A C T A U N I V E R S I T A T I S L O D Z I E N S I S _ _ _ _ _ _ _ _ _ _ FOLIA OECONOMICA 91, 1989
Grażyna Oefin-T omczy k*
MODELLING AND PROJECTION OF THE T'RADE SHARES Of THE IN CMEA FOREIGN TRADE
lj_ Introduction
One of the main problems occurring in the process of building the system of national econometric models for a group of countri is the problem of linkage of these models into one consistent sys-‘ tern. This function can be performed by the trade share matrix a p -proach.
The purpose of this paper is .the present ation and compari son of the application of alternative methods for the linkage of national econometric models through the trade share matrix a p -proach in the CMEA model which has been built in the Institute of E co nometrics and Statistics, University of Łódź [3].
2^. Tirade Share Matrij^
The centerpiece of the trade share matrix approach is a trade share matrix X which permits the calculation of market shares.
let Xjj (i, j a 1, .... n) be the trade flow from country i to j. These elements can be arranged in an n x n matrix of trade flows, where n is the number of countri es to be dis ting uishe d ir\ the model (Table 1). If there are disting uishe d only countries (not regions), the value of diagonal elements X ^ will he equal zero^_____
“ Ph. D., Lecturer in the Institute of Eco nometrics and S t a t i s -tics, University of Łódź.
T a b l e 1 Trade share matrix
The total imports M. of country j are given by the column sums:
* 2 xij. j = 1.... n (1)
l
and the total exports fc^ of country í are found as the row sums:
(
2)
The world exports (imports) cr the total exports (imports) of a given group of countries are given:
= £ E i * Z ■ X. i, j * 1, . v , n (3)
Relation (3) is called the condition of c o n s i s t e n c y 1 .
For the purposes of analysis and prediction of the s t r u c t u -re of trade matrix X it is useful to define certain coefficients С fl] . [5], [6], [7]).
In fact, there exist sta tist ical d is crep ancie s bet ween total exports and imports due to differences in valuation methods (fob, cif), differences in timing etc.
The matrix of import coefficients A ^ . is obtained by d i v i d -ing each column by its column sum (imports):
X . . v->
я , Y A jj = 1, i. j ■ I . . . П (4)
Thus A. ^ is the share of the total imports of country j that is supplied by country, i. The matrix of export coefficients describing the regional distribution of the exports of country i is defined in the following way:
^ i i * = 1. i , j = 1, ..., n (5)
i j
For some purposes other coefficients seem to be interesting:
X, s '
^ i j 1 -- ’ S 2 P 1> i| j s 1. n (6)
X • i i ij
U
X • X
The ft . . coefficients describe the geographical structure of trade, and the á ^ denote the relation of the X tj to the total value of trade, weighted by the shares of total exports of country i and total imports of country j iri the total value of trade X. Using equations (4) and (5) the following balance equations
2 can be written .
2
These equations perform the same role as the balance e q u a -tions in the iriput-output model.
M j 1 £ * i j E i. i. J s 1. . n • (9) i
x = л i j M j 3 2 e i , i. j * i, .... n . d o )
i J . j i
These relations are used to construct export or import p r e d i c -tions.
In most international trade models (e.g. Project LINK [ б ] , EPA [l]) the predictions of trade are constru cted in two main stages:
1° at the first stage the specification of import equations is perfornmtí (at the level of con struction of par ticular country models)
2° at the second stage, having Uie predicted imports obtained as described above and the matrix of import c o e f f icients X .. the
* 3 1J
values of exports are determined by equation (б)4-.
3. Methods of Linkage of Models Thr ough the Trade Share M a trix
The exports of individual country i can be determined for the period t by the following relation:
EI -- ľ A j ] . м] , i, j , 1 . . . n (11) i
where the import coefficients A i ■ and imports must be given for this period. The data connedted with the trade flows are usually lagged to the rest of the data. Hence, the matrix of import c o e f -ficients A ‘a j Is con stru cted for earlier period than the period t, defined further as а Ьазе period.
Of course, it is also pos sible to determine imports having the export equations and export coefficients.
Two possible assumptions can be taken into account: 1 ° A does not change over time,
2 ° Л changes over time.
In the first case, the exports in the period t are equal
EI - • Hj , i, j - 1, ..., n, (12)
i where
- the import c o e ffic ients in the base period.
This met hod is called a "naive method" and is usually applied as a basis for comparisons.
