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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 Columbia, 1973 i . A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF BUSINESS ADMINISTRATION i n the F a c u l t y o f Commerce and Business A d m i n i s t r a t i o n We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1975 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced d e g r e e at the 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 , I a g r e e that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r 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 the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d tha t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . The U n i v e r s i t y o f B r i t i s h Co lumbia V ancouver 8, Canada Date i i Abstract The purpose of t h i s t h e s i s i s to develop a f o r e c a s t i n g model to p r e d i c t demand d e p o s i t s and term de p o s i t s of c r e d i t unions. I t begins w i t h a survey of the 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 equation models and simultaneous equation systems are summarized. The hypo-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 of c r e d i t union f i n a n c i a l behaviour i s a l s o presented. However, a s t r u c t u r a l model cannot be estimated because there are no pu b l i s h e d data on c r e d i t unions' i n t e r e s t rates and there i s a l i m i t e d number of observations f o r the dependent v a r i a b l e . The f o r e c a s t i n g technique that i s being developed i n t h i s t h e s i s i s an a p p l i c a t i o n of time s e r i e s a n a l y s i s . The b a s i c i d e a behind t h i s approach i s to express the time s e r i e s of demand deposits and of term d e p o s i t s as a weighted sum of the past values of d e p o s i t s . The weights i n the sum are de-termined so as to achieve the g r e a t e s t p r e d i c t i v e power by mi n i m i z i n g the mean square e r r o r of the f o r e c a s t s . The data are q u a r t e r l y time s e r i e s f o r demand depo s i t s and f o r term de p o s i t s f o r each of three c r e d i t unions i n the Van-couver Region i n B r i t i s h Columbia from the second q u a r t e r of 1962 to the f o u r t h q u a r t e r of 1974. The data are p r i n t e d i n the Appendix. The s t r e n g t h of the mixed a u t o r e g r e s s i v e moving average process (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 union i s evaluated by u s i n g a l a r g e sample of monthly data of 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 of Canadian chartered banks. The best models f o r each c r e d i t union's demand deposits and term de p o s i t s are matched ag a i n s t the naive model of a random walk process. They are compared w i t h respect to t h e i r minimum mean square e r r o r of p r e d i c t i o n f o r the fou r q u a r t e r s of 1974. i i i For both the three credit unions and the chartered banks, in a l l cases the best ARIMA model outperformed a l l other candidates. i v Table of Contents Page I . I n t r o d u c t i o n ...... 1 I I . Survey o f the L i t e r a t u r e ...... 6 A. S i n g l e Equation Models 6 B. Simultaneous Equation Systems ' ...... 12 C. S t r u c t u r a l Model 19 ( i ) Consumer Behaviour ...... 19 ( i i ) F i n a n c i a l Behaviour of a C r e d i t Union 21 I I I . T h e o r e t i c a l Development of Time S e r i e s A n a l y s i s 24 A. General Class of Models 24 B. I d e n t i f i c a t i o n of a Model 29 C. E s t i m a t i o n of Parameters and D i a g n o s t i c Checking 32 D. F o r e c a s t i n g 34 E. T r a n s f e r Function 35 IV. Data and E m p i r i c a l R e s u l t s 36 A. Data ...... 36 B. Model f o r Chartered Banks' Deposits 37 C. Demand f o r C r e d i t Unions' Demand Deposits ...... 39 D. Demand f o r C r e d i t Unions' Term Deposits 41 E. Forecast E v a l u a t i o n 1974:1-1974:4 43 V. Concluding Remarks 47 B i b l i o g r a p h y 49 Appendix: Data 50 V i t a ...... 55 V L i s t of Tables Page Table 4.1 ARIMA Models f o r Demand Deposits of C r e d i t Unions 41 4.2 ARIMA Models f o r Term Deposits of C r e d i t Unions 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 of Term Deposits of C r e d i t Unions 43 4.4 P r e d i c t i o n E r r o r s , Demand Deposits of C r e d i t Unions 1974:1-1974:4 P r e d i c t i o n s 44 4.5 P r e d i c t i o n E r r o r s , Term Deposits 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 i VI L i s t of Figur e s T.- • Page F i g u r e — e — 1.1 Dynamic F i n a n c i a l Management Process 3 I I I . l Three Stages of Time S e r i e s A n a l y s i s 25 IV. 1 Expected Monthly L e v e l s of Demand Deposits 40 IV.2 P l o t of A c t u a l and P r e d i c t e d Values f o r C r e d i t Unions' Demand Deposits and Term Deposits 1974:1-1974:4 46 1 I . I n t r o d u c t i o n F i n a n c i a l i n t e r m e d i a r i e s are f i r m s t h a t are p r i m a r i l y engaged i n bor-rowing funds (savings) from households and businesses and i n l e n d i n g funds (loans) to other households and businesses. I n the process of 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 withdrawals of savings and the r i s k of d e f a u l t on l o a n s . The o l d approach to t h i s l i q u i d i t y problem was to balance the expected 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 a s s e t s . The percentages of t o t a l funds h e l d i n short-term l o a n s , consumer loans and mortgages would then be s i m i l a r to the percentages of l i a b i l i t i e s i n savings 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 takes advantage of d i v e r s i f i c a t i o n i n the savings p o r t f o l i o . To b e n e f i t from both 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 process s i m i l a r to the one i l l u s t r a t e d i n F i g u r e 1:1 (Cramer and M i l l e r (1973)). Using 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 r a t e s , demand f o r l o a n s , and demand f o r savings d e p o s i t s and share c a p i t a l , the de-c i s i o n maker would apply a n p p t i m i z a t i o n technique to decide on the best mix of loans to i s s u e and on the l e a s t cost combination of savings to a t t r a c t . The d e c i s i o n s to commit funds today f o r one, f i v e , or ten years hence are based on f o r e c a s t s of 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 of these three 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 study of time s e r i e s , models and techniques. I n t h i s t h e s i s we w i l l focus our a t t e n t i o n on the development of a f o r e c a s t i n g t e c h -nique to 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 p r o v i d e s f i n a n c i a l s e r v i c e s s i m i l a r to those of other 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 ch a r -t e r e d banks and t r u s t companies. I t operates as an autonomous u n i t t h a t has few branch operations and deals only w i t h i t s members. The borrowers and the lenders are the shareholders and owners of the a s s e t s . The c r e d i t union faces the same l i q u i d i t y problem of 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 of the c r e d i t union's resources i s made through the same dynamic f i n a n c i a l management process ( 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 union and other f i n a n c i a l i n s t i t u t i o n s i s the former's cooperative p h i l o s o p h y , they a l l have the f o r e c a s t i n g p r o -blem of e s t i m a t i n g f u t u r e l e v e l s of both demand and term d e p o s i t s . The t r a d i t i o n a l approach to the development of a f o r e c a s t i n g model f o r de p o s i t s i s to use economic theory of demand f o r l i q u i d a s s e t s i n order to formulate t h e i r demand equations. This i s c a l l e d a s t r u c t u r a l equation be-cause i t uses predetermined v a r i a b l e s r e p r e s e n t i n g p r i c e o f d e p o s i t s , i n -comes of consumers, t a s t e s and preferences of consumers, and p r i c e s of sub-s t i t u t a b l e l i q u i d a s s e t s . Whereas t h i s demand equation models the behaviour o f households, the s i n g l e equation approach does not h o l d f o r f i n a n c i a l i n -s t i t u t i o n s . The l a t t e r has c o n t r o l over the p r i c e of d e p o s i t s causing the s t r u c t u r a l e quation to have a c u r r e n t endogenous v a r i a b l e and the demand f o r depo s i t s can no longer be estimated by a s i n g l e equation. The s t r u c t u r a l model must be expanded i n t o two or more equations i n o r -der to capture the two-step d e c i s i o n making process of a f i n a n c i a l interme-d i a r y (a c r e d i t u n i o n ) . In the f i r s t s t e p , the c r e d i t union s e t s 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 . In the second s t e p , there i s a s t o c h a s t i c 3 Figure "J lJL . Dynamic Financial Management Process Past Time Series of Input Data Modelling of Data Forecasting 'Feedback" Application of -|>{ Optimization Techniques I Monitoring I New Time Series Input Data "Feedback" Other Factors Decision Making Statistical Analysis "Feedback" Financial Management movement i n the l e v e l of d e p o s i t s i n response to the new i n t e r e s t r a t e s . This adjustment i n the l i a b i l i t i e s w i l l create a feedback to the d e c i s i o n s taken i n the f i r s t stage. The i n t e r e s t r a t e s may have to change again de-pending on competitive market c o n d i t i o n s , consumer p r e f e r e n c e s , or because an unfavourable p o r t f o l i o s t r u c t u r e warrants i t ( i . e . unfavourable l i q u i d i t y p o s i t i o n whereby i n t e r e s t payments are r i s i n g f a s t e r than i n t e r e s t revenues). In other words the i n t e r r e l a t i o n s among the i n t e r e s t r a t e s of the c r e d i t union and the l e v e l of d e p o s i t s must be expressed as a system of simultaneous equations. The s t r e n g t h of e i t h e r the s i n g l e equation model or the simultaneous equation system can only be evaluated e m p i r i c a l l y . To t e s t the hypothesised models, one must have a s u f f i c i e n t number of o b s e r v a t i o n s and time s e r i e s f o r a l l v a r i a b l e s . However, the only data a v a i l a b l e at the time of t h i s study are 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 to 1974. There are no pub-l i s h e d q u a r t e r l y data on c r e d i t unions' i n t e r e s t r a t e s and there are no prox-i e s 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 unions' demand and term d e p o s i t s f o r the 1962-1966 p e r i o d . Therefore 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 of m i s s i n g data and l i m i t e d number of degrees of freedom. Another approach i s to use time s e r i e s a n a l y s i s to formulate a f o r e c a s t -i n g model f o r c r e d i t unions' d e p o s i t s . The method i s data 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 through 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 time s e r i e s of d e p o s i