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http://clevelandfed.org/research/workpaper/index.cfm Best available copy Working Paper 8304 FORECASTING THE MONEY SUPPLY I N TIME SERIES MODELS by Michael L. Bagshaw and William T. Gavin Working papers of the Federal Reserve Bank of C l eve1and are p r e l iminary materials, c i r c u l a t e d t o stimblate discussion and c r i t i c a l comment. The views stated herein are the authors' and not necessarily those o f the Federal Reserve Bank o f Cleveland o r o f the Board o f Governors o f the Federal Reserve Sys tern. December 1983 Federal Reserve Bank o f Cleveland http://clevelandfed.org/research/workpaper/index.cfm Best available copy FORECASTING THE MONEY SUPPLY I N TIME SERIES MODELS Abstract I n t h i s paper, time series techniques are used t o forecast q u a r t e r l y money supply l e v e l s . Results i n d i c a t e t h a t a b i v a r i a t e model i n c l u d i n g an i n t e r e s t r a t e and M-1 p r e d i c t s M-1 b e t t e r than the u n i v a r i a t e model using M-1 only and as well as a 5- variable model which adds prices, output, and c r e d i t . The paper also presents evidence on the issue o f using seasonally adjusted data i n forecasting w i t h time series models. resul t s apply t o a11 econometric model ing. The imp1i c a t i o n s o f these Resul t s support the hypothesis t h a t using seasonally adjusted data can l e a d t o spurious c o r r e l a t i o n i n mu1t i v a r i a t e model s. I. I n t r o d u c t i o n The goal o f t h i s research i s t o b u i l d a s t a t i s t i c a l model r e l a t i n g t h e intermediate targets o f monetary pol i c y t o i n f l a t i o n and output. The Federal Reserve has used both i n t e r e s t r a t e s and the money supply as intermediate targets i n t h e past 20 years. 1 t a r g e t range f o r c r e d i t . I t has j u s t r e c e n t l y adopted an experimental This model would be used t o monitor the economic re1ationships t h a t are assumed (predicted) i n the construction of the intermediate targets and t o devel op t e s t s t h a t woul d suggest when the predicted re1ationships are r e j e c t e d by t h e data. When the assumptions underlying the targets are rejected, the targets should be changed. T h i s paper reports the r e s u l t s o f p r e l iminary work on t h i s project. 5- variate model i s estimated and i t s forecasts o f the money supply are A http://clevelandfed.org/research/workpaper/index.cfm Best available copy compared w i t h forecasts from u n i v a r i a t e and b i v a r i a t e models. Estimation procedures developed by Tiao and Box (1981) are used t o estimate t h e simultaneous equation model (SEM) without p r i o r r e s t r i c t i o n s . Z e l l n e r and Palm (1974) argued t h a t time s e r i e s analysis could be used t o t e s t the assumptions underlying econometric models--assumptions about v a r i a b l e s being exogenous, about lags i n the dynamic s t r u c t u r e o f t h e model, and about the c o r r e l a t i o n s between the random elements o f economic variables. The problem faced by Zel l n e r and Palm i n 1974 was t h a t there were no time series methods a v a i l a b l e by which one could estimate d i r e c t l y the parameters o f an SEM model. The procedures they recommended involved estimating approximations t o appropriate transformations of the time series s t r u c t u r a l model, t h a t i s , the f i n a l form and the t r a n s f e r f u n c t i o n form. This suggestion by Zel l n e r and Palm l e d t o procedures developed by Granger and Newbold (1977), Wall i s (1977), and Chan and Wallis (1978). A l l o f these procedures are computationally burdensome and i n t u i t i v e l y i n f e r i o r t o one t h a t can provide d i r e c t estimates o f the parameters. Because o f computational complexity, these procedures were 1i m i t e d t o models w i t h 2 or, a t most, 3 variables. Sims (1977, 1980) recommended estimating t h e vector autoregressive form o f t h e model. The problem w i t h t h i s approach i s t h a t i t leads t o a plethora of parameters i n m u l t i v a r i a t e models. Sims has solved t h i s problem by a r b i t r a r i l y truncating the order o f the autoregression. Others have used the Akaike (1969, 1970) f i n a l p r e d i c t i o n e r r o r i n p r e l iminary analysis t o specify optional l a g lengths f o r each variable. Fackler 1982. ) (See, f o r example, Hsiao 1982 o r This preliminary analysis i s i n a l i m i t e d