KURODA S METHOD FOR CONSTRUCTING CONSISTENT INPUT-OUTPUT DATA SETS by Peter J. Wlcoxe Ipact Research Cetre, Uversty of Melboure Aprl 1989 Ths paper descrbes a ethod that ca be used to resolve cossteces betwee put-output data fro varous sources. It belogs to the class of algorths ds- cussed by Bacharach (1970) ad has bee used extesvely by Kuroda (1988) ad Wlcoxe (1988). I put-output aalyss t s ofte ecessary to use cosstet data sets orgatg fro dfferet goveret ageces. For exaple, the table of terdustry trasactos created by oe agecy ay ot be cosstet wth value-added ad fal dead vectors produced else- agree: a table of terdustry trasactos, a vector of coodty outputs, ad a vector of where. I ths case, the vestgator wll be cofroted wth three peces of data whch do ot gross outputs by dustry. The task the becoes adustg the trasactos table to atch the coodty ad dustry output vectors. I the past, ths proble has bee solved by usg the RAS ethod. RAS s a teratve algorth whch scales the rows ad colus of the trasactos table up ad dow repeatedly utl the table s row ad colu sus agree wth the target vectors. It has bee show that RAS wll evetually coverge, but the result wll ot ecessarly be close ay ecooc sese to the orgal trasactos table. The purpose of ths paper s to defe a easure of how far a ew trasactos table s fro the orgal, ad to derve a algorth whch wll costruct a table zg that dstace.
-- Gve a atrx X of tal data, defe r ad c to be the shares of each eleo et the row ad colu sus of the orgal atrx: r = o X, c = X o (1) X X = 1 o = 1 o Let R be a vector of target row totals, ad C a vector of desred colu totals. The followg fucto ca the be used to easure the dstace betwee a revsed atrx X ad the orgal (eboded r ad c ), where w ad v are arbtrary sets of weghts: Q = 1 =1= 1 X r w + 1 R = 1= 1 c CX v () It s ow possble to choose X to ze ths fucto subect to the followg costrats: R = X (3) = 1 =1 C = X (4) The Lagraga for ths proble s: L = 1 =1= 1 X r w + 1 R = 1= 1 =1 =1 =1 c CX =1 v + λ (R X ) + µ (C X ) (5 )
-3- Takg frst order codtos gves: L = X X r R w + R X c C v λ µ =0 (6) C Collect ters X, ad for coveece ake the followg deftos: S = w v + (7) R C 1 G = r w + R c v (8) C Ths allows the frst order codtos to be rewrtte as show X = S (λ + µ + G ) (9) Both S ad G deped oly o tal data ad the weghts w ad v, so t s oly ecessary to detere λ ad µ, to fd the optal X. These ay be detered by applyg the co strats: R = X = S (λ + µ + G ) (10) = 1 =1 = 1 =1 C = X = S (λ + µ + G ) (11 )
-4- R C Defe dagoal atrces S ad S as dcated: R =1 S = S, S = S (1) C =1 I atrx otato, the costrats ca ow be expressed as R S. λ+s. µ=r A (13) C S. λ+s. µ=c B (14) where A ad B are vectors defed as follows: A = S G B = S G (15) = 1, = 1 Wth ore apulato, t s possble to derve explct forulae for λ ad µ. For coputa- toal purposes, however, t s better to arrage the equatos to the followg syste, whch ca be solved easly by ay copetet uercal package: R S S SS µ λ = R A C C B (16) Ared wth the values of λ ad µ, the optal choce of X ca be coputed drectly.
-5- It s worthwhle to exae a few of the possble weghtg schees that ca be used. The ost obvous approach s to weght all errors equally, whch eas that w =v =1 for all ad. O the other had, the followg choce of weghts results a drastc splfcato of the revso forula: w = 1 R, v = 1 C (17) Ths eas that S =1 for all ad. Moreover, the followg s true of G : G = 1 (r R + c C ) (18) whch s sply the average of the values obtaed by applyg the orgal shares to the target row ad colu sus. Furtherore, t ca be show that all of the followg are true: =1 λ = 0, µ = 0, (19) =1 S R C =, S =. (0) Ths eas that the revso forula has a partcularly sple for: X = G + 1 (R G ) + 1 (C G ) (1) = 1 = 1 Thus, the revsed X s ust G (whch has the terpretato above) adusted to correct the
-6- row ad colu sus. It s portat to ote that ths ethod does ot guaratee that all eleets of X wll be oegatve. Kuroda proposes a dfferet weghtg schee, wth w ad v detered by the followg equatos: w = 1, v = 1. () r c Ths choce of weghts causes Q to be a fucto of the percetage chages the coeffcets: Q = 1 =1 =1 X /R 1 + r 1 =1=1 X /C 1 (3) c I ost cases, ths wll esure that all eleets of X are postve, sce akg oe egatve would requre a chage of ore tha oe hudred percet, resultg a large value of Q. It s possble, however, for egatve ubers to arse f the row ad colu targets dffer substatally fro the correspodg totals of the tal array.
-7- REFERENCES Bachrach, Mchael (1970), Bproportoal Matrces ad Iput-Output Chage, Cabrdge: Cabrdge Uversty Press. Kuroda, M. (1988), "A Method of Estato for the Updatg Trasacto Matrx the Iput-Output Relatoshps," Statstcal Data Bak Systes, K. Uo ad S. Shshdo ( eds.), Asterda: North Hollad. Wlcoxe, P.J. (1988), "The Effects of Evroetal Regulato ad Eergy Prces o U.S. Ecooc Perforace," Ph.D. Thess, Harvard Uversty.