t s . We assume that there 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 the s e r i e s of demand d e p o s i t s and term d e p o s i t s . This p a t t e r n i s expressed as a weighted sum of past values of these v a r i a b l e s where the weights i n the sum are determined so as to achieve the g r e a t e s t p r e d i c t i v e power ( i . e . minimize 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 three stages: (1) i d e n t i f y the 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 process, a moving average process or 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 ) estimate the parameters 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 data are 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 of three c r e d i t unions i n the Vancouver Region of B r i t i s h Columbia from the second q u a r t e r of 1962 to the f o u r t h q u a r t e r of 1974. The best models f o r each c r e d i t union's demand d e p o s i t s and term d e p o s i t s are matched a g a i n s t the n a i v e model of a random walk process. They are a l l compared w i t h respect to the minimum mean square e r r o r of p r e d i c t i o n f o r the four q u a r t e r s of 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 of c r e d i t union's s e r i e s , but more important to evaluate the s t r e n g t h of the time s e r i e s method used, we a l s o examined a l a r g e r sample of monthly data f o r p e r s o n a l demand dep o s i t s and pe 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 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 begins w i t h a survey of the 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 . The hypothesis f o r a l i k e l y s t r u c t u r a l model of c r e d i t union f i n a n c i a l behaviour i s a l s o presented. Time s e r i e s a n a l y s i s i s de-veloped t h e o r e t i c a l l y i n Chapter I I I u s i n g the n o t a t i o n of Box and Je 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 average process. In Chapter IV the data f o r demand de p o s i t s and term d e p o s i t s of the three c r e d i t unions and f o r p e r s o n a l d e p o s i t s of chartered banks are d i s c u s s e d and the models are estimated and evaluated. Some concluding remarks are presented i n Chapter V. F i n a l l y the Appendix:Data l i s t s the n a t u r a l logarithms and the raw data f o r the d e p o s i t s of both c r e d i t unions and ch a r t e r e d banks. I I . Survey of the L i t e r a t u r e 6 There are v a r i o u s r e p r e s e n t a t i o n s of the demand f o r demand d e p o s i t s and term d e p o s i t s and they depend upon the assumptions made about economic behaviour of i n d i v i d u a l s (or i n s t i t u t i o n s ) and the l e v e l o f aggregation i n the data. 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 quation models, and (b) simultaneous equation systems; and analysed w i t h r e s p e c t to the economic theory, a p p l i c a b i l i t y of s t r u c t u r a l equations to 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 problems-with the data and e s t i m a t i o n . A. S i n g l e Equation Models C l a s s i c a l demand theory s t a t e s t h a t the 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 pre f e r e n c e s , and p r i c e s of s u b s t i t u t e goods or s e r v i c e s . F e i g e (1964) uses t h i s hypothesis to estimate the demand f o r "demand d e p o s i t s , a non-pecuniary flow of s e r v i c e s that provide the 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 convenience. Since the value of the stream o f s e r v i c e s cannot be observed the v a l u e of the stock i s used as a proxy. The assumption i s made t h a t there 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 the stock and the flow o f s e r v i c e s r e n -dered by a gi v e n s t o c k o f demand deposits and hence the demand f o r non-pre-c u n i a r y s e r v i c e s i s eq u i v a l e n t to the 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 the sum of the nominal i n t e r e s t r a t e (zero) and the p o s i t i v e s e r v i c e charges. i s negative and i s d e f i n e d as t o t a l s e r v i c e charges d i v i d e d by the average balance of demand d e p o s i t s . Consumers' incomes are a weighted average of past and present values of p e r s o n a l income, where weights are those developed by Friedman (1957) to represent permanent p e r s o n a l income ( Y ^ ) . Tastes are assumed to be given and to remain constant over time but preferences are p r o x i e d by the per c a p i t a number of o f f i c e s of f i n a n c i a l i n s t i t u t i o n s (#/Pop) to measure convenience (time-space u t i l i t y p r o vided by l o c a t i o n ) . F i n a l l y , the p r i c e s of 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: commercial bank time d e p o s i t s (R t (j) J savings and l o a n a s s o c i a t i o n shares (K-s) and on mutual savings bank d e p o s i t s (R m) • The a c t u a l r a t e i s de 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 the average s i z e of as 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 . Using a l i n e a r form, the demand f u n c t i o n f o r demand d e p o s i t s i s es t i m a -ted by using o r d i n a r y l e a s t squares. The expected s i g n s f o r Y > A, and #/Pop are p o s i t i v e , w h i l e the expected s i g n s f o r the c o e f f i c i e n t s of s u b s t i -tues are n e g a t i v e . The data i s a p o o l i n g of c r o s s - s e c t i o n and time s e r i e s observations from 1949 to 1959 (U.S.). I n the best equation (2.1) Feige found ^ and R ^ to be s i g n i f i c a n t and to have the expected s i g n where ^~p> the dependent v a r i a b l e , i s per c a p i t a commercial bank demand d e p o s i t s ( F e i g e , 1964, p. 24). (2.1) ~ = 535R,, + .365Y - 35R . + 53R + 25R + r e g i o n a l dummies Pop dd p t d s • m ° • (48) (.080) (13) (13) (15) R = ..98 TT) (2.2) =-10lR,, + .122Y + 76R . - 44R - 82R + r e g i o n a l dummies Pop dd p t d s m ° _ (87) (.037) (10) (10) (11) R = .94 (TD) For per c a p i t a commercial bank time d e p o s i t s , R ^ i s now a p r i c e of a s u b s t i t u t e (expected s i g n negative) and R i s the own p r i c e (expected s i g n p o s i t i v e ) . I n t h i s equation (2.2) a l l c o e f f i c i e n t s are 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 the r i g h t s i g n . I n a more recent study, Boyd (1973) uses the same theory but makes an e x p l i c i t assumption about imperfect 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 dep o s i t s of 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 because of minimum balances and minimum terms to earn i n t e r e s t . I n h i s study o f savings 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 along w i t h the c l a s s i c a l determinants of demand. The f u n c t i o n a l form assumes th a t each v a r i a b l e a f f e c t s per 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 "given a change i n the d e s i r e d l e v e l of d e p o s i t s , ( i n d i v i d u a l ) savers w i l l q u i c k l y a d j u s t t h e i r account balance to the new e q u i l i b r i u m " (Boyd, 1973, p. 746). The r e s u l t s f o r the c r o s s - s e c t i o n sample f o r January 1969, u s i n g semi-annual data, are presented below i n equations (2.3) and (2.4) where DD/Pop, demand d e p o s i t s per c a p i t a ; R ^ average DD i n t e r e s t r a t e ; Y/Pop, 12 month average of per 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, per c a p i t a promotional expenses; r a t i o of number of a s s o c i a t i o n s ' branches to number of competitors' branches; R t (j> average r a t e on term d e p o s i t s ; R^, average competitor's savings r a t e (banks); and TD/Pop, term d e p o s i t s per c a p i t a . (2.3)lrtJJD = 5.8 + 3.50 InR,, + .51 lnY - 3.33 InR . + .57 I n A — dd - — t d P ° P (1.24) ~~ (.34) P o p (3.73) (.10) P o p + .55 In , ^  - 2.22 InR + r e g i o n a l dummies , „,N fforrc ,.. ,,x b „9 ,„ (.24) (1.66) R z = .60 (2.4)In TD = -15.93 - 4.92 InR + 17.78 InR. + .69 I n A__ •+ r e g i o n a l dummies Pop (3.20) d a (7.00) t a (.17) Pop 2 , ' K = . O / In the demand de p o s i t equation a l l the c o e f f i c i e n t s have the expected s i g n and o n l y own p r i c e ( R ^ ) » t a s t e s and preferences (|°^^) , 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. "In the c r o s s - s e c t i o n equation f o r TD/Pop Rfc^ and A/Pop are s t a t i s t i c a l l y s i g n i f i c a n t . The other v a r i a b l e s were used but they never entered s i g n i f i c a n t l y i n (2.4) and t h e i r estimates were not 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) admits t h a t was of the wrong s i g n and s i g n i f i c a n t (probably due to m i s s p e c i f i c a t i o n b i a s ) . F i n a l l y , Boyd t e s t s h i s hypothesis that 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 to changes i n d e p o s i t r a t e s . I f t r a n s a c t i o n c o s t s , imperfect 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 assumption, 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 estimates may be b i a s e d . " (Boyd, 1973, p. 746). F i v e year averages are c a l c u l a t e d f o r the independent v a r i a b l e s . They are 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 equation f o r the reason i s t hat 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 over time then the averages should be s i g n i f i c a n t because they represent the values of the independent v a r i a b l e s over time. The r e s u l t s f o r demand de p o s i t s and term d e p o s i t s are not i m p r e s s i v e as only the new average of a d v e r t i s i n g per c a p i t a proves to 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 (probably due to m u l t i -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 concludes that d e p o s i t o r s respond f u l l y t o the changes i n the economic environment t h a t take p l a c e w i t h i n the s i x month i n t e r v a l of h i s data (semi-annual o b s e r v a t i o n p o i n t s ) . I n a paper by Motley (1970), he assumes t h a t households are unable or 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 l e v e l s . The d e s i r e d asset l e v e l i s a f u n c t i o n of expected income ( Y * ) , r a t e s of r e t u r n on a l l a s s e t s (vector R) , and the i m p l i c i t r e n t a l s on durable goods ( u ) . * * (2.5) TD" = f ( Y ~ , R f c d, R d d, Rs,....,u) The form of the f u n c t i o n i s l i n e a r i n the logarithms and the demand f o r a s s e t s (at constant p r i c e s ) i s homogeneous of degree zero i n general 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 n (2.6) l o g TD* = a + a, l o g Y* + £ a„. l o g R. 0 1 j = l J J 10 "A constant p r o p o r t i o n of any r e l a t i v e divergence between a c t u a l and d e s i r e d stock of (term 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 approximated by" (Motley, 1970, p. 236) 2.7 TD TD t-i " [ T V i J - X. * I TD^ X > 0 where X, the d e s i r e d r a t e of adjustment, depends upon the