In the second case (typical in practice) the pr o b l e m of d e t e r -min ing A i ^ in the period t occurs. Some methods of det er m i n a t i o n of the by the direct pre dict ion and e s t imat ion are pre sent ed
below. I
3 . 1. B irpopor tional M et hods of C o n s t r u c t i n g the В a 1anced Predict ions
The iterative met hods of c o n s t r u c t i n g the bal ance d pre dict ions assume that the import c o e ffic ients matrix in the base period A° - [ A y , the vec tor.of the total imports in the per iod t, M*, the vector of the total exports in the period t, E* are kno wn.T he obtained pre d i c t i o n of the matrix A 1 is the bal ance d prediction, i. e . s a t i s f y i n g the c o n d i t i o n that the sum of the p r e dict ions of rows (columns) is equal to the pre d i c t i o n of its sum. This can be w ri tten as follows:
A ł A f
Hence Л = Я 1j sat isfies the cön si s t e n c y c o n d iti on (3). The main role among these met hods is pla yed by the set of met h o d s based on the c l a s s i c a l R A S m e t h o d .
The m o d i f i c a t i o n s of the RAS me t h o d allow to apply the RAS to the part of coe f f i c i e n t s only. There are d i s t i n g u i s h e d "iniportant" c o e f f ic ients from the A 0 matrix ahd d ir ectl y det e r m i n e d for the p re d i c t i o n per iod t. These "im portant" ones can be cho sen using the different criteria, e.g. the c o e f f i c i e n t s i n d i cat ing the s i g nifican t changes over time [2]. The cla ssic al RAS me t h o d is a p -plied to pre d i c t i o n of the rest of coefficients, which n e e d s.t he earlier c o r rect ion of the vectors M t , E t and the unit vector (the vector of sum of coe f f i c i e n t s in columns) to )<eep the c o n s i s t e n -cy con diti on (3). The chosen "im portant" c o e f f i c i e n t s det ermi ne the trends and zeros matrix, T * > n x n - The number of the n on -zer o elements is equal to the number of cho sen coefficients. The classical RAS met hod is app lied to the mat rix A'j j = ^ i j :
U i • h j * 0
1
“ ■í , j = 1, ..., n (14)
Finally, the p r e d i c t i o n of the mat rix A f = С Я * ^ 3 is obtained:
^ ij = *'ij + ł ij* J * 1 . . . n (15)
This m e t h o d is c al led the t r e n d R A S m e t h o d .
The classical RAS me t h o d can be also app lied as the me t h o d which s at isfi es the c o n s i s t e n c y c o n d i t i o n of the p r e d i c t e d matrix A, o bt aine d by the other methods, like the index or the pro ba b i l i t y m e t h o d s .
The i n d e x m e t h o d ass umes that the vector of the total exp orts E and the vector of the total imports M are known in the base and t periods. The p r e d i c t e d ele m e n t s of the ma t r i x A in the per iod t are obtained:
X t(1) = . /ä^bj . А a i + b j о ij . А
ij
version A version В i, j = 1, .... n (16) w h e r e : a i *'í
- export index,M5
b j = -J- - import index. M jIf the obt aine d matrix does not satisfy the con sis t e n c y condition, the classical RAS met hod is applied at the second stage as the cor rect ed method.
The p r o b a b i l i t y m e t h o d assumes that each of the elements of trade matrix X = Cx j j] »is chosen with a certain pro babi lity from the total sum of the f o r eign trade:
E i
p ij = X ' X > i. j = *. •••> n - 117)
Sim ultaneously, this met hod gua r a n t e e s the choice of the most pro-л
bable matrix for the pre di c t e d per iod t . The pre di c t e d trade flow x M 1 ^ is determined: .
4 This met hod can be applied for the sho r t - t e r m p r e dict ion under the a s s umpt ion of the ind ependency of ori gin and des tina tion of trąde flows what in a case of the intra CMEA trade causes the certain und eres t i m a t i o n of "the expexte d" trade flows.
