Using a Weighted Average of Base Period Price Indexes to Approximate a Superlative Index

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1 Journal of Official Statistics, Vol. 25, No. 1, 2009, pp Usin a Weihted Averae of Base Period Price Indexes to Approximate a Superlative Index Janice Lent 1 and Alan H. Dorfman 2 The Lloyd Moulton price index has been advocated as a timely approximation to a superlative price index. We show that a weihted averae of the arithmetic and eometric base-weihted (Laspeyres) indexes can serve as a simple, robust alternative to the Lloyd Moulton. The parameter needed for the weihted averae can be readily and systematically estimated from past data and continuously updated as new data become available. Previous methods of estimatin this parameter have entailed either a trial-and-error process, requirin human judment, or the use of iterative numeric alorithms. An empirical study indicates that we may compute timely, close approximations to a superlative index usin a weihted averae of the arithmetic and eometric Laspeyres indexes with parameters estimated and systematically updated from prior data. Key words: Taylor series; elasticity of substitution; Lloyd Moulton Index; sample survey. 1. Introduction A consumer price index (CPI) is a measure of chane from one time period to another of the purchasin power of a iven population s monetary unit. A cost of livin index (COLI) is the ratio of minimal costs needed in the two time periods to achieve a iven standard of livin. A body of theory suests that certain superlative index formulas ive a ood approximation to a COLI (Diewert 1987). These formulas have been difficult to implement, however, because they require information on consumer expenditure patterns for both of the two reference periods; such information on the more recent period is usually unavailable at the time of index production. The desirability of a COLI is not universally accepted. On rounds of simplicity and transparency, many countries prefer a market basket approach, often choosin the Laspeyres-type index described below (U.S. National Research Council 2002, pp ). Other approaches are discussed in the International Labour Office s Consumer Price Index Manual (2004). In this article however, we present a ready, simple estimation method for applications in which a timely superlative price index is desired. Shapiro and Wilcox (1997) advocated the Lloyd Moulton price index (Lloyd 1975; Moulton 1996) as a timely approximation to a superlative index. Instead of second 1 Enery Information Administration, EI-70, 1000 Independence Ave., Washinton, DC 20585, U.S.A. janice.lent@eia.doe.ov 2 U.S. Bureau of Labor Statistics, 2 Massachusetts Ave., N.E., Suite 1950, Washinton, DC 20212, U.S.A. dorfman.alan@bls.ov. Opinions expressed in this article are those of the authors and do not constitute policy of the Enery Information Administration or the U.S. Bureau of Labor Statistics. q Statistics Sweden

2 140 Journal of Official Statistics period expenditure information, the Lloyd Moulton relies on a parameter representin elasticity of substitution and uses past years data as a basis for evaluatin the parameter estimate. The elasticity of substitution indicates the extent to which consumers chane their buyin patterns in response to chanes in relative prices. Hiher elasticity values indicate reater willinness, on the part of consumers, to substitute cheaper items for more expensive ones. Balk (2000) provides numerical methods of estimatin the elasticity parameter. An alternative approach to estimatin a COLI, providin reater operational simplicity and flexibility, is to compute a weihted averae of the base-weihted arithmetic and eometric (Laspeyres) indexes defined below. We show throuh Taylor series expansions that the arithmetic-eometric averae (or AG Mean ) index closely approximates the Lloyd Moulton and hence the superlative indexes. The weiht applied to the eometric index in the AG Mean may be estimated from prior data throuh a simple formula and then systematically updated with more recent consumer expenditure data. By contrast, Shapiro and Wilcox (1997) used a trial-and-error method to approximate a parameter for the Lloyd Moulton index, and the parameter was then held constant throuhout the time period studied. Because of the systematic updatin, the AG Mean continuously picks up chanes in consumer buyin patterns reflected in the data. The systematic updatin of the AG Mean requires no iterative numerical procedures and can therefore be easily prorammed and automated in a statistical production settin. To illustrate the practicality of this approach, we present findins from an empirical study. 2. Price Index Formulas and Estimators The classic price index formula is the (arithmetic) Laspeyres index, L ¼ X N q j1p j 2 X N q j1 ¼ X N w j1 p j2 where p jt denotes the price of item j at time t, q jt denotes the quantity of item j purchased at time t, w jt ¼ p jt q jt = P N k¼1 p ktq kt, and N denotes the number of items in the taret population. The weiht w jt is the expenditure share for item j in period t [ {1,2}; the ratios p j 2 = are referred to as price relatives. Grounded in the fixed market basket concept, L is the ratio of the total costs, in the two reference periods, of the bundle of oods and services that were purchased in Period 1. The Laspeyres is similar in principle to the Paasche index, iven by P ¼ X N q j2p j2 X N q j2 ¼ " X N # w j2 p j2 = which is based on quantities for Period 2. Estimation of the Laspeyres index is more practical, because estimates of the first period shares w j1 are more likely to be available at Period 2 than are estimates of w j2, which are based on Period 2 expenditure share weihts. On the basis of some postulated desirable properties of

