The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.
http://clevelandfed.org/research/workpaper/index.cfm Best available copy Working paper 8503 FORECASTING GNP USING MONTHLY M1 by Michael L. Bagshaw Thanks a r e due t o B i 11 Gavi n, James Hoehn, and Kim Kowalewski f o r h e l p f u l comments. Working papers o f the F e d e r a l Reserve Bank o f Cleveland are p r e l i m i n a r y materials, circulated 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 t h o s e o f t h e a u t h o r and n o t n e c e s s a r i l y those o f the Federal Reserve Bank o f Cleveland o r the Board o f Governors o f t h e Federal Reserve System. August 1985 Federal Reserve Bank o f C l e v e l a n d http://clevelandfed.org/research/workpaper/index.cfm Best available copy FORECASTING GNP USING MONTHLY MI Key words: Forecasting, m u l t i v a r i a t e time s e r i e s . Abstract I n t h i s paper, we p r e s e n t an a p p l i c a t i o n o f mu1t i v a r i a t e time s e r i e s f o r e c a s t i n g i n which t h e d a t a c o n s i s t of a m i x t u r e o f q u a r t e r l y and m o n t h l y series. I n p a r t i c u l a r , we use monthly s e r i e s o f M1 t o f o r e c a s t q u a r t e r l y values o f t h e nominal gross n a t i o n a l product (GNP). Results from e s t i m a t i n g models o v e r t h e p e r i o d 1959:IQ through 1979:IVQ i n d i c a t e t h a t models i n v o l v i n g o n l y movements i n monthly M I s e r i e s provide approximately t h e same e x p l a n a t o r y power as one u s i n g q u a r t e r l y M I . When these models a r e used t o f o r e c a s t GNP over t h e time p e r i o d 1980:IQ t h r o u g h 1984:IIIQ, t h e r e s u l t s are mixed. For one- quarter- ahead change, four- quarter- ahead change, and one-year change f o r e c a s t s , t h e Root Mean Square E r r o r (RMSE) f o r a l l t h e models ( i n c l u d i n g a u n i v a r i a t e model o f GNP) have approximately t h e same RMSE ( f o r a g i v e n forecast horizon) f o r the e n t i r e period. However, when we examine t h e p e r i o d 1 9 8 3 : I I I Q through 1 9 8 4 : I I I Q , t h e models u s i n g M I p r o v i d e b e t t e r f o r e c a s t s t h a n the u n i v a r i a t e model, i n terms o f RMSE, f o r f o u r - q u a r t e r and one-year change forecasts. Also, the models u s i n g monthly M1 data, p e r f o r m a t l e a s t approximately equal t o the model u s i n g q u a r t e r l y M1 d a t a , and i n some cases substantially better. A l l o f t h e m u l t i v a r i a t e models used i n t h i s s t u d y i n d i c a t e t h a t t h e growth i n GNP was smaller than expected r e l a t i v e t o changes i n M I over t h e e n t i r e p e r i o d . GNP growth had a l a r g e r v a r i a n c e from 1980:IVQ http://clevelandfed.org/research/workpaper/index.cfm Best available copy -2- t o 1983:IIQ than was expected based on a l l models used i n t h i s study. Comparisons o f f o r e c a s t e r r o r s among d i f f e r e n t s t u d i e s i s o f t e n d i f f i c u l t because o f t h e d i f f e r e n t time p e r i o d s i n v o l v e d and because o f t h e d i f f e r e n t amount o f d a t a a v a i l a b l e when the f o r e c a s t s a r e a c t u a l l y made. However, comparisons o f t h e f o r e c a s t s e r r o r s f o r these models t o r e s u l t s from o t h e r s t u d i e s u s i n g S t . L o u i s type equations i n d i c a t e t h a t the models presented i n t h i s s t u d y appear t o perform s l i g h t l y b e t t e r than t h e S t . L o u i s models f o r one- quarter f o r e c a s t s i n terms o f RMSE. Also, r e s u l t s f o r one- year change f o r e c a s t s a r e a p p a r e n t l y b e t t e r than t h e median o f f i v e e a r l y - q u a r t e r f o r e c a s t s by the ASAINBER survey, Chase, Data Resources, I n c . (DRI), Wharton, and BEA. I. Introduction Sometimes d a t a a r e a v a i l a b l e a t d i f f e r e n t p e r i o d i c i t i e s f o r t h e s e r i e s involved i n a m u l t i v a r i a t e forecasting e f f o r t . I t i s d e s i r a b l e t o use t h i s information o p t i m a l l y i n developing f o r e c a s t s . For example, i f p a r t o f t h e data i s a v a i l a b l e monthly and the r e s t q u a r t e r l y , then t h e r e i s a p o s s i b i l i t y o f developing e a r l i e r f o r e c a s t s by u s i n g the monthly