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http://clevelandfed.org/research/workpaper/index.cfm Best available copy Working Paper 8403 FORECASTING USING CONTEMPORANEOUS CORRELATIONS by Michael L. Bagshaw Working papers o f t h e Federal Reserve Bank o f Cleveland a r e 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 those 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 t h e Federal Reserve Bank o f Cleveland o r t h e Board o f Governors o f t h e Federal Reserve System. September 1984 Federal Reserve Bank o f Cleveland http://clevelandfed.org/research/workpaper/index.cfm Best available copy FORECASTING USING CONTEMPORANEOUS CORRELATIONS Key words: Contemporaneous correlations, forecasting, multivariate time -series. Abstract In this paper, we present a forecasting technique that uses contemporaneous correlations for forecasting in a time series model when only a subset of the variables are available for the current period. This method potentially provides more accurate forecasts than the standard time series forecasting method, which does not use contemporaneous data. This procedure is illustrated with an example of forecasting the gross national product (GNP), given current M-1 in a trivariate autoregressive moving average time series model. Results indicate that during the more stable economic period of 1976:IQ through 1979:IVQ, this method indeed provides forecasts with smaller root mean square errors than the standard forecasts. However, the results during the more turbulent 1980s are mixed. This latter result indicates that the relationship between the contemporaneous error terms from M-1 and GNP changed during this period. However, the results for the period 1983:IIQ through 1984:IIQ indicate that the relationship may have returned to pre-1980 form. The forecast errors during this latter period had smaller root mean square errors when the contemporaneous errors were used. http://clevelandfed.org/research/workpaper/index.cfm Best available copy I. I n t r o d u c t i o n When f o r e c a s t i n g i n a m u l t i v a r i a t e framework, t h e r e a r e occasions when d a t a a r e a v a i l a b l e o n l y f o r a subset o f t h e v a r i a b l e s i n t h e model f o r the current time period. I t i s d e s i r a b l e t o use t h e a d d i t i o n a l i n f o r m a t i o n i n t h i s subset o f known v a r i a b l e s t o f o r e c a s t values t h a t a r e n o t c u r r e n t l y available. T h i s i n c l u d e s b o t h values o f t h e unknown v a r i a b l e s i n t h e c u r r e n t p e r i o d and f u t u r e values o f a l l v a r i a b l e s . I n m u l t i p l e t i m e s e r i e s models, any contemporaneous c o r r e l a t i o n s among t h e v a r i a b l e s a r e modeled as p a r t o f the e r r o r structure. Consequently, t h e standard f o r e c a s t s generated b y these models cannot use t h e i n f o r m a t i o n f r o m c u r r e n t d a t a i n developing f o r e c a s t s . However, t h e i n f o r m a t i o n i n t h e contemporaneously c o r r e l a t e d e r r o r terms can be used t o o b t a i n f o r e c a s t s w i t h s m a l l e r e r r o r variances. T h i s paper e x p l a i n s how these f o r e c a s t s can be o b t a i n e d and presents an example where t h e contemporaneous c o r r e l a t i o n between t h e money s u p p l y (M-1) and nominal GNP i s used t o reduce t h e r o o t mean square e r r o r (RMSE) i n f o r e c a s t i n g c u r r e n t and f u t u r e GNP, g i v e n c u r r e n t M-1. While t h i s paper focuses on t h e m u l t i v a r i a t e ARMA t i m e s e r i e s models, t h e r e s u l t s h o l d f o r any m u l t i v a r i a t e models t h a t do n o t e x p l i c i t l y model t h e contemporaneous c o r r e l a t i o n s . 