change i n stock of both term d e p o s i t s and a l l other a s s e t s i n the p o r t f o l i o , r a t i o of c u r r e n t to 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 preference and expe c t a t i o n s parameter. For 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 term d e p o s i t s w i l l i n -crease the demand and l e v e l of TD . This w i l l draw funds away from s e c u r i t i e s t h a t are s u b s t i t u t e s and i t w i l l a f f e c t the 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 the p o r t -f o l i o i s de p i c t e d by the f o l l o w i n g adjustment process: * n * (2.8) l o g TD t-log TD ^ = X ( l o g T D ^ l o g T D ^ ) + E X . ( l o g S . t - l o g S. ) j = l 3 3 3 + y i ( l o g Y t- l o g Y*) * * 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 of assets (TD , S^) i n (2.8) , we o b t a i n n seemingly u n r e l a t e d equations. (2.9) l o g S t = A + Blog Y* + n o g R t-(I-A) l o g S ^ + M l o g (Y f c- Y*) where A, B, and M are n-vectors and T, A are nxm mat r i c e s and i n p a r t i c u l a r the equation f o r term d e p o s i t s i s : n j A n. n (2.10) l o g TD = a + S X c ^ l o g Y + Z X E aJ* l o g R, + ( l - X ) l o g TD ± \ 3 3 k :,\!;: V * - ^ h j l 0 g S j t - 1 + p ( l o g Y - l o g Y ) The c o e f f i c i e n t s are estimated u s i n g n o n - l i n e a r techniques w i t h o n l y f o u r other 11 assets ( i . e . , n = 4 ) . They are: money (M), savings d e p o s i t s (TD), debt (D), and r e a l a s s e t s (RA). Expected 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 weighted average of pe r s o n a l d i s p o s a b l e income where the weights are those used by Friedman (1957). The s i g n i f i c a n t determinants of savings d e p o s i t s i n (2-11) a r e : t r a n s i t o r y income (Y-Y ) f c , and lagged h o l d i n g s (TD^ ^ ) . These r e s u l t s i l l u s t r a t e the p a r t i a l adjustment process but not the r e a l l o c a t i o n of funds i n the p o r t f o l i o , f o r q u a r t e r l y U.S. data between 1953 and 1965. (2.11) l o g TD = 5.72 + .14 l o g Y* + .03 l o g R - .36 l o g TD -.00 l o g M . (.30) C (.04) (.13) (.04) t 1 - .23 l o g D - .14 l o g RA . + .28 l o g (Y-Y*) (.13) (.40) t - i (.08) • • t B a t r a (1973) uses the same f o r m u l a t i o n as Motley. The a s s e t s are interdependent i n a s f a r as they compete w i t h one another i n the 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 at any p o i n t i n time are a l s o a f f e c t e d by the c a p i t a l gains ( l o s s e s ) i n c u r r e d . Adjustments to the d e s i r e d l e v e l s of as s e t s are made I n some propor-t i o n i n a given q u a r t e r . * n * (2.12) ;TDt - TD t_ 1 = X(TD t - (TDfc + G f c)) + I ^ f S j t - ( S ^ ^ + G j t ) ] 3 where G i s the c a p i t a l gains on j t h f i n a n c i a l a s s e t . The demand f o r t h j t e d e s i r e d s t o c k of term d e p o s i t s i s a f u n c t i o n o f expected income (Y ) , expected c a p i t a l gains (G ) , past preferences and h a b i t s (S j), i t s own p r i c e (R t (j) and those o f s u b s t i t u t e s (R(J£J> R , . . . ) . • Assuming a l i n e a r f u n c t i o n (2.13) TD = a n + a, R „ , + a 0Y + a„ G + a. S - + E a_. R. t 0 1 t d 2 3 „ 4 pt-1 ^ 5x l Expected income i s de f i n e d as a l i n e a r f u n c t i o n of c u r r e n t income. Expected c a p i t a l gains are assumed to be a l i n e a r f u n c t i o n of cur r e n t c a p i t a l g a i n s . C a p i t a l gains on asset i are d e f i n e d as: 12 p = i t i t - 1 i t - 1 xt-1 i t C.P.I. ~ C.P.I.,. .. t t-1 where CP.I., implicit price deflator of personal consumption expenditure. By n * * simplifying E X. [S. - (S._ .. + G.J] to (G.D. and substituting for TD i n j J j t Jt-1 j t / J i k 6 (2.12) we get (2.14): (2.14) TDt - T D ^ = 6 Q + 5 ± Y* + ^ Gfc + 63 S p t _ 1 + 6 ^ + 6 ^ + 6 ^ ^ + 6 ^ (2.15) ATD = -7861 + 6922 R + .098Y* - .49 TD + 8564S - .03D t (2100) Z (.03) (.15) t _ 1 (2300) P (.01)k R^  = .92 The empirical results are given after elimination of a l l nonsignificant varia-bles. Data sources for the 1947-1969 time series are not l i s t e d but capital gains (G^) and the price of substitutes (R Q) did not prove to be significant. Since Batra does not state what assets are included i n D^ and does not explain what services are measured by Spt_^, i t i s d i f f i c u l t to conclude that Motley's hypothesis of interdependence i s s t a t i s t i c a l l y important for savings deposits. B. Simultaneous Equation Systems In the previous section, the single equation approach assumed that the variables on the right hand side of the model are predetermined-exogenous, or lagged endogenous- and hence they are a l l independent of the error term and ordinary least squares can give consistent estimates. This assumption is true for the behaviour of individuals but i t cannot be made for financial i n s t i t u -tions, especially at the macroeconomic level because interest rates on deposits are decision variables i n the management of financial intermediaries, the own price becomes a current endogenous variable and single equation least squares w i l l no longer result i n consistent estimates for the coefficients. The econo-metrics of the situation requires that the determinants of own deposit rates 13 be s p e c i f i e d and the equations be estimated s i m u l t a n e o u s l y . Cohan (1973) a p p l i e s a r e c u r s i v e system to determine the 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 of deposit i s s u e d to (a) c o r p o r a t i o n s and (b) households, and the l e v e l o f these 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 to c o r p o r a t i o n s (R- C tj) i s i n f l u e n c e d by: (1) a n t i -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 compe-t i t i v e a s sets ( i . e . Treasury B i l l s R j . ^ ' a n 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 r e s t r i c t i o n s (R^). A p a r t i a l adjustment process i s assumed t o e x p l a i n movements i n R ,. cd ( 2 ' 1 6 ' 1 ) . • A R c d , t " X ( R c d , t " R c d , t ^ (2.16.2) R*dfc - R q - T ( R q / R t b ) where R /R , approximates a cost 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 q tb ( l i m R /R , -* 0 = >R = R ). Assuming t h a t 0 < X < 1 s u b s t i t u t e f o r R , i n q tb q cd (2.16.1) and the f o l l o w i n g equation i s estimated u s i n g q u a r t e r l y data (1961-1967) f o r U.S. commercial banks. (2.17) R , = 2.88 + .83R — 2.64 R + .15 R . , . „ , c d , t ' (.05) q (.20)-5- (.05) c d ' t - 1 R = .99 R t b (2.18) Rp = -4.34 +.89R + .83R + .15R , + .20R, T . 1 7 ) s d (.17) S ( . 0 6 ) c d (.06? R 2 R = .98 The supply 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 of d e p o s i t s (R ) i s determined P by the same f a c t o r s as 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 . R c d ^ s a p r o x y f ° r t^ i e above 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 the p r i c e of s u b s t i t u t e s are: R S (j> savings and l o a n savings d e p o s i t r a t e ; R , bank's savings d e p o s i t r a t e ; R^, short term bank l e n d i n g r a t e . Whereas the author assumed a p a r t i a l adjustment process f o r RC(j» R p i s assumed to a d j u s t 14 f u l l y once R ^ I s s e t and competitors' p r i c e s are known. The l i n e a r f u n c t i o n a l forms are estimated by two stage l e a s t squares and a l l the c o e f f i c i e n t s proved 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 from zero. The s t r u c t u r e of equation (2.17) i s not a p p l i c a b l e to the Canadian f i n a n c i a l system because of the absence o f a l e g a l c e i l i n g r a t e . 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 of deposit (CD) i s based upon the t r a d i t i o n a l demand theory. The dependent v a r i a b l e i s d e f i n e d as the r a t i o of CD to l i q u i d a ssets (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 . L i q u i d a s s e t s c o n s i s t o f corporate 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 and,currency, savings and time d e p o s i t s at commercial banks and at savings i n s t i t u t i o n s , s h o r t term t r e a s u r y s e c u r i t i e s and commercial and f i n a n c e company paper, and s h o r t term U.S. government s e c u r i t i e s . To avoid the 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 the spread between own p r i c e and a s u b s t i t u t e i s used. The estimated demand curve i s l i n e a r and assumes instantaneous adjustment, to exogenous f a c t o r s , r n - • /-<> io\ 7T= - 7 - 6 3 + - 4 2 ( R A ~ + - 4 6 ( R - R J ) +7.61 Y - .03(AY-k) + sea-( 2 ' 1 9 ) (.22) C d t b (.24) P S d (.44) P (.02) sonal, R 2 = .99 where Y , wealth measured weights adopted from Friedman's permanent income P 1 1 i theory Y = .139 Z (.9) GNP , and (AY-k). i s the change i n GNP l e s s the p 1 average q u a r t e r l y growth i n GNP. Th i s v a r i a b l e i s intended t o measure the e f f e c t s of t r a n s i t o r y income. "This type of income i s l i k e l y to be h e l d i n temporary money balances r a t h e r than being s h i f t e d i n t o an i n t e r e s t y i e l d i n g l i q u i d a s s e t . [ I t ] i s expected to be i n v e r s e l y r e l a t e d to the CD's" [Cohan (1973), p. 107]. The spread between i n t e r e s t r a t e s (a^jC^) are of ma 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^ proves to be s i g n i f i c a n t . Cohan's three equation model i s a b l o c k r e c u r s i v e system: two equation 15 supply block (estimated by 2SLS) and a unique demand equation (estimated by OLS). The two d e p o s i t r a t e s are s e t i n t e r d e p e n d e n t l y and then they are p a r t of the f i n a l determinants 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 are such that the determination of supply and demand may be considered s e q u e n t i a l r a t h e r than simultaneous i n nature". (Cohan, (1973), p. 109). DeLeeuw's paper (1965) was the f i r s t simultaneous a n a l y s i s of the monetary s e c t o r . I t i s a model of f i n a n c i a l behaviour i n the many f i n a n c i a l markets i n the U.S. At t h i s l e v e l o f aggregation the market 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 h e l d are interdependent. Of the nineteen equations t h a t make-up the complete model only three equations w i l l be discussed below. They are : demand d e p o s i t h o l d i n g s ; time 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 four assumptions: ( 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 composition and i n t e r e s t r a t e s . The consumer maximizes net worth and w i l l choose those combinations of a s s e t s t h a t w i l l give 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 ) At any p e r i o d there i s a p a r t i a l adjustment to the " d e s i r e d " p o r t f o l i o . Adjustments are 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 are s h o r t - r u n c o n s t r a i n t s that l i m i t behaviour by both consumers and f i n a n c i a l i n t e r m e d i a r i e s . 