d a t a r a t h e r than q u a r t e r l y summary d a t a f o r those s e r i e s . Also, i t m i g h t be p o s s i b l e t o develop b e t t e r f o r e c a s t s u s i n g t h e i n d i v i d u a l monthly s e r i e s r a t h e r t h a n a quarterly- aggregated s e r i e s . I n t h i s study, we a r e i n t e r e s t e d i n t h e p o s s i b l e use of t h e monthly money supply ( M I ) s e r i e s t o f o r e c a s t q u a r t e r l y nominal GNP. We have chosen t o examine t h e r e l a t i o n s h i p between M1 and GNP because t h e instruments o f http://clevelandfed.org/research/workpaper/index.cfm Best available copy -3monetary c o n t r o l a f f e c t t h e money supply and then, i t i s hoped, t h e u l t i m a t e D u r i n g most o f t h e p e r i o d i n which t h e Federal Reserve has t a r g e t GNP. e s t a b l i s h e d e x p l i c i t t a r g e t ranges f o r t h e monetary aggregates, M1 has been regarded as t h e p r i m a r y measure. While t h e r e a r e some questions c o n c e r n i n g the r e c e n t s t a b i l i t y o f t h e r e l a t i o n s h i p between M I and GNP, B a t t e n and Thornton (1983), as a r e s u l t of a comparison of M1 and M2, i n d i c a t e t h a t as o f 1983 t h e r e was no c o n c l u s i v e evidence t h a t t h i s r e l a t i o n s h i p had d e t e r i o r a t e d enough t o j u s t i f y u s i n g M2 i n place of M I . Judd and Motley (1984) agreed with t h i s conclusion. A s we w i l l demonstrate i n t h i s paper, t h e r e l a t i o n s h i p between M I and GNP appears t o have r e s t a b i l i z e d between 1983:IIQ and 1 9 8 4 : I I I Q . This r e s u l t supports t h e study by Judd and Motley (1984) t h a t s t a t e s t h a t t h e change i n v e l o c i t y d u r i n g t h e e a r l y 1980s was caused by t h e sharp d e c l i n e i n nominal i n t e r e s t r a t e s t h a t occurred a t t h a t time. By 1983:IIQ, Judd and M o t l e y p o i n t o u t , the i n t e r e s t r a t e s would no l o n g e r have t h i s impact, and thus, v e l o c i t y and any o t h e r r e l a t i o n s h i p between M1 and GNP, should have r e t u r n e d t o normal. Some o f t h e q u e s t i o n s addressed i n t h i s a n a l y s i s are: 1) can we d e v e l o p forecasts o f GNP u s i n g o n l y t h e f i r s t monthly M1 s e r i e s ( o r f i r s t and second month), which a r e as good as, o r b e t t e r than, those u s i n g the q u a r t e r l y M1 s e r i e s and 2) can we develop f o r e c a s t s o f GNP u s i n g t h e three i n d i v i d u a l monthly M i s e r i e s , which a r e b e t t e r t h a n those developed using t h e q u a r t e r l y M I series. To i n v e s t i g a t e t h i s q u e s t i o n , we use a u t o r e g r e s s i v e moving average (ARMA) and m u l t i v a r i a t e ARMA time s e r i e s methods t o develop models r e l a t i n g 1) GNP and i t s p a s t h i s t o r y , 2) GNP and monthly M1 s e r i e s , and 3) GNP and quarterly M I . http://clevelandfed.org/research/workpaper/index.cfm Best available copy We a l s o a r e i n t e r e s t e d i n d e t e r m i n i n g whether t h e f o r e c a s t s d e r i v e d f r o m time s e r i e s methods are as a c c u r a t e as f o r e c a s t s developed u s i n g o t h e r techniques. This comparison o f o u r r e s u l t s t o o t h e r r e s u l t s i s complicated b y t h e f a c t t h a t o f t e n o t h e r s t u d i e s are done over d i f f e r e n t time p e r i o d s and have d i f f e r e n t amounts o f d a t a a v a i l a b l e when t h e f o r e c a s t s are a c t u a l l y produced. I n t h i s paper, we compare o u r r e s u l t s t o t h e r e s u l t s o f two papers u s i n g S t . Louis type equations. The r e s u l t s should be i n t e r p r e t e d c a r e f u l l y , because these e a r l i e r s t u d i e s were c a r r i e d o u t o v e r a s l i g h t l y d i f f e r e n t time p e r i o d than our study. Also, t h e d a t a a v a i l a b l e a t t h e time o f these s t u d i e s may have been r e v i s e d since then. We a l s o compare o u r r e s u l t s t o a study by McNees and Ries (1983) t h a t used t h e median f o r e c a s t o f a group o f f i v e forecasts--ASA/NBER survey, Chase, D R I , Wharton, and BEA. While t h e d a t a f r o m the McNess and Ries study can be used t o c a l c u l a t e s t a t i s t i c s f o r t h e same p e r i o d as p a r t o f o u r study, t h e r e s u l t s must be i n t e r p r e t e d c a r e f u l l y , because t h e amount o f i n f o r m a t i o n a v a i l a b l e when t h e f o r e c a s t s used i n t h a t study were produced i s most l i k e l y d i f f e r e n t from the i n f o r m a t i o n used i n o u r study. 