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 8405 VELOCITY: A MULTIVARIATE TIME-SERIES APPROACH by Michael L. Bagshaw and W i l l i a m T. Gavin Thanks a r e due t o John B. Carlson, James Hoehn, and Kim Kowal ewski f o r he1p f u l comments. June Gates and David Gaebler provided research assistance f o r t h i s paper. The manuscript was prepared by Veronique L l o y d and Laura Davis. R e s p o n s i b i l i t y f o r any remaining e r r o r s i s , o f course, o u r own. Michael L. Bagshaw i s a s t a t i s t i c i a n and W i l l i a m T. Gavin i s an economist i n t h e Research Department o f t h e Federal Reserve Bank o f Cleveland. Working papers o f t h e Federal Reserve Bank o f Clevelan'd a r e prel iminary 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 s t a t e d h e r e i n a r e those o f t h e authors 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 o f t h e Board o f Governors o f t h e Federal Reserve System. November 1984 Federal Reserve Bank o f C l eve1and http://clevelandfed.org/research/workpaper/index.cfm Best available copy VELOCITY: A MULTIVARIATE TIME-SERIES APPROACH Key words: monetary target, i n t e r e s t r a t e , multivariate time s e r i e s , vel oci ty. Abstract The Federal Reserve announces targets f o r the monetary aggregates that are imp1 i c i t l y condi tiorled on an assumption about future velocity for each of the monetary aggregates. In t h i s paper we present e x p l i c i t models of velocity for constructing rigorous t e s t s t o determine whether the behavior of velocity has changed from what was expected when the targets were chosen. forecasts of velocity. Ne use time-series methods t o develop a1 ternative Mu1 t i v a r i a t e time-series models of velocity t h a t include information about past i n t e r e s t r a t e s produce significantly better out-of-sample forecasts than do univariate methods. Using t h i s multivariate time-series framework, we analyze the Federal Reserve's decisions to change, miss, and switch targets from 1980:IQ t o 1984:IIQ. For this period, we find t h a t when the Federal Reserve deviated from i t s announced target, vel oci ty deviated s i gni ficantly from i t s predicted val ue. http://clevelandfed.org/research/workpaper/index.cfm Best available copy I. I n t r o d u c t i o n I n t h e l a s t two years, i n f l a t i o n f o r e c a s t s have c o n s i s t e n t l y been t o o high, p a r t i c u l a r l y f o r e c a s t s based on t h e q u a n t i t y t h e o r y o f money i n which i n f l a t i o n i s estimated t o be an e x p l i c i t f u n c t i o n o f growth i n M-1 ( t h e narrow d e f i n i t i o n of t h e money s t o c k ) . Throughout 1982 and e a r l y i n 1983, M-1 grew a t d o u b l e - d i g i t r a t e s , w h i l e i n f l a t i o n decelerated t o l e s s than 4 percent. T h i s unexpected s h i f t i n t h e r e l a t i o n between i n f l a t i o n and M-1 has complicated t h e Federal Reserve's monetary t a r g e t i n g approach t o ending inflation. The Federal Reserve began announcing annual t a r g e t s f o r monetary aggregates i n 1975. These t a r g e t s a r e n o t t h e u l t i m a t e goals o f monetary p o l i c y , b u t merely i n t e r m e d i a t e t a r g e t s c o n d i t i o n e d on economic f o r e c a s t s and long- term goals, such as p r i c e s t a b i l i t y and economic growth. The announcements o f monetary t a r g e t s a r e used by t h e p u b l i c as i n d i c a t o r s o f p o l i c y intentions. However, t h e i n t e n t i o n s o f p o l i c y a r e more a c c u r a t e l y d e f i n e d i n terms o f t h e u l t i m a t e o b j e c t i v e s . Each member o f t h e Federal Reserve Open Market Committee (FOMC), t h e d e l i b e r a t i n g body o f t h e Federal Reserve responsible f o r monetary p o l i c y , has a unique model t h a t r e l a t e s t h e i n t e r m e d i a t e t a r g e t s t o t h e f i n a l goals. The i n d i v i d u a l s on t h e FOMC make d e c i s i o n s about t h e monetary t a r g e t s based on f o r e c a s t s (assumptions) about t h e r e l a t i o n s h i p between t h e monetary t a r g e t s and o t h e r economic v a r i a b l e s . As even t h e most casual observer knows, economic f o r e c a s t s a r e s u b j e c t t o l a r g e e r r o r s and f r e q u e n t r e v i s i o n . Understanding t h i s i s b a s i c t o understanding t h e r o l e o f t h e monetary t a r g e t s and why Congress a1lows t h e Federal Reserve so much d i s c r e t i o n i n choosing and changing t h e t a r g e t s . http://clevelandfed.org/research/workpaper/index.cfm Best available copy Imp1 i c i t i n the choice of a monetary target i s an assumption about the expected behavior of velocity- - that i s , the r a t i o of nominal GNP t o the monetary aggregate. Uncertainty about future vel oci t y behavior is one reason that monetary targets are presented as ranges. In the past few years, the Federal Reserve has stated more expl i c i t l y how desi rabl e monetary growth depends upon the unexpected growth of velocity. To quote from a recent Monetary Pol icy Report to Congress, "Growth around the midpoint of the (!