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 con-s i d e r a t i o n s and reserve requirements. ( i v ) The f i n a l assumption s t a t e s t h a t a l l r e l a t i o n s h i p s are homogeneous of 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 of asset x i s a f u n c t i o n of i t s stock i n the previous p e r i o d , r a t e s o f r e t u r n ( i t s own and those of s u b s t i t u t e s ) ( R x > R^, R^, R R) and c u r r e n t and lagged short run c o n s t r a i n t s . ( f ( x ) f c , ( 2 ?m A ( x > t , x t - l A ' fOOt f ( x ) . * (2.20) _ - 0 f l + a i _ + ^ ^ a s R i + _ + a_ _ t + a _ + i 16 The changes i n the quantities demanded are expressed as a percentage of total 19 demand i n the sector. The latter i s measured by the proxy Y _ = 0.114 Z (0.9) i=0 • GNP_^ . It is lagged one period to f a c i l i t a t e simulation. Any measurement error arising from Y^ ^ instead of Y^_ i s assumed to be negligible. The i n t e r -est rates are nominal rates expressed as percentages. The constraints are particular 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 yi e l d on U.S. securities maturing or callable i n ten years or more (R ' yi- eld i n commercial bank time deposits (R t (j) > personal disposable income (Y^), and business gross investment in plant and equipment (1^) plus private nonbusiness, nonresidential construc-tion ( I c ) • The latter two variables serve as a proxy for the expected return on capital goods. Current and lagged values of disposable income represent sources of funds to households. One i s struck by the absence of "own" price i n the above hypothesis. This i s because DeLeeuw assumes that the demand de-posits have their interest rates fixed at zero and his model does not deal with service charges. The results of 2SLS using U.S. quarterly data for the 1948-1962 time period are presented i n equation (2.21) and the coefficients of DD _^ R g b l ' Y,, and (I.+I ) have the right sign and s t a t i s t i c a l significance. Q D C DD .07Y (2.21) § ^ = -.003 - .11 -005R - .002R + ?-V l (.04)Vl (.002)gDJ- (.002) t Q Xt-1 (.03) + .03 Yd,t-1 -.20 V^'V (.02) Yt-2 <' 0 6 ) Y t - 1 17 (2.22) — = _,Q02 - .12 T D t - l - . 0 0 3 R . + .006 R + .02 Y d , t - 1 t-1 (.03) Y t _ 1 (.000) D (.001) (.008)Y f c_ 2 For changes i n term de p o s i t s the f u n c t i o n a l form i s the same as t h a t of 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 (TD ^ ) , the three month t r e a s u r y b i l l r a t e ( R t^) the average y i e l d on bank term d e p o s i t s ' ( R t ( j ) > a n d d i s p o s a b l e income ( Y ^ ) . A l l estimates i n (2.22) are s i g n i f i c a n t . 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 of 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 ' present p o r t f o l i o s toward t h e i r d e s i r e d p o r t -f o l i o s . The q u a r t e r l y change i n R ^ i s assumed to depend on the d e s i r e d r a t e and l a s t q uarter's a c t u a l r a t e . The d e s i r e d r a t e depends upon U.S. s e c u r i t y y i e l d s maturing or c a l l a b l e i n t e n years or.more (R j^)» o n c e i l i n g r e s t r i c -t i o n s ( R g b l - Rq + 1)» a n d o n the r a t i o of loans to t o t a l d e p o s i t s (TL/(DD+TD)) (2.23.1) A R t d , t = X <Vt " Rtd,t-1> ( 2 ' 2 3 - 2 ) < d , t = f ( R g b l ' R g b l - R q + 1 ' Whv) The f o l l o w i n g equation (2.24) approximates t h i s behaviour. A l l the c o e f f i c i e n t s proved to 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. Again the absence of l e g a l c e i l i n g s i n 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 of (2.24) to our c o n t e x t . T T (2.24) AR t d - -1.26 - .39 R ^ ^ •+ .14R gfr.33R Q + 1.02 ( W T 1 D ) _ 2 . A simultaneous model at the m i c r o l e v e l i s the Dhrymes and Taubman (1969) study of the savings and loan a s s o c i a t i o n s i n the U.S. Each f i r m w i l l s e t deposit and l o a n i n t e r e s t r a t e s to a l t e r the demand f o r t h e i r a s s e t s and l i a -b i l i t i e s such t h a t p r o f i t s are maximized i n each time p e r i o d . But the a c t u a l changes i n d e p o s i t s and shares may f a l l s h o r t of the expected l e v e l s and t h i s may f o r c e the f i n a n c i a l manager to 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 . The changes 18 i n i n t e r e s t r a t e s and i n depo s i t s and ioans are interdependent and are d e t e r -mined i n each time p e r i o d simultaneously. T h e i r u n d e r l y i n g theory i s the pa r -t i a l adjustment process i n a competitive framework V a r i a b l e s such as a d v e r t i s -i n g per c a p i t a and number of S & L o f f i c e s per c a p i t a r e f l e c t consumers' t a s t e s and preferences r a t h e r than product d i f f e r e n t i a t i o n . . . .. The S & L have only one type of de p o s i t and the d e s i r e d l e v e l o f term de-p o s i t s i s a f u n c t i o n of own i n t e r e s t r a t e , permanent per c a p i t a income (Y/Pop), number of S & L o f f i c e s per c a p i t a (#/Pop), and p r i c e s of s u b s t i t u t e s ( t r e a s u r y b i l l r a t e R , and a r e g i o n a l r a t e R ). 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 tb e 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 data (1958-65) show logged s t o c k , R t (j> a n d con-venience to be s i g n i f i c a n t . However, the presence of TD t_^ b i a s e s the D u r b i n -Watson s t a t i s t i c and the c o e f f i c i e n t s estimated may be i n c o n s i s t e n t . C o e f f i -c i e n t on income i s s i g n i f i c a n t l y of the wrong s i g n . TD • Y # (2.25) I n — = .37 + .20 I n R. , - .06 I n - — + .04 I n -2 .02 I n R L (.06) t d (.01) P o p (.006) P o p (.014) t b (TD) - .15 I n R + .93 I n —— + seasonals _ (.14) 6 (.005) • 1 . R = .95 (2.26) R = .21 + .94 R , , + .02 R., + .01 ( ^ ) + r e g i o n a l dummies t d (.01) t d ^ _ 1 (.007) t b (.02) t-1 2 „ K — . y x The adjustment 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 l s o expected to f o l l o w a p a r t i a l adjustment process to the opt i m a l l e v e l . I t i s determined by comp e t i t i v e r a t e (three month t r e a s u r y b i l l r a t e R ^) the demand f o r own mortgages (AM). 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 that an excess demand f o r assets w i l l put pressure on S & L f i r m s to a t t r a c t a d d i t i o n a l savings t h a t they . can channel 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 not prove to be s i g n i f i c a n t 19 i n the 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 of the l i t e r a t u r e and research surveyed i s to model a s i m u l -taneous s t r u c t u r a l system of equations f o r the behaviour o f f i n a n c i a l i n t e r -m e d iaries. 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 the f a c t t h a t the f i r m has a number of 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 the manager wants t o know the v a r i o u s responses to those parameters ( i . e . e l a s t i c i t i e s of demand w i t h r e s p e c t to 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 or w e a l t h ) . A hypothesis 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 the 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 union i s developed 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 Behaviour; and ( i i ) F i n a n c i a l Behaviour of a C r e d i t Union. As us u a l the v i a b i l i t y of t h i s approach depends on the a v a i l a b i l i t y of a l a r g e number o f o b s e r v a t i o n s . ( i ) Consumer Behaviour The consumer i s expected t o maximize h i s net worth i n a w o r l d where there 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 time 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 expected to 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 of s e c u r i t i e s . We w i l l c o n sider h i s demand f o r o n l y two such s e c u r i t i e s : demand d e p o s i t s and term d e p o s i t s . The demand f o r demand d e p o s i t s stems from the de-s i r e to h o l d l i q u i d a s s e t s . These assets provide the i n d i v i d u a l 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 , convenience and chequing f a c i l i t i e s . I t i s these s e r v i c e s that the i n d i v i d u a l purchases when he acquires demand d e p o s i t s . The stock of demand d e p o s i t s w i l l serve as a proxy f o r the amount of s e r v i c e s purchased. On the other hand, the consumer purchases a term d e p o s i t i n order to r e -c e i v e a p o s i t i v e r a t e o f r e t u r n . O p t i m i z i n g over time, the consumer attempts 20 to r e c o n c i l e h i s earned income stream 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 to i n v e s t i n f i n a n c i a l s e c u r i t i e s i f h i s expected income i s g r e a t e r than h i s d e s i r e d consumption or i f the r e t u r n from purchasing f i n a n c i a l instruments i s g r e a t e r than the r e t u r n from consump-t i o n of durable and non-durable goods. The stock of term d e p o s i t s i s the mea-sure 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 the consumer does not know a l l the oppor-t u n i t i e s a v a i l a b l e to him and t h e i r c o s t s or p r o f i t s , so h i s behaviour does not a t t a i n the optimum s o l u t i o n i n every time p e r i o d . There i s a time l a g i n executing d e c i s i o n s due to delays i n communications and due to u n c e r t a i n ex-p e c t a t i o n s about the f u t u r e ( i . e . lagged response i n d e p o s i t accounts i n answer to changes i n i n t e r e s t r a t e s ; lagged updating of expected income). Thus house-holds are unable or 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 t o long-run d e s i r e d l e v e l s i n s t a n t a n e o u s l y . Tastes and preferences are i n f l u e n c e d by a d v e r t i s i n g and convenient l o c a -t i o n . There e x i s t many s u b s t i t u t e s among durable goods and f i n a n c i a l instruments, The p r i c e s of s u b s t i t u t e s are a l s o expected to e x p l a i n the s i z e and nature of ( d i s ) e q u i l i b r i u m i n the r e s p e c t i v e markets of these assets i n the p o r t f o l i o . We adopt the assumption that "the demand f o r a s s e t s (at constant p r i c e s ) i s homogeneous of degree zero i n general 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 respect to p o p u l a t i o n . " (Motley, 1970, p. 236). Consumers' income i s an exogenous v a r i a b l e i n the model. Permanent income i s estimated by adapting the weights developed by Friedman (1957) i n h i s consumption study. T r a n s i t o r y income i s expected to have a s i g n i f i c a n t e f f e c t on both types of purchases of l i q u i d a ssets, 21 Therefore the demand f o r a l i q u i d asset i s hypothesized to be an a d j u s t -ment between t h i s period's d e s i r e d stock and l a s t period's a c t u a l s t o c k . The de s i r e d stock i s determined by i t s own p r i c e by the s i z e of the consumers' budget (permanent income and t r a n s i t o r y income), by per c a p i t a a d v e r t i s i n g expenditures and number of o f f i c e s per c a p i t a , and by p r i c e s of s u b s t i t u t e s . ( i i ) F i n a n c i a l Behaviour of a C r e d i t Union The c r e d i t union s e t s 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 to 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 such that the s u r p l u s of revenues minus costs i s maximized i n each time 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 but because of a l a g i n decision-making or an unfavourable l i q u i d i t y p o s i t i o n he i s unable to reach the 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 . The d e s i r e d r a t e on demand depo s i t s i s determined by: s e r v i c e charges ( S g ) ; competitors' r a t e s ( R ^ ) ; c r e d i t union's demand f o r demand de p o s i t s (DD); the 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 the d e s i r e d r a t e on c r e d i t union term d e p o s i t s ( j o i n t d e c i s i o n making on the two r a t e s ) . The d e s i r e d r a t e on term d e p o s i t s i s determined by: competitors' r a t e s ; c r e d i t union's demand f o r term d e p o s i t s ; the mortgage r a t e ; the d e s i r e d r a t e on c r e d i t union demand d e p o s i t s ; and the demand f o r term d e p o s i t s . To ma i n t a i n the d e s i r e d d i s t r i b u t i o n of low co s t funds, we p o s t u l a t e a constant r e l a t i o n s h i p between d e s i r e d demand de p o s i t r a t e and d e s i r e d term d e p o s i t r a t e . To keep a t t r a c t i n g funds i n t o nonchequable sav-ings r a t h e r than i n t o higher cost term de p o s i t s an adaptive e x p e c t a t i o n s equa-t i o n i s used. ( 2 . 