Spatial and sectoral aggregation in the commodity - CiteSeerX

Environment and Planning A, 1990, volume 22, pages 1637-1656

Spatial and sectoral aggregation in the commodityindustry multiregional input-output model R E Miller Regional Science Department, University of Pennsylvania, Philadelphia, PA 19104, USA G Shao Regional Economic Models Incorporated, Amherst, MA 01002, USA Received 18 October 1989

Abstract. In this paper, some experiments on the effect of aggregation in multiregional input-output (MRIO) models are reported. Sectoral aggregation and spatial aggregation are examined, separately and jointly, with use of hypothetical MRIO accounts and also with use of the 1977 US MRIO data. The spatial aggregation experiments constitute an extension and update of similar kinds of research reported on in a previous paper by Blair and Miller, in which hypothetical accounts and the 1963 US MRIO data were used. An important difference between the 1963 and 1977 data sets is that the latter are in the commodity-by-industry format. In the experiments reported here, it has been possible to deal with larger data sets for the least aggregated base cases, and the amount of data that is 'covered up' in the aggregations is therefore much larger. Nevertheless, it is still possible to conclude that spatial aggregation in MRIO models need not generate unacceptable error. The term error suggests that the most disaggregated data are always most accurate; because this position is not embraced by all input-output observers, we can alternatively describe the work in this paper as an investigation into the sensitivity of MRIO model results to sectoral and/or spatial aggregation. 1 Introduction In earlier p a p e r s by o n e of the present authors (Miller), the effects of spatial aggregation in an interregional i n p u t - o u t p u t (IRIO) m o d e l and an industry - industry multiregional i n p u t - o u t p u t (MRIO) m o d e l were investigated (Blair and Miller, 1 9 8 3 ; Miller and Blair, 1980). A general conclusion was that for many realistic situations the errors caused by spatial aggregation are small. (1) T h e latest (1977) U S M R I O m o d e l (US D H H S , 1984) has a structure that differs from the previous version (Polenske, 1980). It is constructed from a set of commodity-by-industry data, a convention that has emerged since the 1 9 6 3 model. T h e 1 9 7 7 U S M R I O m o d e l consists of 51 states (50 states plus Washington, D C ) and 124 commodities (or industries). T h u s the dimensions of total requirements matrices in the m o d e l will b e at least 6 3 2 4 x 6 3 2 4 . T h i s implies a t r e m e n d o u s a m o u n t of c o m p u t e r time and storage, not only to generate the total requirements matrix itself but also to carry out other computations as well. A l t h o u g h c o m p u t e r technology has developed dramatically and many powerful personal and supermainframe computers are available, there is still a high cost associated with use of a mainframe computer. Aggregation may therefore b e a p p r o p r i a t e . For different research purposes, the aggregation can b e over space only, over sectors only, or over both. In this paper, we examine the sensitivity of o u t p u t (1) In this paper, we have followed the general practice in the literature of assuming that less aggregated input-output data are more accurate than more aggregated data. That is, we use the concept of 'aggregation error' throughout. An alternative interpretation is that we are simply measuring 'sensitivity' to levels of aggregation. With that viewpoint, what we have termed 'mean absolute percentage error' might better be thought of as 'mean absolute percentage sensitivity'.

1638

R E Miller, G Shao

results to such aggregations. The paper is arranged as follows: the commodityindustry MRIO model is reviewed in section 2; the construction of hypothetical MRIO accounts and models is described in section 3; the errors associated with aggregation in the hypothetical model are discussed in section 4; presentation and comparison of aggregation errors in the 1977 US MRIO model is in section 5; and conclusions are contained in section 6. 2 The commodity-industry MRIO model In the 1977 US MRIO model, commodities and industries are classified separately.(2) This enables one to account for the multiple outputs that each industry may potentially produce; input and output structures of each industry are recorded in a different matrix. In this section, we review those aspects of the formulation of the model which will be needed in the rest of this paper.(3) The 'make' matrix, VL, is used to represent the output structure of industries in region L ; element v-* in the 'make' matrix records the amount of primary (/ = j) or secondary (/ ^ j) product (commodity) / (= 1,..., n) produced by industry j (= 1,..., JZ) in region L(= 1,..., q). The input structure for industries in region L is represented in the 'use' matrix, U L , in which the element uj- is the amount of intermediate use (input) of commodity / by industry j in region L. The 'use' matrix records the intermediate (industry) demand for commodities; and the vector, EL, accounts for final demand for commodities in region L. In addition, for each commodity /, there is a 'trade' matrix, T z , representing the shipment of commodity /, t™N, from region M to region N (M, N = 1,..., q). The commodity - industry MRIO accounts for each region must satisfy several conditions, (a) Within a single region, the total consumption of (demand for) commodity / is the sum of the zth row in the 'use' matrix in region L plus sales of commodity i to final demand; the total production (supply) of that same commodity is the corresponding row sum of the 'make' matrix in region L. The difference (if any) between these two sums is the net trade flow of commodity / into or out of region L. (b) The total industry output must be equal to all commodity inputs plus any value-added inputs, (c) The output of each commodity in each region must be equal to the sum of the amount of that commodity shipped from the region (including to itself), and the consumption of each commodity should equal the sum of the amounts of that commodity shipped into the region (including from itself). Define L

