EC 936 ECONOMIC POLICY MODELLING
LECTURE 2:
PART I
UPDATING MATRICES:
i: BIPROPORTIONAL (RAS) METHOD
[DEMING & STEPHAN, 1940; LEONTIEF, 1941;
STONE, 1962; BACHARACH, 1970]
MATRIX BALANCING ISSUES
• TYPE I & TYPE II PROBLEMS:
• I: AMENDING MATRIX ENTRIES TO CONFORM TO
NEW ACCOUNT TOTALS
• II: AMENDING MATRIX ENTRIES WHEN ACCOUNT
TOTALS ARE UNKNOWN, BUT ARE GOVERNED BY
KNOWN ACCOUNTING CONSTRAINTS
RAS AND TYPE I PROBLEMS
Input-output transactions matrix, U.S. 2002
1
2
3
4
5
6
7
Agriculture
Mining
Construction
Manufacturing
Trade, Transport
Services
Other
Gross output
1
72028
703
1168
40495
23899
37151
154
271182
7
1653
7909
42456
198600
97013
432413
26458
Gross
Output
271182
184556
1032363
3817360
2683436
9114939
2076198
184556 1032363 3817360 2683436 9114939 2076198
19180034
2
361
8611
6621
17122
12207
46606
1008
3
4
2763 145716
9234 140728
718
12208
253489 1343085
103373 360139
152787 543989
410
31210
5
6
997
7406
63095
2224
15697
74277
132739 424488
220365 239088
508815 2513262
52162
79038
Technical coefficient matrix, U.S. 2002
1
2
3
4
5
6
7
Agriculture
Mining
Construction
Manufacturing
Trade, Transport
Services
Other
1
0.2656
0.0026
0.0043
0.1493
0.0881
0.1370
0.0006
2
0.0020
0.0467
0.0359
0.0928
0.0661
0.2525
0.0055
3
0.0027
0.0089
0.0007
0.2455
0.1001
0.1480
0.0004
4
0.0382
0.0369
0.0032
0.3518
0.0943
0.1425
0.0082
5
0.0004
0.0235
0.0058
0.0495
0.0821
0.1896
0.0194
6
0.0008
0.0002
0.0081
0.0466
0.0262
0.2757
0.0087
7
0.0008
0.0038
0.0204
0.0957
0.0467
0.2083
0.0127
PROBLEM: HOW TO UPDATE A TRANSACTIONS MATRIX WHEN
ONLY THE GROSS OUTPUT FIGURES ARE KNOWN
Gross output
2002
1
2
3
4
5
6
7
Agriculture
Mining
Construction
Manufacturing
Trade, Transport
Services
Other
Gross output
2006
271,182
184,556
1,032,363
3,817,360
2,683,436
9,114,939
2,076,198
319,045
497,343
1,392,907
4,911,868
3,554,657
11,426,365
2,693,408
19,180,034
24,735,592
SOME BASIC DEFINITIONS
Transactions:
zij
Gross output:
Xj
zij
Input-output (technical) coefficients:
aij =
_______
Xj
For rectangular matrices (m≠n)
min
s.t.
