UBC Theses and Dissertations

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UBC Theses and Dissertations

Development of a forecasting model for deposits of credit unions Dobrzanski, Cristobal Tadeusz 1975

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e  8  DEVELOPMENT OF A FORECASTING MODEL FOR DEPOSITS OF CREDIT UNIONS  by CRISTOBAL TADEUSZ DOBRZANSKI B.A., S i r George W i l l i a m s U n i v e r s i t y , 1972 M.A., U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1973  i.  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF BUSINESS ADMINISTRATION  i n the Faculty of Commerce and B u s i n e s s  Administration  We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e required standard  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1975  In  presenting  this  thesis  an a d v a n c e d d e g r e e a t the I  Library  further  for  agree  make  it  partial  University freely  that permission  this  representatives. thesis  for  It  for  V a n c o u v e r 8,  Date  of  financial  British  Canada  of  of  Columbia,  British for  extensive by  the  gain  shall  Columbia  not  the  requirements  reference copying of  Head o f  is understood that  written permission.  The U n i v e r s i t y  fulfilment  available  s c h o l a r l y p u r p o s e s may be g r a n t e d  by h i s of  shall  the  in  I  agree  and this  be a l l o w e d  that  study. thesis  my D e p a r t m e n t  copying or  for  or  publication  w i t h o u t my  ii  Abstract  The p u r p o s e o f t h i s t h e s i s i s t o develop  a f o r e c a s t i n g model t o  p r e d i c t demand d e p o s i t s and term d e p o s i t s o f c r e d i t u n i o n s .  I t begins  with  a s u r v e y o f t h e l i t e r a t u r e on demand f u n c t i o n s f o r l i q u i d a s s e t s .  Both s i n g l e  e q u a t i o n models and s i m u l t a n e o u s  The hypo-  e q u a t i o n systems a r e summarized.  t h e s i s f o r a l i k e l y s t r u c t u r a l model o f c r e d i t u n i o n f i n a n c i a l b e h a v i o u r i s also presented.  However, a s t r u c t u r a l model cannot be e s t i m a t e d because  t h e r e a r e no p u b l i s h e d d a t a on c r e d i t u n i o n s ' i n t e r e s t r a t e s and t h e r e i s a l i m i t e d number o f o b s e r v a t i o n s f o r t h e dependent v a r i a b l e . The  f o r e c a s t i n g technique  t h a t i s b e i n g developed  a p p l i c a t i o n o f time s e r i e s a n a l y s i s . to express  The b a s i c i d e a b e h i n d t h i s approach i s  t h e time s e r i e s o f demand d e p o s i t s and o f term d e p o s i t s as a  w e i g h t e d sum o f the p a s t v a l u e s o f d e p o s i t s . termined  i n t h i s t h e s i s i s an  The w e i g h t s  i n t h e sum a r e de-  so as t o a c h i e v e t h e g r e a t e s t p r e d i c t i v e power by m i n i m i z i n g the mean  square e r r o r o f t h e f o r e c a s t s .  The d a t a a r e q u a r t e r l y time s e r i e s f o r demand  d e p o s i t s and f o r term d e p o s i t s f o r each o f t h r e e c r e d i t u n i o n s  i n t h e Van-  c o u v e r R e g i o n i n B r i t i s h Columbia from t h e second q u a r t e r o f 1962 t o t h e f o u r t h q u a r t e r o f 1974. The  The d a t a a r e p r i n t e d i n t h e A p p e n d i x .  s t r e n g t h o f t h e mixed a u t o r e g r e s s i v e moving average p r o c e s s  (ARIMA)  as a f o r e c a s t i n g t o o l f o r f i n a n c i a l i n t e r m e d i a r i e s such as a c r e d i t u n i o n i s e v a l u a t e d by u s i n g a l a r g e sample o f monthly d a t a o f p e r s o n a l demand d e p o s i t s and p e r s o n a l term d e p o s i t s o f Canadian c h a r t e r e d banks. each c r e d i t u n i o n ' s  The b e s t models f o r  demand d e p o s i t s and term d e p o s i t s a r e matched a g a i n s t  the n a i v e model o f a random w a l k p r o c e s s .  They a r e compared w i t h r e s p e c t t o  t h e i r minimum mean square e r r o r o f p r e d i c t i o n f o r t h e f o u r q u a r t e r s o f 1974.  iii For both the three c r e d i t unions and the chartered banks, i n a l l cases the best ARIMA model outperformed  a l l other candidates.  iv Table o f Contents Page I.  Introduction  ......  1  II.  Survey o f t h e L i t e r a t u r e  ......  6  A.  S i n g l e E q u a t i o n Models  B.  S i m u l t a n e o u s E q u a t i o n Systems  C.  S t r u c t u r a l Model  6 '  ......  19  ( i ) Consumer B e h a v i o u r  ......  ( i i ) F i n a n c i a l Behaviour of a C r e d i t Union III.  IV.  V.  12  19 21  T h e o r e t i c a l Development o f Time S e r i e s A n a l y s i s  24  A.  G e n e r a l C l a s s o f Models  24  B.  I d e n t i f i c a t i o n o f a Model  29  C.  E s t i m a t i o n o f P a r a m e t e r s and D i a g n o s t i c Checking  32  D.  Forecasting  34  E.  Transfer Function  35  D a t a and E m p i r i c a l R e s u l t s A.  Data  B.  Model f o r C h a r t e r e d  C.  Demand f o r C r e d i t U n i o n s ' Demand D e p o s i t s  D.  Demand f o r C r e d i t U n i o n s ' Term D e p o s i t s  E.  Forecast  36 ......  Banks' D e p o s i t s  36 37  ......  39 41  E v a l u a t i o n 1974:1-1974:4  43  C o n c l u d i n g Remarks  47  Bibliography  49  Appendix: Data  50  Vita  ......  55  V  L i s t o f Tables Page  Table 4.1  ARIMA Models f o r Demand D e p o s i t s o f C r e d i t U n i o n s  41  4.2  ARIMA Models f o r Term D e p o s i t s o f C r e d i t U n i o n s  42  4.3  O.L.S. R e s u l t s f o r T r a n s f e r F u n c t i o n o f Term D e p o s i t s o f C r e d i t Unions  43  P r e d i c t i o n E r r o r s , Demand D e p o s i t s o f C r e d i t U n i o n s 1974:1-1974:4 P r e d i c t i o n s  44  P r e d i c t i o n E r r o r s , Term D e p o s i t s o f C r e d i t Unions 1974:1-1974:4 P r e d i c t i o n s  45  4.4 4.5  i  VI  L i s t of Figures  T.Figure  •  Page — — e  1.1  Dynamic F i n a n c i a l Management P r o c e s s  3  III. l  Three S t a g e s o f Time S e r i e s A n a l y s i s  25  IV. 1  E x p e c t e d M o n t h l y L e v e l s o f Demand D e p o s i t s  40  IV.2  P l o t o f A c t u a l and P r e d i c t e d V a l u e s f o r C r e d i t U n i o n s ' Demand D e p o s i t s and Term D e p o s i t s 1974:1-1974:4  46  1  I.  Introduction F i n a n c i a l i n t e r m e d i a r i e s a r e f i r m s t h a t a r e p r i m a r i l y engaged i n b o r -  r o w i n g funds  ( s a v i n g s ) from h o u s e h o l d s and b u s i n e s s e s and i n l e n d i n g funds  ( l o a n s ) to o t h e r h o u s e h o l d s and b u s i n e s s e s .  I n t h e p r o c e s s o f c a r r y i n g out  these t r a n s a c t i o n s they face the l i k e l i h o o d of w i t h d r a w a l s the r i s k of d e f a u l t on l o a n s . was  to b a l a n c e the e x p e c t e d  assets.  The p e r c e n t a g e s  The  of savings  and  o l d approach t o t h i s l i q u i d i t y p r o b l e m  turnover i n l i a b i l i t i e s w i t h the m a t u r i t y of  of t o t a l funds h e l d i n s h o r t - t e r m l o a n s , consumer  l o a n s and mortgages w o u l d then be s i m i l a r t o the p e r c e n t a g e s  of  liabilities  i n s a v i n g s d e p o s i t s , term d e p o s i t s and c a p i t a l funds r e s p e c t i v e l y .  This  method n e i t h e r maximizes the r e t u r n on i n v e s t e d funds nor t a k e s advantage of d i v e r s i f i c a t i o n i n the savings  portfolio.  To b e n e f i t from b o t h f a c t o r s the i n s t i t u t i o n must engage i n a dynamic f i n a n c i a l management p r o c e s s s i m i l a r t o the one (Cramer and M i l l e r (1973)).  i l l u s t r a t e d i n Figure  U s i n g s t a t i s t i c a l i n f o r m a t i o n on i n t e r e s t  1:1 rates,  demand f o r l o a n s , and demand f o r s a v i n g s d e p o s i t s and s h a r e c a p i t a l , t h e c i s i o n maker w o u l d a p p l y a n p p t i m i z a t i o n t e c h n i q u e to d e c i d e on the b e s t  demix  of l o a n s t o i s s u e and on t h e l e a s t c o s t c o m b i n a t i o n o f s a v i n g s t o a t t r a c t . The d e c i s i o n s t o commit funds today f o r one,  f i v e , o r t e n y e a r s hence are  based on f o r e c a s t s o f i n t e r e s t r a t e s , l o a n demand, and d e p o s i t l e v e l s .  To  f o r e c a s t each o f t h e s e t h r e e f a c t o r s f o r a p a r t i c u l a r f i n a n c i a l i n t e r m e d i a r y i n v o l v e s a s i z e a b l e s t u d y o f t i m e s e r i e s , models and t e c h n i q u e s .  In this  t h e s i s we w i l l f o c u s our a t t e n t i o n on the development o f a f o r e c a s t i n g t e c h n i q u e t o p r e d i c t the demand f o r d e p o s i t s f o r c r e d i t  unions.  A c r e d i t union i s a cooperative i n s t i t u t i o n that provides  financial  s e r v i c e s s i m i l a r t o those o f o t h e r f i n a n c i a l i n t e r m e d i a r i e s such as c h a r t e r e d banks and t r u s t companies.  I t o p e r a t e s as an autonomous u n i t t h a t  has few b r a n c h o p e r a t i o n s and d e a l s o n l y w i t h i t s members.  The  and t h e l e n d e r s a r e t h e s h a r e h o l d e r s and owners o f t h e a s s e t s .  borrowers The c r e d i t  u n i o n f a c e s t h e same l i q u i d i t y p r o b l e m o f f i n a n c i a l i n t e r m e d i a t i o n and an o p t i m a l a l l o c a t i o n o f t h e c r e d i t u n i o n ' s r e s o u r c e s i s made t h r o u g h t h e same dynamic f i n a n c i a l management p r o c e s s  ( F i g u r e 1.1).  Although the  e s s e n t i a l d i f f e r e n c e between a c r e d i t u n i o n and o t h e r f i n a n c i a l is  the former's  institutions  c o o p e r a t i v e p h i l o s o p h y , they a l l have t h e f o r e c a s t i n g p r o -  b l e m o f e s t i m a t i n g f u t u r e l e v e l s o f b o t h demand and term d e p o s i t s . The  t r a d i t i o n a l a p p r o a c h t o t h e development o f a f o r e c a s t i n g model f o r  d e p o s i t s i s t o u s e economic t h e o r y o f demand f o r l i q u i d a s s e t s i n o r d e r t o f o r m u l a t e t h e i r demand e q u a t i o n s . cause i t uses p r e d e t e r m i n e d  This i s c a l l e d a s t r u c t u r a l equation be-  variables representing price of deposits, i n -  comes o f consumers, t a s t e s and p r e f e r e n c e s o f consumers, and p r i c e s o f s u b s t i t u t a b l e l i q u i d assets. o f households, stitutions.  Whereas t h i s demand e q u a t i o n models t h e b e h a v i o u r  t h e s i n g l e e q u a t i o n approach does n o t h o l d f o r f i n a n c i a l i n -  The l a t t e r has c o n t r o l o v e r t h e p r i c e o f d e p o s i t s c a u s i n g t h e  s t r u c t u r a l e q u a t i o n t o have a c u r r e n t endogenous v a r i a b l e and t h e demand f o r d e p o s i t s can no l o n g e r be e s t i m a t e d by a s i n g l e The  equation.  s t r u c t u r a l model must be expanded i n t o two o r more e q u a t i o n s i n o r -  d e r t o c a p t u r e t h e two-step d i a r y (a c r e d i t union).  d e c i s i o n making p r o c e s s o f a f i n a n c i a l  interme-  I n the f i r s t s t e p , the c r e d i t union sets the i n t e r e s t  r a t e s on demand and term d e p o s i t s .  I n t h e second s t e p , t h e r e i s a s t o c h a s t i c  3  Figure "JlJL  . Dynamic Financial Management Process  Other Factors  Past Time Series of Input Data  Modelling of Data  -|>{  Forecasting  I  'Feedback"  Application of Optimization Techniques  Decision Making  "Feedback"  Monitoring I  New Time Series Input Data  Financial Management  Statistical Analysis "Feedback"  movement i n the l e v e l o f d e p o s i t s i n response t o the new T h i s adjustment  i n the l i a b i l i t i e s w i l l  t a k e n i n the f i r s t s t a g e .  c r e a t e a feedback  The i n t e r e s t r a t e s may  interest  rates.  t o the d e c i s i o n s  have t o change a g a i n de-  p e n d i n g on c o m p e t i t i v e market c o n d i t i o n s , consumer p r e f e r e n c e s , o r b e c a u s e an u n f a v o u r a b l e p o r t f o l i o s t r u c t u r e w a r r a n t s i t ( i . e .  unfavourable  p o s i t i o n whereby i n t e r e s t payments a r e r i s i n g f a s t e r t h a n i n t e r e s t In  o t h e r words t h e i n t e r r e l a t i o n s among t h e i n t e r e s t r a t e s o f t h e  u n i o n and t h e l e v e l of d e p o s i t s must be e x p r e s s e d as a s y s t e m o f  liquidity revenues). credit  simultaneous  equations. The s t r e n g t h of e i t h e r t h e s i n g l e e q u a t i o n model o r t h e e q u a t i o n system can o n l y be e v a l u a t e d e m p i r i c a l l y .  simultaneous  To t e s t t h e h y p o t h e s i s e d  m o d e l s , one must have a s u f f i c i e n t number o f o b s e r v a t i o n s and time for  a l lvariables.  However, the o n l y d a t a a v a i l a b l e a t t h e t i m e o f t h i s  s t u d y a r e q u a r t e r l y s e r i e s on d e p o s i t s from 1962  t o 1974.  l i s h e d q u a r t e r l y d a t a on c r e d i t u n i o n s ' i n t e r e s t r a t e s and ies  series  There a r e no pubt h e r e a r e no  prox-  f o r i n t e r e s t r a t e s p a i d on c r e d i t u n i o n s ' demand and t e r m d e p o s i t s f o r  the 1962-1966 p e r i o d .  T h e r e f o r e we  cannot m e a n i n g f u l l y t e s t the s t r u c t u r a l  approach because o f m i s s i n g d a t a and l i m i t e d number o f d e g r e e s o f freedom. Another ing  approach  i s to use time s e r i e s a n a l y s i s to f o r m u l a t e a f o r e c a s t -  model f o r c r e d i t u n i o n s ' d e p o s i t s .  The method i s d a t a o r i e n t e d because  i t i n c o r p o r a t e s economic i n f o r m a t i o n t h r o u g h s u b j e c t i v e d e c i s i o n s made i n m o d e l l i n g the t i m e s e r i e s o f d e p o s i t s .  