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BEA's Chain Indexes, Time Series, and Measures of Long-Term Economic Growth

By J. Steven Landefeld and Robert P. Parker

Stephanie L. Howell provided analytical assistance and prepared the tables, charts, and the accompanying note for this article.

Last year, as part of a comprehensive revision of the national income and product accounts (NIPA's), the Bureau of Economic Analysis (BEA) introduced chain-type annual-weighted indexes, also known as Fisher indexes, as its featured measures of real output and prices. These new measures allow for the effects of changes in relative prices and in the composition of output over time, thereby eliminating a major source of bias in the previously featured fixed-weighted, or Laspeyres, measures of real output and prices. The advantages of the new indexes are particularly important for long-term time series, such as those presented in this issue of the SURVEY OF CURRENT BUSINESS, and for analyses of current economic conditions as the base period becomes out of date.

The new indexes are significantly more accurate, but they are also computationally more difficult to use than the old fixed-weighted "constant-dollar" estimates that provided additive and easily manipulated series. To deal with these complexities, BEA introduced dollar-denominated real output series that are based on, and consistent with, the new indexes but that have the computational simplicity of constant-dollar series. As BEA pointed out when these "chained (1992) dollar" series were introduced, they work well for periods close to the 1992 base year, but they may produce increasingly misleading results as one moves away from that year. This article briefly reviews the advantages of BEA's chain-type indexes for various types of analyses, explains the conceptual and empirical problems encountered when using chained-dollar estimates far from the base period, describes the time series BEA will publish, presents several sets of tables and estimates designed to assist analysts in using the NIPA chain-type estimates beginning with 1929, and discusses work that BEA is considering to further improve its chain-type indexes for the most recent quarters.

Problems with fixed-weighted indexes in time series analysis

Chain-type indexes attempt to address one of the most basic problems in measuring real output and prices: The choice of the base period with which all other periods are compared. Quantity and price indexes are analytical devices for decomposing changes in nominal gross domestic product (GDP) into that part due to changes in prices and that part due to changes in quantity. Thus, real GDP is an expression of the changes in output that are associated with changes in quantity and not with changes in prices. The easiest way to calculate real GDP is to specify a single base-period, or constant, set of prices and then value the output in all periods in those prices. Unfortunately, because relative prices and associated patterns of purchases change over time, this measure of real GDP growth will be quite sensitive to the choice of the base year, and a shift in the base year often has a significant impact on the measured growth rates. Indeed, professors of economics delight in illustrating this sensitivity to their students through a series of simple two-good, two-period examples. In these examples, simply shifting the base period, and thus the prices used to value a specified basket of goods, from the first period (known as a Laspeyres index) to the second period (known as a Paasche index) can result in either an increase or a decrease in the value of that basket of goods./1/ Normally, changing the base period does not reverse the direction of change in GDP, but the effect is still quite important. When the base year for real GDP was updated in past comprehensive NIPA revisions, the size of the revisions to the rates of growth in real GDP and its components due solely to updating the base year became topics of debate in discussions of budget projections and monetary policy.

The use of fixed weights not only tends to cause errors and revisions in real GDP and prices when base periods are updated, but the errors themselves are biased. It has been long recognized in the index-number literature that output measures that use fixed weights of a single period tend to misstate growth as one moves further from the base period. This tendency, often called substitution bias, reflects the fact that the commodities for which output grows rapidly tend to be those for which prices increase less than average or decline. Thus, when real GDP is recalculated using more recent-period price weights, the commodities with strong output growth generally receive less weight, and growth in the aggregate measure is reduced./2/ These recalculations result in more accurate measures of growth near the base period because the weights more closely reflect the prices of the economy near the base period. However, the recalculations provide less accurate measures of growth for earlier periods because the price weights are further away from the prices appropriate to those periods. For later periods, even the new weights eventually get out of date, and measures of growth in output become increasingly overstated./3/

Some countries address the long-run distortions caused by fixed-weighted output indexes by updating the base period at 5- or 10-year intervals and then using the new fixed-weighted (Laspeyres) index to extrapolate forward the old fixed-weighted index, thereby creating a series of fixed-weighted indexes that are linked together like a chain. Although this practice does avoid the problems and bias associated with using weights from, for example, the 1980's to value output in the 1950's, the resulting chained Laspeyres indexes is still subject to inaccuracy and bias during periods of extreme price movements.

