Asset Pricing  i t"  R  F t  R  }  the f i r s t  order  Model o f p e r i o d  t  conditions  of the  c a n be w r i t t e n  Capital  i n a  similar  format: E[(a where and  R  - R  t  t  there  the market  The It  that risk  Estimation follows  must s a t i s f y  the market  to the marginal  holds  portfolio  utility  - R  p  t  a,  )).(R  ( 1 1 ) , we  random  such  ±  t  - R  Z  the  and  neglecting (13)  implies  that  p t  see t h a t  variable  of  functions.  of the market p o r t f o l i o ,  (R  constraint  (13)  stationarity  E[(a  (14) t o  on  = 0  utility  a constant,  m t  )]  quadratic  exists  Comparing  b)  premium  -  F t  of return  changing composition  that  - R  i t  investor with  Assuming r i s k  of  (R  i s proportional  mt  representative  the  )  i s the rate  t  ( a - R- )  the  m t  is  )]  =  0  t h e CAPM a  linear  (14) imposes function  premium.  of the Market Kernel  from our a n a l y s i s  the following three  that  (Z(w))  the market k e r n e l  conditions:  Z  9  < 0)  Pr(Z  E[Z]  securities  1  (15)  = R., - R„, i t Ft i  Consider  -  t  X., it  security  0  x. ]  E[Z where  =  i n period  a sample o f  w h e r e N > T.  stationarity, corresponds  i s the excess t .  T monthly excess  return vector  to the r e a l i z a t i o n  under the n u l l  there  exists  N  hypothesis  a particular  premium  f o r a g i v e n month,  of a particular  state.  X , t  More-  o f t h e no a r b i t r a g e p r o p e r t y ,  value  of t h e market k e r n e l  responding  to that state,  successive  s t a t e s are independent,  of  r e t u r n s on  Under t h e assumption o f r i s k  the excess  over,  r e t u r n on  w h i c h we  d e n o t e b y Z^_.  cor-  Since  so a r e s u c c e s s i v e  values  Z . t  Let  e.. it  = Z, X... t i t  E[e^ ] = 0 and  Then under t h e n u l l  i s serially  t  independent  hypothesis J t r  f o r each  security  i.  I f t h e v a r i a n c e o f e., i s a s s u m e d t o b e f i n i t e , t h e n t h e it c o v a r i a n c e m a t r i x o f £., a c r o s s t h e s e c u r i t i e s c a n b e w r i t t e n it as Q where  -CTjN  < aN  If  we  define  a  e. = 1  2  £ t=l  (16)  N  e  i f -  ^  follows that e  with  10  element e^ i s a s y m p t o t i c a l l y normally zero  and covariance  that  theconstraint i snot binding,  be  estimated  matrix  distributed  £ = TO, . U n d e r  Min  Z  mean  the assumption  t h e market k e r n e l can  through a Lagrangian minimization  H G  with  problem.  e - A ( Z ' l - T)  L = e'£  where  The  =  a  e  =  X'Z  x  =  X  i s a Lagrange  linear  t  vector  with  typical  x  X  ]  multiplier  c o n s t r a i n t imposed = 1.  i s  The q u a d r a t i c  a f f e c t e d b y t h e s c a l e o f t h e 1^.  simply  sets  e l e m e n t Z^_  Lxx' 2'••* N TXN  f r o m E(Z)  directly not  Z  T £ Z. t=l  = T which  f o r m o n t h e L.H.S. i s The l i n e a r  the scale t o a convenient  follows  constraint  level.  solution for Zi s  The  Z  u  = T  (1'V  - 1  !)  - 1  V  - 1  1  where V  =  X  Z^ = e s t i m a t e  I'  1  X  1  o f Z without  regard  to the non-negativity  constraint. By  ignoring thenon-negativity  associated procedure kernels  with  quadratic  i ssatisfied  do n o t v i o l a t e  constraint,  programming  provided  that  problems  c a n be a v o i d e d . the estimated  thenon-negativity  This  market  constraint.  Un-  11  fortunately, Z  u  are  i t turns  negative  employed  as  estimation greater  well. of  than  In  and  are  the  90  estimates  of  programming has  to  denoted  constrained  W^,  such that  the  they  must  be  be  i s estimated  , = R . - R„. , mt mt Ft  X..  =  then  (14)  from  becomes:  0  (18)  i t  (18)  the  zero.  X  .) mt  accounted  by  f o r by  h^ ,  the  fraction  security .i ,  and  of  the  market  summing o v e r  i ,  get (a - X A  little  Then the a  scalar  . ) mt  algebra  a = E [X  to  of  CAPM m a r k e t k e r n e l  Define  Premultiplying  we  to  the  (a - X  portfolio  results  equal  (14).  14  market kernels  addition,  equation ^  that  thus quadratic The  the or  out  mt  . mt  will  =  = E[X  mt  0  show  (19) that  ] / E [X . ] ' mt  market kernel  multiple Z  x  under the  CAPM Z  i s given  by  up  by mt  ] / E [X ] - X . ' mt mt  (20)  or Var (m)  "mt  t  Using the ' are  estimated  data.  With  +  (E ( X ^ ) )  above precedences ^ by  the  Brennan use  of  described,  -  Z  u  ,  Z  X  c  mt  ( 2 1 )  and  and  Thompson u s i n g  the  U.S.  these  market kernels  and  equation  Z  m  NYSE (11),  12  they the  show t h a t  t h e no a r b i t r a g e  U.S. m a r k e t .  actually market  kernel  imposes market can  a joint  risk  c)  premium.  Z  Z  t h e CAPM k e r n e l property,  i s a linear  Therefore,  i s i st h e  since  The T e s t i n g  applying  function  rejection of the  o f t h e No A r b i t r a g e  model  Z  u  , Z  c  between  and  their  and Z . m market  i s expected that  o f Brennan  three  hypothesis as the property,  premium.  i n Canada  a n d Thompson by  i n t h e U.S. a n d  kernels will  t h e r e s u l t s prove contradictory, i n market  market  i s no s i g n i f i c a n t d i f ^  conditions  property  This  d i f f e r e n t estimated  I f there  the market  t h e no a r b i t r a g e  differences  Property  of the  t o the Canadian s e c u r i t y market.  p e r f o r m e d by u s i n g  ference  assumption o f the r i s k  paper extends t h e work their  kernels,  t h e CAPM  o r f r o m r e j e c t i o n o f t h e no a r b i t r a g e  the stationarity  This  if  the testing using  t e s t o f whether  the constraint that  i s not refuted i n  e i t h e r c o m e f r o m r e j e c t i o n o f t h e CAPM k e r n e l  given  it  that  and t h e no a r b i t r a g e  market k e r n e l ,  is  Notice  property  imperfections  hold  Canada,  c a n be u s e d as w e l l .  i n Canada However,  i t may b e c a u s e d b y between  t h e two  countries  (e.g. transaction  ferences,  and n o t n e c e s s a r i l y because o f t h e a r b i t r a g e  opportunities  of caution  ignores  dif-  i n Canada.  Before moving word  costs), or tax structure  exchange  on t o t h e a c t u a l  must be s t a t e d . rate  testing procedure, a  The p r o c e d u r e t o be u s e d  f l u c t u a t i o n s between  Canada  and t h e  13  U.S. the  I t assumes t h a t period  ates  and  of interest.  between  create  i n the estimation  t h e CAPM k e r n e l  excess this  since  simplified  expected  procedure  = value  mated w i t h  ~ of  e., . lt  should F  r  o  ).  unconstrained  serious  equation  m  Z  and  problems.  (11) , t h e  E ^ ' S  These  the use o f t h e c o n s t r a i n e d (Z  the  not create  e.^ i t  J  t h e CAPM k e r n e l  f l u c t u a t i o n may  are small empirically,  be z e r o .  Similarly,  u  fluctu-  the results of the test.  stocks  will  during  between exchange r a t e s and  F t ^ f  R  This  t h e use o f t h e unconstrained  denoted  mated w i t h and  the covariance  i s constant  the actual rate  o f both  and t h e r e f o r e  return of individual  Let  are  In reality,  0.9 5 a n d 1.07 C a n / U . S .  a bias  However,  t h e exchange r a t e  a  r  esti-  e  and f o r convenience  u e. i t m  are also  esti(Z )  market kernel  Q  I n a d d i t i o n , t h e means o f a l l  m three  e ^ ' s over  securities e.° 1  and After  taken are  from  used e. 1  time are also i n the data  t h e above  computation,  t h e 15 0 s t o c k  I  U  Q,  ~ -1 where Q , u estimated  I  They  are denoted  £^ / U  30 s t o c k s  sample a t a time  are  and t h e f o l l o w i n g  computed:  e.  e. .  set.  f o r each o f the  respectively. C J.  m  - u'„ - 1 - u  m  computed  z.  l ~ -1 0, c  - c ' " -1 - c  , e.  i  and  variance-covariance  0,  e ~ -1 Q m  e.  1  - m'"  , e.  1  -1 - m e.  m  are the inverse  matrix  u of the  (22)  I  of the c  ,  and  14  Thirty period  of  stocks  used  stocks  are  market kernels  when t h e s e  i s greater  where  T  i s the  degrees of  freedom.  f i g u r e i n the  the  no  The  the  This  small  of  F  the  the  expressions  estimated  resulting  (22)  the  in  distribution t h e n be  test will  even w i t h  T,  with  compared  the  subsequently  by  resulting  table to determine  holds,  test,  however,  statistics.  