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 a u t o r e g r e s s i v e moving average t i m e s e r i e s models (ARMA); 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 . The ARMA models can i n c l u d e seasonal http://clevelandfed.org/research/workpaper/index.cfm Best available copy components; however, t h i s paper uses o n l y examples of nonseasonal models and, thus, p r e s e n t s o n l y a d e s c r i p t i o n o f t h e nonseasonal models. nonseasonal m u l t i v a r i a t e ARMA model o f o r d e r (p,q) The general i s given by where e (B) = I - e B -4 - -1 - ... - -4 e B ~ , where B = b a c k s h i f t o p e r a t o r ( i .e., S B Zi,t = Z i,t - s > 9 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, @ . I s and e . ' s = k x k m a t r i x e s o f unknown parameters, -J -J e = k x 1 v e c t o r o f unknown parameters, and -0 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 j ,t I s a t d i f f e r e n t points 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 t h e elements o f gt a r e independent a t a g i v e n 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 t i m e t ( z t ( n ) ) a r e g i v e n by http://clevelandfed.org/research/workpaper/index.cfm Best available copy where f o r any v a l u e o f t,n,m, [ L ~ +] i~m p-l i~ e s t h e c o n d i t i o n a l expected - e~t. values o f t h e random v a r i a b l e s L ~ +a t~t i m I f n-m i s l e s s t h a n o r equal t o 0, t h e n 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 t h a n 0, t h e n t h e expected v a l u e s a r e t h e b e s t f o r e c a s t s a v a i l a b l e f o r t h e s e random v a r i a b l e s and e r r o r terms a t t i m e t. 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 w i t h p r e s e n t and p a s t i n f o r m a t i o n , t h e b e s t f o r e c a s t s ( i n standard t i m e s e r i e s f o r e c a s t i n g ) o f t h e e r r o r terms f o r n-m g r e a t e r t h a n 0 a r e t h e i r c o n d i t i o n a l means, which a r e 0. The f o r e c a s t s 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 depending 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 l o n g e r - l e n g t h 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 forecasts. 111. Using Contemporaneous C o r r e l a t i o n s I f some o f t h e v a r i a b l e s a t t i m e t+l a r e known, t h e n t h e expected values o f t h e a s s o c i a t e d e r r o r terms a r e n o t n e c e s s a r i l y 0 i f t h e r e i s contemporaneous c o r r e l a t i o n between t h e e r r o r terms. t e r m c u r r e n t stands f o r t i m e t+l, i.e., t h e t i m e p e r i o d f o r which t h e f i n a l v a l u e o f some o f t h e v a r i a b l e s i s known b u t n o t a l l . ) t h e f o l l o w i n g s i m p l e model: Zl,t= d2 1z 2,t- 1 + al,t' ( I n the following, the For example, c o n s i d e r http://clevelandfed.org/research/workpaper/index.cfm Best available copy where each e r r o r t e r m has a v a r i a n c e o f 1 and t h e contemporaneous c o r r e l a t i o n between al and a2 i s 0.5. The standard n- period- ahead f o r e c a s t s a t t i m e t f r o m t h i s model a r e g i v e n by where t h e c o n d i t i o n a l expected values o f t h e e r r o r terms a r e 0 i n t h e standard m u l t i v a r i a