We  assume t h a t t h e r e e x i s t s a b a s i c  u n d e r l y i n g p a t t e r n f o r t h e s e r i e s o f demand d e p o s i t s and t e r m d e p o s i t s . T h i s p a t t e r n i s e x p r e s s e d as a w e i g h t e d where t h e w e i g h t s i n the sum p r e d i c t i v e power ( i . e .  sum  a r e determined  of past v a l u e s of these v a r i a b l e s so as t o a c h i e v e t h e g r e a t e s t  m i n i m i z e the f o r e c a s t i n g e r r o r ) .  5  The  a n a l y s i s i n v o l v e s t h r e e s t a g e s : (1) i d e n t i f y t h e s e r i e s as a  s t a t i o n a r y a u t o r e g r e s s i v e p r o c e s s , a moving a v e r a g e p r o c e s s o r a mixed a u t o r e g r e s s i v e moving average p r o c e s s ; ( i i ) e s t i m a t e t h e p a r a m e t e r s i n the model j u s t i d e n t i f i e d and verJLfy i f i t i s adequate; and ( i i i ) f o r e c a s t f u t u r e values f o r the s e r i e s of d e p o s i t s .  The d a t a a r e q u a r t e r l y time  s e r i e s f o r demand d e p o s i t s and f o r t e r m d e p o s i t s f o r each o f t h r e e unions  credit  i n t h e Vancouver R e g i o n o f B r i t i s h C o l u m b i a from t h e second q u a r t e r  o f 1962 t o t h e f o u r t h q u a r t e r o f 1974. The b e s t models f o r each c r e d i t u n i o n ' s demand d e p o s i t s and t e r m d e p o s i t s a r e matched a g a i n s t t h e n a i v e model o f a random w a l k p r o c e s s .  They a r e a l l compared w i t h r e s p e c t t o t h e  minimum mean s q u a r e e r r o r o f p r e d i c t i o n f o r t h e f o u r q u a r t e r s o f 1974. To o b t a i n "a p r i o r i " i d e n t i f i c a t i o n o f c r e d i t u n i o n ' s important  s e r i e s , b u t more  t o e v a l u a t e t h e s t r e n g t h o f t h e time s e r i e s method u s e d , we a l s o  examined a l a r g e r sample o f m o n t h l y d a t a f o r p e r s o n a l demand d e p o s i t s and p e r s o n a l t e r m d e p o s i t s h e l d i n c h a r t e r e d banks i n Canada (1967:9 - 1974:11, 87 o b s e r v a t i o n s ) . The  t h e s i s b e g i n s w i t h a s u r v e y o f t h e l i t e r a t u r e on demand f u n c t i o n s  for l i q u i d assets.  The h y p o t h e s i s f o r a l i k e l y s t r u c t u r a l model o f c r e d i t  union f i n a n c i a l behaviour veloped  i s also presented.  Time s e r i e s a n a l y s i s i s d e -  t h e o r e t i c a l l y i n C h a p t e r I I I u s i n g t h e n o t a t i o n o f Box and J e n k i n s  (1970) f o r a mixed a u t o r e g r e s s i v e i n t e g r a t e d moving a v e r a g e p r o c e s s . C h a p t e r I V t h e d a t a f o r demand d e p o s i t s and t e r m d e p o s i t s o f t h e t h r e e unions  In credit  and f o r p e r s o n a l d e p o s i t s o f c h a r t e r e d banks a r e d i s c u s s e d and t h e  models a r e e s t i m a t e d and e v a l u a t e d . i n C h a p t e r V.  Some c o n c l u d i n g remarks a r e p r e s e n t e d  F i n a l l y t h e A p p e n d i x : D a t a l i s t s t h e n a t u r a l l o g a r i t h m s and  the raw d a t a f o r t h e d e p o s i t s o f b o t h c r e d i t u n i o n s  and c h a r t e r e d banks.  6 II.  Survey o f t h e L i t e r a t u r e  There a r e v a r i o u s r e p r e s e n t a t i o n s o f t h e demand f o r demand d e p o s i t s and term d e p o s i t s and t h e y depend upon t h e assumptions  made about economic  b e h a v i o u r o f i n d i v i d u a l s ( o r i n s t i t u t i o n s ) and t h e l e v e l o f a g g r e g a t i o n i n the d a t a .  The l i t e r a t u r e i s grouped i n t o (a) s i n g l e e q u a t i o n m o d e l s , and  (b) s i m u l t a n e o u s  e q u a t i o n systems;  and a n a l y s e d w i t h r e s p e c t t o t h e economic  theory, a p p l i c a b i l i t y o f s t r u c t u r a l equations t o f i n a n c i a l i n t e r m e d i a t i o n i n Canada, and p r o b l e m s - w i t h A.  t h e d a t a and e s t i m a t i o n .  S i n g l e E q u a t i o n Models C l a s s i c a l demand t h e o r y s t a t e s t h a t t h e demand f o r a good o r a s e r v i c e  i s determined  by: i t s own p r i c e , consumers' incomes, consumers' t a s t e s and  p r e f e r e n c e s , and p r i c e s o f s u b s t i t u t e goods o r s e r v i c e s .  F e i g e (1964) u s e s  t h i s h y p o t h e s i s t o e s t i m a t e t h e demand f o r "demand d e p o s i t s , a  non-pecuniary  f l o w o f s e r v i c e s t h a t p r o v i d e t h e owner w i t h l i q u i d i t y , s a l a b i l i t y , s a f e t y and c o n v e n i e n c e .  S i n c e t h e v a l u e o f t h e s t r e a m o f s e r v i c e s cannot b e o b s e r v e d  t h e v a l u e o f t h e s t o c k i s used as a p r o x y .  The a s s u m p t i o n  i s made t h a t t h e r e  e x i s t s a f i x e d r e l a t i o n s h i p between t h e s t o c k and t h e f l o w o f s e r v i c e s r e n d e r e d b y a g i v e n s t o c k o f demand d e p o s i t s and hence t h e demand f o r n o n - p r e c u n i a r y s e r v i c e s i s e q u i v a l e n t t o t h e demand f o r demand d e p o s i t s . I t s own p r i c e ( R ^ ) i s t h e sum o f t h e n o m i n a l i n t e r e s t r a t e ( z e r o ) and the p o s i t i v e s e r v i c e c h a r g e s .  i s n e g a t i v e and i s d e f i n e d as t o t a l  charges d i v i d e d b y t h e average b a l a n c e o f demand d e p o s i t s .  service  Consumers' incomes  a r e a w e i g h t e d average o f p a s t and p r e s e n t v a l u e s o f p e r s o n a l income, where w e i g h t s a r e t h o s e d e v e l o p e d b y Friedman (1957) t o r e p r e s e n t permanent p e r s o n a l income ( Y ^ ) .  T a s t e s a r e assumed t o be g i v e n and t o remain c o n s t a n t o v e r  time  but preferences institutions location).  a r e p r o x i e d by t h e p e r c a p i t a number o f o f f i c e s o f f i n a n c i a l  (#/Pop) t o measure c o n v e n i e n c e ( t i m e - s p a c e  u t i l i t y p r o v i d e d by  F i n a l l y , the p r i c e s o f s u b s t i t u t e s are the a c t u a l i n t e r e s t r a t e s  p a i d on: c o m m e r c i a l bank time d e p o s i t s ( R j ) J s a v i n g s and l o a n a s s o c i a t i o n t(  shares  (K- ) and on m u t u a l s a v i n g s bank d e p o s i t s ( R ) • s  m  The a c t u a l r a t e i s  d e f i n e d as t o t a l i n t e r e s t p a i d d i v i d e d by t h e average s i z e o f a s s e t s and i t represents the t r u e o p p o r t u n i t y cost faced by the h o l d e r o f w e a l t h . U s i n g a l i n e a r form, t h e demand f u n c t i o n f o r demand d e p o s i t s i s e s t i m a t e d by u s i n g o r d i n a r y l e a s t s q u a r e s . #/Pop a r e p o s i t i v e , w h i l e t h e e x p e c t e d tues a r e n e g a t i v e .  ^  signs f o r  Y > A, and  signs f o r the c o e f f i c i e n t s of s u b s t i -  The d a t a i s a p o o l i n g o f c r o s s - s e c t i o n and time s e r i e s  o b s e r v a t i o n s from 1949 t o 1959 ( U . S . ) . found  The e x p e c t e d  and R ^  I n the best equation  (2.1) F e i g e  t o be s i g n i f i c a n t and t o have t h e e x p e c t e d  s i g n where  ^~p> t h e dependent v a r i a b l e , i s p e r c a p i t a c o m m e r c i a l bank demand d e p o s i t s ( F e i g e , 1964, p. 2 4 ) .  (2.1)  (2.2)  ~ = 535R,, + .365Y - 35R . + 53R + 25R + r e g i o n a l dummies Pop dd p td s • m ° • (48) (.080) (13) (13) (15) R = ..98 TT)  Pop  =-10lR,, + .122Y + 76R . - 44R - 82R + r e g i o n a l dummies dd p td s m ° _ (87) (.037) (10) (10) (11) R = .94 (TD)  F o r p e r c a p i t a c o m m e r c i a l bank time d e p o s i t s o f a s u b s t i t u t e ( e x p e c t e d s i g n n e g a t i v e ) and R sign positive).  I n t h i s equation  , R^  i s now a p r i c e  i s t h e own p r i c e  (expected  (2.2) a l l c o e f f i c i e n t s a r e s t a t i s t i c a l l y  s i g n i f i c a n t and have t h e r i g h t s i g n . I n a more r e c e n t s t u d y , Boyd (1973) uses t h e same t h e o r y b u t makes an e x p l i c i t a s s u m p t i o n about i m p e r f e c t c o m p e t i t i o n : t h a t t h e r e e x i s t s  product  8  d i f f e r e n t i a t i o n among d e p o s i t s o f v a r i o u s f i n a n c i a l i n t e r m e d i a r i e s b e c a u s e o f minimum b a l a n c e s and minimum terms t o e a r n i n t e r e s t .  I n h i s study o f  s a v i n g s and l o a n a s s o c i a t i o n s , a d v e r t i s i n g i s now i n t r o d u c e d a l o n g w i t h t h e c l a s s i c a l determinants  o f demand.  The f u n c t i o n a l form assumes t h a t e a c h  v a r i a b l e a f f e c t s p e r c a p i t a demand d e p o s i t s as an e x p o n e n t i a l growth (decay) and t h a t " g i v e n a change i n t h e d e s i r e d l e v e l o f d e p o s i t s , ( i n d i v i d u a l ) s a v e r s w i l l q u i c k l y a d j u s t t h e i r account b a l a n c e t o t h e new e q u i l i b r i u m " (Boyd, 1973, p. 7 4 6 ) .  The r e s u l t s f o r t h e c r o s s - s e c t i o n sample f o r J a n u a r y  1969,  data, are presented below i n equations  u s i n g semi-annual  (2.4) where DD/Pop, demand d e p o s i t s p e r c a p i t a ; R ^  (2.3) and  a v e r a g e DD i n t e r e s t  rate;  Y/Pop, 12 month a v e r a g e o f p e r c a p i t a p e r s o n a l d i s p o s a b l e income; A/Pop, p e r c a p i t a p r o m o t i o n a l expenses;  r a t i o o f number o f a s s o c i a t i o n s ' b r a n c h e s  to number o f c o m p e t i t o r s ' b r a n c h e s ;  R j> average r a t e on term d e p o s i t s ; R^, t(  average c o m p e t i t o r ' s s a v i n g s r a t e ( b a n k s ) ; and TD/Pop, term d e p o s i t s p e r capita.  ( 2 . 3 ) l r t J J D = 5.8 + 3.50 I n R , , + .51 l n Y — dd -— ° (1.24) ~~ (.34) P  P  P  +  .55 I n , ^ , „, fforrc (.24) N  ( 2 . 4 ) I n TD = -15.93 - 4.92 InR Pop (3.20)  d a  o  p  - 3.33 I n R . + .57 I n A td (3.73) (.10) P  o  p  - 2.22 I n R + r e g i o n a l dummies ,.. ,,x b „9 ,„ (1.66) R = .60 z  + 17.78 InR. (7.00) t a  + .69 I n A__ •+ r e g i o n a l dummies (.17) Pop 2 , ' K  In  t h e demand d e p o s i t e q u a t i o n a l l t h e c o e f f i c i e n t s have t h e e x p e c t e d  and o n l y own p r i c e ( R ^ ) » t a s t e s and p r e f e r e n c e s  =  . O /  sign  (|°^^) , and a d v e r t i s i n g  A (  ) are s i g n i f i c a n t determinants.  R ^ and A/Pop a r e s t a t i s t i c a l l y fc  " I n t h e c r o s s - s e c t i o n e q u a t i o n f o r TD/Pop  significant.  The o t h e r v a r i a b l e s were used  b u t t h e y n e v e r e n t e r e d s i g n i f i c a n t l y i n (2.4) and t h e i r e s t i m a t e s were n o t  9  p u b l i s h e d , u n f o r t u n a t e l y , because Boyd (1973, p. 741) o f the wrong s i g n and s i g n i f i c a n t  ( p r o b a b l y due  F i n a l l y , Boyd t e s t s h i s h y p o t h e s i s  admits t h a t  was  to m i s s p e c i f i c a t i o n b i a s ) .  t h a t consumers.' i n s t a n t a n e o u s l y  a d j u s t t h e i r d e p o s i t s t o changes i n d e p o s i t r a t e s .  If transaction costs,  i m p e r f e c t i n f o r m a t i o n , e t c . , i n v a l i d a t e t h a t a s s u m p t i o n , the r e g r e s s i o n model i s m i s s p e c i f i e d and e m p i r i c a l e s t i m a t e s may  be b i a s e d . " (Boyd, 1973,  p.  746).  F i v e y e a r a v e r a g e s a r e c a l c u l a t e d f o r the i n d e p e n d e n t v a r i a b l e s . They a r e then added as a d d i t i o n a l v a r i a b l e s i n t o the o r i g i n a l e q u a t i o n f o r the  reason  i s t h a t i f demand f o r d e p o s i t s a d j u s t s p a r t i a l l y o v e r time t h e n the a v e r a g e s s h o u l d be s i g n i f i c a n t b e c a u s e t h e y r e p r e s e n t the v a l u e s o f t h e i n d e p e n d e n t v a r i a b l e s over time.  The r e s u l t s f o r demand d e p o s i t s and t e r m d e p o s i t s a r e  n o t i m p r e s s i v e as o n l y the new  average o f a d v e r t i s i n g p e r c a p i t a p r o v e s t o  be s i g n i f i c a n t and some v a r i a b l e s have the wrong s i g n ( p r o b a b l y due c o l l i n e a r i t y caused by h i s s p e c i f i c a t i o n ) .  Thus Boyd c o n c l u d e s  to m u l t i -  that depositors  r e s p o n d f u l l y t o the changes i n the economic environment t h a t t a k e p l a c e w i t h i n the s i x month i n t e r v a l o f h i s d a t a ( s e m i - a n n u a l I n a p a p e r by M o t l e y  observation points).  (1970), he assumes t h a t h o u s e h o l d s a r e u n a b l e o r  u n w i l l i n g to a d j u s t asset h o l d i n g s i n s t a n t a n e o u s l y to d e s i r e d long-run The  d e s i r e d a s s e t l e v e l i s a f u n c t i o n of expected  levels.  income ( Y * ) , r a t e s o f r e t u r n  on a l l a s s e t s ( v e c t o r R) , and the i m p l i c i t r e n t a l s on d u r a b l e goods ( u ) . (2.5) The  * = f ( Y ~* , R , R ,  TD"  fcd  dd  R ,....,u) s  f o r m o f the f u n c t i o n i s l i n e a r i n the l o g a r i t h m s and the demand f o r a s s e t s  ( a t c o n s t a n t p r i c e s ) i s homogeneous o f degree z e r o i n g e n e r a l p r i c e l e v e l u n i t e l a s t i c w i t h respect to population n (2.6)  l o g TD*  = a  + a, l o g Y* + 0  1  £ a„. l o g R. j=l J  J  and  10  "A c o n s t a n t p r o p o r t i o n o f any r e l a t i v e d i v e r g e n c e between a c t u a l and d e s i r e d s t o c k o f ( t e r m d e p o s i t s ) i s c o r r e c t e d i n each p e r i o d (and) may be a p p r o x i m a t e d by"  ( M o t l e y , 1970, p. 236) - X.  2.7  *  TD TD  I  TD^  t-i  X > 0  " [ ViJ T  where X, t h e d e s i r e d r a t e o f a d j u s t m e n t , depends upon t h e change i n s t o c k o f b o t h t e r m d e p o s i t s and a l l o t h e r a s s e t s i n t h e p o r t f o l i o , r a t i o o f c u r r e n t t o expected  income and some n o n - q u a n t i f i a b l e l i q u i d i t y  parameter.  