= VLi + EL

(1)

L

= vL i ,

(2)

CQ

and PQ

L

L

where CQ and pQ are the vectors of commodity consumption and production in region L, respectively, and i is the unit vector; in general CQ{L ^ pQ{L. [These satisfy LpQL+EL

.

(10) l

q

Define CQ to be the ^ - e l e m e n t vector formed by concatenation of cQ ,..., cQ (the ^-element vectors of commodity consumption in each of the regions) and define pQ and E similarly. Also, define B as a diagonal matrix with elements B 1 , B 2 , . . . , Bq, and D as a diagonal matrix with elements D 1 , D 2 , . . . , D 9 —both are nqXnq block diagonal matrices. Then the parallel equation to equation (10) that expresses the relationships for all regions simultaneously is c

£ = BDpQ + £ .

(11)

In order to generate an expression with commodity production, pQ, on the lefthand side, define trade coefficients, cf" (= t^^QHLet CLM(M = 1,..., q) be an n x n diagonal matrix of the trade coefficients for all n commodities shipped from

1640

R E Miller, G Shao

L to M (many of these elements may be zero). Then the relationship expressed in equation (4) can be written p

e L = [CL1:CL2i - :CL«]C0,

(12)

and, for all regions simultaneously, PQ

= CCQ,

(13)

where

c11 - clq C =

cql - cqq

Last, then, putting equation (11) into equation (13), we obtain pQ

= CBDp2 + C £ ,

and the industry-based technology commodity-by-commodity MRIO model, connecting commodity final demand to commodity output, can be written as pQ

= (l-CBDr'CE

(14)

In place of D L , in the commodity-based technology model the matrix GL is incorporated, defined as GL=\L(XL)~1,

m

(15)

where gtf (= v^\X^) denotes the fraction of total production in industry / in region L that is in the form of commodity /. These fractions are referred to as industry output proportions, and use of the commodity-based technology framework embodies the assumption that the output of each industry in region L is composed of commodities in the fixed proportions given by the elements of G L . In order to arrive at a total-requirements matrix under the assumption of commodity-based technology, use equations (2) and (15) to derive

XL = (G L r i P e L . From equation (7) [as has already been observed in equation (9)], U L i = BL XL; thus ULi = BL(GL)"1peL

(16)

[compare with equation (9)]; and substituting into equation (1), we get L

CQ

= BL(GL )-\QL+EL

(17)

[compare with equation (10)]. Define G _ 1 as the block diagonal matrix with elements (G 1 )" 1 , (G 2 ) - 1 ,..., (G*7)"1, then the parallel to equation (11) that expresses the relationship in equation (17) for all regions simultaneously is cQ

= BG1pQ

+ E.

(18)

Using the expression in equation (13) once again, we have pQ

= CBG\Q + CE ,

from which the commodity-based technology commodity-by-commodity MRIO model is p

6 = (I-CBG"1)"1^

[compare with equation (14)].

(19)

Spatial and sectoral aggregation

1641

For purposes of comparisons that will be made later in this paper, it is useful to contrast the commodity-by-commodity models expressed in equations (14) and (19) with the earlier 1963 US MRIO model. That model is a spatial extension of Leontief's original ^-sector formulation in which, as is well known, X =

Zi+Y

and A=

Z(X)-1

where Z is the nx n transactions matrix, A is the matrix of technical coefficients, and X and Y are ^-element vectors of gross outputs and final demands, respectively. In the notation of this section (but ignoring, for expositional simplicity, the superscript L), in the 1963 US accounts U = Z, B = A, E = F, CQ = pQ = X, V = X, and D = G = I. The fundamental 1963 MRIO model [corresponding to equations (14) and (19)] is X = (I-CA)_1CF.

