Abstract

22.1 The existence of seasonal commodities poses some significant challenges for price statisticians. Seasonal commodities are commodities which are either: (a) not available in the marketplace during certain seasons of the year, or (b) are available throughout the year, but there are regular fluctuations in prices or quantities that are synchronized with the season or the time of the year.1 A commodity that satisfies (a) is termed a strongly seasonal commodity, whereas a commodity that satisfies (b) is called a weakly seasonal commodity. It is strongly seasonal commodities that create the biggest problems for price statisticians in the context of producing a monthly or quarterly consumer price index (CPI) because if a commodity price is available in only one of the two months (or quarters) being compared, then obviously it is not possible to calculate a relative price for the commodity and traditional bilateral index number theory breaks down. In other words, if a commodity is present in one month but not the next, how can the month-to-month amount of price change for that commodity be computed?2 In this chapter, a solution to this problem is presented which “works”, even if the commodities consumed are entirely different for each month of the year.3

Introduction

22.1 The existence of seasonal commodities poses some significant challenges for price statisticians. Seasonal commodities are commodities which are either: (a) not available in the marketplace during certain seasons of the year, or (b) are available throughout the year, but there are regular fluctuations in prices or quantities that are synchronized with the season or the time of the year.1 A commodity that satisfies (a) is termed a strongly seasonal commodity, whereas a commodity that satisfies (b) is called a weakly seasonal commodity. It is strongly seasonal commodities that create the biggest problems for price statisticians in the context of producing a monthly or quarterly consumer price index (CPI) because if a commodity price is available in only one of the two months (or quarters) being compared, then obviously it is not possible to calculate a relative price for the commodity and traditional bilateral index number theory breaks down. In other words, if a commodity is present in one month but not the next, how can the month-to-month amount of price change for that commodity be computed?2 In this chapter, a solution to this problem is presented which “works”, even if the commodities consumed are entirely different for each month of the year.3

22.2 There are two main sources of seasonal fluctuations in prices and quantities: (a) climate, and (b) custom.4 In the first category, fluctuations in temperature, precipitation and hours of daylight cause fluctuations in the demand or supply for many commodities; for example, summer versus winter clothing, the demand for light and heat, holidays, etc. With respect to custom and convention as a cause of seasonal fluctuations, consider the following quotation:

Conventional seasons have many origins - ancient religious observances, folk customs, fashions, business practices, statute law… Many of the conventional seasons have considerable effects on economic behaviour. We can count on active retail buying before Christmas, on the Thanksgiving demand for turkeys, on the first of July demand for fireworks, on the preparations for June weddings, on heavy dividend and interest payments at the beginning of each quarter, on an increase in bankruptcies in January, and so on (Mitchell (1927, p. 237)).

22.3 Examples of important seasonal commodities are: many food items; alcoholic beverages; many clothing and footwear items; water; heating oil; electricity; flowers and garden supplies; vehicle purchases; vehicle operation; many entertainment and recreation expenditures; books; insurance expenditures; wedding expenditures; recreational equipment; toys and games; software; air travel and tourism expenditures. For a “typical” country, seasonal expenditures will often amount to one-fifth to one-third of all consumer expenditures.5

22.4 In the context of producing a monthly or quarterly CPI, it must be recognized that there is no completely satisfactory way of dealing with strongly seasonal commodities. If a commodity is present in one month but missing from the marketplace in the next month, then none of the index number theories that were considered in Chapters 15 to 20 can be applied because all these theories assumed that the dimensionality of the commodity space was constant for the two periods being compared. However, if seasonal commodities are present in the market during each season, then, in theory, traditional index number theory can be applied in order to construct month-to-month or quarter-to-quarter price indices. This “traditional” approach to the treatment of seasonal commodities will be followed in paragraphs 22.78 to 22.90. The reason why this straightforward approach is deferred to the end of the chapter is twofold:

  • The approach that restricts the index to commodities that are present in every period often does not work well in the sense that systematic biases can occur.

  • The approach is not fully representative; i.e., it does not make use of information on commodities that are not present in every month or quarter.

