A. Arithmetic formulas

The Laspeyres price index is given by:

The first term of equation (1) is a standard representation of the Laspeyres formula as a fixed quantity basket index with $p_i^0$ and $q_i^0$ denoting, respectively, prices and quantities in period 0 for i = 1,…, n products/elementary aggregates. In practice CPIs are compiled as a weighted average of price relatives, given by the second and third terms in equation (1), where the weights are the expenditure shares in period 0, $s_i^0$

It takes time to compile and process household expenditure survey data, so there is a lag between the expenditures share survey period, b, and their first use in the index, commencing at the price reference period 0. Thus, in practice, the Laspeyres is generally not used for real time CPI compilation and expenditure shares from the earlier period b may be used to weight period 0 to period t price changes. The resulting Young price index is given by:

More typically, weights are price-updated between period b and the price reference period 0 to effect fixed period-b quantities. The resulting Lowe index is given by:

The expression in square brackets in the first term are the period-b expenditures, $p_i^b{q}_i^b$, price-updated to period 0. The second term shows the Lowe index to be a period-b fixed-quantity basket price index, and the third term to be a weighted average of price changes where the weights are hybrid period 0 prices and period b quantities, with little economic meaning. Price-updating the expenditure shares for price changes is not to make the weights more up-to-date, but to transform the index from a fixed period b expenditure share-weighted index of price changes to a fixed period b quantity basket price index.

Balk and Diewert (2003), from the perspective of the economic theory of index numbers, establish the substitution bias of a Lowe CPI—see also ILO et al., (2004, chapters 15 and 17) and Balk (2010). Not only is the Lowe index shown to have a likely upward substitution bias against a Laspeyres index, but the Laspeyres index has an upward substitution bias against a superlative index. ILO et al., (2004, chapter 16) demonstrates that the Lowe index, however, has good axiomatic properties.⁴

The Young index fails the circularity and time reversal tests (ILO et al., 2004, Appendix 15.3 and chapter 16). The Young index between periods 0 and t will exceed its time antithesis, that is, its inverse between period t and 0, and in this sense is positively biased.⁵ ILO et al. (2004, chapter 15) demonstrate how the discrepancy between Laspeyres and Young is difficult to gauge. It is based on the covariance of the difference between expenditure shares between period b and 0 and the deviations of period 0 to t relative prices from their mean.⁶ A positive covariance would put Young above Laspeyres and negative covariance below Laspeyres, possibly closer to a superlative index. Analytical shortcomings with the Young index are thus the uncertainty a priori about the extent and nature of its deviations from Laspeyres and its relatively poor axiomatic properties.

B. Geometric counterparts

For elementary-level indexes, the CPI Manual recommends the use of the (geometric) Jevons index if weights are not available for individual varieties in the sample (ILO et al., 2004, chapter 20). Using a geometric formula at the higher level would be compatible with the currently widely used Jevons index at the lower level and would have the benefit of maintaining consistency in aggregation.

Formulas (4) and (5) are the geometric counterparts to (2) and (3) and can be readily adopted by statistical offices since they use the same weights and price relatives as the Young and Lowe indexes. The Geometric Young price index is given by:

The geometric version of the Lowe price index with its price-updated weight is given by:

The (superlative) Törnqvist index is given by:

for which current period expenditure shares, $s_i^t$, are not available in real time. It is apparent that for constant expenditure shares over periods 0 and t, consistent with unitary elasticity of substitution, the Geometric Young index given by (4) equals the Törnqvist index given by (6).

Balk (2010) demonstrates that the substitution bias of the Geometric Young index is less than the substitution bias of the currently widely-used Lowe index. The CPI Practical Guide supports the use of the Geometric Young formulas (UNECE et al., 2009, page 160, ff. 50).⁷ The CPI Manual considers the Geometric Young index to be a serious practical possibility for CPI compilation; since the requisite weights are available in real time, and it is less susceptible to bias. With unitary elasticity of substitution, the Geometric Young can be shown to lie within the Laspeyres-Paasche interval. The Geometric Young index, as its name suggests, corresponds to cost-of-living indexes for utility-maximizing households with Geometric Young preferences. The CPI Manual cites as its main concern the unlikelihood of it gaining general acceptance in the foreseeable (then 2004) future since it cannot be interpreted as a fixed quantity basket index. (ILO et al., 2004, chapter 1 paragraphs 1.40 and 9.137).