t e forecasts. I n p a r t i c u l a r , t h e one- period- ahead f o r e c a s t s a r e Thus, i n standard t i m e s e r i e s f o r e c a s t i n g , t h e one- period- ahead f o r e c a s t o f Zz't+l which has a standard e r r o r o f u2 ( t h e standard =12zl,t' e r r o r o f a2), which f o r t h i s model i s assumed t o be 1. i s known and z However, i f z ~ i s ~not, ~t h e n +a b e ~t t e r f o r e c a s t o f zZytcl o b t a i n e d b y u s i n g t h e contemporaneous c o r r e l a t i o n between al i s t r u e because al,t+l, a l,t+1° can be and a2. This which equals t h e f o r e c a s t e r r o r made a t t i m e t f o r z1 a t t i m e t+l (al, t+l = z c o r r e l a t i o n between al 1, t+l l,t+l - z 1,t ( I ) ) , i s known. Thus, t h e and a2 can be used t o e s t i m a t e aZytcl given For t h i s model, t h e r e l a t i o n s h i p ( d e r i v e d from t h e variance- covariance m a t r i x of t h e e r r o r terms) between t h e two e r r o r terms i s given by http://clevelandfed.org/research/workpaper/index.cfm Best available copy where e is the error in predicting a using al t. For this model, 2,t 2,t this error will have a variance of 0.75. Thus, given z ~ , ~ the + ~ best , Y forecast of a2,t+l is given by 0.5 Or * al,t+l; consequently, the forecast Z2,t+l is modified to be This forecast will have an error of e2,t+l, which has a variance of 0.75 compared with the error variance of 1 for the forecast from the original model, using only z to forecast zZyt. Thus, knowing z ~ reduces ~ ~ l,t the error variance in forecasting z ~ , by ~ 25 + ~percent for this example. + Also, the forecast using contemporaneous data will remain unbiased. In this model, because the forecast of zl depends on z2, this reduction in forecast error variance for one-period-ahead forecasts of z2 will lead to a reduction of forecast error variance for forecasts of zl. For example, the one-period-ahead forecast at time t+l for zl using contemporaneous correlations is given by ~ http://clevelandfed.org/research/workpaper/index.cfm Best available copy T h i s f o r e c a s t has an e r r o r el,t+2 S i m i l a r l y , t h e f o r e c a s t e r r o r al,t+l given by (1) for ZlYtc2 without using contemporaneous d a t a i s Thus, t h e one- period- ahead f o r e c a s t o f zl 2 d a t a has a v a r i a n c e o f ol + 2 2 qiZ1*oe2 = 1 + 0.75*qig1 u s i n g contemporaneous while the forecast 2 v a r i a n c e n o t u s i n g contemporaneous d a t a has a v a r i a n c e o f ol + 2 2 @21*02 = 1 + mZ1.2 S i m i l a r l y , i t can be shown t h a t t h e r e i s a r e d u c t i o n i n t h e f o r e c a s t v a r i a n c e f o r f u t u r e p e r i o d s when u s i n g contemporaneous c o r r e l a t i o n s f o r t h i s model. Thus, t h e use o f contemporaneous d a t a reduces n o t o n l y t h e f o r e c a s t e r r o r o f t h e contemporaneous v a l u e o f t h e o t h e r v a r i a b l e i n t h i s model b u t a l s o t h e f o r e c a s t s o f t h e observed v a r i a b l e i n f u t u r e periods. I n general, t h e amount of r e d u c t i o n i n t h e contemporaneous p e r i o d and f u t u r e p e r i o d s w i l l depend on t h e c o r r e l a t i o n s between t h e e r r o r terms i n t h e known v a r i a b l e s and those i n t h e unknown v a r i a b l e s , as w e l l as t h e s t r u c t u r e o f t h e model. http://clevelandfed.org/research/workpaper/index.cfm Best available copy I n t h e general case, t h e v a r i a b l e s can be d i v i d e d i n t o two s e t s -- those f