sense the counterpart o f the i d e n t i f i c a t i o n stage i n the Tiao-Box procedure. A major drawback o f t h i s autoregressive approach i s t h a t one i s constrained t o a subset o f models t h a t are possible using the more general Tiao-Box procedure. http://clevelandfed.org/research/workpaper/index.cfm Best available copy II. The Vector ARIMA Model The f o l l owing i s a very b r i e f d e s c r i p t i o n o f the vector Autoregressive Integrated Moving Average (ARIMA) model. A more d e t a i l e d d e s c r i p t i o n i s given I n the vector ARIMA model, i t i s assumed e i t h e r t h a t i n Tiao and Box (1981 ). each series i s s t a t i o n a r y o r t h a t some s u i t a b l e d i f f e r e n c e o f t h e data i s stationary. Thus, i f z t i s the o r i g i n a l k dimensional vector valued time series, then i t i s assumed t h a t d S Di =(I-B) ( I - B ) Zit 'it i s s t a t i o n a r y f o r each component o f z f o r an appropriate choice o f di and rLt Di where B i s the b a c k s h i f t operator (i.e., seasonal period (e.g., Bzit -- S i s the f o r q u a r t e r l y data, S = 4), and di(Di) nunher o f regul ar (seasonal ) d i fferences necessary t o make wit The model i s presented i n terms o f the s t a t i o n a r y series vector ARIMA model i s given by where O (B) = I %P - - -1 O B - ... - CJ 2'P BP, ct. i s the stationary. The general http://clevelandfed.org/research/workpaper/index.cfm Best available copy the $.Is, @.'s, O.'s,o.'s, *J "JJ SJ "JJ and a. are k x k unknown parameter matrices, and the a 's a r e k x 1 vectors o f random variables which are ,t 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(Oyt ). Thus, i t i s assumed t h a t the z t ' s a t d i f f e r e n t points i n time are independent, b u t n o t necessarily t h a t the elements o f qt are independent a t a given p o i n t i n time. The Tiao-Box procedure a1lows one t o estimate the s t r u c t u r a l parameters o f a mu1t i v a r i a t e simul taneous 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 equation Box-Jenkins model ing. The steps involved 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 autocorrel a t i o n s and cross- correl ations o f t h e series; 2) estimate the parameters o f t h i s model; and 3 ) apply diagnostic checks t o the residuals. I f t h e r e s i d u a l s do n o t pass the diagnostic checks, then the t e n t a t i v e model i s modified and steps two and three are repeated. This process continues u n t i l a s a t i s f a c t o r y model i s obtained. 111. The Empirical Models I n t h i s section the Tiao-Box procedure i s used t o estimate the h i s t o r i c a l r e l a t i o n s h i p s among t h e intermediate targets and the goals o f monetary policy. The model estimated below includes 3 q u a n t i t y variables and 2 p r i c e variables from the markets f o r goods, c r e d i t and money. measure the money supply (M-1). M-1 i s used t o C r e d i t i s measured as funds r a i s e d by the non-f i n a n c i a l sector (NFD) i n c l u d i n g p r i v a t e and government debt. This measure d i f f e r s s l i g h t l y from the actual measure t h a t has been adopted by the Federal Reserve as an experimental and supplemental t a r g e t f o r monetary p o l i c y i n 1983. Our v a r i a b l e i n c l udes e q u i t i e s issued by nonfinancial corporations and funds r a i s e d i n t h e United States by subsidiaries o f foreign corporations. The q u a n t i t y o f goods i s measured as GNP i n constant (1972) d o l l a r s (GNP72). The p r i c e o f output i s the imp1i c i t GNP d e f l a t o r (PGNP). http://clevelandfed.org/research/workpaper/index.cfm Best available copy The p r i c e o f c r e d i t i s measured as the y i e l d on 3-month Treasury s e c u r i t i e s (RTB3). This work i s p r e l iminary i n many ways. F i r s t , we have n o t checked the s e n s i t i v i t y o f our r e s u l t s t o a1t e r n a t i v e measures o f the i n c l uded variables. Certainly, t h e 3-month Treasury b i l l note i s an a r b i t r a r y measure o f the y i e l d on c r e d i t . Second, we have n o t checked the s e n s i t i v i t y o f our r e s u l t s t o the i n c l usion o f other markets. Speci f i c a l ly, much o f the work i n macroeconomics suggests t h a t the 1abor market i s n o t i n continuous e q u i l i b r i u m and t h a t events i n t h a t market a r e important determinants o f f l u c t u a t i o n s i n both output and i n f l a t i o n . Third, one o f the most important t e s t s