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http://clevelandfed.org/research/workpaper/index.cfm Best available copy Working Paper 8506 STOCHASTIC INTEREST RATES I N THE AGGREGATE LIFE CYCLEJPERMANENT INCOME CUM RATIONAL EXPECTATIONS MODEL b y Kim J. Kowalewski Working papers o f t h e Federal Reserve Bank o f C l e v e l a n d are preliminary materials, c i r c u l a t e d t o stimulate d i s c u s s i o n and c r i t i c a l comment. The views expressed h e r e i n a r e those o f t h e a u t h o r and n o t n e c e s s a r l i y those o f t h e Federal Reserve Bank o f C l e v e l a n d o r t h e Board o f Governors o f the Federal Reserve System. James Hoehn, E r i c Kades, and A l a n Stockman p r o v i d e d u s e f u l comments on an e a r l i e r d r a f t . Gordon Schlegel p r o v i d e d e x c e l l e n t r e s e a r c h assistance. O c t o b e r 1985 F e d e r a l Reserve Bank o f Cleveland STOCHASTIC INTEREST RATES I N THE AGGREGATE http://clevelandfed.org/research/workpaper/index.cfm Best available copy LIFE CYCLEIPERMANENT INCOME CUM RATIONAL EXPECTATIONS MODEL B a r r o ' s n e u t r a l i t y h y p o t h e s i s s t a t e s t h a t t h e impacts o f f e d e r a l government spending and t a x p o l i c i e s do n o t depend on how t h e p o l i c i e s a r e f i n a n c e d . An i m p o r t a n t assumption o f t h i s h y p o t h e s i s i s t h a t p r i v a t e agent spending horizons- - the t i m e span o v e r which a permanent i n c r e a s e i n l i f e c y c l e wealthlpermanent income i s consumed--have e s s e n t a i l l y i n f i n i t e l e n g t h , so t h a t (expected) changes i n l i f e c y c l e wealthlpermanent income do n o t a f f e c t c u r r e n t spending. I f h o r i z o n s a r e s h o r t e r , t h e n monetary and f i s c a l s t a b i l i z a t i o n p o l i c i e s may a f f e c t r e a l v a r i a b l e s a t l e a s t i n t h e s h o r t r u n . I n a w o r l d o f p r o p o r t i o n a l income taxes, a necessary c o n d i t i o n f o r i n f i n i t e h o r i z o n l e n g t h s i s t h e e x i s t e n c e o f p e r f e c t c a p i t a l markets. P e r f e c t c a p i t a l markets c o n t a i n no t r a n s a c t i o n s o r o t h e r c o s t s t h a t d r i v e a wedge between b o r r o w i n g and l e n d i n g r a t e s o f i n t e r e s t . They c o n t a i n no i n f o r m a t i o n a l asymmetries o r o t h e r i m p e r f e c t i o n s t h a t a r e c o n t r o l l e d by down payments, c o l l a t e r a l r e q u i r e m e n t s and s e c u r i t y i n t e r e s t s , c o l l a t e r a l maintenence p r o v i s i o n s , q u a n t i t y l i m i t a t i o n s on t h e amount o f d e b t extended t o a s i n g l e borrower, and o t h e r non- price l o a n p r o v i s i o n s . A l l o f these c a p i t a l market i m p e r f e c t i o n s c o n s t r a i n t h e l i q u i d i t y o f consumer l i f e c y c l e w e a l t h 1 permanent income t o be l e s s than i t s f u l l v a l u e ; consumers cannot borrow a g a i n s t t h e f u l l v a l u e o f t h e i r l i f e c y c l e w e a l t h , o r can do so o n l y a t a penalty r a t e o f interest. The absence o f p e r f e c t c a p i t a l markets thus a l l o w s t h e p o s s i b i l i t y t h a t h o r i z o n s a r e shortened. A number o f r e c e n t s t u d i e s have examined consumption behavior f o r evidence about t h e l e n g t h o f consumer spending h o r i z o n s . The approach taken by most r e c e n t s t u d i e s i s t o t e s t some v a r i a n t o f t h e l i f e cyclelpermanent income cum http://clevelandfed.org/research/workpaper/index.cfm Best available copy r a t i o n a l e x p e c t a t i o n s (RE-LCIPI) model assuming p e r f e c t c a p i t a l m a r k e t s . R e j e c t i o n o f t h e RE-LC/PI model i n c o r p o r a t i n g p e r f e c t c a p i t a l m a r k e t s i s t a k e n t o mean t h a t h o r i z o n l e n g t h s may n o t be l o n g enough t o d i m i n i s h t h e power of stabilization policies. H a l l ( 1 9 7 8 ) , F l a v i n (1981,1985>, Hayashi ( 1 9 8 2 ) , M u e l l b a u e r ( 1 9 8 3 ) , Wickens and Molana (1984), Bernanke ( 1 9 8 2 ) , Mankiw (1983), DeLong and Summers ( 1 9 8 4 > , B o s k i n and K o t l i k o f f ( 1 9 8 4 ) , K o t l i k o f f and Pakes (19841, and Mankiw, Rotemberg, and Summers (1985) t e s t t h e model w i t h a g g r e g a t e t i m e s e r i e s d a t a , w h i l e H a l l and M i s h k i n ( 1 9 8 2 > , Bernanke (19841, and Hayashi (1985) use c r o s s - s e c t i o n o r p a n e l d a t a o n i n d i v i d u a l h o u s e h o l d s . O f t h e s t u d i e s e m p l o y i n g m i c r o - d a t a , o n l y Bernanke (1984) c a n n o t r e j e c t t h e model. O f t h e s t u d i e s t h a t employ a g g r e g a t e t i m e s e r i e s d a t a , H a l l ( 1 9 7 8 > , Hayashi (1982), Mankiw (1983), Bernanke ( 1 9 8 4 ) , and D e l o n g and Summers (1984) c a n n o t r e j e c t t h e model d u r i n g t h e p o s t - W o r l d War I 1 p e r i o d . K o t l i k o f f and Pakes (1984) can r e j e c t t h e model, b u t c o n c l u d e t h a t t h e d i f f e r e n c e s f r o m t h e model a r e n o t l a r g e enough t o m a t t e r i n p r a c t i c e . None o f t h e r e c e n t models e s t i m a t e d w i t h U.S. allows for uncertain r e a l i n t e r e s t rates. aggregate time s e r i e s d a t a A l l of t h e models e x c e p t Bernanke (1982) and Mankiw (1983) assume t h a t t h e r e a l i n t e r e s t r a t e i s c o n s t a n t . Bernanke (1982) and Mankiw (1983) a l l o w r e a l i n t e r e s t r a t e s t o v a r y , b u t assume t h a t consumers know a l l f u t u r e r e a l i n t e r e s t r a t e s . It i s rather c u r i o u s t h a t s t o c h a s t i c r e a l i n t e r e s t r a t e s have been i g n o r e d , because t h e r e a l i n t e r e s t r a t e i s a k e y v a r i a b l e i n t h e L C I P I models (and many New C l a s s i c a l models). Changes i n i n t e r e s t r a t e s , e x p e c t e d o r unexpected, s h o u l d l e a d t o a r e a l l o c a t i o n o f consumption s p e n d i n g a c r o s s t i m e . Thus, an a l l o w a n c e f o r s t o c h a s t i c r e a l i n t e r e s t r a t e s s h o u l d p r o v i d e a more p o w e r f u l t e s t o f t h e RE-LCIPI model and i n d i r e c t l y of t h e l e n g t h o f t h e r e p r e s e n t a t i v e consumer's spending h o r i z o n . http://clevelandfed.org/research/workpaper/index.cfm Best available copy The purpose o f t h i s paper i s t o e s t i m a t e a RE-LCIPI model t h a t a l l o w s f o r uncertain future interest rates. The model i s developed by M u e l l b a u e r (1983) and Wickens and Molana (19841, who used U.K. d a t a . I n addition, the Hall (1978) and F l a v i n (1981) models a r e r e e s t i m a t e d t h r o u g h 1984:IVQ. t h e H a l l and F l a v i n models serves a t l e a s t f o u r purposes. Updating F i r s t , t h e updates h e l p p u t t h e r e s u l t s f r o m M u e l l b a u e r ' s model i n p e r s p e c t i v e . Second, b y e s t i m a t i n g t h e models t h r o u g h 1984, t h e s t a b i 1 it y o f these models may be examined. models. T h i r d , i t i s i n t e r e s t i n g t o know how t h e 1980s d a t a f i t t h e s e Real o u t p u t and p r i c e s moved o v e r wide l a t i t u d e s d u r i n g t h e 1980s and, hence, o f f e r macroeconometricians a r i c h s e t o f h i g h i n f l u e n c e d a t a t o h e l p them e s t i m a t e c o e f f i c i e n t s more p r e c i s e l y . I t i s l i k e l y t h a t t h e 1980s d a t a p r o v i d e even s t r o n g e r evidence a g a i n s t t h e RE-LCIPI model t h a n was found by F l a v i n . F i n a l l y , t h e d i f f e r e n t models a r e e s t i m a t e d w i t h d i f f e r e n t i n f o r m a t i o n s e t s (reduced forms) and d i f f e r e n t sample p e r i o d s . It is reasonable t o wonder i f e i t h e r t h e c o n t e n t o f t h e i n f o r m a t i o n s e t o r t h e e s t i m a t i o n p e r i o d has a l a r g e i n f l u e n c e on t h e e s t i m a t e s . The i n t e r e s t i n these models i s n o t o n l y t h a t t h e RE-LCIPI model i s accepted o r r e j e c t e d , though i t i s a v e r y i m p o r t a n t c o n s i d e r a t i o n . I f these models a r e t o be u s e f u l f o r p o l i c y m a k i n g and f o r e c a s t i n g , t h e y should be