X.t(l) ij
E* X° E ° M° X*
ij ' i, J » 1, ( 1 Й )
Tbo consistent matrix X1 [ x K ] is obtained using the classical
RAS m e t h o d . ’ J
3.2. M e t h o d s of L i n e ar Pro g r a m ming
The method of linear pro gram ming needs the same set of informa-tion as the classical RAS method (4). The coefficients Л ? * must be
* f I J
adjusted that the A ( 4 are consistent with the observed values of
f f
exports and imports in the period t, Ej and M ^ , and simultaneously the A^j differ the least from the corresponding
A i l - 1 A °j * min Z i 0 -5 . Ej £ » ‘j и' . H* i . j = 1 . . . n (19)
The objective function (19) can be expressed in the terms of the trade flows in the absolute values:
ľ ľ i j
_ x*
*ij *ij = min, i , j = 1, n ij
(
2 0)
where X^ = Á * M* *ij A ij M j i, j = 1 . . . n \X* = X ° M 1 ij ij j with the conditions
?*íj
JS x t
i, j - 1, , n.
The form of (20) can be solved by the classical simplex algorithm with certain reformulations based on the introduction of active variables Х*^ and XT. which satisfy the relation:
y t y l * ¥ +
*ij = \ j ł ij " "ijXl i i í . J s 1 .
(
2 1)
where.Xjj > 0 and X ’ j > 0,Then the problem can be written:
ľ ľ 1
X ij + X ij ij
j = 1 , (2 2)
with the constraints
Г j E i X + у i j i j x i j + x i j W 'ij
i, J » l,
In the process of computations it turned out that the algorithm did not guarantee the positive signs of the trade flows and the following constraints were added:
X ij = X ij + X ij * X Ij > °* j 1 1 . . . "• (23)
I '■ I
The possibility of introducing the additional information by the new constraints is the feature of the method. However, this method transmits the influence of the "fitness" of the c oe ffic ients of the higher values being usually more stable. This method can be
modified through the changes of the coefficients of the objective function with the introduction of elements (i, j = 1, ..., n) reflecting the stability of trade flows. The new objective func-tion has the form;
+ x i j )rij E m i n . i. j г 1 . . . n (24) with the same set of constraints.
These r ^ have been constructed in the following way:
X° U L X 'ij
r i j " “ 7 " " j ” 1... n ( 2 5 )
where
X ij “ trade fl0w Írom country i to country' j in the p re viou s oer iod p r e c e d i n g the base period.
3•3. Met hods for the E s timation of Import C o e f f l c lents
The next group of methods of the imports coe f f i c i e n t s d e t e r m i -nat ion are the met hods based on the direct est i m a t i o n of these c oe f f i c i e n t s or the trade flows. T. a p 1 i n [ll] assumes that A i ^ is det e r m i n e d by the relation of export and import prices and the e l a s tic ity of country i share in the imports of country j ,is c o n -stant and the same for all cou nt r i e s exp or t i n g to cou ntry j:
A ij = A ij ( P E i/PMj) , i, j = 1 . . . n (26) where:
A i j - constan t term,
PÍT - the average import price of country j, - constant elasticity of imports of country j.
M o r i g u c h i [9] introduced additionally the relation b e -tween the export capacities of country i and the scale of imports of country j:
ftij r i j
X ij * A ij (PEjV P C O Mj j) (SXi/ M j ) i,j = 1, n (27)
where:
SXj - export capacities of country i,
P C O M ^ - export price of country 1 on the J import market, M a r w a h [8j proposed an approach, where A^ . was determined by the relative export prices and the relation between the total exports of country i to the world exports.
Interesting methods, applied in the LINK Projects are the fol-lowing methods satisfying the consistency condition:
- the LES-type method developed by K l e i n and J 0 h n- s 0 n [б] ,
- the Hic kman -Lau met hod [ 5 j .
The method of the LES-type is based on the following export function:
E i = a iH E i + fti £ A i| M j - rf-jPCOMj ♦ á i T R E N D ,
J U = 1 . . . n (20)
w h e r e : о
A j j - import coeffic ient in the base period;
PCOM. - export price of country i; PCOM, = 2 T , A ? • PE. , A i ij KJ К
3 к / j
where A°^ - export coe ffi c i e n t s of country i in the base period. Equ atio n (24) has a similar structu.re to the e x p endi ture fun c-tion in the linear e xp endi ture sys tem (LES). Thus, if the p a r a -meters of equation (24) satisfy the conditions:
S ^ P E I = Г .Г-РСОМ* i l 2 6 t - 0, i, J * 1, . . ., n . (29) i ^ • p i A ij = 1 then * 2 M* i j J
J o n n s o n and K l e i n [б] proposed the application of the RAS method to modify to satisfy the condition:
= 2 AijM j. 1, j = 1 . . . n (30)
because the LES-type method concerns the total export functions
only. У
H i c k m a n and L a u [5] introduced the method based on the CES-type function. The index of imports M* is determined in the following way:
= ^ 3 i i, j = 1, n (31)
where
b ^ - constant parameters;
pj * (1 /<*_}." 1 ). where - elasticity of substitution of im-ports of country j.