Unlike the (arithmetic) Lowe index given by (3), the Geometric Lowe (and like the Geometric Young) indexes have no fixed quantity basket definition. The price updating of the weights has no rationale for the Geometric Lowe. Its standing is so low that neither the CPI Practical Guide nor the CPI Manual mentions it. However, the (arithmetic) Lowe is widely used in practice. It is invariably described in terms of a weighted average of price changes, albeit with little reference to such weights given in the last term of equation (3), which have little economic meaning. There is a prima facie case for some formal and empirical analysis of the geometric counterpart to the arithmetic Lowe.

C. Available empirical work

Given concern about arithmetic formulations and some positive aspects of geometric ones, we consider some of the available, albeit limited, empirical work on how close different formulations lie to a superlative index.

Hansen (2007) using Danish CPI data for 1996 to 2003 found increases for the Young and Lowe indices of 17.49 and 18.01 percent, respectively, compared with an increase in the Törnqvist index of 17.08 percent.⁸ The Geometric Young index was below Törnqvist at 16.51 percent. The differences between Young and Lowe are not always trivial.⁹ The annual inflation rate for 2004/5 and 2005/6 increased from 1.80 to 1.88 percent using Young but decreased from 1.99 to 1.90 percent using Lowe.

Greenlees and Williams (2010), in a major study of the US CPI over December 1990 through December 2008, found Lowe and Young increases to be quite similar, at 18.88 and 18.24 percent respectively, but the (chained) Törnqvist was much lower at 16.78 percent.¹⁰ The Geometric Young index was closer to, and again below, the chained Törnqvist at 15.84 percent.

Pike et al. (2009, Table 10)—using New Zealand CPI data for June 2006 to June 2008 with weights of 2003/4 and 2006/7 respectively price-updated to June 2006 and June 2008 quarters (the New Zealand CPI is quarterly)—found Lowe and Young to differ showing over this period increases of 6.26 and 5.60 percent, respectively. These arithmetic formulations were significantly higher than the 4.83 percent increase for the Geometric Young index which appeared to understate the 5.73 percent increase measured by a retrospective Fisher index.

So while Lowe and Young may generate similar results, their difference from a superlative Törnqvist index is marked and of concern. The Geometric Young index generally falls below (and there is some evidence that it is closer to) the superlative Törnqvist index.

Given the Geometric Young and Geometric Lowe are practical contenders for the CPI aggregation formula, we now present a formal analysis as to why they might differ.

A. Geometric Young vs. Geometric Lowe

Following on from equation (4) we first define a Geometric Young price index as:

where $s_i^b=\frac{p_i^b{q}_i^b}{\underset{i=1}{\overset{n}{\mathrm{\Sigma }}}{p}_i^b{q}_i^b}$ are period b expenditure shares and $y_i\equiv \ln \left(\frac{p_i^t}{p_i^0}\right)$ is the natural logarithm of the i^th price relative.

The difference between the logarithms of a Geometric Lowe and a Geometric Young price index is given by:

where $x_i\equiv \frac{p_i^0}{p_i^b}$. Adopting a Bortkiewicz (1923) decomposition:¹¹

where ${\rho }_{x,y}^{s_i^b}$ is the period-b weighted $\left(s_i^b\right)$ correlation coefficient between price relatives x_i and y_i (that extend respectively from b → 0 and 0 → t); ${cv}_x^{s_i^b}={\sigma }_x^{s_i^b}/{\overline{x}}^{s_i^b}$ is the period-b weighted $\left(w_i^b\right)$ coefficient of variation for x_i, for which ${\sigma }_x^{w_i^b}$ is the standard deviation and ${\overline{x}}^{w_i^b}$ is the $w_i^b$-weighted mean of x_i, that is, a Laspeyres price index between periods b and 0. First, it is apparent from (10) that ${\rho }_{x,y}^{s_i^b}$ dictates whether $I_{GLo}^{0\to t}$ is larger (positive) or smaller (negative) than $I_{GY}^{0\to t}$. For (weighted) price changes between periods b and 0 to be correlated with (weighted logarithms of) price changes between periods 0 and t, there must be some persistent uni-directional long-run price change over period b to t. A priori, a sign cannot be unambiguously attached to this correlation coefficient.