r o b u s t t o d i f f e r e n t assumptions about t h e u n d e r l y i n g s t r u c t u r e used t o d e r i v e t h e reduced forms. The f i r s t s e c t i o n o f t h i s paper r e v i e w s t h e RE-LCIPI model and d e r i v e s t h e model w i t h s t o c h a s t i c i n t e r e s t r a t e s . The second s e c t i o n o u t l i n e s t h e procedures f o l l o w e d i n e s t i m a t i n g t h e t h r e e models and e x p l a i n s t h e r e s u l t s . The l a s t s e c t i o n concludes t h e paper. I. THE LIFE CYCLEIPERMANENT INCOME MODEL WITH RATIONAL EXPECTATIONS T e s t s o f t h e RE-LCIPI model b e g i n w i t h H a l l (1978). The consumer i s assumed t o maximize t h e expected p r e s e n t d i s c o u n t e d v a l u e o f c u r r e n t and http://clevelandfed.org/research/workpaper/index.cfm Best available copy future u t i l i t y . Income i s exogenous and known i n t h e c u r r e n t p e r i o d , b u t unknown t h e r e a f t e r ; t h e consumer's c h o i c e v a r i a b l e i s t h e l e v e l o f consumption each p e r i o d . The h o r i z o n b e g i n s w i t h t h e c u r r e n t p e r i o d and ends a t t h e (known) l a s t p e r i o d o f t h e consumer's l i f e t i m e . c a p i t a l market i m p e r f e c t i o n s . There a r e no bequests and no Expectations are r a t i o n a l - - f u n c t i o n s o f a l l i n f o r m a t i o n a v a i l a b l e i n t h e c u r r e n t p e r i o d ; r e a l i n t e r e s t r a t e s and r a t e s o f t i m e p r e f e r e n c e a r e assumed c o n s t a n t . T-t subject t o 1 where + i=o I - The model i s : J CRiy,+, 1 = A, 6 i s t h e i n v e r s e o f 1 p l u s t h e pure r a t e o f t i m e p r e f e r e n c e , assumed c o n s t a n t , R i s the inverse o f 1 p l u s the r e a l , a f t e r - t a x r a t e o f i n t e r e s t r a l s o assumed c o n s t a n t , (S,R>, C i s r e a l l i f e c y c l e consumption ( n o t N I A personal consumption e x p e n d i t u r e s ) , y i s r e a l l a b o r income, A i s c u r r e n t r e a l nonhuman w e a l t h , U(.) i s t h e instantaneous u t i l i t y f u n c t i o n , and E, i s t h e e x p e c t a t i o n s o p e r a t o r , c o n d i t i o n e d on t h e i n f o r m a t i o n a v a i l a b l e a t t i m e t ( v a r i a b l e s dated t - 1 and e a r l i e r ) . The f i r s t o r d e r c o n d i t i o n s f o r t h i s problem a r e : (2a) EtU1(C,+,> = (R/$>E,U'(C,+,-,> (2b) E t U 1 ( C t + ,> = i=lt o T- t; i n p a r t i c u l a r , f o r i=l, (R/S>U1(C,>. There a r e two t h i n g s t o note about ( 2 b ) . F i r s t , C, can be t h o u g h t o f as a http://clevelandfed.org/research/workpaper/index.cfm Best available copy s u f f i c i e n t s t a t i s t i c f o r C t + l ; t h a t i s , no v a r i a b l e e x c e p t Ct h e l p s p r e d i c t f u t u r e m a r g i n a l u t i l i t y of consumption U ' ( C t + l ) . Second, w i t h t h e assumption o f r a t i o n a l e x p e c t a t i o n s , m a r g i n a l u t i l i t y f o l l o w s t h e r e g r e s s i o n relation: U'(C,+l> (3) yU1(Ct) + = E,,,. c t + , r e p r e s e n t s t h e impact on m a r g i n a l u t i l i t y o f a l l new i n f o r m a t i o n t h a t becomes a v a i l a b l e i n p e r i o d t+l about t h e consumer's l i f e t i m e w e l l - b e i n g . Under r a t i o n a l e x p e c t a t i o n s , E,E,+~ U 1 ( C t > . Moreover, E = 0 and c t + , i s o r t h o g o n a l t o s h o u l d be w h i t e n o i s e , t h a t i s , u n p r e d i c t a b l e u s i n g variables i n the information set. I f t h e u t i l i t y f u n c t i o n i s q u a d r a t i c , o r " t h e change i n marginal u t i l i t y from one p e r i o d t o t h e n e x t i s s m a l l , b o t h because t h e i n t e r e s t r a t e i s c l o s e t o t h e r a t e o f t i m e p r e f e r e n c e and because t h e s t o c h a s t i c change i s s m a l l " ' Ct = yct-1 (4) + Et. That i s , l i f e c y c l e consumption f o l l o w s an AR(1) process- - no o t h e r v a r i a b l e s d a t e d t - 1 o r e a r l i e r a f f e c t C,. random walk. If y = 1 then consumption f o l l o w s a I t i s i m p o r t a n t t o n o t i c e t h a t ( 4 ) i s n o t a s t r u c t u r a l model o f l i f e c y c l e consumption b e h a v i o r . Because i t i s o n l y t h e f i r s t o r d e r c o n d i t i o n f o r u t i l i t y m a x i m i z a t i o n , i t i s o n l y an i m p l i c a t i o n o f t h e l i f e c y c l e model under r a t i o n a l e x p e c t a t i o n s . Indeed, i t i s o n l y a necessary c o n d i t i o n f o r t h i s RE-LC model t o be t r u e . H a l l a l s o shows t h a t l i f e t i m e resources e v o l v e as a random walk w i t h trend. ( 5 ) A, F i r s t , nonhuman w e a l t h f o l l o w s t h e r e l a t i o n : = R - ' ( A ~ - ~+ y t - , - c,-,). Second, human w e a l t h , H t , i s t h e sum o f c u r r e n t l a b o r income and t h e expected p r e s e n t d i s c o u n t v a l u e o f f u t u r e l a b o r income: (6) Ht = T-t 1,=o f r o m which i t f o l l o w s t h a t : [R'Et7t I, where E t y t I = yt, http://clevelandfed.org/research/workpaper/index.cfm Best available copy where p t r e p r e s e n t s t h e p r e s e n t v a l u e o f t h e changes i n e x p e c t a t i o n s o f f u t u r e income t.-h a t o c c u r between p e r i o d t - 1 and t: 1 -t Again under r a t i o n a l e x p e c t a t i o n s , E t - , p t =.-0, and p t should be Under c e r t a i n t y e q u i v a l e n c e , E t r < t b t white noise. at , where i s an a n n u i t y f a c t o r m o d i f i e d t o t a k e account o f t h e f a c t t h a t t h e consumer p l a n s t o make consumption grow a t a p r o p o r t i o n a l r a t e y o v e r h i s Then t h e e q u a t i o n f o r t o t a l w e a l t h i s : remaining l i f e t i m e . (8> At + Ht = + Ht-,) + R-'(l-at-l)(At-l F l a v i n (1981) e s t i m a t e s a d i f f e r e n t pt. v e r s i o n o f t h e permanent income model u s i n g t h e i n s i g h t f r o m ( 7 ) t o e l i m i n a t e t h e unobserved H,. Flavin starts w i t h t h e d e f i n i t i o n t h a t c u r r e n t consumption i s t h e sum o f permanent and t r a n s i t o r y consumption. income (y!), By e q u a t i n g permanent consumption w i t h permanent she has ct (9) = y! + Eat, where c z t i s t r a n s i t o r y consumption. Thus permanent income i s d e f i n e d t o be t h e a n n u i t y v a l u e o f t h e expected p r e s e n t d i s c o u n t e d v a l u e o f human and nonhuman w e a l t h (A, + H t ) , assuming t h e r e a l , a f t e r - t a x r a t e o f i n t e r e s t , r, i s c o n s t a n t : 00 (10) y! = r(At + 1 CR"'Etyt+il). i: 0 F l a v i n shows t h a t Ety!+] = y! u s i n g t h e i n s i g h t b e h i n d ( 7 b ) . S u b s t i t u t i n g (10) i n t o ( 9 ) and u s i n g t h e nonhuman w e a l t h c o n s t r a i n t ; ( 1 1) = R-IA~+ y t - ct. Note t h a t u n l i k e e q u a t i o n ( 5 ) , c u r r e n t p e r i o d s a v i n g does n o t earn i n t e r e s t i n equation (11 ) . http://clevelandfed.org/research/workpaper/index.cfm Best available copy Equation (9) can be used t o s o l v e f o r C t + l i n terms o f C t : Q F l a v i n notes t h a t because the c o e f f i c i e n t o f E~, i s n o t -1, C w i l l n o t evolve as a random walk, unless t h e t r a n s i t o r y consumption term E at is z e r o f o r a1 1 t. Equation (12) contains r e v i s i o n s i n e x p e c t a t i o n s o f f u t u r e r e a l l a b o r income. F l a v i n (1981, p. 998) notes t h a t " ( a h an e m p i r i c a l m a t t e r , however, u n a n t i c i p a t e d c a p i t a l gains and losses on nonhuman wealth p r o b a b l y c o n s t i t u t e a s i g n i f i c a n t f r a c t i o n o f the r e v i s i o n s i n permanent income which t h i s model i s t r y i n g t o capture. She d e f i n e s u n a n t i c i p a t e d c a p i t a l gains as t h e p r e s e n t value o f t h e r e v i s i o n i n t h e expected earnings associated w i t h t h e c u r r e n t nonhuman wealth p o s i t i o n . By then assuming " t h a t changes i n t h e r a t e o f r e t u r n t o c a p i t a l ... are q u a n t i t a t i v e l y more i m p o r t a n t than t h e endogenous changes [ i n nonhuman wealth1 i n d e t e r m i n i n g t h e time- series p r o p e r t i e s o f t h e observed p a t h o f nonlabor income", u n a n t i c i p a t e d c a p i t a l gains can be approximated as t h e present value o f t h e r e v i s i o n i n expected f u t u r e nonlabor income. This p e r m i t s her t o use disposable personal income (YD) i n p l a c e o f l a b o r income ( y ) i n equation ( 1 2 ) . 