HOW THE CHAIN-ADDITIVITY ISSUE IS TREATED IN THE U.S. ECONOMIC ACCOUNTS. Bureau of Economic Analysis, U.S. Department of Commerce

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1 For Official Use STD/NA(2000)25 Organisation de Coopération et de Développement Economiques OLIS : 11-Sep-2000 Organisation for Economic Co-operation and Development Dist. : 12-Sep-2000 Or. Eng. STATISTICS DIRECTORATE STD/NA(2000)25 Or. Eng. For Official Use National Accounts HOW THE CHAIN-ADDITIVITY ISSUE IS TREATED IN THE U.S. ECONOMIC ACCOUNTS Bureau of Economic Analysis, U.S. Department of Commerce Agenda Item 8 OECD MEETING OF NATIONAL ACCOUNTS EXPERTS Château de la Muette, Paris September 2000 Beginning at 9:30 a.m. on the first day Document complet disponible sur OLIS dans son format d origine Complete document available on OLIS in its original format

2 HOW THE CHAIN-ADDITIVITY ISSUE IS TREATED IN THE U.S. ECONOMIC ACCOUNTS by Christian Ehemann, Arnold J. Katz, and Brent R. Moulton Bureau of Economic Analysis, U. S. Department of Commerce Abstract Since the U.S. Bureau of Economic Analysis adopted chain-type indexes using the Fisher formula for its featured measure of real gross domestic product (GDP) in 1996, the currency-denominated real aggregates (known as chained-dollar aggregates) have no longer been exactly additive. We discuss some of the problems with fixed-weighted aggregates, as well as problems that arise when users treat these aggregates as if they were additive. We describe several tools that have been developed for applications that traditionally were based on additivity properties of fixed weighted GDP aggregates. We also describe some tools that were developed for the compilation and review of the chain-type estimates. Finally, we discuss Claude Hillinger s proposed method for creating an additive set of real aggregates and present empirical results that lead us to be skeptical about the effectiveness of his method. When the United States began publishing gross national product in constant dollars on a regular basis in 1951, it was recognized that, theoretically, fixed-weighted estimates were not the best choice. In a summary of methodology released later that year (Office of Business Economics, 1951, p.143), the predecessor agency to the Bureau of Economic Analysis (BEA) stated that theoretical considerations indicate that under these circumstances [non-proportional price changes] comprehensive output comparisons call for calculations in terms of the prices for each year. The fixed-weighted estimates were justified, in part, because the vast additional labor involved... did not seem warranted. Interestingly, additivity was not mentioned as a desirable property of the estimates. In recent years, shortcomings of the fixed-weighted estimates and lower computing costs have led to the use of chain-type index numbers. BEA began publishing experimental chain-type estimates in 1989 (Young, 1989), and the 1993 System of National Accounts recommended chained price and quantity (or volume) indexes (Commission of the European Communities, et al., 1993). This paper takes a look at what, if anything, was lost when we gave up additivity and the implications for economic analysis of having to live without it. Because BEA adopted the Fisher formula, particular attention is given to working with the Fisher index. Tools for analysis are described, including a formula for contributions to percent change of an aggregate that can be utilized when the analyst is interested in decomposing the growth of the aggregate into the contributions of each of its components. We discuss briefly discuss some tools that BEA has developed for compiling and reviewing chained estimates. We conclude by discussing the proposal by Hillinger (1999a, 1999b, 2000) to construct a methodology for national accounts that retains additivity. 2