p r e f e r e n c e and e x p e c t a t i o n s  F o r example more l u c r a t i v e i n t e r e s t r a t e s on t e r m d e p o s i t s w i l l i n -  c r e a s e t h e demand and l e v e l o f TD .  T h i s w i l l draw funds away f r o m s e c u r i t i e s  t h a t a r e s u b s t i t u t e s and i t w i l l a f f e c t t h e l a t t e r ' s market e q u i l i b r i u m ( t h e i r market i n t e r e s t r a t e s and t h e i r q u a n t i t i e s h e l d ) .  The r e a d j u s t i n g i n t h e p o r t -  f o l i o i s d e p i c t e d b y t h e f o l l o w i n g adjustment p r o c e s s :  * (2.8)  l o g T D - l o g TD ^  n  *  = X (log TD^log T D ^ ) + E X.(log S. -log j=l 3 3  t  S.  t  )  3  + y ( l o g Y - l o g Y*) i  t  *  *  S u b s t i t u t i n g f o r d e s i r e d l e v e l s o f a s s e t s (TD , S^) i n ( 2 . 8 ) , we o b t a i n n seemingly (2.9)  unrelated logS  t  equations. = A + B l o g Y* + n o g R - ( I - A ) l o g S ^ t  + M l o g ( Y - Y*) fc  and T, A a r e nxm m a t r i c e s and i n p a r t i c u l a r  where A, B, and M a r e n - v e c t o r s  the e q u a t i o n f o r t e r m d e p o s i t s i s :  (2.10)  l o g TD  = a  n + S X 3  j c^log Y  n. n + Z X E a* 3 k  A  V  - ^ h j The  l o g R,  J  + ( l - X ) l o g TD  ±  \ ,\!:  :  ;  * l  0  g  S  jt-1  +  p  (  l  o  g  Y  - log Y )  c o e f f i c i e n t s are estimated using n o n - l i n e a r techniques w i t h only four other  11  assets ( i . e . , n = 4).  They a r e : money (M), s a v i n g s d e p o s i t s ( T D ) , debt ( D ) ,  and r e a l a s s e t s (RA).  Expected  weighted  income Y ) i s d e f i n e d as a g e o m e t r i c a l l y  a v e r a g e o f p e r s o n a l d i s p o s a b l e income where t h e w e i g h t s  are those  used by Friedman ( 1 9 5 7 ) .  The s i g n i f i c a n t d e t e r m i n a n t s  i n (2-11) a r e : t r a n s i t o r y  income (Y-Y ) , and l a g g e d h o l d i n g s (TD^ ^ ) .  results  of savings deposits  f c  These  i l l u s t r a t e t h e p a r t i a l adjustment p r o c e s s b u t n o t t h e r e a l l o c a t i o n  of  funds i n t h e p o r t f o l i o , f o r q u a r t e r l y U.S. d a t a between 1953 and 1965.  (2.11) l o g TD  = 5.72 + .14 l o g Y* + .03 l o g R (.30) (.04)  - .36 l o g TD (.13)  C  - .23 l o g D (.13)  - .14 l o g RA (.40) t  -  i  B a t r a (1973) uses t h e same f o r m u l a t i o n as M o t l e y . inasfar  -.00 l o g M (.04) t  . 1  . + .28 l o g (Y-Y*) (.08) • • t  The a s s e t s a r e i n t e r d e p e n d e n t  as t h e y compete w i t h one a n o t h e r i n t h e f i n a n c i a l p o r t f o l i o .  Changes i n  the p o r t f o l i o a t any p o i n t i n time a r e a l s o a f f e c t e d b y t h e c a p i t a l g a i n s ( l o s s e s ) incurred.  A d j u s t m e n t s t o t h e d e s i r e d l e v e l s o f a s s e t s a r e made I n some p r o p o r -  t i o n i n a given quarter.  * (2.12) ;TD - T D _ t  t  1  = X(TD  t  *  n  - (TD  fc  + G )) + I ^ f S fc  j  t  - ( S ^ ^ + G  j t  )]  3  where G i s t h e c a p i t a l g a i n s on j jt  t  h  f i n a n c i a l asset.  The demand f o r t e h  d e s i r e d s t o c k o f term d e p o s i t s i s a f u n c t i o n o f e x p e c t e d income (Y ) , e x p e c t e d c a p i t a l g a i n s (G ) , p a s t p r e f e r e n c e s and h a b i t s (S and t h o s e o f s u b s t i t u t e s (R J J> R , . . . ) . • ( £  (2.13)  TD  = a t  Expected  n  0  + a, R „ , + a Y 1 td 2 0  j),  i t s own p r i c e  (R j) t(  Assuming a l i n e a r f u n c t i o n  + a„ G + a. S - + E a_. R. 3 „ 4 p t - 1 ^ 5x l  income i s d e f i n e d as a l i n e a r f u n c t i o n o f c u r r e n t income.  Expected  c a p i t a l g a i n s a r e assumed t o be a l i n e a r f u n c t i o n o f c u r r e n t c a p i t a l g a i n s . Capital  g a i n s on a s s e t i a r e d e f i n e d a s :  12  p  i t it-1 C.P.I. t  =  it  i t - 1 xt-1 C.P.I.,. .. t-1  ~  where C P . I . , i m p l i c i t p r i c e d e f l a t o r of personal  *  n  simplifying (2.12) we  (2.14) TD  E X. [S. - (S._ .. + G.J] j J jt Jt-1 jt  get  (2.15) ATD  J  to  *  (G.D. and s u b s t i t u t i n g f o r TD i k 6  By in  (2.14):  - TD^  t  /  consumption expenditure.  = -7861 t  = 6  Q  +  5  Y* + ^  ±  + 6922 R (2100)  Z  G  fc  + 63 S  p t  + .098Y* - .49 TD (.03) (.15)  _  t _ 1  1 +  6 ^  + 6^  + 8564S (2300) P  + 6^^+  6 ^  - .03D (.01) R^ = k  .92  The empirical r e s u l t s are given a f t e r elimination of a l l n o n s i g n i f i c a n t v a r i a bles.  Data sources for the 1947-1969 time series are not l i s t e d but c a p i t a l  gains (G^) and the p r i c e of substitutes (R ) Q  did not prove to be s i g n i f i c a n t .  Since Batra does not state what assets are included i n D^ and  does not  explain  what services are measured by Sp _^, i t i s d i f f i c u l t to conclude that Motley's t  hypothesis of interdependence i s s t a t i s t i c a l l y important f o r savings deposits.  B.  Simultaneous Equation Systems In the previous section, the s i n g l e equation approach assumed that  the  variables on the r i g h t hand side of the model are predetermined-exogenous, or lagged endogenous- and hence they are a l l independent of the e r r o r term and ordinary l e a s t squares can give consistent estimates.  This assumption i s true  for the behaviour of i n d i v i d u a l s but i t cannot be made for f i n a n c i a l tions, e s p e c i a l l y  institu-  at the macroeconomic l e v e l because i n t e r e s t rates on  are decision variables i n the management of f i n a n c i a l intermediaries,  deposits  the  own  price becomes a current endogenous variable and s i n g l e equation l e a s t squares w i l l no longer r e s u l t i n consistent estimates f o r the c o e f f i c i e n t s . The econometrics of the s i t u a t i o n requires that the determinants of own  deposit  rates  13  be s p e c i f i e d and t h e e q u a t i o n s be e s t i m a t e d  simultaneously.  Cohan (1973) a p p l i e s a r e c u r s i v e s y s t e m t o d e t e r m i n e t h e i n t e r e s t r a t e s on c e r t i f i c a t e s o f d e p o s i t i s s u e d t o (a) c o r p o r a t i o n s and (b) h o u s e h o l d s , and the l e v e l o f t h e s e c e r t i f i c a t e s a c q u i r e d by (a) and ( b ) . The d e s i r e d d e p o s i t r a t e on c e r t i f i c a t e s i s s u e d t o c o r p o r a t i o n s (R- j) i s i n f l u e n c e d by: (1) a n t i Ct  c i p a t e d s t r e n g t h i n l o a n demand, ( i i ) y i e l d s on l o a n s , ( i i i ) y i e l d s on compet i t i v e assets ( i . e . Treasury B i l l s R j . ^ ' restrictions  d c o n s t r a i n e d by ( i v ) c e i l i n g r a t e  a n  ( R ^ ) . A p a r t i a l adjustment p r o c e s s i s assumed t o e x p l a i n movements  i n R ,. cd ( 2  '  '  1 6  . •  1 )  A R  (2.16.2)  cd,t "  R*  - R  dfc  X ( R  q  cd,t "  R  c d , t ^  - T(R /R ) q  t b  where R q /R t,b a p p r o x i m a t e s a c o s t mark-up f a c t o r f o r f i n a n c i a l i n t e r m e d i a t i o n ( l i m R /R , -* 0 = >R q tb  = R ). q  Assuming t h a t 0 < X < 1 s u b s t i t u t e f o r R , i n cd  (2.16.1) and t h e f o l l o w i n g e q u a t i o n i s e s t i m a t e d u s i n g q u a r t e r l y d a t a  (1961-  1967) f o r U.S. c o m m e r c i a l banks.  (2.17)  R ,  = '  c d , t  2.88 + .83R (.05)  q  — 2.64 R + .15 R . , . (.20)-5- (.05) ' tb c d  t  „ R  1  , = .99  R  (2.18)  Rp = -4.34 +.89R T.17)  + .83R (.17)  s d  S  + .15R , + .20R, (.06) (.06?  RR  c d  2  = .98  The s u p p l y p r i c e f o r p e r s o n a l c e r t i f i c a t e s o f d e p o s i t s (R ) i s d e t e r m i n e d P  by t h e same f a c t o r s as R  cd ^  s  a  p  r  o  x  y  f°  r  t  ^  ie  a  as w e l l as r e t u r n s on c o m p e t i t i v e l i q u i d a s s e t s .  b o v e f a c t o r s a f f e c t i n g d e s i r e d d e p o s i t r a t e and t h e  p r i c e o f s u b s t i t u t e s a r e : R j> s a v i n g s and l o a n s a v i n g s d e p o s i t r a t e ; R , S(  bank's s a v i n g s d e p o s i t r a t e ; R^, s h o r t term bank l e n d i n g r a t e . a u t h o r assumed a p a r t i a l a d j u s t m e n t p r o c e s s f o r R j» R C(  p  Whereas t h e  i s assumed t o a d j u s t  14  f u l l y once R ^ I s s e t and c o m p e t i t o r s ' p r i c e s a r e known.  The l i n e a r f u n c t i o n a l  forms a r e e s t i m a t e d b y two s t a g e l e a s t s q u a r e s and a l l t h e c o e f f i c i e n t s p r o v e d t o be s i g n i f i c a n t l y d i f f e r e n t f r o m z e r o .  The s t r u c t u r e o f e q u a t i o n  (2.17) i s  not a p p l i c a b l e t o t h e Canadian f i n a n c i a l system because o f t h e absence o f a legal ceiling rate. The  demand f u n c t i o n f o r c e r t i f i c a t e s o f d e p o s i t (CD) i s based upon t h e  t r a d i t i o n a l demand t h e o r y .  The dependent v a r i a b l e i s d e f i n e d as t h e r a t i o o f  CD t o l i q u i d a s s e t s (LA) h e l d by c o r p o r a t i o n s and i n d i v i d u a l s . c o n s i s t o f c o r p o r a t e and i n d i v i d u a l h o l d i n g s o f demand d e p o s i t s  Liquid  assets  and,currency,  s a v i n g s and time d e p o s i t s a t c o m m e r c i a l banks and a t s a v i n g s i n s t i t u t i o n s ,  short  term t r e a s u r y s e c u r i t i e s and c o m m e r c i a l and f i n a n c e company p a p e r , and s h o r t term U.S. government s e c u r i t i e s .  To a v o i d t h e h i g h c o r r e l a t i o n among i n t e r e s t  r a t e s t h e s p r e a d between own p r i c e and a s u b s t i t u t e i s u s e d . demand c u r v e i s l i n e a r and assumes i n s t a n t a n e o u s rn - • 7T= - -  /-<> io\ ' ( 2  7  6 3  1 9 )  + - ( RA ~ (.22) 4 2  + - ( (.24) 4  C d  t b  6  R P  The e s t i m a t e d  adjustment, t o exogenous f a c t o r s ,  - R J ) +7.61 Y - .03(AY-k) + s e a (.44) (.02) sonal, S d  P  R where Y , w e a l t h measured w e i g h t s P  theory Y  i = .139 Z (.9) GNP 1 1  p  = .99  adopted f r o m Friedman's permanent income  1  , and (AY-k). i s t h e change i n GNP l e s s t h e  a v e r a g e q u a r t e r l y growth i n GNP. e f f e c t s o f t r a n s i t o r y income.  T h i s v a r i a b l e i s i n t e n d e d t o measure t h e  " T h i s type o f income i s l i k e l y t o be h e l d i n  temporary money b a l a n c e s r a t h e r t h a n b e i n g s h i f t e d i n t o an i n t e r e s t l i q u i d asset.  2  [ I t ] i s expected  yielding  t o be i n v e r s e l y r e l a t e d t o t h e CD's" [Cohan  ( 1 9 7 3 ) , p . 1 0 7 ] . The s p r e a d between i n t e r e s t r a t e s ( a ^ j C ^ ) a r e o f m a r g i n a l s t a t i s t i c a l s i g n i f i c a n c e and o n l y Y^ p r o v e s t o be s i g n i f i c a n t . Cohan's t h r e e e q u a t i o n model i s a b l o c k r e c u r s i v e s y s t e m : two e q u a t i o n  15  s u p p l y b l o c k ( e s t i m a t e d by 2SLS) and a u n i q u e demand e q u a t i o n OLS).  The  (estimated  two d e p o s i t r a t e s a r e s e t i n t e r d e p e n d e n t l y and t h e n they a r e p a r t  o f the f i n a l d e t e r m i n a n t s  f o r CD's.  "[The] i n s t i t u t i o n a l arrangements i n t h i s  market a r e s u c h t h a t the d e t e r m i n a t i o n o f s u p p l y and demand may s e q u e n t i a l r a t h e r than simultaneous DeLeeuw's paper (1965) was sector.  i n nature".  be  considered  (Cohan, ( 1 9 7 3 ) , p.  the f i r s t s i m u l t a n e o u s  I t i s a model o f f i n a n c i a l b e h a v i o u r  t h e U.S.  by  109).  a n a l y s i s o f t h e monetary  i n the many f i n a n c i a l markets i n  A t t h i s l e v e l o f a g g r e g a t i o n t h e m a r k e t i n t e r e s t r a t e s and the quan-  t i t i e s of l i q u i d assets held are interdependent.  Of the n i n e t e e n  equations  t h a t make-up the complete model o n l y t h r e e e q u a t i o n s w i l l be d i s c u s s e d below. They a r e : demand d e p o s i t h o l d i n g s ; t i m e d e p o s i t h o l d i n g s , and i n t e r e s t r a t e on time d e p o s i t s .  The model i s based upon f o u r a s s u m p t i o n s : ( i ) There e x i s t s a  " d e s i r e d " r e l a t i o n s h i p between p o r t f o l i o c o m p o s i t i o n and i n t e r e s t r a t e s . consumer maximizes n e t w o r t h and w i l l choose t h o s e c o m b i n a t i o n s  The  of assets that  w i l l g i v e him the h i g h e s t r i s k ' - r e t u r n u t i l i t y .  ( i i ) A t any p e r i o d t h e r e i s a  p a r t i a l a d j u s t m e n t t o the " d e s i r e d " p o r t f o l i o .  A d j u s t m e n t s a r e not immediate  because l a g s i n i n f o r m a t i o n , d e c i s i o n making and p l a n e x e c u t i o n .  ( i i i ) There  a r e s h o r t - r u n c o n s t r a i n t s t h a t l i m i t b e h a v i o u r by b o t h consumers and intermediaries.  financial  These r e f e r to t o t a l s a v i n g s , c u r r e n t income, l i q u i d i t y  s i d e r a t i o n s and r e s e r v e r e q u i r e m e n t s .  ( i v ) The  con-  f i n a l assumption s t a t e s that  a l l r e l a t i o n s h i p s a r e homogeneous o f degree one i n a l l d o l l a r magnitudes.. The  change i n the l e v e l demanded o f a s s e t x i s a f u n c t i o n o f i t s s t o c k  i n the p r e v i o u s p e r i o d , r a t e s o f r e t u r n ( i t s own (R  x >  ?m (2.20) ( 2  R^,  R^,  A  (  _  x  >  R)  and c u r r e n t and l a g g e d s h o r t r u n c o n s t r a i n t s . ( f ( x ) ,  ,  _ t - l  R  t  -  and t h o s e o f s u b s t i t u t e s )  0 f l +  f c  x  a i  +  A  ^  ^'  a  s  R  i  +  _  +  a  _  _  fOO t +  t  a  _  + i  f(x). *  16  The changes i n the quantities demanded are expressed as a percentage of t o t a l 19 demand i n the sector. The l a t t e r i s measured by the proxy Y _ = 0.114 Z (0.9) i=0 • GNP_^.  