o r which i n f o r m a t i o n i s a v a i l a b l e f o r t h e c u r r e n t t i m e p e r i o d (LA) and those f o r which c u r r e n t i n f o r m a t i o n i s n o t a v a i l a b l e (zNA).I n the f o l l o w i n g d i s c u s s i o n , i t i s assumed t h a t t h e f i r s t kA v a r i a b l e s have d a t a a v a i l a b l e f o r t h e c u r r e n t t i m e p e r i o d and t h a t t h e r e m a i n i n g k-kA v a r i a b l e s do n o t . The e r r o r s i n t h e second group t h e n can be f o r e c a s t from t h e known e r r o r s i n t h e f i r s t group. t h e known e r r o r s ( a -4,t (6) ai ,t = b.a -I-A,t The r e l a t i o n s h i p between t h e unknown e r r o r s and - a kA x 1 v e c t o r ) can be r e p r e s e n t e d as + 'i ,t for i = k + A l,k A + 2, ... where bi w i l l be a 1 x k v e c t o r o f e s t i m a t e d c o e f f i c i e n t s . ,k, Because t h e e r r o r terms i n e q u a t i o n 6 g e n e r a l l y w i l l be c o r r e l a t e d across e q u a t i o n s ( t h a t is, e and e w i l l g e n e r a l l y be c o r r e l a t e d ) , these e s t i m a t e s s h o u l d be i ,t j,t determined b y u s i n g a g e n e r a l i z e d l e a s t squares e s t i m a t o r ( T h e i l 1971). Given t h e s e t o f equations p r e s e n t e d i n e q u a t i o n 6, t h e f o r e c a s t s (or, equivalently, ai, t+l (7) t h e c o n d i t i o n a l expected v a l u e s ) o f t h e unknown e r r o r s are aiYt(l) f o r i = kA+ 1, kA+ 2, = bi%,t+l ... ,k. The f o r e c a s t s u s i n g contemporaneous c o r r e l a t i o n s a r e t h u s g i v e n b y e q u a t i o n 3, where [$.t+n-j,~A,t+n-j] I (a, t+l ,bi$, t+l ) i f n - j = l = (1)if n - j i s greater than (versus a f o r e c a s t o f 1 f o r t h e unknown v a r i a b l e s i n normal m u l t i v a r i a t e t i m e s e r i e s f o r e c a s t i n g ) , and cit+n-j i f n- j http://clevelandfed.org/research/workpaper/index.cfm Best available copy i s l e s s t h a n 1. Thus, t h e b a s i c change i n t h e f o r e c a s t s i s t h a t t h e c o n d i t i o n a l expected v a l u e o f ai , t+l (i=kA+l,kA+2, from t h e known e r r o r s ($,t+l ) ... ,k) i s e s t i m a t e d T h i s has an obvious impact on t h e f o r e c a s t s a t t i m e t+l. However, as p r e v i o u s l y i l l u s t r a t e d , i t a l s o may improve t h e f o r e c a s t s f o r l o n g e r t i m e p e r i o d s f o r a l l v a r i a b l e s , because f u t u r e f o r e c a s t s f o r a l l v a r i a b l e s may depend on t h e f o r e c a s t s a t t i m e t+l. IV. F o r e c a s t i n g GNP Using Contemporaneous M-1 To i l l u s t r a t e t h i s method o f f o r e c a s t i n g , we have a p p l i e d i t t o a t h r e e - v a r i a b l e , q u a r t e r l y model t h a t was e s t i m a t e d i n another ongoing r e s e a r c h p r o j e c t (Bagshaw and Gavin 1984). s u p p l y M-1, The v a r i a b l e s i n t h i s model a r e t h e money GNP i n c u r r e n t d o l l a r s , and t h e bond- equivalent y i e l d on Treasury b i l l s w i t h t h r e e months t o m a t u r i t y (RTB3). M-1 and GNP a r e b o t h s e a s o n a l l y a d j u s t e d and measured i n b i l l i o n s of c u r r e n t d o l l a r s ; RTB3 i s n o t s e a s o n a l l y a d j u s t e d and i s measured i n percentage p o i n t s . ln(1 + The model was e s t i m a t e d i n RTB3) and i n changes i n t h e n a t u r a l l o g a r i t h m o f M-1 and GNP, o r vln(M-1) and vln(GNP)). Because of t h e c r e d i t c o n t r o l s d u r i n g 1980, t h