3 Lent and Dorfman: Weihted Averae of Base Period Price Indexes 141 p price indexes, Irvin Fisher (1922) suested that the ideal index would be F ¼ ffiffiffiffiffiffi LP, which has come to be called the Fisher index. Another formula that now plays an important role in the U.S. CPI is a eometric mean of the price relatives, ( G ¼ exp XN w j ln p ) j2 ¼ YN p wj j2 which is a eneralization of the unweihted eometric mean known as the Jevons index, after its oriinator. The weihts w j in G miht be fixed across time or be taken as the first or second period shares defined above. When first period weihts are used, the index G is sometimes referred to as the eometric Laspeyres index. The Törnqvist index is a eometric mean index with weihts based on the arithmetic averae of the expenditure share weihts across the two reference periods, i.e., T ¼ YN p wj;1;2 j2 where w j;1;2 ¼ðw j1 þ w j2 Þ=2. Both the Fisher and Törnqvist indexes are known as superlative indexes, because economic theory suests that, under relatively weak assumptions, they approximate a COLI (Diewert 1987). In practice, overnment aencies apply a iven price index formula to a sample from the taret population, yieldin an index estimator of the selected population index. In most cases, neither estimated quantities nor expenditure share weihts are available for either of the price index reference Periods 1 and 2. Statistical aencies often use expenditure share weihts estimated for some earlier Period 0. Combinin these weihts with price data from Periods 1 and 2, they may estimate a modified Laspeyres index or weihted eometric mean index as ^L 0;1;2 ¼ Xn ^w j0 p j2 or ^G 0;1;2 ¼ Yn ^wj p 0 j2 respectively, where n is the sample size and ^w j0 is an estimated expenditure share for item j in Period 0. In what follows, we put aside this complication. For samplin purposes, overnment aencies often cateorize the population of consumer items into roups defined by item characteristics and/or eoraphic areas and draw a sample of items within each roup. This cateorization ives rise to composite forms of price index formulas. For example, the Laspeyres index (with Period 1 weihts)

4 142 Journal of Official Statistics can be written as L ¼ P w1 L, where w t ¼ X X N q i¼1 itp it X N q i¼1 itp it is the expenditure share for the th roup, and L is the Laspeyres sub-index for roup. The computation of subindexes is called lower-level areation, while the process of combinin subindexes into an overall index, often by a different formula, is called upperlevel areation. In the empirical study we present in Section 4, we focus on an application of the AG Mean at the upper level of areation. 3. The AG Mean Approximations to the Lloyd Moulton For eneral notation, we write I ¼ X Y, where X and Y are the formulas used for upper- and lower-level areation, respectively. A composite price index estimator ^I ^w ; ^I combines subindexes ^I with expenditure weihts ^w, e.., ^w [ ^w 1 ; ^w 2, where ^wt denotes the estimated expenditure share at time t for a stratum. Thus we have the base (Period 1) weihted eometric mean index ^G^I ¼ Q ^I ^w 1 and the base-weihted Laspeyres index ^L^I ¼ P ^w 1^I. The Lloyd Moulton (or CES) index estimator is defined as ^C^I ¼ ( ) 1=ð12tÞ X ^w1^i 12t and has been shown to taret a population COLI, as approximated by a Törnqvist or Fisher index. The parameter t is called the elasticity of substitution. Note that ^C^I! ^G^I as t! 1. (Althouh the elasticity parameter is often denoted by s, we use t here, because s will be used for a different purpose below.) We may approximate ^C^I by a weihted arithmetic or eometric averae of the base-weihted eometric mean ^G ^I and Laspeyres indexes, with t as the weiht assined to ^G^I. Let ~C^I;a ¼ t ^G^I þð1 2 tþ^l^i be the arithmetic AG Mean index. To see that ~C^I;a is an appropriate estimator of ^C^I, we expand ^C^I about a constant vector, usin a eneralization of the approach suested by Dalén (1992). (Other approaches to the use of Taylor series expansions to compare price indexes differ slihtly from Dalén s in both method and purpose; see, for example, Diewert 1987.) For notational simplicity, let w ¼ ^w 1, and let X m a ¼ w^i ; s 2 a ¼ X 2; w ^I 2 m a and a ¼ X w 3 ^I 2 m a