The import demand function can be written as: 1 1
d. n d 1
« и - * * ; p e .j J c s A * k j ] " r 1 . 1 . i . . . . .n
к - 1 or in the linear approximation:
x tj . А » Л - «jXJj (PEjl - ftj). I. i - 1 . . . n < » > where
p"j * X P E ij
The H i c k m a n - L a u m e t h o d sat isfies the c o n s i s t e n c y c o n d i t i o n as w e l l .
4. The C o m p a r i s o n o f the Alt er n a t i v e Met hods *' " -т ~ Яны» «T'A ;д ~ г ; •. ■.- . . . T g r i- r - —г;- я а г г ь т в т м г к т г т ^ г ж а с д а » of the Import C o e fficients P r e d i c t j an
The analysis of the cho sen m e t h o d s of the import c o e f f i c i e n t s p r e d i c t i o n has bee n based on the data c o n c e r n i n g the , com modity g ro ups of the int ra-C MEA trade f l o w s 5 .
There have been d i s t i n g u i s h e d seven Eur o p e a n CME A countries: Bulgaria, Cze chos lovak ia, GDI?, Poland, R o m a n i a . H u n g a r y , the Soviet Union, and the four c o m m o d i t y groups, acc or d i n g to the for eign trade s t a tist ics of the CME A countries:
- fuels, raw m a t e r i a l s and m a t e r i a l s (I), - mac hi n e r y and e q u i pme nt (II),
- c on sume r goods of the industrial or i g i n (III), - food and raw m a t e r i a l s for food p r o d u c t i o n (IV).
The s t a tist ical data of the trade flows have bee n a v a i l a b l e in cur r e n t prices only. The int raC MEA trade flows use d in the c o m -p u t a t i o n s have been based on the e x -p o r t e r s sta t i s t i c s a s s u m i n g the bet ter accuracy of f lo ws6 .
fhe trade share ma t r i x of 1976 was used as the base matrix for the project ion. The p r o j e c t i o n s were c o n s t r u c t e d for the year 1977 (the latest year of the a v a i l a b l e data) and c o m p a r e d wi t h the e m -pir ical trade m a t r i c e s for this year.
At the actual stage of res e a r c h the m e t h o d s based on the d irect e s t i m a t i o n hav e not been tested b e c a u s e of the lack of the f oreign trade price ind ices in the s t a t i s t i c s of the CME A c o u n t -ries. Some r es earc h has bee n p r e s e n t l y u n d e r t a k e n c o n c e r n i n g the s p e c i f i c a t i o n p r o b l e m w i t h o u t the i n t r o d u c t i o n of the p r i ce v a r i a -bles.
The matrices have been projected with the application of the following methods:
- classical R A S ,
- index method (version A), - index method (version B), - probabi lity method,
- linear pro gram ming method,
- modified algorithm of the linear programming.
, The accuracy of the predictions has bean measured by the c o m -putation of the prediction errors for all coefficients:
The mean pre diction errors of ^ h e four commodity groups and the tested methods are presented in Table 2.
The comparison of the applied methods indicates that the obtained results, from the point of view of the accuracy of p r e d i c -tion, measured by the mean pre diction error, does not differ much. In group I fuels, raw materials and materials the mean p r e d i c -tion error (except the linear program ming method) is about 10*, in group II - machinery and equipment - about 6*, in group III - uon- sumer .goods of the industrial ori g i n - about 7%, and in group IV - food and raw mat eria ls for food pro d u c t i o n - about 9*. The least mea n p r e dict ion error in group II is caused by the fact that in the years 1976-1977 the import coe ff i c i e n t s in this group were more stable than in other groups, e.g. in group I the annual c h a n g -es of the c o e ffic ients were equal 100%.
(34)
and the mean p r e dict ion error S for the matrix
T a b l e 2 Comparison of the prediction methods of the elements of the trade share matrix for the four commodity groups in
the intra-CMEA foreign trade
Mean prediction errors for the commodity groups Method materials f u e l s ,ťaw
and m a -terials (I) machinery and e q u i p -ment (II) consumer goods of the in-dustrial origin (III)
food and raw materials for food production (IV) RAS 0.107Ć 0.0590 0.0718 0.0937 Index method v ersion A 0.1076 0.0590 0.0718 0.0937 version В 0 . 107B 0.0598 0.0717 0.0916 P r o b a b i 1 i ty method 0.1075 0.0598 0.0719 0.0935 Linear p r o -gramming 0.1250 - - - ■ Linear p r o -gramming (modi f ied algorithm) 0.1501 - -
-S o u r c e : The author's calculations.