Second, the magnitude of $I_{GLo}^{0\to t}/I_{GY}^{0\to t}$ is determined by

• (a) The magnitude of ${\rho }_{x,y}^{s_i^b}$—smaller ratios of $I_{GLo}^{0\to t}\text{to}{I}_{GY}^{0\to t}$ would be expected from countries with longer time lags in utilizing and updating the weights, that is, longer lags between periods b and 0 and periods 0 and t.

• (b) The dispersion of price changes, ${cv}_x^{s_i^b}\text{and}{\sigma }_y^{s_i^b}$—it is well established in economic theory and empirical work that dispersion in relative prices increases with increases in inflation.¹² The Geometric Lowe will drift above the Geometric Young with higher rates of inflation. Note that ${\sigma }_y^{s_i^b}$ is likely to be the most potent driver of the drift since it is not corrected, as is the coefficient of variation, ${cv}_x^{s_i^b}$, for changes in the mean. ${\sigma }_y^{s_i^b}$ is concerned with the often larger index changes between period 0 to t, than the constant ${cv}_x^{s_i^b}$. over period b to 0.¹³

• (c) The multiplicative nature of terms on the right-hand-side of equation (10)—for example, any chance lowering of ${\rho }_{x,y}^{s_i^b}$ to near zero in a month will lead to the two formula being very similar in spite of higher ${cv}_x^{s_i^b}\text{and}{\sigma }_y^{s_i^b}$.

Third, we do not depict the difference between Geometric Young and Geometric Lowe indexes as substitution bias. It is clear from equation (10) that the differences stem from a correlation between price changes in one period and the (logarithm of) price changes in a subsequent period: not a correlation between price and quantity changes. It is the latter that defines substitution bias.

B. Comparisons with a superlative price index

Having examined how a Geometric Young differs from a Geometric Lowe price index, we turn to consider how both indexes, given by equations (4) and (5), differ from a superlative Törnqvist price index given by equation (6). The ratio of a Geometric Lowe to Törnqvist price index is by extension of equation (10):

and the ratio of a Geometric Young to Törnqvist price index, by definition, equations (4) and (6) by:

The Geometric Young and Törnqvist are equal if the shares in period b are equal to the average of the shares in periods 0 and t, that is, $s_i^b=(s_i^0+s_i^t)/2$. As the index is progressively compiled across periods 0 to t, the implicit assumption is of a price elasticity of substitution of unity for comparisons between b to 0 continuing through between 0 and t. To evaluate the suitability of $I_{GY}^t$ as an estimate of $I_T^t$ we need to evaluate the elasticity of substitution in terms of its proximity to unity and its changes over time. One approach is to use a formula that simply assumes it is constant over time. The Lloyd-Moulton (constant elasticity of substitution—CES) index is given by:¹⁴

for which η is the elasticity of substitution. The formulation is quite flexible: the Young index is consistent with η tending to zero and the Geometric Young index is consistent with η tending to unity. Greenlees (2011) used an approach proposed by Feenstra and Reinsdorf (2007) to estimate η for US data. He found values of η lie between 0 and 1, that is, inelastic substitution, though he also found occasional anomalous years; for 1999/2000 to 2005/6 estimated η varied between 0.521 and 0.655, but was close to (not significantly different from) unity for 2006/7 at 0.981, and close to (not significantly different from) zero for 2007/8 at 0.192; findings are at odds with an assumption of constant elasticity and, moreover, constant unitary elasticity. A finding of inelastic substitution argues against the implicit fixed baskets of (arithmetic) Lowe and Young indexes and implies that items with relatively higher price trends receive less importance in $I_{GY}^{0\to t}$ than in $I_T^{0\to t}$, that is, $I_{GY}^{0\to t}<I_T^{0\to t}$.

