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http://clevelandfed.org/research/workpaper/index.cfm Best available copy Workina P a ~ e r8402 A REGIONAL ECONOMIC FORECASTING PROCEDURE APPLIED TO TEXAS by James G. Hoehn The author g r a t e f u l l y acknowledges many encouragi n g and useful discussions w i t h William C. Gruben. Michael L. Bagshaw, William T. Gavin, and Mark S. Sniderman o f f e r e d h e l p f u l comments on a d r a f t . Diane Mogren (Federal Reserve Bank o f C l evel and) and Frank1i n Berger (Federal Reserve Bank o f D a l l as) provided research assi stance. This manuscript was prepared by Linda Wolner and Laura Davis. Working papers o f t h e Federal Reserve Bank o f C l evel and are p r e l iminary m a t e r i a l s, c i r c u l a t e d t o stimulate discussion and c r i t i c a l comment. The views expressed herein are those o f t h e author and do n o t necessarily r e f l e c t t h e views o f t h e Federal Reserve Bank o f Cleveland o r the Board o f Governors o f the Federal Reserve System. September 1984 Federal Reserve Bank o f Cleveland http://clevelandfed.org/research/workpaper/index.cfm Best available copy Abstract A method f o r b u i l d i n g a time series regional f o r e c a s t i n g model i s proposed and implemented f o r t h e s t a t e o f Texas. The forecasting a b i l i t y o f t h i s method i s subjected t o a number o f diagnostic t e s t s and i s found t o be useful. The method places 1ittl e re1iance on economic theory, i s avai 1abl e-tv any regional economi s t w i t h know1edge of o r d i n a r y 1east squares regression analysis, and provides i n s i g h t s i n t o the regional economic process. This paper compl ements 'Some Time Series Methods of Forecasting t h e Texas Economy," by Hoehn, Gruben, and Fomby, Working Paper No. 8402, Federal Reserve Bank o f Dallas. A Regional Economic Forecasting Procedure Appl i e d t o Texas I n recent years, there has been a r a p i d pro1i f e r a t i o n o f regional model s, fostered by t h e accumulation o f regional economic data. I n t e r e s t i n these I models derives from recognition o f the disparate economic behavior o f d i f f e r e n t regions, t h e desire o f s t a t e and l o c a l governments t o make b e t t e r budget plans and design improved development p o l i c i e s , and t h e desire by business f i r m s t o improve marketing strategies. Unfortunately, t h e i n f a n t i n d u s t r y o f regional model b u i l d i n g has y e t t o prove very useful i n understanding o r forecasting regional economies. Regional model ing presents an i n t r i n s i c a l l y i n t e r e s t i n g f i e l d f o r t h e study o f a l t e r n a t i v e s t a t i s t i c a l modeling methods, p a r t l y because of t h e 1 inkages between the national and regional economies. http://clevelandfed.org/research/workpaper/index.cfm Best available copy In practice, the usefulness of regional forecasts i s l i k e l y t o depend more on seasoned judgment than on access t o formal forecasting procedures. However, formal models can aid and augment judgment, and i n the process of building them, insights into the regional economic process are provided. This paper proposes a method of building a regional forecasting model and applies the method t o construct a model f o r the s t a t e of Texas. A1 though the model b u i l t here i s subjected t o a number of somewhat sophisticated s t a t i s t i c a l t e s t s , the procedure f o r building i t requires only ordinary 1e a s t squares regressions familiar t o a l l economists. consists of two stages: The modelbuilding method f i r s t , "Granger causality" t e s t s are performed t o find variables t h a t provide significant 1eading information about the s e r i e s t o be forecast; second, these variables a r e used t o b u i l d parsimonious forecasting equations. In the second stage, some significant leading I variables are excluded t o achieve parsimony. Parsimony i s needed t o deal w i t h the problems of multicollinearity and the scarceness of degrees of freedom. In e a r l i e r exploratory work by Hoehn, Gruben, and Fomby (1984a, 1984b), i t was found t h a t potentially useful 1eading re1 ations (interactions) existed between seven Texas series and past values of (1 ) t h e i r own, ( 2 ) each other, and (3) certain national variables. A number of exploratory models designed t o assess the potential value of those relations for forecasting were recognized t o be too unparsimonious t o provide e f f i c i e n t forecasts re1 a t i ve t o univariate methods. Among these probing e f f o r t s were a closed-regional model t h a t was essentially a seven-vector autoregression, a "trickle-down" model i n which five national variables were "driving variables" f o r each Texas variable (regional interactions were excl uded), and "Bayesian vector autoregressive" models, such as those advocated by Anderson (1979). The f i r s t two reflected http://clevelandfed.org/research/workpaper/index.cfm Best available copy l i t t l e e f f o r t t o deal w i t h t h e problem o f parsimony and, hence, d i d n o t represent actual forecasting procedures. The t h i r d has been advocated by L i tterman and associates [ f o r example, L i tterman (1979, 1982) ; Doan, Litterman, and Sims (1983)l as, i n effect, a b e t t e r way t o deal w i t h the m u l t i c o l l i n e a r i t y and degrees-of-freedom issues and, therefore, a superior a1t e r n a t i ve t o t h e p r i n c i p l e o f parsimony. The model b u i 1d i n g strategy pursued here empl oys o n l y t h e s i g n i f i c a n t l e a d i n g r e l a t i o n s i n t h e data, and i n a parsimonious way. For a sample o f t e n ex ante forecasts, t h e model b u i l t here provided c o n s i s t e n t and sometimes s i g n i f i c a n t improvements over t h e u n i v a r i a t e methods. These r e s u l t s need t o be i n t e r p r e t e d w i t h some caution, p a r t i c u l a r l y i n view o f t h e smallness o f the sample o f forecasts. Nevertheless, t h e resul t s are o f i n t e r e s t because s i g n i f i c a n t improvement over t h e u n i v a r i a t e methods i s n o t o f t e n achieved by e x i s t i n g mu1t i v a r i a t e models, i n c l u d i n g s t r u c t u r a l econometric model s. 1 Because t h e modelbuilding strategy i s reasonably s t r a i g h t f o r w a r d and easy t o implement, i t may serve as a useful procedure i n forecasting o f o t h e r regional economies o r i n o t h e r applications. This paper i s intended t o complement and extend Hoehn, Gruben, and Fomby (1984a). A number o f r e s u l t s and concepts i n t h a t paper are used here. I d e n t i f i c a t i o n o f a Parsimonious