3 F l a v i n n e x t d e r i v e s an expression for t h e r e v i s i o n i n e x p e c t a t i o n s o f f u t u r e YD by assuming t h a t YD f o l l o w s an ARMA process. r e v i s i o n i n t h e e x p e c t a t i o n o f YD,, She shows t h a t t h e (s>O) between periods t and t - 1 i s t h e p r o d u c t o f t h e moving average e r r o r o f YD i n p e r i o d t ( u t > and t h e s t h c o e f f i c i e n t from t h e corresponding moving average r e p r e s e n t a t i o n f o r YD ( B s > . Then t h e present discounted value o f t h e s e t o f r e v i s i o n s i s : http://clevelandfed.org/research/workpaper/index.cfm Best available copy Thus she demonstrates t h a t t h e r e v i s i o n i n income expectations i s w h i t e noise. The ARMA model f o r YD p l u s t h e equation formed by s u b s t i t u t i n g (13) i n t o (12) i s F l a v i n ' s permanent income consumption model. c o n t a i n s an unobserved v a r i a b l e u t . Note t h a t (13) s t i l l This term i s i n c l u d e d w i t h t h e o t h e r e r r o r terms i n e s t i m a t i o n , making h e r consumption equation very s i m i l a r t o Hall's. The d i f f e r e n c e i s t h a t H a l l ' s model can be viewed as a reduced f o r m o f F l a v i n ' s s t r u c t u r a l model. F l a v i n argues t h a t the e r r o r terms i n t h e two equations a r e c o r r e l a t e d because h e r model i s incomplete. The income equation e r r o r w i l l c o n t a i n a d d i t i o n a l terms, because t h e i n f o r m a t i o n s e t probably c o n t a i n s v a r i a b l e s o t h e r than p a s t income. These o m i t t e d i n f o r m a t i o n s e t v a r i a b l e s w i l l a l s o appear i n t h e consumption equation e r r o r through (131, thus producing the c o r r e l a t i o n between t h e two equation e r r o r s . She dismisses t h i s apparent s p e c i f i c a t i o n b i a s away by assuming t h a t these o m i t t e d i n f o r m a t i o n s e t v a r i a b l e s are s e r i a l l y u n c o r r e l a t e d and u n c o r r e l a t e d w i t h t h e lagged income terms. When t h e i n t e r e s t r a t e i s random, t h e R term i n the f i r s t o r d e r c o n d i t i o n s (2a) cannot be taken o u t s i d e o f t h e expectations operator. Assuming as does Muellbauer (1983) a cash flow c o n s t r a i n t A t = (1/Rt-,> At-l+yt-Ct, the f i r s t o r d e r c o n d i t i o n s become: (1 4) EtU1 ( c ~ + ~ - $ G= E t C ( l / R t + i - l ) U ' ( C t + i ) l , i>=O. A-b Using Muel l b a u e r ' s u t i 1i t y f u n c t i o n U (Ct)=C(C I) -1 ]/(I-b), where b>O, and s e t t i n g i = l , (14) becomes: A t t h i s p o i n t , Muellbauer assumes t h e r e a r e p o i n t expectations about 1/R, c o n d i t i o n a l on t h e i n f o r m a t i o n s e t a t time t, a l l o w i n g him t o p u l l t h e i n t e r e s t - r a t e term o u t s i d e the e x p e c t a t i o n s operator: 4(16) C t b = 6(1+rt) E ~ S ( C ~ + ~ ) - ~ J where (l+r:>i s the p o i n t e x p e c t a t i o n on l / r t . This may be an http://clevelandfed.org/research/workpaper/index.cfm Best available copy i m p o r t a n t assumption, because i t means t h a t income and t h e i n t e r e s t r a t e a r e independent v a r i a b l e s from t h e consumer's p o i n t o f view. Taking logarithms o f b o t h sides and r e a r r a n g i n g terms y i e l d s : ( 17a) EtClnCt+ll - lnCt = ( l / b ) 11-18 + ( l / b ) I n (l+r:) and e l i m i n a t i n g t h e e x p e c t a t i o n s o p e r a t o r w i t h an a d d i t i v e e r r o r y i e l d s : (17b) AlnCt+l = (l/b)ln6 + (l/b)ln(l+r:) + 8t+le The e r r o r term 8 t + 1 represents new i n f o r m a t i o n ( i n n o v a t i o n s ) about l a b o r i n ~ o m e b ; ~ ,and ~ t h e i n t e r e ~ t r a t e c ~a,s~w e l l a s w h i t e n o i s e €++I' Using t h i s and the approximation I n (l+r:) = r: f o r small r:, t h e model estimated by Muel 1bauer i s : (1 8) AlnCt.l = po + t 6, f 1 + 615t+,+ 6z$C+I+ Ct+, . The Wickens and Molana model v a r i e s o n l y s l i g h t l y from t h i s because o f a minor d i f f e r e n c e i n the d a t i n g o f t h e i n t e r e s t r a t e i n the cash f l o w c o n s t r a i n t . + The i n n o v a t i o n and r t terms are g i v e n by simple equations r e l a t i n g income and t h e i n t e r e s t r a t e t o t h e i n f o r m a t i o n s e t v a r i a b l e s , which a r e a l l lagged v a r i a b l e s . The f i t t e d value f r o m the i n t e r e s t r a t e equation serves as the expected r term, and t h e r e s i d u a l s from these equations serve as t h e i n n o v a t i o n terms. Thus (18) p l u s t h e f o r e c a s t i n g equations f o r income and t h e i n t e r e s t r a t e i s Muellbauer's RE-LC model. The idea f o r t h i s f o r m u l a t i o n o f t h e expected values and i n n o v a t i o n s o r i g i n a t e s w i t h B a r r o (1977). The RE-LC/PI models a r e t e s t e d by adding v a r i a b l e s from t h e i n f o r m a t i o n s e t t o the right- hand s i d e o f t h e consumption equation. H a l l uses v a r i o u s lagged income and consumption terms and lagged values o f Standard and Poor's s t o c k p r i c e index. F l a v i n uses t h e c u r r e n t and f i r s t seven lagged changes i n r e a l per c a p i t a YD, and Muellbauer uses t h e explanatory v a r i a b l e s from t h e income and i n t e r e s t - r a t e equations. I f t h e model i s c o r r e c t , then no o t h e r v a r i a b l e i n the i n f o r m a t i o n s e t except Ct-l w i l l h e l p f o r e c a s t Ct. If o t h e r v a r i a b l e s a r e s i g n i f i c a n t l y c o r r e l a t e d w i t h consumption, then e i t h e r t h e http://clevelandfed.org/research/workpaper/index.cfm Best available copy assumption of r a t i o n a l e x p e c t a t i o n s o r t h e LCIPI model ( o r b o t h ) can be U n f o r t u n a t e l y , i t i s n o t p o s s i b l e t o know which assumption i s rejected. O n l y t h e j o i n t h y p o t h e s i s o f b o t h t h e LCIPI model and t h e r a t i o n a l incorrect. e x p e c t a t i o n s assumption can be t e s t e d . 11. UPDATES OF THE AGGREGATE LIFE CYCLE CUM RATIONAL EXPECTATIONS MODEL T h i s s e c t i o n updates t h e e s t i m a t e s and t e s t s o f t h e H a l l and F l a v i n (1981) models and p r e s e n t s e s t i m a t e s and t e s t s o f t h e Muellbauer model u s i n g post- World War 11, U.S. aggregate t i m e s e r i e s d a t a . ' The H a l l and F l a v i n models a r e updated w i t h t h e i r o r i g i n a l samples, s p e c i f i c a t i o n s , and e s t i m a t i o n techniques. Then, t o f a c i l i t a t e t h e comparison o f t h e t h r e e models, we a t t e m p t i s made t o p u t them on an equal f o o t i n g . To do t h i s , a t l e a s t f o u r d e c i s i o n s need t o be made. s p e c i f i c a t i o n o f t h e dependent and independent v a r i a b l e s . One i s t h e H a l l uses p e r c a p i t a PCE-nondurables and s e r v i c e s , F l a v i n uses t h e change i n per c a p i t a PCEnondurables, and M u e l l b a u e r uses t h e change i n t h e l o g a r i t h m o f per c a p i t a (U.K.) PCE-nondurables and s e r v i c e s . The consumption d e f i n i t i o n used i n t h e t e s t s r e p o r t e d here i s p e r c a p i t a PCE-nondurables and s e r v i c e s . F l a v i n argues t h a t o n l y PCE-nondurables should be used as t h e consumption v a r i a b l e , because consumption o f d u r a b l e s e r v i c e s should r e a c t t o changes i n permanent income w i t h a l a g due t o t h e c o s t s o f a d j u s t i n g d u r a b l e s t o c k s . h o u s i n g s e r v i c e s , which f o r m a l a r g e p a r t o f PCE- services. The same i s t r u e o f Althoughthese reasons may be v a l i d , F l a v i n detrends t h e consumption and income data. This s h o u l d e l i m i n a t e , o r a t l e a s t g r e a t l y d i m i n i s h , t h e lagged adjustment problem. I t does n o t m a t t e r which f u n c t i o n a l f o r m i s used because t h e t h e o r y can be expressed i n terms o f l e v e l s o r l o g l e v e l s ; t h e change i n t h e l o g a r i t h m i s used here t o m