3 1. Problems with additivity 1. The goal of additivity may not only be unattainable, but undesirable. We shall argue that, in the past, the additivity of fixed-weighted accounts has misdirected economic analysis. 2. Economists have always been interested in describing aspects of economic performance that do not require additivity. These include changes in the level of real GDP and other real economic aggregates over time (with levels denominated either in index points or the currency of a particular year), percent changes in real aggregates, and comparisons in the rates of change in different real aggregates. In a fixed-weighted system, it appeared that one could also make cross-sectional comparisons of the real magnitudes of different series, and such comparisons were frequently made. However, on further analysis, the validity of such comparisons is generally dubious. An illustration of the problem can be seen in a question asked of BEA staff after our recent redefinition of gross fixed capital formation (investment) to include software. The question was, which is larger, real investment in software or real investment in computers. It turns out that the answer depends on the choice of reference year. For the year 1999, computers and peripheral equipment is larger than software by 29.3 billion chained 1996 dollars. But using our previous reference year, 1992, instead of 1996, software is larger than computers and peripheral equipment by 77.5 billion chained dollars. Clearly, these comparisons are problematical. Other comparisons, in which the differences in dollar-denominated real magnitudes are very large or not so affected by large changes in relative prices may appear to be on firmer ground, but any such comparisons are at best suggestive and prone to error. For almost all purposes, it is better to make these comparisons for aggregates measures in current prices; these are the prices that are most relevant for the period being compared, and restating the aggregates relative to prices for a different (reference) period detracts from the comparison. This is true regardless of the type of index numbers used to create the real estimates. 3. Another question whose answer appeared to be facilitated by fixed-weighted estimates is the share that one real aggregate is of another. However, meaningful shares can be computed only when all the components are measured in the prices of a given period. Almost always, the period whose prices are of interest is the same as that for which the share is required, so the correct share is one measured in current prices. These issues have been described well by Whelan (2000). 4. Old habits of economic analysis are hard to break. It is not sufficient to avoid methods and analytical shortcuts that we now know are flawed. New methods and tools must be developed to take their place. At the Bureau of Economic Analysis, we have attempted to take advantage of the known analytical properties of the Fisher index and to develop new methods of analysis where needed. 2. Analytical tools for Fisher chain-type estimates 5. One of the most common requirements in working with non-additive economic aggregates is the need to obtain new aggregates by forming combinations of existing aggregates; with fixed-weighted aggregates, these new aggregates would have been obtained by simple addition or subtraction. Examples might be an aggregate of expenditures on computer equipment and software, or total imports not including petroleum. Diewert (1978) showed in a classic article that the Fisher index is approximately consistent in aggregation. That means that if Fisher quantity indexes or price indexes are available for all components at some level of aggregation, along with values of the components measured in current prices, these components can be combined in a Fisher index and the result will be approximately the same as a Fisher index built from the most detailed components. Our experience with national accounts data has been that the approximations provided by this Fisher of Fishers approach are usually very accurate. 3

