Deflation of the I-O Series 1959-2. Soe Technical Aspects Giorgio Rapa University of Genoa g.rapa@unige.it April 27 1. Introduction The nuber of sectors is 42 for the period 1965-2 and 38 for the initial period: see the Read e file in the ain page of the website for details. The base year for deflation was chosen to be 1978: this is because 1978 is situated id-way between the two oil shocks (and not ainly because it is idway between 1959 and 2!), thus avoiding the suation of the effects of the two shocks in any of the two tie directions. In addition, 1978 is a year in which the econoic cycle in Italy is not at an extree point, and the growth of the ain acroeconoic variables is in line with the average of the post-war period. The version of I-O tables to be deflated is that expressed in producer prices ( depart usine / depart douane ): this should reduce possible biases iplicit in the deflation of the alternative version ( purchaser prices ), due to the fact that in the latter flows are gross of trade argins and transport costs, and thus the product ix of cells is ore heterogeneous. In particular, this choice should reduce the differences in the deflators as regards interediate and consuption goods (as regards trade argins in consuption goods, se point 2.2.h below). In addition, the deflated tables are those with IBS allocated to interediate costs of sectors (see the Read e file quoted above): in fact this version has a ore natural interpretation in ters of I-O analysis. 2. First estiates of flows: price and quantity indexes, and chaining 2.1 The flows to be deflated were the following for each year: d - interediate flows: a (42 by 42) atrix X of doestically produced goods, and a second (42 by 42) atrix X of iported goods. On the whole, 3528 ites; - final expenditure: private consuption; non-arket services of Governent, and of Social d Institutions; fixed investent; change in stocks; exports. Forally a (42 by 6) atrix Y of produced goods, and a siilar one Y of iported goods, that is 54 ites; - sectors value added (gross of indirect taxes on products), va : 42 ites; - product transfers between sectors (due to siilar products sold by different sectors, according to the ESA1979 standard), t: 42 ites; - sectors doestic output, x: 42 ites; - iports of goods and services (gross of taxes on iports), : 42 ites. In order to exploit soe additional inforation fro National accounts, we inserted also the ten ites coing fro the aggregate equilibriu account of resources and uses: GDP (net of taxes on iports), iports of goods (Mg), iports of services (Ms), household consuption (C), Collective consuption of General Governent (Cg), Collective consuption of social institutions (Cs), Gross fixed capital foration (If), Change in stocks ( s), Exports of goods (Xg), Exports of services (Xs). All these ites are single nubers. The total nuber of ites to be deflated and balanced is then 421 (any of which are equal to zero). However, the Z atrix (as defined by expression 5 of paper #1 of the website: we oit here the left-superscript ) has forally (42+42+4)=88 rows and 48 coluns, and thus we have 1
4224 cells. Thus atrix Z was arranged as follows, with the above sybol definitions (pries denote transposed vectors): d X X va' Z = t x d Y Y e 1, where 2 3 4 e1 GDP 2 = C 3 4 Mg The final square bracket contains 4x6 scalars. The output vector x ust be interpreted as distributed output, that is sectors output plus transfers t. 2.2 The initial estiates of the deflated cells were obtained as follows: a. doestic uses of doestic goods (i.e. excluding exports): deflation by eans of rowunifor price indexes of doestic output; b. exports: deflation by eans of price indexes of exports; c. iports (total, interediate and total): deflation by eans of row-unifor price indexes of iports; d. product transfers: price index of the sector where the transfer appears; e. value added: obtained through quantity indexes of constant-price