2 7 . 1 ) . c 2 - 2 7 - 1 ) 4 - R d d - i =7 ( R t d - ^ d - i ) (2 .27.2) R * D ( l - ( l - y ) L ) = R*J(1-6L) = y R t d where L , l a g operator ( 2- 2 7- 3> R d d = l ^ L R t d 22 P a r t i a l adjustment model f o r term deposit r a t e s (R f c d) (2.28.1) AR t d = X£Rtd - R ^ ) (2.28.2) R* d= f l ( R B d , V TD) S u b s t i t u t i n g f o r R f c d i n (2.28.1) and us i n g the r e s u l t of (2.27.3) (2.28.3) AR t d = A, ( R ^ + R m + ^ R t d + TD - R ^ ) which s i m p l i f i e s to (2.28.4) (2.28.4) R t d = a, R*d - a, R ^ + ^  R m - a 4 R ^ + a 5 TD - ^  TD_( + a7 R t d - 1 + a 8 R t d - 2 P a r t i a l adjustment model f o r demand deposit r a t e ( R d d ) (2.29.1) AR d d = ,.A2 (R* d - P ^ ) (2.29.2) R* d = f 2 ( S ^ , R m, R ^ D D ) S u b s t i t u t i n g the r e s u l t of (2.27.3) i n (2.29.1) and b r i n g R ( i d_ 1 to the r i g h t hand s i d e (2.29.3) R d d = V R t d - (1-X2) l-o l_ (2.29.4) R d d = X 2y Rfcd + ( X ^ y ) R ^ + (1-y) (l-X,,) R d d _ 2 From consumer behaviour the demand demand depo s i t s (DD) and term 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) to form the complete s t r u c -t u r a l model of four simultaneous equations. ' (2.30) DD = ' d n R d d + a 1 2 R t d + a±3^dd f a l 4 fop + : a 1 5 ' f ~ - + a 1 6 \ + a 1 ? ( Y - Y p ) + a l gBD- 1 . 23 (2.31) TD = a n R d d + a 2 2 R t d + + . ^ t ^ M + a 2 6 Y p + a27 ( Y-V " a 2 8 T D - l ( 2 ' 3 2 ) . R d d ' + a 3 1 R t d + a 3 2 R d d - l + ?33 Rdd-2 (2.33) ' R . = a- R + . a R +<*,^KA I +A,,KA o + a/ r R ? , + a / c R B J , t d 41 m -42 m-1 43 td-1 44 td-2 45 t d 46 td-1 + a TD - a TD 4 7 1 U , 4 8 - 1 A l l the equations are over i d e n t i f i e d but they meet the rank c o n d i t i o n (necessary 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 ) . T h i s c o n d i t i o n s t a t e s that the s t r u c t u r a l equation i s i d e n t i f i e d i f and o n l y i f the rank of the m a t r i x formed by the excluded v a r i a b l e s i s equal t o the number of equations l e s s one. U n f o r t u n a t e l y f o r the development of t h i s h y p o t h e s i s , data c o n s t r a i n t are severe. There i s no pu 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 on c r e d i t union i n t e r e s t r a t e s and hence the c r i t i c a l parameters of the model are unobser-v a b l e . 24 I I I T h e o r e t i c a l Development of Time S e r i e s A n a l y s i s This method of f o r e c a s t i n g i s data o r i e n t e d as i t i n c o r p o r a t e s eco-nomic i n f o r m a t i o n through 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 -t i o n . We assume t h a t there 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 the s e r i e s that i s represented by h i s t o r i c a l data and t h i s p a t t e r n can be expressed as a weighted sum of past values of the data. The weights i n the sum are determined so as to achieve the 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 power. This a n a l y s i s i n v o l v e s three stages and n i t e r a t i o n s on these stages, ( i l l u s t r a t e d below i n Fi g u r e I I I ; Box and Jenkins (1970, p. 19). Our concern i s to f i t a s t a t i o n a r y model f o r the s e r i e s of demand de p o s i t s and the s e r i e s of term d e p o s i t s of c r e d i t unions t h a t w i l l be used to 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 . We opt i m i z e the p a t t e r n of a set of data by mi 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 of the time s e r i e s model are: ( i ) a u t o r e g r e s s i v e process where there e x i s t s an a s s o c i a t i o n among values of the same v a r i a b l e but at d i f f e r e n t time p e r i -ods ( s e r i a l or s e a s o n a l ) ; ( i i ) moving average process where there e x i s t s some mutual correspondence among suc c e s s i v e values of r e s i d u a l s (trends or s e a s o n a l ) ; and ( i i i ) a mixture of the above mentioned processes. Each i s presented below. A. General C l a s s of Models S t a t i o n a r y process. A time s e r i e s z i s considered t o be s t a t i o n -ary i f i t has an e q u i l i b r i u m p o i n t about a constant mean and i f the v a r i -2 ances of the observations are the same ( i . e . E(z. ) = E(z ) and E(z ) -t t+n t 2 2 2 E( z. t) = E ( z t + n ) ~ ^ ^ t + n ^ ^* ^ n o n s t a t i ° n a r y Process has no n a t u r a l mean and i t i s assumed that some s u i t a b l e d i f f e r e n c e equation w i l l r e -F i g u r e I I I . l P o s t u l a t e General C l a s s of Models I d e n t i f y Model to be T e n t a t i v e l y E n t e r t a i n e d 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 the model adequate?) No F o r e c a s t i n g Future Values of Model 26 present the process as being s t a t i o n a r y . We i n t r o d u c e V as the backward ; t = z t - k d i f f e r e n c e operator which can be w r i t t e n i n terms of B where B,z = z (3.1.1) Vz f c = z t - z t _ 1 = (1 - B) z t (3.1.2) VdZt - z t - d z ^ + 1/2 d(d - 1) z t _ 2 + ... + ( - l ) d Z t - d A u t o r e g r e s s i v e process. Consider 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 from 1 to T. Assume that 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 (3.2.1) z = <j> z + <J> z _+... + <J> + a_ t 1 t-1 2 t-2 p t-p t where z f c, i s a random o b s e r v a t i o n at p e r i o d t , cf>t i s an a d j u s t a b l e weight, and a f c i s a s e r i e s of random shocks ("white n o i s e " ) . I n t r o d u c i n g the a u t o r e g r e s s i v e operator <}>(B) we now w r i t e (3.2.2) (1 - (j^B 1 - (j) 2B 2 - ... - (j) pB P) z f c = <KB)zt = a f c Moving average process. The time s e r i e s z i s s t a t i o n a r y and can be w r i t t e n as (3.3.2) where 0(B) i s moving average operator (3.3.1) z t = a t - 6 1 a t _ x - 9 2 a f c _ 2 - ... - 9q a t _ q / (3.3.2) z = (1 - 0.B - 0 O B 2 - ... 0 B q) a„ = 0(B) a t t 1 2 q t t Mixed autoregressive-moving average (ARMA) model i s 4>(B)z^_ = 0(B) a t where (f> (B) and 0 (B) are polynomials of degree p and q r e s p e c t i v e l y . ' . This process i s r e f e r r e d to as an ARMA (p,q) process (assumed to be s t a -t i o n a r y ) . (Box-Jenkins (1970), p. 74). (3.4.1) z = <kz 1 + <i> z + ... + fj> z + a. - 0-a^. 0 - ... - 0 t I t - i I t-Z p t-p t 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 process w i t h a pth a u t o r e g r e s s i v e scheme, a dth s t a t i o n a r y d i f f e r -ence, and a qth moving average: ARIMA (p,d,q) .. 27 (3.4.2) <f>(B)Vdzt = • 6 (B) a An example of a (1.1.1) process i s : (3.5.1) (1 - <J» B) Vz = (1 - e^B) a (3.5.2) V z t - ^ V z ^ = a t - 6 ; L a ^ ^ An example of a (0, 2, 2) process i s : (3.6.1) V 2 z t = (1 - 0 1B - 6 2B 2) a t (3.6.2) z t - 2 z t _ 1 + z t _ 2 = a t - Q± a ^ - 8 2 a f c _ 2 Seasonal ARLMA process .Assume that z i s a monthly s e r i e s and t h a t we i n t e n d to l i n k c u r r e n t behaviour f o r month t w i t h behaviour f o r the month i n the previous year t-12 and so on f o r each of the twelve 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 process by d i f f e r e n c i n g i t D times. The seasonal a u t o r e g r e s s i v e process of l e v e l P i s represented by the p o l y -nomial (3.7.1) and the seasonal moving average process of l e v e l Q i s r e p r e -sented by the polynomial (3.7.2), where the seasonal l e n g t h i s denoted by s = 12 ( i n our example of a monthly s e r i e s ) . (3.7.1) <(B 1 2) = 1 - ^ - ¥ l 2 - ... - ^ (3.7.2) 0 ( B 1 2 ) = 1 - ^ B 1 2 - G ^ 2 - ... ^ B J 2 I t i s assumed that the parameters $ and 0 contained i n t h i s monthly model i s approximately the same f o r each month and that the e r r o r s a t's are random. The seasonal ARIMA can thus be w r i t t e n as (P,D,Q)s model s p e c i f i e d i n (3.9.3) (3.7.3) * p ( B s ) V s \ = C Q ( B s ) a t - 28 M u l t i p l i c a t i v e model. In our seasonal s e r i e s d i s c u s s e d above, we r e l a x the assumption about the e r r o r s and now a l l o w - a , a t_2» a t - 2 ' *'* to be c o r r e l a t e d , represented by (3.8.1) (3.8.1) c|> (B)Va t > .6(B) &t where a f c i s white n o i s e . S u b s t i t u t e (3.8.1) i n (3.7.3) and p r e m u l t i p l y i n g the l a t t e r by <J) (B) ' y i e l d s (3.8.2) - the (p,d,q) X (P,D,Q)s model. (3.8.2) * (B) $ p (B s) V d V s D z t = § (B) G Q ( B ^ a f c This type of 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 Nelson (1973) p o i n t s out such 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 to conceive of the p a t t e r n and i n t e n s i t y of seasonal v a r i a t i o n s as undergoing change over time. 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 processes t h a t d i s -p l a y seasonal behaviour as the b a s i s f o r models of seasonal time s e r i e s (Nelson (1973) p. 169). An example of a (0,1,1) x (0,1,1).^ model i s presented below f o r a monthly s e r i e s . To l i n k the monthly z t ' s one year apart we w r i t e (3.9.1) and to l i n k the c o r r e l a t e d a_'s one month apart we w r i t e (3.9.2). The m u l t i p l i c a t i v e model i s presented by (3.9.3). (3.9.1) V 1 2 z t = (1 - G B 1 2 ) a t (3.9.2) ' Va = (1 - GB) a f c (3.9.3) V V 1 2 z t = (1 - G B ) ( 1 - . 