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http://clevelandfed.org/research/workpaper/index.cfm Best available copy Workinq Paper 8507 FORECASTING AND SEASONAL ADJUSTMENT by Mlchael L. Bagshaw Thanks are due to Gordon Schlegel for p r o g r a m i n g support and 8111 Gavln and Klm Kowalewskl for helpful commen t s. Working papers o f the Federal Reserve Bank o f Cleveland are preliminary materials, cfrculated t o stimulate discussion and critical comment. The views expressed herein are those o f the author and not necessarily those o f the Federal Reserve Bank of Cleveland or the Board o f Governors o f the Federal Reserve System. December 1985 Federal Reserve Bank o f Cleveland http://clevelandfed.org/research/workpaper/index.cfm Best available copy FORECASTING AN0 SEASONAL ADJUSTMENT ~ e words: y Seasonal adjustment, forecasting performance, m u l t i v a r i a t e time s e r i e s models. Abstract There have been many studies and papers w r i t t e n about the e f f e c t s of seasonal adjustment on the r e l a t i o n s h i p s among variables. However, there has been a dearth of studies about the effects of seasonal adjustment on the problem o f forecasting. Since the development of time serles models o f t e n has f o r e c a s t i n g as a major product, i t i s essential t o study the e f f e c t s of seasonal adjustment on forecasting i n these models. I n t h i s paper, we present an a p p l i c a t i o n o f mu1 t i v a r i a t e time series forecasting applied t o f l v e economic time series, i n which we compare forecasts developed from seasonally adjusted data w i t h forecasts from seasonally not- adjusted data. o f t h i s exercise are mixed. The r e s u l t s For some forecasting st tuatlons, using not-seasonal l y adjusted data provides b e t t e r forecasts f o r most o f the variables I n t h i s study. However, i n other instances, using seasonally adjusted data provides b e t t e r forecasts for most of the variables i n t h i s study. The r e s u l t s appear t o depend on the length of the forecast period. A 1 so, i t appears t h a t the best s o l u t i o n i n some instances might be t o develop model s f o r both seasonal 1y adjusted data and not-seasonal l y adjus ted data. http://clevelandfed.org/research/workpaper/index.cfm Best available copy I. Introduction The goal of this research is to compare forecasts from two models developed for an earl let study (see Bagshaw and G a v f n C19831) to obtain an indication of whether it Is better to seasonally adjust data when developing mu1 ti variate time series models for forecasting. There have been many stud1 es Indicating that seasonally adjusting data wi 1 1 affect the relatlonshlps among the variables. Bell and Hlllmer (1984) provide references for many of these studi es . However, there has been 1 1 tt 1 e empi tical evi dence concern1 ng the effects o f seasonal adjustment o n forecastlng accuracy. The q u e s t l m of whether to use seasonally o r not-seasonally adjusted data Is especially Important f n time series analysis, because these models are often developed mainly, if not entirely, for forecastlng purposes. Even If the seasonal adjustment procedure changes the relationships a m n g variables, this will not matter for forecasting, if the new relatlonshlps provide as accurate, or even more accurate, forecasts than those developed from not-seasonal ly adjusted data. Ma-kri daki s and Hiban ( 1 979) compared forecasts o f seasonal ly and not-seasonal ly adjusted data us i ng several popular univariate forecasting methods. The1 r conclusion was that us i ng seasonal ly adjusted data provided somewhat better forecasts than using not-seasonally adjusted data. However, these results may have been influenced by their choice of constant seasonal factors in the development o f models for the not-seasonally adjusted data (see Re1 1 and HI 1 lmer t19841). Plosser (1979) forecasts five unadjusted economic time series wi th unlvari ate seasonal autoregressl ve Integrated moving average ( A R I M A ) models and the same series after seasonal adjustment with univariate nonseasonal ARIMA models. He found that the nonseasonal ARIMA models http://clevelandfed.org/research/workpaper/index.cfm Best available copy performed s u b s t a n t i a l l y b e t t e r on two s e r i e s , s l i g h t l y b e t t e r on two s e r i e s , and s i i g h t l y