11. M u l t i v a r i a t e ARMA Time S e r i e s Models The f o l l o w i n g i s a v e r y b r i e f d e s c r i p t i o n o f m u l t i v a r i a t e ARMA s e r i e s models; T i a o and Box (1981) p r o v i d e a more d e t a i l e d d e s c r i p t i o n . general m u l t i v a r i a t e ARMA model o f o r d e r (p,q) i s g i v e n by: The http://clevelandfed.org/research/workpaper/index.cfm Best available copy where (2) where B = backshift operator (i.e., I - = k x k i d e n t i t y matrix, z - = v e c t o r o f k v a r i a b l e s i n t h e model, BSzl,, = z,,,-,), &,Is and 8,'s = k x k m a t r i x e s o f unknown parameters, e0 = k x 1 v e c t o r o f unknown parameters, and a - = k x 1 v e c t o r o f random e r r o r s t h a t a r e i d e n t i c a l l y and independently d i s t r i b u t e d as N(O,C). Thus, i t i s assumed t h a t t h e a,,,'s a t d i f f e r e n t p o i n t s i n time are independent, b u t n o t n e c e s s a r i l y t h a t the elements o f 6, a r e independent a t a given p o i n t i n time. The n- period- ahead f o r e c a s t s from these models a t time t ( z t ( n ) ) a r e given by: (3) -z,(n) = @ l C g t t n -+~ l... + + Cat+,] - ~lCat+,-ll- ..- - gqCgt+n-,l, Cx,+,-,I v a r i a b l e s x,+,-, where f o r any value o f t,n,m, values of t h e random &pCzt+n-pI i m p l i e s t h e c o n d i t i o n a l expected a t time t. I f n-m i s l e s s than o r equal t o zero, then t h e c o n d i t i o n a l expected values a r e t h e a c t u a l values o f t h e random v a r i a b l e s and t h e e r r o r terms. I f n-m i s g r e a t e r than zero, then http://clevelandfed.org/research/workpaper/index.cfm Best available copy the expected values are t h e best f o r e c a s t s a v a i l a b l e f o r these random v a r i a b l e s and e r r o r terms a t time t. w i t h present and past information, Because t h e e r r o r terms a r e u n c o r r e l a t e d t h e b e s t f o r e c a s t s o f t h e e r r o r terms f o r n-m g r e a t e r than zero are t h e i r c o n d i t i o n a l means, which a r e zero. The forecasts can be generated i t e r a t i v e l y w i t h t h e one-period-ahead f o r e c a s t s t h a t depend o n l y on known values o f t h e v a r i a b l e s and e r r o r terms. The longer- length f o r e c a s t s , i n t u r n , depend on t h e s h o r t e r - l e n g t h f o r e c a s t s . 111. Models For Forecasting GNP The v a r i a b l e s i n t h e models developed i n t h i s paper are t h e money supply M1 and GNP i n c u r r e n t d o l l a r s , both s e a s o n a l l y adjusted. The money supply i s represented by f o u r s e r i e s - M1 which i s t h e q u a r t e r l y money supply and M I A , M l B , and M1C which are monthly s e r i e s . MIA i s the f i r s t month o f t h e q u a r t e r , M1B i s the second month of t h e q u a r t e r , and M1C i s t h e t h i r d month of t h e quarter. Thus, models i n v o l v i n g MIA and/or M1B would be models i n v o l v i n g information t h a t would be a v a i l a b l e e i t h e r two months o r one month e a r l i e r than t h e q u a r t e r l y data. Models i n v o l v i n g M1C w i l l be used t o t e s t whether there are more e f f i c i e n t ways o f u s i n g t h e i n f o r m a t i o n w i t h i n a q u a r t e r than j u s t combining t h e i n f o r m a t i o n i n t o one q u a r t e r l y number. The u n i v a r i a t e model used i n t h i s paper was estimated u s i n g Box- Jenkins modeling (Box and Jenkins 1976). The m u l t i v a r i a t e models were e s t i m a t e d u s i n g the Tiao-Box procedure t o e s t i m a t e t h e parameters o f a m u l t i v a r i a t e simultaneous equation model; The procedure i s an i n t e r a c t i v e one s i m i l a r i n p r i n c i p l e t o t h a t used i n s i n g l e Box- Jenkins modeling. See T i a o and Box http://clevelandfed.org/research/workpaper/index.cfm Best available copy -7(1981). The steps i n v o l v e d are: (1) t e n t a t i v e l y i d e n t i f y a model by examining a u t o c o r r e l a t i