approach  distribution  statistic,  periods  the  in  involved,  outcome can  i n the  assumption while  relies  This  sample p r o p e r t i e s c o u l d  Under the  of  30 with  whether use  be  of  U.S.  referred  to  test.  the  more c o n s e r v a t i v e a  The  chi-square  distribution  time  become u n s t a b l e ,  chi-square  chi-square  limited  estimates.  arbitrage property  market k e r n e l s .  the  a  number  a v a i l a b l e , then  follow a chi-square  the  only  I f the  h a l f of  number o f p e r i o d s  will  the  are  m u l t i p l y i n g each of  statistics  as  t h a n one  matrices  e r r o r s i n the By  i s available.  market kernels  variance-covariance large  used each time because  i s to  on  the  i s tenuous  differ  compare the  statistics  way:  that  is jointly  an  asymptotic  because  considerably.  following  having  asymptotic  x?  A to  normal,  distribution, u  has  an  shown  exact  small  sample d i s t r i b u t i o n  that ~ c J ( T - 1) T  J  D  ~  F  J,T-J  as w e l l .  I t can  be  15  where  Fj  T  '  S  =  the  T  =  number o f time s e r i e s o b s e r v a t i o n s  J  =  number of  _ j =  statistic  securities  F - s t a t i s t i c s w i t h J , T-J degrees freedom  of  Another e x t e n s i o n o f t h i s paper comes from equation  (10).  I t i s a c t u a l l y t r u e t h a t the f o l l o w i n g w i l l h o l d : E(Z"  and R £ F  R )  = E(Z  ±t  R )  V  jt  i, j  i s simply a s p e c i a l case of Rj^«  t i o n a l proxy i s used f o r Rj » namely t  Canadian market p o r t f o l i o . Z  (R.  it  -  R  ,)  mtj  R  m t  T h e r e f o r e , an a d d i f  Thus equation  (11) becomes: (24)  i f equation  procedure w i l l  (24) h o l d s .  i t can then be concluded t h a t equation equation  the r e t u r n on the  =0  A s i m i l a r e s t i m a t i o n and computation used to determine  (23)  (2 3)  be  I f i t holds, holds  and  (11) becomes more g e n e r a l s i n c e i t w i l l no longer  be c o n s t r a i n e d to  R_.. Ft F i n a l l y , the Canadian J  The  Z , ut  and  Z . mt  are a l s o  estimated,  former i s estimated from the Lagrangian m i n i m i z a t i o n pro-  cedure w h i l e the l a t t e r from the use o f equation Z £ s 1  are then transformed  (21).  such t h a t they have a mean of  u n i t y c o n s i s t e n t w i t h the other market k e r n e l s .  The  16  Using  t h e above methodology and p e r f o r m i n g  computation it  w i t h t h e use o f t h e Canadian  Z ^  a  similar  and  Z ^ ,  c a n be shown w h e t h e r t h e no a r b i t r a g e p r o p e r t y h o l d s i n  C a n a d a a n d w h e t h e r t h e CAPM k e r n e l i s i n f a c t market  d)  the true  kernel.  Alternative Equation  E  Using  Estimates  of Security Risk  (12) i m p l i e s  (i)  = ~  C  O  V  ( X  E(Z)  '  the convention E(X)  Premium  that: (25)  Z )  t h a t E(Z)  = 1, e q u a t i o n  = - c o v (X,Z)  (26)  where E(X) = E(R. ) - R i s the individual it i t premium. Equation with it  (26) o f f e r s  a new m e a s u r e o f r i s k  t h e u s e o f t h e no a r b i t r a g e p r o p e r t y .  CAPM b e t a .  I f the empirical  evidence  measure i s as good  a s t h e CAPM b e t a ,  will  t o t h e CAPM b e t a  be c o m p a r a b l e  Moreover, obtained As  CAPM e s t i m a t e s o f t h e r i s k  then  assumption  new r i s k  premia  risk  premium sense,  as compared t o  of this this  as a u s e f u l  new  risk risk  risk  measure measure.  premium c a n a l s o be  b y u s i n g t h e CAPM k e r n e l e s t i m a t e this  stock  In this  r e q u i r e s a much w e a k e r s e t o f a s s u m p t i o n s  the  (25) becomes  i n equation (26).  measure o n l y r e q u i r e s t h e  o f t h e no a r b i t r a g e p r o p e r t y , i t i s e s s e n t i a l t o  17  test  f o r i t s explanatory  a n d p r e d i c t i v e power o v e r t i m e , and  to determine i fthe r e l a x a t i o n o f the r e s t r i c t i v e o f CAPM have any d e t r i m e n t a l If  e f f e c t s on t h e new r i s k  t h e answer t o t h e above q u e s t i o n  conclude  i s negative,  that a l o t o f the r e s t r i c t i v e  CAPM c a n be r e l a x e d . Only  assumptions  one c a n  assumptions i n the  The t e s t i n g p r o c e d u r e i s as f o l l o w s :  118 f i r m s i n t h e 1956-1960 a n d 1961-1968  have o b s e r v a t i o n s  across  firm's excess return.  Since  t h e CAPM  (Z ) i s h i g h l y n e g a t i v e l y c o r r e l a t e d w i t h  of determination be  high.  (R ), m  These  procedure.  A r e g r e s s i o n i s r u n between t h e m a r k e t k e r n e l  market measure  subsamples  the e n t i r e p e r i o d o f time.  f i r m s were s e l e c t e d f o r t h e t e s t i n g  individual  measure.  i t i s expected  and each  kernel  t h e r e t u r n on t h e  that the c o r r e l a t i o n  2 (R ) between e a c h o f t h e s e  Moreover, i t i s t h e o r e t i c a l l y  regressions  will  sound t h a t t h e un-  2 constrained the excess The  kernel  ( ? ) has a h i g h u  return of the i n d i v i d u a l  when r e g r e s s e d  for d i f f e r e n t subperiods. the r i s k p r e m i a measure. c  a proxy  r i s k premia  between t h e  f o r each i n d i v i d u a l  f o r the market k e r n e l , t h e r e s u l t On t h e o t h e r  i s essentially  hand, i f t h e u n -  (Z ) i s u s e d , t h e c o v a r i a n c e u ' measure.  kernel  firm  This covariance, o f course, i s I f t h e CAPM k e r n e l (Z ) i s u s e d m  a m u l t i p l e o f t h e CAPM b e t a . constrained  against  f i r m s as w e l l .  second step c a l c u l a t e s c o v a r i a n c e s  excess r e t u r n and t h e market k e r n e l  as  R  i s t h e new  18  Both proxies calculations. form time  are  The  used  i n the  resulting  two  risk  different  premiums a r e  s e r i e s regression to determine  risk  premium  used  i f their  to  per-  explanatory  2 and  p r e d i c t i v e powers are  regressions w i l l  high  show t h e  measures whereas the  over  time.  explanatory  significance  of  The  R  power o f the  in  the  these  risk  t-statistics  will  show t h e i r p r e d i c t i v e p o w e r . Past the  history  CAPM b e t a  power.  of  has  empirical researches  both  E l t o n and  Gruber  association  of betas  They l o o k e d  at Blume's  he  computed betas  data  on  single  stock  portfolios  For  each size  the  betas  period.  [3]  f r o m one The  periods  stock  - June  the  are  forth  of  determination  of  the  1968,  in this  periods.  2  stock  to  which  paper.  where  are  fairly  Moreover, in this  betas  the  4  highly correlated  J u l y 1954  from  generated  portfolios.  betas  comparable  monthly  portfolios,  are  formed  portfolio  the  the  periods.  Blume  5 0 stock  e x a m i n e d how  are d i r e c t l y 2 (R ) o f 0.36  stock  up  interest  essentially  results  single  of  time  s e r i e s r e g r e s s i o n s on  seven year  he  recorded  representative results  time  that  and p r e d i c t i v e  adjacent  p e r i o d were w i t h  periods  examined  portfolios  so  shown  f o r example,  portfolios,  portfolio,  J u l y 1961  fore  b e t w e e n two  using  and  explanatory  [1981],  for non-overlapping  betas  and  a high  has  for a  second  - June  close to 118  time  recorded  the  the  single  p a p e r and to  1961  stock  there-  coefficient  series regression in Elton  and 2  Gruber.  One  would  therefore expect  a  similarly  high  R  19  over  time  since  w h e n t h e CAPM r i s k  this  risk  No p r i o r series  premia  premia  i s regressed  over  i s a m u l t i p l e o f t h e CAPM  e x p e c t a t i o n c a n be formed b e f o r e  r e g r e s s i o n o f t h e no a r b i t r a g e r i s k  time,  beta.  t h e time  premia.  I fthe  2 resulting  R  i s high  o n e may c o n c l u d e explanatory is  rely  risk  i s  t h a t t h e no a r b i t r a g e r i s k  and p r e d i c t i v e  otherwise,  useful  and t h e t - s t a t i s t i c  power over  t h e no a r b i t r a g e r i s k  premia has high  time.  