worse on one s e r i e s . Thus. the r e s u l t s on whether t o seasonal l y a d j u s t o r n o t when developing models f o r f o r e c a s t i n g are mixed and 1 i m i ted. I n p a r t i c u l a r , they are l i m i t e d t o u n i v a r i a t e models. The present study adds t o the information concern1 ng the advi sabi 1 i t y of seasonal adjustment before f o r e c a s t i n g by examining the forecast accuracy o f f i v e economic v a r i a b l e s i n a m u l t i v a r i a t e time s e r i e s model. This i s i n c o n t r a s t t o the abovementioned papers, which deal o n l y w i t h u n i v a r i a t e methods o f forecasting. Because t h e r e is much evidence t h a t seasonal adjustment a f f e c t s the relationships among v a r i a b l e s (see B e l l and H l l l m e r C19841). I t i s c r i t i c a l t o t e s t whether t h i s e f f e c t c a r r i e s over to forecast accuracy. If the seasonal adjustment i s such t h a t the r e l a t i o n s h i p s remain s t a b l e over time i n the seasonally a d j u s t e d data, then seasonally adjusted data might p r o v i d e b e t t e r f o r e c a s t s than not- seasonally a d j u s t e d data. However, i f the seasonal adjustment process i s n o t s t a b l e , then worse forecasts may be obtained u s i n g the seasonally adjusted data. This l a t t e r conclusion was reached by Plosser (1979) i n the u n i v a r i a t e case. 11. M u l t i v a r i a t e ARMA Tlme Serfes Models The f o l l o w i n g i s a v e r y b r i e f d e s c r i p t i o n o f m u l t i v a r i a t e ARMA time s e r i e s models; Tiao and Box (1981) p r o v i d e a more d e t a i l e d d e s c r i p t i o n . general m u l t f v a r l a t e ARMA model o f o r d e r (p,q) i s given by: The http://clevelandfed.org/research/workpaper/index.cfm Best available copy where (2) where s = t h e l e n g t h of t h e seasonal, f o r example, f o r q u a r t e r l y data, s-4, B = b a c k s h i f t o p e r a t o r (i.e., -z = v e c t o r o f k BSzl,, = t i , , , , ) , I = k x k I d e n t i t y matrlx, 9, I s !?o -a - ,&+J Is, eJ' s variables I n t h e model, and 9, ' s = k x k m a t r l xes o f unknown parameters, = k x 1 v e c t o r o f unknown parameters, and k x 1 v e c t o r o f random e r r o r s t h a t are i d e n t f c a l l y and independently d i s t r i b u t e d as N(0.C). Thus, i t i s assumed t h a t the a,, , ' s a t d i f f e r e n t p o i n t s i n time a r e 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 of gt are independent a t a g i v e n p o i n t i n tlm. The n- period- ahead f o r e c a s t s from these models a t time t ( g t ( n ) ) a r e gf ven by: http://clevelandfed.org/research/workpaper/index.cfm Best available copy where, f o r any value of t,n,m, values o f the random v a r i a b l e s C x t + n - m I x,*n-rn i m p l i e s the c o n d i t i o n a l expected a t time t . If n-m i s less than o r equal t o zero, then the condl t i o n a l expected values are the actual values of the random v a r i a b l e s and the e r r o r terms. If n-m i s greater than zero, then the expected values a r e the b e s t forecasts avai l a b l e f o r these random v a r f a b l e s and e r r o r terms a t time t . Because the e r r o r terms are uncorrelated w i t h present and p a s t i n f o r m a t i o n , the b e s t f o r e c a s t s of the e r r o r t e r m s f o r n-m g r e a t e r than z e r o a r e the! r c o n d i t i o n a l means, which are zero. The forecasts can be generated i t e r a t i v e l y w i t h t h e one-period-ahead forecasts t h a t depend o n l y on known values of the v a r i a b l e s and e r r o r terms. longer- length forecasts, The i n t u r n , depend on t h e shorter- length forecasts. . 111. Develo~rnentof Models For Forecastinq The Tlao-Box procedure was used t o e s t i m a t e m u l t i v a r i a t e time series models f o r t h e f o l l o w i n g f i v e v a r i a b l e s : the money supply ( M I ) , credit i s funds r a i s e d by t h e nonfinancial sector (NFD) i n c l u d i n g p r i v a t e and government debt, the q u a n t i t y of goods i s GNP i n constant (1972) d o l l a r s (GNP721, the p r i c e o f o u t p u t i s t h e i m p l i c i t GNP d e f l a t o r (PGNP), and the p r i c e o f c r e d i t i s the y i e l d on three- month Treasury s e c u r i t i e s (RTB3). Two models were estimated, one u s i n g seasonally adjusted data (except f o r RTB3, which i s not- seasonally adjusted) and one w i t h not- seasonally a d j u s t e d d a t a (except f o r , PGNP which i s n o t a v a i l a b l e not- seasonally adjusted). These models were estimated over the time period from the first q u a r t e r o f 1959 through the f o u r t h q u a r t e r of 1979. The r e s u l t s presented here may be s l i g h t l y biased i n f a v o r o f the seasonally adjusted model, because http://clevelandfed.org/research/workpaper/index.cfm Best available copy the latest revised seasonal adjusted data was used i n estimating these models. The seasonal adjustment procedure Is a two-sided f i 1 ter; therefore, some of the data being forecast in thi s study were used in developing seasonal adjustment factors for the data in the estimation period. To be completely comparable, we should really use the seasonal ly adjusted data that were available at the time of the forecast. In this way, the seasonal adjustment factors would not be modified by using data from the forecast period. However, as Young (1968) has indicated, the asymnetrlc f i 1 ters used t o adjust the ends o f a series are chosen wi th the objectlve o f minimizing the revision necessary after new data becomes available. seasonally data should thus be minimal. The effects of using the revtsed The model estimated uslng the not-seasonal ly adjusted data is given in table 1. The model estimated uslng seasonally adjusted data is given in table 2. From the estimation results, we would expect that the seasonally adjusted model would forecast better than the not-seasonally adjusted model for four o f the five variables (PGNP, M I , NFD, GNP72) because the within-sample estimated variances are smaller for the seasonally adjusted model than for the not-seasonal ly adjusted model . Thl s dl fference ranges from 19 percent to 81 percent. For RTB3, which is not seasonally adjusted in either model, the within-sample variance is slJghtly smaller for the not-seasonal ly adjusted data. IV. Forecastinq Results The two models were used t o forecast the levels o f the variables in three different situations:- 1 ) one-quarter ahead, 2) one-year ahead, and 3 ) a http://clevelandfed.org/research/workpaper/index.cfm Best available copy combi nation o f one- through four-quarters ahead. For one-quar ter-ahead forecasts, one-quarter ahead forecasts were generated f o r a given year. resul t l n g forecast e r r o r s were then averaged over the year. The I n t h i s manner, both t h e seasonal l y and not-seasonal l y adjusted model s were forecast! ng the same values because the seasonally adjusted data and the not-seasonal l y adjusted data must sum t o the same value for a year. ahead forecasts were averaged over the year. S i m i l a r l y , the year- That i s , forecasts were generated from the f i r s t quarter of the previous year f o r the f i r s t quarter of the forecast year, from the second quarter f o r the second quarter, etc. These forecast were then averaged. I n the combination forecasts, one-, two-, three-, and four-quarter-ahead forecasts were generated from the f o u r t h quarter o f the year p r i o r t o the forecast year and then the forecast e r r o r s were averaged f o r a given year. I n order t o have consistent forecast periods f o r the three types o f forecast 1ng, one-year-ahead forecasts were generated f o r 1980 s t a r t i n g I n the f i r s t quarter of 1979. Thus, f o r four o f the series (PGNP, M I , NFD, and RTB3) there were f i v e years of forecast e r r o r data. For GNP72, the not-seasonally adjusted data f o r 1984 were not a v a i l a b l e a t the time o f t h e study. To be consistent, the r e s u l t s f o r GNP72 f o r both models i s reported only f o r 1980 through 1983. GNP72 forecast errors. Thus, there are four years o f data f o r Consequently, there are e i t h e r f i v e o r four observations i n the analysis presented I n t h i s paper. The mean e r r o r , mean absolute e r r o r , and the r o o t mean square e r r o r (RMSE) f o r the three forecast horizons and the two models are presented I n tables 3 through 5. The f o l l o w i n g discussion i s based on the analysis of the RMSE from these forecasts. http://clevelandfed.org/research/workpaper/index.cfm Best available copy Examining the one- quarter- ahead forecasts (Presented i n t a b l e 3 1 , we see t h a t the not- seasonal