o n s and c r o s s - c o r r e l a t i o n s o f the s e r i e s , ( 2 ) e s t i m a t e the parameters of t h i s model, and (3) apply d i a g n o s t i c checks t o t h e residuals. These d i a g n o s t i c checks i n c l u d e checks o f c o r r e l a t i o n s i n t h e residuals, n o r m a l i t y o f residuals, etc. If t h e r e s i d u a l s do n o t pass t h e d i a g n o s t i c checks, t h e n t h e t e n t a t i v e model i s m o d i f i e d and steps 2 and 3 a r e repeated. T h i s process continues u n t i l a s a t i s f a c t o r y model i s o b t a i n e d . The models r e s u l t i n g from a p p l y i n g these techniques t o the change i n t h e l o g a r i t h m o f t h e GNP, q u a r t e r l y M I , and monthly M1 s e r i e s from 1959:IQ t h r o u g h 1979:IVQ a r e i n t h e appendix. I n t h i s a n a l y s i s , t h e change i n a monthly series i s d e f i n e d as the d i f f e r e n c e between t h e c u r r e n t value and t h e corresponding v a l u e i n t h e p r e v i o u s q u a r t e r . Table 1 gives the sample standard d e v i a t i o n s f o r t h e GNP e q u a t i o n from t h e w i t h i n sample e s t i m a t i o n of these models. From t a b l e 1, we see t h a t t h e change i n any o f the monthly M1 series has a p p r o x i m a t e l y as much i n f o r m a t i o n concerning the behavior o f t h e change i n GNP as t h e change i n t h e quarterly M1 s e r i e s d u r i n g the e s t i m a t i o n period. These models were then used t o f o r e c a s t from 1980:IQ through 1 9 8 4 : I I I Q . The f o r e c a s t i n g p e r i o d i s broken i n t o two p e r i o d s because o f one- time events i n the e a r l y 1980s (such as t h e i m p o s i t i o n o f c r e d i t c o n t r o l s i n 1980 and t h e Depository I n s t i t u t i o n s D e r e g u l a t i o n and Monetary Control Act o f 1980 and t h e s h i f t i n monetary p o l i c y and h i g h i n t e r e s t r a t e s d u r i n g the 1980s), i n d i c a t i n g t h a t 1980:IQ through 1 9 8 3 : I I Q m i g h t n o t be r e p r e s e n t a t i v e o f the e s t i m a t i o n period. Forecasts were developed f o r t h r e e s i t u a t i o n s : 1) one- quarter- ahead, 2) four- quarter- ahead ( a f o r e c a s t of t h e change i n GNP f o u r q u a r t e r s ahead o f t h e c u r r e n t q u a r t e r ) , and 3) one-year-change ( t h a t i s , the change over t h e http://clevelandfed.org/research/workpaper/index.cfm Best available copy -8next f o u r q u a r t e r s combined). A l l o f these f o r e c a s t s were generated u s i n g only c u r r e n t o r past information. The r e s u l t s a r e presented i n t a b l e s 2, 3, and 4. From t a b l e 2, we see t h a t , i n terms o f RMSE, t h e r e i s e s s e n t i a l l y no difference i n t h e performance o f a l l t h e models used i n t h i s s t u d y for one- quarter f o r e c a s t s . For t h e l a t t e r p e r i o d , t h e u n i v a r i a t e model does have a s m a l l e r RMSE than a l l b u t one of t h e m u l t i v a r i a t e models. Also, we see t h a t there i s a s u b s t a n t i a l d i f f e r e n c e between t h e RMSEs f r o m 1980:IQ t h r o u g h 1983:IHQ and those f r o m 1 9 8 3 : I I I Q through 1 9 8 4 : I I I Q . The RMSEs i n t h e l a t t e r p e r i o d a r e , a t most, 20 p e r c e n t l a r g e r than the corresponding within- sample standard d e v i a t i o n s . I n t h e former p e r i o d , the RMSEs a r e up t o 80 p e r c e n t l a r g e r t h a n t h e standard d e v i a t i o n s . The RMSEs f o r these models can be compared w i t h o t h e r r e s u l t s f o r f o r e c a s t i n g GNP. For example, B a t t e n and Thornton (1983) used a v e r s i o n o f t h e S t . Louis e q u a t i o n i n v o l v i n g a monetary measure ( e i t h e r M I o r M2) and high-employment government e x p e n d i t u r e s . These models were estimated f o r 1962:IIQ through 1979:IVQ and t h e n used t o f o r e c a s t f o r 1980:IQ through 1983:IQ- The r e s u l t i n g RMSEs (when expressed i n u n i t s corresponding t o those used i n t h i s study) were 0.0173 for the model u s i n g M I , and 0.0150 f o r the model u s i n g M2. Both o f these models used contemporaneous values of t h e monetary v a r i a b l e and t h e high-employment government expenditures v a r i a b l e s . A l s o , Hafer (1984) used a v a r i