I f the  premia w i l l  measure and t h e r e f o r e one s h o u l d  o n t h e CAPM b e t a  significant,  as t h e r e l e v a n t r i s k  result  n o t be a  continue  measure.  to  20 r  4.  DATA The  data  data  comes f r o m t h e  of market r e t u r n , r i s k f r e e  returns  of  1950  December 1980.  to  the  391  individual  Wood G u n d y 1 9 8 1  stocks  for this  The 1956  to June  June  of  to  Of  corded  i n the  first  in  second  subperiod.  120 in  stocks the  90  Finally,  the  derived the  from the  CRSP  [4].  For  i n the The  estimates  A l l U.S.  January  were computed  the  study  totally  p r i c e s of  using  the  i s from  391  and  and  into  two  January  1961  a r b i t r a r y and  i t s  have a c t u a l r e t u r n s 15 7 h a v e a c t u a l  convenience,  first  rate  rate  returns first  and  150  stocks  comes f r o m same  the  period.  f i g u r e s f o r 1961-1968  market kernel  s e c u r i t y return data  re-  the  f o r the  B r e n n a n and  over  only  subperiod  riskfree  of  January  r e s u l t s were c o n s i s t e n t 121  market k e r n e l  Canadian unconstrained  Thompson.  only  days Treasury B i l l U.S.  adjusted  is further divided  was  subperiod  second p e r i o d .  period  D e c e m b e r 1960  i f the  stocks,  were used  Canadian  to  division  determine 391  period  1956  time.  the  the  records  [5].  This  This  which  dividend  from the  interest in this  - January  p u r p o s e was  stocks  and  tape which recorded  1968.  1968.  rate  file  These r e t u r n data  period  period  subperiods to  CAPM t a p e  Thompson, was  and  estimated  were taken  were  by  from  21  5.  RESULTS  a)  The No A r b i t r a g e P r o p e r t y Before going i n t o a c t u a l d e t a i l s o f the c h i - s q u a r e t e s t  and the F - t e s t , a comparison was done between the s t a t i s t i c s of Brennan and Thompson similarity  [1983] and those o f t h i s paper.  The  between the two papers i n d i c a t e s t h a t t h i s paper  supports the r e s u l t s o f Brennan and Thompson. Table 5 shows some o f the r e p r e s e n t a t i v e r e s u l t s from Brennan and Thompson [1983] .  The s t a t i s t i c s computed are  based on 30 s e c u r i t y subsamples w i t h 30 degrees o f freedom. These s t a t i s t i c s were chosen p r i m a r i l y f o r comparison purposes s i n c e a s i m i l a r number o f s e c u r i t i e s are used i n the subsample o f - t h i s paper. The f i r s t column o f t a b l e 5 i s w i t h i n sample with the use o f the CAPM k e r n e l .  statistics  T h i s i m p l i e s t h a t the CAPM  k e r n e l e s t i m a t e s and the excess r e t u r n i n e q u a t i o n (11) comes from the same s e t of data over the same p e r i o d o f time. the  In  Brennan and Thompson paper, the s e t o f data i s U.S.  common stock r e t u r n s . An exact comparison can t h e r e f o r e be drawn between t h e i r set  o f s t a t i s t i c s and the s e t o f s t a t i s t i c s i n t a b l e s 1 and  3 under the column o f Can Zm.  These s t a t i s t i c s are d i r e c t l y  comparable t o Brennan and Thompson because they are e s t i mated w i t h the use o f the CAPM k e r n e l .  Moreover, the CAPM  k e r n e l e s t i m a t e i t s e l f and the excess r e t u r n i n e q u a t i o n (11)  22  comes f r o m a the  similar  same p e r i o d The  of  of  since  market data an  A  the  the  of the  and  to  no  Zm,  look  at  t h u s one  of  the can  the As  bution  the  a result,  region with the  any  the  the  be  3 under with  small  this  as  i n Canada,  is  region  is  different  on  the  freedom  i f  f o r the  the  distribution  no  must assumptions  regions.  sample w i l l  to  quite  supports  r e j e c t e d i n C a n a d a , one  the  asymptotic  f o l l o w the  chi-square i s not  90  period  case  no  good p r a c t i c e .  used  at  is  rejection  i s a s s u m e d t o be be  distri-  distribution.  a very  a F-distribution.  statistics  reveals  to whether  the  rejection  U.S.  paper  approach can  5%  the  column  [1983] .  r e l i a n c e on  solely  hand,  that  n o r m a l , a more c o n s e r v a t i v e resulting  the  i s dangerous because there  degrees of  other  estimates  i n both cases  conclusions  earlier,  relying 30  for  CAPM k e r n e l .  r e l e v a n t columns  Thompson  rejection  statistics  that  the  estimated  conclude  should  the  mentioned of  of  1 and  statistics  i m p l i c a t i o n s of  t °  e  guarantee  On  drawing  property  the  use  arbitrage property  f i n d i n g s o*f B r e n n a n a n d  arbitrage  the  tables  U.S.  examination  to  returns  test.  magnitude of  Prior  As  use  to test  brief  similar  of  the  common s t o c k  5 i s out-of-sample  Thompson w i t h  out-of-sample  that  table  column i s comparable  Zm,  Canadian  time.  second column of  f r o m B r e n n a n and This  set of  jointly to  This  compare puts  the  23  F J, T-J and  JCT-1) T-J  x  t h e 5% r e j e c t i o n  F  =  1.65  =  73.425  r e g i o n f o r t h e 60 p e r i o d c a s e  J(T-l) T-J  J,T-J x  89 x ^  1.84  x 59  108.56  An a t t e m p t t o d e t e r m i n e t h e d i s t r i b u t i o n one o f t h e f i r m s i n t h e 150 formed.  The  firm  i s drawn  firm  sample w i t h  of t h i s  variance having  of t h i s  i s then  firm  This  0.25  r e g i o n o f 0.15,  i , a new  data  and w i t h  90  observations  the normality t h e two  r e v e a l s the l a r g e s t distribution  Since  the normality  statistics is  d i f f e r e n c e befunctions.  t h i s , i s w e l l above t h e 5% a s s u m p t i o n must  of the histogram  of t h i s  of  a r e more o b s e r v a t i o n s  i n the r e j e c t i o n  been under t h e n o r m a l i t y level  should  i s fat-tailed,  assumption. be h i g h e r  than  I n o t h e r w o r d s , t h e 5% r e j e c t i o n  For i s comrejection  be r e j e c t e d . sample  firm  implying  r e g i o n than This  dis-  sets of  f i r m , the Kolmogorov-Smirnov s t a t i s t i c [1].  i s per-  t h e mean and  set with  that the d i s t r i b u t i o n  test.  Using  A c o m p a r i s o n between  statistic  f u r t h e r examination  significance  90 p e r i o d s  The mean and v a r i a n c e o f  computed.  sample c u m u l a t i v e  particular t o be  using  p e r f o r m e d and t h e K o l m o g o r o v - S m i r n o v  tween t h e two  puted  i s then  i s generated.  computed.  this  i  t h e same mean and v a r i a n c e  tribution data  firm  of  r a n d o m l y f r o m t h e sample where  has a l r e a d y been c a l c u l a t e d . e^^_  at  that  would  A  shows there have  i m p l i e s that the  t h a t i m p l i e d by t h e F region  should  be  24  well  above  period on  73.425  case.  i n t h e 90 p e r i o d  A l l these  t h e assumption  that  member o f t h e o t h e r Armed w i t h  case  a n d 108.56 i n t h e 60  e x t r a p o l a t i o n s , o f course, the firm  firms  chosen i s a  Canada.  jected  t h e above i n s i g h t ,  I n many c a s e s ,  o n e c a n now  region,  however,  rejection should  based on t h e reasons d e s c r i b e d be  more i n c l i n e d  in  this  the  light,  unconstrained  U.S. k e r n e l  time period  f o r any subsample.  of can  a l l firms conclude  trage  i n t h e sample that  property Of  t h e examined  This  too seriously,  Instead,  one  should  region.  subsample w i t h  Viewed  the use o f  c a n be r e j e c t e d i n a n y  t o the extreme, given the  firm  above  i s representative  i n terms o f d i s t r i b u t i o n ,  i n none o f t h e c a s e s  should  t h e no  one arbi-  be r e j e c t e d .  a l l the proxies  Canadian unconstrained proxy  i s re-  o f 43.77.  rejection  No o t h e r s  argument i s taken  assumption that  be r e j e c t e d  (Zu) c a n be r e j e c t e d i n one  (1961-1968).  this  region  above.  o n e 30 s t o c k  time period  If  should  n o t be t a k e n  t o use the F - t e s t  only  determine  t h e no a r b i t r a g e p r o p e r t y  under t h e chi-square  rejection  representative  i n t h e sample.  