22.5 In the next section, a modified version of Tur-vey’s (1979) artificial data set is introduced. This data set will be used in order to evaluate numerically all the index number formulae suggested in this chapter. It will be seen in paragraphs 22.63 to 22.77 that very large seasonal fluctuations in volumes, combined with systematic seasonal changes in price, can make month-to-month or quarter-to-quarter price indices behave rather poorly.

22.6 Even though existing index number theory cannot deal satisfactorily with seasonal commodities in the context of constructing month-to-month indices of consumer prices, it can deal satisfactorily with seasonal commodities if the focus is changed from month-to-month CPIs to CPIs that compare the prices of one month with the prices of the same month in a previous year. Thus, in paragraphs 22.16 to 22.34, year-over-year monthly CPIs are studied. Turvey’s seasonal data set is used to evaluate the performance of these indices and they are found to perform quite well.

22.7 In paragraphs 22.35 to 22.44, the year-over-year monthly indices defined in paragraphs 23.16 to 23.34 are aggregated into an annual index that compares all the monthly prices in a given calendar year with the corresponding monthly prices in a base year. In paragraphs 22.45 to 22.54, this idea of comparing the prices of a current calendar year with the corresponding prices in a base year is extended to annual indices that compare the prices of the last 12 months with the corresponding prices in the 12 months of a base year. The resulting rolling year indices can be regarded as seasonally adjusted price indices. The modified Turvey data set is used to test out these year-over-year indices, and they are found to work very well on this data set.

22.8 The rolling year indices can provide an accurate gauge of the movement of prices in the current rolling year compared to the base year. This measure of price inflation can, however, be regarded as a measure of inflation for a year that is centred around a month six months prior to the last month in the current rolling year. Hence for some policy purposes, this type of index is not as useful as an index that compares the prices of the current month to the previous month, so that more up-to-date information on the movement of prices can be obtained. In paragraphs 22.55 to 22.62, it will nevertheless be shown that under certain conditions, the year-over-year monthly index for the current month, along with the year-over-year monthly index for last month, can successfully predict or forecast a rolling year index that is centred around the current month.

22.9 The year-over-year indices defined in paragraphs 22.16 to 22.34, and their annual averages studied in paragraphs 22.35 to 22.54, offer a theoretically satisfactory method for dealing with strongly seasonal commodities; i.e., commodities that are available only during certain seasons of the year. These methods rely on the year-over-year comparison of prices and hence cannot be used in the month-to-month or quarter-to-quarter type of index, which is typically the main focus of a consumer price programme. Thus there is a need for another type of index, which may not have very strong theoretical foundations, but which can deal with seasonal commodities in the context of producing a month-to-month index. In paragraphs 22.63 to 22.77, such an index is introduced and it is implemented using the artificial data set for the commodities that are available during each month of the year. Unfortunately, because of the seasonality in both prices and quantities of the always available commodities, this type of index can be systematically biased. This bias shows up for the modified Turvey data set.

22.10 Since many CPIs are month-to-month indices that use annual basket quantity weights, this type of index is studied in paragraphs 22.78 to 22.84. For months when the commodity is not available in the marketplace, the last available price is carried forward and used in the index. In paragraphs 22.85 and 22.86, an annual quantity basket is again used but instead of carrying forward the prices of seasonally unavailable items, an imputation method is used to fill in the missing prices. The annual basket type indices defined in paragraphs 22.78 to 22.84 are implemented using the artificial data set. Unfortunately, the empirical results are not satisfactory in that the indices show tremendous seasonal fluctuations in prices, so they would not be suitable for users who wanted up-to-date information on trends in general inflation.

22.11 In paragraphs 22.87 to 22.90, the artificial data set is used in order to evaluate another type of month-to-month index that is frequently suggested in the literature on how to deal with seasonal commodities; namely the Bean and Stine Type C (1924) or Rothwell (1958) index. Again, this index does not get rid of the tremendous seasonal fluctuations that are present in the modified Turvey data set.