5 Lent and Dorfman: Weihted Averae of Base Period Price Indexes 143 When we approximate ^C^I by a Taylor series expansion about the point ^I ¼ m a for all, and express the third-order expansion in terms of the above moments, we have ^C^I < m a 2 ts2 a 2m a þ t ðt þ 1Þ a 6m 2 a ð1þ (The relevant derivatives are iven in the Appendix.) Note that m a ¼ ^L^I. Settin t ¼ 1, we obtain the third-order expansion of the base-weihted Geometric mean index ^G^I ¼ Q ^I w : ^G^I < m a 2 s 2 a 2m a þ a 3m 2 a ð2þ So from (1), we have, to third order, so ^C^I;a < m a 2 ts 2 a 2m a þ t a 3m 2 a ~C^I;a 2 ^C^I < t ð1 2 tþ a 6m 2 a With t [ ½0; 1Š, t ð1 2 tþ a 6m 2 # a a 24m 2 a so the difference ~C^I;a 2 ^C^I should be small provided a is reasonably small. As the empirical results in the next section illustrate, this third order difference is often neliible. When ~C^I;a is used to approximate the Fisher index ^F^I, the appropriate weiht of ^G^I in the averae is simply u F;a ¼ ^L^I 2 ^F^I ^L^I 2 ^G^I ð3þ obtained by settin ~C^I;a equal to ^F^I with t ¼ u F;a and solvin for u F;a. Similarly, u T;a ¼ ^L^I 2 ^T^I = ^L^I 2 ^G^I is appropriate when the estimation taret is a Törnqvist index. Note that u F,a and u T,a rely on the index estimates ^F^I and ^T^I, respectively. Because the elasticity of substitution does not normally chane rapidly over time, however, we can use the index estimates calculated from prior data to obtain current values of u F,a and u T,a. These parameter estimates can then be continuously and systematically updated. The closed alebraic forms of u F,a and u T,a are convenient for analysis purposes and can be easily prorammed, e.., as part of a data processin system used for scheduled production computations. We also consider a eometric AG Mean, ~C^I;j ¼ ^G t^i ^L 12t ^I, which can also be used as an alternative to the Lloyd Moulton. Törnqvist (1936, quoted by Vartia 1978) provides the

6 144 Journal of Official Statistics followin third-order expansion of lo ^C^I : lo ^C^I < mj þ ð1 2 tþs 2 j þ ð1 2 tþ2 j 2 6 where ð4þ and m j ¼ j ¼ X w lo ^I ; s 2 j ¼ X 3 w lo ^I 2 mj Settin t ¼ 0 in (4), we have lo ^L^I < mj þ s 2 j 2 þ j 6 So we may write t lo ^G^I ~C^I;j X 2 w lo ^I 2 mj þð12tþlo ^L^I < mj þ ð1 2 tþs 2 j þ ð1 2 tþ j 2 6 ¼ mj ). From (4) and (5), we have, to third-order ^G^I (since lo lo 2 lo ^C^I t ð1 2 tþ j < # j 6 24 when t [ ½0; 1Š. Thus ~C^I;j should also serve as a ood approximation to ^C^I. To estimate the parameter t for ~C^I;j, we may use 21 u T;j ¼ lo ^L^I 2 lo ^T^I lo ^L^I 2 lo ^G^I or 21 u F;j ¼ lo ^L^I 2 lo ^F^I lo ^L^I 2 lo ^G^I which are analoous to (3) above. Usin our estimates of t from (3) or (6), we could calculate a Lloyd Moulton index, ^C^I, but ~C^I;a and ~C^I;j are computationally simpler. In the case t ¼ 1, all three are equivalent to ^G^I, but ^C^I cannot be directly computed. For very lare values of t (approachin infinity), ~C^I;a and ~C^I;j may not be ood approximations, but very lare values of t are unlikely to be appropriate for consumer price indexes. ð5þ ð6þ 4. Empirical Results To test the usefulness of the AG Mean, we performed an empirical study usin airfare data from the Passener Oriin and Destination (O&D) Survey, a quarterly survey conducted by the U.S. Bureau of Transportation Statistics (BTS). Our data comprised unit value subindexes ^I (averae price in time t divided by averae price in time t 2 1) for detailed cateories of airline itineraries, alon with correspondin expenditure share weihts w.