w h e t h e r o r n o t t h e no a r b i t r a g e p r o p e r t y in  depend  f o rt h e market k e r n e l kernel  ( C a n Zu) s e e m s t o b e t h e b e s t  f o rt h e Canadian market k e r n e l ,  C a n a d i a n CAPM k e r n e l  ( C a n Zm)  the  kernel  U.S. c o n s t r a i n e d  used, t h e  followed  by t h e  a n d t h e U . S . CAPM k e r n e l ( Z m ) ,  (Zc) and l a s t l y  t h e U.S. u n -  25  constrained relative  kernel  (Zu).  The c r i t e r i a  a t t r a c t i v e n e s s as b e i n g  the best  of  the s t a t i s t i c s .  it  i s f o r t h e Canadian market k e r n e l .  recorded  i n tables  The s m a l l e r  i n this  To  the better  a proxy  These s t a t i s t i c s a r e  as t h e d i f f e r e n t  proxies  paper f o r Canadian market  f r o m J a n u a r y 1956 t o J u n e 1968. 6 a n d 7.  their  proxy i s the t o t a l  the t o t a l ,  1 t o 4, a s w e l l  which a r e estimated  used t o judge  They a r e p r e s e n t e d  i n tables  complete t h e a n a l y s i s , t h e c o r r e l a t i o n m a t r i c e  of d i f f e r e n t proxies  over the e n t i r e period  periods  i n tables  are reported  and b o t h s u b -  8 t o 10.  Some a d d i t i o n a l i n s i g h t s c a n be drawn f r o m t h e s e tics.  kernels  statis-  F o r e x a m p l e , a l t h o u g h one c a n c o n c l u d e t h a t t h e  C a n a d i a n m a r k e t i s d i f f e r e n t f r o m t h e U.S. m a r k e t i n t e r m s o f market i m p e r f e c t i o n s , for  the difference i s not s u f f i c i e n t  i n d i v i d u a l s t o make a r b i t r a g e p r o f i t s  This  observation  recorded  c a n be c o n f i r m e d by l o o k i n g  i n tables  markedly g r e a t e r  1 t o 4.  i n m a g n i t u d e when u s i n g  n o t r e j e c t e d o n c e a more a p p r o p r i a t e  is  assumed. i n t e r e s t i n g observation  market k e r n e l , arbitrage the  i s not the best  the inherent  opportunity  test statistics  t h e U.S.  t h e no a r b i t r a g e  is  C a n a d i a n CAPM k e r n e l  at the s t a t i s t i c s  proxies  and t h u s show e v i d e n c e o f a  i n market c o n d i t i o n s ,  Another  the border.  Although the s t a t i s t i c s are  f o r t h e Canadian market k e r n e l difference  across  property  5% r e j e c t i o n r e g i o n  i s that  although the  proxy f o r the Canadian  assumption that  there  i s no  u n d e r t h e CAPM c a n n o t be r e j e c t e d a s  are well within  the F-test  5% r e j e c t i o n  ~\  26  region. Finally,  the testing  form o f equation results  place  the  region  Canadian  i n the risk  premia  CAPM k e r n e l  excess  return  correlated  with  would expect kernel  slope  each  a high  results  Cov(X,Z)  significant  over  l i e  time  outside  as a Risk  (Zm) i s t h e f i r s t Before  Measure  kernel this  risk  a n d t h e CAPM k e r n e l  are performed.  negatively  correlated with the  (Rm) a n d t h e Rm m e a s u r e i s h i g h l y individual negative  stocks'  correlation  excess  excess  r e t u r n , one  b e t w e e n t h e CAPM  return.  of the regressions  coefficient  both  the individual  and t h e stocks'  The  (24) i s u s e d i n  between  i s highly  on t h e market  vary  by t h e F - t e s t .  measure o f Cov(X,Zm).  t h e CAPM k e r n e l  return  of Using  i s calculated, regressions  stocks'  the statistics  under no subsample over  as defined  general  based on the t e s t  a case where t h e s t a t i s t i c s  The A p p l i c a b i l i t y  used  As  (11),  i s there  The  Although  i n m a g n i t u d e , when e q u a t i o n  5% r e j e c t i o n  b)  2 a n d 4.  o f equation  periods  (23) a s a more  (11) c a n n o t be r e j e c t e d ,  i n tables  considerably  of equation  of the regressions  are as expected. are highly  The  negatively  time i n almost a l l o f the cases.  The  average  2 regression to the  verify  R  i s around  that  CAPM k e r n e l  time.  0.35.  the excess  This  returns  do, i n f a c t ,  procedure  i s performed  of individual  stocks  come f r o m t h e same p e r i o d  and of  27  The return for  covariance  between t h e i n d i v i d u a l  a n d t h e CAPM k e r n e l  different  subperiods  to  16.  Notice  of  t h e CAPM b e t a  expect  that these measure.  t h a t when t h e s e  the  coefficient  and  comparable  securities'  ( i . e . cov(X,Zm))  of time.  calculated  They a r e shown i n t a b l e s 11  covariances Viewed  are i n fact multiples  i n this  covariances  light,  are regressed  one would across  2 (R ) w o u l d b e f a i r l y  of determination to the results  i s then  excess  o f Blume,  as recorded  time,  high  i n Elton  2 and  Gruber.  low  and t h e t - s t a t i s t i c s  sions  are performed  interval. sion  However, i t t u r n s o u t t h a t R  over  the entire  insignificant.  It  appears data  beta  that either  these  over  a  short  power o f t h e r e g r e s and t h e t - s t a t i s t i c  are reported  something p e c u l i a r  examination  118 s e c u r i t i e s ,  there  and Northgate  w h i c h have excess Since  periods  when t h e r e g r e s -  i n Table  17.  has happened t o  r e g r e s s i o n s o r t h a t t h e CAPM  measure over  this  data  setof  returns.  Further  the  These r e s u l t s  i s not a useful risk  Dufault  time  period i s close t o zero  s e t which biases  security  the  f o r adjacent  significant  As a whole, t h e e x p l a n a t o r y  is  the  are only  i s consistently  of the data  set reveals that out of  a r e two p e c u l i a r  stocks,  Lake  E x p l o r a t i o n ( o b s e r v a t i o n s 9 and 1 0 ) ,  r e t u r n s o f 1100% and 250% r e s p e c t i v e l y .  the observations  a r e s e v e r a l standard d e v i a t i o n s from 2 mean, t h e r e g r e s s i o n R and t - s t a t i s t i c s a r e g r e a t l y  affected.  28  A  second  regression  again  performed,  tions  which  this  between  the covariances  time without  have phenomenal  t h e two  excess  over  outlying  returns.  The  time i s  observa-  results  2 are  shown i n t a b l e  18.  Notice  that  the R  has markedly  and a l l t - s t a t i s t i c s a r e s i g n i f i c a n t o v e r t i m e . 2 whole i n t e r v a l , the R i s 0.226 a n d t h e t - s t a t i s t i c  im-  proved  Over  the  i s  highly  significant.  One  can therefore  conclude  that  although  2 the  R  i s n o t as h i g h  interval, the  as t h a t  t h e CAPM b e t a  Canadian  stocks  o f Blume  i s s t i l l  within  over  a useful  the period  a s i m i l a r time  risk  measure f o r  of interest i n this  study. Exactly  t h e same p r o c e d u r e  of  the Canadian unconstrained  in  the r i s k  factory. of  measure  kernel  of cov(X,Zu).  In general,  regressions  the use  (Zu) a s t h e k e r n e l  The  r e s u l t s are not  between  the excess  between  the individual securities'  and t h e u n c o n s t r a i n e d lated the  kernel  ( i . e . cov(X,Zu))  f o r d i f f e r e n t subperiods  new  risk  regression high  excess  As one  of time.  premia measure . ( t a b l e s  between return  the covariances, stocks, 2  can see, the R  This  19-24) . with  return calcu-  i s essentially time  and w i t h o u t  t o zero  satis-  The  i s then  The  i s shown i n t a b l e s  's a r e c l o s e  excess  used  return  d i f f e r e n t s e c u r i t i e s a n d t h e Zu a r e i n s i g n i f i c a n t .  covariance  are  i s repeated, but with  25 a n d  and t h e  series  the  two  26.  t-statistics  insignificant. An  tions  explanation  i s that  that  i s consistent  the unconstrained  kernel  with  t h e above  observa-  (Zu) i s s a m p l e s e n s i t i v e .  29  Since  t h e two o u t l y i n g  have been entered has been b i a s e d An set  interesting  into  securities with  extension  t h e above r e s u l t s  i s to reperform  t h e two  returns  i n the computations.  results  will  improve.  excess  t h e c o m p u t a t i o n o f Zu, t h e  and t h e r e f o r e  o f Zu, e x c l u d i n g  high  securities One  high  can speculate  estimation  are  the test with  returns  observed.  with  a  excess  that  the  new  30  6.  CONCLUSION With respect  arbitrage  property  seems t h a t rejected tion  to the results, should  of the s t a t i s t i c s .  time period  Once  kernel  kernel.  