M u l t i v a r i a t e Autoregressive Model The seven Texas variables t o be forecast are (1 ) t h e Texas I n d u s t r i a l Production Index ( T I P I ), (2) the Dal 1as- Fort Worth Consumer P r i c e Index (CPIDFW), ( 3 ) employment according t o t h e survey o f business es ,abl ishments (PAYROLL), (4) employment according t o the household survey (TEMP), ( 5 ) the l a b o r f o r c e (TLF), ( 6 ) personal income (TPY), and ( 7 ) r e t a i l sales (TRET). http://clevelandfed.org/research/workpaper/index.cfm Best available copy These are the same as in Hoehn, Gruben, and Fomby (1984a), except that here personal income and retail sales have not been deflated. , In Hoehn, Gruben, and Fomby (1984a), they were deflated by CPIDFW. The data series used began with 1969:IQ and ended in 1983:IIQ. The sample period for model construction ended in 1980:IVQ, preserving ten quarters for out-of-sample simulation. The size of the samples--both the within-sample period of model construction and the out-of-sample period of forecast performance evaluation--were rather small and require some justification. The lengths of available data series vary, but all were available from 1969. Using the entire length of an available series where possible in an equation might have given the univariate equations an advantage over the multivariate equations if the structure was stable over time. This is an advantage of autoregressive integrated moving averages (ARIMAs) that forecasters would want to exploit. Truncating the series to begin in 1969 preserves, in a sense, a "level playing field" for comparing forecasting accuracy of the two kinds of models. A better justification for beginning with 1969 is the problem of structural change. Such change, due either to real changes in the regional economy or to changes in data collection and assimilation, make data in the distant past less relevant. Hol t and Olson (1982) examined the improvement in forecasting accuracy from exponentially weighting data used to estimate a transfer function model for Texas personal income. This procedure involved weighting the observations k periods in the past by a factor of Ak For quarterly . data, they found that a x value of around 0.95, depending on the forecast horizon, produced the best forecasting model. After ten years, the weight would be about 0.13 (that is, 0.95 40) of that on the current observation. In addition, Holt and Olsen found that merely reducing the sample length from 18 to 13 years was sufficient to deliver most of the forecasting improvement http://clevelandfed.org/research/workpaper/index.cfm Best available copy relative t o the model estimated over the f u l l sample and without weighting. In the estimation of the present model, there are 12 years of data t o estimate the i n i t i a l model, and t h a t period i s effectively expanded up t o 14 years i n updating the estimates during the forecasting period. While the results of Holt and Olsen suggest t h a t e a r l i e r data may be of s l i g h t value, data more recent than 1980 would s t i l l help. Other than u s i n g the l a t e r data i n updating the coefficient estimates, we cannot extend the sample forward without reducing the period of forecast performance eval uati on. The forecasting period might be too small f o r very powerful evaluation of forecasts, as 1a t e r r e s u l t s will show. B u t lengthening the forecasting period > would reduce the sample f o r model construction, which would render the primary objective of uncovering useful forecasting and structural re1 ationships more d i f f i c u l t t o achieve. A1 1 Texas and national variables are transformed t o natural logarithms and differenced once t o achieve stationari ty. evaluation are 1ogari thmic 1eve1 s employed. Only i n forecast performance Performance of forecasting methods i s evaluated by root means of squared e r r o r s (RMSEs), where the e r r o r i s the forecast (logarithmic) level of the s e r i e s minus the actual (logarithmic) level of the series. Although forecast horizons extend a s f a r as ten quarters ahead, emphasis i s placed on the accuracy of one-quarter-ahead t o six-quarter-ahead forecasts. The model was used t o generate a sample of ten one-quarter-ahead forecasts, nine two-quarter-ahead forecasts, and so on. The form of the model i s t h a t of a mu1 t i v a r i a t e autoregression (MAR): 2 http://clevelandfed.org/research/workpaper/index.cfm Best available copy 3 where yt i s t h e (7x1) v e c t o r o f l o g a r i t h m i c f i r s t d i f f e r e n c e s o f Texas variables, 2 t i s a ( k x l ) vector o f logarithmic f i r s t differences o f national variables, 2 et i s a (7x1) v e c t o r o f disturbances, 3 ut i s a ( k x l ) v e c t o r o f disturbances, where L i s t h e l a g operator ( L k zt = z ~ - ~ ) , 0 otherwise, 0 otherwise, and i f k = O 0 otherwise. The model can a l s o be represented as a s e t o f equations, one f o r each o f t h e seven y - v a r i a b l e s p l u s one f o r each o f t h e k x - v a r i a b l e s . r e p r e s e n t a t i o n w i l l be u s e f u l below. Such a http://clevelandfed.org/research/workpaper/index.cfm Best available copy An important feature o f t h e model i s t h a t the x- vector i s exogenous w i t h respect t o t h e y- vector. While national v a r i a b l e s may have i n t e r a c t i o n s , and while they i n f l u e n c e the regional variables, they are themselves assumed t o be unaffected by t h e regional variables. A s i g n i f i c a n t l i m i t a t i o n o f t h e model i s t h a t the disturbances a r e assumed t o be non- autocorrelated. This assumption can be made t o be reasonably p l a u s i b l e through s u f f i c i e n t l y l a r g e powers o f L i n the bi c o e f f i c i e n t matrices. A more f l e x i b l e model b u i 1d i n g s t r a t e g y woul d a11ow t h e disturbance vectors t o be moving average processes, as i n t h e m u l t i v a r i a t e ARIMA models. The added f l e x i b i l i t y can reduce t h e number of parameters needed t o adequately characterize t h e data, y e t i d e n t i f i c a t i o n o f mu1t i v a r i a t e ARIMAs i s q u i t e problematic. A1 so, t h e more r e s t r i c t i v e MAR form imposed here w i l l be more transparent t o most regional economists. The ordinary l e a s t squares estimation technique used i s a l s o much more famil i a r . 3 Hence, t h e model- b u i l d i n g procedure w i l l be easy f o r others t o i m i t a t e . Model i d e n t i f i c a t i o n e n t a i l s t h e choice o f t h e v a r i a b l e s t o be included i n a x and t h e imposition o f appropriate zero r e s t r i c t i o n s i n t h e b. .