Lent and Dorfman (2009) derive their formulation from a Taylor approximation to CES and superlative indexes. They find that a weighted average of arithmetic Laspeyres, $I_{Las}^{0\to t}$, and Geometric Laspeyres, $I_{GLas}^{0\to t}$, indexes (called an AG Mean index) can approximate a superlative target index:

The weights are not restricted to be constant. The authors demonstrate that the AG Mean can provide a close approximation to a superlative (Fisher) price index when 0 ≤ η ≤ 1. Estimators of η vary to be compatible with the target index number. For a Fisher price index as the target, $I_{AG}^{0\to t}=I_F^{0\to t}$, equation (14) is given by:

Solving (15) for η^τ:

The estimated weights in equation (16) and used in equation (15) can vary. Lent and Dorfman (2009) suggest that a moving average of η^τ be used over τ = t - T, t to smooth any volatility. We consider in the next (empirical) section how well the currently used arithmetic Lowe and Young indexes compare to (the bounds of) a superlative index. We then look at whether their geometric counterparts do any better and, if so, why they differ, leading to a real-time Lent-Dorfman approach.

A. The data

The data used are the elementary aggregate indexes for the U.S. Urban CPI and their weights over the period December 1997 to December 2010, provided by the U.S. Bureau of Labor Statistics (BLS). The elementary aggregate indexes are for about 211 item strata (product groups).. We stress that the compilation of the U.S. Urban CPI is based on 211 item strata (product groups) for 38 area strata, that is, 8,018 cells. The indexes for the individual item/area strata are for the large part derived using weighted geometric means, while the aggregation across areas uses the Lowe formula.¹⁵ Our analysis is, for simplicity, of the effect of using a different formula to measure the US Urban CPI if only the 211 weights for product groups were available, as is the case with many countries. The results of our estimates of a chained Törnqvist index are very close to the BLS’,¹⁶ a finding in itself of interest.¹⁷

The dates of the weights used over this period are given below. Following BLS procedures for their aggregation at the higher level, they were price updated from the expenditure period to the December prior to their use in the index. Note that the mean annual 1993-95 urban US expenditures for the 211 CPI item strata were the basis of the CPI weights for the four years from January 1998 through December 2001. Unlike subsequent expenditure weights, these expenditures were (i) from a 3-year period (not a 2-year period), (ii) were used in the CPI for a 4-year period (not a 2-year period), and (iii) were price-updated to December 1997 from about 3½ years (not 2 years) earlier—from the midpoint, June-July 1994.

Mean-annual expenditures	Basis of weights for:
1993-1995	Jan98-Dec01
1999-2000	Jan02-Dec03
2001-2002	Jan04-Dec05
2003-2004	Jan06-Dec07
2005-2006	Jan08-Dec09
2007-2008	Jan10-Dec11
2008-2009	Jan11-Dec12

View Table

Mean-annual expenditures	Basis of weights for:
1993-1995	Jan98-Dec01
1999-2000	Jan02-Dec03
2001-2002	Jan04-Dec05
2003-2004	Jan06-Dec07
2005-2006	Jan08-Dec09
2007-2008	Jan10-Dec11
2008-2009	Jan11-Dec12

B. Results

Figure 1 shows the standard arithmetic price indexes: Lowe, Laspeyres, and Young and the harmonic Paasche. The target index is a superlative index Fisher price index, a symmetric (geometric) average of Laspeyres and Paasche price indexes that lies between them.¹⁸ The (arithmetic) Lowe price index, widely used by many countries for their CPI, is above Laspeyres. It performs poorly against the Young. Young is much closer to Laspeyres and, thus, to the desirable Laspeyres-Paasche interval.