I t i s lagged one period to f a c i l i t a t e simulation. Any measurement  error a r i s i n g from Y^ ^ instead of Y^_ i s assumed to be n e g l i g i b l e . est rates are nominal rates expressed as percentages.  The i n t e r -  The constraints are  p a r t i c u l a r to the demand equation. In DeLeeuw's condensed model (1969), the change i n demand deposits i s determined by i t s previous stock (DD^_ ^ ) , average y i e l d on U.S. s e c u r i t i e s maturing or c a l l a b l e i n ten years or more (R  ' yi- ld i e  n  commercial bank time  deposits (R j) > personal disposable income (Y^), and business gross investment t(  i n plant and equipment (1^) plus private nonbusiness, n o n r e s i d e n t i a l construct i o n ( I ) • The l a t t e r two variables serve as a proxy f o r the expected r e t u r n c  on c a p i t a l goods.  Current and lagged values of disposable income represent  sources of funds to households. i n the above hypothesis.  One i s struck by the absence of "own" p r i c e  This i s because DeLeeuw assumes that the demand de-  posits have t h e i r i n t e r e s t rates fixed at zero and h i s model does not deal with service charges. The r e s u l t s of 2SLS using U.S. quarterly data f o r the 1948-1962 time period are presented i n equation (2.21) and the c o e f f i c i e n t s of DD _^ Y,, and (I.+I ) have the right sign and s t a t i s t i c a l s i g n i f i c a n c e . Q  D C  DD (2.21)  § ^  Vl  = -.003 - .11  (.04)Vl  -005R - .002R (.002) - (.002) gDJ  + .03 d , t - 1 -.20  ^V'V  Y  (.02) t - 2 Y  <'  06)  Y  t-1  + tQ  .07Y ?t-1 (.03) X  R  gbl'  17  (2.22)  —  = _,Q02  - .12 (.03)  t-1  T  t - l Y _  - . 0 0 3 R . + .006 R (.000) (.001)  D  t  + .02 d , t - 1 (.008)Y _ Y  D  1  fc  2  F o r changes i n t e r m d e p o s i t s t h e f u n c t i o n a l form i s t h e same as t h a t o f ADD  and  the determinants  are l a s t p e r i o d stock of term d e p o s i t s  t h r e e month t r e a s u r y b i l l (  R t (  j) >  d  a n  r a t e ( ^)  (TD  ^ ) , the  the a v e r a g e y i e l d on bank t e r m d e p o s i t s '  R  t  d i s p o s a b l e income ( Y ^ ) .  A l l estimates  i n (2.22) a r e  significant.  The.change i n the i n t e r e s t r a t e on term d e p o s i t s i s a r e s u l t o f a d j u s t i n g the f i n a n c i a l i n t e r m e d i a r i e s ' p r e s e n t p o r t f o l i o s t o w a r d t h e i r d e s i r e d p o r t folios. and  The  q u a r t e r l y change i n R ^ i s assumed t o depend on t h e d e s i r e d r a t e  l a s t quarter's actual rate.  y i e l d s maturing tions ( R  (2.23.1)  ( 2  '  The  2 3  d e s i r e d r a t e depends upon U.S.  o r c a l l a b l e i n t e n y e a r s or.more (R j^)»  - Rq + 1)»  g b l  The  A R  - )  a  n d  <d,t  2  f o l l o w i n g equation  =  X  =  f  (  R  <Vt  "  gbl'  R  R  restric(TL/(DD+TD))  td,t-1>  g b l - q R  +  1  Whv)  '  (2.24) a p p r o x i m a t e s t h i s b e h a v i o u r .  p r o v e d t o s t a t i s t i c a l l y s i g n i f i c a n t by 2SLS. in  ceiling  o n  the r a t i o of loans to t o t a l d e p o s i t s  o n  td,t  security  A l l the  coefficients  A g a i n t h e absence o f l e g a l  Canada on R ^ l i m i t s any d i r e c t a p p l i c a t i o n o f (2.24) t o o u r  ceilings  context.  T T  (2.24)  AR  t d  - -1.26  A simultaneous  - .39 R ^ ^  •+ . 1 4 R f r . 3 3 R g  Q  + 1.02  (  W  T  1  D ) _  2  .  model a t the m i c r o l e v e l i s t h e Dhrymes and Taubman (1969)  s t u d y o f the s a v i n g s and l o a n a s s o c i a t i o n s i n t h e U.S.  Each f i r m w i l l  set  d e p o s i t and l o a n i n t e r e s t r a t e s t o a l t e r t h e demand f o r t h e i r a s s e t s and b i l i t i e s s u c h t h a t p r o f i t s a r e maximized i n each time p e r i o d . changes i n d e p o s i t s and s h a r e s may may  f a l l s h o r t of the expected  But  the a c t u a l  l e v e l s and  f o r c e the f i n a n c i a l manager t o f u r t h e r r e a d j u s t i n t e r e s t r a t e s .  lia-  The  this changes  18  in  i n t e r e s t r a t e s and i n d e p o s i t s  and i o a n s a r e i n t e r d e p e n d e n t and a r e d e t e r -  mined i n each time p e r i o d s i m u l t a n e o u s l y .  Their underlying  t i a l adjustment process i n a competitive ing  theory i s the par-  framework V a r i a b l e s s u c h as a d v e r t i s -  p e r c a p i t a and number o f S & L o f f i c e s p e r c a p i t a r e f l e c t c o n s u m e r s ' t a s t e s  and p r e f e r e n c e s r a t h e r t h a n p r o d u c t d i f f e r e n t i a t i o n . The  S & L have o n l y one t y p e o f d e p o s i t  . . ..  and t h e d e s i r e d l e v e l o f t e r m de-  p o s i t s i s a f u n c t i o n o f own i n t e r e s t r a t e , permanent p e r c a p i t a income ( Y / P o p ) , number o f S & L o f f i c e s p e r c a p i t a (#/Pop), and p r i c e s o f s u b s t i t u t e s b i l l r a t e R , and a r e g i o n a l r a t e R ) . tb e  For a d i s t r i c t , the e s t i m a t e s l i n e a r i n  n a t u r a l l o g s u s i n g q u a r t e r l y d a t a (1958-65) show l o g g e d s t o c k , R j>  a  n  t(  v e n i e n c e t o be s i g n i f i c a n t .  (treasury  However, t h e p r e s e n c e o f TD _^ b i a s e s t  d con-  the Durbin-  Watson s t a t i s t i c and t h e c o e f f i c i e n t s e s t i m a t e d may be i n c o n s i s t e n t .  Coeffi-  c i e n t on income i s s i g n i f i c a n t l y o f t h e wrong s i g n . TD  (2.25) I n —  •Y # = .37 + .20 I n R. , - .06 I n - — + .04 I n -2 .02 I n R (.06) (.01) (.006) (.014) t d  P  o  p  (TD) - .15 I n R + .93 I n —— (.14) (.005) • 6  (2.26) R t d  The  = .21 + .94 R , (.01) ^ t  d  _  1  P  1  o  p  + seasonals _ . R = .95  , + .02 R., + .01 ( ^ ) + r e g i o n a l dummies (.007) (.02) t-1 2 „ K — .y x t b  a d j u s t m e n t i n t h e i n t e r e s t r a t e on term d e p o s i t s  f o l l o w a p a r t i a l adjustment process t o the optimal competitive  level.  r a t e ( t h r e e month t r e a s u r y b i l l r a t e R ^ )  mortgages (AM).  L  t b  i s a l s o expected to I t i s d e t e r m i n e d by t h e demand f o r own  The l a t t e r v a r i a b l e i l l u s t r a t e s t h a t an e x c e s s demand f o r  a s s e t s w i l l p u t p r e s s u r e on S & L f i r m s t o a t t r a c t a d d i t i o n a l s a v i n g s t h a t t h e y . can c h a n n e l i n t o h i g h e r y i e l d i n g l o a n s , however i t d i d n o t p r o v e t o be s i g n i f i c a n t  19  i n t h e 1960-66 p e r i o d .  C.  S t r u c t u r a l Model The t h r u s t o f t h e l i t e r a t u r e and r e s e a r c h s u r v e y e d i s t o model a s i m u l -  taneous s t r u c t u r a l system o f e q u a t i o n s f o r t h e b e h a v i o u r o f f i n a n c i a l mediaries.  inter-  The m o t i v a t i o n f o r t h i s approach r e s t s w i t h t h e f a c t t h a t t h e f i r m  has a number o f p o l i c y v a r i a b l e s a t i t s d i s c r e t i o n and t h e manager wants t o know t h e v a r i o u s r e s p o n s e s t o t h o s e parameters  (i.e.  e l a s t i c i t i e s o f demand  w i t h r e s p e c t t o i n t e r e s t r a t e s , a d v e r t i s i n g , and exogenous i n t e r e s t r a t e s , i n comes o r w e a l t h ) .  A h y p o t h e s i s f o r such a model f o r demand d e p o s i t s and term  d e p o s i t s , and t h e r e s p e c t i v e i n t e r e s t r a t e s f o r a c r e d i t u n i o n i s d e v e l o p e d below.  The d i s c u s s i o n i s d i v i d e d i n t o two p a r t s : ( i ) Consumer B e h a v i o u r ; and  ( i i ) F i n a n c i a l Behaviour of a C r e d i t Union.  As u s u a l t h e v i a b i l i t y o f t h i s  approach depends on t h e a v a i l a b i l i t y o f a l a r g e number o f o b s e r v a t i o n s . ( i ) Consumer B e h a v i o u r The consumer i s e x p e c t e d t o maximize h i s n e t w o r t h i n a w o r l d where t h e r e e x i s t s l i m i t e d i n f o r m a t i o n and a t i m e l a g i n r e a l i z a t i o n s .  In p a r t i c u l a r the  consumer i s e x p e c t e d t o maximize r i s k - r e t u r n u t i l i t y f o r h i s p o r t f o l i o o f securities.  We w i l l c o n s i d e r h i s demand f o r o n l y two s u c h s e c u r i t i e s :  demand  d e p o s i t s and t e r m d e p o s i t s .  The demand f o r demand d e p o s i t s stems from t h e d e -  s i r e to h o l d l i q u i d a s s e t s .  These a s s e t s p r o v i d e t h e i n d i v i d u a l w i t h  s a l a b i l i t y , s a f e t y , c o n v e n i e n c e and c h e q u i n g f a c i l i t i e s .  liquidity,  I t i s these services  t h a t t h e i n d i v i d u a l p u r c h a s e s when he a c q u i r e s demand d e p o s i t s .  The s t o c k o f  demand d e p o s i t s w i l l s e r v e as a p r o x y f o r t h e amount o f s e r v i c e s p u r c h a s e d . On t h e o t h e r hand, t h e consumer p u r c h a s e s a term d e p o s i t i n o r d e r t o r e ceive a p o s i t i v e rate of return.  O p t i m i z i n g o v e r t i m e , t h e consumer a t t e m p t s  20 t o r e c o n c i l e h i s earned income s t r e a m w i t h h i s d e s i r e d consumption p a t t e r n i n every p e r i o d .  He may  have s u r p l u s funds t o i n v e s t i n f i n a n c i a l  securities  i f h i s e x p e c t e d income i s g r e a t e r t h a n h i s d e s i r e d c o n s u m p t i o n o r i f the r e t u r n from p u r c h a s i n g f i n a n c i a l i n s t r u m e n t s i s g r e a t e r t h a n the r e t u r n from consumpt i o n o f d u r a b l e and n o n - d u r a b l e  goods.  The  s t o c k o f term d e p o s i t s i s t h e mea-  s u r e f o r q u a n t i t y purchased. L i m i t e d i n f o r m a t i o n i m p l i e s t h a t t h e consumer does n o t know a l l t h e o p p o r t u n i t i e s a v a i l a b l e t o him and t h e i r c o s t s o r p r o f i t s , so h i s b e h a v i o u r n o t a t t a i n the optimum s o l u t i o n i n e v e r y t i m e p e r i o d .  There i s a t i m e l a g i n  e x e c u t i n g d e c i s i o n s due to d e l a y s i n communications and due p e c t a t i o n s about the f u t u r e ( i . e . l a g g e d r e s p o n s e  does  t o u n c e r t a i n ex-  i n d e p o s i t a c c o u n t s i n answer  t o changes i n i n t e r e s t r a t e s ; l a g g e d u p d a t i n g o f e x p e c t e d i n c o m e ) .  Thus house-  holds are unable or u n w i l l i n g to a d j u s t a s s e t h o l d i n g s t o long-run d e s i r e d levels instantaneously. T a s t e s and p r e f e r e n c e s a r e i n f l u e n c e d by a d v e r t i s i n g and c o n v e n i e n t tion.  loca-  There e x i s t many s u b s t i t u t e s among d u r a b l e goods and f i n a n c i a l i n s t r u m e n t s ,  The p r i c e s o f s u b s t i t u t e s a r e a l s o e x p e c t e d t o e x p l a i n t h e s i z e and n a t u r e o f ( d i s ) e q u i l i b r i u m i n t h e r e s p e c t i v e m a r k e t s o f t h e s e a s s e t s i n the p o r t f o l i o . We  adopt t h e a s s u m p t i o n  t h a t " t h e demand f o r a s s e t s ( a t c o n s t a n t p r i c e s ) i s  homogeneous o f degree z e r o i n g e n e r a l p r i c e l e v e l and u n i t - e l a s t i c w i t h r e s p e c t to p o p u l a t i o n . " ( M o t l e y , 1970, v a r i a b l e i n the model.  p. 236).  Consumers' income i s an exogenous  Permanent income i s e s t i m a t e d by a d a p t i n g the w e i g h t s  developed by Friedman (1957) i n h i s consumption s t u d y .  T r a n s i t o r y income i s  expected t o have a s i g n i f i c a n t e f f e c t on b o t h t y p e s o f p u r c h a s e s assets,  of l i q u i d  21 T h e r e f o r e t h e demand f o r a l i q u i d a s s e t i s h y p o t h e s i z e d ment between t h i s p e r i o d ' s  d e s i r e d s t o c k and l a s t p e r i o d ' s  t o be an a d j u s t a c t u a l stock.  The  d e s i r e d s t o c k i s d e t e r m i n e d b y i t s own p r i c e by t h e s i z e o f t h e consumers' budget (permanent income and t r a n s i t o r y i n c o m e ) , b y p e r c a p i t a a d v e r t i s i n g expenditures  and number o f o f f i c e s p e r c a p i t a , and by p r i c e s o f s u b s t i t u t e s .  ( i i ) F i n a n c i a l Behaviour o f a C r e d i t Union The  c r e d i t u n i o n s e t s i n t e r e s t r a t e s on demand and t e r m d e p o s i t s t o  a l t e r t h e i r r e s p e c t i v e l e v e l s s u c h t h a t t h e s u r p l u s o f r e v e n u e s minus c o s t s i s maximized i n each t i m e p e r i o d .  The d e c i s i o n maker has a d e s i r e d r a t e b u t  because o f a l a g i n d e c i s i o n - m a k i n g o r a n u n f a v o u r a b l e l i q u i d i t y p o s i t i o n he i s unable t o reach t h e d e s i r e d r a t e i n s t a n t a n e o u s l y . deposits  The d e s i r e d r a t e o n demand  i s d e t e r m i n e d by: s e r v i c e c h a r g e s ( S ) ; c o m p e t i t o r s ' g  c r e d i t u n i o n ' s demand f o r demand d e p o s i t s  rates ( R ^ ) ;  (DD); t h e mortgage r a t e (R ) , m  l i q u i d i t y c o n s t r a i n t ) ; and t h e d e s i r e d r a t e on c r e d i t u n i o n t e r m ( j o i n t d e c i s i o n making on t h e two r a t e s ) . i s determined by: competitors'  deposits  The d e s i r e d r a t e o n t e r m  deposits  r a t e s ; c r e d i t u n i o n ' s demand f o r t e r m d e p o s i t s ;  t h e mortgage r a t e ; the d e s i r e d r a t e on c r e d i t u n i o n demand d e p o s i t s ; a n d t h e demand f o r t e r m d e p o s i t s .  