(L) matrices !J o f polynomials i n t h e l a g operator L. choice o f l a g lengths. The l a t t e r e s s e n t i a l l y represents a The method proposed here f o r i d e n t i f i c a t i o n proceeds i n two stages. F i r s t , "Granger causal ity" t e s t s were performed t o f i n d s i g n i f i c a n t 1eading re1a t i o n s h i ps. follows: l e t yit Formal l y, these causal ity t e s t s were performed as A be the i t h element o f yt. For each i = 1, 2, . . ., 7, run the f o l 1owing regressions and determine t h e i r sums o f squardd e r r o r s : http://clevelandfed.org/research/workpaper/index.cfm Best available copy + b3jJ'j,t-l + b4jyj,t-I + e2it for all j f; i. where xlt= hlnLEADt and LEAD= U.S. Index of Leading Indicators. + d5k 'j,t-l + d6k 'j,t-2 + e4it for 15 kil4, Expression (a) represents a single regression, a second-order univari ate autoregression. For example, for i = 1, the growth rate of TIP1 is regressed on its first two own-lags. The results of regression (a) could be used to http://clevelandfed.org/research/workpaper/index.cfm Best available copy - 9 - establ i s h t h e p o t e n t i a l value of p a s t own-lags i n forecasting, when compared w i t h a random walk model. However, o t h e r w e l l developed procedures f o r assessing t h e importance o f autocorrelation, i n v o l v i n g a u t o c o r r e l a t i o n functions and the f i t t i n g and t e s t i n g o f ARIMA models, were given primary focus. Expression (b) represents s i x d i f f e r e n t regressions. Pursuing t h e example, the growth r a t e o f TIP1 i s regressed n o t only on i t s f i r s t two own-lags, b u t a l s o on two lagged growth r a t e s of CPIDFW; then T I P I ' s growth r a t e i s regressed on two own-lags p l u s two lagged growth r a t e s o f PAYROLL; and so on. Results from (a) and ( b ) can be used t o c o n s t r u c t b i v a r i a t e " c a u s a l i t y " t e s t s among t h e regional variables by using t h e F - s t a t i s t i c t o t e s t t h e n u l l hypothesis t h a t b3j=b4j=0. I n three o f the s i x such t e s t s i n v o l v i n g TIP1 as t h e 1eft- hand- side variable, t h e nu11 hypothesis was r e j e c t e d a t the 0.05 l e v e l of significance. These three cases involved growth r a t e s of TEMP, PAYROLL, and TLF as right- hand- side variables. I n addition t o t h e F - t e s t o r " c a u s a l i t y test," t h e standard e r r o r o f each o f t h e regression equations i n (b) was compared w i t h t h a t o f equation (a). The reduction o r increase i n the standard e r r o r from i n c l u s i o n o f a variable, defined here as t h e " information gain," provides a q u a n t i t a t i v e assessment o f t h e p o t e n t i a l usefulness o f t h e v a r i a b l e i n forecasting. For example, t h e standard e r r o r o f t h e equation f o r TIP1 was lowered by about 10 percent by i n c l u d i n g TEMP as a right- hand- side variable, by about 7 percent by i n c l u d i n g PAYROLL, and by about 6 percent by i n c l u d i n g TLF. Regression ( c ) employs two lagged growth r a t e s o f t h e U.S. index o f l e a d i n g i n d i c a t o r s as right- hand- side variables, i n a d d i t i o n t o two own-lags. Together ( c ) and ( a ) can be used t o construct t e s t s o f " c a u s a l i t y " running - http://clevelandfed.org/research/workpaper/index.cfm Best available copy - 10 from t h e l e a d i n g i n d i c a t o r index t o t h e regional variables. For example, t h e growth r a t e o f TIP1 was found t o be s i g n i f i c a n t l y r e l a t e d t o p a s t growth r a t e s i n t h e l e a d i n g index. I n addition, i t was found t h a t i n c l u s i o n o f t h e l e a d i n g index reduced t h e standard e r r o r by about 12 percent. I n ( d ), the regional variable, yi t, i s regressed on two own-1ags, two lagged growth r a t e s o f the l e a d i n g index, and two lagged growth r a t e s o f one o f t h i r t e e n other national variables. A c a u s a l i t y t e s t f o r each o f these 13 o t h e r variables i s performed using t h e r e s u l t s o f ( d ) and (c), and the i n f o r m a t i o n gain (reduction i n standard e r r o r ) i s assessed. I n t h e example o f TIPI, i t was found that, once the l e a d i n g index was included, none o f the o t h e r 13 national variables provided s i g n i f i c a n t information gain ( t h e hypothesis t h a t the dSj and dGj were zero could n o t be r e j e c t e d ) . The b a t t e r y o f c a u s a l i t y t e s t s j u s t described was repeated f o r each o f t h e regional variables and reported i n Hoehn, Gruben, and Fomby (1984a). These r e s u l t s c o n s t i t u t e t h e f i r s t stage o f model i d e n t i f i c a t i o n and provide candidates f o r i n c l u s i o n i n the equations o f the MAR. I n the second stage, a search was undertaken t o determine t h e b e s t s p e c i f i c a t i o n o f each equation. I n each equation, two lagged growth r a t e s o f each o f the candidate right- hand- side variables were t r i e d a1 1 a t once, then i n more l i m i t e d combinations. specification: Two c r i t e r i a were used t o s e l e c t t h e f i n a l low standard e r r o r of t h e equation and parsimony. Judgment was necessary, since the s p e c i f i c a t i o n t h a t met one o f the c r i t e r i a d i d n o t always meet the other. L i k e most o t h e r i d e n t i f i c a t i o n methods f o r time series model s, the model i d e n t i f i c a t i o n procedure i s n e i t h e r d e t e r m i n i s t i c nor replicable. For example, the i n i t i a l unparsimonious treatment o f the TIP1 equation included two lags each o f TIP1 i t s e l f , a l l three Texas l a b o r series, http://clevelandfed.org/research/workpaper/index.cfm Best available copy - 11 - and the U.S. leading index. That equation's standard error was found to be reduced by excluding the labor force and establishment-survey employment from the equation, and including only the first lag of the leading index and TIP1 itself. Some other combinations were tried. The objective was to find an equation with only a few p-arameters and a relatively low standard error. Specification of the Model The specification finally chosen for the first equation in the parsimonious MAR is: see = ~2 = - .01308 Q(18) = 11.9 I = 24.7 Values in parentheses are standard errors of parameter estimates. The .44 standard error of the equation (SEE) is 0.01308. This standard error can be compared with the standard deviation of AlnTIPI, the latter essentially representing the standard error of the random walk model. This comparison is formalized by the I-statistic: 1 - standard I error of MAR equation x 100 standard deviation http://clevelandfed.org/research/workpaper/index.cfm