i n i m i z e h e t e r o s k e d a s t i c i t y problems. i s r e a l d i s p o s a b l e income p e r c a p i t a . The income d e f i n i t i o n The l o g r e a l p e r c a p i t a income and http://clevelandfed.org/research/workpaper/index.cfm Best available copy consumption d a t a a r e d e t r e n d e d by t h e i r average g r o w t h t r e n d s f r o m t h e 1947:IQ t o 1984:IVQ. When t h e same dependent v a r i a b l e i s used, F l a v i n ' s consumption e q u a t i o n i s f o r a l l p r a c t i c a l purposes, t h e same as M u e l l b a u e r ' s w i t h c o n s t a n t i n t e r e s t rates. The d i f f e r e n c e models. i s t h e s p e c i f i c a t i o n of t h e v a r i a b l e s used t o t e s t t h e T h i s d i f f e r e n c e i s i m p o r t a n t , because i t means t h a t F l a v i n must e s t i m a t e h e r model d i f f e r e n t l y t h a n Muel l b a u e r does. F l a v i n ' s model i s simultaneous because she adds t h e c u r r e n t and f i r s t seven l a g s o f AYD t o h e r consumption e q u a t i o n . When d e r i v i n g t h e reduced f o r m o f h e r two=equation system, t h e e q u a t i o n f o r YD i s used t o s u b s t i t u t e o u t t h e c u r r e n t YD t e r m i n t h e r e v i s i o n t o permanent income due t o t h e new i n f o r m a t i o n p r o v i d e d AYD,; b y c u r r e n t YD (13) cannot be i d e n t i f i e d and i s t h u s t h r o w n i n t o t h e e r r o r term. Because M u e l l b a u e r uses o n l y lagged v a r i a b l e s t o t e s t t h e model, t h e income and i n t e r e s t - r a t e i n n o v a t i o n s remain i d e n t i f i e d b y t h e income and i n t e r e s t - r a t e e q u a t i o n s and t h e i r c o e f f i c i e n t s can be e s t i m a t e d . I g n o r i n g t h e i n t e r e s t - r a t e terms i n M u e l l b a u e r ' s model, i t i s n o t c l e a r t h a t h i s t e s t i s more p o w e r f u l t h a n F l a v i n ' s . The presence o f AYDt i n t h e consumption e q u a t i o n g i v e s F l a v i n a d i r e c t t e s t o f t h e impact o f c u r r e n t income on c u r r e n t consumption. I f t h e RE-LCIPI model i s r e j e c t e d , t h e r e i s some knowledge about what d i r e c t i o n t h e search f o r t h e c o r r e c t a l t e r n a t i v e might take. But F l a v i n cannot t e s t f o r t h e i m p a c t o f t h e income i n n o v a t i o n term, an i m p o r t a n t v a r i a b l e of t h e n u l l h y p o t h e s i s . By n o t a d d i n g any c u r r e n t income terms, M u e l l b a u e r cannot t e s t f o r a d i r e c t e f f e c t o f c u r r e n t income on c u r r e n t consumption, b u t he does have a d i r e c t t e s t o f t h e i m p a c t o f income innovations. A second c h o i c e concerns seasonal a d j u s t m e n t o f t h e d a t a . M u e l l b a u e r uses s e a s o n a l l y u n a d j u s t e d d a t a , w h i l e H a l l and F l a v i n use s e a s o n a l l y a d j u s t e d data. A l t h o u g h t h e r e a r e good reasons for u s i n g s e a s o n a l l y u n a d j u s t e d d a t a , http://clevelandfed.org/research/workpaper/index.cfm Best available copy i t was decided t o use s e a s o n a l l y a d j u s t e d d a t a i n o r d e r t o m a i n t a i n c o m p a r a b i l i t y w i t h o t h e r U.S. consumption r e s u l t s . A t h i r d c h o i c e concerns e s t i m a t i o n t e c h n i q u e . H a l l uses o r d i n a r y l e a s t squares (OLS), F l a v i n uses maximum l i k e l i h o o d t o e s t i m a t e her consumption e q u a t i o n j o i n t l y w i t h h e r income f o r e c a s t i n g e q u a t i o n , and Muellbauer uses a two- step OLS procedure p o p u l a r i z e d by B a r r o (1977). The o r i g i n a l e s t i m a t i o n techniques used by H a l l and F l a v i n w i l l be used t o update t h e i r models w i t h t h e most r e c e n t d a t a . However, t h e p r e f e r r e d e s t i m a t i o n technique f o r M u e l l b a u e r ' s model i s maximum l i k e l i h o o d because, as i n F l a v i n ' s model, t h e r e a r e n o n l i n e a r c r o s s - e q u a t i o n parameter c o n s t r a i n t s , and because maximum likelihood i s asymptotically e f f i c i e n t . Moreover, Pagan (1984) shows t h a t t h e c o e f f i c i e n t s f r o m a two- step procedure may n o t have t h e same l i m i t i n g d i s t r i b u t i o n as those f r o m maximum l i k e l i h o o d , meaning t h a t t h e two- step c o e f f i c i e n t s may n o t be a s y m p t o t i c a l l y e f f i c i e n t . More i m p o r t a n t l y , he a l s o shows t h a t t h e c o r r e c t OLS c o e f f i c i e n t standard e r r o r formulas may d i f f e r f r o m t h e usual OLS f o r m u l a s employed b y computer r e g r e s s i o n programs, meaning t h a t computed t - s t a t i s t i c s may be i n c o r r e c t . U n f o r t u n a t e l y , Pagan does n o t examine t h e p r o p e r t i e s o f two- step e s t i m a t o r s f o r M u e l l b a u e r ' s model. Extending Pagan's Theorem 4 t o M u e l l b a u e r ' s model w i t h o u t t h e i n t e r e s t r a t e terms i n t h e consumption e q u a t i o n , shows t h a t t h e two- step e s t i m a t o r w i 11 have t h e same 1 i m i t i n g d i s t r i b u t i o n as maximum l i k e l i h o o d i f t h e t e s t v a r i a b l e s used i n t h e consumption e q u a t i o n a r e i n c l u d e d i n t h e i n f o r m a t i o n s e t , which i s t h e case. However, e x t e n d i n g Pagan's Theorem 5 shows t h a t the usual OLS formula f o r t h e standard e r r o r s i s wrong. However, t h e usual OLS standard e r r o r s a r e g r e a t e r than t h e c o r r e c t OLS e r r o r s , so t h a t t h e usual OLS t - s t a t i s t i c s a r e u n d e r s t a t e d . Similar r e s u l t s a r e o b t a i n e d f o r M u e l l b a u e r ' s model, i n c l u d i n g t h e i n t e r e s t - r a t e terms i n t h e consumption e q u a t i o n by an e x t e n s i o n of Pagan's Theorems 2 and 8 . http://clevelandfed.org/research/workpaper/index.cfm Best available copy These problems can be avoided by e s t i m a t i n g M u e l l b a u e r ' s model w i t h maximum l i k e l i h o o d u s i n g a stacked n o n l i n e a r l e a s t squares t e c h n i q u e s i m i l a r t o M i s h k i n ( 1 983). A f o u r t h choice i s t h a t o f t h e d e f i n i t i o n o f the r e a l i n t e r e s t - r a t e . I n s t e a d of u s i n g an ex p o s t r e a l i n t e r e s t r a t e , Muellbauer uses something l i k e an ex a n t e r a t e - - a nominal i n t e r e s t r a t e minus an expected i n f l a t i o n r a t e . He computes t h i s r e a l r a t e b y s u b t r a c t i n g f r o m t h e nominal r a t e a f i t t e d v a l u e f r o m an i n f l a t i o n e q u a t i o n . T h i s c h o i c e o f r e a l r a t e i s r a t h e r odd, f o r i t means t h a t i n s t e a d o f u s i n g an expected r e a l i n t e r e s t r a t e as h i s t h e o r y r e q u i r e s , he i s u s i n g something l i k e an expected expected r e a l i n t e r e s t r a t e i n h i s consumption e q u a t i o n . I t a l s o means t h a t he i s u s i n g a t h r e e - s t e p e s t i m a t i o n process, w i t h t h e e s t i m a t i o n o f t h e i n f l a t i o n e q u a t i o n as t h e f i r s t step. Moreover, t h e i n f l a t i o n e q u a t i o n uses a i n f o r m a t i o n s e t d i f f e r e n t f r o m t h a t used f o r t h e income and i n t e r e s t - r a t e e q u a t i o n s . A l o g i c a l e x t e n s i o n and c o r r e c t i o n o f h i s model would be t o s p e c i f y separate f o r e c a s t i n g e q u a t i o n s f o r t h e nominal r a t e and t h e i n f l a t i o n r a t e , use t h e same i n f o r m a t i o n s e t f o r a l l o f t h e e q u a t i o n s , and use t h e f i t t e d values and r e s i d u a l s f r o m b o t h e q u a t i o n s t o compute t h e expected r e a l r a t e and i t s i n n o v a t i o n . An e q u i v a l e n t t e c h n i q u e i s t h e use o f an ex p o s t r a t e , as i n Wickens and Molana (1984). three- month U.S. This r e q u i r e s o n l y one f o r e c a s t i n g e q u a t i o n . The ex p o s t r e a l Treasury b i l l r a t e (nominal r a t e minus c u r r e n t q u a r t e r compounded annual a c t u a l growth r a t e i n t h e PCE-nondurables and s e r v i c e s d e f l a t o r ) i s used as t h e r e a l i