To m a i n t a i n  f u n d s , we p o s t u l a t e a c o n s t a n t and  t i o n i s used. c 2  2 7  - )  4  1  - - > 2 7  3  c o s t term d e p o s i t s an a d a p t i v e  R*  R  - dd-i R  D  dd=  7  =  (  R  expectations  t d- ^ d - i )  ( l - ( l - y ) L ) = R*J(1-6L) = y R l ^ L  rate  To keep a t t r a c t i n g f u n d s i n t o n o n c h e q u a b l e s a v -  (2.27.1).  (2.27.2)  ( 2  r e l a t i o n s h i p between d e s i r e d demand d e p o s i t  d e s i r e d term d e p o s i t r a t e .  ings r a t h e r than i n t o higher  the d e s i r e d d i s t r i b u t i o n o f low cost  R  t d  t d  where L , l a g o p e r a t o r  equa-  22  Partial  a d j u s t m e n t model f o r t e r m d e p o s i t r a t e s  (2.28.1)  AR  (2.28.2)  = X£R  t d  R* = f d  Substituting for R (2.28.3)  AR  -  td  l ( R  B , d  fcd  )  R ^ ) TD)  V  i n (2.28.1) and u s i n g t h e r e s u l t o f (2.27.3)  fcd  = A, ( R ^  t d  (R  R  +  +  m  ^  R  t d  +  TD -  R ^ )  w h i c h s i m p l i f i e s t o (2.28.4) (2.28.4) R  t d  = a, R*  +  a  - a, R ^  d  7 td-1 R  +  a  + ^  R  m  - a  4  R ^  + a  AR  d d  (2.29.2)  R*  d  (  8 td-2 R  P a r t i a l a d j u s t m e n t model f o r demand d e p o s i t r a t e ( R (2.29.1)  TD - ^ TD_  5  d d  )  = ,.A ( R * - P ^ ) 2  = f  2  d  ( S ^ , R , R^DD) m  S u b s t i t u t i n g t h e r e s u l t o f (2.27.3) i n (2.29.1) and b r i n g R  (id  _  1  t o the r i g h t  hand s i d e (2.29.3)  R  d d  = V  (2.29.4)  R  d d  = X y R  R  t d  -  (1-X ) 2  l-o l_ 2  fcd  + (X^y) R ^  + (1-y) (l-X,,) R  From consumer b e h a v i o u r t h e demand demand d e p o s i t s  d d  _  2  (DD) and t e r m d e p o s i t s  (TD) can be w r i t t e n w i t h (2.28.4) and (2.29.4) t o form t h e c o m p l e t e s t r u c t u r a l model o f f o u r s i m u l t a n e o u s ' (2.30) DD = ' d R n  +a  1 ?  d d  + a  1 2  R  + a ^  t d  ±3  (Y-Y ) + a BDp  equations.  l g  1  fa  dd  .  l 4  f  op  + a :  1 5  'f~-  + a  1 6  \  23 (2.31)  TD = a  +  ( 2  '  3 2 )  (2.33)  n  a  R  d  d  26 p Y  +  a  +a  . dd ' R  2  2  R  t  +  d  27 -V (Y  +  a  31 td R  " 28 a  +  a  T D  +  ^  ^  t  M  -l  32 dd-l R  .  +  ?33 dd-2 R  ' R . = a- R + . a R +<*,^KA I ,,KA o+ / rR?, + td 41 m -42 m-1 43 t d - 1 44 t d - 2 45 t d +A  a  a  R , 46 t d - 1 B  / c  J  + a TD - a TD 47 , 4 8 - 1 1 U  A l l t h e e q u a t i o n s a r e o v e r i d e n t i f i e d b u t t h e y meet t h e r a n k ( n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r i d e n t i f i c a t i o n ) . s t a t e s t h a t the s t r u c t u r a l of  This condition  e q u a t i o n i s i d e n t i f i e d i f and o n l y i f t h e r a n k  the m a t r i x formed b y t h e e x c l u d e d v a r i a b l e s i s e q u a l  equations l e s s  condition  t  o  the number o f  one.  U n f o r t u n a t e l y f o r t h e development o f t h i s h y p o t h e s i s , d a t a c o n s t r a i n t are severe. interest vable.  There i s no p u b l i s h e d q u a r t e r l y i n f o r m a t i o n o n c r e d i t  r a t e s and hence t h e c r i t i c a l p a r a m e t e r s  o f the m o d e l a r e  union unobser-  24 III  T h e o r e t i c a l Development o f Time S e r i e s A n a l y s i s T h i s method o f f o r e c a s t i n g i s d a t a o r i e n t e d as i t i n c o r p o r a t e s e c o -  nomic i n f o r m a t i o n t h r o u g h s u b j e c t i v e d e c i s i o n s made i n model s p e c i f i c a tion.  We assume t h a t t h e r e e x i s t s a b a s i c u n d e r l y i n g p a t t e r n f o r t h e  s e r i e s t h a t i s r e p r e s e n t e d by h i s t o r i c a l d a t a and t h i s p a t t e r n can be expressed  as a w e i g h t e d  the sum a r e d e t e r m i n e d power.  sum o f p a s t v a l u e s o f t h e d a t a .  The w e i g h t s i n  so as t o a c h i e v e t h e g r e a t e s t p o s s i b l e p r e d i c t i v e  T h i s a n a l y s i s i n v o l v e s t h r e e s t a g e s and n i t e r a t i o n s on t h e s e  s t a g e s , ( i l l u s t r a t e d below i n F i g u r e I I I ; Box and J e n k i n s (1970, p. 1 9 ) . Our c o n c e r n i s t o f i t a s t a t i o n a r y model f o r t h e s e r i e s o f demand d e p o s i t s and t h e s e r i e s o f t e r m d e p o s i t s o f c r e d i t u n i o n s used t o f o r e c a s t t h e i r r e s p e c t i v e v a l u e s .  t h a t w i l l be  We o p t i m i z e t h e p a t t e r n o f  a s e t o f d a t a by m i n i m i z i n g i t s f o r e c a s t i n g e r r o r .  The components o f t h e  time s e r i e s model a r e : ( i ) a u t o r e g r e s s i v e p r o c e s s where t h e r e e x i s t s an a s s o c i a t i o n among v a l u e s o f t h e same v a r i a b l e b u t a t d i f f e r e n t t i m e ods  peri-  ( s e r i a l o r s e a s o n a l ) ; ( i i ) moving average p r o c e s s where t h e r e e x i s t s  some m u t u a l correspondence  among s u c c e s s i v e v a l u e s o f r e s i d u a l s ( t r e n d s  o r s e a s o n a l ) ; and ( i i i ) a m i x t u r e o f t h e above m e n t i o n e d p r o c e s s e s . Each i s presented  A.  below.  G e n e r a l C l a s s o f Models Stationary process.  A time s e r i e s z  i s c o n s i d e r e d t o be s t a t i o n -  a r y i f i t has an e q u i l i b r i u m p o i n t about a c o n s t a n t mean and i f t h e v a r i 2 t h e o b s e r v a t i o n s a r e t h e same ( i . e . E(z. ) = E ( z ) and E(z ) t t+n t 2 2 2 E ( . ) = E ( z ) ~ ^ ^ t + n ^ ^* ^ i° y P r o c e s s has no n a t u r a l mean and i t i s assumed t h a t some s u i t a b l e d i f f e r e n c e e q u a t i o n w i l l r e ances o f  n o n s t a t  z  t  t + n  n a r  Figure  III.l  Postulate General Class of Models I d e n t i f y Model t o be Tentatively Entertained  v Estimate Parameters i n T e n t a t i v e l y E n t e r t a i n e d Model  D i a g n o s t i c Checking ( i s t h e model adequate?)  No  Forecasting Future Values of Model  26  p r e s e n t t h e p r o c e s s as b e i n g s t a t i o n a r y .  We i n t r o d u c e V as t h e backward  d i f f e r e n c e o p e r a t o r w h i c h can be w r i t t e n i n terms o f B where B, (3.1.1)  Vz  = z  t  - z _  (3.1.2)  VZ - z  t  - d z ^ + 1/2 d ( d - 1) z _  fc  t  d  t  = (1 - B) z  1  2  + ... +  (-l)  (3.2.1)  z  1  zz  t-k  Z  t-d  C o n s i d e r a time s e r i e s z w i t h o b s e r v a t i o n s  Assume t h a t i t i s s t a t i o n a r y and can be w r i t t e n as  t  =  t  d  from 1 t o T.  = <j> z  =  ;  t  t  Autoregressive process.  z  + <J> t-1  z  2  _+...  + <J>  t-2  + a_  p  t-p  t  where z , i s a random o b s e r v a t i o n a t p e r i o d t , cf> i s an a d j u s t a b l e w e i g h t , fc  and a  fc  t  i s a s e r i e s o f random shocks ("white n o i s e " ) .  a u t o r e g r e s s i v e o p e r a t o r <}>(B) we now w r i t e (3.2.2) (1 - ( j ^ B - (j) B - ... - (j) B ) z 1  2  P  2  Moving average p r o c e s s .  p  fc  Introducing the  = <KB)z = a t  fc  The t i m e s e r i e s z i s s t a t i o n a r y and c a n  be w r i t t e n as (3.3.2) where 0(B) i s moving average o p e r a t o r (3.3.1)  z  (3.3.2)  z  t  = a  t  - 6  1  a _ t  - 9  x  = (1 - 0.B - 0 B O  t  1  2  2  a _ f c  2  - ... - 9  q  a _ t  /  q  - ... 0 B ) a„ = 0(B) a q  2  q  t  t  t  M i x e d a u t o r e g r e s s i v e - m o v i n g average (ARMA) model i s 4>(B)z^_ = 0(B) a  t  where (f> (B) and 0 (B) a r e p o l y n o m i a l s o f degree p and q r e s p e c t i v e l y . ' . T h i s p r o c e s s i s r e f e r r e d t o as an ARMA (p,q) p r o c e s s (assumed t o be t i o n a r y ) . ( B o x - J e n k i n s ( 1 9 7 0 ) , p. 7 4 ) . (3.4.1) z = <kz + <i> z + ... + fj> z 1  t  I t-i  I t-Z  + a. - 0-a^.  p t-p  t  0  sta-  - ... - 0  2 t-2  q t-q  A complete model i s c a l l e d the a u t o r e g r e s s i v e i n t e g r a t e d moving average p r o c e s s w i t h a p t h a u t o r e g r e s s i v e scheme, a d t h s t a t i o n a r y ence, and a q t h moving average: ARIMA (p,d,q)  differ-  .. 27  (3.4.2)  <f>(B)V z = •• 6 (B) a d  t  An example o f a (1.1.1) p r o c e s s i s : (3.5.1)  (1 - <J» B) Vz  (3.5.2)  Vz  - ^  t  = (1 - e^B)  V z ^  = a  -  t  a a ^  6;L  ^  An example o f a ( 0 , 2, 2) p r o c e s s i s : (3.6.1)  V z  t  (3.6.2)  z  - 2z _  2  t  = (1 - 0 B - 6 B ) a 2  1  t  1  2  + z _ t  2  = a  t  - Q  t  S e a s o n a l ARLMA p r o c e s s .Assume t h a t z  a ^  ±  - 8  2  a _ f c  2  i s a m o n t h l y s e r i e s and t h a t we  i n t e n d t o l i n k c u r r e n t b e h a v i o u r f o r month t w i t h b e h a v i o u r f o r t h e month i n t h e p r e v i o u s y e a r t-12 and so on f o r each o f t h e t w e l v e months.  The  s e r i e s can be w r i t t e n as a s t a t i o n a r y p r o c e s s by d i f f e r e n c i n g i t D t i m e s . The s e a s o n a l a u t o r e g r e s s i v e p r o c e s s o f l e v e l P i s r e p r e s e n t e d by t h e p o l y n o m i a l (3.7.1) and t h e s e a s o n a l moving average p r o c e s s o f l e v e l Q i s r e p r e s e n t e d by the p o l y n o m i a l ( 3 . 7 . 2 ) , where t h e s e a s o n a l l e n g t h i s d e n o t e d by s = 12 ( i n our example o f a m o n t h l y s e r i e s ) . (3.7.1)  <(B ) = 1 - ^  (3.7.2)  0(B  12  I t i s assumed  1 2  ) = 1 - ^ B  1  -  ¥  2  -  - ... -  2 l  ... ^  G^ 2  t h a t t h e p a r a m e t e r s $ and  ^  B  J  2  0 c o n t a i n e d i n t h i s m o n t h l y model  i s a p p r o x i m a t e l y the same f o r each month and t h a t t h e e r r o r s a ' s a r e t  random. in  The s e a s o n a l ARIMA can thus be w r i t t e n as (P,D,Q)s model  (3.9.3)  (3.7.3)  * (B ) V p  s  s  \  =  C (B )a Q  s  t  specified  - 28  M u l t i p l i c a t i v e model.  I n o u r s e a s o n a l s e r i e s d i s c u s s e d above, we  r e l a x t h e a s s u m p t i o n about t h e e r r o r s and now a l l o w - a , _ 2 » a  t  t o be c o r r e l a t e d , r e p r e s e n t e d by (3.8.1) where a  c|> (B)Va >  fc  i s white noise.  the l a t t e r by <J) (B) (3.8.2)  t - 2 ' *'*  (3.8.1)  .6(B) &  t  a  t  Substitute  (3.8.1) i n (3.7.3) and  premultiplying  ' y i e l d s (3.8.2) - t h e (p,d,q) X (P,D,Q)s model.  * (B) $  (B ) V V d  p  s  D s  z  = § (B)  t  G  Q  (B^ a  fc  This type o f s p e c i f i c a t i o n d i f f e r s from the t r a d i t i o n a l approach to t r e a t s e a s o n a l i t y as an a d d i t i v e component i n a time s e r i e s .  As N e l s o n  (1973) p o i n t s out s u c h methods as dummy v a r i a b l e s assume d e t e r m i n i s t i c s e a s o n a l i t y whereas i t i s more p l a u s i b l e t o c o n c e i v e o f t h e p a t t e r n i n t e n s i t y o f s e a s o n a l v a r i a t i o n s as u n d e r g o i n g change o v e r t i m e .  and  We  con-  s i d e r i n s t e a d a p a r t i c u l a r c l a s s of l i n e a r s t o c h a s t i c p r o c e s s e s t h a t  dis-  p l a y s e a s o n a l b e h a v i o u r as t h e b a s i s f o r models o f s e a s o n a l t i m e s e r i e s ( N e l s o n (1973) p. 1 6 9 ) . An example o f a (0,1,1) x ( 0 , 1 , 1 ) . ^ model i s p r e s e n t e d b e l o w f o r a monthly s e r i e s .  To l i n k t h e m o n t h l y z ' s one y e a r a p a r t we w r i t e t  and t o l i n k t h e c o r r e l a t e d a_'s one month a p a r t we w r i t e m u l t i p l i c a t i v e model i s p r e s e n t e d by (3.9.1)  V  (3.9.2)  ' Va  (3.9.3)  VV  1 2  z  = (1 -  t  = (1 -  1 2  z  t  = (1 +9  1 2  a  The  (3.9.3).  GB ) a  GB)  (3.9.2).  (3.9.1)  t  fc  GB)(1-.0B ) 1 2  0a _ t  1 3  a  fc  = a  fc  -• •/B a ^ -  8 . ^  29 The  moving average o p e r a t o r i s now o f o r d e r q + sQ = 13 and we  have t h i r t e e n a d j u s t a b l e and  c o e f f i c i e n t s ( i . e . t w e l v e monthly  one y e a r l y c o n t r i b u t i o n ) .  picked  contributions  We o b s e r v e t h a t t h e s e a s o n a l b e h a v i o u r i s  up by t h e w e i g h t e d e r r o r terms on t h e r i g h t hand s i d e o f e q u a t i o n  (3.9.3)  B.  I d e n t i f i c a t i o n o f a Model The  i n d i v i d u a l t i m e s e r i e s o f d e p o s i t s w i l l be i d e n t i f i e d as an ARIMA  (p,d,q) o r m u l t i p l i c a t i v e ARIMA (p,d,q) X (P,D,Q)s. i s b r o k e n up i n t o t h r e e s t a g e s :  The i d e n t i f i c a t i o n  ( i ) t o i d e n t i f y t h e degree o f d i f f e r e n c i n g  t o o b t a i n a s t a t i o n a r y s e r i e s e x p r e s s e d as a t r a n s f o r m  of the o r i g i n a l  s e r i e s z^_; ( i i ) t o i d e n t i f y t h e r e s u l t a n t s t a t i o n a r y s e r i e s as an ARMA p r o c e s s , and ( i i i ) t o i d e n t i f y t h e absence o r p r e s e n c e o f s e a s o n a l i t y i n t h e ARIMA. I f t h e t h e o r e t i c a l a u t o c o r r e l a t i o n f u n c t i o n d e f i n e d b e l o w i n (3.10) does n o t d i e o u t f a i r l y r a p i d l y f o r t h e raw d a t a , Vz^,  o r some h i g h e r  difference.  t h e n one may  consider  " I t i s assumed t h a t t h e degree o f d i f f e r -  e n c i n g d, n e c e s s a r y t o a c h i e v e s t a t i o n a r i t y , has been r e a c h e d when t h e a u t o c o r r e l a t i o n f u n c t i o n p, o f V^z K  dies out f a i r l y q u i c k l y .  d i s n o r m a l l y e i t h e r 0, 1 o r 2 The  autocorrelation f o rz  ( B o x - J e n k i n s ( 1 9 7 0 ) , p. 1 7 5 ) . = <J> z  at l a g k i s  1  £  (3.10)  In practise  t  X  p =  U.— iC  t  2  k  a z The  s t a t i o n a r y AR p r o c e s s o f o r d e r k (f> (B) z  = a , p r e m u l t i p l i e d by z  can be e x p r e s s e d as ( 3 . 1 1 . 1 ) , and t a k i n g e x p e c t a t i o n s  7  and d i v i d i n g t h r o u g h  30 2 by a'•. P  k  we g e t an e q u a t i o n (3.11.2) t h a t p r e s e n t s t h e k t h a u t o c o r r e l a t i o n sum o f 4' s and k-1 a u t o c o r r e l a t i o n s .  as a w e i g h t e d  = <!