Best available copy Because the standard error is 24.7 percent below the standard deviation of AlnTIPI, we say that the information gain associated with the equation, I, is 24.7. The Q-statistic reported is the sample size times the sum of squared autocorrelations in the residuals, for the first 18 lags. Prior to fitting equations for (nominal) personal income and retail sales, it was necessary to perform the sets of "causality tests," as these were performed in Hoehn, Gruben, and Fomby (1984a) only for their deflated counterparts. The results indicated that lagged growth rates in CPIDFW and TRET were promising candidates for inclusion in the equation for TPY, and that TEMP and TPY belonged in the equation for TRET. There was also evidence that the (national) finished goods producer price index was a significant aid to predicting TPY, but that price index was eventually excluded in the process of choosing a parsimonious model. The other equations for Texas variables were derived in a similar manner. Only once was a right-hand-side variable excluded on a priori grounds. The U.S. Consumer Price Index and the GNP deflator were excluded from the equation for the Texas labor force, even though they significantly improved the fit (lowered the standard error) of the equation. derived from purely statistical criteria. Otherwise, all equations were It can be regarded as a favorable result that the equations arrived at, listed below, appear quite reasonable in view of available rough prior notions about the regional economy. see = .007027 Q(18) = 12.5 http://clevelandfed.org/research/workpaper/index.cfm Best available copy see = .004174 - Q(18) = ~2 = .55 see = .007557 - I = 32.5 Q(18) = 16.7 I = 5.8 ~2 = -14 see = .006417 i2 = .02 see = .009878 Q(18) = 13.1 I see = .01616 ~2 = .30 = 0.4 Q(18) = 18.7 I = 18.4 i 2 = .36 - 7.9 ~ ( 1 8 )= 14.7 I = 17.3 http://clevelandfed.org/research/workpaper/index.cfm Best available copy These equations i n c l u d e f o u r n a t i o n a l v a r i a b l e s : t h e Index o f Leading Economic I n d i c a t o r s (LEAD), t h e Index of Roughly Coincident I n d i c a t o r s (COIN), t h e Producers P r i c e Index f o r A l l Finished Goods (PPI), and r a t e (RFF). t h e f e d e r a l funds I n order t o c o n s t r u c t f o r e c a s t s f o r more than one q u a r t e r ahead, t h e model must be able t o generate forecasts f o r those n a t i o n a l v a r i a b l e s . This i s accomplished by appending t o t h e MAR t h e f o l l o w i n g equations, which t r e a t t h e n a t i o n a l v a r i a b l e s as block exogenous: see = .01681 i2 Q(18) = 4.5 = .64 see = .0416 Q(18) = 22.6 - ~ 2 = .52 see = .009672 - .Q(18) = 15.1 ~ 2 = .38 see = .I695 -~ 2 = .ll Q(18) = 12.0 http://clevelandfed.org/research/workpaper/index.cfm Best available copy These four equations were built using criteria similar to that of the Candidates for right-hand-side variables were confined to earlier seven. lagged values of the four national variables themselves, and simple equations were chosen with low standard errors. Further improvements might be made by searching a larger set of national variables for promising right-hand-side variables for these equations. The I-measures of information gain suggest substantial gains may be available from the use of the model relative to a naive model. Table 1 compares the model's standard errors with those of three alternatives: (i ) The random walk model Alnyt (i i ) v + et The second order autoregression, or ARIMA(2,1,0) Alnyt (iii) = = v + $l Alnyt-l + $2 Aln~t-2+ et ARIMAs identified by the methods of Box and Jenkins, or ARIMA(p,l,q) Alnyt-l = v + Alnyt-, + . . . + 4P Alnyt-P The identified and estimated Box-Jenkins ARIMAs (i i i ) are described in appendix A. The I-measure reported for equations (1) through (7) above , epresents the reduction from the first to the fourth column of table 1. The fourth column can be compared with the second and third columns to determine the degree of improvement relative to univariate equations. Such a comparison indicates quite substantial improvement in the equations for personal income, industrial http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 1 Comparison o f Standard E r r o r s o f Equations (1 1 Right-handside v a r i abl es Random wal k (21 ARIMA(2,1,0) (3 1 Box-Jenkins (4 Model CPIDFW .01093 .00768 .00769 .00703 PAYROLL .00618 .00442 .00432 .00417 TEMP .00802 .00825 .00802 .00756 TLF .00644 .00641 .00644 .00642 TPY .01211 .01200 .01176 .00988 TRET .01953 .01919 .01809 .01616 http://clevelandfed.org/research/workpaper/index.cfm Best available copy production, and r e t a i l sales; s i g n i f i c a n t b u t l e s s e r improvement f o r consumer p r i c e s and household-survey employment; and no gain f o r t h e l a b o r force. (The l a b o r f o r c e i s exogenous i n the model ' s equation, a f i r s t - o r d e r u n i v a r i a t e autoregression. It should be noted t h a t t h e procedure f o r s e l e c t i n g t h e model ensured t h a t i t would have favorable comparisons against u n i v a r i a t e equations i n terms o f standard errors. A more important issue i s whether t h e m u l t i v a r i a t e model provides b e t t e r out-of-sample forecasts. We should n o t expect a selected model ' s degree of s u p e r i o r i t y re1a t i v e t o ARIMAs t o h o l d up out-of-sampl e. Nevertheless , unless a model provides b e t t e r w i thin-sampl e performance, i t i s u n l i k e l y t o do as w e l l as ARIMAs o u t o f t h e sample. Out-of-Sample S t a b i l i t y o f t h e Model The c o e f f i c i e n t s o f t h e model were re- estimated each quarter during t h e post-sampl e forecasting period. As one might expect, t h e c o e f f i c i e n t s did, i n some cases, change s u b s t a n t i a l l y as new data were incorporated i n estimation. However, t h e equations d i d n o t d i s p l a y marked i n s t a b i l i t y . Indeed, t h e range o f v a r i a t i o n i n t h e c o e f f i c i e n t s over time seems r a t h e r modest i n view o f t h e severe economic c o n d i t i o n s during the post-sampl e period. Tab1e 2 displays t h e i n i t i a l , lowest, highest, and f i n a l values o f t h e c o e f f i c i e n t s f o r each equation. The model as f i n a l l y estimated using data through 1983: IIQ is presented i n appendix B. Somewhat s u r p r i s i n g l y , the measures o f f i t o f t h e equations and the r a t i o s ( t - s t a t i s t i c s ) o f estimated c o e f f i c i e n t s t o t h e i r -2 2 standard e r r o r s d i d n o t d e t e r i o r a t e over time. R (R corrected f o r degrees o f freedom) rose f o r f i v e o f the seven equations f o r Texas v a r i