(20)

3 Experiments with random matrices When considering possible applications of the MRIO model, one may be concerned with economic changes in only one or a small number of regions. For example, in order to keep the number of regions small, the United States of America might be separated into only two regions, one of which consists of one or more states (the region of interest) and the other of which is the aggregation of the rest of the states. Alternatively, an analyst might want to keep all of the regions in the model, but aggregate the sectors in each region, again to reduce the size of the model; or one may want to aggregate some regions and some sectors simultaneously. Most research on input-output sectoral aggregation has been done on the national (or single-region) model rather than on a connected spatial model of the MRIO sort. Miller and Blair (1980) and Blair and Miller (1983) investigated spatial aggregation for the IRIO model and the industry-industry version of the MRIO model. In the present paper, we investigate aggregation of regions only, of sectors only, and of regions and sectors simultaneously in the commodity-bycommodity version of the MRIO model under the assumption of both a commoditybased and an industry-based technology. To this end we use hypothetical data as well as the 1977 US MRIO accounts. 3.1 Generating hypothetical MRIO models A specific IRIO or MRIO model may contain special characteristics for a certain nation's (multiregional) economy or certain economic situations particular to the year in which the model was built. Therefore, empirical research for the aggregation problem can be carried out by using a hypothetical model (created via random numbers) and by using the particular real model as well. This can help to alleviate the acceptability problem (Leontief, 1951), which is that an acceptable aggregation for one application may not be equally acceptable in another case. Although the hypothetical model can provide some insights into the aggregation issue, it may underestimate the error in an MRIO model, as Blair and Miller (1983) indicated. A total of 40 hypothetical commodity - industry MRIO models were constructed; 20 are industry-based technology models and the other 20 are commodity-based technology models. Each model has 5 regions and 10 commodities (industries). Construction of these models is based on the 20 artificial MRIO accounts described below.

1642

R E Miller, G Shao

The 'use' and the 'make' matrices for all regions were created first. Elements of those matrices were randomly drawn from a uniform distribution on the unit interval, then multiplied by 100 (or by 1000 for the diagonal elements of the 'make' matrices). This was done to help ensure that the output of primary product was at least 50% larger than any secondary product, represented by off-diagonal elements in any column. (When the '50% criterion' was not met by a given element, it was discarded.) After a value-added vector was created for each region, the condition on industry output, that a column sum of the 'make' matrix be equal to the corresponding column sum of the 'use' matrix plus value added [as in equations (5) and (6)], was examined. If the two were not equal, the appropriate column in the 'use' matrix was adjusted, using the 'RAS' method on columns only. Some 10 initial 5 x 5 'trade' matrices were constructed, also by using numbers drawn randomly from a uniform distribution on the unit interval and then multiplied by 100. After the vectors of regional commodity final demand were created, total commodity consumption {CQL) and production {pQL) in each region were derived, with use of equations (1) and (2). Then the conditions in equations (3) and (4) were examined by using the randomly generated 'trade' matrices. Generally, these conditions were not met, and an RAS-type of method was used to modify the 'trade' matrices. Each RAS iteration forced first a row balance and then a column balance. After the first half of the twentieth iteration, the row-sums condition [equation (4)] was satisfied and the residuals that were left for the column sums of the 'trade' matrices were assumed to be part of the regional final demand. At the conclusion of all of these steps, one artificial commodity - industry MRIO account had been established. A total of 20 such accounts were generated. Then, with use of the models expressed in equations (14) and (19), the 'total-requirements' matrices were calculated under the assumption of industry-based and commodity-based technology. Thus, 20 commodity-industry MRIO models for each of the technology assumptions were created. One can classify studies in which a MRIO model structure might be used into two broad groups. In one case, the analyst is interested in the effects in one region of an exogenous (demand) change in that one region only (for example, changes in federal trade policy that decrease exports of commercial airliners made by a manufacturer located in one region). Here the connected-region input-output model would be appropriate if it was felt that interregional feedbacks were important. [Of course, the analyst might be interested in impacts also on other regions (interregional spillovers) that were generated by the change in final demand of that one region; in that situation, too, some sort of multiregional model is needed.] In the second group of studies, the exogenous demand changes occur simultaneously in several regions (for example, changes in federal agricultural policy that increase government purchases of agricultural outputs from farmers in many states). In this kind of situation, an MRIO model formulation might be used to connect the demand changes and consequent economic activity in the several states. We have employed two kinds of experiments in order to investigate aggregation sensitivity in these two cases—that is, by estimating the change in total commodity output owing to a change in final demand in one region only or owing to changing final demand in all regions simultaneously. These are referred to as 'region-1' experiments and 'all-region' experiments, respectively (in the spatial aggregation simulations, 'region 1' always denotes the unaggregated region). In each case, 5 different randomly generated final demand vectors were created for the 20 hypothetical accounts.(4) (4