o f any model i s how well i t does i n forecasting out-of-sample. I n the l a s t section we compare out-of-sample forecasts f o r M-1 from a1t e r n a t i v e time series models, b u t we do n o t evaluate forecasts o f the other variables nor do we provide a comprehensive comparison o f our model's p r e d i c t i o n s w i t h non-time series 2 procedures. Using t h e n o t a t i o n from the introduction, w i s a vector o f the 5 economic (\I variables. T h i s vector has an associated random vector, at. The model i s ".A estimated twice, once using seasonally adjusted datq and once w i t h not-seasonally adjltsted data. 1ogari thms o f each variable. The w vector includes appropriately differenced The estimates using not-seasonal l y adjusted data should be considered superior a p r i o r i because the seasonal f a c t o r s are estimated j o i n t l y w i t h the other parameters o f the model. This i s i n c o n t r a s t t o using seasonally adjusted data where the seasonal f i l t e r s applied t o the data are d i f f e r e n t f o r each v a r i a b l e and the seasonal adjustment procedures do -, n o t take account o f c o r r e l a t i o n between series. Wall i s (1974) has shown t h a t using data t h a t has been seasonally adjusted w i t h conventional procedures may l e a d t o i n c o r r e c t inference i n dynamic models. http://clevelandfed.org/research/workpaper/index.cfm Best available copy The model estimated using the not- seasonally adjusted data i s given i n t a b l e 1. The model estimated using seasonally adjusted data i s given i n tab1e 2. When the models are i n the general form, they are d i f f i c u l t t o i n t e r p r e t because there may be i n t e r a c t i o n s among t h e autoregressive and moving average operators. Consequently, we express the models i n the moving average form as shown i n t a b l e 3. The P r i c e o f Goods. This leads t o the f o l l o w i n g i n t e r p r e t a t i o n s . For the not-seasonal l y adjusted data, the imp1i c i t d e f l a t o r i s independent o f the r e s t o f t h e model i n c l u d i n p contemporaneous correlations. According t o these estimates, i n f l a t i o n can be modeled as a u n i v a r i a t e ARIMA model w i t h a f i r s t - o r d e r autoregressive and a f i r s t - o r d e r moving average term. This model suggests t h a t information from the money supply, c r e d i t aggregates, t h e i n t e r e s t r a t e and r e a l output w i l l n o t h e l p p r e d i c t changes i n the p r i c e l e v e l once we have taken account o f information i n the h i s t o r y o f the p r i c e l e v e l . This s i t u a t i o n changes dramatically when we examine the same equation from the model estimated w i t h seasonally adjusted data. I n t h i s model, i n f l a t i o n responds p o s i t i v e l y t o 1agged money supply, negatively t o 1agged c r e d i t , and (a1though weakly) negatively t o lagged i n t e r e s t rates. A l l o f these r e l a t i o n s h i p s i n v o l v e decaying lagged patterns because o f the autoregressive terms i n the model. While the p o s i t i v e dependence o f i n f l a t i o n on money supply growth w i l l be encouraging t o some, we would have more confidence i n t h i s sesul t i f i t was evident i n the not-seasonally adjusted model. Part o f the model n o t captured i n the parameter matrices i s the estimate o f t h e c o r r e l a t i o n s between contemporaneous errors. I n neither case i s there a s i g n i f i c a n t c o r r e l a t i o n between the e r r o r s from the i n f l a t i o n equation and the other errors. 3 M-1. - The second equation determines t h e money supply. I n t a b l e 1 we can http://clevelandfed.org/research/workpaper/index.cfm Best available copy see t h a t the seasonal p a r t o f the model r e q u i r e d a f o u r t h d i f f e r e n c e and a fourth- order moving average t o represent t h e seasonal movement i n the ~ e r i e s . ~The money supply i s determined by a moving average o f t h e e r r o r from the M-1 equation and a second-order moving average o f the e r r o r from the i n t e r e s t r a t e equation. The sign o f t h e moving average parameter on the i n t e r e s t r a t e e r r o r i s consistent w i t h the money demand l i t e r a t u r e . The s i g n i f i c a n c e o f a "scale" variable, u s u a l l y income o r wealth, i n almost every model o f money demand suggests t h a t there should be s i g n i f i c a n t c o r r e l a t i o n between M-1 and output. I n t a b