4 6. Because chained-dollar aggregates may be approximately additive for periods close to the base year, in some cases it may be beneficial to rebase the chained-dollar aggregates to a reference year close to the period being studied. Landefeld and Parker (1997) explain how to rebase chained-dollar aggregates. In addition to BEA s featured data that use 1996 as the reference year, the agency also publishes major aggregates from the national income and product accounts (NIPA) using four alternative reference years (1937, 1952, 1972, and 1982; see NIPA tables 1.2A-1.2D). 7. A third tool for working with Fisher indexes, developed at BEA, has probably been the most important of these tools in practice a method for decomposing the percent change in an aggregate into contributions of its components. These contributions to percent change are strictly additive. BEA now publishes five tables presenting quarterly and annual contributions to percent change of GDP and of its major final expenditures components by detailed components (NIPA tables ), as well as annual contributions to percent change of GDP by industry. BEA s written analyses of national accounts data in its news releases and in the Survey of Current Business have focus on contributions to percent change, rather than on the less satisfactory chained-dollar estimates, for explaining the relative importance of GDP components to overall GDP growth. 8. Our method of calculating contributions to percent change was revised slightly in the comprehensive revision of the U.S. NIPA released in October Previously, we used the formula (1) CPC t i = 100 (p t-1 i + p t i /PPI t )(x t i x t-1 i ) / (p t-1 + p t /PPI t )x t-1, where CPC t i is the contribution of the ith good to the percent change in the aggregate from period t 1 to period t, p t i is its price in period t, x t i is its quantity, p t-1, p t, and x t are vectors of prices and quantities, and PPI t is the (bilateral) Paasche price index from period t-1 to period t for the aggregate. This formula did not sum exactly to the percent change in the aggregate, although it came extremely close. However, Yuri Dikhanov 1 of the World Bank observed that if PPI t is replaced by the Fisher price index (denoted by FPI t ), the decomposition of percent change is exact. Thus, the revised formula is (2) CPC t i = 100 (p t-1 i + p t i /FPI t )(x t i x t-1 i ) / (p t-1 + p t /FPI t )x t A proof of exact additivity by one of us (Ehemann) is provided in the Appendix Improvements in other computational details were also made in the 1999 comprehensive NIPA revision. BEA presents its estimates of quarterly percent change at an annual rate, and that transformation causes the contributions formula to no longer be exactly additive. An algorithm for distributing the difference between compound and simple annualization of the aggregate to the components was developed that does not change the sign of contributions that happen to be very small, and which works reliably for percent changes of any magnitude. 11. Our decomposition of the percent change in an aggregate has the attractive properties that the contribution of each component is proportional to its quantity change, and the prices and quantities of all other goods enter symmetrically. However, the formula for contributions to percent change used by BEA is not the only possible one that has these properties. The strategy for deriving the contributions formula used in the appendix is to let the weight on current-period prices be an unknown and show that 1 2 Private correspondence, October 30, For many purposes, these estimates of contributions to percent change can be approximated satisfactorily by the product of the percent change in quantity and the current-dollar share that the component is of the aggregate in the preceding period. This approximation is used by analysts who do not have access to BEA s detailed database. Another method for approximating contributions to percent change for multiple periods is given by Landefeld and Parker (1997). 4

5 the solution value that makes contributions exactly additive is the reciprocal of the Fisher price relative. A different solution is obtained if, in the denominator of the right-hand side of equation (A.1), x t-1 is replaced by the arithmetic mean of period t and period t 1 quantities (or any other function of these two amounts) 3. A different one-parameter function of current- and previous-period prices could also be used. Nevertheless, the present choice of specification is justified; because the contribution of a component to the movement in an aggregate from one period to the next is of interest, it appears reasonable to use previous period quantities as the base and not an expression that involves current period quantities. 12. The percentage change in Fisher price indexes can also be decomposed into contributions to percent change. However, it is not clear that the requirements of price analysis exactly parallel those of quantity analysis. Accordingly, we are experimenting with alternative approaches in our internal analysis before releasing estimates of price contributions. We expect to release such estimates in the future. 3. Tools for compilation and review of estimates 13. The adoption of chained estimates required changes not only in the way economists use the data but changes in how BEA produces and reviews the data before release. One area that has received attention is the checking of Fisher estimates, which is more difficult because quantities can no longer be added to replicate a total. We have replaced older validation methods with methods that use the properties of Fisher indexes. Thus, real GDP can be replicated within a predetermined tolerance by constructing a Fisher of Fishers. The product of a Fisher quantity index denominated in dollars and a Fisher price index is current-dollar expenditures, and this relationship can be checked. Estimates of quantity change expressed as chained dollars and as indexes must have the same growth rates. These tests are used in computerized checking procedures that produce diagnostic reports identifying any test failures, which can then be investigated. 14. New methods are also needed to review revisions to Fisher estimates. Usually, the sources of revisions to Fisher estimates are the expected ones: revisions to detailed quantity components lead to revisions in quantity aggregates and revisions to detailed price components lead to revisions in price aggregates. In the Fisher formula, however, interactions are possible between prices and quantities, and between data revisions and changes from the previous period. In order to validate occasional unexpected revisions, a method for pinpointing the source of the revisions has been developed that uses Taylor s series approximations to simplify the Fisher formula (Ehemann, 2000). These Taylor s series approximations have been found to be extremely accurate. 4. The Hillinger approach to additivity 15. Finally, we wish to comment on the proposal by Hillinger (1999b) to construct a methodology for national income accounting that retains additivity. 4 The system that Hillinger proposes for national economic accounts is indeed additive. Consider first a two-period example; the change in the real value of a top-level aggregate such as GDP (which Hillinger refers to as aggregate quantity variation ), from base period 0 to period 1 is defined as (3) AQV = ½ (p 0 + λ 1 p 1 )(x 1 x 0 ), 3 4 The authors are indebted to Erwin Diewert for this observation. Dikhanov (1997) presents another proposal to retain additivity that is based on the Iklé index. 5