>  (3.11.1)  z_z  (3.11.2)  p =  t  k  t  + Vt-k t-2 Z  lVkVl  +  ... + Vt-k t-k" z  + <j, p _ + . . . + <f,  k  2  k  2  k  To e s t i m a t e t h e a u t o c o r f e l a t i o n f u n c t i o n , r e p l a c e t h e p's'by t h e e s t i m a t e d u a u t o c o r r e l a t i o n s / ^s."define'"d 'by (3.10); . We.: can =write=k> l i n e a r 'equations f of:  'the p^s- '"in .befms'~&f~ty. s'^and.l p ^s'.T.They-.are g i v e n b y t h e Y u l e - W a l k e r  equations  :<  (3.12) where t h e k t h e q u a t i o n i s s i m p l y (3.11.2).(Box p. 54-55) (3.12)  ' -•"'-'"."-  ;  :  p  l  *1  =  +  p = 2  P  k  ''  :  =  <f,  (1970),  ' \'«  *2 1 P  lPl  and J e n k i n s  +  +  Vk-1  + <f, + , . . + 2  * P _ k  k  2  V k - 1 ' * 2 k - 2 + ••• + * ' ' +  p  k  '  We have assumed t h a t t h e AR p r o c e s s i s o f o r d e r k and hence can be e x p r e s s e d i n terms o f k n o n - z e r o a u t o c o r r e l a t i o n p a r a m e t e r s .  To be s u r e o f t h e l e n g t h  o f t h e AR p o l y n o m i a l we examine t h e p a r t i a l a u t o c o r r e l a t i o n p a r a m e t e r s i n equations  (3.12) - t h e kth" <j> i n t h e k t h e q u a t i o n o f (3.12) i s c a l l e d  <J>. , , p a r t i a l a u t o c o r r e l a t i o n . ktc  These (}>..;. c o e f f i c i e n t s a r e found by s o l v i n g JJ 1  the Y u l e W a l k e r e q u a t i o n s f o r j = 1,2, e q u a t i o n o f an A R ( k + l ) be z e r o ( ^  -^  =  k + 1k +  k. Should t h e  "*- ^  0)_> then we c o n c l u d e  that p  a t c u t - o f f p o i n t a t k+1 and t h e p r o c e s s i s o f l e n g t h k.  nt  i e  k  ^"*" +  has  This i s our iden-  t i f i c a t i o n r u l e f o r an AR(p) p r o c e s s : i f t h e a u t o c o r r e l a t i o n f u n c t i o n t a i l s o f f and t h e p a r t i a l a u t o c o r r e l a t i o n s have a c u t - o f f p o i n t a f t e r l a g p then p i s t h e expected  order o f the autoregressive process.  F o r t h e MA p r o c e s s ^of o r d e r q z = 8 (B) a^, i t s v a r i a n c e and a u t o c o v a r i a n c e s a r e g i v e n b y (3.13.1) and (3.13.2) r e s p e c t i v e l y .  Given that  31  E(a a  ) ='0 kjO  k greater  i s o b v i o u s t h a t the a u t o c o v a r i a n c e y, i s z e r o f o r a l l  t h a n the l e n g t h o f the p o l y n o m i a l q .  (3.13.1)  Y  0  = E [(a  - O.a^  t  2  2  = c  (3.13.2)  Y  - 0  q = a (2  U s i n g our  ...+•• e  1 t  k  =  t  2  q  )  ^  q  t  q  t  k  -  ^ a ^ - . . .  t-k-q' 0  f u n c t i o n f o r the MA(q)  p  q  q - ... - e a _ ) ( a _  k  +  e 0 1  k  +  1  +  ... + e _ 0 ) q  definition of autocorrelation  (3.14)  e a _ )]  2  ( i +• e , +  a = E [(a  l  - ... -  k  k<  q  (3.10) we d e f i n e  q  the  autocorrelation  p r o c e s s b y (3.14)  - 0 + 0 0 +. ..+ 0 . 0 k . 1 k+1 . q-k q 1 + 0 2 +...+• 0 2 q  k = 1,2,. . .q  0  k >q  T h i s r e s u l t y i e l d s an o b v i o u s i d e n t i f i c a t i o n r u l e f o r the MA(q) if  the a u t o c o r r e l a t i o n has  a c u t - o f f p o i n t a f t e r l a g q and t h e  process for partial  a u t o c o r r e l a t i o n f u n c t i o n t a i l s o f f , then q i s t h e e x p e c t e d o r d e r o f the moving average p r o c e s s . A mixed p r o c e s s ARMA (p,q)  i s suggested i f b o t h the  f u n c t i o n and the p a r t i a l a u t o r e g r e s s i v e l a t i o n f u n c t i o n has off  according  autocorrelation  function t a i l o f f .  The a u t o c o r r e -  an i r r e g u l a r p a t t e r n a t l ^ g s 1 t h r o u g h q , t h e n i t t a i l s  to i t s functional values.  C o n v e r s e l y the p a r t i a l  l a t i o n f u n c t i o n i s dominated b y an i r r e g u l a r p a t t e r n  autocorre-  a t l a g s p-q and  on-  ward. The searching  p r e s e n c e o f any  s e a s o n a l i n f l u e n c e i n t h e d a t a can be o b s e r v e d b y  f o r peaks i n the a u t o c o r r e l a t i o n f u n c t i o n and i n the  partial  32  a u t o c o r r e l a t i o n f u n c t i o n t h a t appear a t r e g u l a r i n t e r v a l s . o f t h e s e a s o n a l i s s u g g e s t e d by t h e d e f i n i t i o n o f t h e d a t a  The l e n g t h (i.e.,  monthly,  q u a r t e r l y , e t c ) , and t h e l e v e l s o f (P,D,Q) w i l l be t h o s e i n w h i c h t h e s p i k e s a t t - s l a g d i e out q u i c k l y f o r both f u n c t i o n s .  C.  E s t i m a t i o n o f P a r a m e t e r s and D i a g n o s t i c  Checking  Once we have i d e n t i f i e d t h e s e r i e s as an ARIMA (p,d,q) X (P,D,Q)s, the n e x t s t e p i s t o e s t i m a t e t h e p a u t o r e g r e s s i v e p a r a m e t e r s , a u t o r e g r e s s i v e parameters,  q moving average p a r a m e t e r s and Q s e a s o n a l  moving average p a r a m e t e r s . minimize  P seasonal  The c r i t e r i o n f o r e f f i c i e n t e s t i m a t i o n i s t o  t h e s q u a r e d - ;dfference between t h e a c t u a l v a l u e o f z^ and i t s  e s t i m a t e d v a l u e z^.  We p r o c e e d  t o maximize t h e l i k e l i h o o d f u n c t i o n o f t h e  j o i n t n o r m a l d i s t r i b u t i o n o f p(V V z / <j>, <f>,9, Bp  ) i n the m u l t i p l i c a t i v e 2  -model.  Thus we. t r y o u t a l l c o m b i n a t i o n s  of. v e c t o r s (J) , $ ,8, 0 and a  s u c h t h a t t h e y maximize t h e l i k e l i h o o d o f t h e s e p a r a m e t e r s b y m i n i m i z i n g the sum o f squares i n ( 3 . 1 5 ) . (3.15) exp [ - ^ £ U / <f>, * , 6 , 0 , z ] ) 2  2  2  t  To b e g i n t h e e s t i m a t i o n p r o c e d u r e we must g i v e s t a r t - u p v a l u e s t o t h e p a r a m e t e r s i n t h e f o u r p o l y n o m i a l s and t o g e t t h e a l g o r i t h m s t a r t e d we must calculate the a  fc q  ' s w h i c h s p e c i f y t h e moving a v e r a g e p a r t o f t h e m o d e l .  The v a l u e s a r e e s t i m a t e d by u s i n g t h e g i v e n p a r a m e t e r s b y b a c k f o r e c a s t i n g on z  ( o r i t s d i f f e r e n c e d l e v e l ) ( s e e Box and J e n k i n s , 1970, p. 212-220).  Assume t h a t t h e model i s an ARMA (.0,0,1), i n i t i a l i z e t h e v a l u e o f 0 and express  t h e model i n terms o f t h e f o r w a r d s h i f t o p e r a t o r i?' where Fe^=e , fc  n  33 (3.16)  z  = (1-0B) a  t  and z  t  = (l^-0F)e  fc  Set E [ e _ ^ 0 ,z J. = 0 and s o l v e f o r Z T  (3.17)  t  E[e |G z ] T  E [ e  3  T-l  1  E[e J  =  t  '• ) t Q  Z  E [z ] T  ]  =  J )  [ Z  T-1  = E [ Z  0 z ]  o  E  +  t  q  by back  q  0E ]  I0I<1 forecasting.  t e ^ l S y z , . ] ••'•= z  +  G  ]  E  [e  T  1  + GE [  G  ^  Z  e  t  ±  ]  =  Z  \  Gz^  T  T-1  +  0  T  2  MO  T h i s l a s t e q u a t i o n i n (3.17) g i v e s us an i n i t i a l v a l u e f o r E [ Z ] =-0 ,:[e^( Q  GjZ ] from w h i c h we can s t a r t our f o r w a r d f o r e c a s t i n g f o r a's on  0 and z .  R e c a l l that  dependently of z  (3.18)  E  fc  Q  | G  z ]  =  E [Z ]  +  E [a  1  | 0:0z ]  =  E [Z^  + ' 0E  T  t  t  | 0  Our i n p u t s a r e now  i Z t  ]  =  q  E [z ] T  +.  GE  [  a;L  [a  | Q z^]  =E  [z ]  | ' 0, z ]  =E  [z^  y  GEt[a _ ( 0 , J T  1  Z {  =E  o  + 0 E [z  [z ]+0E[z T  o  T  complete and we can s t a r t t h e i t e r a t i o n s f o r t h e non-  l i n e a r e s t i m a t i o n o f the parameters function.  conditional  I G , z ] = 0 since i t i s distributed i n -  E [a  E [a  :  t h a t w i l l m i n i m i z e t h e sum o f s q u a r e s  The a l g o r i t h m used i n the computer program i s Marquandt's  iter-  a t i v e p r o c e d u r e w h i c h i s a compromise between t h e methods o f Gauss-Newton and s t e e p e s t d e s c e n t ( N e l s o n (1974), p. 8 ) . The e s t i m a t e d c o e f f i c i e n t s a r e t h e n examined f o r t h e i r  significance  l e v e l s and the model i s d i a g n o s e d w i t h r e s p e c t t o the e s t i m a t e d r e s i d u a l s . The former i s done by t e s t i n g the h y p o t h e s i s t h a t any p a r a m e t e r i s d i f f e r e n t from z e r o ( t - t e s t ) .  I n p a r t i c u l a r the h y p o t h e s i s that f = 1 i s a t e s t f o r  34 n o n - s t a t i o n a r i t y and s h o u l d we a c c e p t t h e n u l l h y p o t h e s i s we w o u l d t a k e f i r s t d i f f e r e n c e s o f t h e d a t a and t e s t i f cf) = 1 i n <j> ^  z t  _ » etc.  If  1  the c o n s t a n t term o f t h e raw d a t a i s s i g n i f i c a n t l y n o n - z e r o t h e n we a l s o c o n c l u d e . t h a t t h e r e i s a d r i f t i n t h e s e r i e s and t h a t i t s mean i s n o t independent of time.  The p r e s e n c e  o f s e a s o n a l i t y can be a s c r i b e d t o t h e  s i g n i f i c a n c e o f e i t h e r a u t o r e g r e s s i v e o r moving average c o e f f i c i e n t s o f o r d e r P and Q r e s p e c t i v e l y . The  e s t i m a t i o n programs a l s o y i e l d two model c h a r a c t e r i s t i c s .  The  f i r s t i s t h e h y p o t h e s i s t h a t t h e p o p u l a t i o n r e p r e s e n t e d b y t h e model and the p o p u l a t i o n i l l u s t r a t e d by t h e d a t a come from t h e same p o p u l a t i o n . non-parametric  t e s t i s t h e Kolmogorov-Smirnov S t a t i s t i c w h i c h g i v e s  This  con-  f i d e n c e bands f o r t h e d i f f e r e n c e s between 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 b o t h populations.  The second i s t h e h y p o t h e s i s t h a t we have r e d u c e d  to w h i t e n o i s e .  t h e model  T h i s i s a c h i - s q u a r e t e s t known as t h e B o x - P i e r c e  Statis-  t i c and i f t h e sum o f t h e f i r s t k sample a u t o c o r r e l a t i o n s o f t h e e r r o r s i s l e s s than the c h i - s q u a r e v a l u e w i t h (k-p-q-P-Q) degrees o f freedom, t h e n t h e r e s i d u a l s a r e s a i d t o be random.  D.  Forecasting The  ,  c o n c e r n f o r e f f i c i e n c y i n e s t i m a t i o n and s i g n i f i c a n c e o f c o e f f i -  c i e n t s stems from t h e d e s i r e t o have e f f i c i e n t f o r e c a s t s f o r t h e s e r i e s z ^ ^ where 1 i s some p e r i o d i n t o t h e f u t u r e . +  in  t h e same way as i n e s t i m a t i o n : m i n i m i z e  E f f i c i e n c y i s defined here  t h e mean s q u a r e e r r o r .  When  the model i s i d e n t i f i e d and p a r a m e t e r e s t i m a t e s a r e o b t a i n e d , t h e a l g o r i t h m once a g a i n g e n e r a t e s  t h e d i s t u r b a n c e terms (a^.'s) and we c a n e a s i l y f o r e c a s t  next p e r i o d ' s value f o r z ^  c o n d i t i o n a l on t h i s p e r i o d ' s a u t o r e g r e s s i v e  35 parameters  and moving average terms.  I t s h o u l d be n o t e d t h a t i f t h e  r a w - d a t a has been t r a n s f o r m e d t o n a t u r a l l o g a r i t h m s t h e n t h e f o r e c a s t s a r e l o g - n o r m a l l y d i s t r i b u t e d and the a n t i l o g s must be a d j u s t e d b a s e d on the l o g - n o r m a l d i s t r i b u t i o n ( N e l s o n ( 1 9 7 3 ) , p. 161-163).  E.  Transfer Function Throughout t h i s c h a p t e r we have shown t h a t a p a r t i c u l a r t i m e s  can be r e p r e s e n t e d by an ARIMA p r o c e s s b u t i t may some e x t e r n a l s h o c k s . nous v a r i a b l e X  fc  I f our s e r i e s z  In  z  t  =-u \ 1  a l s o be s e n s i t i v e t o  i s dependent on t h e c u r r e n t exoge-  fc  o r i t s p a s t v a l u e s X _ ^ ' s t h e n e q u a t i o n (3.19) i s t h e t  transfer function for z (3.19)  series  + o-  2  ( i . e . a structural equation). X_ t  1  + ... +  or  n  X _ t  n  + a  fc  =  a (B) X  fc  + a  fc  our c o n t e x t o f time s e r i e s f o r demand d e p o s i t s and t e r m d e p o s i t s , the  economic t h e o r y r e v i e w e d by C h a p t e r I I s u g g e s t l i k e l y  c a n d i d a t e s f o r the  X's.  independent  I n p a r t i c u l a r f o r term d e p o s i t s t h e most l i k e l y  i s t h e i n t e r e s t r a t e on t e r m d e p o s i t s .  The e m p i r i c a l p a r t o f t h i s  variable study  w i l l examine t h e s i g n i f i c a n c e o f t h i s v a r i a b l e and the e q u a t i o n s p e c i f i e d in  ( 3 . 1 9 ) , w i l l be compared w i t h t h e ARIMA models f o r i t s power t o p r e d i c t  f u t u r e v a l u e s of time d e p o s i t s .  The former i s measured by the t - t e s t  the l a t t e r i s measured by t h e mean square  error.  and  36 IV  D a t a and E m p i r i c a l R e s u l t s A.  Data  A c r e d i t u n i o n i s a f i n a n c i a l i n t e r m e d i a r y t h a t i s an autonomous ' entity.  I t h a s i t s own o p e r a t i n g c h a r t e r , i t s own Board o f D i r e c t o r s  and management, and i t s own b o r r o w i n g and l e n d i n g p o l i c i e s .  The common  bond.-; o f a s s o c i a t i o n may v a r y from employees i n one f i r m t o a community of a m i l l i o n people.  The most homogeneous c r e d i t u n i o n s were t o be c o n -  s i d e r e d i n o r d e r t o compare t h e e s t i m a t e d ARIMA models.  The c r e d i t  unions  t h a t had t h e l o n g e s t h i s t o r y of p r o v i d i n g b o t h demand d e p o s i t s and t e r m d e p o s i t s t o t h e i r members would p r o v i d e the l a r g e s t number o f o b s e r v a t i o n s to evaluate forecasts.  The i n t e r s e c t i o n o f t h e s e two c r i t e r i a  resulted  i n t h e c h o i c e o f t h r e e o f t h e l a r g e s t c r e d i t u n i o n s i n t h e Vancouver p o l i t a n R e g i o n i n B r i t i s h Columbia and t h e y had t h e f o l l o w i n g tics:  ( i ) t h e y f a c e t h e same e x t e r n a l market;  Metro-  characteris-  ( i i ) they have t h e same com-  mon bond o f community a s s o c i a t i o n ; ( i i i ) t h e y a r e m u l t i - b r a n c h o p e r a t i o n s ; and  ( i v ) t h e y each have o v e r t h i r t y - m i l l i o n d o l l a r s i n a s s e t s , and b o t h  demand d e p o s i t s and term d e p o s i t s a r e o f f e r e d t o t h e i r members. The two a c c o u n t s examined a r e t o t a l demand d e p o s i t s and t o t a l deposits.  