n t e r e s t r a t e i n t h e e s t i m a t i o n s o f M u e l l b a u e r ' s model shown below. Because t h e r e i s no reason t o t h i n k t h a t U.S. r e a l i n t e r e s t r a t e s have behaved as random walks d u r i n g t h e post- World War I 1 p e r i o d , an assumption M u e l l b a u e r made f o r U.K. r e a l r a t e s , t h e r e a l i n t e r e s t - r a t e e q u a t i o n f o r M u e l l b a u e r ' s model e s t i m a t e d here w i l l have i n f o r m a t i o n s e t v a r i a b l e s as http://clevelandfed.org/research/workpaper/index.cfm Best available copy regressors. These w i l l be t h e same as t h o s e used f o r t h e income e q u a t i o n - - t h e f i r s t t w o l a g s o f income, t h e f i r s t t w o l a g s o f t h e r e a l i n t e r e s t r a t e , and t h e f i r s t l a g o f consumption. This i s simple extension o f Muellbauer's o r i g i n a l i n f o r m a t i o n s e t , w h i c h c o n s i s t e d o f t h e f i r s t t w o l a g s of income and t h e f i r s t l a g o f consumption. The e s t i m a t i o n r e s u l t s a r e shown i n t a b l e s 1 t o 5. The d a t a used f o r t h e c o m p u t a t i o n s c o n t a i n r e v i s i o n s t h r o u g h t h e second r e v i s e d e s t i m a t e s f o r 1984:IVQ, d a t e d March 1985. The models i n t a b l e s 1 t o 3 were e s t i m a t e d o v e r t h e i r o r i g i n a l samples and o v e r 1 9 4 9 : I I I Q t o 1984:IVQ. o f H a l l ' s model, t h e d a t a were n o t d e t r e n d e d . For t h e r e - e s t i m a t e s For t h e r e - e s t i m a t e s o f F l a v i n ' s model, t h e consumption and income d a t a were d e t r e n d e d , u s i n g t h e i r a v e r a g e g r o w t h r a t e s f r o m t h e 1 9 4 7 : I Q t o 1979:IQ. When t h e two models a r e u p d a t e d w i t h t h e d a t a t h r o u g h 1984:IVQ, t h e consumption and income d a t a a r e d e t r e n d e d u s i n g t h e i r average g r o w t h r a t e s f r o m t h e 1 9 4 7 : I Q t o 1984:IVQ, and a dummy v a r i a b l e i s added t o c o n t r o l f o r t h e c r e d i t c o n t r o l s o f 1 9 8 0 : I I Q . D e t r e n d i n g b i a s e s t h e t e s t i n f a v o r of t h e random w a l k h y p o t h e s i s , because i t removes t h e main s o u r c e of c o r r e l a t i o n f r o m t h e s e v a r i a b l e s . I t a l s o may remove s t r u c t u r a l c o r r e l a t i o n between C and YD, a g a i n f a v o r i n g t h e walk hypothesis. I t u n f o r t u n a t e l y leaves the t r e n d unexplained. random The dummy v a r i a b l e i s p a r t o f t h e m a i n t a i n e d h y p o t h e s i s and i s n o t i n c l u d e d among t h e v a r i a b l e s i n t h e t e s t o f t h e RE-LCIPI model. The f i r s t t a b l e c o n t a i n s OLS e s t i m a t e s o f H a l l ' s model. were e s t i m a t e d w i t h t h e OLSQ o p t i o n i n PEC v e r s i o n 9.1. These e q u a t i o n s The f i r s t e q u a t i o n shows t h e r e e s t i m a t e s of H a l l ' s model, w i t h o n l y one lagged income t e r m . The c o e f f i c i e n t s , though d i f f e r e n t from H a l l ' s p u b l i s h e d numbers, y i e l d t h e same apparent inference: t h e RE-LC/PI model c a n n o t be r e j e c t e d . shows t h e o r i g i n a l H a l l model u p d a t e d t h r o u g h 1984:IVQ. The n e x t e q u a t i o n Note t h a t t h e a d d i t i o n o f t h e 1980s d a t a d i d n o t change t h e c o n c l u s i o n o f t h e h y p o t h e s i s http://clevelandfed.org/research/workpaper/index.cfm Best available copy t e s t - - t h e c o e f f i c i e n t on lagged personal income i s s m a l l , has t h e wrong s i g n , However, t h e D u r b i n h - s t a t i s t i c r e j e c t s and i s s t a t i s t i c a l l y i n s i g n i f i c a n t . the hypothesis o f p o s i t i v e s e r i a l l y uncorrelated e r r o r s a t b e t t e r than a 5 percent significance l e v e l using a one- tailed t e s t . Because t h e t h e o r y p r e d i c t s t h a t t h e e r r o r s h o u l d be w h i t e n o i s e , t h e a d d i t i o n o f t h e 1980s d a t a may be s i g n a l i n g a breakdown o f t h e model. The t h i r d e q u a t i o n c o n t a i n s t h e change i n t h e detrended l o g of p e r c a p i t a PCE-nondurables and s e r v i c e s as t h e dependent v a r i a b l e and t h e detrended l o g a r i t h m o f r e a l p e r c a p i t a d i s p o s a b l e personal income as t h e income v a r i a b l e . 1977:IQ. The e s t i m a t i o n p e r i o d i s 1948:IQ t o N e i t h e r c o e f f i c i e n t i s l a r g e , t h e t - s t a t i s t i c s a r e v e r y low, and t h e adjusted R2 i s negative. The r e s u l t s change v e r y l i t t l e when t h e e s t i m a t i o n p e r i o d i s extended t h r o u g h 1984:IVQ; a l l of t h e e x p l a n a t o r y power o f t h e r i g h t - h a n d s i d e v a r i a b l e s comes f r o m t h e dummy v a r i a b l e . Thus H a l l ' s model can f i n d no evidence t o r e j e c t t h e RE-LCIPI model. The r e s u l t s f o r F l a v i n ' s model a r e shown i n t a b l e s 2 and 3 . The general s p e c i f i c a t i o n i s : (12a) Y D t = p l + ~ I Y D , - ~+ a 2 Y D t - 2 + * * * + a 8 Y D t - 8 + q l t Act. p2 + fio(u1 + - 1 Y D + a2YDt-2 + * * * + a 8YDt-8) + BlAYDt-1 + l32AYDt-2 + * * * + 87AYDt-7 + where q at c o n t a i n s cazand (13). q 2 t 3 The R ' s a r e "measures o f t h e 'excess s e n s i t i v i t y ' o f consumption t o c u r r e n t income, t h a t i s , s e n s i t i v i t y i n excess o f t h e response a t t r i b u t a b l e t o t h e new i n f o r m a t i o n c o n t a i n e d i n c u r r e n t income. F l a v i n (1981, p . 990). Thus, a t e s t o f t h e j o i n t s t a t i s t i c a l s i g n i f i c a n c e o f t h e R ' s i s a t e s t o f t h e RE-PI model. The r e s u l t s i n t a b l e 2 a r e t h e r e - e s t i m a t e s o f F l a v i n ' s model, u s i n g her d e f i n i t i o n s o f consumption and income. Table 3 c o n t a i n s t h e r e s u l t s of her model u s i n g t h e change i n t h e l o g o f PCE-nondurables and s e r v i c e s . TSP v e r s i o n - The m u l t i v a r i a t e r e g r e s s i o n t e c h n i q u e i n 4.OE was used t o e s t i m a t e these e q u a t i o n s . T h i s i s t h e same http://clevelandfed.org/research/workpaper/index.cfm Best available copy e s t i m a t o r F l a v i n used. I t allows j o i n t estimation o f nonlinear equations w i t h n o n l i n e a r c r o s s - e q u a t i o n c o n s t r a i n t s and w i t h contemporaneous c o r r e l a t i o n a c r o s s e q u a t i o n e r r o r terms. I t should y i e l d r e s u l t s c l o s e t o F l a v i n ' s i n d i r e c t l e a s t squares e s t i m a t e s a p a r t f r o m t h e i m p a c t o f d a t a r e v i s i o n s . The f i r s t e q u a t i o n i n t a b l e 2 shows t h e r e - e s t i m a t e s o f h e r o r i g i n a l L i k e t h e updates of H a l l ' s model, t h e s e c o e f f i c i e n t s a r e n o t specification. q u a n t i t a t i v e l y t h e same as t h e o r i g i n a l e s t i m a t e s b u t , qua1i t a t i v e 1 . y t h e y a r e very similar. The c o e f f i c i e n t on A Y t , R o , though f a i r l y l a r g e , has a v e r y low t - s t a t i s t i c o f t h e R ' s , o n l y i s s i g n i f i c a n t a t b e t t e r than 5 The second e q u a t i o n d r o p s t h e AYD terms t o percent using a one- tailed t e s t . test t h e i r j o i n t significance. 81 R e c a l l t h a t t h e s e terms must be j o i n t l y s t a t i s t i c a l l y d i f f e r e n t f r o m z e r o i n o r d e r t o r e j e c t t h e model. Surprisingly t h e RE-LCIPI model cannot be r e j e c t e d a t t h e o r i g i n a l s i g n i f i c a n c e l e v e l . F l a v i n ' s o r i g i n a l l i k e l i h o o d r a t i o s t a t i s t i c (LRS), which i s a s y m p t o t i c a l l y d i s t r i b u t e d as X 2 ( 8 ) , i s 27.0, s i g n i f i c a n t a t b e t t e r t h a n 0.5 p e r c e n t . The LRS f o r t h e t e s t u s i n g e q u a t i o n s ( 1 ) and ( 2 ) i s o n l y 1 1 . 8 - - s i g n i f i c a n t a t s l i g h t l y b e t t e r t h a n 25.0 p e r c e n t . I d e n t i c a l t e s t r e s u l t s are o b t a i n e d by e s t i m a t i n g o n l y t h e consumption reduced f o r m e q u a t i o n w i t h OLS and t e s t i n g f o r t h e j o i n t s i g n i f i c a n c e o f t h e lagged income t e r m s . 