a n t o f t h e S t , Louis model u s i n g M I o r a debt measure ( t o t a l domestic n o n f i n a n c i a l d e b t ) and high-employment f e d e r a l expenditures, r e l a t i v e p r i c e o f energy, and a s t r i k e variable. These models were estimated f o r 1960:IQ through 1981:IVQ and then used t o f o r e c a s t 1982:IQ t h r o u g h 1983:IVQ. models were 0.0148 and 0.0155. The r e s u l t i n g RMSEs f o r these two Again, these models used contemporaneous values o f t h e independent v a r i a b l e s . Although i t i s d i f f i c u l t t o compare t h e http://clevelandfed.org/research/workpaper/index.cfm Best available copy -9- r e s u l t s o f t h e c u r r e n t study w i t h these e a r l i e r s t u d i e s because of d i f f e r e n t time p e r i o d s , the r e s u l t s o f t h i s s t u d y do compare f a v o r a b l y w i t h p r e v i o u s results. The l a r g e s t RMSE o f any of the models i n t h i s study f o r one- quarter f o r e c a s t i s 0.0139. Also, t h e models presented i n t h e contemporaneous st.udy d i d n o t use c u r r e n t values of M I . From t a b l e 3, we see t h a t a g a i n a l l o f t h e models p r o v i d e r o u g h l y equal f o r e c a s t s f o r the e n t i r e time p e r i o d f o r four- quarter- ahead f o r e c a s t s . However, a l l o f t h e models i n v o l v i n g M I have s l i g h t l y smaller RMSEs than t h e u n i v a r i a t e model. For t h e l a t t e r p e r i o d , a l l o f t h e models u s i n g t h e M1 series have RMSEs t h a t a r e moderately smaller t h a n the u n i v a r i a t e model's RMSE. The model w i t h o n l y M I A does s l i g h t l y worse than the o t h e r models. This r e s u l t i n d i c a t e s t h a t once we know t h e M1 value f o r the second month o f the q u a r t e r , we can f o r e c a s t t h e four- quarter- ahead change i n the l o g o f GNP j u s t as w e l l as i f we knew and used t h e q u a r t e r l y M1 value. There i s a s l i g h t i n d i c a t i o n t h a t f o r t h i s l a t t e r p e r i o d , we can o b t a i n a b e t t e r f o r e c a s t when we have an e n t i r e q u a r t e r ' s i n f o r m a t i o n on M1 by u s i n g the i n d i v i d u a l monthly data s e r i e s i n s t e a d o f t h e q u a r t e r l y s e r i e s . However, t h i s d i f f e r e n c e i s v e r y small, and g i v e n t h e small sample ( f i v e q u a r t e r s ) , t h e r e s u l t c o u l d be due t o random e f f e c t s . When we examine t h e one year change f o r e c a s t s ( t a b l e 4). we see t h a t again t h e r e i s no s u b s t a n t i a l differences period. among t h e models i n the e n t i r e t i m e However, t h e u n i v a r i a t e model does have a smaller RMSE then most o f the models. This does n o t c o n t i n u e i n t h e l a t t e r p e r i o d . I n f a c t , the u n i v a r i a t e model has t h e l a r g e s t RMSE i n t h i s l a t t e r period. I n contrast t o the four- quarter- ahead f o r e c a s t s , t h e f o r e c a s t u s i n g o n l y M I A has a much smaller RMSE then any o f t h e o t h e r models. Also, a l l t h e models u s i n g m o n t h l y http://clevelandfed.org/research/workpaper/index.cfm Best available copy M I data, except f o r t h e f o u r - v a r i a t e model, have smaller RMSEs t h a n t h e q u a r t e r l y model i n t h i s l a t t e r p e r i o d . As a comparison t o these f o r e c a s t s , McNees and Ries (1983) p r e s e n t e d t h e e r r o r s made i n t h e median o f e a r l y - q u a r t e r f o r e c a s t s by t h e ASAINBER survey, Chase, DRI, Wharton, and BEA. These f o r e c a s t s had a RMSE o f 0.0476 and a mean e r r o r o f 0.0213 f r o m 1980:IVQ through 1983:IIQ. The l a r g e s t RMSE o v e r t h i s time p e r i o d f o r t h e models presented i n t h i s s t u d y was 0.0428. mean e r r o r was -0.0146. The l a r g e s t Thus, the f o r e c a s t s g i v e n by these models compare f a v o r a b l y w i t h t h e median f o r e c a s t s as r e p o r t e d i n McNees and R i e s . This c o n c l u s i o n must be made i n t h e knowledge t h a t t h e f o r e c a s t e r s used i n t h e McNess and Ries study would have had a d i f f e r e n t s e t o f i n f o r m a t i o n t h a n used i n t h e models developed i n t h i s study. I n p a r t i c u l a r , these f o r e c a s t e r s would have