22.12 Paragraphs 22.78 to 22.84 show that the annual basket type indices with carry forward of missing prices or imputation of missing prices do not get rid of seasonal fluctuations in prices. However, in paragraphs 22.91 to 22.96, it is shown how seasonally adjusted versions of these annual basket indices can be used successfully to forecast rolling year indices that are centred on the current month. In addition, the results show how these annual basket type indices can be seasonally adjusted (using information obtained from rolling year indices from prior periods or by using traditional seasonal adjustment procedures), and hence these seasonally adjusted annual basket indices could be used as successful indicators of general inflation on a timely basis.

22.13 Paragraph 23.97 outlines some conclusions.

A seasonal commodity data set

22.14 It is useful to illustrate the index number formulae defined in subsequent sections by computing them for an actual data set. Turvey (1979) constructed an artificial data set for five seasonal commodities (apples, peaches, grapes, strawberries and oranges) for four years by month so that there are 5 x 4 x 12 = 240 observations in all. At certain times of the year, peaches and strawberries (commodities 2 and 4) are unavailable, so in Tables 22.1 and 22.2 the prices and quantities for these two commodities are entered as zeros.6 The data in Tables 22.1 and 22.2 are essentially the same as the data set constructed by Turvey except that a number of adjustments have been made to it in order to illustrate various points. The two most important adjustments are:

Table 22.1

An artificial seasonal data set: Prices

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  • The data for commodity 3 (grapes) have been adjusted so that the annual Laspeyres and Paasche indices (defined in paragraphs 22.35 to 22.44) would differ more than in the original data set.7

  • After the above adjustments were made, each price in the last year of data was escalated by the monthly inflation factor 1.008 so that month-to-month inflation for the last year of data would be at an approximate monthly rate of 1.6 per cent per month compared to about 0.8 per cent per month for the first three years of data.8

Table 22.2

An artificial seasonal data set: Quantities

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22.15 Ralph Turvey sent his artificial data set to statistical agencies around the world, asking them to use their normal techniques to construct monthly and annual average price indices. About 20 countries replied, and Turvey (1979, p. 13) summarized the responses as follows: “It will be seen that the monthly indices display very large differences, e.g., a range of 129.12–169.50 in June, while the range of simple annual means is much smaller. It will also be seen that the indices vary as to the peak month or year.”

The above (modified) data are used to test out various index number formulae in subsequent sections.

Year-over-year monthly indices

22.16 It can be seen that the existence of seasonal commodities that are present in the marketplace in one month but not the next causes the accuracy of a month-to-month index to fall.9 A way of dealing with these strongly seasonal commodities is to change the focus from short-term month-to-month price indices and instead focus on making year-over-year price comparisons for each month of the year. In the latter type of comparison, there is a good chance that seasonal commodities that appear, say, in February will also appear in subsequent Februarys so that the overlap of commodities will be maximized in these year-over-year monthly indices.

22.17 For over a century, it has been recognized that making year-over-year comparisons10 provides the simplest method for making comparisons that are free from the contaminating effects of seasonal fluctuations. According to W. Stanley Jevons (1884, p. 3):

In the daily market reports, and other statistical publications, we continually find comparisons between numbers referring to the week, month, or other parts of the year, and those for the corresponding parts of a previous year. The comparison is given in this way in order to avoid any variation due to the time of the year. And it is obvious to everyone that this precaution is necessary. Every branch of industry and commerce must be affected more or less by the revolution of the seasons, and we must allow for what is due to this cause before we can learn what is due to other causes.

22.18 The economist A.W. Flux and the statistician G. Udny Yule also endorsed the idea of making year-over-year comparisons to minimize the effects of seasonal fluctuations:

Each month the average price change compared with the corresponding month of the previous year is to be computed…. The determination of the proper seasonal variations of weights, especially in view of the liability of seasons to vary from year to year, is a task from which, I imagine, most of us would be tempted to recoil (Flux (1921, pp. 184-185)).

My own inclination would be to form the index number for any month by taking ratios to the corresponding month of the year being used for reference, the year before presumably, as this would avoid any difficulties with seasonal commodities. I should then form the annual average by the geometric mean of the monthly figures (Yule (1921, p. 199)).