+I-1 ) range would appear appropriate on the assumption of re1 a t i vely normal velocity growth; i f velocity growth remains weak compared w i t h historical experience, M-1 growth might appropriately be higher i n the range" (Board of Governors of the Federal Reserve System 1984, p. 72). While monetarists such a s Karl Brunner (1983) have argued t h a t the Federal Reserve shoul d ignore temporary deviations of velocity i n imp1 ementi ng monetary policy, no one would deny t h a t the targets should be changed when there i s a fundamental change i n the behavior of velocity growth. In t h i s paper, the expected behavior of velocity i s defined as the forecast from a time-series model. We use a recent development i n time-series modeling by Tiao and Box (1981 ) t o construct mu1 t i v a r i a t e models of velocity. Univariate Box-Jenkins (1976) models are used a s the standard against which we compare these mu1 t i v a r i a t e model s. The time-series model s are reduced-form models t h a t may be consistent w i t h many different structural models of the economy. Our goal i n this paper i s 1imi ted: t o develop models of velocity f o r constructing rigorous t e s t s t o determine whether velocity behavior has changed. A by-product of this exercise i s a better forecasting model for velocity. Although we use reduced-form time-series models, we must rely on economic theory t o decide which variables to include, Row t o measure them, and http://clevelandfed.org/research/workpaper/index.cfm Best available copy -4g e n e r a l l y how they a r e expected t o be r e l a t e d i n a s t r u c t u r a l model. These decisions a r e necessary f o r s e t t i n g up a mu1t i v a r i a t e t i m e - s e r i e s model because t h e way one transforms t h e v a r i a b l e s (whether one takes d i f f e r e n c e s , logarithms, e t c . ) a f f e c t s t h e processes t h a t generate t h e e r r o r terms. Also, t h e choice o f t h e sample p e r i o d may depend on knowledge about t h e economic structure. While one g e n e r a l l y uses a l l a v a i l a b l e i n f o r m a t i o n , knowledge about special circumstances o r s t r u c t u r a l changes may suggest u s i n g l e s s than t h e f u l l p e r i o d f o r which data a r e a v a i l a b l e . I n t h i s e m p i r i c a l study o f v e l o c i t y , we s e l e c t a sample t h a t s t a r t s i n 1959. T h i s y e a r marked t h e beginning o f t h e Federal Reserve's h i s t o r i c a l data s e t on t h e most r e c e n t v e r s i o n s o f M-1 and Fli-2. We assume t h a t t h e r e was a stab1 e s t o c h a s t i c process generating v e l o c i t y from 1959 through 1979. The e s t i m a t i o n p e r i o d ends i n 1979:IVQ, because i n t h a t q u a r t e r t h e Federal Reserve announced i t s d e t e r m i n a t i o n t o r e s t r a i n monetary growth and adopted a new o p e r a t i n g procedure t o 1end c r e d i b i l i t y t o t h e announcement. This change i n procedures was t h e f i r s t o f several events t h a t may have induced a s t r u c t u r a l change i n t h e economy and i n t h e s t o c h a s t i c process generating velocity. Other events t h a t may have induced a s t r u c t u r a l change i n t h e economy were t h e i m p o s i t i o n and subsequent r e l a x a t i o n o f c r e d i t c o n t r o l s i n 1980; d e r e g u l a t i o n o f i n t e r e s t - r a t e r e s t r i c t i o n s i n d e p o s i t markets i n 1981, 1982, and 1983; and another change i n o p e r a t i n g procedures i n l a t e 1982. We use u n i v a r i a t e Box-Jenkins (1976) models and t h e Tiao-Box (1981) mu1t i v a r i a t e procedure t o meas c o n s t r u c t expl i c i t model s o f nominal GNP, and i n t e r e s t r a ncludc money and nom h a v i o r o f v e l o c i t y growth. Me 1 as t r i v a r i a t e models o f money, which a v e l o c i t y f o r e c a s t can be eparately, because both money and nominal GNP a r e endogenous v a r i a b l e s i n periods as s h o r t as one quarter. By http://clevelandfed.org/research/workpaper/index.cfm Best available copy -5- including these two variables separately, we hope t o s o r t out t h e i r dynamic behavior, which may become obscured i f we look a t the r a t i o of the two. We use the quantity theory of money a s the analytical framework for selecting and scaling variables in this study. We s e t aside the problem of sorting out nominal versus real e f f e c t s of monetary growth and look only a t nominal GMP. Growth rates of nominal GNP and the money stock are approximated by changes i n the logarithm. Previous research suggests t h a t past interest rates contain important information about future money growth (see Bagshaw and Gavin 1983). Studies i n money demand also suggest t h a t the i n t e r e s t r a t e should be an important determinant of the r a t i o of income to money. In section 11, we present univariate and mu1 t i v a r i a t e model s of velocity growth. We include models f o r M-1 and M- 2 velocity growth because the Federal Reserve has alternately used one or the other of these aggregates as i t s primary target. The Federal Reserve makes use of both aggregates i n the policy process. Section I11 includes a comparison of the out-of-sample forecasting properties of the different models. In section IVYwe use the estimated time-series models t o monitor whether and when the actual behavior of velocity deviated from what was expected during the period from 1380:IQ t o 1984:IIQ. Section V contains a summary and concluding comments. 