based t h e i r f o r e c a s t s on d a t a t h a t has s i n c e been r e v i s e d . The f o r e c a s t s developed i n o u r study used t h e l a t e s t d a t a a v a i l a b l e . To examine t h e r e s u l t s o f t h e one-period- ahead forecasts f u r t h e r , we examine t h r e e s t a t i s t i c s t h a t t e s t whether t h e estimated models p r o v i d e an adequate r e p r e s e n t a t i o n f o r t h e post- sample p e r i o d s . I f t h e model remains constant over time, then t h e f o l l o w i n g s t a t i s t i c s have the i n d i c a t e d approximate d i s t r i b u t i o n s : 1 0, C air JT t - N(O,l), and http://clevelandfed.org/research/workpaper/index.cfm Best available copy where o , i s t h e e s t i m a t e d within- sample standard d e v i a t i o n f o r t h e i t h model, T i s t h e number o f observations i n t h e post-sample p e r i o d b e i n g t e s t e d , and Xi i s t h e mean f o r e c a s t e r r o r i n t h e post- sample p e r i o d . Equation ( 4 ) i s t h e sum o f the square o f t h e f o r e c a s t e r r o r s standardized by t h e a p p r o p r i a t e within- sample variance. I f e i t h e r t h e mean o r the variance o f t h e change i n t h e l o g of GNP has changed, then t h i s s t a t i s t i c w i l l be a f f e c t e d . This s t a t i s t i c t h u s t e s t s f o r changes i n b o t h t h e v a r i a n c e and the mean o f t h e s e r i e s . This s t a t i s t i c can a l s o be used t o t e s t whether the RMSE i s s t a t i s t i c a l l y l a r g e r than t h e within- sample standard d e v i a t i o n , because i t i s t h e mean square e r r o r . Equation 5 i s t h e sum o f t h e f o r e c a s t e r r o r s s t a n d a r d i z e d by the within- sample standard d e v i a t i o n f r o m t h e appropriate model. I f t h e mean o f t h e change i n t h e log o f GNP has s h i f t e d r e l a t i v e t o t h e e s t i m a t e d models, t h e n t h i s s t a t i s t i c w i l l be a f f e c t e d . Equation 6 i s t h e sum o f the square of t h e d e v i a t i o n o f the i n d i v i d u a l f o r e c a s t e r r o r s f r o m t h e i r mean, standardized by t h e a p p r o p r i a t e within- sample variance e s t i m a t e . T h i s s t a t i s t i c w i l l be a f f e c t e d i f the variance o f t h e change i n t h e l o g o f GNP changes i n t h e post-sample p e r i o d r e l a t i v e t o t h e models. The r e s u l t s o f a p p l y i n g these t e s t s t o each o f the models estimated i n t h i s paper a r e i n t a b l e s 5 through 7. From t a b l e 5, we see t h a t f o r t h e e n t i r e post-sample p e r i o d and the 1980:IVQ t o 1 9 8 3 : I I Q p e r i o d , a l l t h e t e s t s are s i g n i f i c a n t a t the 5 p e r c e n t level a t least. T h i s i m p l i e s t h a t e i t h e r the mean o r t h e variance ( o r b o t h ) of t h e GNP s e r i e s has changed r e l a t i v e t o a l l o f t h e models being used i n t h i s study. For t h e p e r i o d 1 9 8 3 : I I I Q t o 1 9 8 4 : I I I Q , none o f t h e models has significant results. Examining t a b l e 6, we see t h a t the mean forecast e r r o r http://clevelandfed.org/research/workpaper/index.cfm Best available copy -1 2f o r t h e u n i v a r i a t e model i s n o t s i g n i f i c a n t l y d i f f e r e n t f r o m z e r o for any o f the p e r i o d s being s t u d i e d here. However, t h e r e s t o f t h e models have a s i g n i f i c a n t n e g a t i v e mean f o r e c a s t e r r o r f o r the e n t i r e post- sample p e r i o d and f o r t h e e a r l i e r subperiod. Also, i n t h e second subperiod, the mean e r r o r s f o r a l l t h e m u l t i v a r i a t e models a r e n e g a t i v e , a l t h o u g h n o t s i g n i f i c a n t . This means t h a t on average a l l o f the m u l t i v a r i a t e models a r e o v e r f o r e c a s t i n g t h e change i n GNP for t h e e n t i r e post- sample p e r i o d . Thus, the models a r e i n d i c a t i n g t h a t GNP has n o t grown as r a p i d l y as expected r e l a t i v e t o growth i n MI. Table 7 i n d i c a t e s t h a t a l l o f t h e models have s i g n i f i c a n t l y l a r g e r out- of- sample variances r e l a t i v e t o in- sample variances. Thus, t h e growth o f GNP i n t h i s p e r i o d has been more v a r i a b l e than expected. IV. Summary The r e s u l t s