In periods such as the energy crisis of 1973–75, relative price and consumption patterns can change rapidly, and significant bias can creep into fixed-weighted measures even during periods close to the base period. Moreover, chain-type Fisher indexes are superior to chain-type Laspeyres indexes even during periods when price movements are less extreme./4/ In addition, when chain-type Laspeyres indexes are used, the corresponding dollar-denominated real series are not additive in periods before the most recent base period, and series breaks cause the years adjacent to the base year to be noncomparable.

Finally, because fixed-weighted output and price indexes use different weights than those contained in current-period output and prices, the product of the output and price indexes for GDP does not equal the index for current-dollar GDP, a desirable characteristic for data users interested in decomposing and analyzing current-period growth and in forecasting future growth and inflation./5/ Instead, implicit price deflators, which are derived by dividing real GDP into nominal GDP and are simply the average price of goods and services in GDP, have been used for this purpose because the product of the implicit price index and the fixed-weighted quantity index does equal the index for current-dollar GDP. However, these implicit price deflators can be distorted by temporary shifts in the composition of output; for example, if consumers shift enough of their purchases from goods and services with relatively high price indexes to those with relatively low price indexes, the implicit price deflator will fall even though the price of every good and service, including those with relatively low price indexes, increased./6/

Improvements in accuracy associated with chain-type indexes

BEA introduced the chain-type Fisher index into its measures of real output and prices to address these problems. This index, developed by Irving Fisher, is a geometric mean of the conventional fixed-weighted Laspeyres index (which uses the weights of the first period in a two-period example) and a Paasche index (which uses the weights of the second period)./7/

Changes in this measure are calculated using the weights of adjacent years. These annual changes are "chained" (multiplied) together to form a time series that allows for the effects of changes in relative prices and in the composition of output over time. Thus, BEA is able to calculate an index that uses weights appropriate for each period and thereby avoids the rewriting of economic history that results from updating the base period of a fixed-weighted index as well as the substitution bias that is inherent in fixed-weighted indexes. The chain-type indexes also provide more accurate measures of current-period output during periods of significant price changes. Finally, they provide real output and price indexes whose product equals the index for current-dollar GDP without the distortions caused by shifts in the composition of output associated with the old implicit price deflator./8/

The improvement in accuracy associated with the new indexes is significant. The new indexes produce more accurate estimates of growth in GDP, components of GDP, and GDP by industry.

Other U.S. statistical agencies have moved to, or are considering a move to, various types of chain-type indexes. For example, The Federal Reserve Board switched to Fisher indexes for the industrial production and capacity utilization indexes earlier this year, and the Bureau of Labor Statistics recently released an experimental CPI that is based on a geometric mean index. Internationally, the new System of National Accounts recommends the use of Fisher indexes for computing output and price indexes.

Chained-dollar measures and their limitations

As with most improvements, there is a cost to the new chain-type indexes. Although the annual weights provide more accurate estimates, the chained (1992) dollars are not strictly additive, especially for periods far away from the base period. Previously, the use of the same base period for all time periods produced a set of indexes that converted to dollar-denominated measures in which the components were valued in the same prices over all time periods and added up precisely to the totals. BEA had featured such measures partly because many users consider this additive property to be useful; for example, it facilitates analyses of contributions to growth and provides flexibility in aggregating the detailed components. (It also facilitates verification of calculations using these detailed components.)

In order to assist users, BEA introduced several series as part of the recent comprehensive NIPA revision. In particular, the new chained (1992) dollar estimates provide users with real estimates for current-period analysis and for macro-modeling that are approximately additive and are free of upper-level substitution bias.

The chained (1992) dollars are constructed by setting 1992 as the base year and by using the percent changes in the annual chain-type indexes to extrapolate the real chained-dollar estimates for GDP and its components from their 1992 current-dollar levels. Although the resulting estimates are not precisely additive, for years close to the 1992 base year (when the price weights of the chain-type index are not too far from the prices of the base year), the "residual" is small, and the contributions to growth obtained from the chained (1992) dollars are reasonable approximations to those calculated by BEA from the detailed chain-type indexes./11/ However, for periods far from the base period, the residual in chained dollars becomes large, and contributions to GDP growth computed from the chained-dollar components can differ significantly from those produced by the chain-type indexes.