No o t h e r s  only  i s used  property  I t  i n Canada can o n l y  can only  subsample  as a proxy  c a n be r e j e c t e d  i s there  no a r b i t r a g e  well.  This  be  distribu-  be r e j e c t e d i n i f t h e U.S.  f o r the Canadian i n any time  exchange r a t e  no a r b i t r a g e  property  implies  abnormal r e t u r n  uncon-  market  period f o r  that  kernel  proxy  across  a U.S. i n v e s t o r  within  i s constant  over  the border  cannot  earn  time. that  the Canadian  t h e CAPM k e r n e l  assumption  i n this  Finally,  i n t h e CAPM b a s e d  On  refute the  on t h e  resulting  paper.  as f o r t h e v a l i d i t y o f using  between t h e excess  return  constrained  a s a new r i s k  kernel  unconstrained  i s not the best  f o r t h e C a n a d i a n market k e r n e l , one c a n n o t  statistics  as  an  f o r the Canadian market k e r n e l .  hand, although  no a r b i t r a g e  Canada,  by i n v e s t i n g i n Canada, assuming t h a t t h e  i s the best proxy  other  opportunity  seems t o h o l d  Moreover, one c a n c o n c l u d e  the  i n Canada.  subsample. Not  the  t h e no  t h e normal d i s t r i b u t i o n i s  f o r o n e 30 s t o c k  strained  any  property  that  i f we a s s u m e t h e a s y m p t o t i c  assumed, t h e no a r b i t r a g e one  n o t be r e j e c t e d  t h e no a r b i t r a g e  i n some c a s e s  i tappears  of individual  the  stocks  covariances and the. un-  measure, t h e r e s u l t  i s i n -  31  conclusive. sensitive of  Unfortunately, this  and t h e r e  abnormally  obscures  high  measure i s extremely  a r e two o u t l y i n g excess  the results  return  observations  data  i n terms  i n t h e sample s e t .  of the experiment.  As t h i s  new  This risk  premia measure does n o t r e q u i r e t h e s t r o n g assumptions, in  t h e case  o f t h e CAPM b e t a ,  sound, more t h o r o u g h future with is  t o reperform  which  high  to this  the testing  i s estimated  extremely for  respect  testing  without  excess  future interested  and thus should  i s more  theoretically  be p e r f o r m e d  measure.  One o b v i o u s  procedure  with  using  returns.  a new  i n the extension s e t o f Zu  t h e two s e c u r i t i e s This  individuals.  as  extension w i l l  with be  left  32  Zm  Can  Zm  Can  Zu  1 s t 30 s t o c k s  39.5039  42.4921  40.3003  2 n d 30 s t o c k s  35.9834  34.2566  28.2166  3 r d 30 s t o c k s  52.3820*  47.5471*  25.8718  4 t h 30 s t o c k s  27.1945  28.2151  12.0332  155.0438  152.5109  106.4219  Total  C h i - s q u a r e t e s t w i t h 30 degrees o f freedom @  a = 0.05  = 43.77  No a r b i t r a g e p r o p e r t y r e j e c t e d under the c h i - s q u a r e  test.  F - t e s t w i t h 30 s e c u r i t i e s and 60 p e r i o d s . @  a = 0.05  = 108.56  t No a r b i t r a g e p r o p e r t y r e j e c t e d under the F - t e s t s .  Table 1 F - t e s t and C h i - s q u a r e t e s t by u s i n g R January 1956 - December 1 9 6 0  F  for  33  Can  Zm  Zm  Can  Zu  1 s t 30 s t o c k s  43.3581  45.3584*  48.1337*  2nd  37.7123  36.8242  31.5761  3rd 30 s t o c k s  100.7400*  99.9700*  49.5979*  4th 30 s t o c k s  41.6479  41.0963  21.1218  223.4383  223.2489  150.4294  30 s t o c k s  Total  Chi-square  t e s t w i t h 30 degrees o f freedom @  a = 0.05 = 43.77  * No a r b i t r a g e p r o p e r t y r e j e c t e d under the c h i - s q u a r e  test.  F - t e s t w i t h 30 s e c u r i t i e s and 60 p e r i o d s @  a = 0.05 = 108.56  t No a r b i t r a g e p r o p e r t y r e j e c t e d under t h e F - t e s t s .  Table 2 F - t e s t and Chi-square  t e s t by u s i n g R  January 1956 - December 1960  for  34  Zu  Zc  Zm  Can Zm  Can Zu  1 s t 30 s t o c k s  55.3063"  50.6075  43.5416  42.0168  18.1670  2nd  33.8524  33.4258  30.8356  31.8311  9.6170  3rd 30 s t o c k s  66.9030*  65.905i"  36.6577  36.2457  19.4040  4 t h 30 s t o c k s  81.9035*+  72.1916"  61.8829"  69.8073"  18.2704  5th 30 s t o c k s  53.6266*  57.5594*  63.4275"  61.0484"  17.6316  30 s t o c k s  Total  291.5918  279.6894  Chi-Square t e s t w i t h @  a = 0.05  236.3453  240.9493  30 degrees o f freedom =  43.77  * No a r b i t r a g e p r o p e r t y r e j e c t e d under the c h i - s q u a r e F - t e s t with @  test.  30 s e c u r i t i e s and 90 p e r i o d s a = 0.05  =  73.425  ^ No a r b i t r a g e p r o p e r t y r e j e c t e d under t h e F - t e s t .  Table 3 F - t e s t and Chi-square  t e s t by u s i n g  January 1961 - June 1968  for  83.0900  35  Zu  Zc  Zm  Can  Zm  Can  Zu  1 s t 30 s t o c k s  59.5018'  52.2760  44.5338  41.7573  23.7071  2nd  35.5870  34.0172  30.2311  30.6456  8.2553  3rd 30 s t o c k s  66.7238*  64.8256*  36.8240  36.8847  21.7404  4th 30 s t o c k s  74.6817*  63.4206*  57.8114*  65.6502*  18.3011  5th 30 s t o c k s  46.2267*  55.1691"  64.8513*  61.8867*  14.8161  30 s t o c k s  Total  282.7210  269.7085  234.2516  236.8245  Chi-Square t e s t w i t h 30 degrees o f freedom @  a = 0.05 = 43.77  * No a r b i t r a g e p r o p e r t y r e j e c t e d under t h e c h i - s q u a r e F - t e s t with @  test.  30 s e c u r i t i e s and 90 p e r i o d s a = 0.05  =  73.425  t No a r b i t r a g e p r o p e r t y r e j e c t e d under t h e F - t e s t .  Table 4 F - t e s t and C h i - s q u a r e  t e s t by u s i n g  January 1961 - June 1968  for  86.8200  36  30 S e c u r i t y  Subsamples  W i t h i n Sample Zm  Out o f Sample Zm  1  44.55  40.80  2  47.78  74.10  3  31.76  41.29  4  46.19  51.41  5  38.22  6  16.73  Subsample  W i t h i n and Out-of sample C h i - s q u a r e t e s t o f the no a r b i t r a g e p r o p e r t y i n U.S. w i t h 30 degrees o f freedom by u s i n g  R  p  Table 5 30 s e c u r i t y subsamples s e l e c t e d from t h e Brennan Si Thompson paper f o r comparison purpose  37  Month  US ZM  CAN ZU  CAN ZM  Month  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30  1 .18525 0 .84690 0 .73497 1 .03221 1 .30427 0 .92188 0 .80809 1 .16204 1 .28025 1 .01600 0 .99077 0 .94668 1 .03806 1 .20299 0 .92092 0 .89883 0 .92609 1 .10274 1 .02313 1 .36933 1 .38508 1 .47093 0 .93165 1 .36151 0 .39775 1 .12490 0 .82626 0 .86190 0 .80601 0 .84838  1 .55040 1 .58268 1 .43148 2 .37984 2 .11710 2 .35374 0 .52326 1 .00164 1 .17204 1 .08150 0 .15258 1 .91526 1 .63434 0 .82308 1 .06578 1 .78032 0 .94518 •3.03582 1 .79148 2 .64396 1 .36140 1 .53318 0 .48792 0 .18264 2 .18250 0 .67206 1 .59558 -0 .00816 2 .05824 1 .80126  1 .01728 0 .96350 0 .92230 1 .00822 1 .03064 0 .96667 0 .93164 1 .03686 1 .04970 1 .02378 1 .04918 0 .92854 0 .99633 1 .04229 0 .96872 0 .96459 0 .96840 1 .02396 1 .03353 1 .08896 1 .07283 1 .07359 0 .96088 1 .05557 0 .96581 1 .01348 0 .97247 1 .00620 0 .95496 0 .97107  31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60  US ZM 0 .71047 0 .86205 0 .72833 0 .85150 0 .79096 0 .81322 0 .79939 0 .88610 0 .97656 0 .89741 1 .01116 1 .03021 0 .87077 1 .14963 1 .34700 0 .91680 0 .96764 0 .92771 1 .30831 1 .01161 1 .21607 1 .17645 0 .90191 0 .92355 1 .16515 0 .83610 1 .41748 1 .19954 0 .76260 ' 0 .81868  Table 6 A l t e r n a t i v e E s t i m a t e s o f Market K e r n e l January 1956 - December 1960  CAN ZU  CAN ZM  2 .73534 0 .75978 0 .82128 0 .48180 0 .04986 -1 .16658 -0 .00222 -0 .14064 -0 .50166 0 .73410 0 .18210 0 .65820 -0 .11514 0 .36990 1 .78140 0 .35004 0 .71094 1 .30818 2 .56254 1 .08276 0 .37842 1 .47282 -0 .57942 -0 .52752 0 .13848 1 .93428 1 .84788 -0 .28860 0 .18594 -0 .07242  0 .95367 0 .98994 0 .96779 0 .99829 0 .99177 0 .98523 0 .97556 0 .99709 1 .00906 0 .99364 1 .00345 0 .99350 0 .96417 1 .05145 1 .05413 1 .00586 1 .01143 0 .96748 1 .04980 1 .03834 0 .99519 1 .02234 0 .97376 1 .02300 1 .02508 0 .93723 1 .04337 1 .00187 0 .96257 0 .95228  38  Month 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45  CAN ZU 0.86886 0.71469 -0.16506 0.55125 0.88488 0.28629 -0.29241 0.41814 1.06488 0.86868 0.24984 0.31356 0.59049 -0.23427 0.75528 1.42191 2.71395 0.92889 0.33138 1.77300 1.91682 5.19282 1.24245 0.03366 1.80486 1.10934 3.20661 0.75924 -1.12356 -0.47322 1.19916 1.85445 3.26043 0.23598 2.95146 1.11465 0.19791 1.18179 0.24687 2.27772 1.67868 4.28130 2.78361 1.44621 -0.89757  CAN ZM  Month  CAN ZU  CAN