11. Models of Velocity Growth We begin by estimating univariate autoregressive integrated moving average (ARIMA) models of velocity growth f o r M-1 and M-2 (see table 1 ). For the 1959:IIQ t o 1979:IVQ period, M-1 velocity growth can be represented by a constant growth trend (3.1 percent annually) plus a white noise process. http://clevelandfed.org/research/workpaper/index.cfm Best available copy -6Brunner (1983) has used t h i s r e s u l t t o support the case f o r a constant money-growth rul e. M- 2 velocity growth i s identified as a first- order moving average process. There i s a 3 percent information gain over the naive model.' naive model i s j u s t the average growth r a t e f o r the sample period. The (We saw above t h a t the univariate model for M-1 velocity was the naive model. ) Bivariate model s of velocity are estimated using procedures developed i n Tiao and Box (1981 ). These procedures are used t o estimate the parameters of a mu1 t i v a r i a t e simultaneous equation model. This method i s interactive, simil a r i n principle to t h a t of sing1 e-equation Box-Jenkins model ing. steps are: (1 ) The tentatively identify a model by examining autocorrel a t i ons and cross- correlations of the series, ( 2 ) estimate the parameters of t h i s model, and ( 3 ) apply diagnostic checks t o the residuals. I f the residuals do not pass the diagnostic checks, then the tentative model i s modified, and steps 2 and 3 are repeated. obtained. This process continues until a satisfactory model i s This i s basically a forecasting procedure; contemporaneous correlation among the variables i s not explained or taken into account, b u t relegated t o the error matrix. The time-series procedure effectively f i l t e r s out autocorrel a t i on and dynamic cross-correl ation among the errors. For a more detailed description of how t o identify and estimate the vector autoregressive moving average (ARMA) model , see Tiao and Box ( 1981 ) . There i s a controversy about the amount of differencing t h a t should be used in mu1 t i v a r i a t e time-series analysis. In ur~ivariate procedures, the variable i s differenced i f the s e r i e s i s not stationary. In multivariate procedures, Tiao and Box (1981) suggest not differencing t o avoid specification error. Clowever, this does not r u l e out differencing i f economic theory suggests a relationship i n the differenced data. In t h i s paper, we http://clevelandfed.org/research/workpaper/index.cfm Best available copy -7difference the monetary variables and GMP, b u t not the i n t e r e s t r a t e , t o conform with a priori economic theory. From one period of equilibrium t o the next, we expect money growth t o be proportional t o income growth and approximately proportional t o the logarithm of 1 plus the nominal yield on short-term r i s k l e s s assets. Therefore, the raw data are taken t o be f i r s t differences i n the natural logarithm of velocity and the logarithm of 1 plus the quarterly bond-equivalent yield on Treasury b i l l s w i t h three months t o maturity. The bivariate M-1 velocity growth model includes a lagged e r r o r from the interest- rate equation (see table 2). Like the univariate model, t h i s model includes a constant equal to the average growth of velocity during the sample period. The information gain from the inclusion of the interest- rate variable i s 3.4 percent. The mu1 t i v a r i a t e M-2 velocity growth model a1 so includes the lagged e r r o r from the interest- rate equation. M-2 velocity growth i s more sensitive t o deviations of the i n t e r e s t r a t e from trend than i s M-1 velocity growth. The information gain i n the M-2 velocity growth equation i s 7.2 percent, somewhat larger than f o r f4-1 velocity. These multivariate velocity growth models represent an improvement over the univariate models, although they may not detect a systematic dynamic relationship between money and nominal GMP t h a t would help explain the velocity trend. We t r y t o do this by using t r i v a r i a t e model s t h a t include money and GNP separately. The t r i v a r i a t e models a r e shown i n table 3. la?-1 growth i s estimated to depend on past M-1 growth and the lagged error from the interest- rate equation. The i n t e r e s t r a t e i s estimated t o be a function of the lagged i n t e r e s t r a t e and the error i n the previous period ' s i nterest- rate forecast. According to t h i s equation, a s e t of information t h a t excludes past values o f http://clevelandfed.org/research/workpaper/index.cfm Best available copy -8M-1 and nominal GNP appears s u f f i c i e n t t o p r e d i c t f u t u r e i n t e r e s t r a t e s . The c o e f f i c i e n t on t h e lagged i n t e r e s t r a t e 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 from 1. GNP i s estimated t o be a f u n c t i o n o f p a s t M-1 growth and t h e lagged e r r o r from t h e M-1 equation. The Treasury b i l l r a t e i n f l u e n c e s GNP through i t s e f f e c