l y a d j u s t e d model forecasts b e t t e r f o r three of the series (PGNP, RTB3 and GNP72). and the seasonally adjusted model forecas'ts b e t t e r f o r t h e o t h e r two s e r i e s ( M I and NFD). The differences i n the RMSE are v e r y s u b s t a n t i a l f o r several of these s e r i e s . The r a t i o s o f the not- seasonally a d j u s t e d models RMSE t o the seasonally a d j u s t e d models RMSE are 0.60 f o r PGNP, 1.16 f o r H I , 1.32 f o r NFD, 0.65 f o r RTB3, and 0.58 f o r GNP72. Given t h a t t h e w i thin- sample standard d e v i a t i o n r a t i o s were 1.09, 1.22, 1.19, 0.98, and 1.35 ( I n terms o f logarithms of PGNP, M I , NFO, RT83, and GNP72, r e s p e c t i v e l y ) , t h i s r e s u l t i s somewhat unexpected. The seasonal l y adjusted model provides a b e t t e r w i thin- sample f i t for four of the f i v e series. The f i f t h s e r i e s i s e s s e n t i a l l y t i e d , w h i l e I t provides b e t t e r f o r e c a s t f o r o n l y two s e r i e s . This appears t o imply t h a t the r e l a t f o n s h i p among seasonally adjusted d a t a may n o t be as s t a b l e as t h a t among not- seasonally adjusted data. When we examine the year-ahead forecasts (presented i n t a b l e 4 1 , we obtain different results. Here, the seasonally a d j u s t e d model forecasts four o f the s e r i e s (PGNP,MI, NFD, and GNP72) b e t t e r than the not- seasonally adjusted model. However, t h r e e o f these four have e s s e n t i a l l y the same RMSEs f o r the two models. The r a t i o s o f the corresponding RMSEs are 1.30, 1.01, 1.01, 0.59, 1.02 f o r PGNP, M I , NFD, RTB3, and GNP72, r e s p e c t i v e l y . Thus, "on average", these two models perform r o u g h l y the same f o r the f i v e s e r i e s considered as a group when f o r e c a s t i n g one year ahead. This may be r e l a t e d t o the f a c t t h a t we are f o r e c a s t f n g here one season ahead. Thus, the seasonal 1 y adjusted model may have a b u i 1t - i n advantage f o r t h i s forecast length. E i m i n i ng the combined one- t o four- quarters- ahead f o r e c a s t s (presented i n t a b l e 5). we again a r r i v e a t a d i f f e r e n t r e s u l t . Here, the not- seasonally http://clevelandfed.org/research/workpaper/index.cfm Best available copy adjusted model f o r e c a s t four of the f i v e series b e t t e r than the seasonal l y adjusted model. The corresponding RMSE r a t i o s are 0.83, 0.81, 1.01, 0.81, and 0.80, f o r PGNP, M i , NFD, RT83, and GNP72, respect ivel y . The on 1 y ser i e s f o r which the seasonally adjusted model had a smaller RMSE than the not- seasonally adjusted model f o r t h i s combination forecast was NFD, a s e r i e s constructed such t h a t ( f o r the technique used i n t h i s paper o f averaging f o r e c a s t over a year), the combination forecast r e s u l t i s the same as the one-year-ahead forecast r e s u l t. the seasonal model ' s Thus, t h l s r e ~ utl may agal n be a t t r i b u t e d t o advantage i n f o r e c a s t i n g one season ahead. I n t h i s study, we have examined whether one should seasonally a d j u s t data before developing m u l t i v a r i a t e time s e r i e s models t o provide f o r e c a s t s . The r e s u l t s are mixed; t h a t i s , performance of each model seemed t o depend on the l e n g t h o f the f o r e c a s t . For one-period-ahead forecasts, the evidence o f t h i s study suggests t h a t perhaps i t would be best t o develop models f o r both seasonal 1 l y adjusted and not- seasonal l y adjusted data. The forecasts from these models would then be evaluated t o determine which series are b e t t e r f o r e c a s t us1ng t h e seasonal l y adjusted model , and which using the not-seasonal l y adjusted model. The w i thin- sample f i t i s n o t a good d e c i d i n g f a c t o r I n t h i s choice. since the w i thin- sample f i t s i n d i c a t e d t h a t the seasonally adjusted mode1 provided a b e t t e r f i t f o r four o f the f i v e s e r i e s ( w i t h a v i r t u a l t i e f o r t h e f i f t h ) , w h i l e forecasts i n d i c a t e t h a t the not- seasonally adjusted model d i d b e t t e r f o r three of the f i v e series. http://clevelandfed.org/research/workpaper/index.cfm Best available copy .. - =&.;---. - . r t he.*/:hez-.toforecait'for. -.- .-. - - - i .' A: b i than%ie ~ period a h-w-, - . .