In group I (fuels, raw materials and materials) relatively the "best" results have been o bt aine d by the a p p l i c a t i o n of the p r o -b a-b ilit y method. The linear p r o g r a m m i n g met hod and the m o d i f i e d a l g o rit hm of the linear p r o g r a m m i n g had been tested, but the res ults hav e been u n s a t i s f a c t o r y (the me a n p r e d i c t i o n err ors 12- 15X). The m o d i f i e d algorithm, in spite of exp ecta tions , has p r o -duced the worse results than the cla ssical ver s i o n of the algorithm. "The adjusted" influence of this me t h o d has been c o n c e n t r a t e d on the least stable trade flows. The high .value of the mea n p r e d i c -tion error seems to be c au sed by the lack of c o n s t r a i n t s which w o u l d allow to exist the zero values of flows in the set of the constraints.
The results obtained for group II (machinery and equipment), as the group of relatively stable coefficients, are the same for all of the tested methods.
In group III - (consumer.goods of the industrial origin), relatively the best fitted projections have been obtained by the index method, version B.
The results of computations, presenting the mean prediction errors for the methods of projection of the matrix coefficients of the intra CMEA trade allow to apply the foreign trade flows model to determine the exports of the countries in thť aggregation for the four commodity groups, substituting the stochastic e q u a -tions in this model. The introduction of the foreign trade flows model allows to observe and analyze the multilateral connections in the CMEA region, which can be important for the process of c on struction of the forecasting and simulation scenarios.
References
[1] A m a n o A , , K u r i h a r a E., S a m u e 1 s o n L. (1980), Trade Linkage Submodel of the EPA World Economic Model, "Ec onomic Bulletin", No. 19, Eco nomi c P la nnin g Agency, Jap anes e Government.
[2] D e f i n-T o m c z -y k G. (1980), Emp iryczna w e r yfik acja iteracy jnych metod budowy prognoz zbi l a n s o w a n y c h (Empirical V a l idat ion of Iterative M e t hods of the C o n s t r u c t i o n pf B a -lanced Predictions), Łódź (mimeo):
[3] G a j d a J. et al. (1981), Model of the CMEA Countries, paper p r e s e n t e d at the C o n f e r e n c e "Problems of Bui l d i n g and Es t imat ion of Large Econometric Models", Łódź.
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[11] T a p 1 i n G. 8, (1973), A Model of W orld Trade, [in:] R. J. Ball, 1 he International L in kdge of National E co nomi c Models, N o r t h - H o lla n d, Amsterdam.
Gr a ż y n a O e f in-Tomczyk
MOD ELOWANE 1 PRO G NOZ O WANI E MAC I E R Z Y U D Z I AŁÓ W H AN DLU W WYMIANIE W ZA JE MN EJ KRA JÓ W RWPG
C e l em ar t ykuł u jest p r e z ent a cja i po r ó w n a n i e z a s t o s o wani a a l -te r n a t y wny ch m e t od łączenia e k o n o m e t r y c z n y c h modeli p oprzez m a -cierz u dz ia łó w handlu w e k o n o m e t r y c z n y m mod e lu k r a j ów RWPG.
Omó wio n e zos ta ły teo ret y c zne p o d s taw y metody RAS, p r o g r a m o w a -nia l in io w e g o oraz specyfi k acj i ró wnań w zak resi o w s p ó ł c z y n n i k ó w importowych.
W m ac ie rz y han dlu w za j e m n e g o w y r ó ż n i o n o cztery grupy w yr obów (kl a syf i kacj a RWPG). Na p od st aw ie m etod b i p r o p o r c j o n a l n y c h oraz p r o g r a m o w a n i a m a t e m a t y c z n e g o s k o n s t r u o w a n o prognozy m ac i e r z y w s p ó ł czynników. O t r z ym a ne rez u ltaty z a n a l i z o w a n o pod kąt em ich p r z y d a t -ności do pr o g n o z o w a n i a ma c ierz y u d z i a ł ó w h an dl u w z a j e m n e g o k rabów RWPG, o p i e r a j ą c się na u z y s k a n y c h b ł ę d a c h prognoz.