for i = 1,2,…,m
for j = 1,2,…,n
Technical coefficient matrix, U.S. 2006
1
2
1 Agriculture
0.2441
0.0000
2 Mining
0.0019
0.1312
3 Construction
0.0035
0.0002
4 Manufacturing
0.1867
0.0961
5 Trade, Transport
0.0801
0.0390
6 Services
0.0905
0.1340
7 Other
0.0008
0.0036
3
0.0014
0.0066
0.0010
0.2686
0.1100
0.1305
0.0016
5
0.0001
0.0310
0.0039
0.0580
0.0721
0.1908
0.0140
6
0.0018
0.0001
0.0072
0.0554
0.0341
0.2987
0.0090
7
0.0006
0.0062
0.0242
0.1028
0.0455
0.2098
0.0145
RAS results derived from 2002 technical coefficient matrix
1
2
3
4
1 Agriculture
0.2451
0.0015
0.0026
0.0362
2 Mining
0.0046
0.0672
0.0169
0.0672
3 Construction
0.0033
0.0221
0.0006
0.0025
4 Manufacturing
0.1442
0.0728
0.2526
0.3494
5 Trade, Transport
0.0877
0.0535
0.1061
0.0965
6 Services
0.1220
0.1828
0.1404
0.1305
7 Other
0.0006
0.0043
0.0004
0.0082
5
0.0003
0.0423
0.0045
0.0485
0.0830
0.1716
0.0194
6
0.0009
0.0005
0.0076
0.0549
0.0319
0.3001
0.0104
7
0.0008
0.0076
0.0174
0.1036
0.0521
0.2080
0.0140
Absolute error in technical coefficients
1
2
1 Agriculture
0.0010
0.0015
2 Mining
0.0027
0.0640
3 Construction
0.0002
0.0219
4 Manufacturing
0.0424
0.0233
5 Trade, Transport
0.0076
0.0145
6 Services
0.0316
0.0487
7 Other
0.0003
0.0007
4
0.0012
0.0073
0.0006
0.0167
0.0071
0.0025
0.0067
5
0.0003
0.0114
0.0006
0.0094
0.0109
0.0191
0.0054
6
0.0009
0.0004
0.0004
0.0004
0.0022
0.0013
0.0014
7
0.0002
0.0014
0.0068
0.0008
0.0067
0.0018
0.0005
Percentage absolute errors in technical coefficients
1
2
3
4
1 Agriculture
0.0042 640.2892
0.9113
0.0350
2 Mining
1.4270
0.4880
1.5545
0.0980
3 Construction
0.0529 104.1330
0.4334
0.3293
4 Manufacturing
0.2274
0.2425
0.0594
0.0503
5 Trade, Transport
0.0945
0.3712
0.0355
0.0688
6 Services
0.3492
0.3637
0.0763
0.0196
7 Other
0.3116
0.2055
0.7411
0.4469
5
3.3301
0.3667
0.1615
0.1625
0.1512
0.1003
0.3864
6
0.4928
3.0003
0.0497
0.0078
0.0642
0.0044
0.1599
7
0.2794
0.2276
0.2806
0.0079
0.1468
0.0084
0.0365
3
0.0013
0.0103
0.0004
0.0160
0.0039
0.0100
0.0012
4
0.0350
0.0745
0.0019
0.3327
0.1037
0.1280
0.0149
Technical coefficient matrix, U.S. 2005
1
2
1 Agriculture
0.2292
0.0000
2 Mining
0.0017
0.1438
3 Construction
0.0051
0.0002
4 Manufacturing
0.1965
0.0879
5 Trade, Transport
0.0847
0.0434
6 Services
0.0872
0.1319
7 Other
0.0008
0.0033
3
0.0015
0.0062
0.0010
0.2603
0.1046
0.1268
0.0016
5
0.0001
0.0367
0.0037
0.0546
0.0728
0.1821
0.0133
6
0.0017
0.0001
0.0071
0.0563
0.0346
0.2880
0.0086
7
0.0007
0.0066
0.0215
0.1011
0.0504
0.2097
0.0157
RAS results derived from 2003 technical coefficient matrix
1
2
3
4
1 Agriculture
0.2116
0.0000
0.0015
0.0350
2 Mining
0.0015
0.1756
0.0084
0.0416
3 Construction
0.0018
0.0001
0.0009
0.0011
4 Manufacturing
0.1285
0.0950
0.2520
0.2344
5 Trade, Transport
0.0591
0.0491
0.0858
0.0645
6 Services
0.0521
0.1309
0.1050
0.0714
7 Other
0.0005
0.0044
0.0015
0.0072
5
0.0002
0.0631
0.0040
0.0737
0.0949
0.2039
0.0152
6
0.0037
0.0002
0.0075
0.0824
0.0426
0.3083
0.0104
7
0.0016
0.0089
0.0212
0.1377
0.0674
0.2010
0.0172
Absolute error in technical coefficients
1
2
1 Agriculture
0.0176
0.0000
2 Mining
0.0002
0.0317
3 Construction
0.0033
0.0000
4 Manufacturing
0.0680
0.0071
5 Trade, Transport
0.0257
0.0058
6 Services
0.0352
0.0010
7 Other
0.0003