value added (derived for National accounts) applied to the current-price value added of the base-year table; f. sectors output: a siple ean of (a) deflation with output price indexes and (b) coputation through quantity indexes applied to the base-year table; g. ites of the aggregate resources/uses account: quantity indexes for national accounts; h. trade and transportation argins: since they appear explicitly on proper rows of the table (Trade, Inland transport, Maritie and air transport), they were row-uniforly deflated with the price indexes of the corresponding selling sectors. A distinction was however ade as regards trade argin on interediate and on consuption goods, since it is reasonable to think that the price dynaics of the latter ay diverge fro that of wholesale trade. This was ade possible by coparison between the price index of goods as expressed in producer prices, and that of goods as derived fro household consuption statistics. When the indexes, derived fro various sources of National accounts, were ore aggregated than 42 sectors, the sae index was utilised for sectors belonging to the sae acro-sector. 2.3 Deflation was ipleented through a sort of chained quantity index ethod. That is, we did not deflate flows directly to 1978 prices, using 1978-based indexes. We followed instead the following procedure. Start fro 1978 and ove towards 2 (when oving in the other direction, towards 1959, siply change t 1 into t+1 in what follows). Consider the table of year t > 1978, and assue you already deflated (and balanced) the table of year t 1 in 1978 prices: of course, when dealing with t=1979, the current-price 1978 table is already deflated as such! Copute price and quantity indexes of year t based on year t 1, and deflate all ites of the table of year t, thus obtaining a provisional table of year t in (t 1) prices. Then divide each cell of this table by the corresponding cell of the current-price table of year t 1: you get quantity indexes of year t based on year t 1. Finally, ultiply these quantity indexes by the correspond cells of the 1978-price table of year t 1, which is already balanced by assuption. This is the first estiate of the 1978-price table of year t. 2 Cg Ms Cs If s Xg Xs (1)
3. Constraints and final balancing The constraints to be fulfilled are basically the following (a suffix to vectors indicates their diension; k is a null k-vector; u is a vector of ones): C.1. the su of each row (interediate plus final uses) of the doestic block ust be equal to d d distributed output, in sybols X u42 + Y u6 x = 42 : 42 constraints; C.2. the su of interediate costs plus value added plus product transfers ust be equal to output, in sybols u 42 X d + u 42 X + va + t x = 42 : 42 constraints; C.3. the su of each row (interediate plus final uses) of the iport block ust be equal to iported resources, in sybols X u42 + Y u6 = 42 : 42 constraints; C.4. the su of product transfers ust be equal to zero, in sybols u t : 1 constraint Up to now nuber of constraints is 127. However, as we said, we added ten aggregated variables fro the resources/uses account, in order to take ore inforation into account. This adds ten ore constraints to be satisfied: su of sectors value added = GDP, su if iported goods = Mg, su of iported services = Ms, and so on (see definitions at the end of page 1). Thus we have a total of 137 constraints. 