6 where λ is the centered price index, (4) λ = (x 0 + x 1 )p 1 / (x 0 + x 1 )p 0, which is also known as the Edgeworth-Marshall index. In this system, the value of real GDP in period 1 is (5) p 0 x 0 + ½ (p 0 + λ 1 p 1 )(x 1 x 0 ). 16. Because the system is additive, a subaggregate is obtained using conformably partitioned price and quantity vectors. Hillinger s preferred definition of real subaggregates, however, is unclear to us. For example, Hillinger (1999b, paragraphs 33 and 35) refers to the real magnitude for sector k as (6) q 1 k = p 0 k x 0 k + ½ (p 0 k + λ 1 p 1 k )(x 1 k x 0 k ), where the value of λ corresponding to the aggregate in (4) is retained. On the other hand, Hillinger (2000, paragraphs 59, 60) refers to current-price expenditures deflated by the common price index, λ, as real magnitudes : (7) p 1 k x 1 k / λ. 17. The real expenditure variation is then decomposed into a normalized quantity variation and a relative price variation. Expression (6) is the result of chaining the normalized quantity variations beginning with a base-period value for period Expression (7), which divides the value of each subaggregate in current prices by a common, aggregate price index, appears to provide data users with very little information beyond what is already provided in the aggregates valued at current prices. Most cross-sectional comparisons focus on shares or relative magnitudes, which are not affected by use of a common deflator. As we argue above, the measures valued at current prices are generally more appropriate for this type of analysis anyway. Consequently, the following comments will on the properties of the subaggregates defined by (6). 19. A serious difficulty with Hillinger s system is that the level of subaggregate k need not be positive, even if all of its components and their prices are positive. This possibility can be shown by a numerical example. The economy is assumed to contain four goods. The prices and quantities of each good are shown in the following table: Table 1 Hypothetical Prices and Quantities Price of good i in period 0 Quantity of good i in period 0 Price of good i in period 1 Quantity of good i in period 1 Good # , ,940 Good # , ,600 Good # , Good # , Given these assumptions, λ takes the value 3,547.5/5,685 = Computing the real value of each good in period 1 for goods i = 1, 2,...,4 using (6), one obtains, respectively, 2,045.3, , , and These values sum to 3,677.8, which is the Hillinger estimate of real GDP. Now suppose that goods 1 and 2 are the components of the subaggregate A and goods 3 and 4 are the components of that the subaggregate B. Summation gives A = 3,717.1 and B = Thus, the level of B is 6