Arithmetic indexes

Citation: IMF Working Papers 2012, 105; 10.5089/9781475502954.001.A001

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Arithmetic indexes

Citation: IMF Working Papers 2012, 105; 10.5089/9781475502954.001.A001

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Arithmetic indexes

Citation: IMF Working Papers 2012, 105; 10.5089/9781475502954.001.A001

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The differences between the results of the formulas are not large. Some of this is due to the more frequent updating of weights undertaken by the US Bureau of Labor Statistics than in many other countries. Yet, the differences are not insubstantial, especially given the CPI is used extensively to escalate payments for rents, wages, alimony, child support and other such obligations. The Fisher price index increased in 2010 compared with 1998 at a (compound) annual average rate of 2.27 percent, compared with a Lowe price index increase of 2.44 percent.

As outlined in section II, arithmetic Lowe and Young indexes have counterpart geometric averages as practical alternatives, the Geometric Lowe and Geometric Young indexes respectively. Both formulas use the same data and can be compiled in real time.

Figure 2 shows results for the Geometric-Lowe, Geometric Young, and Törnqvist price indexes. At the beginning of the series, all indexes seem to track each other quite closely. After December 2003, the Geometric Lowe and Geometric Young indexes drift apart. The Geometric Lowe is closer to the Törnqvist from December 2003 to December 2007, providing evidence, at least for this data set, in its favor against the Geometric Young. Bear in mind that the Geometric Lowe has little conceptual support. The arithmetic Lowe had conceptual support as a fixed quantity basket price index and price updating took place with this in mind. However, price updating the weights for a geometric formulation does not yield a fixed basket index. It is the Geometric Young price index that has a conceptual foundation as a period-b weighted average of price relatives. But for these data, the Geometric Young price index has a marked downward bias against the Törnqvist index.

Geometric vs. standard price indexes

Citation: IMF Working Papers 2012, 105; 10.5089/9781475502954.001.A001

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Geometric vs. standard price indexes

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Geometric vs. standard price indexes

Citation: IMF Working Papers 2012, 105; 10.5089/9781475502954.001.A001

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Two questions arise. First, what factors underlie the difference between the two geometric formulas and second, is the nature of the bias such that an average of formulas may be more suitable, similar to the averaging of Laspeyres and Paasche indexes. By inspection in Figure 2, Lowe overstates and Geometric Young understates the Törnqvist index and an average of the two may be suitable. We consider below, for US inflation rates, this and other combinations of formulas using the Lent-Dorfman (2009) framework, outlined in section IIIB above.

C. The Geometric formulas: differences and adjustments

The factors underlying the differences between the Geometric Lowe and Geometric Young are of interest as are averages of formulas that can make them better track superlative indexes. We consider the results for both in turn.

What factors underlie the difference between the two formulas?

Factors underlying the differences between the Geometric Lowe and Geometric Young can be understood based on the decomposition in equation (10). This decomposition is considered in Table 1 below. For brevity the results are only given from December 2007, re-referenced to December 2007=1.0000.

Table 1:

Decomposition of Geometric-Lowe to Geometric Young ratio

Table 1:

Decomposition of Geometric-Lowe to Geometric Young ratio

	Geometric- Young	Geometric Lowe	Geometric Lowe/Geometric Young	${\rho }_{x,y}^{s_i^b}$	${\sigma }_y^{s_i^b}$	${cv}_x^{s_i^b}$	$\exp \left({\rho }_{x,y}^{s_i^b}{cv}_x^{s_i^b}{\sigma }_y^{s_i^b}\right)$
2007 Dec	1	1	1
2008 Jan	1.0048	1.0052	1.0004	0.1962	0.0133	0.1712	1.0004
2008 Feb	1.0077	1.0081	1.0003	0.1039	0.0184		1.0003
2008 Mar	1.0147	1.0170	1.0023	0.4543	0.0292		1.0023
2008 Apr	1.0194	1.0232	1.0038	0.6069	0.0366		1.0038
2008 May	1.0250	1.0316	1.0065	0.7142	0.0530		1.0065
2008 Jun	1.0321	1.0414	1.0091	0.7324	0.0719		1.0091
2008 Jul	1.0367	1.0467	1.0096	0.6892	0.0809		1.0096
2008 Aug	1.0356	1.0434	1.0075	0.6390	0.0685		1.0075
2008 Sep	1.0355	1.0423	1.0066	0.6256	0.0612		1.0066
2008 Oct	1.0294	1.0315	1.0021	0.2621	0.0469		1.0021
2008 Nov	1.0139	1.0064	0.9927	-0.5619	0.0767		0.9927
2008 Dec	1.0031	0.9900	0.9869	-0.6632	0.1159		0.9869
2009 Jan	1.0077	0.9959	0.9883	-0.6329	0.1089		0.9883
2009 Feb	1.0131	1.0026	0.9897	-0.6119	0.0987		0.9897
2009 Mar	1.0156	1.0051	0.9897	-0.6074	0.0998		0.9897
2009 Apr	1.0178	1.0085	0.9909	-0.5676	0.0943		0.9909
2009 May	1.0203	1.0133	0.9932	-0.3853	0.0845		0.9944
2009 Jun	1.0275	1.0248	0.9974	-0.2102	0.0725		0.9974
2009 Jul	1.0259	1.0226	0.9968	-0.2383	0.0779		0.9968
2009 Aug	1.0276	1.0255	0.9979	-0.1581	0.0766		0.9979
2009 Sep	1.0284	1.0256	0.9973	-0.2025	0.0794		0.9973
2009 Oct	1.0298	1.0266	0.9969	-0.2312	0.0785		0.9969
2009 Nov	1.0300	1.0281	0.9981	-0.1457	0.0747		0.9981
2009 Dec	1.0280	1.0259	0.9979	-0.1589	0.0766		0.9979

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Table 1:

Decomposition of Geometric-Lowe to Geometric Young ratio

	Geometric- Young	Geometric Lowe	Geometric Lowe/Geometric Young	${\rho }_{x,y}^{s_i^b}$	${\sigma }_y^{s_i^b}$	${cv}_x^{s_i^b}$	$\exp \left({\rho }_{x,y}^{s_i^b}{cv}_x^{s_i^b}{\sigma }_y^{s_i^b}\right)$
2007 Dec	1	1	1
2008 Jan	1.0048	1.0052	1.0004	0.1962	0.0133	0.1712	1.0004
2008 Feb	1.0077	1.0081	1.0003	0.1039	0.0184		1.0003
2008 Mar	1.0147	1.0170	1.0023	0.4543	0.0292		1.0023
2008 Apr	1.0194	1.0232	1.0038	0.6069	0.0366		1.0038
2008 May	1.0250	1.0316	1.0065	0.7142	0.0530		1.0065
2008 Jun	1.0321	1.0414	1.0091	0.7324	0.0719		1.0091
2008 Jul	1.0367	1.0467	1.0096	0.6892	0.0809		1.0096
2008 Aug	1.0356	1.0434	1.0075	0.6390	0.0685		1.0075
2008 Sep	1.0355	1.0423	1.0066	0.6256	0.0612		1.0066
2008 Oct	1.0294	1.0315	1.0021	0.2621	0.0469		1.0021
2008 Nov	1.0139	1.0064	0.9927	-0.5619	0.0767		0.9927
2008 Dec	1.0031	0.9900	0.9869	-0.6632	0.1159		0.9869
2009 Jan	1.0077	0.9959	0.9883	-0.6329	0.1089		0.9883
2009 Feb	1.0131	1.0026	0.9897	-0.6119	0.0987		0.9897
2009 Mar	1.0156	1.0051	0.9897	-0.6074	0.0998		0.9897
2009 Apr	1.0178	1.0085	0.9909	-0.5676	0.0943		0.9909
2009 May	1.0203	1.0133	0.9932	-0.3853	0.0845		0.9944
2009 Jun	1.0275	1.0248	0.9974	-0.2102	0.0725		0.9974
2009 Jul	1.0259	1.0226	0.9968	-0.2383	0.0779		0.9968
2009 Aug	1.0276	1.0255	0.9979	-0.1581	0.0766		0.9979
2009 Sep	1.0284	1.0256	0.9973	-0.2025	0.0794		0.9973
2009 Oct	1.0298	1.0266	0.9969	-0.2312	0.0785		0.9969
2009 Nov	1.0300	1.0281	0.9981	-0.1457	0.0747		0.9981
2009 Dec	1.0280	1.0259	0.9979	-0.1589	0.0766		0.9979