7 Apparently the r e s u l t s a r e s e n s i t i v e t o r e v i s i o n s i n t h e d a t a and t h e use o f d i f f e r e n t t r e n d v a l u e s for PCE-nondurables and YD. E q u a t i o n s ( 3 ) and ( 4 ) i n t a b l e 2 update F l a v i n ' s o r i g i n a l model t h r o u g h 1984:IVQ. The 1947:IQ t o 1979:IQ t r e n d v a l u e s a r e used t o d e t r e n d t h e post- 1979:IQ d a t a . I n t e r e s t i n g l y , t h e model can now be r e j e c t e d a t b e t t e r t h a n a 5.0 p e r c e n t s i g n i f i c a n c e l e v e l . The LRS i s 17.1, w h i l e t h e X 2 ( 8 > c u t - o f f v a l u e i s 15.5, a t 5.0 p e r c e n t . The c o e f f i c i e n t R o i s now s m a l l e r , b u t i t s t - s t a t i s t i c i s l a r g e r ; t h e c o e f f i c i e n t and t - s t a t i s t i c o n AlnYD,-, are larger also. Moreover, t h e f i t of t h e two e q u a t i o n s i s improved o v e r t h e http://clevelandfed.org/research/workpaper/index.cfm Best available copy l o n g e r p e r i o d ; t h e s t a n d a r d e r r o r s o f t h e two e q u a t i o n s a r e s m a l l e r i n t h e l o n g e r sample. Thus, as was expected, t h e 1980s d a t a appear t o t i g h t e n up c o e f f i c i e n t standard e r r o r s and h e l p r e j e c t t h e RE-LCIPI model. Equations ( 5 ) t h r o u g h ( 8 ) i n t a b l e 3 use t h e change i n t h e l o g a r i t h m o f p e r c a p i t a r e a l PCE-nondurables and s e r v i c e s as t h e dependent v a r i a b l e , and t h e l o g p e r c a p i t a consumption and income d a t a a r e detrended o v e r t h e 1947:IQ t o 1984:IVQ p e r i o d . 1. They compare t o t h e H a l l e q u a t i o n s ( 3 ) and ( 4 ) i n t a b l e The f i f t h e q u a t i o n shows t h e u n c o n s t r a i n e d r e s u l t s o v e r t h e 1 9 4 9 : I I I Q t o 1979:IQ sample p e r i o d . N o t i c e t h a t t h e y a r e q u a l i t a t i v e l y s i m i l a r t o t h o s e of e q u a t i o n ( 1 ) ; R o i s about 0.3 and s t a t i s t i c a l l y i n s i g n i f i c a n t , and 81 i s l a r g e and s t a t i s t i c a l l y s i g n i f i c a n t . Testing t h i s equation against the s i x t h e q u a t i o n , which drops t h e AlnYD terms, y i e l d s an LRS o f 27.1, which i s s i g n i f i c a n t a t b e t t e r than 0.5 p e r c e n t , F l a v i n ' s o r i g i n a l s i g n i f i c a n c e l e v e l . Note t h a t t h i s r e s u l t i s much s t r o n g e r t h a n F l a v i n ' s o r i g i n a l r e s u l t , because t h e consumption v a r i a b l e i n c l u d e s PCE- services, which F l a v i n argued would b i a s t h e r e s u l t s a g a i n s t t h e RE-LCIPI model. Equations ( 7 ) and ( 8 ) show t h e e s t i m a t i o n r e s u l t s o v e r t h e 1 9 4 9 : I I I Q t o 1984:IVQ sample. Q u a l i t a t i v e l y , these r e s u l t s a r e s i m i l a r t o those of e q u a t i o n s ( 5 ) and ( 6 ) . The LRS o f t h e t e s t o f t h e lagged AlnYD terms i s now 29.4, g r e a t e r than t h e LRS o v e r t h e 1 9 4 9 : I I I Q t o 1979:IQ sample; t h e s t a n d a r d e r r o r s o f t h e e q u a t i o n s a l s o a r e s m a l l e r i n t h e l o n g e r sample. Again i t appears t h a t t h e 1980s d a t a p r o v i d e a d d i t i o n a l s t r o n g e r evidence a g a i n s t t h e RE-LC/PI model . Tables 4 and 5 c o n t a i n t h e e s t i m a t e s of M u e l l b a u e r ' s model. A l l o f the e q u a t i o n s a r e e s t i m a t e d w i t h t h e stacked maximum l i k e l i h o o d t e c h n i q u e u s i n g PEC v e r s i o n 9.1. t h e LSQ o p t i o n o f - The dependent v a r i a b l e i s t h e change i n t h e l o g a r i t h m o f r e a l p e r c a p i t a PCE-nondurables and s e r v i c e s ; d e t r e n d i n g o f t h e l o g r e a l per c a p i t a consumption and income d a t a occurs o v e r t h e 1947:IQ t o http://clevelandfed.org/research/workpaper/index.cfm Best available copy 1984:IVQ p e r i o d . r a t e terms Table 4 c o n t a i n s t h e e s t i m a t e s of (18) w i t h o u t t h e i n t e r e s t and o,,. C: + a21nYt-, + a31nCt-l + asDUM802, + InY, = p i . r a l l n Y t - l AlnCt = p, The e s t i m a t e d model i s : p1 - allnYt-l + 6,<lnYt - el, - a21nYt-, - a31nCt-l - a9DUM80Zt) + B l l n Y t - l + B21nYt-, + B s l n C t - l + B9DUM802, + C , , . The c o e f f i c i e n t 61 should be p o s i t i v e because p o s i t i v e i n n o v a t i o n s i n c u r r e n t income should l e a d t o upward r e v i s i o n s i n l i f e c y c l e w e a l t h l p e r m a n e n t income and hence i n consumption. The f i r s t two e q u a t i o n s show t h e r e s u l t s u s i n g t h e 1 9 4 9 : I I I Q t o 1979:IQ sample. e q u a t i o n s ( 5 ) and (6) i n t a b l e 3. statistically significant. These e q u a t i o n s compare t o F l a v i n ' s The c o e f f i c i e n t 61 i s p o s i t i v e and S u r p r i s i n g l y , t h e RE-LCIPI model cannot be r e j e c t e d by t h i s f o r m o f M u e l l b a u e r ' s model, even though F l a v i n ' s model could. The LRS i s o n l y 3.8, s i g n i f i c a n t a t s l i g h t l y l e s s t h a n 30 p e r c e n t . Apparently the r e s u l t s are sensitive t o the specification o f the t e s t . The t h i r d and f o u r t h e q u a t i o n s i n t a b l e 4 update M u e l l b a u e r ' s model w i t h o u t t h e i n t e r e s t r a t e terms o v e r t h e 1 9 4 9 : I I I Q t o 1984:IVQ sample. As was t r u e o f F l a v i n ' s model, Muel 1b a u e r ' s model w i t h o u t t h e i n t e r e s t r a t e terms f i t s b e t t e r w i t h t h e 1980s d a t a . a t b e t t e r than 1 percent. o f t h e RE-LC/PI model. Moreover, t h e LRS i s now 14.2, significant Again t h e 1980s d a t a l e a d t o a c o n v i n c i n g r e j e c t i o n Note t h a t t h e B c o e f f i c i e n t s a r e o f t h e same o r d e r of magnitude and s t a t i s t i c a l s i g n i f i c a n c e i n e q u a t i o n s ( 1 ) and (3). The d i f f e r e n c e i s t h a t t h e models f i t b e t t e r w i t h t h e 1980s d a t a . Table 5 c o n t a i n s t h e e s t i m a t e s of M u e l l b a u e r ' s model i n c l u d i n g t h e i n t e r e s t r a t e terms. The model i s : l n Y t = p 1 + a l l n Y t - ] + a21nYt-, + a3rt-l + a 4 r t - *+ a s l n C t - , + a9DUM802, + c l t r =.)I, + + y ~ l n Y , - ~+ / 9DUM802t + ( z t I 21nYt-2 1 +r / + y 4 r t - r+ Y 51nCt-l http://clevelandfed.org/research/workpaper/index.cfm Best available copy AlnCt = p2 + 61(lnYt - p, - a l l n Y t - l - a s l n C t - l - a9DUM80Zt) + b Z ( r t - - y 3 r t - , - r'r - yslnCt-, I + y ~ l n Y +~ Y- 7~ r t - L + ~ - - a ~ l n Y -~ -a 3 ~rt-) yo - - a4rt-2 llnYt-l -y21nY+-Z y~DUM802,) + &,(Go + YllnY+,-, ~+ y 5 1rn C t - Z~- y 9 D U-M 8 0 2 ~ t-1) + R l l n Y t - I + B21nYt-r + B 3 r t - l + B 4 r t - , + B s l n C t - l + B9DUM802, + E3t R e c a l l f r o m (18) t h a t 6 ? i s a p o s i t i v e f u n c t i o n o f t h e r a t i o o f 1 p l u s t h e i n t e r e s t r a t e t o 1 p l u s t h e r a t e o f t i m e p r e f e r e n c e , and hence, s h o u l d be positive. Presumably t h e c o e f f i c i e n t 62 on t h e i n t e r e s t - r a t e i n n o v a t i o n i s n e g a t i v e , s i n c e a higher- than- expected i n t e r e s t r a t e should cause consumers t o save more i n t h e c u r r e n t p e r i o d . o v e r t h e 1 9 4 9 : I I I Q t o 1979:IQ sample. Equations ( 1 ) and ( 2 ) a r e t h e r e s u l t s The two i n t e r e s t - r a t e c o e f f i c i e n t s appear t o be small i n magnitude, b u t t h i s i s s i m p l y a s c a l i n g d i f f e r e n c e because i n t e r e s t r a t e s a r e measured i n percentage p o i n t s ; t h e i n t e r e s t r a t e i n n o v a t i o n c o e f f i c i e n t 62 i s s t a t i s t i c a l l y i n s i g n i f i c a n t i n b o t h e q u a t i o n s , w h i l e 63 i s s i g n i f i c a n t a t s l i g h t l y b e t t e r t h a n 10 p e r c e n t i n e q u a t i o n ( 1 ) and b e t t e r t h a n 5 p e r c e n t i n e q u a t i o n (2), u s i n g t w o - t a i l e d tests. The LRS f o r t h e t e s t o f t h e RE-LC/PI model i s 27.8, which i s a s y m p t o t i c a l l y d i s t r i b u t e d as X 2 ( 5 ) , and i s s i g n i f i c a n t a t b e t t e r t h a n 1 percent. Compared w i t h e q u a t i o n s ( 1 ) and ( 2 ) i n t a b l e 4, t h e a l l o w a n c e for s t o c h a s t i c i n t e r e s t r a t e s now