term  The c r e d i t u n i o n s p r i m a r i l y d e a l w i t h n o n - c o r p o r a t e b o d i e s and  economic t h e o r y s u g g e s t s two d i f f e r e n t b e h a v i o u r s by consumers t o t h e two types of d e p o s i t s ;  ( i ) demand d e p o s i t s a r e p u r c h a s e d by i n d i v i d u a l s f o r  l i q u i d i t y and c o n v e n i e n c e i n c a r r y i n g o u t t r a n s a c t i o n s ; and ( i i ) term dep o s i t s a r e p u r c h a s e d as an i n v e s t m e n t - o f s a v i n g s i n l o w r i s k  securities.  The d a t a a r e g a t h e r e d q u a r t e r l y and date back t o t h e second q u a r t e r o f 1962.  From 1962 2to 1974:4 we have 52 o b s e r v a t i o n s f o r demand d e p o s i t s and ;  37  no l e s s t h a n 36 o b s e r v a t i o n s f o r term d e p o s i t s (some c r e d i t u n i o n s n o t s t a r t t o o f f e r t h e s e s e r v i c e s u n t i l 1966)^ examined b e l o w a r e r e f e r r e d t o as C.U.I, C.U.2  The  three c r e d i t  and C.U.3  did unions  and t h e y  are  o r d e r e d w i t h r e s p e c t to t h e i r a s s e t s i z e ( T h i s d a t a s e r i e s a r e l i s t e d i n Appendix:Data).  B.  Model f o r C h a r t e r e d Banks' D e p o s i t s To o b t a i n "a p r i o r i " i d e n t i f i c a t i o n o f t h e c r e d i t u n i o n s '  b u t more i m p o r t a n t be used, we  series,  t o e v a l u a t e the s t r e n g t h o f the t i m e s e r i e s method t o  f i r s t l o o k a t the b e h a v i o u r  o f demand d e p o s i t s and  t e r m depo-  s i t s f o r a s i m i l a r f i n a n c i a l i n t e r m e d i a r y f o r w h i c h we have a l a r g e r number of data p o i n t s .  The  t i m e s e r i e s are the t o t a l o f p e r s o n a l demand  de-  p o s i t s and o f p e r s o n a l term, d e p o s i t s h e l d i n c h a r t e r e d b a n k s i n Canada as p u b l i s h e d monthly i n the Bank o f Canada S t a t i s t i c a l Review (1967:9 11, 87 o b s e r v a t i o n s ) . The  .  Box and J e n k i n s t e c h n i q u e s u g g e s t s  demand d e p o s i t s .  The  1974:  a m o n t h l y s e a s o n a l model f o r  s i m p l e s t model whose c o e f f i c i e n t s p r o v e d t o be  sta-  t i s t i c a l l y s i g n i f i c a n t i s (0,1,0) x (0,1,1) 12 m u l t i p l i c a t i v e model where the d a t a has been t r a n s f o r m e d  to n a t u r a l logarithms.  T h i s model i n d i c a t e s  t h a t the s e r i e s i s s t a t i o n a r y a f t e r f i r s t d i f f e r e n c e s have b e e n t a k e n  and  t h a t t h e r e i s a s i g n i f i c a n t s e a s o n a l moving average term t h a t d e t e r m i n e s the l e v e l o f demand d e p o s i t s i n banks. seasonal troughs  Indeed t h i s a c c u r a t e l y d e p i c t s t h e  o f June and December„when the a b s o l u t e l e v e l s o f  these  d e p o s i t s d e c r e a s e w i t h r e s p e c t t o the b a l a n c e h e l d i n the p r e v i o u s month. The Kolmogrov-Smirnov S t a t i s t i c and t h e B o x - P i e r c e S t a t i s t i c b o t h  suggest  38  t h a t the  (4.1)  f i t t e d model a d e q u a t e l y r e p r e s e n t s  Vln-DD. - V l n DD  =  . The  .008 (.001)  Z  a i l  e  the r e s i d u a l s a r e w h i t e n o i s e V l n TD  = Z  The  .128  a  (.012)  Z  t r a n s f o r m e d s e r i e s of t e r m d e p o s i t s was  model ( d a t a t r a n s f o r m e d t o l°g )  (4.2)  +  the d a t a ( e q u a t i o n  .87Vln TD (.06) ~ Z  t r a n s f e r f u n c t i o n was  -  = 23  n-k  =  L l  i d e n t i f i e d t o be  (1,1,1)  .34 (.12)  4.2). a  t  ~  B&P  = 13  n-k  =22  1  a l s o t e s t e d f o r term d e p o s i t s  (TD) , where t h e  exogenous v a r i a b l e s were i n t e r e s t r a t e on 90 day bank term d e p o s i t s time (T).  B o t h c o e f f i c i e n t s had  statistically  (although  t h e r i g h t s i g n and were  vealed  TD  l a g g e d one  (R)  and  significant  the s t a n d a r d e r r o r s a r e u n d e r e s t i m a t e d due  s e r i a l a u t o c o r r e l a t i o n , i t i s u n l i k e l y t h a t they a r e not the e q u a t i o n w i t h R and  22  d the d i a g n o s t i c checks suggest t h a t  (Equation X  B&P  4.1)  to  high  significant).  For  p e r i o d , the m o n t h l y model a g a i n r e -  the e x p e c t e d s i g n s arid s i g n i f i c a n c e , however, the Durbin-Watson  S t a t i s t i c i n d i c a t e s a u t o c o r r e l a t i o n among the r e s i d u a l s  (4.3.1)  InTD = 7.05 (.07)  +  .07R (.01)  (4.3.2)  InTD =  +  .01R + (.002)  .20 (.04)  +  ..02T (.001) .97 l n T D _ (.005)  Thus our d i g r e s s i o n on banks has of the two  1  •; '  •  D.W.  = .07  R  2  =  .93  D.W.  = .54  R  2  =  .99  c o n f i r m e d our "a p r i o r i "  s e r i e s i n r e l a t i o n t o consumers' c h o i c e s  behaviours  and p r e f e r e n c e s ,  i t has  s u g g e s t e d an o r d e r o f d i f f e r e n c i n g f o r the model and  the s i g n i f i c a n c e o f  t r a n s f e r f u n c t i o n s , and  been shown t o f i t the  data.  the time s e r i e s a n a l y s i s has  39 C.  Demand f o r C r e d i t U n i o n s ' Demand D e p o s i t s Upon i n s p e c t i o n o f the q u a r t e r l y s e r i e s from 1962-1974 f o r each o f  the t h r e e c r e d i t u n i o n s i t was e v i d e n t t h a t t h e r e i s an e x p o n e n t i a l growth t r e n d i n the d a t a .  The b e s t model o f demand d e p o s i t s o f C.U. 1  and C.U.3 ( t r a n s f o r m e d i n t o n a t u r a l l o g a r i t h m s ) was t h e second  difference  f i r s t o r d e r moving average model.For C.U.2 t h e r e were two m o d e l s , n o t q u i t e comparable, whose c o e f f i c i e n t s were s i g n i f i c a n t l y d i f f e r e n t f r o m z e r o : ( i ) s e a s o n a l model (1,1,1) x ( 0 , 1 , 1 ) 4 ; and ( i i ) second d i f f e r e n c e model ( 0 , 2 , 1 ) . The  s t u d y o f banks' p e r s o n a l demand d e p o s i t s showed a s i g n i f i c a n t  seasonal  t r o u g h a t t h e end o f the s i x t h and t w e l f t h months ( i d e n t i c a l t o o u r second and f o u r t h q u a r t e r o b s e r v a t i o n s ) , and we e x p e c t a s i m i l a r s e a s o n a l p a t t e r n f o r demand d e p o s i t s o f c r e d i t u n i o n s .  However, i f one i s u s i n g q u a r t e r l y  d a t a as we a r e i t i s n o t o b v i o u s t h a t the o b s e r v a t i o n s s u g g e s t a s e a s o n a l pattern.  As f i g u r e I V . l s h o w s , t h e d o t t e d l i n e i s t h e e x p e c t e d m o n t h l y l e v e l  w i t h June and December b e i n g s i g n i f i c a n t l y l o w e r t h a n May and November ( r e s p e c t i v e month p r i o r ) .  Whereas the m o n t h l y d a t a s u g g e s t  t h a t D D ^ ^ D D ^ and  DD. <DD , they do n o t suggest t h a t DD ( 4 t h q u a r t e r ) < DD ( 3 r d q u a r t e r ) o J xz y C  0  o r t h a t DDg (2nd q u a r t e r < DD^ ( 1 s t q u a r t e r ) , s o t h a t o u r q u a r t e r l y  data  does n o t p i c k up t h e s e a s o n a l demand f o r c r e d i t u n i o n demand d e p o s i t s — e x c e p t i n t h e case o f C.U.2. The b e s t e q u a t i o n s and t h e i r parameters  a r e l i s t e d b e l o w i n T a b l e 4.1.  We are s a t i s f i e d t h a t the u n e x p l a i n e d v a r i a n c e i n t h e s e models i s w h i t e noise.  41  TABLE ARIMA MODELS FOR  DEMAND DEPOSITS OF CREDIT UNIONS  (0,2,1)  Vln DD  C.U.I  9^ = .81  C.U.2  Q%  .88  C.U. 3  6^=  .84  (l,l,l)x(0,1,1)4  :VlnDD  t  C.U.2.  D.  4.1  fc  t  - VlnDD _ t  1  = a^ + 1)  a ^  = .09  B&P  = 22  n-k  c = .07  B&P  = 17  n-k  =  23  = .09  B&P  = 11  n-k  =  23  CT  CT  - V l n D D _ = ^VlnDD _ +a .-9 ( a ^ ^ t  ^ = -.77 \ = .46 0 = -1.10  4  CT CT  t  1  0 a _  t  = .05 a = .14 . = .05  B&P  f c  = 26  n-k  =23  4  +  6 0  =  &_ t  5  21  Demand f o r C r e d i t U n i o n s ' Term D e p o s i t s The s e r i e s a r e t r a n s f o r m e d  to natural  l o g a r i t h m s and  the f i r s t  dif-  f e r e n c e f i r s t o r d e r a u t o r e g r e s s i v e scheme i s t h e b e s t model w i t h  the  s m a l l e s t number o f c o e f f i c i e n t s f o r C.U.I  signifi-  cantly  d i f f e r e n t from z e r o and o n e ) .  cant and the r e s i d u a l s  and C.U.2  For C.U.3  (The  (1,1,1) m o d e l i s s i g n i f i -  of t h i s model a r e l e s s c o r r e l a t e d  (1,1,0) model f o r the time s e r i e s .  4>'s a r e  than those i n  S i n c e the a u t o r e g r e s s i v e parameter i n  the former model i s .92 - s i g n i f i c a n t l y d i f f e r e n t f r o m z e r o b u t just within  the 95% c o n f i d e n c e i n t e r v a l ( o n e - t a i l t e s t ) - the s e c o n d  f e r e n c e model was  fitted.  The  (0,2,1) s p e c i f i c a t i o n was  the e v a l u a t i o n o f the t h r e e models f o r C.U.3 section Table  on p r e d i c t i o n . 4.2  lieing  The  significant  difand  i s postponed u n t i l the l a t e r  e s t i m a t e d ARIMA e q u a t i o n s a r e l i s t e d b e l o w i n  42  TABLE 4.2 •ARIMA MODELS FOR TERM DEPOSITS OF CREDIT UNIONS  (1,1,0) C.U.I  VlnTD  t  = cJ>VlnTD  t 1  . (J) = .62 A  + a o  — .06  B&P = 28  n-k = 23  C.U.2  •*  = .32  a  = .09  B&P = 16  n-k = 23  C.U.3  ?  = '.64  a  = .12  B&P = 23  n-k = 23  (1,1,D  VlnTD  C.U.3  $  = .92  c  = .04  B&P = 18  . n-k = 22  •%  = .70  a  = .13  ( .,2,1)  VlnTD  C.U.3  §  ;  The  t  t  =  cJiVlnTD  - ln'TD _  = .64  V  t  + a  1  = a  fc  + 0a  +  n  B a ^ .14  B&P = 18 .  t r a n s f e r f u n c t i o n f o r demand f o r c r e d i t u n i o n s '  n-k = 23  term d e p o s i t s  was a l s o f i t t e d . R e c a l l t h a t i n t h e example f o r c h a r t e r e d banks t h e explanat o r y v a r i a b l e s were t h e i n t e r e s t r a t e on 90 day term d e p o s i t s time ( T ) .  (R) and  The a p p r o p r i a t e i n t e r e s t r a t e f o r t h e time s e r i e s o f c r e d i t  u n i o n s i s t h e r a t e p a i d by c r e d i t u n i o n s o n a comparable s e c u r i t y .  How-  e v e r , as s t a t e d i n C h a p t e r I I t h e i n t e r e s t r a t e s p a i d by c r e d i t u n i o n s a r e not p u b l i s h e d on a q u a r t e r l y b a s i s hence t h e r a t e p a i d by c h a r t e r e d banks  43 i s used as a p r o x y . significance  O r d i n a r y l e a s t squares was used t o e s t i m a t e t h e  o f R and T.  F o r each o f t h e t h r e e c r e d i t u n i o n s  the former  v a r i a b l e was n o t s i g n i f i c a n t l y d i f f e r e n t from z e r o ( s e e . T a b l e 4 . 3 ) . A g a i n i t i s i n t e r e s t i n g t o s p e c u l a t e as t o why t h i s r e s u l t i s s o d i f f e r e n t from t h a t o f the c h a r t e r e d banks. change i n p r o m o t i o n  Either  t h e r e has been a s t r u c t u r a l  and p r e f e r e n c e o f term d e p o s i t s i n t h e 1962-1966 and  1967-1974 p e r i o d s o r t h a t i n t h e q u a r t e r l y too d i s c r e t e  t o be p o s i t i v e l y c o r r e l a t e d  d a t a t h e changes i n R may be  w i t h t h e new l e v e l o f t e r m d e -  posits.  TABLE 4.3  . . . ...  O.L.S. RESULTS FOR TRANSFER FUNCTION OF TERM DEPOSITS OF CREDIT UNIONS  lnTD  t  '= c  +  a R. x  C.U.I  12.49 (.16)  -.01 '(.04)  .12 (.004)  D.W.  = .29  R  .97  C.U.2  11.18 (.18)  -.02 • (.04)  .13 (.005)  D.W.  = .59  R  = .97  C.U.3  8.53 (.20)  -.03 (.03)  .18 (.005)  D.W  ,26  E.  z  2 R'  F o r e c a s t E v a l u a t i o n 1974:1-1974:4 The b e s t ARIMA model f o r t h e demand f o r demand d e p o s i t s o f c r e d i t  unions i s e v a l u a t e d a g a i n s t a n a i v e random w a l k model ( z = 0z + a ") t t-1 t/ v  .98  and a g a i n s t any o t h e r models t h a t p r o v e d t o be s i g n i f i c a n t i n . t h e e s t i m a t i o n stage. random w a l k .  T a b l e 4.4 The  p r e s e n t s the r e s u l t s and t h e ARIMA o u t p e r f o r m  (0,2,1) model i s the b e s t s p e c i f i c a t i o n  the  for forecasting  c r e d i t u n i o n s ' demand d e p o s i t s .  TABLE  4.4  PREDICTION ERRORS, DEMAND DEPOSITS"OF CREDIT UNIONS 1974:1 - 1974:4 PREDICTIONS  Average Absolute E r r o r  Root Mean Square E r r o r  C.U.I (0,2,1) (1,0,0)  0.29 0.33  0.30 0.39  (0,2,1) (l,l,l)x(0,1,1)4 (1,0,0)  0.07 0.09 0.12  0.09 0.11 0.20  (0,2,1) (1,0,0)  0.15 0.51  0.21 0.80  C.U.2  C.U.3  2 !' n o t e : t h e root-mean s q u a r e - e r r o r i s (Za In) , where a's a r e the e r r o r s , the summation i s o v e r a l l o b s e r v a t i o n s , and n i s the number o f observations 2  S i m i l a r l y the v a r i o u s models o f term d e p o s i t s s e r i e s a r e  evaluated  w i t h r e s p e c t t o the minimum mean s q u a r e e r r o r c r i t e r i a f o r t h e q u a r t e r l y f o r e c a s t s i n 1974. f u n c t i o n s and  I n a l l cases the ARIMA models o u t p e r f o r m e d the t r a n s f e r  the random w a l k e q u a t i o n  ( T a b l e 4.5)  illustrates  t h a t the  b e s t models f o r demand of c r e d i t u n i o n s ' TD a r e the (1,1,0) model f o r  45  C.U.I and C.U.2 and (1,1,1) model f o r C.U.3.  F i g u r e IV. 2 shows t h e p l o t  of t h e c a l c u l a t e d and a c t u a l v a l u e s o f demand d e p o s i t s and terra d e p o s i t s f o r the f o r e c a s t i n t e r v a l .  I n a l m o s t a l l t h e c a s e s our f o r e c a s t s were  too o p t i m i s t i c , o v e r s t a t i n g t h e a c t u a l b a l a n c e s o f d e p o s i t s h e l d b y C.U.I, C.U.2 o r C.U.3.  