Box: Note on Computing Alternative Chained Dollar Indexes and Contributions to Growth

Users can easily prepare close approximations of contributions to real GDP growth or to the growth of other aggregates using chain-type annual-weighted indexes. The table below shows how to estimate these contributions to real GDP growth, using the cyclical expansion from the second quarter of 1954 to the third quarter of 1957 as an example. This methodology provides a close approximation of the component contributions, as indicated by the "residual" line in the table below. The table of component contributions regularly released by BEA (table 8.2) use exact formulas for attributing growth to the components of GDP or of other aggregates but these tables are limited to contributions to changes from the preceding year or quarter.

The contributions are calculated as follows: First, the levels of real GDP and its major components at a midpoint of the period—in this example, the fourth quarter of 1955—are set equal to the published current-dollar levels. Second, corresponding dollar series for the beginning and end of the period—the second quarter of 1954 and the third quarter of 1957—are computed by extrapolating (multiplying) the fourth-quarter 1955 level for each component by the percent change in the chain-type output index for that component; that is, the percent change in the index from the second quarter of 1954 to the fourth quarter of 1955 and from the fourth quarter of 1955 to the third quarter of 1957. Finally, the contribution of each component to the change in GDP is calculated as the ratio of the dollar change in each component to the dollar change in GDP.