The sign of ${\rho }_{x,y}^{s_i^b}$ dictates which of the two formulas’ growth exceeds the other. From January 2008 to October 2008 it was positive, leading to the Geometric Lowe growing faster than the Geometric Young and from November 2008 onward it was negative leading to the reverse position, as shown in Table 1. These empirical runs in signs reflect long-run trends in price change between periods b to 0 (2004/5 to December 2007) being continued in sub-periods 0 to t (December 2007 to months up to October 2008) and then being reversed in subsequent sub-periods (to months after October 2008). This illustrates the dependency on long-run trends for the relative positioning of the two formulas.

The magnitude of the difference is determined by the magnitude of three factors. The correlation coefficient is not expected to be strong given price changes in one period are to be related to the logarithms of price changes in a subsequent one; ${cv}_x^{s_i^b}$ is a one-off factor for period b to 0—the higher it is, other factors equal, the larger the difference. If such dispersion increases over time, then lags between introducing weights from the survey period into the rebased index, between b and 0, will accentuate the difference between the two formulas. Finally, ${\sigma }_y^{s_i^b}$ and thus the difference between the two formulas, can be expected to increase over time. Of note is that the three factors are multiplicative: minimize any one factor, such as ${cv}_x^{s_i^b}$ by minimizing the time lag in the introduction of weights, and the difference between formulas becomes smaller. The results from Table 1 confirm this.

Averages of formulas that better track superlative indexes

An approach based on the Lent-Dorfman (2009) (hereafter L-D) framework uses averages of two formulas to more closely correspond to a superlative index.¹⁹

In Table 2 we consider average monthly percentage differences between the target indexes and alternative measures using the simulated US CPI data for January 2004 to December 2010. Lowe has the largest bias of about 1.7 percent from a superlative index and Young performs much better, reducing the bias to about 0.5 percent. Their geometric equivalents show mixed results with the Geometric Lowe being, on average, 0.2 percent above the Törnqvist while the Geometric Young is about 1.0 percent below. We also consider averaging of these indexes along with some variants of the L-D approach.

Table 2,

Average Monthly Percentage Differences between Alternative vs. Target Indexes:^*

^*Covers January 2004 to December 2010

Table 2,

Average Monthly Percentage Differences between Alternative vs. Target Indexes:^*

	Target Indexes
	Fisher	Törnqvist
Arithmetic formulas
Lowe	1.712	1.689
Young	0.466	0.443
Geometric formulas
Geometric Lowe (GLowe)	0.265	0.242
Geometric Young	-0.959	-0.981
Geometric means of formulas
GYoung-Young	-0.249	-0.272
GYoung-Lowe	0.368	0.345
GLowe:Young	0.366	0.343
GLowe:Lowe	0.986	0.963
Lent-Dorfman (η using lag)
GYoung:Young	-0.339	-0.375
GYoung:Lowe	-0.196	-0.219
GLowe:Young	0.339	0.316
GLowe:Lowe	0.581	0.558

^*Covers January 2004 to December 2010

View Table

Table 2,

Average Monthly Percentage Differences between Alternative vs. Target Indexes:^*

	Target Indexes
	Fisher	Törnqvist
Arithmetic formulas
Lowe	1.712	1.689
Young	0.466	0.443
Geometric formulas
Geometric Lowe (GLowe)	0.265	0.242
Geometric Young	-0.959	-0.981
Geometric means of formulas
GYoung-Young	-0.249	-0.272
GYoung-Lowe	0.368	0.345
GLowe:Young	0.366	0.343
GLowe:Lowe	0.986	0.963
Lent-Dorfman (η using lag)
GYoung:Young	-0.339	-0.375
GYoung:Lowe	-0.196	-0.219
GLowe:Young	0.339	0.316
GLowe:Lowe	0.581	0.558

^*Covers January 2004 to December 2010

First, use is made of approximations to Laspeyres and Geometric Laspeyres in equation (14) for which real-time data would not be available in practice. We use, in turn, Lowe and Young formulas to approximate Laspeyres and Geometric Lowe and Geometric Young formulas to approximate Geometric Laspeyres indexes in (14); for example, one variant would be the geometric average of the Young and Geometric Young and another, the geometric average of the Lowe and the Geometric Young.