In more recent times, Victor Zarnowitz (1961, p. 266) also endorsed the use of year-over-year monthly indices:

There is of course no difficulty in measuring the average price change between the same months of successive years, if a month is our unit “season”, and if a constant seasonal market basket can be used, for traditional methods of price index construction can be applied in such comparisons.

22.19 In the remainder of this section, it is shown how year-over-year Fisher indices and approximations to them can be constructed.11 For each month m = 1, 2,…, 12, let S(m) denote the set of commodities that are available in the marketplace for each year t = 0, 1,…, T. For t = 0, 1,…,T and m = 1, 2,…,12, let and qm denote the price and quantity of commodity n that is in the marketplace in month m of year t, where n belongs to S(m). Let pt,m and q t,m denote the month m and year t price and quantity vectors, respectively. Then the year-over-year monthly Laspeyres, Paasche and Fisher indices going from month m of year t to month m of year t+ 1 can be defined as follows:

PL(pt,m,pt+1,m,qt,m)=ΣnS(m)pnt+1,mqnt,mΣnS(m)pnt,mqnt,mm=1,2,,12(22.1)
PP(pt,m,pt+1,m,qt+1,m)=ΣnS(m)pnt+1,mqnt+1,mΣnS(m)pnt,mqnt+1,mm=1,2,,12(22.2)
PF(pt,m,pt+1,m,qt,m,qt+1,m)PL(pt,m,pt+1,m,qt,m)PP(pt,m,pt+1,m,qt+1,m)m=1,2,,12.(22.3)

22.20 The above formulae can be rewritten in price relative and monthly expenditure share form as follows:

PL(pt,m,pt+1,m,st,m)=ΣnS(m)snt,m(pnt+1,m/pnt,m)m=1,2,…12(22.4)
PL(pt,m,pt+1,m,st+1,m)=[ΣnS(m)snt+1,m(pnt+1,m/pnt,m)1]1m=1,2,…12(22.5)
PL(pt,m,pt+1,m,st,m,st+1,m)PL(pt,m,pt+1,m,st,m,st+1,m)PP(pt,m,pt+1,m,st,m,st+1,m)=ΣnS(m)snt,m(pnt+1,m/pnt,m)×[ΣnS(m)snt,m(pnt+1,m/pnt,m)1]1m=1,2,,12(22.6)

where the monthly expenditure share for commodity nS(m) for month m in year t is defined as:

snt,m=pnt,mqnt,mΣiS(m)pit,mqit,mm=1,2,,12nS(m)t=0,1,,T(22.7)

and st,m denotes the vector of month m expenditure shares in year t, [snt,m] for nS(m).

22.21 Current period expenditure shares snt,m are not likely to be available. Hence it will be necessary to approximate these shares using the corresponding expenditure shares from a base year 0.

22.22 Use the base period monthly expenditure share vectors s0,m in place of the vector of month m and year t expenditure shares st,m in equation (22.4), and use the base period monthly expenditure share vectors s0,m in place of the vector of month m and year t + 1 expenditure shares in equation (22.5). Similarly, replace the share vectors st,m and st+l,m in equation (22.6) by the base period expenditure share vector for month m, s0,m. The resulting approximate year-over-year monthly Laspeyres, Paasche and Fisher indices are defined by equations (22.8) to (22.10):12

PAL(pt,m,pt+1,m,s0,m)=ΣnS(m)sn0,m(pnt+1,m/pnt,m)m=1,2,,12(22.8)
PAL(pt,m,pt+1,m,s0,m)=[ΣnS(m)sn0,m(pnt+1,m/pnt,m)1]1m=1,2,,12(22.9)
PAF(pt,m,pt+1,m,s0,m,s0,m)PAL(pt,m,pt+1,m,s0,m)PP(pt,m,pt+1,m,s0,m)=ΣnS(m)snt,m(pnt+1,m/pnt,m)×[ΣnS(m)snt,m(pnt+1,m/pnt,m)1]1m=1,2,,12(22.10)

22.23 The approximate Fisher year-over-year monthly indices defined by equation (22.10) will provide adequate approximations to their true Fisher counterparts defined by equation (22.6) only if the monthly expenditure shares for the base year 0 are not too different from their current year t and t + 1 counterparts. Hence, it will be useful to construct the true Fisher indices on a delayed basis in order to check the adequacy of the approximate Fisher indices defined by equation (22.10).