The residuals arise because the chained (1992) dollar indexes are inconsistent in that the growth rates of the chain-type indexes for real GDP and its components are calculated using annual weights for each year, whereas the chained (1992) dollar levels are based both on these annual weights and on the "weights" from the 1992 base year. Therefore, the chained (1992) dollars produce estimates, such as the contributions of components to GDP growth, that are inconsistent with those produced by the chain-type indexes. These inconsistencies become more apparent as the estimates move farther from the base period. Examples of these errors include the following:

For analyses of changes over time in an individual component, the chained dollars do produce the same results as the chain-type indexes. The percent changes in chained (1992) dollars are based on—and therefore equal to—the percent changes in the chain indexes; the chained dollars are simply indexed to the level of current-dollar GDP and its components during 1992, while the indexes are all indexed to 100.00 in 1992.

For analyses of contributions to GDP growth, however, the problems with using chained (1992) dollars have led BEA to prepare a special table of component contributions (NIPA table 8.2) for periods far from the base period—especially for periods prior to 1982, when both the overall residual and the errors in contributions to growth become quite large. The annual and quarterly indexes and the contributions tables provided by BEA offer a significantly more accurate basis for assessing contributions to growth in the economy, both in the aggregate and by component, than do chained dollars indexed to a single base year./13/

For users who rely on real estimates that are denominated in dollars, the July 1995 SURVEY contained a sample table that demonstrated how to prepare close approximations of contributions to real growth or relative changes for any period./14/ That example is reproduced in the note accompanying this article for the period from the second quarter of 1954 to the third quarter of 1957. In effect, users can compute a chained-dollar series for any period by using the percent changes in the chain-type annual-weighted indexes to compute chained-dollar series indexed to the current dollars of whatever base period is appropriate for the analysis. In addition, in this article, BEA has provided a number of chained-dollar series over frequently cited time periods, such as decades and business cycles. In computing these series, BEA used different base periods, depending upon the time period analyzed; for example, for decades and business cycles, BEA used the midpoints of these periods. However, users should be aware that these tables of contributions are approximations and may produce misleading results for periods far from the base period or when prices are changing rapidly, such as during the energy crisis of 1973–75.

Presentation of the estimates

Consistent with this discussion, BEA is providing users with the following measures of real output and prices:

The chain-type quantity and price indexes, in combination with the current-dollar GDP estimates, provide users with the basic data series for the NIPA's. All other analytical tables and presentations are derived from these base data. The chained (1992) dollars provide accurate estimates of percent changes of GDP and its components; they also provide comparisons of levels over time for a single aggregate as well as reasonable approximations of the relative importance, and the contributions to growth, of components for 1982 to the present. The chained (1992) dollars provide data on levels for computing certain key aggregates, such as per capita GDP. The contributions-to-growth tables provide appropriately weighted approximations of the contributions to growth for frequently used components over common intervals—decades and economic expansions. The chained 1952, 1972, and 1992 dollar series for GDP and its major expenditure components provide appropriately weighted estimates for users that want them for all periods. Users interested in chained dollars for specific detailed components or for specific subperiods are referred to the note accompanying this article.

Further work

For recent quarters, BEA's chain-type annual-weighted measure differs from that used for earlier periods: The most recent quarterly values are calculated using as weights the annual prices for only the most recently available year. When the next full year of data becomes available, the weights are updated to incorporate the prices from the 2 adjacent years. For example, as part of the annual revision of the NIPA's in August 1996, annual weights for 1995 were incorporated: The quarterly changes from the third quarter of 1994 to the second quarter of 1995 and the annual change for 1995 were recalculated using the weights of the adjacent years 1994 and 1995. Previously, the changes for these periods were calculated using only 1994 weights.