-tri . , .? . . .. :* .-.z.- - .-.: .-.-;*- .1.; r e r u l tr iut .; .k & .~ --b -a - n & : t e - e -. .-. - - ..&-. . ...9nr-*-* --*- &- - - *-?!;--• . ,, . d ,-*-: the' - i u tr .-. l ~ & eiha&f r ~ . . . 7 : <&- 5 l y ~- ~ - * e * - y . >?. *:.I-.---serf es -for u h i i h th'c n o t - s e a ~ o n a ~ ~ ~ted d j umso ~ p i d v . i d e d - a ' g e t t e r forecast .= -- s- - . -... . . . - :; , .. r..- L , f o r one year ahead was RTB3, which I s not-seasonally adjusted. -25 Relationships among variables change more d r a s t l c a l l y if some series are seasonally adjusted and others are not, than if a l l serles are treated equally, which could explafn t h i s result. For the case where I t 1s desirable t o forecast a comb1nation o f 1engths ahead, the resul t s appear t o 1ndl cate t h a t the not- seasonally adjusted data are the best cholce, because the not- seasonally adjusted model forecast four of the f i v e serles b e t t e r . The f t f t h was a special case, which n a t u r a l l y favored using seasonally adjusted data. Because o f the small out-of-sample forecast period used here, and the small number o f serles studied. there f s obviously no way t h a t the r e s u l t s presented here can be conclusive. area i s c a l l e d f o r . Thus, more study of t h i s very important http://clevelandfed.org/research/workpaper/index.cfm Best available copy http://clevelandfed.org/research/workpaper/index.cfm Best available copy http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 3 Out-of-Sample Forecasts: One-Quarter-Ahead Forecast Errors Series Mean error - Mean absol u te error PGNP - Seasonally adjusted Not-seasonal ly adjusted Seasonally adjusted Not-seasonally adjusted NFD - Seasonal ly adjusted Not-seasonally adjusted Seasonally adjusted Not-seasonal ly adjusted Seasonally adjusted Not-seasonal 1 y adjusted *RMSE Is the root mean square error of the forecast. RMS E - http://clevelandfed.org/research/workpaper/index.cfm Best available copy L Tab1 e 4 Out-of-Sampl e Forecasts : Year-Ahead Forecast Errors J - Mean Serl es PGNP Seasonally adjusted Not-seasonal ly adjusted Seasonal 1 y a d j w t a d - - ..Not-seasonal ly adjusted Seasonally adjusted Not-seasonally adjusted Seasonally adjusted Not-seasonal 1 y adjusted Seasonally adjusted Not-seasonal 1y adjusted error - - . &,--.? .A .--.I . ; - ' Mean absolute -. error - ------- RMS E - . . http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 5 Out-of-Sample Forecasts: Combined One- to Four-Quarters Forecast Errors Mean Mean absolute error error RMS E - .0071 .0190 .0426 .0362 .0489 .0404 15.6510 11.2780 15.6510 11.2780 18.9070 15.3530 169.3800 150.5500 169.3800 1 50.5500 205.6400 207.3700 Not-seasonally adjusted -1.5847 -. 1101 2.4767 2.4485 3.1615 2.5517 Seasonally adjusted Not-seasonally adjusted 31.4150 -1.5364 49.0840 48.6520 64.7170 51.4900 Series - PGNP - Seasonally adjusted Not-seasonally adjusted Seasonally adjusted Not-seasonally adjusted NFD Seasonally adjusted Not-seasonal ly adjusted Seasonally adjusted http://clevelandfed.org/research/workpaper/index.cfm Best available copy Bagshaw, Hichael L . , and William T . Gavin. "Forecasting the Money Supply i n Time Series Models," Workinq Paper 8304, Federal Reserve Bank of Cleveland, December 1983. Bell, William R., and Steven C. Hillmer. "Issues Involved with the Seasonal Adjustment of Economic Time Series," Journal of 8usiness & Economic Statistics, vol. 2, no. 4 (October 19841, pp. 291-320. B o x , George E.P., and Gwilym M. Jenklns. Time Series Analysis: Forecastinq and Control. San Francisco: Makridakis, Spyros, and Mfchele Hibon. Holden-Day, 1976. "Accuracy of Forecasting: An Empirical Investigation," Journal of the Royal Statistical Society, Series A (General), Plosser, Charles I. vol. 142, part 2 (19791, pp. 97-125. "Short-term Forecasting and Seasonal ~ d j u s t m e n t," Journal of the American Statisticial Association, vol. 74, no. 365 (March 1979), pp. 15-24. Tiao, G.C., and G.E.P. Box. "Modeling Multiple Time Series with Applicatlons," Journal of the American Statistical Assocfatlon, vol. 76, no. 376 (December 19811, pp. 802-16. Young, Allan H. "Linear Approximations t o the Census and BLS Seasonal Adjustment Methods," Journal o f the American Statlsticial Association, vol . 63, no. 322 (June 19681, pp. 445-71 .
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