> The number of final demand vectors generated for these experiments was limited by the requirements of computer time and storage.

Spatial and sectoral aggregation

1643

3.2 Spatial aggregation In the spatial aggregation experiments, the first aggregation combined regions 4 and 5 in each of the original hypothetical MRIO accounts by combining the 'use' and the 'make' matrices of these 2 regions and adding the elements in the fifth row and column to the fourth row and column in every 'trade' matrix. This reduces the accounts to 4 regions and the dimensions of the 'trade' matrices from 5 x 5 to 4 x 4 . Recalculating the total requirements matrices, we obtained a group of new aggregated commodity-industry MRIO models. Regions 4 and 5 elements in the final demand vectors were aggregated and then used to compute total commodity output. Since there were 5 final demand vectors and 20 MRIO models for each technology, there were 100 calculated vectors of total commodity output for 'region-1' and for 'all-region' experiments. The same procedure was repeated to generate a 3-region MRIO account system, in which the 'use' and the 'make' matrices of region 3 were combined with those matrices of regions 4 and 5, and the third row and column in the 'trade' matrices were combined with the fourth and fifth rows and columns to form new 3 x 3 'trade' matrices. Once this is completed, another new aggregated vector of total commodity outputs can be computed. Last, a 2-region set of MRIO accounts was created in an exactly parallel way. 3.3 Sectoral aggregation One of the standard results on sectoral aggregation is that the distortion caused by aggregation of 2 sectors is minimal in an industry - industry input-output model if the 2 sectors have the same production functions—that is, columns in the A matrix (Morimoto, 1970). In the commodity-industry MRIO model, one is faced with both 'use' and 'make' matrices, and commodities (industries) may have different consumption and/or production patterns. Thus, aggregation effects are not likely to be mitigated in the commodity - industry input-output model, compared with the industry - industry version (for example, the 1963 US MRIO model and the US national models prior to 1972). As there is no reason in the hypothetical models to consider which commodities (or industries) are most appropriately aggregated, commodities (or industries) 5 through 10 were chosen as candidates for the aggregation. The sectoral aggregations were performed five times in our experiments. Initially, commodities and industries 9 and 10 were combined together in the 'use' and 'make' matrices of each region. The 'trade' matrices for commodities 9 and 10 are simply summed. The 'totalrequirements' matrix for the resulting MRIO model has dimensions 45 x 45. Next, one additional commodity and industry join the aggregation—commodities and industries 8 in the second run, 7 in the third run, and so on, until there were only 5 commodities and industries in the models. Commodities and industries 1 to 4 always remained disaggregated in all exercises. The dimensions of the totalrequirements matrices in the MRIO models were thus reduced from 50 x 50 to 25 x 25, in 5-unit intervals. 3.4 Spatial and sectoral aggregation A combination of spatial and sectoral aggregation is probably what most analysts would use in reducing the size of a many-region, many-sector model. We have investigated 3 cases of such combined aggregation—commodities and industries 9 and 10 were combined together in the first case, 8 to 10 in the second case, and 5 to 10 in the third case. For all cases, the same regional aggregations as discussed in section 3.2 were used; that is, combining regions 4 and 5, 3 to 5, and 2 to 5.