BEA is considering replacing this method for recent quarters with a Fisher chain-type measure that uses weights from the two adjacent quarters. Although weights based on quarterly data are likely to be less stable and subject to more statistical noise and revision than weights based on annual data, there are a number of advantages to the use of adjacent quarterly weights when the adjacent annual weights are not available. First, the use of quarterly weights within a Fisher formula would put the estimates for the most recent quarters on the same conceptual basis as those for earlier periods. (As a result, the product of the real output and price indexes for recent quarters would equal the index for nominal output; it does not with the current weighting system.) Second, based on a review of past revisions, the introduction of changes in weights on a more gradual—quarterly—basis will produce more accurate estimates and reduce the size of revisions when the annual weights are introduced (use of the geometric mean between the adjacent quarters should help smooth out quarterly instability in the estimates). The use of a quarterly chain-type index will also reduce the differences for recent quarters between the chain-price index and the implicit price deflator based on the chained-dollars. Finally, the use of a quarterly chain index will make it easier to model, analyze, and forecast current-period estimates.

In addition to possible changes in the method for calculating real GDP for recent quarters, BEA plans to continue its efforts to develop more accurate estimates of contributions to growth over longer periods to replace the approximations presented in this article.


1. Boskin, Michael J., Ellen R. Dulberger, Zvi Griliches, Robert J. Gordon, Dale Jorgensen. "Toward a More Accurate Measure of the Cost of Living: Final Report to the Senate Finance Committee from the Advisory Commission to Study the Consumer Price Index." December 4, 1996.

2. Bureau of Labor Statistics, Department of Labor. March 1997 CPI Detailed Report. (forthcoming).

3. Corrado, Carol, Charles Gilbert, and Richard Raddock. "Industrial Production and Capacity Utilization: Historical Revision and Recent Developments." Federal Reserve Bulletin 83 (February 1997): 67–91.

4. Ferguson, C.E. Microeconomic Theory. Homewood, Illinois: Richard D. Irwin, Inc., 1972, 78–83.

5. Fisher, Irving. The Making of Index Numbers. New York: Houghton Mifflin Company, 1922.

6. "Improved Estimates of the National Income and Product Accounts for 1959–95: Results of the Comprehensive Revision." SURVEY OF CURRENT BUSINESS 76 (January/February 1996): 1–31.

7. Landefeld, J. Steven and Robert P. Parker. "Preview of the Comprehensive Revision of the National Income and Product Accounts: BEA's New Featured Measures of Output and Prices." SURVEY OF CURRENT BUSINESS 75 (July 1995): 31–38.

8. Parker, Robert P. and Jack E. Triplett. "Chain-Type Measures of Real Output and Prices in the U.S. National Income and Product Accounts: An Update." Business Economics (October, 1996).

9. Parker, Robert P. "Gross Product by Industry, 1977–90." SURVEY OF CURRENT BUSINESS 73 (May 1993): 33–54.

10. System of National Accounts 1993. Brussels: Commission of the European Communities, International Monetary Fund, Organisation for Economic Co-operation and Development, United Nations, and World Bank, 1993.

11. Triplett, Jack E. "Economic Theory and BEA's Alternative Quantity and Price Indexes." SURVEY OF CURRENT BUSINESS 72 (April 1992): 49–52.

12. Young, Allan H. "Alternative Measures of Change in Real Output and Prices, Quarterly Estimates for 1959–92." SURVEY OF CURRENT BUSINESS 73 (March 1993): 31–37.

13. Yuskavage, Robert E. "Improved Estimates of Gross Product by Industry, 1959–94." SURVEY OF CURRENT BUSINESS 76 (August 1996): 133–155.

Table 4

Table 5

Table 6

Table 7


1. For example, see C.E. Ferguson [4].

2. The substitution bias in GDP relates to shifts in the composition of GDP across broad categories of goods and services, such as from new autos to used autos or from engines and turbines to computers. It should not be confused with possible biases in the detailed consumer price indexes (CPI's) used to deflate the components of consumer spending in GDP. This second type of substitution bias relates to shifts in consumer spending within a given type of good or service, such as from romaine to iceberg lettuce or from Coke to Pepsi. BEA's use of chain indexes in computing GDP, personal consumption expenditures (PCE), and other GDP components addresses what the Bureau of Labor Statistics (BLS) and the "Final Report of the Advisory Commission to Study the Consumer Price Index"—"the Boskin report"—have described as upper-level substitution bias, but it does not address the lower, or component, level bias contained in the detailed CPI's that BEA uses to deflate components that account for about three quarters of consumer spending.