l e 1, t h e c o r r e l a t i o n between e r r o r s i n the money and output equations i s not significant. However, there i s a s t r o n g contemporaneous c o r r e l a t i o n between the e r r o r i n the M-1 equation and t h e e r r o r i n the c r e d i t equation. Using seasonally adjusted data resul t s i n changes t h a t support t r a d i t i o n a l money demand models. The major differences a r e a s i g n i f i c a n t p o s i t i v e c o r r e l a t i o n between the e r r o r s from the M-1 and output equations and a 50 percent increase i n the estimated i n t e r e s t r a t e e l a s t i c i t y . There i s a l s o a s i g n i f i c a n t e f f e c t from c r e d i t s t a r t i n g a t l a g one. Credit. The t h i r d equation determines c r e d i t , t h a t i s , t h e amount o f funds r a i s e d by the nonfinancial sector. I n t a b l e 3, we see t h a t not-seasonally adjusted c r e d i t depends on lagged M-1 growth, o n the i n t e r e s t r a t e 1agged 3 quarters and on a f i r s t - o r d e r moving average error. I n a1 1 these "quantity" equations, M-1, NFD, and GNP72, the seasonal model involved a fourth- order difference and a fourth- order moving average parameter. The contemporaneous e r r o r i n the c r e d i t equation was s i g n i f i c a n t l y c o r r e l a t e d w i t h the e r r o r s from the M-1 and the r e a l output equations. The c r e d i t equation estimated using seasonally adjusted data d i f i e r s from the equation i n t a b l e 1 i n t h a t c r e d i t does n o t depend on past M-1 o r past http://clevelandfed.org/research/workpaper/index.cfm Best available copy i n t e r e s t rates. Using seasonally adjusted data we f i n d t h a t M-1 depends on p a s t c r e d i t b u t t h a t c r e d i t does n o t depend on past M-1. This i s e x a c t l y opposite t o our findings when we used not-seasonally adjusted data. The I n t e r e s t Rate. The f o u r t h equation determines the i n t e r e s t r a t e , the y i e l d on Treasury b i l l s w i t h 3 months t o maturity. I n the not- seasonally adjusted model, changes i n t h e i n t e r e s t r a t e depend only on past e r r o r s from the M-1 equation and on past e r r o r s from the i n t e r e s t r a t e equation. There i s no s i g n i f i c a n t contemporaneous c o r r e l a t i o n between the e r r o r from the i n t e r e s t r a t e equation and any o f the e r r o r s from the other equations. I n the seasonally adjusted model the i n t e r e s t r a t e depends on past M-1 and credit. I n both models the r e l a t i o n s h i p between the i n t e r e s t r a t e and M-1 i s p o s i t i v e i n d i c a t i n g a supply re1ationship. These models suggest t h a t s i n g l e equation money demand models i n c o r r e c t l y t r e a t the i n t e r e s t r a t e as exogenous. Again, t h e e r r o r from the i n t e r e s t r a t e equation i s n o t s i g n i f i c a n t l y c o r r e l a t e d w i t h contemporaneous e r r o r s from any o f the other equations. Real Output. I n the not-seasonally adjusted model r e a l output depends on lagged M-1 growth, i n f l a t i o n and i n t e r e s t rates. These estimates c l e a r l y r e j e c t the hypothesis t h a t r e a l output i s independent o f a n t i c i p a t e d changes i n the money supply. There i s a weak c o r r e l a t i o n between contemporaneous e r r o r s i n M-1 and output, b u t i t i s n o t s i g n i f i c a n t a t the 5-percent l e v e l . When seasonally adjusted data i s used output depends on past i n f l a t i o n , M-1 , c r e d i t , and i n t e r e s t rates. This equation i s consistent w i t h the hypothesis t h a t accel e r a t i n g i n f l a t i o n has a s i g n i f i c a n t depressing e f f e c t on t h e trend i n output growth. The e r r o r s i n output are s i g n i f i c a n t l y c o r r e l a t e d w i t h the errors from the money and c r e d i t equations. Sumnary o f Estimated Model s. I n every equation, d i f f e r e n t variables http://clevelandfed.org/research/workpaper/index.cfm Best available copy were s i gni f icant depending on whether not-seasonal l y o r seasonal l y adjusted data was used. The contemporaneous c o r r e l a t i o n s between e r r o r s were very s i m i l a r i n both models. The strongest contemporaneous c o r r e l a t i o n s were between M-1 and c r e d i t and between r e a l output and c r e d i t . The contemporaneous c o r r e l a t i o n between output and money was j u s t b a r e l y s i g n i f i c a n t i