0.0010
4
0.0040
0.0247
0.0007
0.0895
0.0382
0.0499
0.0065
5
0.0001
0.0264
0.0003
0.0192
0.0222
0.0218
0.0020
6
0.0020
0.0001
0.0004
0.0261
0.0080
0.0203
0.0018
7
0.0009
0.0023
0.0002
0.0366
0.0170
0.0087
0.0015
Percentage absolute errors in technical coefficients
1
2
3
4
1 Agriculture
0.0767
1.0755
0.0087
0.1030
2 Mining
0.0942
0.2205
0.3565
0.3730
3 Construction
0.6401
0.2473
0.1413
0.3743
4 Manufacturing
0.3463
0.0806
0.0321
0.2762
5 Trade, Transport
0.3029
0.1333
0.1797
0.3719
6 Services
0.4031
0.0077
0.1720
0.4113
7 Other
0.3426
0.3076
0.0251
0.4768
5
0.6248
0.7209
0.0859
0.3511
0.3046
0.1198
0.1471
6
1.1582
0.7438
0.0499
0.4645
0.2320
0.0703
0.2064
7
1.4291
0.3567
0.0114
0.3621
0.3379
0.0416
0.0962
3
0.0000
0.0022
0.0001
0.0084
0.0188
0.0218
0.0000
4
0.0390
0.0663
0.0018
0.3239
0.1026
0.1212
0.0137
MAD =
MAPE =
RAS AND TYPE II PROBLEMS
Use Diagonal Similarity Scaling (DSS) Method
s.t.
EC 936 ECONOMIC POLICY MODELLING
LECTURE 2:
PART I
UPDATING MATRICES:
ii: CROSS-ENTROPY METHOD
[GOLAN ET AL, 1994; ROBINSON ET AL, 2001]
The entropy problem is to find a new set of A coefficients which minimize the so-called
Kullback-Leibler (1951) measure of the ‘cross entropy’ (CE) distance between the prior A
and the new estimated coefficient matrix A*.
subject to
Source: Bwanakare (2006), p. 240
RAS vs CROSS ENTROPY METHODS
(McDougall, 1999)
•
The RAS is an entropy optimization method, and has long been known to
be so.
•
For the matrix filling problem, in general, the entropy optimization method of
choice is proportional allocation.
•
For the matrix balancing problem, in general, the entropy optimization
method of choice is the RAS.
•
If, following the GCE approach, we treat matrix elements as expected
values of discrete random variables, the method of choice (in the absence
of distributional data) is equivalent to the RAS.
•
Entropy theory may fruitfully be used, not in attempting to supplant the RAS,
but in extending and adapting it to problems that do not well fit the
traditional matrix balancing framework.
EC 936 ECONOMIC POLICY MODELLING
LECTURE 2:
PART II
CALIBRATING SAMS:
ERRORS, INCOMPLETE &
INCONSISTENT INFORMATION
[STONE, CHAPERNOWNE, MEADE 1942;
STONE 1977; BYRON 1978]
G =
1
0
0
1
-1
0
0
0
1
0
0
1
0
0
0
0
x=
1
0
0
1
0
0
0
0
1
0
0
1
0
-1
0
0
220
390
100
50
750
130
200
350
30
725
420
250
240
175
560
50
140
70
60
130
130
-1
0
0
0
0
0
0
0
0
1
0
-1
1
0
0
0
0
1
0
0
1
0
0
0
Gx =
0
1
0
0
1
0
0
0
0
1
0
0
1
-1
0
0
10
-15
0
-30
65
0
0
-100
0
-1
0
0
0
0
0
0
0
0
0
-1
0
0
0
1
v=
0
0
0
0
-1
0
0
1
0
0
0
-1
0
0
0
0
0
0
0
0
-1
0
0
0
13.75
3.9
6.25
0.03125
7.5
0.125
2
21.875
0.1875
7.25
4.2
2.5
2.4
1.75
35
0.03125
9.375
0.04375
0.0375
0.08125
0.08125
0
0
0
0
0
0
0
-1
0
0
0
0
0
-1
0
0
x** =
0
0
0
0
0
0
0
-1
0
0
1
0
0
1
0
-1
0
0
1
0
0
1
0
0
253.969
384.742
91.573
49.968
780.252
129.615
199.211
341.365
29.985
700.176
417.082
264.523
233.555
181.683
490.328
50.010
121.338
69.940
60.023
129.962
129.962
0
0
-1
0
0
0
-1
0
0
0
0
0
0
0
1
0
Incomplete, inconsistent Social Accounting Matrix
1a
1a Production Activities
Manuf
0
1b
Non-manuf
130
2a Factors of Production
Labour
420
2b
Capital
240
3a Instititutions
Low-income HH
0
3b
High-inc HH
0
3c
Government
0
TOTAL
.