3. For example, the published chain-type measure of real GDP growth in the first quarter of 1997 is 5.6 percent at an annual rate; the fixed (1992) weighted measure of real GDP growth in the first quarter is 6.3 percent, an overstatement of 0.7 percentage point.

4. See Robert P. Parker and Jack E. Triplett [8].

5. This characteristic also means that—discounting the effects of rounding and of interaction terms—the sum of the growth rates of real output and prices is approximately equal to the growth rate in nominal output.

6. Effective with the recent comprehensive NIPA revision, real output is calculated using the chain-type index, with the result that the implicit price deflator is the equivalent of the chain-type price index and, thus, is not subject to the limitations discussed in this paragraph.

7. Laspeyres quantity index (L):

I_i,o_={∑ P_o_Q_i_∑ P_o_Q_o_}

Paasche quantity index (P):

I_i,o_={∑ P_i_Q_i_∑ P_i_Q_o_}

Fisher quantity index:

I_i,o_=L * P

The Fisher Ideal index was one of many index formulas examined by Irving Fisher [5].

8. Chain indexes address shifts over time in the composition of output that cause substitution bias by using weights that are updated annually. Chain-price indexes moderate the distortions associated with implicit price deflators by using the average (geometric mean) of the weights in two adjacent periods. In any given quarter or year, chain-type price indexes reflect the change in prices, whereas implicit price deflators reflect changes in prices and in the composition of output. In addition, implicit price deflators that are based on fixed-weighted output indexes tend to exaggerate the impact of shifts in the composition of output by using outdated weights (for example, 1987 $=$ 100) that exaggerate the effects of temporary shifts in the composition of output on prices. Finally, as pointed out by Triplett (see [11]), Fisher indexes are superior to other superlative indexes—such as Tournquist indexes—because for those indexes, the product of the price and real output indexes does not equal the nominal output index.

9. Although the substitution bias in fixed-weighted measures causes them to understate real GDP growth for most periods, there are instances in which both the quantities and prices of some components have risen rapidly. In these instances, the use of fixed-weighted indexes can overstate growth. For example, in the 1949–53 economic expansion, which included the Korean war buildup, rapid growth in government spending accounted for a very large share of real GDP growth; the use of 1987 relative prices for government—which were quite high relative to the prices of the early 1950's—weights the contribution of government even more heavily and results in an even higher overall real GDP growth during this period.

The expansion from the third quarter of 1982 to the second quarter of 1990 is included in the calculation of the average growth during postwar expansions before 1987 because the bulk of this expansion is before 1987.

10. Because of the large bias in real GDP-by-industry estimates, BEA switched to a type of chain-weighted measure—a benchmark-weighted index—in 1993, 2 years before the switch to chain-type annual-weighted measures for real GDP. See Robert P. Parker [9] and Robert E. Yuskavage [13].

11. Because of the formula used for calculating real GDP, the chained (1992) dollar estimates for the detailed GDP components do not add to the chained-dollar value of GDP or to any intermediate aggregate. In the NIPA tables, the residual is the difference between GDP and the sum of the most detailed components shown in each table. However, the residuals shown in the special tables accompanying this article are the difference between GDP and the sum of the major aggregates (see the footnotes to the special tables).

12. The "contemporaneous" weights used here are taken from the midpoint of the period being analyzed. For example, the contributions for the second quarter of 1954 to the third quarter of 1957 expansion are derived from real estimates that are based on the percent changes in the quarterly chain indexes from the current-dollar levels at the midpoint of the expansion, the fourth quarter of 1955. Other tables in this article also use the midpoints of the period as the base period (see the accompanying "Note on Computing Alternative Chained-Dollar Indexes and Contributions to Growth"). It is possible, however, that the midpoint of a period is not the most appropriate base period; for example, if the middle year of a decade is a recession year and the price weights are taken from that year, the picture of the economy over that decade may be distorted.

13. As a result of the increased emphasis on chain-type indexes, BEA is now showing them with an additional decimal place to provide the same level of precision for calculating changes in the indexes as that provided by the chained-dollar estimates.

14. See [7], table 1, page 37.