n the seasonally adjusted model and j u s t marginally i n s i g n i f i c a n t i n the not-seasonall y adjusted model . One i n t e r e s t i n g r e s u l t was t h a t f o r the seasonally adjusted data, twelve o f the twenty off- diagonal terms o f the moving average representation were non-zero, w h i l e only seven were non-zero f o r not-seasonally adjusted data. T h i s r e s u l t supports the ( W a l l i s (1974) c l a i m t h a t the o f f i c i a l (Census X-11 v a r i a n t ) seasonal adjustment procedure can induce spurious dynamic c o r r e l a t i o n between variables. Using not- seasonally adjusted data r e s u l t s i n a forecasting model t h a t i s block recursive w i t h two independent leading blocks, the p r i c e equation b y i t s e l f , and the money and i n t e r e s t r a t e equations. depends on the money and i n t e r e s t r a t e block. both leading blocks. The c r e d i t equation The output equation depends on This r e s u l t suggests t h a t a b i v a r i a t e model i n c l u d i n g j u s t the i n t e r e s t r a t e and M-1 would p r e d i c t M-1 as well as t h e 5- variate model. Both should outperform a u n i v a r i a t e model o f the money supply process. Using seasonally adjusted data r e s u l t s i n a block recursive forecasting model i n which the c r e d i t equation forms the leading block, t h e money and i n t e r e s t equations form t h e second block, the i n f l a t i o n equation i s t h e t h i r d block, and the output equation i s the f i n a l block. I n t h i s case the forecasts of M-1 from the 5- variable model should outperform both t h e b i v a r i a t e , i n c l u d i n g M-1 and the i n t e r e s t rate, and u n i v a r i a t e models. http://clevelandfed.org/research/workpaper/index.cfm Best available copy IV. Forecasting the Money Supply i n Time Series Models Three time series models o f the money supply were estimated using both seasonally and not-seasonal l y adjusted data over t h e period from the f i r s t quarter o f 1959 t o the f o u r t h quarter o f 1979, and forecasts were generated over the period from the f i r s t quarter o f 1980 t o the t h i r d quarter o f 1982. The 3 models are a u n i v a r i a t e model o f M-1 , a b i v a r i a t e model o f M-1 i n c l u d i n g t h e y i e l d on 3-month Treasury b i l l s , and the 5 - v a r i a t e model shown i n t a b l e 1 o f section 1. The r e s u l t s i n t a b l e 1 show t h a t f o r not- seasonally adjusted data, t h e money supply and the i n t e r e s t r a t e form a 1eading recursive block i n the forecasting model. Therefore, we would expect t h e b i v a r i a t e model t o do b e t t e r than the u n i v a r i a t e and as well as t h e 5- variate model. M-1 are displayed i n t a b l e 4. their similarity. The models f o r An i n t e r e s t i n g feature o f these three models i s The f i r s t - and fourth- order moving average terms are almost i d e n t i c a l i n a l l t h r e e cases. The estimated i n t e r e s t r a t e e l a s t i c i t y i s s i m i l a r i n the m u l t i v a r i a t e models. I n the b i v a r i a t e model the f i r s t - o r d e r moving average parameter on the i n t e r e s t r a t e e r r o r 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 zero, b u t i t s excl usion 1eads t o s i g n i f i c a n t s e r i a l c o r r e l a t i o n between e r r o r s i n the i n t e r e s t r a t e and M-1. The r e s u l t s o f the forecasting experiment are given i n t a b l e 5. o f t a b l e 5 shows t h e r e s u l t s o f one-step-ahead forecasts. Panel a. The r e s u l t s show t h a t the forecasts became s l i g h t l y b e t t e r as more variables were added t o the model. The differences are small, however, and the Root Mean Square E r r o r s (RMSEs) are disappointingly large. One reason f o r t h i s may have been the c r e d i t c o n t r o l s imposed i n the second quarter o f 1980 and removed i n the t h i r d quarter o f the .same year. We attempted t o a b s t r a c t from the e f f e c t o f these http://clevelandfed.org/research/workpaper/index.cfm Best available copy controls i n two ways. F i r s t , we ran n-step-ahead forecasts, which d i d n o t use any actual data a f t e r the f o u r t h quarter of 1979. The r e s u l t s were much b e t t e r and they favored the mu1t i v a r i a t e model s (see panel b. 1. However, the confidence i n t e r v a l s are so wide on these forecasts t h a t we must ascribe the good performance t o coincidence. I n panel c. we repeated