R E Miller, G Shao

1644

4 Results from the trials with random matrices 4.1 Measurement of error (or sensitivity) The mean absolute percentage error, i, is used to measure the differences in outputs in the aggregated and disaggregated models. Let Qk and Qk* be the vectors of total commodity output in, respectively, the disaggregated and aggregated model for the kih (k = 1,..., r) experiment, and let S be the aggregation matrix containing zeros and ones. Then

.

ioo ' HSA-gii

' - " £

nsaii '

(21)

where r is the number of simulations in an experiment (r = 100 in all of our experiments), and ||.|| is the vector norm that denotes the sum of the absolute values of elements in the vector. [Because we found in our trials both negative and positive differences in the vectors of equation (21), the absolute values of these differences were used to prevent cancellation of negative and positive elements.] 4.2 Data content of the models We believe that it is useful to view the results on e in light of the reduced requirements for data that accompany more aggregated (smaller) models. Therefore, we need a figure that reflects the number of pieces of data required for each particular model. We refer to this as the 'data content' and denote the measure for a matrix M as dc(M). In the industry-based technology model, of equation (14), the building blocks are matrices C, B, and D; in the commodity-based technology model, of equation (19), they are C, B, and G (from which G _ 1 follows, provided that G is nonsingular). From equation (7), if there are n commodities (and industries), B L requires n2 pieces of transaction information (in the 'use' matrix, U L ) and the n elements in

XL. From equation (8), D L requires the data in the nXn 'make' matrix, V L , as well as the n elements of pQL. However, elements in XL are derived as column sums of VL [as in equation (6)] and the elements in pQL are row sums of VL [as in equation (2)]. Thus, only the information contained in U L and VL is needed. Although both are nXn matrices, the 'use' matrix is likely in general to contain fewer zero cells than the 'make' matrix. 'Make' matrices tend very much toward diagonality, the number (and size) of off-diagonal elements indicating the extent of secondary production.^ For our rough measure of data content, we therefore use n2 pieces of data for each U L (and hence for each B L ) and n for each VL (and hence for each D L ). For a ^-region model, our data-content measure for B and D in equation (14) will be given by dc(B) = n2q and dc(D) = nq, respectively. Last, C is made up of q2 (5)

Of course the level of sectoral aggregation makes a difference. In general, a more aggregated 'use' matrix will be less sparse than a less aggregated one for the same economy. For 'make' matrices it is not so clear how the tendency toward diagonality is affected by matrix size. Blair and Wyckoff (1989) present some empirical evidence that bears on this question, based on 'make' matrices for the US national economy in 1972 and 1977. They define the 'secondary production ratio' as the sum of off-diagonal elements in the 'make' matrix as a percentage of the total of all elements in that matrix (which equals total gross output). For 1972, this ratio is 3.44 for the 85-sector data set and it decreases monotonically with sectoral aggregation to 1.51 for a 7-sector model. Similarly, for 1977, the ratio is 5.47 for the 533-sector data set and it is 1.56 for a 7-sector model. It is likely that the smaller this ratio, the closer the 'make' matrix is to diagonal, although the exact relationship will depend not only on the total amount of nonprimary production (the sum of the off-diagonal elements) but also on the distribution of this sum over those elements—that is, on how many of the off-diagonal elements are nonzero.

1645

Spatial and sectoral aggregation

diagonal matrices, each containing n trade coefficients, so dc(C) = nq2S6) Thus our measure of data content for the industry-based technology model in equation (14) is dc(B) + dc(D) + dc(C) = n2q + nq + nq2 = nq(n + q + l). L

