# Contributions to growth

6.34 In the dissemination of quarterly national accounts, contributions to growth play a prominent role - a role that has become more important with the loss of additivity that has accompanied the introduction of chain volume estimates. While the chain volume estimates of the components of an aggregate do not generally add up to the chain volume estimate of the aggregate, it is possible to calculate the contributions of each component to the growth rate of the aggregate. These growth rates are additive, which will be explained below.

6.35 Deriving contributions to growth from additive data, such as constant price estimates, is straightforward. Deriving the contributions to growth of quarterly chain volume estimates is more complex and unlike constant price estimates there is no one formula that can be applied in all cases. Rather, the methods that can be used depend on how the chain volume estimates have been derived, which include:

- the index formula used (e.g. Laspeyres or Fisher);
- annual or quarterly base years;
- method of linking in the case of annual base years;
- the period over which the contributions to growth are calculated (e.g. quarter-to-quarter or quarter on same quarter of previous year); and
- special features of a component (e.g. changes in inventories).

6.36 The method used in the ASNA compromises the additivity of chain Laspeyres volume indexes in the year following the reference year. This phenomenon arises because the chain volume estimates in this year are in effect values in the prices of the previous year.

6.37 The quarterly chain volume estimates of the components and the aggregates in year y-1 and year y are re-referenced to their respective annual current price values in year y-1 by multiplying them by their implicit price deflators for year y-1. This amounts to dividing each time series of quarterly chain volume estimates by the annual value of the chain volume estimates in year y-1 and then multiplying the result by the current price value in year y-1. The resulting quarterly chain volume estimates are additive in year y, and so the contributions to growth for quarters within year y are exactly additive.

6.38 To determine the quarterly contribution to growth of a component of an aggregate, the following calculation occurs:

\(\large Contrib.{\left( {{x_i},X} \right)^{c,y}} = \frac{{P_{{x_i}}^{y - 1}}}{{P_X^{y - 1}}} \times \frac{{\left( {x_{CVi}^{c,y} - x_{CVi}^{c - 1,y}} \right)}}{{X_{CVi}^{c - 1,y}}}\)

where

- \(X_{CV}^{c,y}\) is the chain volume estimate of an aggregate, such as GDP, in the \(c^{th}\) quarter of year \(y\) and \(P_X^{c,y}\) is the corresponding implicit price deflator; and
- \(x_{C{V_i}}^{c,y}\) is the chain volume estimate of the \(i^{th}\) component of the aggregate in the \(c^{th}\) quarter of year \(y\) and \(P_{{x_i}}^{c,y}\) is the corresponding implicit price deflator.

6.39 During the 2012-13 annual compilation cycle, improvements were made to the method by which pre-1985-86 volume components of GDP(E) are calculated. These components were previously constant price estimates, and not 'true' chain volume measures. This break in series dated from the initial introduction of chain volume measures to the set of compilation methods underpinning the Australian national accounts. Chain volume measures were originally only implemented back to 1985-86. Prior year estimates were calculated as backcasts of historic constant price estimates.

6.40 Implementation of chain volume measures for pre-1985-86 estimates of GDP(E) was not carried through the complete aggregation structure, but headline components (consumption, investment and trade) are all now calculated as chain volume measures, as well as GDP(E) itself, back to 1959-60. Owing to difficulties in recalculating change in inventories estimates in chain volume terms prior to 1985-86, this component is calculated residually for this part of the time series. The result is that percentage point contributions to chain volume GDP(E) growth are now additive for the full time series. Additionally, real income measures such as real gross domestic income (RGDI) are now fully consistent with the terms of trade series across the full time series.

## Effects of benchmarking

6.41 As described earlier, the ABS benchmarks its quarterly chain volume estimates to their annual counterparts. Prior to benchmarking, quarterly estimates in the prices of the previous year are additive, but after benchmarking and re-referencing they are usually not quite additive. This phenomenon arises because each quarterly chain volume series is independently benchmarked to its annual counterpart and the adjustments made to the quarterly estimates of the components are unlikely to be exactly consistent with the adjustments made to the aggregate. Contributions to growth are also unlikely to be perfectly additive after benchmarking, however they can be expected to be sufficiently close to being additive for practical purposes.

## Data that are not strictly positive

6.42 The above method cannot be applied to data that are not strictly positive because meaningful implicit price deflators cannot be derived for them, and so the contributions to growth of such variables are derived residually by taking advantage of the fact that quarter-to-quarter contributions to growth are additive (or nearly so). For example, the contribution to growth in GDP of changes in inventories is derived as the difference between the contribution of gross capital formation and the contribution of gross fixed capital formation.