t on M-1. A f o r e c a s t f o r v e l o c i t y can be d e r i v e d from these t r i v a r i a t e models. For M-1 we g e t the f o l l o w i n g equation: The d i f f e r e n c e between t h i s model and t h e b i v a r i a t e M-1 v e l o c i t y model i s t h e i m p l i c a t i o n f o r the behavior o f v e l o c i t y . I n t h e b i v a r i a t e M-1 model of t a b l e 2, t h e t r e n d i n M-1 v e l o c i t y growth i s a constant growth rate- - 3.1 percent annually. I n t h e derived- vel o c i ty model, v e l o c i t y i s determined by M-1 growth. In the steady s t a t e , h i g h e r M-1 growth i m p l i e s f a s t e r v e l o c i t y growth. This i m p l i c a t i o n i s c o n s i s t e n t w i t h a standard economic model t h a t in c l udes non-i n t e r e s t - b e a r i n g money. When money growth exceeds r e a l economic growth, i n f l a t i o n and h i g h e r i n t e r e s t r a t e s r a i s e t h e o p p o r t u n i t y c o s t o f h o l d i n g money, and people devise ways t o manage money balances more c l o s e l y . T h i s model i s a l s o c o n s i s t e n t w i t h t h e hypothesis s t a t e d i n Me1t z e r (1983) t h a t a pol icy- induced supply shock t o rnoney growth i s associated w i t h a temporary d e c l i n e i n v e l o c i t y . The reason i s simply t h a t a shock t o money growth a f f e c t s GNP growth w i t h a l a g . The M-2 v e l o c i t y equation d e r i v e d from t h e M-2 model i s shown below: The c o e f f i c i e n t on lagged M-2 growth i s very small. http://clevelandfed.org/research/workpaper/index.cfm Best available copy 111. Forecast Performance Post-sample f o r e c a s t s from t h e models shown i n t a b l e s 1, 2, and 3, a r e used t o examine t h e advantages o f these models i n p r e d i c t i n g v e l o c i t y from 1980:IQ t o 1984:IIQ. d i f f e r e n t model s. The s t a t i s t i c s i n t a b l e 4 compare v e l o c i t y f o r e c a s t s o f C l e a r l y , t h e b i v a r i a t e vel o c i t y model produces t h e b e s t f o r e c a s t s f o r M-1 v e l o c i t y . The r o o t mean square e r r o r (RMSE) i s reduced from 1.73 p e r c e n t i n t h e u n i v a r i a t e model t o 1 .17 percent i n t h e b i v a r i a t e model. The RMSE o f the v e l o c i t y f o r e c a s t s d e r i v e d from t h e t r i v a r i a t e M-1 model i s 1.55 percent, b e t t e r than t h e u n i v a r i a t e v e l o c i t y f o r e c a s t b u t s u b s t a n t i a1l y worse than f o r e c a s t s from t h e b i v a r i a t e vel o c i ty model .2 A l l of t h e M-1 v e l o c i t y growth f o r e c a s t s a r e badly biased. occurs i n t h e f o r e c a s t s f o r 1982 and 1983. The b i a s The b i v a r i a t e model i n c l u d e s a l a r g e e f f e c t from t h e lagged e r r o r i n t h e i n t e r e s t - r a t e equation t h a t causes t h e model t o t r a c k movement i n v e l o c i t y w i t k o u t b i a s through 1981 :I V Q . The RMSE from t h i s b i v a r i a t e model i s 0.88 percent f o r t h e f i r s t e i g h t q u a r t e r s o f o u r post-sample period. This i s o n l y one-ha1 f t h e RMSE from t h e u n i v a r i a t e model (1.62) and about equal t o t h e in-sample e r r o r f o r the b i v a r i a t e model. The accuracy o f t h e M-1 v e l o c i t y growth f o r e c a s t i n 1980 and 1981 i s s u r p r i s i n g , because i n t e r e s t r a t e s were more v o l a t i l e i n t h e post-1979 p e r i o d than d u r i n g any comparable p e r i o d i n t h e sample. u s i n g t h e t r i v a r i a t e M-1 model. S i m i l a r r e s u l t s a r e obtained Furthermore, t h e contemporaneous c o r r e l a t i o n between t h e PI-1 and i n t e r e s t - r a t e f o r e c a s t e r r o r s from t h e t r i v a r i a t e model i s s t r o n g and p o s i t i v e (0.41 ) , w h i l e t h e in-sample c o r r e l a t i o n i s weak and n e g a t i v e (-0.14). The change i n monetary pol i c y o p e r a t i n g procedures i s most http://clevelandfed.org/research/workpaper/index.cfm Best available copy -1 01ikely responsible f o r the high positive correlation between the forecast errors (see Hoelln 1983). The negative correlation between contemporaneous values of M-1 and i n t e r e s t rates during the period before 1979 has been interpreted as a money demand relationship and was most l i k e l y caused by the Federal Reserve's shifting the money supply curve t o smooth i n t e r e s t rates. As a r e s u l t , the scatter of points i n the interest- rate M-1 space tended t o trace a relatively stable demand curve. In October 1979, the Federal Reserve adopted a nonborrowed-reserve operating procedure in which the nonborrowed-reserve path was constructed on a stable money-supply path. When money demand took M-1 above (below) path, i n t e r e s t rates were forced up (down) by the nonborrowed-reserve operating procedure. Under this regime, the s c a t t e r of points i n the i n t e r e s t - r a t e b!