a b l e s and f e l l f o r two. (x2 d i d f a l l f o r three o f the f o u r n a t i o n a l v a r i a b l e . http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 2 Range o f C o e f f i c i e n t s as Estimation Period Extended Left- hand- side variabl e R i ght-hand-side v a r i abl e Initial Low - PAY ROLLt-1 .63 .62 .74 . .74 TEMPt-1 .19 .16 .20 .16 High Final http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 2 - Continued Range o f C o e f f i c i e n t s as Estimation Period Extended Left-hand-si de variable Right-hand-side v a r i abl e I n i ti a1 Low - High Final http://clevelandfed.org/research/workpaper/index.cfm Best available copy equations.) The standard e r r o r s o f f o u r o f t h e equations f o r Texas v a r i a b l e s rose, and t h r e e f e l l . However, the standard e r r o r s generally decreased r e l a t i v e t o t h e standard d e v i a t i o n o f growth rates: f i v e o f seven equations. the I - s t a t i s t i c rose i n There are no compelling reasons f o r a l t e r i n g t h e model from i t s o r i g i n a l s p e c i f i c a t i o n , a1though de novo analysis might l e a d t o some improvement. inadequacy. The Box-Pierce s t a t i s t i c s do n o t i n d i c a t e any serious model (The equation f o r t h e U.S. c o i n c i d e n t index, as f i n a l l y estimated, does d i s p l a y marginally s i g n i f i c a n t autocorrel a t i o n o f errors, however. ) The re1a t i o n between c o e f f i c i e n t s t a b i 1ity and s t a b i 1ity o f t h e model ' s forecasting p r o p e r t i e s i s n o t very precise. Nevertheless, c o e f f i c i e n t i n s t a b i lity would be a negative i n d i c a t i o n f o r a model. The reasonable s t a b i l i t y o f t h e model reinforces t h e n o t i o n t h a t the model i s f a i r l y robust and t h a t t h e underlying s t r u c t u r e o f the regional economy d i d n o t change r a d i c a l l y d u r i n g the weakness o f the e a r l y 1980s. Out-of-Sample Performance o f t h e Model The RMSE serves as t h e absolute measure o f forecast accuracy. It i s s t r i c t l y appropriate i f t h e costs o f f o r e c a s t e r r o r s a r e quadratic i n the errors. This i s a reasonable assumption, i s a n a l y t i c a l l y most t r a c t a b l e , and d i r e c t l y r e l a t e s t o the l e a s t squares estimation procedure [Granger and Newbold (1977, p. 280)l. The performance o f the model i s evaluated by r e l a t i v e e f f i c i e n c y and conditional e f f i c i e n c y . Re1a t i v e e f f i c i e n c y i s defined here as the a b i l i t y o f the model t o produce forecasts w i t h 1ower RMSEs than u n i v a r i a t e ARIMAs. Conditional e f f i c i e n c y , as defined by Granger and Newbold (1 977, p. 283), i s a somewhat stronger c r i t e r i o n . I f a model produces forecasts w i t h RMSEs t h a t cannot be s i g n i f i c a n t l y reduced by combining i t s http://clevelandfed.org/research/workpaper/index.cfm Best available copy forecasts w i t h u n i v a r i a t e Box-Jenkins forecasts, then i t i s c o n d i t i o n a l l y e f f i c i e n t w i t h respect t o t h e Box-Jenkins forecasts. I n t h i s section we examine r e l a t i v e e f f i c i e n c y ; i n t h e next section, c o n d i t i o n a l e f f i c i e n c y . I n examining re1a t i v e e f f i c i e n c y, t h e two uni v a r i a t e benchmark model s were again employed--ARIMAs i d e n t i f i e d by the methods o f Box and Jenkins and a r b i t r a r i l y s p e c i f i e d ARIMA(2,1,0) times. equations. Each model was updated t e n J u s t as f o r the MAR, i n i t i a l estimation o f ARIMAs used t h e sample from 1969:IQ t o 1980:IVQ; the second estimation used the sample from 1969:IQ t o 1981:IQ, and so f o r t h , u n t i l a t e n t h estimation used t h e sample from 1969:IQ t o 1983:IQ. A f t e r each estimation, forecasts were generated f o r t h e seven Texas variables f o r t h e quarter f o l l o w i n g t h e end o f t h e estimation sample u n t i l 1983:IIQ. Hence, the f i r s t forecast provided one f o r e c a s t f o r each horizon from one t o t e n quarters; the second produced one f o r e c a s t f o r each horizon up t o nine quarters, and so f o r t h . RMSEs f o r the 1981:IQ t o 1983: IIQ out-of-sample f o r e c a s t p e r i o d are presented i n t a b l e 3 f o r t h e MAR, i n t a b l e 4 f o r t h e Box-Jenkins ARIMAs, and i n t a b l e 5 f o r t h e a r b i t r a r i l y s p e c i f i e d ARIMA(Z,l,O)s. Table 6 presents f o r e c a s t accuracy rankings f o r t h e MAR, Box-Jenkins ARIMAs and an unweighted average o f the two t o be discussed i n the next section. The model performed r a t h e r we1 1 when compared w i t h t h e Box-Jenkins ARIMAs, outperforming them i n 30 o f t h e 42 possible comparisons, and i n 20 o f 21 one- t o three-step-ahead f o r e c a s t comparisons. The MAR a1so generally performed we1 1 r e l a t i v e t o t h e a r b i t r a r i l y s p e c i f i e d ARIMA(2,1,0) equations. For the one-, two-, and http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 3 Root Mean Square Errors f o r Parsimonious MAR Step TIP1 - CPIDFW PAYROLL TEMP TLF - TPY TRET http://clevelandfed.org/research/workpaper/index.cfm Best available copy Step TIP1 CPIDFW PAYROLL TEMP TLF - TPY TRET http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 5 Root Mean Square E r r o r s f o r ARIMA(Z,l,O)s Step TIP1 - CPIDFW PAYROLL TEMP TLF - TPY - TRET - http://clevelandfed.org/research/workpaper/index.cfm Best available copy - 25 - Table 6 Ranking o f ~ b r e c a s tAccuracya Step a. TIP1 - CPIDFW PAYROLL TEMP TLF - TPY TRET 1 MCA MCA ACM MCA MC A CMA MCA 2 MCA MCA MCA MCA CMA MCA MCA 3 MCA MCA MCA MCA MCA MCA MCA 4 ACM MCA MCA MCA ACM MCA ACM 5 ACM MCA MCA ACM CAM MCA ACM 6 ACM MCA MCA ACM MCA ACM ACM M = Model, A = Box-Jenkins ARIMA, C = Average Sum o f RMSEs f o r 7 Texas Variables Step 1 Model .0840 Box-Jenki ns ARIMA .0960 Average .0847 http://clevelandfed.org/research/workpaper/index.cfm Best available copy three-quarter-ahead forecasts, t h e model very c l e a r l y outperformed Box-Jenkins ARIMAs. The RMSE f o r each o f these t h r e e horizons and each o f t h e seven Texas variables was lower f o r t h e MAR i n every case, except f o r t h e one-quarter-ahead forecasts o f PAYROLL where t h e d i f f e r e n c e was very s l i g h t . The four-, five- , and six-quarter-ahead forecasts presented a mixed p i c t u r e . The model ' s sum o f RMSEs across the seven v a r i a b l e s was smaller f o r t h e f o u r and five-step- ahead forecasts, b u t very s l i g h t l y higher f o r the six-step-ahead forecasts ( f o r which we had o n l y a sample o f f i v e f o r each variable). The