TABLE 4.5 PREDICTION ERRORS, TERM DEPOSITS OF CREDIT UNIONS 1974:1 - 1974:4 PREDICTIONS  Average Absolute Error  Root Mean Square E r r o r  C.U.I (1,1,0) (1,0,0) f(R,T)  0.23 0.32 0.26  0.23 0.35 0.27  (1.1.0) (1,0,0) f(R,T)  0.09 0.38 0.61  0.11 0.44 0.65  (1.1.1) (1,1,0) (0,2,1) (1,0,0) f(R,T)  0.11 0.12 0.28 0.37 0.67  0.11 0.18 0 30 0.41 0 69  C.U.2  C.U.3  •  n o t e : s e e note i n t a b l e 4.4; f ( R , T ) i s t h e t r a n s f e r f u n c t i o n where R i s t h e i n t e r e s t r a t e and T i s t i m e as d e f i n e d above  IPililit^  47 V.  C o n c l u d i n g Remarks  We have found t h a t t h e ARIMA models f u r n i s h t h e b e s t f o r e c a s t s f o r demand d e p o s i t s and t e r m d e p o s i t s o f c r e d i t u n i o n s d e p o s i t s and term d e p o s i t s o f c h a r t e r e d b a n k s .  and f o r p e r s o n a l demand  F o r demand d e p o s i t s , t h e  (0,2,1) model b e s t d e s c r i b e s t h e demand f o r c r e d i t u n i o n s ' d e p o s i t s and t h e (0,1,0) x (0,1,1) 12 p r o c e s s  i s t h e one we i d e n t i f i e d f o r c h a r t e r e d b a n k s .  F o r t e r m d e p o s i t s , CJJ 1 and CIL 2 d a t a f o l l o w a (1,1,0) p r o c e s s w h i l e t h e t  (1,1,1) model i s t h e b e s t f o r m u l a t i o n f o r CU. 3 and c h a r t e r e d b a n k s .  In a l l  the cases we a r e s a t i s f i e d t h a t t h e u n e x p l a i n e d v a r i a t i o n i n t h e s e r i e s o f demand d e p o s i t s o r term d e p o s i t s i s w h i t e n o i s e . The p r e d i c t i o n s f o r 1974 p r o v e d t o be t o o o p t i m i s t i c . t h a t c o n s t a n t f e e d b a c k must be m a i n t a i n e d and t o m o n i t o r  suggests  i n order t o update the f o r e c a s t s  the t u r n i n g p o i n t s i n the s e r i e s .  v a l u e s o f t h e p a r a m e t e r s may change as t h e s e data p o i n t s .  This  I t i s l i k e l y t h a t the  ARIMA models a r e f i t t e d t o new  F u t u r e r e s e a r c h s h o u l d t r y t o use m o n t h l y d a t a b e c a u s e t h e r e  i s a l i k e l y s e a s o n a l p a t t e r n t h a t i s n o t b e i n g p i c k e d up b y t h e q u a r t e r l y data.  T h i s w i l l a l s o g i v e more o b s e r v a t i o n s t o t h e t i m e s e r i e s and s t r e n g t h -  en t h e model i d e n t i f i c a t i o n and e s t i m a t i o n . There i s a n o t h e r p r o b l e m w i t h h a v i n g used t h e q u a r t e r l y s e r i e s f o r t h e y e a r s between 1962 and 1974.  T h i s p e r i o d i s b y no means a homogeneous one  f o r f i n a n c i a l i n t e r m e d i a t i o n i n Canada o r f o r c r e d i t u n i o n s i n B r i t i s h Columbia.  The market s t r u c t u r e was q u i t e d i f f e r e n t p r i o r t o 1967 a t w h i c h  time  the Bank A c t was changed and c h a r t e r e d banks i n c r e a s e d t h e i r a c t i v i t i e s i n the consumer m a r k e t .  There has been a marked s h i f t i n t h e growth r a t e o f  c r e d i t u n i o n s ' a s s e t s i n B r i t i s h Columbia s i n c e 1970 and p e r h a p s t h e u n d e r l y i n g p a t t e r n o f t h e d a t a i s n o t t h e same as f o r t h e 1962-69 p e r i o d .  If  48 t h i s s h i f t s h o u l d be s i g n i f i c a n t t h e n our 1962-74 ARIMA models may  have  i n t r o d u c e d an a r c h a i c p a t t e r n i n t o the model and i n t o t h e f o r e c a s t s .  As  the number o f o b s e r v a t i o n s w i l l i n c r e a s e w i t h time i t w i l l be p o s s i b l e t o t e s t the homogeneity o f t h e time s e r i e s f o r c r e d i t u n i o n  deposits.  Thus o u r t h e s i s has s u c c e s s f u l l y m o d e l l e d t h e time s e r i e s f o r demand d e p o s i t s and t e r m d e p o s i t s o f a c r e d i t u n i o n .  The f i n a n c i a l manager i n a  c r e d i t u n i o n can g e n e r a t e t h e f o r e c a s t s f o r d e p o s i t s u s i n g o u r ARIMA model and w i t h f o r e c a s t s o f i n t e r e s t r a t e s and o f l o a n demand he can implement them i n an o p t i m i z a t i o n  technique.  49  BIBLIOGRAPHY  B a t r a , H. (1973) . "Dynamic I n t e r d e p e n d e n c e i n Demand f o r S a v i n g s D e p o s i t s " , Journal of Finance, May, 1973, v o l X X V I I I , No. 2, p. 507-514. Box,  G. and G. J e n k i n s  control,  Time Series  (1970).  Analysis:  forecasting  and  H o l d e n Day, San F r a n c i s c o , 1971.  Boyd, J . (1973).  "Some Recent Developments i n t h e S a v i n g s and Loan  Deposit Markets",  Journal  of Money,  Credit  and Banking,  August,  1973, V o l . v , No. 3, p . 733-750. Cohan, S.  (1973).  "The D e t e r m i n a n t s o f S u p p l y and Demand f o r C e r t i -  f i c a t e s o f D e p o s i t " , Journal  of Money,  Credit  and  Banking,  F e b r u a r y , 1973, V o l . v , No. 1, p. 100-112. Cramer, R. and R. M i l l e r (1973). "Development o f a D e p o s i t F o r e c a s t i n g P r o c e d u r e f o r Use i n Bank F i n a n c i a l Management", Journal of Bank Research, Summer, 1973, p. 122-138. DeLeeuw, F.  (1965).  Qv.aj?terly  "A Model o f F i n a n c i a l B e h a v i o r " , i n Brookings  Econometric  Model of  the  United  States,  e d s . J.B.  Duesenberry e t a l . , Rand-McMally, C h i c a g o , 1965, p. 465-532. DeLeeuw, F.  (1969).  The Brookings  "A Condensed Model o f F i n a n c i a l B e h a v i o r , " i n  Model.  Some Further  Results,  eds. J.B. Duesenberry  e t a l . , R a n d - M c N a l l y , C h i c a g o , 1969, p. 270-316 Dhrymes, P. and P. Taubman Savings  (1969).  "An E m p i r i c a l A n a l y s i s o f t h e  and Loan I n d u s t r y " i n Study  of  the  Savings  and Loan  Indus-  try, e d . I . F r i e n d , F e d e r a l Home Loan Bank B o a r d , W a s h i n g t o n , D.C., 1969, p . 67-182. F e i g e , E.  Section  (1964).  Analysis,  The Demand for  Liquid  Assets:  A Temporal  Cross-  P r e n t i c e - H a l l , New J e r s e y , 1964.  F e i g e , E. ( 1 9 7 4 ) , " A l t e r n a t i v e Temporal C r o s s - S e c t i o n S p e c i f i c a t i o n s o f t h e Demand f o r Demand D e p o s i t s " , Journal of Finance, June, 1974, v o l . XXIX, n o . 3, p . 923-940. Friedman, M. (1957). A Theory of the Consumption Function, Bureau o f Economic R e s e a r c h , P r i n c e t o n , 1957. M o t l e y , B.  (1970).  Adjustments",  Vol. N e l s o n , C.  National  "Household Demand f o r A s s e t s : A Model o f Short-Run  Review of  Economics^ and Statistics,  A u g u s t , 1970,  X I I , No. 3, p. 236-241.  casting,  (1973).  Applied  Time Series  Analysis  Holden-Day, San F r a n c i s c o , 1973.  for  Managerial  Fore-  DEMAND DEPOSITS C .U.I 290200. 283728. 288410. 277601. 579923. 328629. 397636. 483302. 859983. 670283. 664819. 700115. 1315540. 1145867. 1286335. 1168532. 1623500. 1570730. 1361218. 1561367. 3069123. 1976460. 2667036. 2467194. 3451364. 3778215. 4433987. 4638152. 5338262. 6508993. 6116369. 6064195. 6573045. 7154165. 8641735. 9423604. 13240078. 16313127. 17623152. 19440843. 21584896. 24559728. 27228336. 32179760. 38420704. 37876336. 32794640. 35345552. 32716080. 30331968. 29406736. DEMAND DEPOSITS LN C .U.I 12.556 12.572 12.534 12.578 13.271 12.703 12.893 13.088 13.665 13.415 13.407 13.459 14.090 13.952 14.067 13.971 14.300 14.267 14.124 14.261 14.937 14.497 14.796 14.719 15.350 15.054 15.145 15.305 15.618 15.490 15.689 15.626 16.059 15.698 15.783 15.972 16.7 83 16.399 16.607 16.685 17.287 16.887 17.017 17.120 17.381 17.464 17.450 17.306 17.303 17.228 17.197 TERM DEPOSITS C .U.I 390500. 1. 85500. 231000. 800000. 507000. 616500. 695000. 863500. 949723. 1130535. 1278997. 2412211. 1591329. 1727357. 2198997. 2568230. 3113829. 3496095. 39469 60 . 4101678. 4466207. 5006250. 5546293. 5940851. 6321382. 6540961. 7158364. 7795252. 8087713. 8395023. 8454867. 9291387. 10108465. 13068507. 15679535. 19320080. 21975056. 29106528. 32807952. 35523872. 40226720. 49773856. 58248880. 66961040. 76191184.104511584. 107059568. 121612864.120154240.131267616. TERM DEPOS ITS LN C . U . I 0.0 11. 356 12.350 12.875 13.13 6 13. 332 13.452 13.592 13.669 13. 764 13.938 14.062 14.696 14.280 14. 362 14.604 15.188 15.067 14.759 14. 951 15.426 15.529 15 .227 15. 312 15.784 15.694 15 .597 15. 659 15.95 0 15 .869 15.943 15. 906 16. 129 16.386 16.568 16.045 17.306 16.777 17.186 16.905 17.386 17.880 17. 510 17.723 18.465 18.489 18 .020 18. 149 18.616 604 18 . 18.693  DEMAND DEPOSITS C .U.2 42566. 41730. 49070. 48420. 81171. 94573. •112242. 125489. 113742. 125952. 147022. 167439. 210250. 271566. 263801. 444177. 493942 . 378563. 626290. 632440. 782152. 882650. 1054217. 1272136. 1465475. 1468787. 1783469. 1930989. 2200238 . 2257850. 2234457. 2313589. 2615 548. 2710551. 2880538. 3182165. 4333078. 4990218. 6097890. 6852933. 7941170 . 9464295. 9860277. 11098798. 14047351 . 16122663. 15874823. 16234764. 23868592. 22500016. 25584384. DEMAND DEPOSITS LN C .U.2 10 .659 10.639 10.801 10 .788 11.304 11.457 11.628 11.740 11 .642 11.744 11.898 12.028 12.256 12.512 12 .483 13.004 13.110 12.844 13.348 13.357 13.570 13.691 13.868 14.056 14.198 14.200 14.394 14.474 14.604 14. 630 14.620 14.654 14.777 14.813 14.873 14.973 15 .282 15.423 15.623 15.740 15.888 16.063 16.104 16.222 16.45 8 16.596 16.580 16.6 03 16.988 16.929 17.057 TERM DEPOSITS C,.U.2 1 . 1. 1. 1 . 1. 1. 34800. 135900. 204200 . 250900. 316375. 375375. 411908. 447175. 501977. 361300. 449200. 846244. 729636. 834899. 991779. 1214559. 1395449. 1493096. 1796670. 2141400. 2566803. 2808933. 3575329. 3928884. 4180953. 4638880. 5071935. 5619708. 6515980. 7376612. 8505891. 9221807. 9 992 905. 10870671. 11891481. 12879057. 14714702. 16288378. 17563008. 18394128. 23261632. 28583872. 22601120. 23471504. 26045296. TERM DEPOSITS LN C.U.2 0.0 0.0 0.0 0.0 0 .0 0.0 10.457 11.820 12.227 12.433 12.665 12.836 12.929 13.011 13.126 12.797 13.015 13.649 13.500 13.635 13.807 14.010 14.149 14.216 14.401 14.577 14.758 14.848 15.090 15.184 15.246 15.350 15 .43 9 15.542 15 .690 15.814 15.956 16.037 16.117 16.2 0-2 16 .291 16.371 16.504 16.606 16.681 16.728 16.962 17.168 16.934 16.971 17.075  DEMAND DEPOSITS C.U.3 30211. 29922. 35076. 36119. 47415. 40146. 58095. 54531. 54238. 50398. 61514. 67321. . 70655. 102755. 85070. 83590. 104508. 129995 . 145659. 193789. 247533 . 286000 . 300776. 373105. 470602. 486069. 719561. 588993. 890417. 948713. 969893. 1210128. 1489363. 1501195. 2392084. 2063821. 3401836. 3854162. 4818345. 5859625. 7983876. 8676195. 10 504561. 12132317. 15228363 . 15218569. 13645977. 15597718. 17014160. 16019679. 15431547. DEMAND DEPOSITS LN C.U.3 10.316 10.306 10.465 10.495 10.767 10.600 10.907 10.970 10.901 10.828 11.02 7 11.117 11 .166 11.351 11.540 11.334 11 .557 11.775 11.889 12 .175 12-419 12.564 12.614 12.830 13.094 13.286 13.486 13 .062 13 .699 14.006 13.763 13.785 14.214 14.222 14.5 40 14.688 15.040 15. 165 15.388 15.584 16.167 15-893 15.976 16.311 16.539 16.538 16.429 16.563 16.650 16.589 16.552 TERM DEPOSITS C.U.3 1 . 1. 1. 1 . 1. 1. 1. 1. 1 . 1. 1. 1. ' 1 . 1. 1. 1. 59000. 97000. 113500. 132500. 176500. 186000. 205500. 363500. 419500. 569700. 858700. 1013400. 1470231. 1607531. 1696131. 1835406. 2150876. 2425786. 29329263343537. 4428 624. 5866774. 7293652. 8168730. 8880908. 9554125. 11617550. 12667943. 14149311. 17151568. 22396832. 23459472. 26597312. 29519456. 30429056. TERM DEPOSITS LN C.U.3 0.0 0.0 0.0 0.0 0.0 0.0 0.0. 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 11.640 11.794 10.985 11.482 12 -081 12.804 12.134 12.233 12.947 13.829 13.663 13.253 14.201 14.290 14.344 14.423 14.581 14.892 14.702 15.023 15.304 15.585 15 .803 15.916 15.999 16.355 16.072 16.268 16 .465 16.971 16.658 16.924 17.096 17.231 17.201  DEMAND DEPOSITS -X10**6*CANADIAN BANKS 1040 . 1083. 1174. 1261. 1326. 1400. 1506 . 1640. 1853. 2099. 2293. 2408. 2450. 2487. 2502. 2539. 2634. 2772. 2875 . 2950. 3048 . 3140. 3243. 3389. 3508. 3570. 3579. 3594. 3636. 3711. 3781. 3873. 4005. 4104. 4202. 4306. 4391 . 4428. 4465. 4481. 4551. 4648. 4706 . 4602. 4442. 4328. 4235. 4198. 4182. 4207. 4150. 4127. 4234. 4324. 4416 . 4493. 4595. 4697. 4788. 4922. 5058. 5130. 5114. 5191. 5349. 5544. 5675. 5789. 5989. 6 27 3. 6537. 6796. 7034. 7384. 8117. 8579. 8987. 9457. 9785 . 10000. 10504. 1 1170. 11751. 12360. 12739. 13038. 12490. DEMAND DEPOSITS.LN X10**6 CANADIAN BANKS 6.947 6.987 7.068 7.140 7.190 7.244 7.317 7.402 7.525 7.649 7.738 7.787 7.804 7.819 7.825 7.840 7. 876 7.927 7.964 7.990 8.022 8.052 8. 084 8.128 8.163 8.180 8.183 8.187 8. 199 8.219 8.238 8.262 8.295 8. 320 8.343 8.368 8.387 8 .396 8.404 8.408 8.423 8.444 8.457 8 .434 8.399 8.373 8. 351 8.342 8.339 8.345 8.331 8.325 8.351 8.372 8.393 8 .410 8 .433 8.455 8.474 8.501 8.529 8 .543 8.540 8.555 8.585 8.620 8.644 8.664 8.698 8.744 8.785 8.824 8.859 8.907 9.002 9.057 9. 104 9. 155 9.189 9 .210 9.260 9.321 9.372 9.422 9.452 9.476 9.433  TERM DEPOSITS 10443 . 10587. 10865 . 11136. 1142611543 . 12002 . 12367. 13879 . 14161 . 14791. 15016. 15829 . 16204. 17343 .  TERM  9.254 9.267 9.293 9.318 9.344 9.354 9.393 9.423 9.53 8 9.558 9.602 9.617 9.670 9.693 9.761  X10**6 CANADIAN 10535. 10532. 10367. 10461. 10694. 10767. 10702. 10768. 11031. 11038. 10979. 11021. 11281. 11302. 11296. 11357. 11516. 11473. 1 1297. 11355. 11696. 11871. 11742. 11716. 12125. 12101. 11987 . 12106. 12782. 12945. 13156. 13418. 14075. 13655. 13406. 13625. 14301. 14408. 14379. 14531. 14962. 14797. 14540. 14781. 15214. 15323. 15381. 15589. 15966 . 15834. 15699. 15867. 16601. 16940. 16860. 17043. 17625. 17562. DEPOSITS LN X10**6 CANADIAN 9. 246 9.255 9.262 9.262 9.278 9. 284 9.277 9.284 9.308 9.309 9.304 9.308 9. 338 9.332 9.331 9.333 9. 337 9.348 9.332 9.351 9.371 9. 369 9.367 9.382 9.401 9.401 9.392 9.403 9. 504 9.468 9.485 9.456 9.520 9 .552 9.522 9.503 9.584 9.568 9.576 9.574 9.602 9. 585 9.601 9.613 9.654 9.641 9.637 9.630 9.672 9.670 9.661 9.678 9.717 9.737 9. 733 9. 743 9.777 9.773  BANKS 10539. 10200. 11077. 11394. 11463. 11888. 12239. 13654. 13834. 14625. 14932. 15742. 16029. 17167.  BANKS 9.263 9. 230 9.313 9.341 9.347 9.3 83 9.412 9.522 9.535 9.590 9.611 9.664 9.682 9.751  

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