0 B 1 2 ) a f c = a f c -• •/B a ^ - 8 . ^ + 9 0 a t _ 1 3 29 The moving average operator i s now of order q + sQ = 13 and we have t h i r t e e n a d j u s t a b l e c o e f f i c i e n t s ( i . e . twelve monthly c o n t r i b u t i o n s and one y e a r l y c o n t r i b u t i o n ) . We observe t h a t the seasonal behaviour i s pi c k e d up by the weighted e r r o r terms on the r i g h t hand s i d e of equation (3.9.3) B. I d e n t i f i c a t i o n of a Model The i n d i v i d u a l time s e r i e s of de 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) or m u l t i p l i c a t i v e ARIMA (p,d,q) X (P,D,Q)s. The i d e n t i f i c a t i o n i s broken up i n t o three stages: ( i ) to i d e n t i f y the degree of d i f f e r e n c i n g to o b t a i n a s t a t i o n a r y s e r i e s expressed as a transform of the o r i g i n a l s e r i e s z^ _; ( i i ) to i d e n t i f y the 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 pr o -cess, and ( i i i ) to i d e n t i f y the absence or presence of s e a s o n a l i t y i n the ARIMA. I f the 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 defined below i n (3.10) does not d i e out f a i r l y r a p i d l y f o r the raw data, then one may consider Vz^, or some h i g h e r d i f f e r e n c e . " I t i s assumed th a t the degree of d i f f e r -encing d, necessary to achieve s t a t i o n a r i t y , has been reached when the a u t o c o r r e l a t i o n f u n c t i o n p, of V^z dies out f a i r l y q u i c k l y . In p r a c t i s e K t d i s normally e i t h e r 0, 1 or 2 (Box-Jenkins (1970), p. 175). The a u t o c o r r e l a t i o n f o r z = <J>1 z at l a g k i s £ X U.— iC t (3.10) p k = 2 a z The s t a t i o n a r y AR process of order k (f> (B) z = a , p r e m u l t i p l i e d by z 7 can be expressed as (3.11.1), and t a k i n g expectations and d i v i d i n g through 30 2 by a'•. we get an equation (3.11.2) t h a t presents the k t h a u t o c o r r e l a t i o n P k as a weighted sum of 4' s and k-1 a u t o c o r r e l a t i o n s . (3.11.1) zt_kzt = <!> l V k V l + Vt -k Z t-2 + . . . + Vt-k z t -k" (3.11.2) p k = + <j,2pk_2 + . . .+ <f,k To estimate the 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 the p's'by the estimated 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 given by the Yule-Walker equations (3.12) where the k t h equation i s simply (3.11.2).(Box and Jenkins (1970), p. 54-55) ;' -•"'-:'".:"- '' ' \'« (3.12) p l = *1 + *2 P1 + + V k-1 p 2= <f, l P l + <f,2 + , . . + * k P k _ 2 P k = V k-1 + ' * 2 p k - 2 + ••• + * k ' ' ' We have assumed that the AR process i s of order k and hence can be expressed i n terms of k non-zero a u t o c o r r e l a t i o n parameters. To be sure o f the le n g t h of the AR polynomial we examine the p a r t i a l a u t o c o r r e l a t i o n parameters i n equations (3.12) - the kth" <j> i n the k t h equation of (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 . These (}>..;. c o e f f i c i e n t s are found by s o l v i n g ktc J J 1 the Yule Walker equations f o r j = 1,2, k. Should the "*-n t^ i e ^+"*" equation of an AR(k+l) be zero (^ k + 1 k+-^ = 0)_> then we conclude t h a t p k has at c u t - o f f p o i n t a t k+1 and the process i s of le n g t h k. T h i s i s our i d e n -t i f i c a t i o n r u l e f o r an AR(p) process: i f the 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 the 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 the expected order of the au t o r e g r e s s i v e process. For the MA process ^of order q z = 8 (B) a^, i t s v a r i a n c e and auto-covariances are given by (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 i s obvious t h a t the autocovariance y, i s zero f o r a l l k g r e a t e r than the l e n g t h of the polynomial q. (3.13.1) Y 0 = E [ ( a t - O . a ^ - ... - e q a t _ q ) ] 2 2 2 2 = c ( i +• e , + ...+•• e ) a 1 q (3.13.2) Y l = E [(a t - 0 ^ - ... - e q a t _ q ) ( a t _ k - ^ a ^ - . . . q t-k-q' = a 2 ( - 0 k + e 1 0 k + 1 + ... + e q _ k 0 q ) k < q Using our d e f i n i t i o n of a u t o c o r r e l a t i o n (3.10) we def i n e the a u t o c o r r e l a t i o n f u n c t i o n f o r the MA(q) process by (3.14) (3.14) p k = - 0 + 0 0 +. ..+ 0 . 0 k . 1 k+1 . q-k q k = 1,2,. . .q 1 + 0 2 +...+• 0 2 q 0 k > q This r e s u l t y i e l d s an obvious i d e n t i f i c a t i o n r u l e f o r the MA(q) process f o r i f 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 the p a r t i 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 t a i l s o f f , then q i s the expected order of the moving average process. A mixed process ARMA (p,q) i s suggested i f both 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 the p a r t i a l a u t o r e g r e s s i v e f u n c t i o n t a i l o f f . The au t o c o r r e -l a t i o n f u n c t i o n has an i r r e g u l a r p a t t e r n at l^g s 1 through q, then i t t a i l s o f f according to i t s f u n c t i o n a l v a l u e s . Conversely the p a r t i 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 i s dominated by an i r r e g u l a r p a t t e r n at l a g s p-q and on-ward. The presence of any seasonal i n f l u e n c e i n the data can be observed by sear c h i n g 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 p a r t i a l 32 a u t o c o r r e l a t i o n f u n c t i o n that appear at r e g u l a r i n t e r v a l s . The l e n g t h of the seasonal i s suggested by the d e f i n i t i o n of the data ( i . e . , monthly, q u a r t e r l y , e t c ) , and the l e v e l s of (P,D,Q) w i l l be those i n which the spi k e s at 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 of Parameters 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 the s e r i e s as an ARIMA (p,d,q) X (P,D,Q)s, the next step i s to estimate the p a u t o r e g r e s s i v e parameters, P seasonal a u t o r e g r e s s i v e parameters, q moving average parameters and Q seasonal moving average parameters. 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 to minimize the squared - ;dfference between the a c t u a l value of z^ and i t s estimated value z^. We proceed to maximize the l i k e l i h o o d f u n c t i o n o f the j o i n t normal d i s t r i b u t i o n of 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 out a l l combinations of. v e c t o r s (J) , $ ,8, 0 and a such t h a t they maximize the l i k e l i h o o d of these parameters by m i n i m i z i n g the sum of squares i n (3.15). (3.15) exp [ - 2 ^ 2 £ U t / <f>, * , 6 , 0 , z ] 2 ) To b e g i n the e s t i m a t i o n procedure we must g i v e s t a r t - u p v a l u e s to the par a -meters i n the f o u r polynomials and t o get the a l g o r i t h m s t a r t e d we must c a l c u l a t e the a f c q ' s which s p e c i f y the moving average p a r t of the model. The values are estimated by u s i n g the given parameters by back f o r e c a s t i n g on z (or i t s d i f f e r e n c e d l e v e l ) (see Box and J e n k i n s , 1970, p. 212-220). Assume that the model i s an ARMA (.0,0,1), i n i t i a l i z e the valu e of 0 and express the model i n terms of the forward s h i f t o p e r ator i?' where Fe^=efc, n 33 (3.16) z t = ( 1 - 0 B ) a t and zfc = ( l ^ - 0 F ) e I0I<1 Set E [ e T _ ^ 0 ,z tJ. = 0 and s o l v e f o r Z q by back f o r e c a s t i n g . (3.17) E [ e T | G 3 z t ] = E [z T] + 0E t e ^ l S y z , . ] ••'•= z T E [ e T - l 1 '• Q ) Z t ] = E [ Z T - 1 ] + G E [ eT 1 G ^ Z t ] = ZT-1 + 0 2 T E [ e o J 0 J )z t] = E [ Z q ] + GE [ e± \ Gz^ MO This l a s t equation i n (3.17) gives us an i n i t i a l v alue f o r E [ Z Q ] =-0:,:[e^( GjZ ] from which we can s t a r t our forward f o r e c a s t i n g f o r a's c o n d i t i o n a l on 0 and z . R e c a l l that E I G , z ] = 0 s i n c e i t i s d i s t r i b u t e d i n -dependently of zfc E [ a Q | G z t ] = E [Z q ] + GE [ a ; L | Q y z^] = E [ z o ] (3.18) E [ a 1 | 0:0z t] = E [ Z ^ + ' 0 E [a | ' 0, z ] = E [ z ^ + 0 E [ z o E [ a T | 0 i Z t ] = E [z T] +. G E t [ a T _ 1 ( 0 , Z { J =E [ z T ] + 0 E [ z T Our i n p u t s are now complete and we can s t a r t the i t e r a t i o n s f o r the non-l i n e a r e s t i m a t i o n of the parameters that w i l l minimize the sum of squares f u n c t i o n . The a l g o r i t h m used i n the computer program i s Marquandt's i t e r -a t i v e procedure which i s a compromise between the methods of Gauss-Newton and steepest descent (Nelson (1974), p. 8 ) . The estimated c o e f f i c i e n t s are then examined f o r t h e i r s i g n i f i c a n c e l e v e l s and the model i s diagnosed w i t h respect to the estimated r e s i d u a l s . The former i s done by t e s t i n g the hypothesis that any parameter i s d i f f e r e n t from zero ( t - t e s t ) . I n p a r t i c u l a r the hypothesis 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 should we accept the n u l l hypothesis we would take f i r s t d i f f e r e n c e s of the data and t e s t i f cf) = 1 i n <j> ^ z t _ 1 » e t c . I f the constant term of the raw data i s s i g n i f i c a n t l y non-zero then we a l s o conclude.that there i s a d r i f t i n the s e r i e s and that i t s mean i s not independent of time. The presence of s e a s o n a l i t y can be a s c r i b e d to the s i g n i f i c a n c e of e i t h e r a u t o r e g r e s s i v e or moving average c o e f f i c i e n t s of order 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 the hypothesis t h a t the p o p u l a t i o n represented by the 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 the data come from the same p o p u l a t i o n . This non-parametric t e s t i s the Kolmogorov-Smirnov S t a t i s t i c which giv e s con-f i d e n c e bands f o r the d i f f e r e n c e s between the d i s t r i b u t i o n f u n c t i o n of both p o p u l a t i o n s . The second i s the hypothesis t h a t we have reduced the model to white n o i s e . This i s a chi-square t e s t known as the Box-Pierce S t a t i s -t i c and i f the sum of the 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 the e r r o r s i s l e s s than the chi-square value w i t h (k-p-q-P-Q) degrees of freedom, then the r e s i d u a l s are s a i d to be random. D. F o r e c a s t i n g , The concern 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 of c o e f f i -c i e n t s stems from the d e s i r e to have e f f i c i e n t f o r e c a s t s f o r the s e r i e s z^ +^ where 1 i s some p e r i o d i n t o the f u t u r e . E f f i c i e n c y i s de f i n e d here i n the same way as i n e s t i m a t i o n : minimize the mean square e r r o r . When the model i s i d e n t i f i e d and parameter estimates are o b t a i n e d , the a l g o r i t h m once again generates the disturbance terms (a^.'s) and we can e a s i l y f o r e c a s t next period's value f o r z ^ c o n d i t i o n a l on t h i s period's a u t o r e g r e s s i v e 35 parameters and moving average terms. I t should be noted t h a t i f the raw-data has been transformed to n a t u r a l logarithms then the f o r e c a s t s are 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 based on the log-normal d i s t r i b u t i o n (Nelson (1973), p. 161-163). E. T r a n s f e r Function Throughout t h i s chapter we have shown t h a t a p a r t i c u l a r times s e r i e s can be represented by an ARIMA process but i t may a l s o be s e n s i t i v e to some e x t e r n a l shocks. I f our s e r i e s z f c i s dependent on the c u r r e n t exoge-nous v a r i a b l e Xfc or i t s past values X t_^'s then equation (3.19) i s the t r a n s f e r f u n c t i o n f o r z ( i . e . a s t r u c t u r a l e q u a t i o n ) . (3.19) z t =-u1\ + o-2 Xt_1 + ... + or n X t _ n + afc = a (B) Xfc + a f c In our context of time s e r i e s f o r demand d e p o s i t s and term d e p o s i t s , the economic theory reviewed by Chapter I I suggest l i k e l y candidates f o r the X's. I n p a r t i c u l a r f o r term d e p o s i t s the most l i k e l y independent v a r i a b l e i s the i n t e r e s t r a t e on term 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 study w i l l examine the 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 equation s p e c i f i e d i n (3.19), w i l l be compared w i t h the ARIMA models f o r i t s power to p r e d i c t f u t u r e values of time d e p o s i t s . The former i s measured by the t - t e s t and the l a t t e r i s measured by the mean square e r r o r . 