1b
220
0
250
175
0
0
0
.
2a
0
0
0
0
560
140
70
.
2b
0
0
0
0
.
.
60
.
3a
390
200
0
0
0
0
0
.
3b
100
350
0
0
0
0
0
.
3c
50
30
0
0
50
0
0
130
TOTAL
750
725
.
.
.
.
130
Solving for inconsistencies
1a
1a Production Activities
1b
2a Factors of Production
2b
3a Instititutions
3b
3c
TOTAL
Manuf
Non-manuf
Labour
Capital
Low-income HH
High-inc HH
Government
1b
2a
2b
3a
3b
3c
0 253.969
0
0 384.742 91.573 49.968 780.252
129.615
0
0
0 199.211 341.365 29.985 700.176
417.082 264.523
0
0
0
0
0
.
233.555 181.683
0
0
0
0
0
.
0
0 490.328
.
0
0 50.010
.
0
0 121.338
.
0
0
0
.
0
0 69.940 60.0229
0
0
0 129.962
.
.
.
.
.
.
129.962
Solving for incompleteness
1a
1a Production Activities
1b
2a Factors of Production
2b
3a Instititutions
3b
3c
TOTAL
Manuf
Non-manuf
Labour
Capital
Low-income HH
High-inc HH
Government
0
129.615
417.082
233.555
0
0
0
780.252
1b
2a
2b
3a
3b
3c
253.969
0
0 384.742 91.573 49.968
0
0
0 199.211 341.365 29.985
264.523
0
0
0
0
0
181.683
0
0
0
0
0
0 490.328 43.614
0
0 50.010
0 121.338 311.601
0
0
0
0 69.940 60.0229
0
0
0
700.176 681.606 415.238 583.952 432.939 129.962
780.252
700.176
681.606
415.238
583.952
432.939
129.962
Initial (diagonal) variance matrix (V)
1
2
3
4
5
6
1 13.75
0
0
0
0
0
2
0
3.9
0
0
0
0
3
0
0
6.25
0
0
0
4
0
0
0 0.0313
0
0
5
0
0
0
0
7.5
0
6
0
0
0
0
0 0.125
7
0
0
0
0
0
0
8
0
0
0
0
0
0
9
0
0
0
0
0
0
10
0
0
0
0
0
0
11
0
0
0
0
0
0
12
0
0
0
0
0
0
13
0
0
0
0
0
0
14
0
0
0
0
0
0
15
0
0
0
0
0
0
16
0
0
0
0
0
0
17
0
0
0
0
0
0
18
0
0
0
0
0
0
19
0
0
0
0
0
0
20
0
0
0
0
0
0
21
0
0
0
0
0
0
Adjusted variance matrix (V**)
1
2
3
4
5
1 4.044 -1.142 -1.830 -0.010 1.062
2 -1.142 3.106 -1.272 -0.005 0.687
3 -1.830 -1.272 4.211 -0.008 1.100
4 -0.010 -0.005 -0.008 0.028 0.004
5 1.062 0.687 1.100 0.004 2.853
6 0.031 0.010 0.017 0.000 0.058
7 0.190 -0.057 -0.091 0.001 0.042
8 2.075 -0.624 -0.999 0.012 0.464
9 0.012 0.003 0.004 -0.021 -0.002
10 2.307 -0.667 -1.069 -0.008 0.563
11 0.668 0.407 0.652 0.003 1.730
12 -1.013 0.264 0.422 0.001 -0.326
13 0.363 0.269 0.431 0.002 1.065
14 -0.723 0.211 0.338 0.001 -0.173
15 -0.273 0.529 0.848 0.002 1.106
16 -0.001 0.001 0.002 -0.004 -0.001
17 -0.073 0.142 0.227 0.000 0.296
18 0.000 0.000 0.000 0.002 0.001
19 0.000 -0.001 -0.001 0.001 0.000
20 0.001 -0.001 -0.002 0.003 0.001
21 0.001 -0.001 -0.002 0.003 0.001
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0 21.875
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0.1875
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
7.25
0
0
4.2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2.5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2.4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1.75