leads t o t h e r e j e c t i o n o f t h e RE-LCIPI model. Again, t h e s p e c i f i c a t i o n o f t h e t e s t has an i m p o r t a n t e f f e c t on t h e r e s u l t s . The t h i r d and f o u r t h e q u a t i o n s i n t a b l e 5 show t h e e s t i m a t e s o f M u e l l b a u e r ' s model w i t h t h e i n t e r e s t - r a t e terms f r o m 1 9 4 9 : I I I Q t o 1984:IVQ. A l l o f t h e c o e f f i c i e n t s a r e e s t i m a t e d more p r e c i s e l y , b u t u n l i k e t h e p r e v i o u s r e s u l t s , b o t h equations f i t t h e l o n g e r p e r i o d l e s s w e l l . The c o e f f i c i e n t s 6Z and 63 have t h e c o r r e c t s i g n s , b u t a r e t h e same magnitude o r s m a l l e r and m o s t l y s t a t i s t i c a l l y i n s i g n i f i c a n t . The LRS s t a t i s t i c f o r t h e http://clevelandfed.org/research/workpaper/index.cfm Best available copy t e s t o f t h e RE-LCIPI model i s 26.2, r e j e c t i n g t h e model a t b e t t e r t h a n a 1 p e r c e n t s i g n i f i c a n c e l e v e l , b u t i t i s a b i t s m a l l e r t h a n t h e LRS f r o m t h e s h o r t e r sample p e r i o d . The worse f i t u s i n g t h e 1980s d a t a i s e x p l a i n e d by t h e i n t e r e s t - r a t e equation f i t t i n g less well i n the longer period. This i s n o t s u r p r i s i n g , since ~ i n t e r e s t r a t e s behaved so d i f f e r e n t l y i n t h e 1980s t h a n b e f ~ r e . Does this mean t h a t t h e t e s t i s i n v a l i d because t h e e q u a t i o n g e n e r a t i n g t h e i n t e r e s t - r a t e e x p e c t a t i o n s i s wrong? T h i s does n o t seem l i k e l y . Although the t - s t a t i s t i c s on 6 2 and 6 3 a r e m o s t l y v e r y low, t h e LRS o f t h e j o i n t s i g n i f i c a n c e o f t h e two i n t e r e s t - r a t e terms i n e q u a t i o n 7 i s 46.1. Thus, t h e i n t e r e s t - r a t e terms a r e undoubtedly i m p o r t a n t , even i f t h e y a r e p o o r l y computed. Moreover, i t i s n o t c l e a r how q u i c k l y i n t e r e s t - r a t e f o r e c a s t i n g models were a d j u s t e d i n t h e 1980s. Given t h e l a g i n t h e l e a r n i n g process, t h e number o f q u a r t e r s f o r which t h e i n t e r e s t - r a t e e q u a t i o n may be wrong i s Even if t h e i n t e r e s t - r a t e e q u a t i o n i s wrong, i t i s p r o b a b l y s m a l l e r t h a n 20. not necessarily i r r a t i o n a l . F i n a l l y , t h e f i t o f t h e model d i d n o t worsen so much t h a t t h i s i s l i k e l y t o be t h e s o l e reason t h e RE-LCIPI model i s r e j e c t e d . 111. WHAT HAS BEEN LEARNED? The e s t i m a t i o n r e s u l t s p r o v i d e ample evidence t o r e j e c t t h i s f o r m o f t h e RE-LCIPI model d u r i n g t h e post- war p e r i o d , e s p e c i a l l y when t h e 1980s d a t a a r e included. Even though H a l l ' s s p e c i f i c a t i o n cannot r e j e c t t h e model, m i n o r g e n e r a l i z a t i o n s of F l a v i n and M u e l l b a u e r can, and M u e l l b a u e r ' s s p e c i f i c a t i o n i n c l u d i n g u n c e r t a i n i n t e r e s t r a t e s can r e j e c t t h e model w i t h o r w i t h o u t t h e 1980s d a t a . I t would appear t h a t an i m p o r t a n t assumption f o r B a r r o ' s n e u t r a l i t y hypothesis does n o t h o l d . U n f o r t u n a t e l y t h i s r e j e c t i o n of t h e RE-LCIPI model does n o t o f f e r an e x p l i c i t a l t e r n a t i v e as a replacement. As mentioned e a r l i e r , t h e s e t e s t s c a n n o t d i s t i n g u i s h t h e assumption of r a t i o n a l e x p e c t a t i o n s f r o m t h a t o f t h e http://clevelandfed.org/research/workpaper/index.cfm Best available copy l i f e cyclelpermanent income model. A l l t h a t can be i n f e r r e d from these t e s t s i s t h a t t h e j o i n t h y p o t h e s i s can be r e j e c t e d . F l a v i n (1985) a t t e m p t s t o d i s t i n g u i s h whether t h e r e j e c t i o n o f t h e RE-LCIPI model i s due t o t h e assumption o f p e r f e c t c a p i t a l markets o r t o t h a t of t h e permanent income model. She uses her o r i g i n a l model augmented w i t h an e q u a t i o n f o r t h e unemployment r a t e , which i s a p r o x y f o r t h e number o f l i q u i d i t y - c o n s t r a i n e d consumers. However, t h e r e a r e many problems u s i n g such a crude v a r i a b l e f o r such a complex h y p o t h e s i s ; h e r t e s t s undoubtedly have l i t t l e power. Nor do these t e s t s p r o v i d e many c l u e s about t h e e x a c t l e n g t h o f consumer spending h o r i z o n s , o r how t h e d i s t r i b u t i o n o f h o r i z o n l e n g t h s changes as i n t e r e s t r a t e s , t h e d i s t r i b u t i o n o f income, o r t h e s u p p l y o f consumer c r e d i t changes. That t h e d i s t r i b u t i o n o f consumer h o r i z o n l e n g t h s may v a r y o v e r t i m e i s suggested by t h e i n c r e a s e d s i g n i f i c a n c e o f t h e l i k e l i h o o d r a t i o t e s t s when t h e 1980s d a t a a r e i n c l u d e d . I n t h e e a r l y 1980s, a p p a r e n t l y , t h e d i s t r i b u t i o n o f h o r i z o n s l e n g t h s was skewed toward t h e s h o r t e r end, i n c r e a s i n g t h e c o r r e l a t i o n o f aggregate consumption t o c u r r e n t d i s p o s a b l e income. Additional evidence about changes i n t h e d i s t r i b u t i o n o f consumer spending h o r i z o n s i s p r o v i d e d by Kowalewski (1982>, who s t u d i e s t h e t i m e s e r i e s b e h a v i o r o f aggregate personal b a n k r u p t c y f i l i n g s i n t h e U n i t e d S t a t e s . Personal b a n k r u p t c y f i l i n g s a r e c o u n t e r c y c l i c a l , i n c r e a s i n g i n r e c e s s i o n s and f a l l i n g i n recoveries. For a v a r i e t y o f reasons discussed i n t h e a r t i c l e , i t i s l i k e l y t h a t j u s t b e f o r e t h e y f i l e f o r bankruptcy, personal bankrupts have a b o u t t h e s h o r t e s t spending h o r i z o n s o f a l l consumers. Thus increases i n t h e number o f personal bankruptcy f i l i n g s may i n d i c a t e a s h i f t i n t h e d i s t r i b u t i o n o f consumer spending h o r i z o n s towards s h o r t e r l e n g t h s . I n a regression e x p l a i n i n g p e r c a p i t a personal bankruptcy f i l i n g s , t r a n s i t o r y income had a much l a r g e r impact than permanent income, s u g g e s t i n g t h a t l i q u i d i t y i s v e r y i m p o r t a n t f o r these f i n a n c i a l l y d i s t r e s s e d consumers. The c o m p o s i t i o n o f http://clevelandfed.org/research/workpaper/index.cfm Best available copy consumer p o r t f o l i o s was a l s o s i g n i f i c a n t l y r e l a t e d t o t h e b e h a v i o r o f p e r s o n a l bankruptcy f i l i n g s . the distribution. U n f o r t u n a t e l y , t h i s i s o n l y evidence about one t a i l o f I t i s c l e a r t h a t much work remains t o be done b e f o r e t h e t i m e s e r i e s b e h a v i o r o f aggregate consumption i s understood. http://clevelandfed.org/research/workpaper/index.cfm Best available copy TABLE 1 Hal 1 E s t i m a t e s #1 Ct NDSIPOP yt-I YD72lPOP Sampl e 48Q1-7741 #2 NDSIPOP YD72lPOP 4801-84Q4 #3 #4 Chg i n d e t r e n d e d Chg i n d e t r e n d e d l o g of log o f NDS 1POP NDSIPOP Detrended l o g o f YD72lPOP 48Q1-7741 - 0.0068 Detrended l o g o f YD72lPOP 4943-8444 a d j R2 0.9989 0.9994 0.0263 Durbi n h 1 .358 1 -7327 1 .7752" 1 .7460a SER 0.01 36 0.0290 0.0058 0.0056 a. D u r b i n - Watson s t a t i s t i c s Variables: NDS = PCE- nondurables + s e r v i c e s , 1972 d o l l a r s YD72 = d i s p o s a b l e p e r s o n a l income, 1972 d o l l a r s POP = n o n i n s . t i t u t i o n a l i z e d , c i v i 1 i a n p o p u l a t i o n Notes: The v a r i a b l e s ( 3 ) and ( 4 ) a r e d e t r e n d e d f r o m 1947:IQ t o 1984:IVQ. T - s t a t i s t i c s a r e shown below t h e c o e f f i c i e n t e s t i m a t e s . http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 2 F l a v i n Reestimates Sarnpl e AC=chg i n d e t r e n d per capita PCE-Nondurabl es 1972$ lJ 2 B0 Y = detrended per capita d i sposabl e personal income , 1972$ lJ 1 a1 a2 DUM802 = 1, 198002 0, e l s e a3 Detrending over 194701-1979Q1 a4 a5 a6 a7 a8 a9 81 