model outperformed ARIMAs f o r f o u r of the seven variables i n t h e four-step-ahead forecasts, b u t f o r o n l y three v a r i a b l e s e i n the f i v e - and six-step-ahead forecasts. One might consider t h e s i z e o f the e r r o r s t o be q u i t e l a r g e i n economic terms, e s p e c i a l l y a t t h e longer forecast horizons. This may be a r e s u l t o f the unusual weakness o f t h e regional economy during the period. With t h e exception o f t h e l a b o r force, most of t h e f o r e c a s t e r r o r s were negative (actual values t y p i c a l l y f e l l below predicted values), and the e r r o r s over 1onger f o r e c a s t horizons tended t o accumulate as t h e recession continued. This accumulation o f negative e r r o r occurred f o r both the ARIMA equations and t h e model. As one would expect, the RMSEs of one-period-ahead forecasts were generally l a r g e r than within-sample standard e r r o r s , both f o r the MAR and f o r the ARIMAs. I n some cases, t h e d i f f e r e n c e was q u i t e large. For example, t h e RMSE f o r TIPI, using t h e model, was 61 percent higher than t h e within-sample standard e r r o r , and f o r the ARIMA was 56 percent higher. This could r e s u l t from t h e unusual turbulence o f t h e regional economy i n t h e simulation period, a changing economic structure, o r model inadequacy. http://clevelandfed.org/research/workpaper/index.cfm Best available copy - 27 - Although one m i g h t have expected the ARIMAs t o be more robust because of t h e i r r e l a t i v e parsimony, the MAR displayed no greater increase i n RMSEs r e l a t i v e t o standard errors. Indeed, while the ARIMA1s RMSEs were above standard errors f o r a l l variables except TEMP, the MAR'S RMSEs were lower than standard errors f o r both TEMP and CPIDFW, and stayed the same f o r TLF. The average increase in RMSE r e l a t i v e t o standard errors across the seven Texas variables was 26 percent using the model and 28 percent using the ARIMAs. In view of the smallness of the out-of-sample forecasting period, i t i s natural t o ask how significant, i n a s t a t i s t i c a l sense, the evidence i s t h a t the model can outperform ARIMAs. A t e s t designed t o detect "causality 1 ' as described i n Ashley, Granger, and Schmalensee (1980) can be adapted f o r t h i s purpose. Essentially, the t e s t involves regressing dt on st, where and e t and e: are forecast errors f o r the ARIMA and model forecasts, respectively. The regression i s of the form: I f the mean square e r r o r of the MAR i s 1ower than t h a t of the ARIMAs, either or 6 or both must be nonzero. The null hypothesis, t h a t the model does not provide better forecasts, i s rejected i f the F - s t a t i s t i c f o r and i s sufficiently large, and i f estimates of and have appropriate http://clevelandfed.org/research/workpaper/index.cfm Best available copy signs. One ambiguity of the test involves the signs of coefficients; it is in essence a four-tailed test. The true significance level of the F-statistic is something less than one-half that found in tables of the F-distribution, if estimated regression coefficients are of the correct signs. The correct sign for 6 is always positive. The correct sign for a is negative if the mean errors are negative, as they are for all variables and horizons, except for TLF, whose mean errors are positive for all horizons. The F-statistics of the Ashley-Granger-Schmalensee tests are displayed in table 7. The F-statistics can be judged against critical values from distribution tables. For one-step-ahead forecasts, the relevant distribution has 2 numerator and 8 denominator degrees of freedom; for two-step-ahead, 2 numerator and 7 denominator degrees of freedom; etc. Halving the significance level from the F-distribution tables, and assuming correct signs of coefficients, an F-statistic in table 7 is significant at the 0.05 level (or lower) if above 3.1 1 for one-step-ahead forecasts, 3.26 for two steps ahead, 3.46 for three steps ahead, and 3.78 for four steps ahead. The results suggest significant improvement in MAR forecasts of consumer prices and personal income beyond one quarter ahead, compared with ARIMA forecasts, and significant improvement also in one-quarter forecasts of household-survey employment. None of the other improvements is significant, using the test criterion. However, the test has low power due to the smallness of the sample. The results for Texas personal income are considerably stronger than for an alternative forecasting equation studied by Ashley (1980 and 1983). He reports some evidence, that the growth rate in forecasts of personal income http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 7 Statistics for Significance of Model-Forecasting Efficiency (Ash1ey-Granger-Schmal ensee Test) Forecast horizon Forecast variable 1 -Step 2-Step TIP1 1.28a .20a CPIDFW 1.26 PAYROLL TEMP .12C 5.57 3-Step .06 3.76 3.34 .35 1.31b 1.40 .42b .32 2.18a TLF .94a TPY .29 3.64 TR ET 1.31 3.03 3.27 .36a 4-Step . 83C 3.43 .81b .16 .9lb 4.57 1.14C a. a was of wrong sign, but not significantly different from zero. b. B was of wrong sign, but not significantly different from zero. c. Both a and B were of wrong sign, but not significantly different from zero. http://clevelandfed.org/research/workpaper/index.cfm Best available copy could be improved s l i g h t l y r e l a t i v e t o an ARIMA(4,1,0) by using a b i v a r i a t e model w i t h three own-lags and one c u r r e n t growth r a t e i n n a t i o n a l GNP. The improvement was s l i g h t , was measured over a s i n g l e f o r e c a s t o f one t o e i g h t quarters, and depended on h i gh-qua1 ity, judgmental l y adjusted s t r u c t u r a l econometric forecasts o f GNP. i n f e r i o r t o the ARIMA(4,1,0) The b i v a r i a t e equation produced forecasts when GNP forecasts were generated using a s t r i c t l y formal method (a f i r s t - o r d e r autoregression). MAR reported here are considerably stronger. The r e s u l t s f o r the The forecasts f o r t h e growth r a t e f o r Texas personal income were considerably b e t t e r than those o f ARIMAs, and s i g n i f i c a n t l y so. Furthermore, our model does n o t r e q u i r e as an i n p u t any judgmental forecasts o f exogenous variables. Combination Forecasts Another approach t o improving forecast accuracy i s t h a t o f combining forecasts o f d i f f e r e n t methods. Given the two methods we have constructed, i t i s easy t o combine them by, f o r example, averaging them. forecasts are shown i n t a b l e 8. RMSEs o f t h e average The simple average was never 1ess accurate than both the model and t h e ARIMA, f o r any