-1 space tended to trace out a relatively stable supply curve. While the change i n monetary control procedures was associated with a different contemporaneous correlation between M-1 and the i n t e r e s t rate, the change does not seem t o have affected the relationship between the interest- rate error lagged one quarter and M-1 velocity growth. In table 4, we show t h a t the forecasts from the bivariate velocity model are better than the forecasts from the univariate models. This r e s u l t implies t h a t the preferred specification of a velocity model should include information about i n t e r e s t rates. In a recent paper, Ashley, Granger, and Schmalensee (1980) describe i n detail a t e s t s t a t i s t i c t h a t we use t o determine whether the bivariate model i s significantly better than the univariate model. Because time-series procedures require mining the data t o identify the model, in-sample s t a t i s t i c s are inappropriate for specification testing. The proposed specific forecasting performance is based on ou t-of-sampl e ion f o r performance i s the mean http://clevelandfed.org/research/workpaper/index.cfm Best available copy square e r r o r (MSE) of the forecast. -11The t e s t s t a t i s t i c s are calculated by regressing the difference between the out-of-sample forecast errors on a constant and the sum of out-of-sample forecast errors. In particular, we construct a t e s t of the bivariate model, as follows: Let: dt = u t - St = U t + bt, and: bt, where u t i s the forecast error from the univariate model, and b t i s the forecast error +from the bivariate model. Estimate the following regression: where et i s treated as i f i t were independent of s t and F i s the mean of the The difference between MSEs i s equal t o the sum of two components: the difference between the mean of the errors squared and the difference between the variances. This regression provides a t e s t of whether the difference between MSEs i s significant. estimate, co, The ordinary 1e a s t squares (OLS) i s an estimate of the difference between the mean of the error terms from each model. The OLS estimate, c, , i s proportional to the difference between the variances of the error terms from each nodel. The mean of errors 5s negative f o r both univariate and bivariate models of 11-1 and M-2 (see table 4 ) . Therefore, we can reject the bivariate node1 i f positive and significant, or i f to i s il i s negative and significant. If io< 0, 0 , we can use an F- test of the joint hypothesis that both to and c1 are n o t significantly di Fferent than zero. Ashley, Granger, and Scf~malensee(1980) note t h a t t h i s F- test i s four- tailed; i t does not take into account the signs of the estimated coefficients. When the signs are taken into account, the appropriate http://clevelandfed.org/research/workpaper/index.cfm Best available copy -1 2significance 1eve1 i s one-ha1 f t h a t obtained from the tables. The regression r e s u l t s using one-step-ahead e r r o r s from 1980:IQ t o 1984:IIQ a r e shown i n table 5. forecasts: In both cases, taking i n t e r e s t r a t e s i n t o account improves the f o r M-1 the improvement i s highly s i g n i f i c a n t a t a 0.2 percent c r i t i c a l level ; f o r M-2, the improvement i s not s t a t i s t i c a l l y s i g n i f i c a n t . IV. Monitoring the Vel oci ty Assumption The monetary t a r g e t s announced each year by the Federal Reserve a r e implicitly conditioned on an assumption about the expected behavior of velocity. Given a goal f o r i n f l a t i o n and an assumption about the trend i n real output growth, whether money grows on average along the midpoint of the t a r g e t range should depend on whether new information indicates t h a t t h e assumption about velocity i s accurate. To make t h a t judgment, we must have a model of velocity and a notion about the probabil i t y d i s t r i b u t i o n describing deviations of velocity from i t s expected value. the model f o r t h e FOMC's i m p l i c i t forecast of Since we cannot know velocity, we assume t h a t the predicted value from our time- series model i s the same a s the FOMC expectation. Under t h i s assumption, t e s t s about model adequacy provide a method of monitoring the velocity assumptions t h a t were made when the t a r g e t s were chosen. To see whether t h i s i s a reasonable assumption, we compare the four-step-ahead forecast f o r velocity growth w i t h the ex ante M-1 velocity assumption implied by the FOMC forecasts of nominal GNP and the midpoints of the M-1 t a r g e t ranges (information presented t o Congress by t h e Federal Reserve Chairman i n February of each year, 1980 through 1984). A summary of the forecasts and the implied velocity http://clevelandfed.org/research/workpaper/index.cfm Best available copy -1 3assumptions a r e l i s t e d i n table 6 w i t h the four-step-ahead M-1 velocity forecast (using the bivariate models from table 2 i n the t e x t ) . The four-step-ahead forecast of bl-1 velocity growth f a l l s w i t h i n the range predicted by the FOMC in three of the f i v e years shown. In 1981 :IVQ, the actual i n t e r e s t r a t e was 1 percent ( a t quarterly r a t e s ) below the forecast. This led t o a much lower velocity forecast i n early 1982. The actual velocity growth i n 1982 was -5.7 percent, we1 1 below both the FOMC and the time-series forecast. I n s p i t e of some obvious differences between the FOMC's implied assumption of M-1 velocity and our time-series