e s h i f t i n monetary p o l i c y d u r i n g t h e 1980s, and t h e D e p o s i t o r y 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 C o n t r o l A c t o f 1980, t h e r e i s some q u e s t i o n whether t h e l a s t t h r e e o r f o u r y e a r s would be adequately represented b y a model e s t i m a t e d over an e a r l i e r t i m e p e r i o d . Consequently, we developed t i m e s e r i e s models c o v e r i n g two o v e r l a p p i n g t i m e p e r i o d s t o t e s t f o r t h i s problem. The two t i m e p e r i o d s were from 1959:IQ through 1976:IVQ and 1959:IQ t h r o u g h 1979:IVQ. These a l l o w f o r e c a s t s t o be produced f o r t h e t i m e p e r i o d s f r o m http://clevelandfed.org/research/workpaper/index.cfm Best available copy 1977:IQ t h r o u g h 1979:IVQ and 1980:IQ t h r o u g h 1984:IIQ. I f t h e s e changes d i d indeed have an impact, t h e n t h e e a r l i e r p e r i o d should be more s t a b l e t h a n t h e l a t t e r p e r i o d and t h e model should p e r f o r m b e t t e r d u r i n g t h i s p e r i o d r e l a t i v e t o t h e l a t t e r period. The models were e s t i m a t e d u s i n g t h e 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 e q u a t i o n 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 (Box and Jenkins 1976). The s t e p s i n v o l v e d a r e (1) t e n t a t i v e l y i d e n t i f y a model b y 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 t h e series; ( 2 ) e s t i m a t e t h e parameters of t h i s model; and ( 3 ) a p p l y d i a g n o s t i c checks t o t h e r e s i d u a l s . 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 r e s i d u a l s , n o r m a l i t y of r e s i d u a l s , e t c . If the residuals 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 s t e p s 2 and 3 a r e repeated. T h i s process c o n t i n u e s u n t i l a s a t i s f a c t o r y model i s obtained. When a p p l i e d t o t h e two t i m e p e r i o d s , t h i s t e c h n i q u e r e s u l t e d i n t h e same f u n c t i o n a l f o r m f o r t h e model, w i t h s l i g h t l y d i f f e r e n t e s t i m a t e d parameters. The r e s u l t i n g model was A l l o f t h e e s t i m a t e d parameters o f t h e model a r e s i g n i f i c a n t a t t h e 0.001 l e v e l (see t a b l e 1 ) . There a r e no s i g n i f i c a n t c o r r e l a t i o n s r e m a i n i n g i n t h e http://clevelandfed.org/research/workpaper/index.cfm Best available copy r e s i d u a l s , except f o r t h e contemporaneous c o r r e l a t i o n between t h e e r r o r terms i n M-1 and GNP, and t h e models passed t h e usual d i a g n o s t i c t e s t s . Because o n l y t h e contemporaneous c o r r e l a t i o n between t h e e r r o r terms i n GNP and M-1 was s i g n i f i c a n t , we developed a f o r e c a s t o f t h e e r r o r i n t h e GNP equation given o n l y t h e e r r o r i n t h e M-1 equation. That i s , even though t h e d a t a f o r RTB3 were a v a i l a b l e even b e f o r e t h e d a t a f o r M-1, these d a t a do n o t p r o v i d e u s e f u l i n f o r m a t i o n i n f o r e c a s t i n g e i t h e r contemporaneous GNP o r M-1. ( I n f a c t , we estimated regressions u s i n g b o t h contemporaneous e r r o r s i n M-1 and RTB3 as independent v a r i a b l e s and t h e e r r o r i n GNP as t h e dependent variable. The RTB3 term was n o t s i g n i f i c a n t a t t h e 0.05 percent l e v e l , and t h e a d d i t i o n o f t h e RTB3 term d i d n o t