the n-step-ahead forecasts using the i n i t i a l values from the f i r s t quarter o f 1980. The resul t s were much worse, a1 though the mu1t i v a r i a t e model s s t i l l outperformed the u n i v a r i a t e model. The second method we used t o intervene i n the model t o c o r r e c t for c r e d i t controls was t o rep1ace actual values o f M-1 and the i n t e r e s t r a t e i n the second quarter o f 1980 and t h i r d quarter o f 1980 w i t h predicted values. e l iminated e r r o r s i n those quarters. This Panel d. 1i s t s the mean e r r o r and RMSE f o r the 8 quarters beginning i n the f i r s t quarter o f 1980. I n t h i s case, t h e mean e r r o r was s ligh t l y 1arger than i n panel a. , b u t the RMSE was much small e r and more i n 1i n e w i t h the e r r o r normal l y found i n regression models o f t h e money supply. For the seasonally adjusted data, t h e models f o r M-1 are given i n t a b l e 6. The b i v a r i a t e and 5- variate models are s i m i l a r i n t h a t t h e autoregressive terms are close and t h e f i r s t - o r d e r moving average terms on the i n t e r e s t r a t e are roughly the same. The non- significance o f the constant i n the 5- variate model i s due t o the a d d i t i o n o f t h e c r e d i t term. The u n i v a r i a t e model i s a c t u a l l y closer t o the other two models than i t a t f i r s t appears. This can be seen by transforming t h i s model as follows: (1-.414B-.363b 2 - 1 A 2 lnM1 = (1-.238B 8 )a2t ) o r by d i v i d i n g the f i r s t operator i n t o one o f the 1-B factors, http://clevelandfed.org/research/workpaper/index.cfm Best available copy Also, the r e s i d u a l s from both the b i v a r i a t e and 5- variate models f o r M-1 had j u s t barely nonsignificant correlations a t l a g 8 . Thus, these models would have a moving average term o f l a g 8, which would n o t d i f f e r s u b s t a n t i a l l y from t h a t o f t h e u n i v a r i a t e model i f t h i s parameter were included. Thus, t h e models are q u i t e s i m i l i a r . The r e s u l t s o f forecasting using t h e seasonally adjusted models are presented i n tab1e 7. The resul t s f o r the one-step-ahead forecasts agree w i t h the statement t h a t the u n i v a r i a t e model should be outperformed by both the b i v a r i a t e and the 5- variate models and t h a t the 5- variate model shoul d do b e t t e r than t h e b i v a r i a t e model. A1 so, these RMSEs are smaller than those o f the not-seasonally adjusted models. T h i s may be due t o the f a c t t h a t when the data was seasonally adjusted, an attempt was made t o a d j u s t f o r the e f f e c t s o f c r e d i t c o n t r o l .5 from above. We repeated the t h r e e a d d i t i o n a l forecasting experiments The r e s u l t s f o r the n-step-ahead forecasts from the f o u r t h quarter o f 1979 are r a t h e r strange i n t h a t the u n i v a r i a t e model i s much b e t t e r than t h e other two models. This r e s u l t i s n o t t r u e when forecasting from the f i r s t quarter o f 1980 where the 5 - v a r i a t e model i s much b e t t e r . Examining the f i n a l r e s u l t , we see t h a t indeed, even the seasonally adjusted models forecast b e t t e r past the c r e d i t control period. Overall, these forecasting r e s u l t s from t h i s s h o r t period do n o t d i s t i n g u i s h sharply between the three time series models. This may r e f l e c t , i n part, the p a r t i c u l a r l y v o l a t i l e p e r i o d over which the forecasts were run. Besides the c r e d i t controls, there was a1so a change i n Federal Reserve operating procedures j u s t before the s t a r t o f the forecasting period. This change has been associated w i t h higher variance i n both i n t e r e s t r a t e s and M-1 One way t o get around t h i s problem would be t o "backcast" i n t o t h e 1950s using t h e estimated parameters o f the model. look a t d i f f e r e n t variables. I t may also be i n s t r u c t i v e t o Forecasting output may be more useful i n determining the advantage o f 1arger time series model s because output depends on more variables i n the system than does M-1. . http://clevelandfed.org/research/workpaper/index.cfm Best available copy Conclusion V. I n t h i s paper we have used the Tiao-Box procedure t o i d e n t i f y and estimate a dynamic simultaneous equation model. The procedure leads t o a parsimonious representation o f a model i n c l u d i n g markets f o r goods, money, and c r