I3 2 B3 B4 05 86 B7 139 C SER C D-W Y SER Y D-W Log Lklhood 0.01 03 2.0101 0.0329 2.0008 0.0108 1.8921 0.0332 1.8834 0.0104 2.0521 0.0325 2.0009 0.01 10 1 .8748 0.0327 1.8461 622.413 616.536 742.652 734.078 http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 3 F l a v i n E s t i m a t e s U s i n g Logs Samp 1e AlnC=chg, d e t r e n d l o g per c a p i t a PCE-Nondur+Serv 1972$ 1nY = d e t r e n d l o g per capita d i sposabl e personal income, 1972$ DUM802 = 1, 198002 0, e l s e Detrending over 194701-198444 C SER C D-W Y SER Y D-W Log Lklhood 0.0051 1.8636 0.0102 1.9942 0.0057 1.7396 0.0104 1 .8075 0.0050 1 .go03 0.0098 2.0022 850.000 836.462 1020.09 0.0056 1 -7458 0.01 0 0 1.8012 1005.41 http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 4 Muellbauer Estimates Without Interest Rate Samp 1 e AC = detrended per capi ta PCE-Nondur+Serv 1972$ Y C SER Y SER iLog Lkl hood 0.0053 0.0104 ) 1175.6 0.0053 0.0104 1173.7 0.0050 0.0104 1412.3 detrended per capita di sposabl e personal income, 1972$ = 0.0052 0.0103 1405.2 The log likelihood statistic is computed using -(n/2>ln(detZ> where n is sample size and 1 is the system covariance matrix. http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 5 Samp 1 e Muellbauer Estimates Interest Rate AlnC=chg, detrend log per capita PCE-Nondur+Serv 1972$ 1nY = detrended log per capita disposable personal income, 1972$ DUMB02 r = = fi(0,, else 198042 real, ex post 3-mo T-bi 1 1 rate C SER Y SER r SER Log Lkl hood http://clevelandfed.org/research/workpaper/index.cfm Best available copy Appendi x A A d i f f e r e n t t e c h n i q u e can be used t o o b t a i n an e s t i m a b l e e q u a t i o n f r o m (15). The e x p e c t a t i o n s o p e r a t o r can be removed by i n c l u d i n g t h e e r r o r term EXP[e,+ll, t h e e x p o n e n t i a l of e,,~, on t h e l e f t - h a n d s i d e o f ( 1 5 ) : ( C T ~ )EXPCet+,3=6[(1/Rt) ( C t + l ) -bl. (16) T h i s e r r o r t e r m r e p r e s e n t s t h e r a t i o o f t h e r i g h t - h a n d s i d e t e r m o f (16a>, t h e a c t u a l v a l u e o f t h e p r o d u c t , t o t h a t of ( 1 5 ) , t h e expected v a l u e o f t h e product. ln(l/R,> Taking l o g a r i t h m s o f b o t h s i d e s , u s i n g t h e a p p r o x i m a t i o n t h a t = r, f o r small r,, and r e a r r a n g i n g terms, (16a) becomes: ( 1 7 ~ )AlnC,+, = (l/b> [ln6 + lnr,] - (l/b> et+l. The f o r m u l a t i o n ( 1 7 ~ )has t h e advantage of a v o i d i n g t h e assumption o f p o i n t e x p e c t a t i o n s f o r t h e i n t e r e s t r a t e ; ( 1 7 ~ ) i s i d e n t i c a l t o (17b) except t h a t t h e i n t e r e s t - r a t e term i n ( 1 7 ~ )i s n o t t h e p o i n t e x p e c t a t i o n o f r,, b u t t h e a c t u a l v a l u e o f r,. I n a way ( 1 7 ~ )makes more sense t h a n (17b), because (17b) says t h a t consumption n e x t p e r i o d depends on what consumers expected t h e i n t e r e s t r a t e t o be t h i s p e r i o d . U s u a l l y f u t u r e a c t i o n s depend upon what i s e x p e c t e d i n t h e f u t u r e , n o t what was expected i n t h e p a s t , u n l e s s t h e r e a r e s p e c i f i c l e a r n i n g b e h a v i o r assumptions i n t h e model. The e r r o r t e r m EXPCet+ll i s a c o m p l i c a t e d f u n c t i o n o f t h e i n n o v a t i o n s i n income and i n t e r e s t r a t e s t h r o u g h t h e j o i n t d i s t r i b u t i o n o f income and t h e interest rate. Assuming t h a t e t + ! can be approximated by e,t+l + e y r t * ! - '++% , where e ,,+, +erg +L i s t h e income i n n o v a t i o n , e r t + l i s t h e i n t e r e s t - r a t e i n n o v a t i o n , e,, i s t h e i n t e r a c t i o n term, and i s t h e remainder o f t h e e r r o r and i s w h i t e n o i s e , ( 1 7 ~ )i s s i m i l a r t o (18) w i t h o u t t h e p o i n t e x p e c t a t i o n on r,, b u t w i t h e,r,+l term. http://clevelandfed.org/research/workpaper/index.cfm Best available copy The one major problem f o r e s t i m a t i n g ( 1 7 ~ )i s t h e e s t i m a t i o n o f e,,,,,. M u e l l b a u e r uses an i n t e r a c t i o n t e r m "because o f t h e way i n t e r e s t r a t e s and incomes a r e i n t e r t w i n e d i n human c a p i t a l " (Muellbauer [1983, p. 451) and e s t i m a t e s i t w i t h t h e p r o d u c t o f t h e income and i n t e r e s t - r a t e i n n o v a t i o n s . The reason f o r an i n t e r a c t i o n t e r m here i s t o account f o r t h e j o i n t d i s t r i b u t i o n o f income and i n t e r e s t r a t e s used by consumers t o f o r m t h e i r expectations. M u e l l b a u e r ' s s t r a t e g y i s easy t o implement when OLS i s used. However, i t was argued i n t h e t e x t t h a t u s i n g OLS w i t h a model t h a t does n o t i n c l u d e an i n t e r a c t i o n t e r m i s i n c o r r e c t because t h e t - s t a t i s t i c s a r e b i a s e d downward; an i n t e r a c t i o n t e r m may o n l y r e i n f o r c e t h i s r e s u l t and c o m p l i c a t e the d i r e c t i o n o f bias. E s t i m a t i n g t h e model w i t h maximum l i k e l i h o o d i s v e r y d i f f i c u l t when t h e p r o d u c t o f t h e e r r o r s i s i n c l u d e d because t h e c r o s s - e q u a t i o n c o e f f i c i e n t c o n s t r a i n t s add 21 messy terms t o t h e consumption equation. An e a s i e r procedure would be t o e s t i m a t e an e q u a t i o n f o r t h e p r o d u c t ( l n Y t > r t and use t h e r e s i d u a l f o r t h e i n t e r a c t i o n t e r m i n t h e consumption e q u a t i o n . http://clevelandfed.org/research/workpaper/index.cfm Best available copy FOOTNOTES 1. M u e l l b a u e r (1983) and Wickens and Molana (1984) r e j e c t t h e model, u s i n g U. K. consumption and income d a t a . 2. See H a l l (1978, p. 975). 3 . T h i s assumption i s n o t unreasonable, g i v e n t h a t h e r model i s e x p l a i n i n g s h o r t - r u n changes i n consumption. However, i n h e r l a t e r paper, F l a v i n (1985), uses annual d a t a , and i t seems l e s s l i k e l y t h a t changes i n t h e r a t e o f r e t u r n t o c a p i t a l dominate endogenous changes i n w e a l t h a c c u m u l a t i o n . 4. M u e l l b a u e r says t h a t t h i s assumption was n o t i m p o r t a n t f o r h i s f i n d i n g s on t h e U. K . economy. See appendix f o r t h e a l t e r n a t i v e d e r i v a t i o n o f M u e l l b a u e r ' s model. 5 . The Wickens and Molana model was n o t updated, because i t i s s i m i l a r t o M u e l l b a u e r ' s , a p a r t f r o m some a d d i t i o n a l terms t h a t c o m p l i c a t e t h e e s t i m a t i o n procedure. 6. S e r i a l l y c o r r e l a t e d e r r o r s may n o t s i g n a l a breakdown of t h e model i f consumers t a k e more t h a n one q u a r t e r t o a s s i m i l a t e new i n f o r m a t i o n and a c t upon a changed e x p e c t a t i o n o f l i f e c y c l e w e a l t h . 7. The e q u i v a l e n c e o f t h e s e two procedures i s p r o v e d i n F l a v i n (1981, Appendix 11). 8. When t h e consumption and income v a r i a b l e s a r e d e t r e n d e d w i t h t h e i r average g r o w t h r a t e s between 1947:IQ and 1984:IVQ, t h e LRS f o r t h e j o i n t t e s t o f t h e AlnYD terms becomes 13.5, which i m p l i e s t h e r e j e c t i o n o f t h e n u l l h y p o t h e s i s a t about a 10 p e r c e n t s i g n i f i c a n c e l e v e l . 9. The s t a n d a r d e r r o r of t h e consumption e q u a t i o n a l s o i n c r e a s e d , b u t t h i s i s p r o b a b l y due t o t h e p o o r e r f i t o f t h e i n t e r e s t - r a t e e q u a t i o n t h r o u g h t h e cross- equation c o n s t r a i n t s . http://clevelandfed.org/research/workpaper/index.cfm Best available copy References B a r r o , R o b e r t J . " U n a n t i c i p a t e d Money Growth and Unemployment i n t h e U n i t e d S t a t e s , " American Economic Review, v o l . 67, no. 2 (March 1977) pp. 101-15. . " Are Government Bonds Net Wealth?" J o u r n a l o f P o l i t i c a l Economy, v o l . 82 no. 6 (November/December 19741, pp. 1095- 1117. Bernanke, Ben, S. "Permanent Income, L i q u i d i t y , and E x p e n d i t u r e on Automobiles: Evidence f r o m Panel Data," The Q u a r t e r l y J o u r n a l of Economics, August 1984, pp. 587-614. . 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