s i g n i f i c a n t l y improve t h e regression.) The r e s u l t i n g r e l a t i o n s h i p was where bl was estimated t o be 0.792 i n t h e p e r i o d through 1976:IVQ and 0.768 i n t h e p e r i o d through 1979:IVQ. The standard d e v i a t i o n o f el was 0.0078 versus a standard d e v i a t i o n o f 0.0089 f o r a3 i n t h e p e r i o d through 1976:IVQ and 0.0080 versus 0.0090 i n t h e p e r i o d through 1979:IVQ. Thus, we would expect approximately a 12 percent r e d u c t i o n i n t h e RMSE f r o m f o r e c a s t i n g GNP when contemporaneous M-1 i s a v a i l a b l e , compared w i t h u s i n g o n l y lagged M-1 . To t e s t t h e r e s u l t s on out- of- sample forecasts, we used t h e model f i t t e d through 1976:IVQ t o f o r e c a s t GNP over t h e p e r i o d 1977:IQ through 1979:IVQ. I n one case, we used o n l y lagged M-1 and i n t h e second case, we used contemporaneous M-1. Also, we used t h e model f i t t e d through 1979:IVQ t o http://clevelandfed.org/research/workpaper/index.cfm Best available copy f o r e c a s t over t h e p e r i o d 1980:IQ t h r o u g h 1984:IIQ. I n a l l cases, t h e models were used t o f o r e c a s t b o t h contemporaneous changes i n ln(GNP) and changes i n ln(GNP) from t i m e t t o t i m e t+2, t h a t i s , t h e change f r o m t h e l a s t known v a l u e o f ln(GNP) t o t h e f o r e c a s t v a l u e f o r two q u a r t e r s a f t e r t h a t known v a l u e o c c u r r e d (see t a b l e 2 ) . From these r e s u l t s , we see t h a t d u r i n g t h e p e r i o d 1977:IQ through 1979:IVQ t h e r e was a decrease i n RMSE o f r o u g h l y 8 p e r c e n t u s i n g contemporaneous M-1 i n f o r e c a s t i n g c u r r e n t GNP, as compared w i t h u s i n g o n l y lagged M-1. percent. T h i s agrees q u i t e w e l l w i t h t h e t h e o r e t i c a l r e d u c t i o n o f 12 However, d u r i n g t h e p e r i o d 1980:IQ through 1 9 8 4 : I I Q t h e RMSE was a c t u a l l y m a r g i n a l l y l a r g e r u s i n g contemporaneous M-1. For t h e two- quarter change f o r e c a s t s , t h e r e was a r e d u c t i o n of r o u g h l y 4 p e r c e n t i n t h e f i r s t p e r i o d and an i n c r e a s e o f r o u g h l y 3 p e r c e n t i n t h e l a t t e r p e r i o d . As mentioned e a r l i e r i n t h i s paper, t h e 1980s i s a t i m e when many events would p o t e n t i a l l y a f f e c t t h e r e l a t i o n s h i p s among M-1, GNP, and RTB3. However, i t i s t h o u g h t t h a t t h e s e events have changed again f r o m r o u g h l y 1 9 8 3 : I I I Q t o t h e p r e s e n t i n such a way t h a t t h e r e l a t i o n s h i p s would be back t o pre-1980 form. Table 3 p r e s e n t s t h e contemporaneous f o r e c a s t e r r o r s from 1980:IQ t o 1984:IIQ. From these e r r o r s , we see t h a t d u r i n g t h e p e r i o d o f 1 9 8 3 : I I Q through 1 9 8 4 : I I Q t h e f o r e c a s t u s i n g contemporaneous M-1 has done much b e t t e r than t h e one u s i n g o n l y lagged M-1. I n f a c t , t h e RMSE f o r t h e f o r e c a s t s u s i n g o n l y lagged M-1 was 0.0131 over t h i s period, w h i l e t h a t f o r t h e f o r e c a s t s u s i n g contemporaneous M-1 was 0.0101, a p p r o x i m a t e l y 22 percent. a reduction o f For t h e two- quarter change f o r e c a s t s , t h e t i m e p e r i o d over which t h e l a t e s t change would a f f e c t t h e outcome i s f r o m 1983:IVQ http://clevelandfed.org/research/workpaper/index.cfm Best available