forecasts, we proceed a s i f our tine- series model forecasts of velocity were the same as the FOMC's assumption. We use the bivariate velocity models of M-1 and M-2 t o evaluate the behavior of velocity over the period 1980:IQ to 1983:IVQ. This evaluation i s based on the one-step-ahead forecasts from the model estimated f o r the period 1959:IIQ t o 1979:IVQ. Under the null hypothesis t h a t the estimated model i s an adequate representation f o r the post-sample period, the one-step-ahead forecasts are distributed randomly w i t h zero mean and covariance matrix, i. The sum of e r r o r s i s approximately distributed as: The sum of the squared errors i s approximately distributed as: Tables 7 and 8 include s t a t i s t i c s f o r testing the hypothesis t h a t the forecast errors of velocity growth from the bivariate models have zero mean and variance equal to the estimated variance of the sample errors. The t e s t s are calculated for forecast errors accumulated over four quarters, beginning http://clevelandfed.org/research/workpaper/index.cfm Best available copy -1 4w i t h the forecast e r r o r i n the f i r s t quarter of each year. This t e s t can be constructed from any point i n time t o examine the s t a b i l i t y of velocity growth. I n table 7 , we compare the univariate and bivariate forecast errors f o r M-l velocity. If we had used the univariate model, we would have rejected the hypothesis t h a t velocity was stable in 1981. The e r r o r was posi ti we; the Federal Reserve elected t o aim a t the low end of the t a r g e t ranges 1). (see chart I f we had used the bivariate model, we would not have rejected the hypothesis t h a t M-l velocity was stable. A decision t o r e s t r a i n bl-1 growth a t the end of 1980 was implemented by choosing a lower path f o r reserves and, consequently, inducing an unexpected r i s e i n the i n t e r e s t rates. This unpredicted j u m p in i n t e r e s t rates explains the subsequent r i s e i n velocity i n the bivariate model. Taking i n t e r e s t rates into account does not completely explain the large decl ine i n velocity in 1 9 8 2 . ~ Preliminary information about velocity i n the 1982:IQ was available i n March, b u t was not finalized until June 1 9 8 2 . ~ By t h a t time, however, the evidence was convincing, and a t i t s July meeting, the FOMC voted t o a1 low M-l growth t o exceed the upper 1imit of the target range. The M-1 velocity model continued t o produce large negative forecast errors throughout the f i r s t quarter of 1983. Since then the e r r o r s have been small and offsetting. Clearly, the bivariate model failed t o explain M-1 velocity growth i n 1982. Whether the breakdown was permanent or temporary i s a subject of continuing research. The end-of-year cumulative M-1 errors shown i n chart 1 are important because they are incorporated permanently into the base f o r the next y e a r ' s t a r g e t range. The Federal Reserve has been c r i t i c i z e d f o r t h i s practice, b u t s h i f t s i n the base for the target since 1979 can be j u s t i f i e d because they o f f s e t an unexpected d r i f t i n velocity. http://clevelandfed.org/research/workpaper/index.cfm Best available copy -15The f o r e c a s t e r r o r s f o r M-2 v e l o c i t y a r e shown i n t a b l e 8. Using t h e u n i v a r i a t e model l e d us t o r e j e c t t h e hypothesis t h a t t h e v e l o c i t y t r e n d was stab1 e i n 1981 and 1982. Using t h e b i v a r i a t e M-2 v e l o c i t y growth model, we c o u l d n o t r e j e c t t h e hypothesis t h a t t h e v e l o c i t y t r e n d was s t a b l e u n t i l The s t a b i l i t y o f M-2 v e l o c i t y through 1982 l e d t h e FOMC t o s w i t c h 1983: IQ. i t s primary emphasis from M-1 t o i'4-2 i n October 1982. T h i s change i n emphasis occurred j u s t b e f o r e t h e o n l y s i g n i f i c a n t f o r e c a s t e r r o r f o r M-2 v e l o c i t y growth, which was associated w i t h t h e i n t r o d u c t i o n o f money market d e p o s i t accounts (MMDAs). tlowever, i n a n t i c i p a t i o n o f t h i s e r r o r , t h e FOMC chose t h e 1983 February-to-March average as t h e base f o r t h e 11-2 t a r g e t range. V. Concl u s i o n I n t h i s paper, we have shown t h a t mu1t i v a r i a t e t i m e - s e r i e s procedures produce s i g n i f i c a n t l y b e t t e r f o r e c a s t s o f M-1 v e l o c i t y than u n i v a r i a t e procedures do. The b e s t model o f M-1 v e l o c i t y growth i s a b i v a r i a t e model t h a t i n c l u d e s M-1 v e l o c i t y growth and t h e Treasury b i l l r a t e . T h i s model, estimated from a p e r i o d d u r i n g which t h e Federal Reserve used an i n t e r e s t - r a t e o p e r a t i n g t a r g e t , d i d an e x c e p t i o n a l l y good j o b o f f o r e c a s t i n g v e l o c i t y i n 1980 and 1981 and continued t o produce f o r e c a s t s t h a t v a r i e d w i t h a c t u a l values i n 1982 and 1983. The f o r e c a s t s were b a d l y biased i n these l a s t two years, although n o t as b a d l y b i a s e d as t h e f o r e c a s t s from t h e u n i v a r i a t e model o r t h e d e r i v e d v e l o c i t y model. The b e s t model o f M-2 v e l