of t h i s paper a r e mixed -- t h a t i s , i f we examine a l l t h e 1980s, t h e conclusions are d i f f e r e n t from those o b t a i n e d i f we examine o n l y 1 9 8 3 : I I I Q through 1 9 8 4 : I I I Q . I n t h e e n t i r e p e r i o d , the u n i v a r i a t e model o f GNP f o r e c a s t s as w e l l as, i f n o t b e t t e r than, any o f the m u l t i v a r i a t e models, d e s p i t e t h e f a c t t h a t m u l t i v a r i a t e models p r o v i d e d b e t t e r - f i t t i n g models during the estimating period. We b e l i e v e t h a t t h i s i s due t o t h e one-time events t h a t o c c u r r e d d u r i n g t h e e a r l y 1980s. Events o f t h i s s o r t would n a t u r a l l y a f f e c t r e l a t i o n s h i p s among v a r i a b l e s more than they would a f f e c t the r e l a t i o n s h i p o f one v a r i a b l e t o i t s own p a s t . http://clevelandfed.org/research/workpaper/index.cfm Best available copy -1 3The evidence from 1 9 8 3 : I I I Q through 1 9 8 4 : I I I Q appears t o i n d i c a t e t h a t these disturbances have worked t h e i r way through the economy, and t h a t t h e models estimated through 1979:IVQ a r e once a g a i n a p p l i c a b l e for f o r e c a s t i n g . The r e s u l t s f o r t h i s p e r i o d seem t o i n d i c a t e t h a t indeed, i f we wish t o f o r e c a s t nominal GNP f o r more t h a n one- quarter ahead, i t i s w o r t h w h i l e t o consider adding a measure of M1 t o t h e f o r e c a s t i n g model. small number o f observations ( f i v e ) Because o f t h e i n t h i s period, t h i s c o n c l u s i o n i s weak, and f u r t h e r study i s necessary when more d a t a become a v a i l a b l e . The r e s u l t s i n t h i s l a t t e r p e r i o d do appear t o i n d i c a t e , t h a t b y u s i n g monthly M1 data, we can f o r e c a s t q u a r t e r l y GNP as w e l l as, o r b e t t e r than by u s i n g q u a r t e r l y M1 data. The f o r e c a s t s f r o m the f i r s t two monthly M1 s e r i e s would be a v a i l a b l e before t h e q u a r t e r l y M I s e r i e s , p r o v i d i n g us e a r l i e r f o r e c a s t s t h a t a r e a t l e a s t as a c c u r a t e . For t h e one-year-change f o r e c a s t s , the f o r e c a s t s u s i n g monthly M I d a t a a r e a c t u a l l y s u b s t a n t i a l l y b e t t e r than those f r o m t h e q u a r t e r l y model. T h i s c o n c l u s i o n must be f u r t h e r t e s t e d as more d a t a become a v a i l a b l e because o f t h e small sample s i z e i n t h i s l a t t e r period. The r e s u l t s i n t h i s s t u d y a l s o i n d i c a t e t h a t t h e growth i n M I d u r i n g t h i s t i m e was slower than would have been expected, r e l a t i v e t o models i n v o l v i n g t h e growth o f M I . T h i s seems t o have l e v e l e d o f f i n t h e second subperiod s t u d i e d , b u t the d i f f e r e n c e i s s t i l l s l i g h t l y negative, a l t h o u g h n o t s i g n i f i c a n t l y so. Also, t h e v a r i a n c e of t h e growth i n GNP was s i g n i f i c a n t l y l a r g e r f r o m 1980:IVQ t o 1983:IIQ, r e l a t i v e to the in- sample variance o f a l l t h e models used i n t h i s s t u d y . http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 1 Within-Sample Standard D e v i a t i o n s o f GNP Samp 1 e standard deviation Mode 1 Uni v a r i a t e -0095 Bivariate with quarterly M l B i v a r i a t e w i t h MIA,- t- .0081 I .0082 B i v a r i a t e w i t h MI B, - I .0082 B i v a r i a t e w i t h MICt- -0080 Bivariate with and M I B t - , Four- variate w i t h M I A t - , , M I B t - , , and MIC,-I .0082 -0079 http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 2 One-Quarter Forecasts Time p e r i o d Mode 1 Mean error RMSE Mean error RMSE Mean error RMSE Univariate .0004 .0122 -.0004 -0136 .0024 .0071 - .0051 -0125 -.0056 -0136 -.0037 .0089 -. 0041 .0116 -.0047 -0129 -.0025 .0069 Bi v a r i a t e w i t h MlBt-1 - .0048 -0125 -.0055 .0135 -.0028 .0092 Bi v a r i a t e w i t h MlCt- I - .0046 .0121 -.0055 -0128 -.0023 .0098 Trivariate with M I A t - l and M I B t - l -.0047 .0135 -.0049 -0148 -.0043 ,0083 Four- vari a t e w i t h M l A t - I , MlBt-1, and M1Ct-, -. 