copy t o 1984:IIQ, because t h e t w o - q u a r t e r change f o r 1983:IIIQ depends on t h e d a t a f r o m 1983:IIQ. The RMSE f o r t h e p e r i o d 1983:IIQ t h r o u g h 1984:IIQ u s i n g o n l y lagged M-1 was 0.0210; t h a t f o r t h e f o r e c a s t u s i n g contemporaneous M-1 was 0.0188. Thus, u s i n g contemporaneous M-1 reduced t h e RMSE f o r t h i s p e r i o d b y r o u g h l y 10 p e r c e n t . V. Summarv Because m u l t i v a r i a t e t i m e s e r i e s models i n c l u d e t h e contemporaneous c o r r e l a t i o n s i n t h e e r r o r s t r u c t u r e , t h e s e models do n o t use contemporaneous data t o f o r e c a s t v a r i a b l e s t h a t are n o t a v a i l a b l e f o r t h e c u r r e n t period. However, as demonstrated i n t h i s paper, t h e s e d a t a can s u c c e s s f u l l y be used t o o b t a i n more a c c u r a t e f o r e c a s t s f o r t h e c u r r e n t and f u t u r e t i m e p e r i o d s . Indeed, f o r e c a s t s of GNP u s i n g contemporaneous M-1 had r o u g h l y an 8 p e r c e n t s m a l l e r RMSE d u r i n g t h e p e r i o d 1977:IQ t h r o u g h 1979:IVQ t h a n t h a t u s i n g o n l y lagged M-1. mixed. The r e s u l t s d u r i n g t h e p e r i o d 1980:IQ t h r o u g h 1984:IIQ a r e However, t h e r e s u l t s i n d i c a t e t h a t u s i n g contemporaneous M-1 p r o v i d e d more a c c u r a t e f o r e c a s t s d u r i n g 1983:IIQ t h r o u g h 1984:IIQ, s u g g e s t i n g t h a t u s i n g contemporaneous M-1 d a t a would p r o v i d e more a c c u r a t e f o r e c a s t s i n t h e future. http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 1 Estimated Parameters Time period Parameter Standard devi a t i ons and c o r r e l a t ions 1959:IQ - 1976:IVQ 1959:IQ - 1979:IVQ http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 2 Forecast Error Statistics for GNP using M-1 Time ~ e r i o d Contemporaneous forecasts Lagged Contemporaneous Lagged Contemporaneous Mean error 0.0009 -0.0003 -0.0072 -0.0080 RMSE 0.0095 0.0087 0.0142 0.0143 MAE 0.0068 0.0068 0.0122 0.0129 Two-quarter change forecasts Lagged Contemporaneous Lagged Mean error 0.0036 0.0010 -0.0155 -0.0157 RMSE 0.0114 0.0109 0.0260 0.0267 MAE 0.0093 0.0093 0.0212 0.0214 NOTE: RMSE - root mean square error of the forecast. MAE - mean absolute error of the forecast. Contemporaneous http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 3 Forecast Errors for GNP using M-1 for 1980:IQ through 1982:IVQ Contemporaneous forecast Two-quarter change forecast Time peri od Lagged Contemporaneous Lagged Contemporaneous 1980: I I IQ 0.0163 -0.0115 0.0038 0.0242 1980: I V Q 0.0080 0.0125 0.0243 -0.0035 1981:I I I Q 0.0046 0.0107 -0.0003 -0.0040 1982: I1 IQ 1982: I VQ 1983: IQ 1983: I I Q 1983: I I IQ 1983: I V Q 1984: IQ 1984: I IQ http://clevelandfed.org/research/workpaper/index.cfm Best available copy References Bagshaw, Michael L., and William T. Gavin. S e r i e s Approach, " Working Paper. " Velocity: A M u l t i v a r i a t e Time Federal Reserve Bank o f C l eve1 and, forthcoming. Box, G.E.P., and G.M. Jenkins. Time S e r i e s Analysis, Forecasting and Control, San Franci sco: Hol den-Day, 1 976. T h e i l , Henri. P r i n c i p l e s o f Econometrics, New York: John Wiley & Sons, Inc., 1971. Tiao, G.C., and G.E.P. Box. "Modeling Multiple Time S e r i e s with Applications," Journal o f t h e American S t a t i s t i c a l Association, vol. 76, no. 376 (1 981 1, pp. 802-1 6.
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