7 negative. An accounting system that yields a negative value for a subaggregate composed entirely of positive quantities would appear to be unacceptable. 21. The aspect of the example that contributes the unacceptable result is price dispersion; in this case one price increases rapidly while the others fall, and the good with a large price increase also exhibits a large decline in quantity purchased. One might wonder whether negative values for a major aggregate could occur in practice. A second example suggests that the problem is very real. 22. Using Hillinger s method, we constructed estimates with U. S. historical data using 1996 as the base year. To simplify the calculations we calculated final sales to domestic purchasers (that is, gross domestic final expenditures less the change in inventories) as our highest level aggregate, rather than GDP, and constructed the measures using only six components: four components of private fixed investment, personal consumption expenditures, and government consumption expenditures and gross investment. The quantity data were Fisher chained dollars; the price data were Fisher implicit price deflators. The period of calculation extended to 1966, for a span of 30 years. The data for the beginning and ending periods are shown in Table 2. Table 2 Fisher Prices and Quantities for Components of U.S. Final Sales to Domestic Purchasers Expenditure category Price in 1966 Quantity in 1966 Price in 1996 Quantity in 1996 Private investment in computers and peripheral equipment Private investment in nonresidential equipment and software, except computers and peripherals Private investment in residential equipment Private investment in structures Personal consumption expenditures.240 2, , Government consumption expenditures and gross investment , Because of the spectacular decline that has occurred in computer prices, Table 2 looks qualitatively much like Table 1 with the direction of change reversed that is, one price shows a large decrease and other prices increase. In the simulation, annual data were used and chaining was done as in (Hillinger 2000). The results were as anticipated, based on the preceding example. The lower panel of Table 3 shows the Hillinger chains at ten-year intervals. (For comparison, the table also shows the Fisher chains and Hillinger bilateral estimates for the same years.) By 1986, the Hillinger chained value of computers turns negative. By 1966, the negative investment in computers is larger than nonresidential private investment in equipment and software excluding computers, so the subaggregate nonresidential private investment in equipment and software is negative. Nevertheless, the most important result of the simulation may be that illustrated by private investment in residential equipment. The price of investment in residential equipment increased between 1966 and 1996 at a rate that was only moderately slower than the rate of increase in the general price level. However, the chained Hillinger quantity value turns negative for 1976 and earlier. Evidently, any price that differs from the general price level will lead to anomalous results using the Hillinger deflation method if the chaining is continued over a sufficiently long interval. 7

8 24. We note some other features of the Hillinger method in this simulation. The overall growth of the highest-level Hillinger aggregate (in this case, final sales to domestic purchasers) is remarkably similar to the growth of the chained-fisher aggregate. This similarity probably reflects Hillinger s use of a chain-type, superlative index (the Edgeworth-Marshall price index) for deflation and the well-known empirical tendency of alternative superlative index numbers to closely approximate each other. We also note that Hillinger presents a parallel treatment of quantity indexes, which presumably should be used for the calculation of growth rates and rates of change (perhaps the most common use of real aggregates in national accounts). These indexes would give different growth rates than the additive quantity values, which suggests that the usefulness of the additive measures is fairly limited. 25. In view of the results of this simulation, we are highly skeptical that an approach using variations rather than relative changes can solve issues in national income accounting. In the U.S., we have found that Fisher indexes provide a firm basis for the real components of the national income and product accounts, both conceptually and computationally. We believe that much has been gained and little lost by our switch from additive, fixed-weighted estimates of real quantities. We have had to reeducate both ourselves and our users, but the quality of both our estimates and the economic analysis of the estimates has been greatly improved. Table 3 - Estimated Components of Final Sales to Domestic Purchasers (In billions of 1996 dollars) NIPA chained-dollar value Private investment in computers and peripheral equipment Private investment in nonresidential equipment and software, except computers and peripherals Private investment in residential equipment Private investment in structures Personal consumption expenditures 2, , , , Government consumption expenditures and gross investment , , Residual Final sales to domestic purchasers 3, , , , Chained Hillinger value Private investment in computers and peripheral equipment Private investment in nonresidential equipment and software, except computers and peripherals Private investment in residential equipment Private investment in structures Personal consumption expenditures 2, , , , Government consumption expenditures and gross investment , , Final sales to domestic purchasers 3, , , ,