42 = Define, as in section 3 of paper #1 of the website, the vector z vec( Z) = : it has 4224 eleents. The constraints can be written as Gz = 137, where G is a (137 by 4224) atrix whose cells are equal to, 1 or 1 in due positions. Appendix 1 contains the precise definition of G. It was built by eans of a MATLAB progra, whose script is contained in Appendix 2; translation into different languages should not be difficult. The first estiates contained in vector z do not fulfil the constraints. The SWLS balancing algorith, as discussed in paper #1, is thus the following: 1 [ GVG' ] [ Gz] z* = z VG' (2) where V is the subjective variances/co-variances atrix of the ites in vector z, that is it contains subjective evaluations of the reliability of first estiates. The atrix V was assued to be diagonal, i.e. co-variances were assued null. Variances were coputed dividing the size of each cell (taken in absolute value) by an index of its reliability: the reliability indexes were set in a very siple-inded anner. A unifor reliability was assigned to interediate flows; this index is lower than that of final uses (excluding inventory changes, which were assigned a reliability index even lower than that of interediate uses). Aong final uses, the higher reliability was that of exports, since they and their prices are presented in uch ore detail in published statistics. The sae arguent was applied to iported resource (not for their disaggregated uses). The figures coputed by eans to quantity indexes (output, value added and aggregate resources/uses account, excluding aggregate change in stocks) are deeed to be ore reliable than the previous ones, since they are the core of national accounts. Of course, all cells which are equal to zero after the initial estiate should reain such also after balancing: then their reliability should be infinite. However, in order to avoid singularities in 5 the atrix inversion appearing in (2), their variance was not set equal to zero, but equal to 1. The balancing algorith was ipleented by eans of the MATLAB package for Windows (progra available on request). 3
Appendix 1 Definition of atrix G Matrix G can be thought of as ade up of 14 row-blocks, one for each set of constraints described in section 3: C.1 C.4, plus the ten additional constraints coing fro the aggregate account of uses/resources equilibriu. In sybols: G = G G G Now, recall the definition (1) of Z (in particular: the ordering of the last 6 coluns of the ites of final deand, and of the last four rows), and the eaning of the operation z = vec( Z). One ust reeber the sequence of the eleents in vector z: is contains sequentially all coluns of atrix Z. I now define the 14 blocks of G. They ust be defined in such a way that they operate on vector z (not on atrix Z); then the use of the Kronecker product is convenient. Given any two atrices A and B, where A is (n by ) and B is (k by l), one has: a11b a12b a1b = a21b a22b a2b A B, a (nk by l) atrix. an1b an2b anb Define in addition the following ites: I l is the identity atrix of size l; n, is the (n by ) null atrix (in particular it could be a vector); u k is the vector containing k ones. 1 2 14 Block G 1 Constraints C.1. For each of the first 42 rows of Z, su all the 48 eleents on that row and then subtract fro this the corresponding eleent of x (distributed output, lying on row 87 of Z, coluns 1-42): the result ust be zero. Now translate this into constraints on vector z. Define the (42 by 88) atrix H x = [ I 42 42,46 ], and the 88-vector f x = [ 1,86 1 ]. Soe G = u H + I f is the atrix required reflection will convince you that {[ ] [ ]} 1 48 x 42 x 42,528 for representing constraints C.1: G 1z = 42, 1. It is a (42 by 4424) atrix. Block G 2 Constraints C.2. For each of the first 42 coluns of Z, su all the first 86 eleents on that colun (interediate costs plus value added plus transfers) and then subtract fro this the corresponding eleent of x (distributed output, lying on row 87 of Z, coluns 1-42): the result ust be zero. To translate this into constraints on vector z, define the 88-vector s = [ u 86 1 ] and the (x by y) atrix K I s G = K is the atrix required for representing = 42. Then [ ] 2 42,528 constraints C.2: G 2z = 42, 1. It is again a (42 by 4424) atrix. 4