e d i t . r e s u l t s from the forecasting experiment were mixed. The I n 5 o f the 8 experiments, the 5- variate model gave b e t t e r forecasts than t h e small e r models. I n two o f t h e other cases the r e s u l t s were very close. turbulent period for monetary pol icy. operating procedure i n October 1979. This was a The Federal Reserve adopted a new That change i n regimes was followed by unpredicted swings i n the i n t e r e s t r a t e and more v o l a t i l e growth i n the money supply. I n s p i t e o f t h i s , the out-of-sample q u a r t e r l y p r e d i c t i o n e r r o r o f M-11 was on the order o f 1 percent when we intervened f o r the period o f c r e d i t controls. This e r r o r i s o f the same magnitude as t h a t which has been found when standard econometric models are used. d i f f e r e n c e between the d i f f e r e n t model s. Overall, there was n o t much Perhaps as we gather more information we w i l l be b e t t e r able t o choose between these models. I n the not-seasonal l y adjusted model, i n f l a t i o n was independen't o f a11 the intermediate targets. This suggests t h a t a d i f f e r e n t s p e c i f i c a t i o n o f t h e model w i l l be needed t o represent the transmission mechanism going from monetary pol i c y t o i n f l a t i o n . Using seasonally adjusted data leads t o a model t h a t i s more useful f o r p o l i c y evaluation. However, i f the dynamic c o r r e l a t i o n s are spurious, caused by an inappropriate seasonal mode?, then we cannot r e l y on t h i s model e i t h e r . One possible approach t h a t we plan t o investigate, i s t o combine i n f l a t i o n and output i n t o nominal GNP and b u i l d a model r e l a t i n g nominal GNP t o the intermediate targets. I n practice, much o f the discussion surrounding monetary pol i c y goals i s couched i n terms o f nominal GNP. http://clevelandfed.org/research/workpaper/index.cfm Best available copy Footnotes 1. Fackler and S i l v e r (1982-83) discuss the issues involved i n use o f c r e d i t as an intermediate t a r g e t f o r monetary policy. Friedman (1981 ) and Fackler use vector autoregressive methods w i t h seasonally adjusted data t o examine the dynamic r e l a t i o n s h i p s among i n f l a t i o n , output, i n t e r e s t rates, M-1 and c r e d i t . 2. O ' R e i l l y e t al. (1981) r e p o r t s t h a t u n i v a r i a t e ARIMA models d i d n o t forecast as well as the D R I l a r g e model. The 1arge model forecasts had a r o o t mean square e r r o r average 73 percent lower than ARIMA models. They present a m u l t i v a r t a t e model b u t do n o t present comparative s t a t i s t i c s f o r t h i s model. I n general, 1arge model forecasts t h a t "do we1 1 " do so because o f judgmental adjustments t o t h e model forecasts. The vector ARIMA model can be expected t o beat non-judgemental forecasts from l a r g e . econometric model s 3. Throughout t h i s work, we have used a 5-percent c r i t i c a l region t o define significance. 4. I n p r e l iminary work, we found t h a t i f a Tourth-order autoregressive term was included i n the model, then i t s estimate was close t o 1. Consequently, t h e data were seasonal l y d i fferenced. 5. Pierce and Cleveland (1981) discuss the method used by the Federal Reserve t o a d j u s t f o r the e f f e c t s o f c r e d i t control. http://clevelandfed.org/research/workpaper/index.cfm Best available copy http://clevelandfed.org/research/workpaper/index.cfm Best available copy D D D D D http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 3 Moving Average Representation Not-seasonally adjusted data i Seasonally adjusted data - . r 1 http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 4 Time Series Models of MlNS* (Sample Period: 1959:IQ to 1979: IVQ) UNIVAR IATE: BIVARIATE: 5- VARIATE: vv 4 l n M I N S =~ (1 + .504B) (1 - * - .558B4)a2t .012B2 qt Ml NS i s M-1 n o t seasonally adjusted a2 = Random component o f 1n Ml NS aq = Random component from the i n t e r e s t r a t e equation not shown i n t h i s paper http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 5 Out-of-Sample Forecasts f o r MINS (Bi 11ions of do1 1a r s ) Mean Error a. One-s tep-ahead forecast 1980: IQ t o 1982 : I I IQ Univariate Bivariate 5-Vari a t e b. d. 7.629 7.526 7.200 -4.871 -0.381 -1.367 6.994 3.968 4.179 n-Step-ahead forecast from 1980: IQ to 1982: I I IQ Uni v a r i a t e Bivariate 5- Variate - -0.582 -0.277 -0.380 n-Step-ahead f o r e c a s t from 1979:IVQ t o 1982:IIIQ Univariate Bivariate 5- Variate c. RMS E -9.103 -5.939 -5.411 - - 10.611 7.393 6.919 - - - One-s tep-ahead f o r e c a s t wi t h intervention from 1979:IVQ t o 1982:IIIQ Univariate B i vari a t e 5- Variate -0.967 -0.547 -0.574 4.735 4.845 4.934 http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 6 Time Series Models of M- l* (Sample Period : 1959: IQ t o 1979: IVQ) UNIVARIATE: BIVAR IATE: 2 l n ~ l = (1 (1 - .648B) - .414B - . 