9 References Bureau of Economic Analysis, Implementing Chain-Type Indexes Using a Fisher Formula: Educating the Data User in the United States, paper presented at OECD Meeting of National Accounts Statistics, September (Available from Commission of the European Communities, International Monetary Fund, Organisation for Economic Cooperation and Development, United Nations, and World Bank, System of National Accounts 1993, Brussels/Luxembourg, New York, Paris, Washington DC, Diewert, W. E., Superlative Index Numbers and Consistency in Aggregation, Econometrica 46(4), 1978, pp Dikhanov, Y., Neutralizing Substitution Bias while Retaining Additivity in U.S. National Accounts, mimeo, World Bank, Ehemann, C., Price and Quantity Effects in Revisions to Chain-Type Estimates, unpublished paper, Bureau of Economic Analysis, U.S. Department of Commerce, Washington, July Hillinger, C., Aggregate Real Consumption and the Cost-of-Living: Theory and Measurement, paper presented at OECD Meeting of National Accounts Statistics, September 1999a. (Available from On Chained Price and Quantity Measures that are Additively Consistent, unpublished paper, SEMECON, University of Munich, München, revised December 1999b. (Available from Consistent Aggregation and Chaining of Price and Quantity Measures, paper to be presented at OECD Meeting of National Accounts Statistics, September (Available from Landefeld, J.S., and R.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 (Available from BEA s Chain Indexes, Time Series, and Measures of Long-Term Economic Growth, Survey of Current Business, 77, May (Available from Lasky, M.J., Chain-Type Data and Macro Model Properties: The DRI/McGraw-Hill Experience, Journal of Economic and Social Measurement, 24(2), OECD, Chain Volumes Based on the Hillinger Aggregation Method, paper to be presented at OECD Meeting of National Accounts Statistics, September (Available from Office of Business Economics, U.S. Department of Commerce, National Income and Product Accounts of the United States: , U.S. Government Printing Office, Washington,

10 Varvares, C., J. Prakken, and L. Guirl, Macro Modeling with Chain-Type GDP, Journal of Economic and Social Measurement, 24(2), Whelan, K., A Guide to the Use of Chain Aggregated NIPA Data, unpublished paper, Board of Governors of the Federal Reserve System, Washington, April Young, A.H., Alternative Measures of Real GNP, Survey of Current Business, 69, April

11 Appendix: Derivation of Contributions to Percent Change To derive equation (2), let FQI t denote the Fisher quantity index. It is sufficient to show that the equation (A.1) FQI t = x t (p t-1 + δp t ) / x t-1 (p t-1 + δp t ) when solved for δ, has the solution δ = 1/ FPI t. Rearranging terms gives (A.2) δ = (FQI t x t-1 p t-1 x t p t-1 ) / (x t p t FQI t x t-1 p t ) Now multiply numerator and denominator by FPI t / x t p t = 1 / (FQI t x t-1 p t-1 ), obtaining (A.3) δ = (1 FPI t x t p t-1 / x t p t ) / (FPI t x t-1 p t / x t-1 p t-1 ) = (1 FPI t /PPI t ) / (FPI t LPI t ), where LPI t is the bilateral Laspeyres price index. Finally, multiply numerator and denominator by PPI t and observe that PPI t LPI t = (FPI t ) 2 ; this yields the desired result (A.4) δ = (PPI t FPI t ) / FPI t (PPI t FPI t ) = 1 / FPI t. The Fisher quantity relative thus has the additive form, (A.5) FQI t = x t (p t-1 + p t /FPI t ) / x t-1 (p t-1 + p t /FPI t ). To obtain the contributions to percent change, subtract one from both sides of (A.5) and on the right-hand side make the substitution (A.6) 1 = x t-1 (p t-1 + p t /FPI t ) / x t-1 (p t-1 + p t /FPI t ) The terms corresponding to the ith component, when multiplied by 100, give the contribution to percent change of that component, and may be written (A.7) CPC t i = 100 (p t-1 i + p t i /FPI t )(x t i x t-1 i ) / (p t-1 + p t /FPI t )x t-1. 11