horizon o r variable. It always came i n a t l e a s t second among t h e three possible methods, and i n f i v e o f forty- two cases, i t came i n f i r s t . Furthermore, the average forecast tended o v e r a l l t o be nearly as accurate as t h e model f o r one-period-ahead forecasts and those a t the longer horizons as well. The sum o f t h e seven v a r i a b l e s ' RMSEs f o r t h e combined forecast, as shown a t t h e bottom o f t a b l e 6, was a c t u a l l y lower than t h a t o f t h e model a t the six-period-ahead horizon, and was always lower than t h a t o f ARIMAs. http://clevelandfed.org/research/workpaper/index.cfm Best available copy - 31 - Table 8 Root Mean Square Errors f o r Average Forecasts Step TIP1 CPIDFW PAYROLL TEMP TLF TPY TRET http://clevelandfed.org/research/workpaper/index.cfm Best available copy The simple average forecast need n o t be t h e b e s t weighting scheme. ' It seems that, because t h e model f o r e c a s t general l y outperformed t h e average forecast, more weight should be given t o t h e model than the ARIMA. But t h e weights do n o t have t o be t h e same f o r a l l v a r i a b l e s o r f o r e c a s t horizons. In an attempt t o determine appropriate weights e m p i r i c a l l y , the RMSE-minimizing weights were calculated, subject t o the c o n s t r a i n t t h a t they summed t o one? I n t a b l e 9, these weights are presented f o r one- and four- quarter forecast horizons. The r e s u l t s o f t h i s exercise are n o t very encouraging. In only three cases o u t o f fourteen are the weights w i t h i n t h e i n t e r v a l from zero t o unity. The sample i s probably too small. Probably t h e b e s t conclusion t o be drawn from t h e study o f combinations, simple and weighted, i s t h a t t h e r e i s no strong evidence t h a t model forecasts can be much improved by combining them w i t h those o f ARIMAs. Hence, we may p r o v i s i o n a l l y regard t h e MAR as c o n d i t i o n a l l y e f f i c i e n t w i t h respect t o t h e Box-Jenkins ARIMAs. Conclusion The r e s u l t s must be i n t e r p r e t e d w i t h caution, p a r t i c u l a r l y i n view o f t h e smallness o f t h e sample o f forecast errors. However, the evidence presented suggests t h a t the model can provide r e l a t i v e l y e f f i c i e n t forecasts, i n t h e sense t h a t the magnitude o f forecast e r r o r s tends t o be l e s s f o r t h e model than f o r u n i v a r i a t e ARIMAs. The r e s u l t s are stronger than those i n o t h e r studies o f regional forecasting. Models o f f e r i n g systematic forecasting improvements over u n i v a r i a t e ARIMAs are n o t common i n practice. The re1a t i v e l y s t r a i g h t f o r w a r d model b u i l ding procedure applied here t o the Texas economy could be employed t o forecast other regional economies as well. http://clevelandfed.org/research/workpaper/index.cfm Best available copy Table 9 Optimal Weights ( ARIMA/model ) Vari abl e . One-guarter-ahead Four-quarter-ahead TIP1 CPIDFW PAYROLL TEMP TLF TPY TRET -.78/1.78 1 .65/-. 65 http://clevelandfed.org/research/workpaper/index.cfm Best available copy Indeed, the methodology might be useful i n any forecasting problem where there a r e numerous potential sources of information f o r forecasting, b u t incl uding a1 1 of them i s inappropriate due t o issues of degrees of freedom and mu1 tic01 1ineari ty. Mu1 t i v a r i a t e ARIMA methods, such as those proposed by Tiao and Box (1981 ), are more flexible than the MAR method proposed here, and might provide further gains i n forecasting accuracy. However, many practical forecasters will find the MAR much easier t o imp1ement. Other 1ess time-consumi ng mu1 t i v a r i a t e methods e x i s t , such a s the "vector autoregressions" of Anderson and of Kuprianov and Lupoletti (19841, which can be implemented w i t h a single computer r u n and no diagnostic efforts.6 However, there i s no evidence that they can provide e f f i c i e n t regional forecasts relative t o univariate methods. Neither i s there any c l e a r evidence t h a t structural econometric model s of regions can provide e f f i c i e n t forecasts i n any systematic way. A further advantage or byproduct of the method here proposed i s t h a t , i n performing the two stages of MAR modelbuilding, i n s i g h t s into the regional economic process may be generated t h a t are not generated by other methods. course, there will always be a place f o r a number of different methods. Of In the final analysis, many kinds of models can shed l i g h t on the forecasting problem and on economic relationships. An ideal forecast might take a l l into account i n an optimally weighted combination. The author has begun work a t the Federal Reserve Bank of Cleveland on forecasting the Ohio economy, which is structurally very different from the Texas economy. Aside from f u l f i l l i n g i n s t r i n s i c i n t e r e s t i n forecasting Ohio, the r e s u l t s of this study will be compared w i t h those f o r Texas i n the http://clevelandfed.org/research/workpaper/index.cfm Best available copy - 35 - foll owing respects: (1 ) the persistence of autocorrelation in growth rates of regional series, ( 2 ) the importance of linkages to the national economy in providing useful forecasting relationships, and ( 3 ) the value of certain regional series, particularly the employment series, in forecasting other regional series. http://clevelandfed.org/research/workpaper/index.cfm Best available copy Footnotes See Granger and Newbol d (1977, pp. 289-302) f o r an assessment o f t h e 1. comparative accuracy o f time s e r i e s versus econometric macro forecasts. See Nel son (1 984) f o r a comparison o f u n i v a r i a t e ARIMAs and judgmental l y adjusted econometric model s in r e a l - time macroeconomic forecasting 2. . The name and acronym f o r t h e model form are the t r a d i t i o n a l ones, and those p r e f e r r e d by Granger (1982). "It now seems o b l i g a t o r y t o provide an acronym, o r catchy abbreviation, whenever a new time s e r i e s model, technique, o r computer program i s introduced. . . . As t h i s pro1i f e r a t i o n continues i t seems 1i k e l y t h a t soon, competing i n i t i a l s f o r t h e same model, o r t h e same i n i t i a l s f o r d i f f e r e n t models, w i l l arise. . . . It can be . . . argued t h a t unnecessary p r o l i f e r a t i o n of these abbreviations should n o t be encouraged .. 3. .'I (p. 103). The ordinary l e a s t squares estimation technique ignored t h e c o r r e l a t i o n between e r r o r s i n d i f f e r e n t equations. The "seemingly unrel ated regression" e s t i m a t i o n technique might have provided s l i g h t l y b e t t e r f o r e c a s t i n g equations. 