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
35
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0.0313
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 9.375
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0.0438
0
0
0 0.0375
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0.0813
0
0
0
0
0
0
0
0
0
0
0
0
0
0
10
11
14
15
16
17
18
19
20
21
0.031 0.190
2.075
0.012
2.307
0.668
-1.013
0.363
-0.723
-0.273
-0.001
-0.073
0.000
0.000
0.001
0.001
0.010 -0.057 -0.624
0.003
-0.667
0.407
0.264
0.269
0.211
0.529
0.001
0.142
0.000
-0.001
-0.001
-0.001
0.017 -0.091 -0.999
0.000 0.001 0.012
0.004
-0.021
-1.069
-0.008
0.652
0.003
0.422
0.001
0.431
0.002
0.338
0.001
0.848
0.002
0.002
-0.004
0.227
0.000
0.000
0.002
-0.001
0.001
-0.002
0.003
-0.002
0.003
0.058 0.042 0.464
0.123 -0.008 -0.084
-0.002
0.000
0.563
0.031
1.730
-0.040
-0.326
0.001
1.065
-0.025
-0.173
-0.001
1.106
-0.031
-0.001
0.000
0.296
-0.008
0.001
0.000
0.000
0.000
0.001
0.000
0.001
0.000
-0.008 1.859 -1.542
-0.004
0.305
0.029
0.066
0.021
0.050
0.075
0.002
0.020
-0.001
-0.001
-0.001
-0.001
-0.043
0.060
3.337
0.013
0.313
-0.001
0.718
0.001
0.235
-0.001
0.544
0.000
0.819
-0.008
0.017
-0.021
0.219
-0.002
-0.007
0.010
-0.007
0.008
-0.014
0.018
-0.014
0.018
6
7
8
-0.084
0.000
-1.542
-0.004
5.007
-0.043
9
12
13
0 0.0813
0.031
0.305
3.337
0.013
3.687
0.301
0.786
0.231
0.593
0.856
-0.003
0.229
0.002
0.001
0.003
0.003
-0.040
0.001
0.029
0.066
0.313
0.718
-0.001
0.001
0.301
0.786
2.575
-0.269
-0.269
2.047
-0.804
-0.058
-0.098
-0.247
1.817
1.401
-0.001
-0.001
0.487
0.375
0.002
0.002
0.000
0.000
0.001
0.001
0.001
0.001
-0.025
0.021
0.235
-0.001
0.231
-0.804
-0.058
1.894
-0.074
-0.680
-0.001
-0.182
0.000
0.000
0.000
0.000
-0.001
-0.031
0.050
0.075
0.544
0.819
0.000
-0.008
0.593
0.856
-0.098
1.817
-0.247
1.401
-0.074
-0.680
1.563
-0.272
-0.272
9.947
-0.001
-0.002
-0.073
-6.711
0.000
-0.019
0.000
0.011
0.001
-0.008
0.001
-0.008
0.000
0.002
0.017
-0.021
-0.003
-0.001
-0.001
-0.001
-0.001
-0.002
0.028
-0.001
0.002
0.001
0.003
0.003
-0.008
0.000
0.020
-0.001
0.219
-0.007
-0.002
0.010
0.229
0.002
0.487
0.002
0.375
0.002
-0.182
0.000
-0.073
0.000
-6.711
-0.019
-0.001
0.002
7.578
-0.005
-0.005
0.027
0.003
-0.014
-0.002
0.013
-0.002
0.013
0.000
0.000
-0.001
-0.001
-0.007
-0.014
0.008
0.018
0.001
0.003
0.000
0.001
0.000
0.001
0.000
0.000
0.000
0.001
0.011
-0.008
0.001
0.003
0.003
-0.002
-0.014
0.013
0.025
0.011
0.011
0.024
0.011
0.024
0.000
-0.001
-0.014
0.018
0.003
0.001
0.001
0.000
0.001
-0.008
0.003
-0.002
0.013
0.011
0.024
0.024
EC 936 ECONOMIC POLICY MODELLING
LECTURE 2:
PART III
INPUT-OUTPUT
TECHNIQUES
&
CONSISTENCY MODELLING
z11 +
z12 +
z13 +
..... +
z1n +
f1
=
x1
z21 +
.