36 IV Data 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 union i s a f i n a n c i a l i ntermediary that i s an autonomous ' e n t i t y . I t has 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 of D i r e c t o r s and management, and i t s own borrowing and l e n d i n g p o l i c i e s . The common bond.-; of a s s o c i a t i o n may vary from employees i n one f i r m to a community of a m i l l i o n people. The most homogeneous c r e d i t unions were to be con-s i d e r e d i n order to compare the estimated ARIMA models. The c r e d i t unions t h a t had the lo n g e s t h i s t o r y of p r o v i d i n g both demand d e p o s i t s and term de p o s i t s to t h e i r members would provide the l a r g e s t number of o b s e r v a t i o n s to e v a l u a t e f o r e c a s t s . The i n t e r s e c t i o n of these two c r i t e r i a r e s u l t e d i n the choice of three of the l a r g e s t c r e d i t unions i n the Vancouver Metro-p o l i t a n Region i n B r i t i s h Columbia and they had the f o l l o w i n g c h a r a c t e r i s -t i c s : ( i ) they f a c e the same e x t e r n a l market; ( i i ) they have the same com-mon bond of community a s s o c i a t i o n ; ( i i i ) they are m u l t i - b r a n c h o p e r a t i o n s ; and ( i v ) they each have over 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 both demand dep o s i t s and term d e p o s i t s are o f f e r e d to t h e i r members. The two accounts examined are t o t a l demand dep o s i t s and t o t a l term d e p o s i t s . The c r e d i t unions p r i m a r i l y d e a l w i t h non-corporate bodies and economic theory suggests two d i f f e r e n t behaviours by consumers to the two types of d e p o s i t s ; ( i ) demand d e p o s i t s are purchased 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 convenience i n c a r r y i n g out t r a n s a c t i o n s ; and ( i i ) term de-p o s i t s are purchased as an investment-of savings i n low r i s k s e c u r i t i e s . The data are gathered q u a r t e r l y and date back to the second q u a r t e r of 1962. From 1962;2to 1974:4 we have 52 observations f o r demand d e p o s i t s and 37 no l e s s than 36 observations f o r term d e p o s i t s (some c r e d i t unions d i d not s t a r t to o f f e r these s e r v i c e s u n t i l 1966)^ The three c r e d i t unions examined below are r e f e r r e d t o as C.U.I, C.U.2 and C.U.3 and they are ordered w i t h respect to t h e i r asset s i z e (This data s e r i e s are l i s t e d i n Appendix:Data). B. Model f o r Chartered Banks' Deposits 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 of the c r e d i t unions' s e r i e s , but more important to evaluate the s t r e n g t h of the time s e r i e s method t o be used, we f i r s t l ook at the behaviour of demand d e p o s i t s and term 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 which we have a l a r g e r num-ber of data p o i n t s . The time s e r i e s are the t o t a l of p e r s o n a l demand de-p o s i t s and of 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 banks i n Canada as p u b l i s h e d monthly i n the Bank of Canada S t a t i s t i c a l Review (1967:9 - 1974: 11, 87 o b s e r v a t i o n s ) . . The Box and Jenkins technique suggests a monthly s e a s o n a l model f o r demand d e p o s i t s . The s i m p l e s t model whose c o e f f i c i e n t s proved to be s t a -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 data has been transformed to n a t u r a l l o g a r i t h m s . 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 been taken and that there i s a s i g n i f i c a n t seasonal moving average term t h a t determines the l e v e l of demand deposits i n banks. Indeed t h i s a c c u r a t e l y d e p i c t s the seasonal troughs of June and December„when the absolute l e v e l s of these d e p o s i t s decrease w i t h r e s p e c t to the balance 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 the Box-Pierce S t a t i s t i c both suggest 38 t h a t the f i t t e d model adequately represents the data (equation 4.1) (4.1) Vln-DD. - V l n DD = .008 + .128 a B&P = 23 n-k = 22 . Z (.001) (.012) Z L l The transformed s e r i e s of term d e p o s i t s was i d e n t i f i e d to be (1,1,1) model (data transformed t o l°g e) a i l d the d i a g n o s t i c checks suggest t h a t the r e s i d u a l s are w h i t e n o i s e (Equation 4.2). (4.2) V l n TD = .87Vln TD - .34 a - B&P = 13 n-k = 2 2 Z (.06) Z ~ X (.12) t ~ 1 The t r a n s f e r f u n c t i o n was a l s o t e s t e d f o r term d e p o s i t s (TD) , where the 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 (R) and time (T). Both c o e f f i c i e n t s had the r i g h t s i g n and were s i g n i f i c a n t s t a t i s t i c a l l y (although the standard e r r o r s are underestimated due to h i g h 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 are not s i g n i f i c a n t ) . For the equation w i t h R and TD lagged one p e r i o d , the monthly model again r e -vealed the expected sig 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 + .07R + ..02T D.W. = .07 R 2 = .93 (.07) (.01) (.001) (4.3.2) InTD = .20 + .01R + .97 lnTD_ 1 D.W. = .54 R 2 = .99 (.04) (.002) (.005) Thus our d i g r e s s i o n on banks has confirmed our "a p r i o r i " behaviours of the two s e r i e s i n r e l a t i o n to consumers' choices and p r e f e r e n c e s , i t has suggested an order of 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 of t r a n s f e r f u n c t i o n s , and the time s e r i e s a n a l y s i s has been shown to f i t the data. 39 C. Demand f o r C r e d i t Unions' Demand Deposits Upon i n s p e c t i o n of 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 three c r e d i t unions i t was evident t h a t there i s an e x p o n e n t i a l growth trend i n the data. The best model of demand d e p o s i t s of C.U. 1 and C.U.3 (transformed i n t o n a t u r a l logarithms) was the second d i f f e r e n c e f i r s t order moving average model.For C.U.2 there were two models, not 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 from z e r o : ( i ) seasonal 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 study of banks' per s o n a l demand deposits showed a s i g n i f i c a n t seasonal trough at the end of the s i x t h and t w e l f t h months ( i d e n t i c a l to our 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 expect a s i m i l a r seasonal p a t t e r n f o r demand dep o s i t s of c r e d i t unions. However, i f one i s u s i n g q u a r t e r l y data as we are i t i s not obvious t h a t the observations suggest a seasonal p a t t e r n . As f i g u r e IV.lshows, the dotted l i n e i s the expected monthly l e v e l w i t h June and December being s i g n i f i c a n t l y lower than May and November ( r e -s p e c t i v e month p r i o r ) . Whereas the monthly data suggest t h a t DD^^DD^ and DD. <DDC, they do not suggest t h a t DD 0 (4th q u a r t e r ) < DD (3rd q u a r t e r ) o J xz y or t h a t DDg (2nd q u a r t e r < DD^ (1st q u a r t e r ) , so that our q u a r t e r l y data does not p i c k up the seasonal demand f o r c r e d i t union demand d e p o s i t s — e x -cept i n the case of C.U.2. The best equations and t h e i r parameters are l i s t e d below i n Table 4.1. We are s a t i s f i e d that the unexplained v a r i a n c e i n these models i s white n o i s e . 41 TABLE 4.1 ARIMA MODELS FOR DEMAND DEPOSITS OF CREDIT UNIONS (0,2,1) Vln tDD f c - VlnDD t_ 1 = a^ + 1) a ^ C.U.I 9^  = .81 CT = .09 B&P = 22 n-k = 2 3 C.U.2 Q% .88 c = .07 B&P = 17 n-k = 23 C.U. 3 6^ = .84 CT = .09 B&P = 11 n-k = 23 ( l , l , l ) x ( 0 , 1 , 1 ) 4 :VlnDD t - VlnDD t_ 4= ^VlnDD t_ 1+a t.-9 ( a ^ ^ 0 a f c _ 4 + 6 0 &t_5 C.U.2. ^ = -.77 CT = .05 B&P = 26 n-k = 21 \ = .46 a = .14 . 0 = -1.10 CT = .05 D. Demand f o r C r e d i t Unions' Term Deposits The s e r i e s are transformed t o n a t u r a l logarithms and the f i r s t d i f -ference f i r s t order a u t o r e g r e s s i v e scheme i s the b e s t model w i t h the s m a l l e s t number of c o e f f i c i e n t s f o r C.U.I and C.U.2 (The 4>'s are s i g n i f i -c a n t l y d i f f e r e n t from zero and one). For C.U.3 (1,1,1) model i s s i g n i f i -cant and the r e s i d u a l s of t h i s model are l e s s c o r r e l a t e d than those i n (1,1,0) model f o r the time s e r i e s . Since 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 from z e r o but l i e i n g j u s t w i t h i n the 95% confidence i n t e r v a l ( o n e - t a i l t e s t ) - the second d i f -ference model was f i t t e d . The (0,2,1) s p e c i f i c a t i o n was s i g n i f i c a n t and the e v a l u a t i o n of the three models f o r C.U.3 i s postponed u n t i l the l a t e r s e c t i o n on p r e d i c t i o n . The estimated ARIMA equations are l i s t e d below i n Table 4.2 42 TABLE 4.2 •ARIMA MODELS FOR TERM DEPOSITS OF CREDIT UNIONS (1,1,0) VlnTD t = cJ>VlnTDt 1 + a C.U.I . (J) = .62 o — .06 B&P = 28 n-k = 23 C.U.2 A • * = .