ZM  0.72828 0.82480 0.90636 0.79857 0.88363 1.01521 0.94077 0.93043 1.12137 0.95918 0.82790 0.89883 1.19288 0.98979 1.05161 1.18908 1.49833 1.42793 0.93505 0.90268 1.26679 0.98617 0.65998 0.91735 0.79839 1.21421 0.85255 0.79693 0.92272 1.25213 1.18575 1.00703 0.85723 1.00902 1.06148 0.81977 0.90362 1.08392 0.78659 0.85197 0.83047 1.05897 0.90811 1.06195 0.81422  46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90  1.48716 0.18612 1.66401 -0.42426 0.66321 1.71666 0.78381 3.20967 3.24738 2.63016 1.54134 2.37555 -0.57096 2.82393 0.97758 -1.54386 0.50850 0.61641 1.01916 1.16208 1.93761 0.58221 2.01429 2.25288 2.97945 -0.53064 2.43594 2.09736 -0.72783 -1.38015 -0.16245 1.19646 0.53820 1.32759 0.89388 0.83295 -0.92034 -0.94734 -0.56565 1.02987 0.94275 0.42282 -1.57356 -0.36558 0.08955  1.00288 0.99482 0.99751 0.75148 1.05231 1.09157 0.92354 1.05587 1.43003 1.10822 0.88346 0.93990 0.94171 1.11880 0.99832 0.83491 1.15339 1.14252 0.98249 1.23880 1.05132 1.15076 1.44844 1.17226 0.92950 1.05447 0.89678 0.65198 0.98255 0.83076 0.91805 1.24815 0.86161 0.86015 1.10320 0.92237 1.37301 0.88708 0.99172 1.22005 1.29162 1.24406 0.51684 1.08713 0.73383  Table 7 A l t e r n a t i v e E s t i m a t e s o f Market K e r n e l January 1961 - June 1968  39  CZU  1.0000  ZM  .1685  CZM  .0596  .8068  1.0000  CZU  ZM  CZM  1.0000  Table 8 C o r r e l a t i o n matrix o f kernels January 1956 - December 1960  40  zu  1.0000  zc  .9711  1.0000  ZM  .3202  .3406  1.0000  CZU  .3544  .3499  .2681  1.0000  CZM  .3114  .3331  .8069  .2131  1.0000  ZU  ZC  ZM  CZU  CZM  Table 9 C o r r e l a t i o n Matrix of Kernels J a n u a r y 1 9 6 1 - June 1 9 6 8  czu  1.0000  ZM  .2396  1.0000  CZM  .1855  .7384  1.0000  CZU  ZM  CZM  Table  10  C o r r e l a t i o n matrix o f kernels January 1956 - June 1968  42  :curity  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39  Cov(Xi t ,Zmt)  Security  -0.00193 -0.00134 -0.00224 -0.00129 -0.00175 -0.00154 -0.00182 -0.00218 -0.00377 -0.00276 -0.00299 -0.00032 -0.00108 -0.00041 -0.00140 -0.00125 -0.00115 -0.00180 -0.00147 -0.00034 -0.00030 -0.00036 -0.00240 -0.00246 -0.00243 -0.00237 -0.00144 -0.00193 -0.00304 -0.00172 -0.00234 -0.00190 -0.00111 -0.00112 -0.00173 -0.00179 -0.00160 -0.00115 -0.00239  40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78  Cov(X  it  ,Z  mt  -0.00171 -0.00131 -0.00065 -0.00072 -0.00026 -0.00097 -0.00107 -0.00073 -0.00094 -0.00067 0.00002 -0.00136 -0.00074 -0.00080 -0.00071 -0.00068 -0.00094 -0.00129 -0.00094 -0.00111 -0.00169 -0.00096 -0.00157 -0.00139 -0.00242 -0.00121 -0.00113 -0.00180 -0.00060 -0.00195 0.00004 -0.00001 -0.00158 -0.00151 -0.00145 -0.00151 -0.00141 -0.00052 -0.00105  )  Security  79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118  T a b l e 11 C o v a r i a n c e s between the 118 s e c u r i t i e s r i s k premium and the CAPM k e r n e l Z f o r t h e p e r i o d January 1956 - December 1960 m  Cov(X. .Z i t ml -0.00159 -0.00127 -0.00161 -0.00196 -0.00208 -0.00147 -0.00142 -0.00036 -0.00061 -0.00046 -0.00022 -0.00070 -0.00068 -0.00105 -0.00148 -0.00069 -0.00094 -0.00138 -0.00181 -0.00094 -0.00155 -0.00133 -0.00169 -0.00151 -0.00072 -0.00042 -0.00071 -0.00031 -0.00086 -0.00110 -0.00062 -0.00193 -0.00110 -0.00059 -0.00135 -0.00059 -0.00172 -0.00011 -0.00159 -0.00154  43  i curity  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39  Cov(X  it  ,Z  ) mt  -0.00242 -0.00165 -0.00292 -0.00187 -0.00237 -0.00228 -0.00300 -0.00378 -0.00670 -0.00510 -0.00527 -0.00049 -0.00207 -0.00035 -0.00151 -0.00163 -0.00186 -0.00217 -0.00195 -0.00031 -0.00079 -0.00042 -0.00328 -0.00309 -0.00379 -0.00333 -0.00209 -0.00254 -0.00371 -0.00245 -0.00421 -0.00255 -0.00152 -0.00211 -0.00253 -0.00240 -0.00211 -0.00170 -0.00298  Security  Cov(X. .Z ) it mt  Security  40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78  -0.00155 -0.00192 -0.00112 -0.00117 0.00002 -0.00066 -0.00048 -0.00096 -0.00098 -0.00122 0.00026 -0.00183 -0.00105 -0.00135 -0.00081 -0.00050 -0.00053 -0.00153 -0.00116 -0.00107 -0.00239 -0.00144 -0.00260 -0.00189 -0.00353 -0.00190 -0.00211 -0.00243 -0.00093 -0.00281 -0.00037 -0.00010 -0.00170 -0.00193 -0.00170 -0.00231 -0.00156 -0.00084 -0.00090  79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118  T a b l e 12 C o v a r i a n c e s between t h e 118 s e c u r i t i e s r i s k premium and t h e CAPM k e r n e l Z f o r t h e p e r i o d January 1956 - June 1958 m  Cov(X. .Z it mt -0.00237 -0.00215 -0.00254 -0.00229 -0.00264 -0.00186 -0.00191 -0.00048 -0.00073 -0.00042 -0.00065 -0.00055 -0.00073 -0.00117 -0.00207 -0.00128 -0.00150 -0.00188 -0.00299 -0.00106 -0.00199 -0.00163 -0.00234 -0.00178 -0.00056 -0.00037 -0.00068 -0.00009 -0.00100 -0.00139 -0.00082 -0.00262 -0.00158 -0.00104 -0.00216 -0.00034 -0.00249 0.00045 -0.00196 -0.00198  44  icurity  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39  Cov(X. .Z ) it mt  Security  -0.00140 -0.00098 -0.00152 -0.00070 -0.00110 -0.00079 -0.00066 -0.00058 -0.00081 -0.00028 -0.00075 -0.00013 -0.00006 -0.00042 -0.00127 -0.00082 -0.00050 -0.00142 -0.00105 -0.00033 0.00021 -0.00032 -0.00158 -0.00190 -0.00109 -0.00150 -0.00078 -0.00136 -0.00248 -0.00102 -0.00052 -0.00127 -0.00067 -0.00005 -0.00090 -0.00116 -0.00109 -0.00059 -0.00174  40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78  Cov(X  it  ,Z  ) mt  -0.00183 -0.00069 -0.00016 -0.00030 -0.00053 -0.00127 -0.00157 -0.00048 -0.00086 -0.00015 -0.00020 -0.00087 -0.00044 -0.00026 -0.00059 -0.00085 -0.00136 -0.00102 -0.00071 -0.00115 -0.00097 -0.00042 -0.00046 -0.00083 -0.00132 -0.00055 -0.00018 -0.00114 -0.00026 -0.00109 0.00047 0.00013 -0.00142 -0.00102 -0.00119 -0.00067 -0.00127 -0.00022 -0.00115  •  Security  79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118  T a b l e 13 C o v a r i a n c e s between t h e 118 s e c u r i t i e s r i s k premium and t h e CAPM k e r n e l Z f o r t h e p e r i o d J u l y 1958 - December 1960 m  Cov(X. .Z it mt -0.00080 -0.00040 -0.00067 -0.00161 -0.00152 -0.00107 -0.00096 -0.00022 -0.00046 -0.00049 0.00021 -0.00079 -0.00065 -0.00090 -0.00089 -0.00013 -0.00033 -0.00090 -0.00063 -0.00079 -0.00108 -0.00102 -0.00104 -0.00120 -0.00086 -0.00045 -0.00071 -0.00049 -0.00070 -0.00080 -0.00040 -0.00124 -0.00061 -0.00018 -0.00055 -0.00080 -0.00096 -0.00064 -0.00124 -0.00113  45  >. c u r i t y  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39  Cov(X. .Z J it mt  Security  Cov(X..,Z ) it mt  Security  -0.00811 -0.00553 -0.00643 -0.00529 -0.0065,3 -0.00517 -0.00633 -0.00462 0.00923 0.00079 -0.00662 -0.00798 -0.00648 0.00085 -0.00472 -0.00374 -0.00519 -0.00598 -0.00616 0.00094 0.00101 -0.00073 -0.00620 -0.00623 -0.00601 -0.00892 -0.00264 -0.00494 -0.00727 -0.00492 -0.00729 -0.00709 -0.00365 -0.00380 -0.00811 -0.00430 -0.00550 -0.00629 -0.00541  40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78. 79  -0.00689 -0.00729 -0.00139 -0.00635 -0.00445 -0.00436 -0.00444 -0.00474 -0.00613 -0.00399 -0.00270 -0.00749 -0.00440 -0.00425 -0.00405 -0.00572 -0.00465 -0.00599 -0.00627 -0.00685 -0.00397 -0.00738 -0.00772 -0.00549 -0.00679 -0.00701 -0.00908 -0.00835 -0.00749 -0.00787 -0.00439 -0.00367 -0.00700 -0.00827 -0.00620 -0.00626 -0.00634 -0.00590 -0.00566 -0.00680  80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118  T a b l e 14 C o v a r i a n c e s between t h e 118 s e c u r i t i e s r i s k premium and t h e CAPM k e r n e l Z f o r t h e p e r i o d January 1961 - June 1968 m  Cov(X. .Z i t mi -0.00574 -0.00588 -0.00774 -0.00307 -0.00521 -0.00558 -0.00288 -0.00394 -0.00349 -0.00357 -0.00550 -0.00397 -0.00626 -0.00577 -0.00333 -0.00318 -0.00552 -0.00740 -0.00660 -0.00682 -0.00545 -0.00603 -0.00642 -0.00502 -0.00642 -0.00493 -0.00791 -0.00515 -0.00481 -0.00101 -0.00766 -0.00638 -0.00546 -0.00532 -0.00667 -0.00724 -0.00425 -0.00633 -0.00203  46  icurity  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39  "  Cov(X. .Z ) it mt  Security  Cov(X. .Z ) it mt  Security  -0.00745 -0.00494 -0.00697 -0.00391 -0.00807 -0.00496 -0.00560 -0.00675 0.02656 0.01185 -0.00777 -0.00874 -0.00623 -0.00026 -0.00332 -0.00433 -0.00370 -0.00601 -0.00453 0.00144 0.00193 -0.00089 -0.00302 -0.00578 -0.00431 -0.01081 -0.00287 -0.00352 -0.00521 -0.00411 -0.00664 -0.00534 -0.00263 -0.00290 -0.00840 -0.00256 -0.00412 -0.00305 -0.00362  40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78  -0.00452 -0.00651 0.00089 -0.00595 -0.00353 -0.00377 -0.00328 -0.00411 -0.00433 -0.00317 -0.00092 -0.00926 -0.00612 -0.00302 -0.00508 -0.00531 -0.00260 -0.00562 -0.00599 -0.00411 -0.00484 -0.00195 -0.01117 -0.00466 -0.00587 -0.00564 -0.00804 -0.00624 -0.00413 -0.00642 -0.00165 -0.00458 -0.00667 -0.00537 -0.00532 -0.00551 -0.00466 -0.00566 -0.00280  79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118  T a b l e 15 C o v a r i a n c e s between the 118 s e c u r i t i e s r i s k premium and t h e CAPM k e r n e l Z f o r t h e p e r i o d January 1961 - September 1964 m  Cov(X., ,Z i t ml -40.00742 -0.00665 -0.00436 -0.00610 -0.00193 -0.00451 -0.00467 -0.00331 -0.00325 -0.00283 -0.00087 -0.00470 -0.00274 -0.00604 -0.00528 -0.00292 -0.00300 -0.00466 -0.00820 -0.00511 -0.00490 -0.00463 -0.00414 -0.00496 -0.00464 -0.00442 -0.00434 -0.00765 -0.00529 -0.00541 0.00076 -0.00858 -0.00696 -0.00254 -0.00407 -0.00438 -0.00469 -0.00389 -0.00465 0.00044  47  Security  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39  Cov(X  ,Z  -0.00859 -0.00548 -0.00562 -0.00629 -0.00482 -0.00486 -0.00684 -0.00203 -0.00156 -0.00793 -0.00527 -0.00676 -0.00661 0.00167 -0.00605 -0.00331 -0.00707 -0.00568 -0.00796 0.00019 -0.00010 -0.00113 -0.00929 -0.00688 -0.00788 -0.00649 -0.00217 -0.00635 -0.00936 -0.00568 -0.00803 -0.00903 -0.00413 -0.00431 -0.00690 -0.00559 -0.00624 -0.00899 -0.00666  )  Security  Cov(X. ,Z ) xt mt  Security  Cov(X. ,Z ) xt mt  40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78  -0.00863 -0.00778 -0.00352 -0.00670 -0.00533 -0.00496 -0.00502 -0.00556 -0.00795 -0.00458 -0.00445 -0.00547 -0.00259 -0.00558 -0.00287 -0.00604 -0.00678 -0.00613 -0.00615 -0.00857 -0.00254 -0.01197 -0.00472 -0.00625 -0.00687 -0.00807 -0.00972 -0.00987 -0.00982 -0.00894 -0.00725 -0.00229 -0.00651 -0.00997 -0.00665 -0.00608 -0.00742 -0.00592 -0.00893  79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118  -0.00643 -0.00445 -0.00717 -0.00968 -0.00363 -0.00591 -0.00655 -0.00215 -0.00437 -0.00461 -0.00628 -0.00636 -0.00498 -0.00742 -0.00613 -0.00423 -0.00360 -0.00602 -0.00644 -0.00776 -0.00869 -0.00631 -0.00799 -0.00789 -0.00510 -0.00797 -0.00499 -0.00756 -0.00451 -0.00400 -0.00229 -0.00666 -0.00544 -0.00923 -0.00611 -0.00854 -0.00987 -0.00401 -0.00721 -0.00421  T a b l e 16 C o v a r i a n c e s between t h e 118 s e c u r i t i e s r i s k premium and t h e CAPM k e r n e l Z f o r t h e p e r i o d October 1964 - June 1968 m  48  Dependent V a r i a b l e  Period J a n . 1956 to June 1958  b R t  J u l y 1958 to r-i X! Dec. 1960 (8  R t  <D  • H M  2  J a n . 1961 to -d Sept. 1964 fl Q) ft 0)  T3 fl H  R t  J a n . 1956 to Dec. 1960  R t  0.38735 0.03590 2.0782  1.5546 0.89695 31.775  0.026938 0.00016 0.13752  1.5922 0.10955 3.7777  0.45616 0.42754 9.3077  1.5813 0.10146 3.6191  0.17689 0.08919 3.3703  -0.015059 0.00728 -0.92232  0.53024 0.75253 18.782  0.079646 0.07145 2.9876  0.74449 0.52046 11.22  O c t . 1964 to June 1968  0.16044 0.14251 4.3906  -0.60697 0.03092 -1.9239 1.4961 0.03393 2.0186  R t  O c t . 1964 to June 1968  J a n . 1961 to Dec. 1968  J a n . 1961 to Sept. 1964  >  -P C (U  Jan. 1956 to Dec. 1960  J u l y 1958 to Dec. 1960  0.40324 0.01356 1.2626  T a b l e 17 2  R e g r e s s i o n c o e f f i c i e n t , R and t - s t a t i s t i c s o f t h e r e g r e s s i o n between c o v a r i a n c e s o f t h e 118 s e c u r i t i e s and Zm o v e r d i f f e r e n t p e r i o d s o f time  49  Dependent V a r i a b l e  Period J a n . 1956 to June 1958 J u l y 1958 to H A Dec. 1960 rd  IT t  0)  •H  t  J u l y 1958 to Dec. 1960  J a n . 1961 to Sept. 1964  O c t . 1964 to June 1968  J a n . 1956 to Dec. I960  J a n . 1961 to Dec. 1968  0.22479 0.21577 5.6006  0.99464 0.19329 5.2263  0.64459 0.07884 3.1237  0.60581 0.88796 30.059  0.83421 0.18896 5.1536  1.2103 0.06702 2.8616  1.6452 0.12028 3.9480  0.97425 0.53779 11.517  1.4842 0.14007 4.3092  0.35709 0.12385 4.0143  0.12432 0.19141 5.1947  0.69291 0.66725 15.12  0.096456 0.11863 3.9171  0.68532 0.67204 15.284  U  rd > J a n . 1961 •P  C 0) T3 C 0)  to Sept. 1964  R  t  a <u O c t . 1964  Tl C  H  to June 1968  R  J a n . 1956 to Dec. 1960  b R t  t  1.4200 0.22630 5.7744  2  T a b l e 18 o  R e g r e s s i o n c o e f f i c i e n t , R and t - s t a t i s t i c s o f t h e r e g r e s s i o n between c o v a r i a n c e s o f t h e 116 s e c u r i t i e s and Zm over d i f f e r e n t p e r i o d s o f time  50  scurity  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39  Cov(X. ,Z )  Security  Cov(X. .Z ) it ut  Security  0.00216 0.00333 -0.00881 0.00372 -0.01058 0.00160 -0.00364 -0.00449 0.01760 0.01050 0.00264 -0.00165 0.01157 -0.00991 0.00084 -0.00087 0.00005 0.00108 0.00670 -0.01261 -0.00721 0.00542 0.01087 -0.00152 0.00934 0.00638 0.00556 0.00250 0.01881 -0.00020 0.01089 -0.00315 -0.00499 -0.00411 0.00055 0.00084 -0.00256 0.00316 -0.00181  40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78  0.00259 0.00268 -0.01020 -0.00033 -0.00293 -0.00806 -0.01074 -0.01059 -0.00485 0.00032 -0.00402 -0.01232 -0.00304 0.00169 -0.00719 -0.00822 -0.00733 -0.00457 -0.00496 -0.00913 -0.02727 -0.00953 0.00066 -0.00135 0.00114 0.00452 -0.00048 -0.00616 -0.01113 -0.00434 0.00728 -0.00541 -0.00878 -0.01051 0.00822 0.00195 0.00123 0.00585 -0.02382  79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118  t  u t  T a b l e 19 C o v a r i a n c e s between the 118 s e c u r i t i e s r i s k premium and the -unconstrained .kernel Z . f o r the p e r i o d January 1956 - December 1960 u  Cov(X. ,Z i t ul -0.01416 0.00181 -0.00147 0.00828 -0.01244 -0.00709 -0.00741 -0.00227 -0.00179 -0.00536 -0.00592 -0.00513 -0.00554 -0.00410 -0.00784 -0.00911 -0.00680 -0.00523 -0.00155 -0.00863 -0.00724 -0.00678 -0.00356 -0.00843 -0.00512 -0.00850 -0.00842 -0.00910 -0.00262 -0.00203 -0.00655 -0.00366 0.00236 0.00459 0.00689 0.00205 -0.00466 -0.00684 -0.00061 0.00340  51  i curity  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39  Cov<X ,Z > i t  u t  0.01203 0.01636 -0.00478 0.01079 0.00011 0.00177 0.00280 -0.00101 0.04752 0.03346 0.00865 0.01191 0.02190 0.00225 0.00555 0.00540 0.00451 0.00600 -0.00026 -0.01547 0.00008 0.01424 0.00931 -0.01371 0.00774 -0.00054 0.00051 -0.00253 -0.00795 0.00016 0.00953 0.00209 0.00308 0.00524 0.01132 0.01069 0.00355 0.00474 0.00833  Security  Cov(X..,Z ) i t ut  40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78  0.00847 0.00833 -0.00345 0.00201 -0.00313 0.00058 0.01267 0.01379 0.00698 -0.00205 -0.00388 -0.01596 -0.00974 0.00616 -0.00039 0.00010 -0.00928 -0.00398 -0.00151 -0.00490 -0.02013 0.00929 0.02058 0.01013 0.00315 -0.00166 -0.00466 -0.00243 -0.00095 -0.00584 0.00857 0.00931 -0.00402 0.01227 0.01373 -0.00170 -0.00088 0.00168 -0.00891  Security  79 80 81 82 83 84 85 86 87 88 89 90 21 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 ' 111 112 113 114 115 116 117 118  T a b l e 20 C o v a r i a n c e s between t h e 118 s e c u r i t i e s r i s k premium and the 'unconstrained k e r n e l Z f o r the p e r i o d January 1956 - June 1958 u  Cov(X. ,Z i t ut -0.01198 0.00090 -0.00523 -0.01027 -0.00875 0.00001 -0.01608 -0.00280 0.00151 -0.00224 -0.01243 -0.00478 -0.00918 0.00951 -0.00777 -0.00259 -0.00447 -0.00552 -0.00713 -0.00215 0.00117 0.00315 -0.00589 -0.00688 -0.00366 -0.00664 0.00115 -0.00219 -0.00077 -0.00480 -0.02043 0.00549 0.00097 -0.00752 0.00911 0.00244 0.00865 0.00092 -0.00196 0.02521  52  icurity  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39  Cov(X. . Z J it ut  Security  Cov(X. . Z ) it ut  Security  0.00011 -0.00104 -0.00421 -0.00050 -0.01470 0.00416 -0.01436 -0.00925 -0.00615 0.01517 -0.01323 -0.00803 0.00764 -0.01102 0.00036 0.00240 -0.01616 -0.00241 0.00307 -0.00217 -0.01045 -0.00885 0 .00288 -0.00409 0.00623 -0.00836 0.01043 0.00145 0.02211 -0.00491 -0.00106 -0.01346 -0.00601 -0.00016 -0.00324 -0.00566 -0.00778 0.00237 -0.00063  40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78  0.00568 -0.00089 -0.01162 -0.00782 -0.00322 -0.01523 -0.01749 -0.02938 -0.00847 -0.00401 -0.00191 -0.00264 -0.00059 -0.00576 -0.00875 -0.01378 -0.00633 0.00100 -0.00735 -0.01275 -0.03498 -0.01924 -0.00092 -0.00096 -0.00403 0.00810 -0.00263 -0.00485 -0.01828 -0.00302 0.01102 -0.00821 -0.00691 -0.02099 0.00307 0.01208 0.00156 0.00487 -0.02986  79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118  T a b l e 21 C o v a r i a n c e s between t h e 118 s e c u r i t i e s r i s k premium and the u n c o n s t r a i n e d k e r n e l Z f o r the p e r i o d J u l y 1958 - December 1960 u  Cov(X.. , Z i t ut -0.01577 -0.00084 -0.00058 0.02526 -0.01943 -0.01339 -0.00880 0.00262 -0.00133 -0.00675 0.00063 0.00632 -0.00956 -0.01420 -0.00949 -0.01944 -0.00083 -0.01353 0.00016 -0.01021 -0.01201 -0.01482 -0.00248 -0.00478 -0.00403 -0.00494 -0.01331 -0.00971 -0.00219 0.00158 0.01132 -0.01307 0.00425 0.00733 0.00383 0.00830 -0.02053 -0.00471 -0.00525 -0.02116  53  icurity  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39  Cov(X.. , Z J it ut  Security  Cov(X. . Z ) it ut  Security  -0.00453 -0.00199 -0.01418 -0.00899 -0.01209 -0.00696 -0.00965 -0.00594 0.00837 -0.06562 -0.00920 -0.01596 -0.00758 -0.00504 -0.00481 -0.00133 -0.02317 -0.01172 -0.02180 -0.01087 -0.01339 -0.00742 -0.01809 -0.01179 -0.01984 -0.01533 -0.00971 -0.00737 -0.01904 -0.00650 -0.01956 -0.01166 0.00102 0.00255 0.00051 0.00709 -0.00088 -0.00554 -0.01049  40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78  -0.00395 -0.00449 -0.00563 -0.01058 -0.01277 -0.00739 -0.02316 -0.01856 -0.00929 -0.00663 -0.00686 -0.00492 -0.00416 -0.00763 -0.00561 -0.01073 -0.01829 -0.01174 -0.00911 -0.00204 -0.00154 -0.01579 -0.01545 -0.01228 -0.00383 -0.01110 -0.00584 -0.00792 -0.01671 -0.00669 -0.01066 -0.00948 -0.00672 -0.01573 -0.00384 -0.01044 -0.00759 -0.01437 -0.01266  79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118  T a b l e 22 C o v a r i a n c e s between t h e 118 s e c u r i t i e s r i s k premium and t h e u n c o n s t r a i n e d k e r n e l Z f o r t h e p e r i o d January 1961 - June 1968 u  Cov(X..,Z , i t ut -0.01290 -0.00736 -0.00778 -0.01623 -0.00477 -0.01054 -0.01118 -0.00238 -0.00411 -0.01035 -0.02765 -0.01121 -0.00777 -0.01974 -0.01291 -0.01554 0.00002 -0.00880 -0.00570 -0.00437 -0.00495 -0.00512 -0.00913 -0.00799 -0.00569 -0.01164 -0.00382 -0.01007 -0.00891 -0.00825 -0.00318 -0.00375 -0.00092 0.01133 -0.00575 0.00641 -0.01177 -0.00222 -0.01344 -0.01478  54  Security  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39  Cov.(X. .Z ) it ut  Security  -0.00107 -0.00448 -0.01258 -0.00559 -0.01640 -0.01262 -0.01053 -0.00749 -0.02677 -0.12394 -0.00912 -0.00812 -0.00699 -0.02486 -0.00444 0.00158 -0.02686 -0.01187 -0.03126 -0.01387 -0.00935 -0.00194 0.00410 -0.00626 -0.02505 -0.02197 -0.01195 -0.00491 0.00063 -0.00532 -0.01017 -0.00598 -0.00445 -0.01057 0.00196 0.00228 0.00212 -0.01920 -0.00720  40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78  Cov(X. ,Z  -0.00031 0.00006 -0.00568 0.01267 -0.00683 -0.00742 -0.01033 -0.00596 -0.00489 -0.00554 -0.01167 -0.00255 -0.00103 ^0.01017 -0.00092 0.00160 -0.00290 -0.00512 -0.01082 -0.00413 -0.01055 -0.00300 -0.01030 0.00244 0.00456 -0.01742 -0.00136 -0.00660 -0.00933 -0.00410 0.00445 -0.01867 -0.00294 -0.01466 -0.00374 0.00009 -0.00124 -0.00669 0.00694  )  Security  79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118  T a b l e 23 C o v a r i a n c e s between t h e 118 s e c u r i t i e s r i s k premium and the u n c o n s t r a i n e d k e r n e l Z f o r t h e p e r i o d January 1961 - September 1964 u  Cov(X  ,Z  -0.00979 -0.00739 -0.00342 0.00143 -0.00934 -0.00422 -0.00771 -0.00167 -0.00067 -0.00179 -0.01761 -0.00511 -0.00050 0.00002 -0.01308 -0.01748 0.01651 -0.00577 0.00027 -0.00320 0.00700 0.01017 0.00059 0.00545 -0.00952 -0.01567 -0.01443 -0.00363 -0.00422 -0.01056 -0.00122 0.00044 0.00066 0.04619 0.00173 0.01386 0.01234 -0.00804 -0.00514 0.00537  )  55  icurity  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38  Cov(X. .Z J it ut  Security  Cov(X..,Z ) it ut  Security  -0.00907 -0.00342 -0.01737 -0.01463 -0.00885 -0.00447 -0.01013 -0.00717 0.00382 -0.02157 -0.01051 -0.02655 -0.00892 0.01657 -0.00567 -0.00327 -0.01704 -0.01321 -0.01127 -0.00632 -0.01628 -0.00952 -0.04088 -0.01605 -0.01361 -0.01196 -0.00890 -0.00990 -0.03847 -0.00798 -0.02837 -0.01616 0.00320 0.01325 -0.00650 0.00908 -0.00784 0.00488  39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78  -0.01703 -0.01136 -0.01089 -0.00656 -0.03407 -0.01894 -0.00728 -0.03951 -0.03000 -0.01358 -0.00913 -0.00221 -0.00878 -0.00784 -0.00440 -0.01122 -0.02359 -0.03315 -0.01969 -0.00971 -0.00607 0.00409 -0.03373 -0.01780 -0.02738 -0.01731 -0.00659 -0.01273 -0.01280 -0.03036 -0.01155 -0.02497 -0.00310 -0.01547 -0.02404 -0.00655 -0.02650 -0.01751 -0.02333 -0.02972  79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118  T a b l e 24 C o v a r i a n c e s between t h e 118 s e c u r i t i e s r i s k premium and t h e u n c o n s t r a i n e d k e r n e l Z f o r the p e r i o d October 1964 - June 1968 u  Cov(X..,Z i t ut -0.01441 -0.00952 -0.01352 -0.03205 -0.00367 -0.01673 -0.01433 -0.00490 -0.00906 -0.01603 -0.03756 -0.01687 -0.01628 -0.03375 -0.01346 -0.01058 -0.01502 -0.01392 -0.01256 -0.00754 -0.01713 -0.02015 -0.01843 -0.02135 -0.00373 -0.01020 0.00370 -0.02009 -0.01649 -0.00712 -0.00791 -0.00849 -0.00465 -0.01816 -0.01599 -0.00341 -0.03528 0.00002 -0.02650 -0.03668  56  Dependent V a r i a b l e J u l y 1958 to Dec. 1960  Period  Si  <ti  •H U  > c c a c  J a n . 1956 to June 1958  R t  J u l y 1958 to Dec. 1960  R' t  J a n . 1961 to Sept. 1964  R t  Oct. 1964 to June 1968  R t  J a n . 1956 to Dec. 1960  R t  0.011445 0.00014 0.12551  J a n . 1961 to Sept. 1964 -0.4245 0.0822 -3.2232 -0.096361 0.00409 -0.68990  O c t . 1964 to June 1968  J a n . 1956 to Dec. 1960  -0.032302 0.00083 -0.31114  0.37681 0.27246 6.5910  -0.12282 0.02114 -1.5827  0.043286 0.00144 0.40966  0.51453 0.49014 10.560  -0.019484 0.00051 -0.24407  -0.055318 0.01287 -1.2299  0.38869 0.46414 10.024  -0.045112 0.00489 -0.75474  0.43713 0.33507 7.6455  -0.11464 0.02303 -1.6535  J a n . 1961 to Dec. 1968  -0.10136 0.00750 -0.93643  T a b l e 25 R e g r e s s i o n c o e f f i c i e n t , R and t - S t a t i s t i c s o f t h e r e g r e s s i o n between c o v a r i a n c e s o f t h e 118 s e c u r i t i e s and Zu o v e r d i f f e r e n t p e r i o d s o f time  57  Dependent V a r i a b l e J u l y 1958 to Dec. 1960  Period J a n . 1956 to June 1958 J u l y 1958 to Dec. 1960  a  0)  xs a  Jan. 1961 to S e p t . 1964  R'  t  R t  -0.057382 0.00254 -0.53864  J a n . 1961 to Sept. 1964 0.003544 0.00001 0.033581 0.12440 0.01581 1.3531  O c t . 1964 to June 1968 -0.11688 0.00803 -0.96072 0.061628 0.0029 0.57543 -0.32729 0.07997 -3.1478  t  Jan. 1956 to Dec. .1960  Jan. 1961 to Dec. 1968  0.35780 0.19513 5.2571  -0.044827' 0.00357 -0.63929  0.51172 0.51769 11.062  0.090297 0.01880 1.4780  0.0594 0.00683 0.88537  0.31982 0.23091 5.8504  a T3  C  H  Oct. 1964 to June 1968  R'  Jan. 1956 to Dec. 1960  R"  -0.065233 0.01103 -1.1277  t  0.39519 0.47226 10.100 -0.039025 0.00178 -0.45040  t  T a b l e 26 R e g r e s s i o n c o e f f i c i e n t , R and t - s t a t i s t i c s o f the r e g r e s s i o n between c o v a r i a n c e s o f t h e 116 s e c u r i t i e s and Zu over d i f f e r e n t p e r i o d s o f time 2  58  Bibliography [1]  B i c k e l , P . J . a n d K.A. D o k s u m . Mathematical S t a t i s t i c s : B a s i c Ideas and S e l e c t e d T o p i c s . Holden-Day,I n c . , 1st e d . , 1977.  [2]  B r e n n a n , M . J . a n d R. T h o m p s o n . T h e No A r b i t r a g e P r o p e r t y of S e c u r i t y M a r k e t s . U n i v e r s i t y o f B r i t i s h Columbia, u n p u b l i s h e d p a p e r , November 1983.  [3]  B l u m e , M. Journal  [4]  Centre f o rResearch i n Security Prices. CRSP M a s t e r File: Monthly and D a i l y Data [machine-readable data file]. Conducted by Graduate School o f B u s i n e s s , University o f Chicago. CRSP e d . , M a r c h , 1 9 8 3 . Graduate School o f Business, U n i v e r s i t y o f Chicago [producer and d i s t r i b u t o r ] , 1983. 2 data f i l e s (5,980 l o g i c a l r e c o r d s / 685 l o g i c a l r e c o r d s ) a n d accompanying codebook.  [5]  C o p e l a n d , T.E. a n d J . F . W e s t o n . F i n a n c i a l Theory and Corporate Policy. 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