.
.
zm1 +
z22 +
z23 +
..... +
z2n +
f2
=
zm2 +
zm3 +
..... +
zmn +
fm
=
x2
.
.
.
xm
z11 +
z12 +
z13 +
..... +
z1n +
f1
=
x1
z21 +
.
.
.
zn1 +
z22 +
z23 +
..... +
z2n +
f2
=
zn2 +
zn3 +
..... +
znn +
fn
=
x2
.
.
.
xn
since m = n
a11 x1 + a12 x2 + a13 x3 + . . . . . +
a1n xn + f1
=
x1
a21 x1 + a22 x2 + a23 x3 + . . . . . +
.
.
.
an1 x1 + an2 x2 + an3 x3 + . . . . . +
a2n xn + f2
=
ann xn + fn
=
x2
.
.
.
xn
x1 -
a11 x1
- a12 x2
-
a13 x3
-
..... -
a1n xn
=
f1
x2 .
.
.
x3 -
a21 x1
- a22 x2
-
a23 x3
-
..... -
=
f2
an1 x1 - an2 x2 -
an3 x3
-
..... -
a2n xn
.
.
.
ann xn
=
fn
(1 - a11) x1 a21 x1
.
.
.
an1 x1
a12 x2
-
a13 x3
-
..... -
a1n xn
=
f1
- (1 - a22) x2 -
a23 x3
-
..... -
a2n xn
=
-
an3 x3 -
..... -
f2
.
.
.
fn
an2 x2 -
(1 - ann) xn =
x=
Z=
xi
.
.
.
.
xn
f=
z11
z21
.
.
.
z n1
z12
z22
z13
z23
z n2
z n3
fi
.
.
.
.
fn
. . . . . z1n
. . . . . z2n
.
.
.
. . . . . znn
A =
I =
a11
a21
.
.
.
an1
a12
a22
a13
a23
an2
a n3
1
0
.
.
.
0
0
1
.
0
0
.
0
0
[I–A]x = f
x = [ I – A ] -1 f
. . . . . a1n
. . . . . a2n
.
.
.
. . . . . ann
..... 0
..... 0
.
.
.