L

L

(22) 1

In the commodity-based technology model, G = \ {X )~ [as in equation (15)] and, as above, we will count n pieces of data for each VL (and therefore for G L , as the XL are column sums of V L ). Thus the data-content measure for the model in equation (19) is also given by equation (22), as dc(D) = dc(G). It will also be the same for the fundamental 1963 US MRIO model expressed in equation (20), as dc(A) = dc(Z) + dc(Z) = n2q + nq , = dc(B) + dc(D), = dc(B) + dc(G) . 4.3 Evaluation of the results Values for the mean absolute percentage error, s, in gross commodity outputs in the industry-based technology and the commodity-based technology commodity-bycommodity hypothetical MRIO models are presented in three tables in this section, for various experiments. The general layout of all tables is the same. Results are presented first for the 'region-1' experiments, then for the 'all-region' experiments. For each experiment, results are given for the industry-based technology model [as expressed in equation (14)] and for the commodity-based technology model [as given in equation (19)]. For each of these models, the first results are labelled 'region 1'. This means that the e-values [as in equation (20)] were calculated for pQl and (pQ1)* (the first n elements in pQ and pQ*), the vectors of commodity output for region 1 only. The second set of results, labelled 'other regions', contain f-values calculated for the remaining n(q — l) elements in pQ and pQ*, containing commodity outputs for each of the other regions. Last, results for the 'total MRIO system' contain ^-values for the output vectors of the entire multiregional system (pQ and (The results for the 'total MRIO system' lie between the other two in value, pQ*). as the figures in these rows are weighted averages of the entries in the preceding two rows; the weights are the relative outputs of region 1 and all other regions.) Also given is the data-content measure for the aggregated model as a percentage of the data-content measure for the most disaggregated model; we term this the 'relative data content'. For example, a data content of 25 would indicate that the data requirements of that particular aggregated model are one quarter of those of the most disaggregated alternative. Specifically, these values of data content are calculated by taking the data content of the 'total-requirements' matrix for the aggregated model and dividing it by that for the most disaggregated model, and then multiplying this figure by 100. From the three tables we can see that: (1) the overall values of the mean absolute percentage errors, i, are small, (2) the errors for industry-based technology models are generally (sometimes not much) smaller than those for commodity-based technology models, and (3) with some exceptions, values of e are larger for 'region1' results than for those in the 'other regions'. Table 1 contains the results from the spatial aggregation experiments. As might be expected, errors tend to increase with aggregation; there are a few exceptions in the last column, which contains the results for the greatest amount of spatial As CiLM = tiLM/cQiM, the underlying information needed for a particular CLM (diagonal) matrix are the n trade figures (^LM,..., t„M) and the ^-element vector CQM. However, given a full set of ttLM for all sectors, i, and all regions of origin, L, CQM is derived from column sums of the th as in equation (3).

R E Miller, G Shao

1646

aggregation. For the 'region-1' experiments, the values of e for commodity-based technology experiments tend to be larger than those for the industry-based technology experiments by a factor of 3 or more. We have no explanation for this difference. However, in the 'all-region' experiments, the figures are very similar (the commoditybased technology results are consistently but only marginally larger). The largest value of s in the 'region-1' experiments is 0.88 for the industry-based technology model and 2.58 for the commodity-based technology experiments. These occur under the most extreme spatial aggregation (in which only one-third of the disaggregated model data are needed), and we would argue that the e values are not distressingly large. For the 'all-region' experiments, the maximum values of e are 7.69 and 7.73 for these same two models, respectively—certainly a less satisfactory outcome. However, to 'miss' by 8% when using only one-third as much information, may not be so outrageously bad. The random-matrix experiments reported in Blair and Miller (1983) are for a model built on the structure of the 1963 US MRIO framework, in which no distinction is made between commodities and industries. The spatial aggregation results in the present paper are thus not strictly comparable with those in Blair and Miller (1983). We nonetheless offer a few potentially comparable figures. The Blair and Miller random-matrix experiments also began with a 5-region, 10-sector set of accounts, and spatial aggregation proceeded as in the present paper. Results of those earlier experiments are shown in table 2.(7) We have no specific explanations for particular kinds of differences, other than the general Table 1. Mean absolute percentage error in gross outputs for different levels of spatial aggregation (hypothetical MRIO model). Region or system

Number of regions outside region 1 3

2

1

Data content (%) 75 53 33 'Region-1' experiments: industry-based technology Region 1 0.43 0.71 0.88 Other regions 0.36 0.52 0.50 Total MRIO system 0.37 0.55 0.57 'Region-1' experiments: commodity-based technology Region 1 1.36 2.02 2.58 Other regions 1.24 1.71 1.92 Total MRIO system 1.26 1.75 2.04 All-region' experiments: industry-based technology Region 1 3.17 5.49 7.69 Other regions 2.73 2.86 1.89 Total MRIO system 2.81 3.37 3.03 All-region' experiments: commodity-based technology Region 1 3.25 5.57 7.73 Other regions 2.84 3.00 2.08 Total MRIO system 2.91 3.50 3.19 Note: for 3 regions, 2 regions, and 1 region outside region 1, regions 4 and 5, 3 - 5 , and 2 - 5 , respectively, are aggregated.