Second, we tailor estimates of η^τ in (15) to be based on a selected superlative benchmark, say the Törnqvist index. The Törnqvist index cannot be estimated in real time so the most recently available estimates are used to enable a real-time computation. For example, for January 08 to December 2009, using our US data, the most recent estimates of Törnqvist indexes are used, that is those available starting January 2008.

Third, the average η^τ over January 2004 to December 2005 are used in equation (15) for January 08 to December 2009, and similarly over other periods. The Lent-Dorfman formulation has the advantage of allowing η^τ to change on a monthly basis. However, we constrain such changes to the period of the rebasing of the index and then hold η^τ constant until the next rebasing. This has practical advantages: the timing of such weight changes concurs with the rebasing of the index and CPI changes are not affected by changes in the weight η^τ given in (14) to the different formulas used. Table 2 shows the results from using, in equations (15) and (17), different formulas as approximations to Laspeyres and Geometric Laspeyres benchmarked on the two superlative indexes, Fisher and Törnqvist. All formulations can be calculated in real time using the existing prices database.

Fourth, estimates of the η^τ weights using equation (16) may be negative depending on the relative positioning of the Geometric Young (as a proxy for the Geometric Laspeyres) and the Young (for Laspeyres) indexes to the Fisher (superlative) index and that 0 ≥ η^τ ≤ 1. Instead of (16) we use an adaptation, where ABS are absolute values:

$\begin{array}{cc}{\eta }^{\tau }=\frac{ABS(I_F^{0\to t}-I_{CD}^{0\to t})}{ABS(I_F^{0\to t}-I_{CD}^{0\to t})+ABS(I_F^{0\to t}-I_Y^{0\to t})}\mathrm{.}& \left(17\right)\end{array}$

In Table 2 four different variants of the L-D formulation are given each using the Fisher and Törnqvist indexes as benchmarks. This is preceded by the geometric mean of the two formulas—one arithmetic one geometric—used as an approximation.

These simple geometric means are highly successful (especially for Geometric Young:Lowe and GLowe:Young) in cutting the bias from using the Young index. The L-D approximations have similar effects in reducing the bias. The Lowe formula is used by the US and many other countries for CPI compilation. From Table 2 we can see that in real time using the same database as used by Lowe, we can cut Lowe’s bias from 1.7 percent to 0.3 percent. The Geometric Young:Lowe formulation has much to commend it as an adjustment, via the Geometric Young index, of the existing Lowe formula.

From our analysis in Figure 3, the L-D approximations (average of Lowe: Geometric Young and average of Young:Geometric Young) to the Törnqvist and Fisher indexes appear to work: they successfully track the Törnqvist index and alleviate the downward bias in the Geometric Young and the upward bias in the Lowe or Young indexes. The L-D index has a conceptual basis as a Taylor approximation to the Törnqvist (or Fisher) index that can be calculated in real time. As Lent and Dorfman (2009) note: the systematic updating of the index continuously picks up changes in consumer buying patterns as reflected in the data, while requiring no iterative numerical procedures and can therefore be easily programmed and automated in a statistical production setting.

Target indexes vs. L-D proxies

Citation: IMF Working Papers 2012, 105; 10.5089/9781475502954.001.A001

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Target indexes vs. L-D proxies

Citation: IMF Working Papers 2012, 105; 10.5089/9781475502954.001.A001

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Target indexes vs. L-D proxies

Citation: IMF Working Papers 2012, 105; 10.5089/9781475502954.001.A001

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• Download figure as PowerPoint slide