..... 1
2
3
∞
4
∆Y = X + cX + c X + c X +c X + . . . . +c X =
1
______
(1 – c)
2
3
4
∞
x = f + Af + A f + A f + A f + . . . . + A f
= (I + A + A2 + A3 + A4 + . . . . + A∞) f
≈ [ I – A ] -1 f
L matrix for U.S. 2006
3
0.02380
0.05656
1.00585
0.46496
0.18254
0.32938
0.03230
4
0.07353
0.14697
0.00814
1.59722
0.20128
0.38284
0.05247
5
0.00745
0.05240
0.00791
0.14236
1.10757
0.33446
0.03590
6
0.01006
0.01613
0.01199
0.14369
0.07183
1.46603
0.03424
7
0.01176
0.03049
0.02863
0.21724
0.09100
0.36981
1.03824
L (8 iterations)
2
3
0.00995 0.02360
1.17140 0.05622
0.00355 1.00580
0.20551 0.46356
0.08192 0.18197
0.27876 0.32756
0.02383 0.03212
4
0.07326
0.14650
0.00808
1.59531
0.20051
0.38037
0.05223
5
0.00735
0.05223
0.00789
0.14165
1.10729
0.33353
0.03581
6
0.00995
0.01595
0.01196
0.14293
0.07152
1.46504
0.03414
7
0.01164
0.03027
0.02860
0.21635
0.09064
0.36866
1.03812
Percent error of power-series approximation
1
2
3
1 Agriculture
0.0206 1.2221 0.8447
2 Mining
0.9686 0.0180 0.6076
3 Construction
0.6806 0.7979 0.0046
4 Manufacturing
0.4455 0.4179 0.3020
5 Trade, Transport
0.4429 0.4215 0.3092
6 Services
0.8090 0.4004 0.5530
7 Other
0.7058 0.4638 0.5592
5.0729 5.7417 6.1804
4
0.3718
0.3175
0.7736
0.1194
0.3807
0.6455
0.4671
7.0756
5
6
1.3542 1.0813
0.3304 1.1575
0.2970 0.2116
0.4975 0.5318
0.0257 0.4278
0.2755 0.0678
0.2542 0.2877
8.0346 9.7655
mean error
7
1.0773
0.7120
0.1027
0.4086
0.3921
0.3119
0.0110
10.0155
1.0589
1
2
3
4
5
6
7
Agriculture
Mining
Construction
Manufacturing
Trade, Transport
Services
Other
1
1.33652
0.04813
0.00921
0.42750
0.17271
0.30407
0.03461
Power-series approximation to
1
1 Agriculture
1.33624
2 Mining
0.04766
3 Construction
0.00915
4 Manufacturing
0.42560
5 Trade, Transport 0.17194
6 Services
0.30161
7 Other
0.03437
2
0.01007
1.17161
0.00358
0.20637
0.08226
0.27988
0.02394
Leontief inverse matrix, U.S. 2006
1
1 Agriculture
1.3436
2 Mining
0.0464
3 Construction
0.0087
4 Manufacturing
0.4298
5 Trade, Transport
0.1792
6 Services
0.3150
7 Other
0.0132
2
0.0103
1.1719
0.0034
0.2061
0.0848
0.2900
0.0113
3
0.0243
0.0546
1.0055
0.4664
0.1890
0.3416
0.0147
4
0.0751
0.1451
0.0077
1.5994
0.2083
0.3968
0.0313
5
0.0075
0.0520
0.0076
0.1412
1.1110
0.3467
0.0212
6
0.0102
0.0153
0.0117
0.1419
0.0740
1.4835
0.0168
7
0.0119
0.0295
0.0283
0.2165
0.0940
0.3835
1.0230
L using RAS results derived from 2002 technical coefficient matrix
1
2
3
4
5
1 Agriculture
1.3422
0.0111
0.0257
0.0789
0.0072
2 Mining
0.0407
1.0899
0.0592
0.1274
0.0608
3 Construction
0.0099
0.0279
1.0068
0.0115
0.0097
4 Manufacturing
0.3518
0.1743
0.4486
1.6266
0.1277
5 Trade, Transport
0.1821
0.0991
0.1826
0.2030
1.1222
6 Services
0.3602
0.3525
0.3539
0.4092
0.3261
7 Other
0.0113
0.0119
0.0118
0.0225
0.0268
6
0.0087
0.0147
0.0126
0.1422
0.0708
1.4859
0.0183
7
0.0122
0.0291
0.0224
0.2172
0.0997
0.3829
1.0222
Absolute error in Leontief coefficients
1
2
1 Agriculture
0.0014
0.0008
2 Mining
