Table 2.12
Contributions to growth for period 3, a sample calculation - The case of volume estimates based on chain fisher formula

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  Data Empty cell 
t0 t1 t2 t3
Aggregates – current dollars1 Empty cell 
A 80 90 72 48
B 180 210 240 255
C 250 450 455 504
D 100 150 150 135
AB 260 300 312 303
CD 350 600 605 639
Total 610 900 917 942
Aggregates – price indexes2 Empty cell 
A 100.0 75.0 50.0 25.0
B 100.0 116.7 125.0 125.0
C 100.0 150.0 130.0 140.0
D 100.0 120.0 120.0 100.0
Aggregates – volume measures (unlinked indexes)3 Empty cell 
AB (Paasche) 100.0 111.1 109.5 109.8
CD (Paasche) 100.0 121.2 112.0 103.9
Total (Paasche) 100.0 117.6 111.2 105.7
AB (Laspeyres) 100.0 115.4 110.7 112.5
CD (Laspeyres) 100.0 121.4 112.5 104.1
Total (Laspeyres) 100.0 118.9 111.9 107.0
AB (Fisher) 100.0 113.2 110.1 111.1
CD (Fisher) 100.0 121.3 112.3 104.0
Total (Fisher) 100.0 118.2 111.5 106.3
Aggregates – volume measures (linked indexes)4 Empty cell 
AB (Fisher) 100.0 113.2 124.6 138.5
CD (Fisher) 100.0 121.3 136.2 141.7
Total (Fisher) 100.0 118.2 131.9 140.2
Aggregates -- volume measures (real dollars, reference period = t1) and calculation of contribution to growth Volume measures Growth rate Calculation using the published formula5 Calculation illustrating the weighting6
t0 t1 t2 t3 (t3 vs t2) Results in t3 %Δxi,t W1 W2 Results in t3
A (C/P * 100) 60.0 90.0 108.0 144.0 33.33% 1.98% 33.33% 0.08 0.76 1.98%
B (C/P * 100) 210.0 210.0 224.0 238.0 6.25% 1.66% 6.25% 0.26 1.01 1.66%
C (C/P * 100) 375.0 450.0 525.0 540.0 2.86% 1.49% 2.86% 0.50 1.05 1.49%
D (C/P * 100) 120.0 150.0 150.0 162.0 8.00% 1.22% 8.00% 0.16 0.93 1.22%
AB (Fisher) 265.0 300.0 330.2 367.0 11.13% Empty cell  Empty cell  Empty cell  Empty cell  Empty cell 
CD (Fisher) 494.6 600.0 673.6 700.7 4.02% Empty cell  Empty cell  Empty cell  Empty cell  Empty cell 
TOTAL (Fisher) 761.1 900.0 1,003.7 1,067.4 6.35% Empty cell  Empty cell  Empty cell  Empty cell  Empty cell 
Sum of aggregates (A+B+C+D) Empty cell  Empty cell  Empty cell  Empty cell  Empty cell  6.35% Empty cell  1.00 3.75 6.35%
Sum of sub-aggregates (A+B) Empty cell  Empty cell  Empty cell  Empty cell  Empty cell  3.64% Empty cell  0.34 1.77 3.64%
Sum of sub-aggregates (C+D) Empty cell  Empty cell  Empty cell  Empty cell  Empty cell  2.71% Empty cell  0.66 1.98 2.71%

1. Series A, B, C, D are the lowest level of aggregation, in current dollars. Series AB and CD are sub-aggregates while the Total series is the sum of series A, B, C, D. In current dollars, aggregates and sub-aggregates are perfectly additive for each period.

2. Price indexes at the lowest level of aggregation. These prices take the value 100 in t0.

3. The indexes on the left side are unlinked. They represent growth between two adjacent periods. Paasche and Laspeyres indexes are the basic indexes used to construct the Fisher volume index. The Paasche index is weighted using the relative value (current dollars) of the final period while the Laspeyres index uses the relative value of the initial period. The Fisher index is the geometric mean of the Paasche and Laspeyres indices.

4. Chained indexes represent the growth in volume between the current period and the reference period (t0 in this case).

5. Published formula used to calculate the contribution of each aggregate to total growth

Published contribution formula

6. The following formula illustrates that the contribution is the growth rate of the aggregate (ΔX) weighted by a factor W. In the case of volume estimates, this factor corresponds to the relative value between the aggregate and the total at period t-1 in current dollars, as in the calculation in current dollars. An additional factor of correction (W) is then applied.

Contribution formula
, where:
Weight factor formula
and
Weight factor formula
.

Source: Guide to the Income and Expenditure Accounts, Statistics Canada, catalogue no.13-017-X