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 = t cJiVlnTD + a + 0 a n C.U.3 $ = .92 • % = .70 c = .04 a = .13 B&P = 18 . n-k = 22 (;.,2,1) C.U.3 VlnTD t - V l n ' T D t _ 1 = a f c + B a ^ § = .64 .14 B&P = 18 . n-k = 23 The 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 unions' term de p o s i t s was a l s o f i t t e d . R e c a l l that i n the example f o r chartered banks the explana-t o r y v a r i a b l e s were the i n t e r e s t r a t e on 90 day term d e p o s i t s (R) and time (T). The appr o p r i a t e i n t e r e s t r a t e f o r the time s e r i e s of c r e d i t unions i s the r a t e p a i d by c r e d i t unions on a comparable s e c u r i t y . How-ever, as s t a t e d i n Chapter I I the i n t e r e s t r a t e s p a i d by c r e d i t unions are 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 the r a t e p a i d by cha r t e r e d banks 43 i s used as a proxy. Ordinary l e a s t squares was used to es t i m a t e the s i g n i f i c a n c e of R and T. For each of the three c r e d i t unions the former v a r i a b l e was not s i g n i f i c a n t l y d i f f e r e n t from zero (see.Table 4.3). Again i t i s i n t e r e s t i n g to speculate as to why t h i s r e s u l t i s so d i f f e r e n t from t h a t of the ch a r t e r e d banks. E i t h e r there has been a s t r u c t u r a l change i n promotion and preference o f term d e p o s i t s i n the 1962-1966 and 1967-1974 p e r i o d s o r th a t i n the q u a r t e r l y data the changes i n R may be too d i s c r e t e to be p o s i t i v e l y c o r r e l a t e d w i t h the new l e v e l of term de-p o s i t s . TABLE 4.3 . . . ... O.L.S. RESULTS FOR TRANSFER FUNCTION OF TERM DEPOSITS OF CREDIT UNIONS C.U.I C.U.2 C.U.3 lnTD t '= c 12.49 (.16) 11.18 (.18) 8.53 (.20) + a xR. -.01 '(.04) -.02 • (.04) -.03 (.03) .12 (.004) .13 (.005) D.W. = .29 R .97 .18 D.W (.005) D.W. = .59 R z = .97 2 ,26 R' .98 E. Forecast E v a l u a t i o n 1974:1-1974:4 The best ARIMA model f o r the demand f o r demand d e p o s i t s of c r e d i t unions i s evaluated a g a i n s t a naive random walk model (z = 0z + a ") v t t - 1 t / and a g a i n s t any other models that proved to be s i g n i f i c a n t i n . t h e estima-t i o n stage. Table 4.4 presents the r e s u l t s and the ARIMA outperform the random walk. The (0,2,1) model i s the best s p e c i f i c a t i o n f o r f o r e c a s t i n g c r e d i t unions' demand d e p o s i t s . TABLE 4.4 PREDICTION ERRORS, DEMAND DEPOSITS"OF CREDIT UNIONS 1974:1 - 1974:4 PREDICTIONS C.U.I C.U.2 (0,2,1) (1,0,0) (0,2,1) ( l , l , l ) x ( 0 , 1 , 1 ) 4 (1,0,0) C.U.3 (0,2,1) (1,0,0) Average Absolute E r r o r 0.29 0.33 0.07 0.09 0.12 0.15 0.51 Root Mean Square E r r o r 0.30 0.39 0.09 0.11 0.20 0.21 0.80 2 !- ' note: the root-mean square-error i s (Za In)2, where a's are the e r r o r s , the summation i s over a l l o b s e r v a t i o n s , and n i s the number of observations S i m i l a r l y the v a r i o u s models of term d e p o s i t s s e r i e s are evaluated w i t h respect to the minimum mean square e r r o r c r i t e r i a f o r the q u a r t e r l y f o r e c a s t s i n 1974. In a l l cases the ARIMA models outperformed the t r a n s f e r f u n c t i o n s and the random walk equation (Table 4.5) i l l u s t r a t e s t hat the best models f o r demand of c r e d i t unions' TD are 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 the p l o t of the c a l c u l a t e d and a c t u a l values of 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 almost a l l the cases 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 the a c t u a l balances of de p o s i t s h e l d by C.U.I, C.U.2 or C.U.3. TABLE 4.5 PREDICTION ERRORS, TERM DEPOSITS OF CREDIT UNIONS 1974:1 - 1974:4 PREDICTIONS Average Root Mean Absolute E r r o r Square E r r o r C.U.I (1,1,0) 0.23 0.23 (1,0,0) 0.32 0.35 f(R,T) 0.26 0.27 C.U.2 (1.1.0) 0.09 0.11 (1,0,0) 0.38 0.44 f(R,T) 0.61 0.65 C.U.3 (1.1.1) 0.11 0.11 (1,1,0) 0.12 0.18 (0,2,1) 0.28 0 30 (1,0,0) 0.37 0.41 f(R,T) 0.67 • 0 69 note: see note i n t a b l e 4.4; f(R,T) i s the t r a n s f e r f u n c t i o n where R i s the i n t e r e s t r a t e and T i s time as d e f i n e d above IPililit^ 47 V. Concluding Remarks We have found t h a t the ARIMA models f u r n i s h the 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 term d e p o s i t s of c r e d i t unions and f o r p e r s o n a l demand de p o s i t s and term de p o s i t s of chartered banks. For demand d e p o s i t s , the (0,2,1) model best describes the demand f o r c r e d i t unions' d e p o s i t s and the (0,1,0) x (0,1,1) 12 process i s the one we i d e n t i f i e d f o r c h a r t e r e d banks. For term d e p o s i t s , CJJt 1 and CIL 2 data f o l l o w a (1,1,0) process w h i l e the (1,1,1) model i s the best 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 banks. In a l l the cases we are s a t i s f i e d t h a t the unexplained v a r i a t i o n i n the 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 white n o i s e . The p r e d i c t i o n s f o r 1974 proved to be too o p t i m i s t i c . T h i s suggests t h a t constant feedback must be maintained i n order to update the f o r e c a s t s and to monitor the t u r n i n g p o i n t s i n the s e r i e s . I t i s l i k e l y t h a t the values of the parameters may change as these ARIMA models are f i t t e d to new data p o i n t s . Future research should t r y to use monthly d a t a because there i s a l i k e l y seasonal p a t t e r n that i s not being p i c k e d up by the q u a r t e r l y data. This w i l l a l s o give more observations to the time s e r i e s and s t r e n g t h -en the 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 another problem w i t h having used the q u a r t e r l y s e r i e s f o r the years between 1962 and 1974. This p e r i o d i s by 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 or f o r c r e d i t unions i n B r i t i s h Co-lumbia. 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 at which time the Bank Act was changed and chartered 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 market. There has been a marked s h i f t i n the growth r a t e of c r e d i t unions' assets i n B r i t i s h Columbia s i n c e 1970 and perhaps the under-l y i n g p a t t e r n of the data i s not the same as f o r the 1962-69 p e r i o d . I f 48 t h i s s h i f t should be s i g n i f i c a n t then our 1962-74 ARIMA models may have in 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 the f o r e c a s t s . As the number of observations 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 to t e s t the homogeneity of the time s e r i e s f o r c r e d i t union d e p o s i t s . Thus our t h e s i s has s u c c e s s f u l l y modelled the time s e r i e s f o r demand d e p o s i t s and term d e p o s i t s of a c r e d i t union. The f i n a n c i a l manager i n a c r e d i t union can generate the 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 our 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 of 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 Interdependence i n Demand f o r Savings D e p o s i t s " , Journal of Finance, May, 1973, v o l XXV I I I , No. 2, p. 507-514. Box, G. and G. Jenkins (1970). Time Series Analysis: forecasting and control, Holden Day, San F r a n c i s c o , 1971. Boyd, J . (1973). "Some Recent Developments i n the Savings 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 Determinants of Supply and Demand f o r C e r t i -f i c a t e s of Deposit", Journal of Money, Credit and Banking, February, 1973, V o l . v, No. 1, p. 100-112. Cramer, R. and R. M i l l e r (1973). 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(1974), " A l t e r n a t i v e Temporal Cr o s s - S e c t i o n S p e c i f i c a t i o n s of the Demand f o r Demand Deposi t s " , Journal of Finance, June, 1974, v o l . XXIX, no. 3, p. 923-940. Friedman, M. (1957). A Theory of the Consumption Function, N a t i o n a l Bureau of Economic Research, P r i n c e t o n , 1957. Motley, B. (1970). "Household Demand f o r A s s e t s : A Model o f Short-Run Adjustments", Review of Economics^ and Statistics, August, 1970, V o l . X I I , No. 3, p. 236-241. Nelson, C. (1973). Applied Time Series Analysis for Managerial Fore-casting, Holden-Day, San F r a n c i s c o , 1973. 328629. 670283. 1145867. 1570730. 1976460. 3451364. 5338262. 6573045. DEMAND DEPOSITS C 283728. 288410. 277601. 397636. 483302. 664819. 700115. 1286335. 1168532. 1361218. 1561367. 2667036. 2467194. 3778215. 4433987. 6508993. 6116369. 7154165. 8641735. 13240078. 16313127. 17623152. 21584896. 24559728. 27228336. 38420704. 37876336. 32794640. 32716080. 30331968. 29406736. DEMAND DEPOSITS LN C 12.556 12.572 12.534 12.703 12.893 13.088 13.415 13.407 13.459 13.952 14.067 13.971 14.267 14.124 14.261 14.497 14.796 14.719 15.054 15.145 15.305 15.490 15.689 15.626 15.698 15.783 15.972 16.399 16.607 16.685 16.887 17.017 17.120 17.464 17.450 17.306 17.303 17.228 17.197 TERM DEPOSITS C 1. 85500. 231000. 507000. 616500. 695000. 863500. 949723. 1130535. 1591329. 1727357. 2198997. 2568230. 3113829. 3496095. 4101678. 4466207. 5006250. 5940851. 6321382. 6540961. 7795252. 8087713. 8395023. 9291387. 10108465. 13068507. 19320080. 21975056. 29106528. 35523872. 40226720. 49773856. 66961040. 76191184.104511584. 121612864.120154240.131267616. TERM DEPOS ITS LN C .U.I 290200. 579923. 859983. 1315540. 1623500. 3069123. 4638152. 6064195. 9423604. 19440843. 32179760. 35345552. .U.I 12.578 13.271 13.665 14.090 14.300 14.937 15.350 15.618 16.059 16.7 83 17.287 17.381 .U.I 390500. 800000. 1278997. 2412211. 39469 60 . 5546293. 7158364. 8454867. 15679535. 32807952. 58248880. 107059568. .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 14.280 14. 362 14.604 14.696 14.759 14. 951 15.067 15.188 15 .227 15. 312 15.426 15.529 15 .597 15. 659 15.694 15.784 15 .869 15. 906 15.943 15.95 0 16.045 16. 129 16.386 16.568 16.777 16. 905 17.186 17.306 17.386 17. 510 17.723 17.880 18 .020 18. 149 18.465 18.489 18.616 18 . 604 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. 54531. 58095. 54238. 50398. 61514. 67321. . 70655. 102755. 85070. 83590. 104508. 129995 . 145659. 193789. 247533 . 286000 . 300776. 373105. 470602. 486069. 588993. 719561. 890417. 948713. 969893. 1210128. 1489363. 1501195. 2063821. 2392084. 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 11.540 11.351 11.334 11 .557 11.775 11.889 12 .175 12-419 12.564 12.614 12.830 13 .062 13.094 13.286 13.486 13 .699 13.763 13.785 14.006 14.214 14.222 14.5 40 14.688 15.040 15. 165 15.388 15.584 15-893 15.976 16.167 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. 2932926- 3343537. 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 10.985 11.482 11.640 11.794 12 -081 12.134 12.233 12.804 12.947 13.253 13.663 13.829 14.201 14.290 14.344 14.423 14.581 14.702 14.892 15.023 15.304 15.585 15 .803 15.916 15.999 16.072 16.268 16.355 16 .465 16.658 16.924 16.971 17.096 17.201 17.231 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 X10**6 CANADIAN BANKS 10443 . 10535. 10532. 10367. 10461. 10539. 10587. 10694. 10767. 10702. 10768. 10200. 10865 . 11031. 11038. 10979. 11021. 11077. 11136. 11281. 11302. 11296. 11357. 11394. 11426- 11516. 11473. 1 1297. 11355. 11463. 11543 . 11696. 11871. 11742. 11716. 11888. 12002 . 12125. 12101. 11987 . 12106. 12239. 12367. 12782. 12945. 13156. 13418. 13654. 13879 . 14075. 13655. 13406. 13625. 13834. 14161 . 14301. 14408. 14379. 14531. 14625. 14791. 14962. 14797. 14540. 14781. 14932. 15016. 15214. 15323. 15381. 15589. 15742. 15829 . 15966 . 15834. 15699. 15867. 16029. 16204. 16601. 16940. 16860. 17043. 17167. 17343 . 17625. 17562. TERM DEPOSITS LN X10**6 CANADIAN BANKS 9.254 9.262 9.262 9. 246 9.255 9.263 9.267 9.277 9.284 9.278 9. 284 9. 230 9.293 9.308 9.309 9.304 9.308 9.313 9.318 9.331 9.333 9.332 9. 338 9.341 9.344 9.351 9.348 9.332 9. 337 9.347 9.354 9.367 9.382 9.371 9. 369 9.3 83 9.393 9.403 9.401 9.392 9.401 9.412 9.423 9.456 9.468 9.485 9. 504 9.522 9.53 8 9 .552 9.522 9.503 9.520 9.535 9.558 9.568 9.576 9.574 9.584 9.590 9.602 9.613 9.602 9. 585 9.601 9.611 9.617 9.630 9.637 9.641 9.654 9.664 9.670 9.678 9.670 9.661 9.672 9.682 9.693 9.717 9.737 9. 733 9. 743 9.751 9.761 9.777 9.773 

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