0055 .0129 -.0060 -0139 -.0043 .0095 Bi v a r i a t e w i t h M l t- I Bivari ate with MlAt'-, NOTE: RMSE i s t h e r o o t mean square e r r o r o f the f o r e c a s t . http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 3 Four-Quarter-Ahead Forecasts Time period 1980:IVQ-1984:IIIQ Mode 1 Mean error Univariate -001 2 Bivariate with M1 t-1 -.0012 Bivariate with MlAt-1 .0004 Bivariate with MlBt-1 - .0005 Bivariate with MICt_, - .0000 Trivariate with and MIBt-l -.0013 Four-vari ate wi th MlAt-1, MlBt-1, and MICt-l -.0017 1980:IVQ-1983:IIQ Mean error RMS E .0147 1983:IIIQ-1984:IIIQ Mean error RMSE -0082 http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 4 One-Year-Change Forecasts Time period 1980:IVQ-1984:IIIQ Mode 1 Univariate B i vari ate with Ml t - 1 Bi vari ate wi th MlAt-1 Bi variate with MlBt-1 Bi variate with MlCt-1 Trivariate with MIAt-] and MIBt Four-variate with MlAt-, , MlBt-I, and M I C t - , Mean error RMS E - 1980:IVQ-1983:IIQ Mean error RMSE 1983:IIIQ-1984:IIIQ Mean error RMSE http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 5 T e s t s F o r RMSE Changes 1 980: IVQ1984: IIIQ Model Univariate Biv a r i a t e w i t h M1 , - I Bivariate with MIA,-I Bi v a r i a t e w i t h M1Bt-1 Bivariate w i t h MlCt- I Trivariate with MIA t - 1 and M I B t - , Four- variate w i t h M I B t - l and MlCt-I a. b. S i g n i f i c a n t a t 0.05 l e v e l . S i g n i f i c a n t a t 0.01 l e v e l . 1 980 : IVQ1983: 110 http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 6 Tests For Mean Changes 1980: IVQ1984:IIIQ Mode 1 Univariate Bi v a r i a t e w i t h Mlt- 1 Bi v a r i a t e w i t h MIA,- I Bivariate with MlBt-1 Bivariate w i t h MlCt- l Trivariate with MIAt-land MIBthl Four - vari a t e w i t h M I B t - l and MlCt - 1 a. b. S i g n i f i c a n t a t 0.05 l e v e l . S i g n i f i c a n t a t 0.01 l e v e l . 1 980: IVQ1983: I I Q -0.16 1983: IIIQ1984: I I I Q http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 7 Tests For Variance Changes 1 980: IVQ1984: IIIQ Model Uni v a r i a t e Bivariate with M I ,-I Bivariate with MlAt-1 Bivari ate w i t h MlBt-1 Bivariate with MlCt-1 T r i variate with MIAt-land M I B t - ! Four- variate w i t h MlAt-1, MlBt-1 and MICt-I a. b. S i g n i f i c a n t a t 0.05 l e v e l . S i g n i f i c a n t a t 0.01 l e v e l . 1980: IVQ1983: IIQ 28-67" 1983: IIIQ1984: IIIQ http://clevelandfed.org/research/workpaper/index.cfm Best available copy Append i x Univariate model (1-.3098B)Vln(GNPt) = -0137 + a t Bivariate model with quarterly GNP and M1 Bivariate model with quarterly G N P and first month o f quarter MI (MIA) Vln(GNPt) = .429V1n(M1At-l) + .318V1n(M1Ar-,> + a l +.0110 Bivariate model with quarterly GNP and second month of quarter M1 (MlB) Vln(GNPt) = .334V1n(M1Bt-1) + .475Vln(M1Bt-,> + a l t +.0103 Bivariate model with quarterly GNP and third month o f quarter MI (MlC) Vln(GNPt) = .334V1n(M1Cr-l) + .482Vln(M1C,-2) + a l e +.0102 http://clevelandfed.org/research/workpaper/index.cfm Best available copy -22Appendix continued T r i v a r i a t e model w i t h q u a r t e r l y GNP and f i r s t and second month of quarter M I F o u r - v a r i a b l e model w i t h q u a r t e r l y GNP and f i r s t , second, and t h i r d month o f q u a r t e r M1 http://clevelandfed.org/research/workpaper/index.cfm Best available copy References Batten, D.S., and Daniel L. Thornton. " M I or M2: Which I s t h e B e t t e r Monetary Target?" Review, Federal Reserve Bank of S t . Louis, v o l . 65, no. 6 (June- July 19831, pp. 36-42. Box, George E.P., and Gwilym M. Jenkins. Time S e r i e s A n a l y s i s : F o r e c a s t i n q and C o n t r o l . San Francisco: Holden-Day, 1976. Hafer, R. W. "Money, Debt, and Economic A c t i v i t y , " Review, Federal Reserve Bank o f S t . Louis, v o l . 66, no. 6 (June- July 19841, pp. 18-25. Judd, John P., and B r i a n M o t l e y . "The ' G r e a t V e l o c i t y D e c l i n e ' o f 1982-83: A Comparative A n a l y s i s o f MI and M 2 , " Economic Review, Federal Reserve Bank o f San Francisco, no. 3 (Summer 1984). pp. 56-74. McNees, Stephen K., and John Ries. "The Track Record o f Macroeconomic England Economic Review, Federal Reserve Bank o f Boston (November-December 19831, pp. 5-18. ore casts ," New Tiao, G.C., and G.E.P. Box. " Modeling M u l t i p l e Time S e r i e s w i t h A p p l i c a t i o n s , " J o u r n a l of t h e American S t a t i s t i c a l A s s o c i a t i o n , v o l . 76, no. 376 (December 19811., pp. 802-16.
@FedFRASER