o c i t y i s d e r i v e d from t h e t r i v a r i a t e model t h a t i n c l u d e s M-2, nominal GNP, and t h e Treasury b i l l r a t e . The estimated e f f e c t of t h e lagged i n t e r e s t - r a t e e r r o r on M- 2 v e l o c i t y growth i s approximately o n e - t h i r d l a r g e r than t h e impact on M-1 v e l o c i t y . Taking i n t e r e s t r a t e s i n t o http://clevelandfed.org/research/workpaper/index.cfm Best available copy account does improve the out-of-sample forecast f o r M- 2 velocity, but the ir,~provementi s not s t a t i s t i c a l l y s i g n i f i c a n t . The b i v a r i d t e model i s s i m i l a r t o the velocity model derived from the t r i v a r i a t e model and leads t o similar out-of-sample forecasts. The n-step-ahead forecast f o r changes i n !.I-2 velocity i s zero f o r n g r e a t e r than 1 i n the bi v a r i a t e model, and very close t o zero f o r the t r i v a r i a t e model. The unusual behavior of velocity i n 1982 and 1983 has been a t t r i b u t e d t o deregul a t i on and the rapid decl i ne of i nfl a t i on. Constructing and imp1 enenting monetary t a r g e t s during t l ~si period required several major changes i n tile monetary t a r g e t s . In the absence of a complete structural model of the economy, we will never be able t o p r e d i c t a l l the s h i f t s i n velocity, b u t we have presented evidence t h a t re1 a t i v e l y simp1e model s of vel oci ty t h a t incorporate information about i n t e r e s t r a t e s yiel d s i gni f icantly b e t t e r forecasts than do univariate models. i n t h e l a s t four years, these models would have warned of a s h i f t in velocity. Furthermore, f o r the period since 1980, they sllow t h a t deviations of the money stock from announced t a r g e t s have o f f s e t unexpected s h i f t s i n velocity. http://clevelandfed.org/research/workpaper/index.cfm Best available copy Footnotes 1. The information gain of model B over model A i s calcul ated as : where SEE i s the standard e r r o r of the equation. models \,/as suggested by James Hoehn. 2. This method of comparing See tloehn, Gruben, and Fomby (1984.). Our univariate f o r e c a s t e r r o r s a r e comparable in s i z e t o the univariate forecast e r r o r s presented i n Hein and Veugelers (1983). 3. There a r e several explanations f o r the decline i n velocity. One i s t h a t there was a s h i f t in money demand associated w i t h the introduction o f interest- bearing checkable deposits (see Simpson 1984). Judd (1983) argues t h a t the s h i f t i n money demand was caused by a sudden 1oweVing of i n f l a t i o n expectations. See the proceedings from a conference held a t the Federal Reserve Bank of San Francisco (1983), f o r other papers attempting t o explain the unusual behavior of velocity i n 1982 and early 1983. 4. These data have been revised. However; the money supply and GNP data t h a t were available a t the time resulted in an even more dramatic breakdown i n a1 1 the M-1 velocity models. http://clevelandfed.org/research/workpaper/index.cfm Best available copy Refererices Ashley, Richard A., C. W. J . Granger, and R. Schmal ensee. Aggregate Consumption: "Advertising and An Analysis of Causal i t y , " Econornetrica, vol . 48, no. 5 (July 1980), pp. 1149-67. Bagshaw, Michael L . , and William T. Gavin. "Forecasting the Money Supply i n Tine Series l~lodels,"Working Paper 8304, Federal Reserve Bank of Cl eve1 and, December 1 983. Board of Governors of the Federal Reserve System. the congress," Federal Reserve Bulletin, vol . "Monetary Policy Report t o 7C1, no. 2 (February 1984), pp. 63-86. Box, George E.P., and Control Brunner, Karl. . and Gwilyrn M. Jenkins. Time Series Analysis: Kevised Edition, San Francisco, CA: Forecasting Hol den-Day Inc. , 1976. "Monetary Policy as a Random Walk Through History," i n Shadow Open Market Committee: Pol icy Statement and Position Papers, PPS-83-2. Center f o r Research i n Government Pol icy and Business, Graduate School of Management, University of Rochester (March 6-7, 1983), pp. 7-1 6. Federal Reserve Bank of San Francisco. Monetary Targeting and Velocity : Conference Proceedings, San Francisco, C A Y December 4-6, 1983. http://clevelandfed.org/research/workpaper/index.cfm Best available copy Hein, S c o t t E . , and Paul T. W. H. Veugelers. "Predicting Velocity Growth: A Time S e r i e s Perspective," Review, Federal Reserve Bank of S t . Louis, vol. 65, no. 8 (October 19831, pp. 34-43. "Recent Monetary Control Procedures and t h e Response of Hoehn, James G . I n t e r e s t Rates t o Fluctuations i n Monetary Growth," Economic Review, Federal Reserve Bank of Dallas, September 1983, pp. 1-1 0. , and William C. Gruben, with Thomas B. Fomby. S e r i e s Methods of Forecasting the Texas Economy," "Some Time Norking Paper No. 8402, Federal Reserve Bank of Dallas, April 1984. "The Recent Decline i n Velocity: Judd, dohn P. I n s t a b i l i t y i n Money Demand o r I n f l a t i o n ? " Economic Review, Federal Reserve Bank of San Francisco, no. 2 (Spring 19831, pp. 12-19. Meltzer, Allan H. "Recent Behavior of Base Velocity," i n Shadow Open Market Pol icy Statement and Position Papers, PPS-83-5. Center f o r Committee: Research i n