4. This use o f t h e term causal i t y i s controversial. "It i s doubtful t h a t - p h i 1osophers woul d completely accept t h i s d e f i n i t i o n , and p o s s i b l y cause i s t o o strong a term, o r one too emotionally laden, t o be used. A b e t t e r term http://clevelandfed.org/research/workpaper/index.cfm Best available copy might be temporally related, b u t since cause i s such a simple term we s h a l l continue t o use it. " Granger and Newbold (1977, p. 225). 5. Nelson (1984) conducts s i m i l a r analysis o f optimal weights f o r ARIMA and judgmental l y adjusted macroeconometric model forecasts. However, he does n o t enforce the requirement t h a t the weights sum t o u n i t y . Granger and Ramanathan (1984) show t h a t a l i n e a r combination forecast w i t h weights n o t constrained t o add t o one and w i t h a constant term can l e a d t o improved forecast accuracy r e l a t i v e t o a combination w i t h t h e sum of weights constrained t o one and w i t h o u t a constant, as i n t h i s paper. The method o f Granger and Ramanathan requires estimation o f three f r e e parameters, compared w i t h o n l y one i n t h e t r a d i t i o n a l method, employed i n t h i s paper. As t h e reader w i l l note, t h e sample was e v i d e n t l y r a t h e r small even f o r t h e estimation o f a sing1e parameter. The author d i d try estimating t h r e e parameters, b u t t h e r e s u l t s were uni n f o m a t i ve. 6. Kuprianov and L u p o l e t t i (1984) b u i l d a " vector autoregression" ( n o t t h e "Bayesian" v a r i e t y ) f o r q u a r t e r l y employment and d e f l a t e d personal income f o r f i v e states and t h e D i s t r i c t o f Columbia, w i t h two exogenous national variables, and a l a g 1ength o f s i x quarters. The method here d i f f e r s i n t h e method o f choosing variables t o be included and i n the method o f choosing t h e appropriate l a g lengths. The longest l a g i n t h e MAR was three quarters, and t h a t occurred i n o n l y one equation. http://clevelandfed.org/research/workpaper/index.cfm Best available copy Appendix A: Box- Jenkins ARIMA Models see = .01538 I = 11.1 x2(18) = 19.4 see = .00769 1, = 2 9 . 7 x2(18) = 9.1 see = .00432 I = 30.1 x2(18) = 9.1 see = .00802 I = o x 2 ( 1 8 ) = 18.4 see = .00644 I = o x 2(18) = 17.9 see = .01163 I = 4.0 xz(18) = 12.0 see = .01787 I = 8.5 xZ(18) = 10.3 http://clevelandfed.org/research/workpaper/index.cfm Best available copy Appendix A: Continued Autocorrelation functions k! TPY TRET - 1 .30 .18 2 .13 -.I4 3 .18 -.38 4 .01 .10 5 -.03 .22 6 .14 .27 7 .27 -.09 8 -.04 -.28 9 .14 .07 10 .ll .19 X 2 Test for white noise TPY To lag 6 x2 8.3 TRET Significance .21 x2 17.2 Significance .O1 Note: Autocorrel ation functions and x 2 tests for nonautocorrelation for the other five Texas series are found in Hoehn, Gruben, and Fomby (1984a). http://clevelandfed.org/research/workpaper/index.cfm Best available copy Appendix B: F i n a l E s t i m a t e o f Model see = .01467 see = .006910 c2 = .61 see = .004846 R2 = .61 see = .007212 i2 = .17 Q(21) = 8.2 Q(21) = 22.4 I = 36.5 Q(21) = 4.8 I = 36.7 Q(21) = 17.8 I = 7.8 http://clevelandfed.org/research/workpaper/index.cfm Best available copy Continued Appendix B: see = .006389 Q(21) = 17.4 see = .01016 Q(21) = 15.3 see = .01773 - ~2 = .26 see = .01833 - ~2 = .56 Q(21) = 16.8 I = 13.8 Q(21) = 10.2 http://clevelandfed.org/research/workpaper/index.cfm Best available copy Appendix B: Continued see = .01425 k2 = .51 see = .009475 - ~2 Q(21) = 26.2 Q(21) = 20.0 = .40 see = .I672 k 2 = .07 Q(21) = 13.2 http://clevelandfed.org/research/workpaper/index.cfm Best available copy References 1979. Anderson, Paul A. Autoregression," VOI. "Help f o r the Regional Forecaster: Vector Federal Reserve Bank o f Minneapolis Q u a r t e r l y Review, 3, no. 3 (Summer), pp. 2-7. Ash1ey, Richard A. Analysis," 1980. "Texas Forecasting Studies: Time Series BP 80-3 Study f o r the L e g i s l a t i v e Budget Board by the Bureau o f Business Research, U n i v e r s i t y o f Texas a t Austin (September 3). . 1983. "On t h e Usefulness o f ~acroeconomicForecasts As Inputs t o Forecasting Models," Journal o f Forecasting, vol . 2, no. 3 (July-September), pp. 211 -23. , C.W.J. Granger, and R. Schmalensee. Aggregate Consumption: 1980. " Advertising and An Analysis o f Causal ity ," Econometrica, vol . 48, no. 5, pp. 1149-67. and G w i l y m M. Jenkins. Box, George E.P., Forecasting and Control. Doan, Thomas A., 1970. San Francisco: Time Series Analysis: . Hol den-Day Robert B. Litterman, and Christopher A. Sirns. 1983. "Forecasting and Conditional P r o j e c t i o n Using Real is t i c P r i o r Distributions, 1 202. " National Bureau o f Economic Research Working Paper No. , http://clevelandfed.org/research/workpaper/index.cfm Best available copy Granger, C.W.J. 1982. Time Series Analysis, , and New York: "Acronyms i n Time Series Analysis (ATSA)," vol . 3, Journal o f no. 2, pp. 103-07. Paul Newbold. 1977. Forecasting Economic Time Series. Academic Press. , and Forecasts," Hoehn, James G., R. Ramanathan. 1984. "Improved Methods o f Combining Journal o f Forecasting, vol . 3, no. 2 (Apri 1-June), pp. and W i l 1iam C. Gruben, w i t h Thomas B. Fomby. "Some Time Series Methods o f Forecasting the Texas Economy," 1984a. Federal Reserve Bank o f Dallas Working Paper No. 8402 ( A p r i l ). . Economy: 1984b. A Comparison, " "Time Series Forecasting Models o f the Texas Economic Review, Federal Reserve Bank o f D a l l as (May), pp. 11-23. Hol t, Charles C., and Jerome A. 01sen. 1982. " Structural Change and Forecasting by Exponentially Weighted Regressions," Bureau o f Business Research, U n i v e r s i t y o f Texas a t Austin, BP 82-6 ( J u l y 16). http://clevelandfed.org/research/workpaper/index.cfm Best available copy Kuprianov, Anatol i, and Will iam Lupol e t t i . Fifth D i s t r i c t States i n 1984: Models," 1984. "The Economic Out1ook f o r Forecasts from Vector Autoregression Economic Review, Federal Reserve Bank of Richmond vol. 70/1 (January/February 1, pp. 12-23. L i tterman, Robert B. 1 979. "Techniques of Forecasting Using Vector Federal Reserve Bank of M i nneapol i s Working Paper No. Autoregressions," 115. . 1 982. "Specifying Vector Autoregressi ons f o r Macroeconomic Forecasting," Federal Reserve Bank of M i nneapol i s Working Paper No. 208 (July). Nelson, Charles R. 1984. Forecasts of GNP," " A Benchmark for the ~ c c h r a cof~ Econometric Business Economics, vol . 19, no. 3 (April ), pp. 52-58. Tiao, G.C., and G.E.P. Box. 1981. 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