7 Lent and Dorfman: Weihted Averae of Base Period Price Indexes 145 Both the ^I and the w were computed from the O&D survey data, at the most detailed level of areation allowed by the data. Each roup comprises itineraries flown durin the reference quarter on the same sequence of carriers with identical trip routes. The dates and fliht times for the itineraries may differ within a roup, but the trip route, sequence of class of service cateories (e.., coach, first class), and sequence of air carriers are the same for all itineraries within a roup. (For details on the survey and the estimation method, see Lent and Dorfman BTS is now publishin the Air Travel Price Index series, computed by this method, as a quarterly research series.) We computed superlative index estimates usin the formula ^F^I. We then estimated elasticity parameters based on Formula (3) above and applied the AG Mean approximations. Our results illustrate the use of the AG Mean, with systematic parameter updates, in the case of chained index series. The fiures below show comparisons between the quarterly chained Fisher, Laspeyres, and arithmetic AG Mean index series for the period between the 4th quarter of 1998 and the 3rd quarter of In this application, the differences between the Laspeyres and Fisher index series are, to some extent, due to chain drift in the Laspeyres series (see Lent 2003). We computed the AG Mean indexes, ~C^I;a, usin two different estimators of the elasticity parameter u F,a both based on four-quarter movin averae estimates of parameters computed from previous quarters. The unsmoothed quarterly parameter estimates are iven in Table 1. The series labeled AG Mean 1 was computed assumin a one-quarter la between the availability of price data and expenditure share weihts, while AG Mean 2 was computed assumin a two-quarter la. We use laed data, even thouh contemporary expenditure estimates are available for the airline data, because, typically in Table 1. Unsmoothed quarterly elasticity estimates u F,a Quarter New York Denver Manchester San Juan 1998 Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q

8 146 Journal of Official Statistics practice, only laed expenditure data are available. For k [ {1,2}, the elasticity parameter for AG Mean k is u ðmkþ F;a;t ¼ X t2k l¼t2k23 u F;a;l 4 where u F,a,l is the parameter estimate computed usin data from quarters l 2 1 and l, and u ðmkþ F;a;t is the movin averae parameter used in the AG Mean index measurin chane between quarters t 2 1 and t. The parameters u F,a,l measure the elasticity of substitution between itineraries with different destinations as well as between itineraries with flihts operated by different air carriers. The parameters enerally run between 0 and 0.10, indicatin little substitution between itinerary destinations. Fiure 1 shows the four index series for itineraries oriinatin in New York City or Newark (John F. Kennedy, LaGuardia, or Newark Liberty International Airport). We see very little difference between AG Mean 1 and AG Mean 2, indicatin only very radual chane in the movin averae elasticities over the five-year period. The AG Mean series do rise very slihtly above the Fisher durin the later quarters of the series, due to the cumulative effect of a radual rise in the u F,a,l elasticity estimates. These parameters increase from rouhly 0.05 to 0.09 durin this period, which was marked by the expansion of low-cost air carriers and increased competition in the air travel service market. In spite of the low elasticities and the radual chane in the elasticity estimates over the period, both of the AG Mean series provide much better approximations to the Fisher index than does the Laspeyres. Fiures 2 and 3 provide results for itineraries oriinatin in Denver, Colorado and Manchester, New Hampshire (a much smaller market), respectively. In eneral, the Denver and Manchester series are similar to those for the New York City area. Beinnin in 2002, with the expansion of low-cost carrier service from the Denver airport, the elasticity parameter for the Denver AG Mean series increases, causin the AG Mean indexes to run slihtly above the Fisher, thouh well below the Laspeyres. For itineraries oriinatin in Manchester, both of the AG Mean series closely approximate the Fisher. The AG Mean series for the San Juan market, shown in Fiure 4, run slihtly below the Fisher, due to decreases in the elasticity parameters, which were not seen in the data for any of several U.S. cities examined. The elasticities for San Juan were initially hiher than Fi. 1. Alternative airfare indexes for New York, New York and Newark, New Jersey 99Q4 ¼ 100

9 Lent and Dorfman: Weihted Averae of Base Period Price Indexes 147 Fi. 2. Alternative airfare indexes for Denver, Colorado 99Q4 ¼ 100 Fi. 3. Alternative airfare indexes for Manchester, New Hampshire 99Q4 ¼ 100 Fi. 4. Alternative airfare indexes San Juan, Puerto Rico 99Q4 ¼ 100 those for the U.S. cities, however, enerally runnin between 0.06 and 0.20 over the test period. In this case also, both of the AG Mean series dramatically outperform the Laspeyres by correctin albeit with a la for the elasticity of substitution between air travel itineraries and carriers. 5. Conclusion The weihted arithmetic or eometric of the base-weihted Geometric mean and Laspeyres indexes, with simple, continuously updated estimates of elasticity based on laed data, provides a ood on-time approximation to a superlative index. Like the