22.24 The year-over-year monthly approximate Fisher indices defined by equation (22.10) will normally have a certain amount of upward bias, since these indices cannot reflect long-term substitution of consumers towards commodities that are becoming relatively cheaper over time. This reinforces the case for computing true year-over-year monthly Fisher indices defined by equation (22.6) on a delayed basis so that this substitution bias can be estimated.

22.25 Note that the approximate year-over-year monthly Laspeyres and Paasche indices, PAL and PAP defined by equations (22.8) and (22.9) above, satisfy the following inequalities:

PAF(pt,m,pt+1,m,s0,m)PAL(pt+1,m,pt,m,s0,m)1m=1,2,,12(22.11)
PAP(pt,m,pt+1,m,s0,m)PAP(pt+1,m,pt,m,s0,m)1m=1,2,,12(22.12)

with strict inequalities if the monthly price vectors pt,m and pt+1,m are not proportional to each other.13 The inequality (22.11) says that the approximate year-over-year monthly Laspeyres index fails the time reversal test with an upward bias, while the inequality (22.12) says that the approximate year-over-year monthly Paasche index fails the time reversal test with a downward bias. Hence the fixed weight approximate Laspeyres index PAL has a built-in upward bias and the fixed weight approximate Paasche index PAP has a built-in downward bias. Statistical agencies should avoid the use of these formulae. The formulae can, however, be combined as in the approximate Fisher formula (22.10) and the resulting index should be free from any systematic formula bias (but there still could be some substitution bias).

22.26 The year-over-year monthly indices defined in this section are illustrated using the artificial data set given in Tables 22.1 and 22.2. Although fixed base indices are not formally defined in this section, these indices have similar formulae to the year-over-year indices except that the variable base year t is replaced by the fixed base year 0. The resulting 12 year-over-year monthly fixed base Laspeyres, Paasche and Fisher indices are listed in Tables 22.3 to 22.5.

22.27 Comparing the entries in Tables 22.3 and 22.4, it can be seen that the year-over-year monthly fixed base Laspeyres and Paasche price indices do not differ substantially for the early months of the year, but that there are substantial differences between the indices for the last five months of the year by the time the year 1973 is reached. The largest percentage difference between the Laspeyres and Paasche indices is 12.5 per cent for month 10 in 1973 (1.4060/1.2496 = 1.125). However, all the year-over-year monthly series show a smooth year-over-year trend.

22.28 Approximate fixed base year-over-year Laspeyres, Paasche and Fisher indices can be constructed by replacing current month expenditure shares for the five commodities by the corresponding base year monthly expenditure shares on the five commodities. The resulting approximate Laspeyres indices are equal to the original fixed base Laspeyres indices so there is no need to present the approximate Laspeyres indices in a table. The approximate year-over-year Paasche and Fisher indices do, however, differ from the fixed base Paasche and Fisher indices found in Tables 22.4 and 22.5, so these new approximate indices are listed in Tables 22.6 and 22.7.

22.29 Comparing Table 22.4 with Table 22.6, it can be seen that, with a few exceptions, the entries correspond fairly closely. One of the bigger differences is the 1973 entry for the fixed base Paasche index for month 9, which is 1.1664, while the corresponding entry for the approximate fixed base Paasche index is 1.1920, for a 2.2 per cent difference (1.1920/1.1664 = 1.022). In general, the approximate fixed base Paasche indices are somewhat bigger than the true fixed base Paasche indices, as could be expected, since the approximate indices have some substitution bias built into them as their expenditure shares are held fixed at the 1970 levels.

Table 22.3

Year-over-year monthly fixed base Laspeyres indices

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Table 22.4

Year-over-year monthly fixed base Paasche indices

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Table 22.5

Year-over-year monthly fixed base Fisher indices

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Table 22.6

Year-over-year approximate monthly fixed base Paasche indices

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22.30 Turning now to the chained year-over-year monthly indices using the artificial data set, the resulting 12 year-over-year monthly chained Laspeyres, Paasche and Fisher indices, PL, PP and PF, where the month-to-month links are defined by equations (22.4) to (22.6), are listed in Tables 22.8 to 22.10.