Government P ~ l i c yand Business, Graduate School of Management, University of Rochester (September 7 8-1 9, 1983), pp. 19-24. . "Base Velocity-- the Trend Continues," i n Shadow Open Market Commi t t e e Pol icy Statement and Position Papers, PPS-84-1. Center f o r Research i n Government Pol icy and Business, Graduate School of Management, b University of Rochester (March 11 -1 2, 1984), pp. 29-30. http://clevelandfed.org/research/workpaper/index.cfm Best available copy Simpson, Thomas D. "Change i n t h e ~ i n a n c i a lSystem: I m p l i c a t i o n s f o r Monetary P o l i c y , " i n W i l l i a m C. Brainard, and George L. Perry, eds., Brookings Papers on Economic A c t i v i t y , 1 :1984, pp. 249-72. Tiao, G.C., and G.E.P. Box. "blodeling 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 , " Journal of t h e American S t a t i s t i c a l Association, v o l . 76, no. 376 (December 1981 ) , pp. 802-1 6. http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 1 U n i v a r i a t e V e l o c i t y Models f o r 1959:IIQ t o 1979:IVQ M-1 v e l o c i t y v l n V W l t = .0077 SEE + at = .0087 M-2 v e l o c i t v v1n VMLt = .270 at-l SEE + at = .0097 I(U,N) = 3.0 NOTE: SEE i s t h e standard e r r o r o f t h e equation. I(U,M) i s the i n f o r m a t i o n g a i n o f t h e u n i v a r i a t e model over t h e n a i v e model. The M-1 v e l o c i t y model i s t h e naive model ; t h a t i s , v e l o c i t y growth f o r e c a s t i s equal t o t h e mean growth r a t e o f t h e sample. http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 2 Bivariate Velocity Models f o r 1959:IIQ t o 1979: IVQ M-1 velocity model Error correlation matrix = P 14-2 velocity model Error correlation matrix = NOTE: The standard deviations of the e r r o r term a r e shown i n parentheses on the diagonal of the e r r o r correlation matrix. http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 3 T r i v a r i a t e Models f o r Nominal GNP, t h e Treasury B i l l Rate, and the Money Stock f o r 1959: I I Q t o 1979: IVQ M-1 model Vln G N P t = 1.553 vln Mlt-1 -.898 a l , t - 1 + a 3 , t ( .0055) Error c o r r e l a t i o n matrix = (.0014) .45 -. 21 ( .0092) M- 2 model Vln F12t = .973 Vln M2t-1 + a l , t + .329 a l , t - ~ Error c o r r e l a t i o n matrix = 1 -.29 (.0014) I NOTE: The standard deviation of the e r r o r tetm i s l i s t e d on t h e diagonal of t h e e r r o r c o r r e l a t i o n matrix. http://clevelandfed.org/research/workpaper/index.cfm Best available copy http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 5 Ashley, Granger, and Schmalensee S p e c i f i c a t i o n Tests Dependent v a r i abl e !.I-1 vel o c i ty growth Estimation r e s u l t s -. 075 .217 7.943 M-2 v e l o c i t y growth NOTE: The t - s t a t i s t i c s are shown i n parentheses. a. The F - s t a t i s t i c r e j e c t s t h e hypothesis t h a t z0 and cl are n o t s i g n i f i c a n t l y d i f f e r e n t from zero a t t h e 0.002 c r i t i c a l l e v e l . http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 6 Year a. M-1 V e l o c i t y : Imp1i e d Assumptions and Tirne-Series Forecasts a GNP f o r e c a s t c e n t r a l tendency M-1 m i d p o i n t Imp1i e d velocity assumption 4-Step-ahead vel oci ty forecast A l l f i g u r e s i n percent growth r a t e s . b. The 1.1-1 m i d p o i n t was adjusted f o r expected growth i n n e g o t i a b l e o r d e r o f withdrawal (NOW) accounts by the s t a f f o f t h e Board o f Governors o f t h e Federal Reserve System. SOURCE: "Monetary 'Pol i c y Report t o Congress," Federal Reserve B u l l e t i n , v a r i o u s issues. http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 7 Tests for Changes i n the Trend of F-1-1 Velocity Growth Vel oci ty forecast error cumul ated over the year Year: t q Univariate Bi variate N(Oy1) test for change in mean growth rate Univariate B i variate t e s t for a change i n the variance of the error Univariate Bivariate NOTE: The errors are it1 percent a t quarterly rates cumulated from tlie f i r s t to the fourth quarter. a. Using the 5 percent c r i t i c a l region, we can r e j e c t the null iiypothesis t h a t t h e process generating velocity has not changed. http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 8 Tests f o r Changes i n the Trend of M-2 Velocity Growth Vel oci t y f o r e c a s t e r r o r cunul ated over the yeara Year: t q Univariate Bivariate M(0,1) t e s t for change i n mean growth r a t e Univariate Bivariate t e s t for a change i n the variance of the e r r o r Univariate Bivariate The e r r o r s a r e i n percent a t quarterly r a t e s cumulated from the f i r s t t o the fourth quarter. NOTE: a. Using a 5 percent c r i t i c a l region, we can r e j e c t the null hypothesis t h a t the process generating velocity had not changed. http://clevelandfed.org/research/workpaper/index.cfm Best available copy Chart 1 Deviations of M-1 and Velocity from Expected values a Percent M-1 growth Vel oci ty growth NOTE: a. Quarterly deviations are cumulated over the calendar year. Expected values of M-1 growth are based on the midpoint of the M-1 t a r g e t range. Expected values of velocity are one-step-ahead forecast errors from the bivariate model.
@FedFRASER