Block G 3 Constraints C.3. For each of the 42 rows of the second block (rows 43 to 84) of Z, su all the 48 eleents on that row and then subtract fro this the corresponding eleent of (iports, lying on row 88 of Z): the result ust be zero. In order to translate this into constraints on vector z, define the (42 by 88) atrix I = 1. Then the (42 by 4224) atrix H = [ 42,42 42 42,4 ] and the 88-vector f [ 1,87 ] G u H + {[ I f ] [ ]} = is the one required for representing constraints C.3: 3 48 42 42,528 G z =. 3 42,1 Block G 4 Constraint C.4. The su of all transfers (row 86 of Z, coluns 1-42) ust be equal to zero. w = 1. Then To translate this into constraints on vector z, define the 88-vector [ 1,85 ] the 4224-vector G {[ u w ] [ ]} G 4 z = (scalar). = is the one required for representing constraint C.4: 4 42 1,528 Block G 5 Constraint C.5. The su of all sectors value added (row 85 of Z, coluns 1-42) inus GDP (cell [85,43] of Z) ust be equal to zero. b = 1 ; then To translate this into constraints on vector z, define the 88-vector [ 1,84 ] the 4224-vector G [ u b b ] G 5 z = (scalar). = is the one required to represent constraint C.5: 5 42 1, 44 Block G 6 Constraint C.6. The su of all iports of goods (lying on row 88 of Z) inus Mg (cell [88,43] of Z) ust be equal to zero: given our product classification (see the Read e file in the ain page of the website), it turns out that goods are the products fro 1 to 28 (while services are the reaining 14 ones). To translate this into constraints on vector z, define the 2464-vector c = 1, 28 ( f ), where f was defined under block G 3. Then the 4224-vector G 6 = [ c 1,1232 f 1,44 ] is the one required to represent constraint C.6: G z (scalar). 6 = Block G 7 Constraint C.7. The su of all iports of services (lying on row 88 of Z) inus Mg (cell [88,44] of Z) ust be equal to zero: given our product classification (see the Read e file in the ain page of the website), it turns out that services are the 14 products fro 29 to 42. To translate this into constraints on vector z, define the 1232-vector h = 1, 14 ( f ). Then G = h f is the one required to represent constraint the 4224-vector [ ] C.7: G z (scalar). 7 = 7 1,2464 1,88 1,352 5
Block G 8 Constraint C.8. The su of all ites of household consuption (lying on colun 43 of Z) inus C (cell [86,43] of Z) ust be equal to zero. To translate this into constraints on vector z, define the 88-vector d = [ u 84 1 ]. G = d is the one required to represent constraint C.8: Then the 4224-vector [ ] G 8 z = (scalar). 8 1,3696 1,44 Block G 9 Constraint C.9. The su of all ites of Collective consuption of General Governent (lying on colun 44 of Z) inus Cg (cell [86,44] of Z) ust be equal to zero. To translate this into constraints on vector z, recall the definition of the 88-vector d fro block G 8. Then the 4224-vector G = [ d ] is the one required to represent constraint C.9: G z (scalar). 9 = 9 1,3784 1,352 Block G 1 Constraint C.1. The su of all ites of Collective consuption of social Institutions (lying on colun 45 of Z) inus Cs (cell [86,45] of Z) ust be equal to zero. To translate this into constraints on vector z, recall the definition of the 88-vector d fro block G 8. Then the 4224-vector G 1 = [ 1,3872 d 1,264 ] is the one required to represent constraint C.1: G z (scalar). 1 = Block G 11 Constraint C.11. The su of all ites of Fixed investent (lying on colun 46 of Z) inus Is (cell [86,46] of Z) ust be equal to zero. To translate this into constraints on vector z, recall the definition of the 88-vector d fro block G 8. Then the 4224-vector G 11 = [ 1,396 d 1,176 ] is the one required to represent constraint C.11: G z (scalar). 11 = Block G 12 Constraint C.12. The su of all ites of Changes in stocks (lying on colun 47 of Z) inus s (cell [86,47] of Z) ust be equal to zero. To translate this into constraints on vector z, recall the definition of the 88-vector d fro block G 8. Then the 4224-vector G = [ d ] is the one required to represent constraint C.12: G z (scalar). 12 = 12 1,448 1,88 6