3 6 3 ~ 2 ) (1 InM12 = a 2 t 5- VAR IATE : *a2 = Random component o f lnMl a4 = Random component o f lnRTB3 - - .0194aqt,l .238~8)a~~ + .00431 http://clevelandfed.org/research/workpaper/index.cfm Best available copy T a b l e 7 Out- of Sample F o r e c a s t s f o r M-1 ( B i 11i o n s o f do1 1a r s ) - Mean E r r o r a. One-step-ahead f o r e c a s t 1980:IQ t o 1 9 8 2 : I I I Q Univariate Bi variate 5- Variate b. 6.532 5.644 5.274 -2.245 11.536 8.220 4.639 13.418 9.920 n-Step-ahead f o r e c a s t f r o m 1980: IQ t o 1982 :IIIQ Univariate Bivariate 5- Vari a t e d. -0.422 0.118 0.206 n-Step-ahead f o r e c a s t f r o m 1979:IVQ t o 1 9 8 2 : I I I Q Univariate Bivariate 5- Variate c. RMS E -5.810 4.573 1.868 7.296 7.119 4.774 One-step-ahead f o r e c a s t w i t h i n t e r v e n t i o n from 1979 : I V Q t o 1982 :IIIQ Univariate Bivariate 5- Vari a t e -0.846 -0.242 -0.070 4.541 4.880 4.160 -- http://clevelandfed.org/research/workpaper/index.cfm Best available copy References Akaike, H. " S t a t i s t i c a l p r e d i c t o r i d e n t i f i c a t i o n , " Annal s o f the I n s t i t u t e o f S t a t i s t i c a l Mathematics, 21 (1969a), pp. 203-217. . " F i t t i n g Autoregressions f o r Prediction," Annal s o f the I n s t i t u t e o f S t a t i s t i c a l Mathematics, 21 (1969b), 243-247. Box, G. E.P. , and G. C. Tiao. " I n t e r v e n t i o n Analysis w i t h Appl i c a t i o n s t o Economic and Environmental Problems, " Journal o f the American S t a t i s t i c a l Association, vol. 70 (March 1975), pp. 70-79. and Kenneth F. Wallis. " M u l t i p l e Time-Series Modelling: Chan, W.-Y.T., Another Look a t the Mink-Muskrat Interaction," Applied S t a t i s t i c s , vol. 27, no. 2 (1978), pp. 168-75. Fackler, James. "An Empirical Model o f the Markets f o r Money, C r e d i t and Output. " Processed. Federal Reserve Bank o f New York, r e v i sed December 1982. , and Andrew S i l v e r . " Credit Aggregates as P o l i c y Targets," Q u a r t e r l y Review, Federal Reserve Bank o f New York, v o l . 7, no. 4 (Winter 1982-83), pp. 2-9. Friedman, Benjamin M. "The Re1a t i v e S t a b i l i t y o f Money and C r e d i t V e l o c i t i e s i n the United States: Evidence and Some Speculations," Working Paper No. 645, National Bureau o f Economic Research, March 1981. Granger, C. W. J. , and Paul Newbold. Academic Press, 1977. Forecasting Economic Time Series, New York: http://clevelandfed.org/research/workpaper/index.cfm Best available copy Hsiao, Cheng. "Autoregressive Model i n g and Causal Ordering o f Economic Variables," Journal o f Economic Dynamics and Control, vol. 4, no. 3 (August 1982), 243-259. O'Reilly, Dan, -e t al. "Macro Forecasting w i t h Mu1t i v a r i a t e ARIMA," Resources Review o f the U.S. Pierce, David A. , Data Economy, August 1981, pp. 1.29-1.33. and W i l l iam P. Cleveland "Seasonal Adjustment Methods f o r the Monetary Aggregates ,I1 Federal Reserve B u l l e t i n , Board o f Governors of t h e Federal Reserve System, Washington, DC, vol. 67, no. 12 (December 1981), pp. Sims, Christopher A. "Exogeneity and Causal Ordering i n Macroeconomic Models," New Methods i n Business Cycle Research: Proceedings from a Conference, Federal Reserve Bank o f Minneapol i s , 1977, pp. 23-44. . "Macroeconomics and Real ity," Econometri cay v o l. 48, no. 1 (January 1980), pp. 1-48. Tiao, G.C., and G.E.P. Box "Modeling M u l t i p l e ,Time Series w i t h Applications," Journal o f the American S t a t i s t i c a l Association, vol. 76, no. 376 (December 1981), pp. 802-16. Wall is, Kenneth F. "Seasonal Adjustment and Relations Between Variables," Journal o f the American S t a t i s t i c a l Association, v o l . 69, no. 345 (March 1974), pp. 18-31. . "Mu1 t i p l e Time Series Analysis and the F i n a l Form o f Econometric Models," Econometrica, vol. 45, no. 6 (September 1977), pp. 1481-97. Zel 1ner, Arnol d, and Franz Palm. "Time Series Analysis and Simul taneous Equation Econometric Model s," Journal o f Econometrics, vol . 2, no. 1 (May
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