10 148 Journal of Official Statistics Lloyd Moulton index, the AG Mean relies on a parameter that is intuitively meaninful and useful in its own riht. Other methods of usin data from prior time periods to approximate a superlative index, e.., calculatin a simple ratio adjustment factor of the Fisher to the Laspeyres index, lack this property. Unlike the Lloyd Moulton parameter, however, the AG Mean parameter has a closed alebraic form. It can therefore be more easily analyzed, and its update computations can be readily automated for use in lare-scale data processin and statistical production systems. Appendix To derive expression (1), we expand the function ^C^I around the point ^I ¼ m ¼ðm; :::;mþ, where m ¼ m a. The eneral formula for the Taylor expansion is f ^I ¼ f m X þ f 0 m ^I 2 m þ 1 6 XXX f 000 þ 1 2 1;2;3 m XX f 00 1;2 m ^I 1 2 m ^I 2 2 m ^I 1 2 m ^I 2 2 m ^I 3 2 m The necessary derivatives of ^C^I, evaluated at ^I ¼ m are as follows. ^C^I ¼ w ^I ^I¼m 2 ^C^I ^I 2 ¼ m 21 t w ðw 2 1Þ ^I¼m 2 ^C^I ¼ m 21 t w 1 w 2 ^I 1 ^I 2 ^I¼m 3 h i ^C^I ^I 3 ¼ m 22 t ð1 2 2tÞw 3 2 3t 2 w 2 þ t ðt þ 1Þw ^I¼m 3 h i ^C^I ¼ m 22 t ð1 2 2tÞw 2 ^I 2 1 1^I w 2 þ t w 1 w 2 2 ^I¼m 3 ^C^I ¼ m 22 t ð1 2 2tÞw 1 w 2 w 3 ^I 1^I 2^I 3 ^I¼m 6. References Balk, B.M. (2000). On Curin the CPI s Substitution and New Good Bias. Research Paper no RSM, Statistics Netherlands, Voorbur.

11 Lent and Dorfman: Weihted Averae of Base Period Price Indexes 149 Dalén, J. (1992). Computin Elementary Areates in the Swedish Consumer Price Index. Journal of Official Statistics, 8, Diewert, W.E. (1987). Index Numbers. The New Palrave: A Dictionary of Economics, J. Eatwell, M. Milate, and P. Newman (eds). London: MacMillan. Fisher, I. (1922). The Makin of Index Numbers: A Study of Their Varieties, Tests, and Reliability. New York: Sentry Press. International Labour Office (ILO) (2004). Consumer Price Index Manual; Theory and Practice. Geneva: ILO Publications. Lent, J. (2003). Chain Drift in Experimental Air Travel Price Index Series. Proceedins of the American Statistical Association, Section on Survey Research Methods. Lent, J. and Dorfman, A. (2005). A Transaction Price Index for Air Travel. Monthly Labor Review, Lloyd, P.J. (1975). Substitution Effects and Biases in Nontrue Price Indices. The American Economic Review, Moulton, B. (1996). Constant Elasticity Cost of Livin Index in Share-Relative Form. Unpublished U.S. Bureau of Labor Statistics manuscript. Shapiro, M.D. and Wilcox, D.W. (1997). Alternative Strateies for Areatin Prices in the CPI. Federal Reserve Bank of St. Louis Review, , May/June. Törnqvist, L. (1936). Levnadskostadsindexerna i Finland och Sverie, deras tillförlitlihet och jämförbarhet. Ekonomiska Samfundets Tidskrift, 37, 1 35, (Reference obtained from Vartia 1978). [In Swedish] U.S. National Research Council (2002). At What Price? Conceptualizin and Measurin Cost-of-Livin and Price. Committee on National Statistics, Division of Behavioral and Social Sciences and Education, Charles L. Schultz and Christopher Mackie (eds). Washinton, DC: National Academy Press. Vartia, Y.O. (1978). Fisher s Five Tines and Other Quantum Theories of Index Numbers. Theory and Applications of Economic Indices, W. Eichhorn et al. (eds). Würzbur: Physica-Verla. Received November 2005 Revised April 2008