0.0057
0.0820
3 Construction
0.0012
0.0245
4 Manufacturing
0.0780
0.0318
5 Trade, Transport
0.0028
0.0144
6 Services
0.0453
0.0625
7 Other
0.0019
0.0007
3
0.0014
0.0047
0.0013
0.0177
0.0064
0.0123
0.0029
4
0.0039
0.0177
0.0038
0.0272
0.0052
0.0123
0.0088
5
0.0003
0.0087
0.0020
0.0135
0.0112
0.0207
0.0056
6
0.0014
0.0005
0.0009
0.0003
0.0032
0.0025
0.0015
7
0.0003
0.0003
0.0060
0.0007
0.0057
0.0006
0.0008
Percentage absolute errors in Leontief coefficients
1
2
3
1 Agriculture
0.0010
0.0779
0.0579
2 Mining
0.1227
0.0699
0.0855
3 Construction
0.1332
7.3057
0.0013
4 Manufacturing
0.1816
0.1544
0.0380
5 Trade, Transport
0.0158
0.1695
0.0337
6 Services
0.1437
0.2156
0.0360
7 Other
0.1468
0.0609
0.1991
4
0.0519
0.1218
0.4943
0.0170
0.0251
0.0310
0.2820
5
0.0392
0.1681
0.2685
0.0957
0.0101
0.0596
0.2625
6
0.1384
0.0356
0.0765
0.0022
0.0435
0.0017
0.0925
7
0.0259
0.0112
0.2111
0.0030
0.0609
0.0016
0.0008
Absolute error in technical coefficients
1
2
1 Agriculture
0.0191
0.0036
2 Mining
0.0048
0.1282
3 Construction
0.0005
0.0543
4 Manufacturing
0.0574
0.0226
5 Trade, Transport
0.0066
0.0155
6 Services
0.0410
0.0785
7 Other
0.0006
0.0012
3
0.0026
0.0193
0.0014
0.0189
0.0128
0.0114
0.0031
4
0.0061
0.0281
0.0004
0.0101
0.0121
0.0130
0.0074
5
0.0007
0.0204
0.0016
0.0086
0.0240
0.0535
0.0153
6
0.0029
0.0010
0.0004
0.0095
0.0049
0.0137
0.0022
7
0.0003
0.0058
0.0113
0.0023
0.0094
0.0188
0.0060
Absolute error in Leontief coefficients
1
2
1 Agriculture
0.0014
0.0008
2 Mining
0.0057
0.0820
3 Construction
0.0012
0.0245
4 Manufacturing
0.0780
0.0318
5 Trade, Transport
0.0028
0.0144
6 Services
0.0453
0.0625
7 Other
0.0019
0.0007
3
0.0014
0.0047
0.0013
0.0177
0.0064
0.0123
0.0029
4
0.0039
0.0177
0.0038
0.0272
0.0052
0.0123
0.0088
5
0.0003
0.0087
0.0020
0.0135
0.0112
0.0207
0.0056
6
0.0014
0.0005
0.0009
0.0003
0.0032
0.0025
0.0015
7
0.0003
0.0003
0.0060
0.0007
0.0057
0.0006
0.0008
Percentage absolute errors in technical coefficients
1
2
3
1 Agriculture
0.2623 640.2814
1.8850
2 Mining
2.3241
0.1177
2.3954
3 Construction
0.1070 204.7931
0.3873
4 Manufacturing
0.0315
0.3883
0.3272
5 Trade, Transport
0.1502
0.7700
0.4707
6 Services
0.5113
0.8906
0.3227
7 Other
0.2568
0.4101
0.7167
4
0.2056
0.8940
0.5080
0.3735
0.2391
0.2594
0.1711
5
4.3610
0.6867
0.5959
0.3008
0.5520
0.2427
0.9605
6
0.4329
4.7054
0.3268
0.2151
0.2244
0.3250
0.4240
7
0.0599
0.7717
0.0331
0.2307
0.1366
0.2656
0.0398
Percentage absolute errors in Leontief coefficients
1
2
3
1 Agriculture
0.0010
0.0779
0.0579
2 Mining
0.1227
0.0699
0.0855
3 Construction
0.1332
7.3057
0.0013
4 Manufacturing
0.1816
0.1544
0.0380
5 Trade, Transport
0.0158
0.1695
0.0337
6 Services
0.1437
0.2156
0.0360
7 Other
0.1468
0.0609
0.1991
4
0.0519
0.1218
0.4943
0.0170
0.0251
0.0310
0.2820
5
0.0392
0.1681
0.2685
0.0957
0.0101
0.0596
0.2625
6
0.1384
0.0356
0.0765
0.0022
0.0435
0.0017
0.0925
7
0.0259
0.0112
0.2111
0.0030
0.0609
0.0016
0.0008