Block G 13 Constraint C.13. The su of all Exports of goods (lying on colun 48 of Z, rows 1 to 28 [doestic products] and rows 43 to 7 [re-exports of iports]) inus Xg (cell [86,48] of Z) ust be equal to zero. G = u u 1 is the One can check that 4224-vector [ ] one required to represent constraint C.13: 13 1,4136 28 1,14 28 1,14 G 13 z = (scalar). Block G 14 Constraint C.14. The su of all Exports of services (lying on colun 48 of Z, rows 29 to 42 [doestic products] and rows 71 to 84 [re-exports of iports]) inus Xs (cell [87,48] of Z) ust be equal to zero. G = u u 1 is the One can check that 4224-vector [ ] one required to represent constraint C.14: 13 1,4136 1,28 14 1,28 14 G 14 z = (scalar). Putting all together, one obtains: as said in section 3 above. G1z 42,1 G 2z 42,1 G3z 42,1 G 4z G5z G 6z G 7z Gz = = = 137,1 = G8z G9z G1z G11z G12z G13z G14z 137 7
Appendix 2 A progra for building atrix G The progra is written in MATLAB language, but it should not be difficult to translate it into other languages. It is highly recoended that sparse atrices are used (if this is allowed by your package) in building and using atrix G in progras, e.g. when ipleenting the balancing algorith (2). This is in fact uch ore efficient than using the full version, given the large diensions involved. Hence, the use of the sparse instruction is everywhere present in the following script. Other details: sparse(x) defines atrix X as sparse speye(n) = sparse identity atrix of size n ones(n,) = (n,) atrix all eleents of which are one zeros(n,) = (n,) atrix all eleents of which are zero [a b] = horizontal concatenation of eleents a and b: a and b ight be scalars, vectors, or atrices: but the nuber of their rows ust be the sae [a;b] = vertical concatenation of eleents a and b: a and b ight be scalars, vectors, or atrices: but the nuber of their coluns ust be the sae kron(a,b) = Kronecker product = a b % construction of sparse 42-identity atrix I42=speye(42); % construction of various sparse vectors of ones (subsequently I will oit sparse ) u14=sparse(ones(1,14)); u28=sparse(ones(1,28)); u42=sparse(ones(1,42)); u48=sparse(ones(1,48)); u84=sparse(ones(1,84)); u86=sparse(ones(1,86)); % construction of various atrices and vectors of zeros O424=sparse(zeros(42,4)); O4242=sparse(zeros(42,42)); O4246=sparse(zeros(42,46)); O42528=sparse(zeros(42,528)); z1232=sparse(zeros(1,1232)); z14=sparse(zeros(1,14)); z176=sparse(zeros(1,176)); z2464=sparse(zeros(1,2464)); z264=sparse(zeros(1,264)); z28=sparse(zeros(1,28)); z352=sparse(zeros(1,352)); z3696=sparse(zeros(1,3696)); z3784=sparse(zeros(1,3784)); z3872=sparse(zeros(1,3872)); z396=sparse(zeros(1,396)); z448=sparse(zeros(1,448)); z4136=sparse(zeros(1,4136)); z44=sparse(zeros(1,44)); z528=sparse(zeros(1,528)); z85=sparse(zeros(1,85)); z86=sparse(zeros(1,86)); z87=sparse(zeros(1,87)); z84=sparse(zeros(1,84)); z88=sparse(zeros(1,88)); % construction of soe vectors of ixed ones, inus ones and zeros x=[z86-1 ]; y=[u86-1 ]; z=[z87-1]; w=[z85 1 ]; b=[z84 1 ]; c=[z87 1]; d=[u84-1 ]; 8
% construction of G1 H=[I42 O4246]; G11=kron(u48,H); G121=kron(I42,x); G12=[G121 O42528]; G1=G11+G12; clear G11 G121 G12 H; % construction of G2 G21=kron(I42,y); G2=[G21 O42528]; clear G21; % construction of G3 H=[O4242 I42 O424]; G31=kron(u48,H); G321=kron(I42,z); G32=[G321 O42528]; G3=G31+G32; clear H G31 G321 G32; % construction of G4 G41=kron(u42,w); G4=[G41 z528]; clear G41; % construction of G5 H=kron(u42,b); G5=[H -b z44]; clear H; % construction of G6 H=kron(u28,c); G6=[H z1232 -c z44]; clear H; % construction of G7 H=kron(u14,c); G7=[z2464 H z88 -c z352]; clear H; % construction of G8 G14 G8=[z3696 d z44]; G9=[z3784 d z352]; G1=[z3872 d z264]; G11=[z396 d z176]; G12=[z448 d z88]; G13=[z4136 u28 z14 u28 z14-1 ]; G14=[z4136 z28 u14 z28 u14-1 ]; % final construction of G G=[G1;G2;G3;G4;G5;G6;G7;G8;G9;G1;G11;G12;G13;G14]; % reove soe large atrices fro eory clear G1 G2 G3 G4 G5 G6 G7 G8 G9 G1 G11 G12 G13 G14 I42 O4246 O42528; After this, store atrix G in soe file, so that it can be recovered when needed (MATLAB would use save and load respectively). 9