22.31 Comparing the entries in Tables 22.8 and 22.9, it can be seen that the year-over-year monthly chained Laspeyres and Paasche price indices have smaller differences than the corresponding fixed base Laspeyres and Paasche price indices in Tables 22.3 and 22.4. This is a typical pattern, as found in Chapter 19: the use of chained indices tends to reduce the spread between Paasche and Laspeyres indices compared to their fixed base counterparts. The largest percentage difference between corresponding entries for the chained Laspeyres and Paasche indices in Tables 22.8 and 22.9 is 4.1 per cent for month 10 in 1973 (1.3593/1.3059= 1.041). Recall that the fixed base Laspeyres and Paasche indices differed by 12.5 per cent for the same month, so that chaining does tend to reduce the spread between these two equally plausible indices.

22.32 The chained year-over-year Fisher indices listed in Table 22.10 are regarded as the “best” estimates of year-over-year inflation using the artificial data set.

22.33 The year-over-year chained Laspeyres, Paasche and Fisher indices listed in Tables 22.8 to 22.10 can be approximated by replacing current period commodity expenditure shares for each month by the corresponding base year monthly commodity expenditure shares. The resulting 12 year-over-year monthly approximate chained Laspeyres, Paasche and Fisher indices, PAL, PAP and PAF, where the monthly links are defined by equations (22.8) to (22.10), are listed in Tables 22.11 to 22.13.

22.34 The year-over-year chained indices listed in Tables 22.11 to 22.13 approximate their true chained counterparts listed in Tables 22.8 to 22.10 very closely. For the year 1973, the largest discrepancies are for the Paasche and Fisher indices for month 9: the chained Paasche is 1.2018, while the corresponding approximate chained Paasche is 1.2183 for a difference of 1.4 per cent, and the chained Fisher is 1.2181, while the corresponding approximate chained Fisher is 1.2305 for a difference of 1.0 per cent. It can be seen that for the modified Turvey data set, the approximate year-over-year monthly approximate Fisher indices listed in Table 22.13 approximate the theoretically preferred (but in practice unfeasible in a timely fashion) Fisher chained indices listed in Table 22.10 quite satisfactorily. Since the approximate Fisher indices arc just as easy to compute as the approximate Laspeyres and Paasche indices, it may be useful to ask that statistical agencies make available to the public these approximate Fisher indices along with the approximate Laspeyres and Paasche indices.

Table 22.7

Year-over-year approximate monthly fixed base Fisher indices

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Table 22.8

Year-over-year monthly chained Laspeyres indices

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Table 22.9

Year-over-year monthly chained Paasche indices

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Table 22.10

Year-over-year monthly chained Fisher indices

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Table 22.11

Year-over-year monthly approximate chained Laspeyres indices

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Table 22.12

Year-over-year monthly approximate chained Paasche indices

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Table 22.13

Year-over-year monthly approximate chained Fisher indices

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Year-over-year annual indices

22.35 Assuming that each commodity in each season of the year is a separate “annual” commodity is the simplest and theoretically most satisfactory method for dealing with seasonal commodities when the goal is to construct annual price and quantity indices. This idea can be traced back to Bruce D. Mudgett in the consumer price context and to Richard Stone in the producer price context:

The basic index is a yearly index and as a price or quantity index is of the same sort as those about which books and pamphlets have been written in quantity over the years (Mudgett (1955, p. 97)).

The existence of a regular seasonal pattern in prices which more or less repeats itself year after year suggests very strongly that the varieties of a commodity available at different seasons cannot be transformed into one another without cost and that, accordingly, in all cases where seasonal variations in price are significant, the varieties available at different times of the year should be treated, in principle, as separate commodities (Stone (1956, pp. 74-75)).

22.36 Using the notation introduced in the previous section, the Laspeyres, Paasche and Fisher annual (chain link) indices comparing the prices of year t with those of year t + 1 can be defined as follows:

PL(pt,1,,pt,12;pt+1,1,,pt+1,12;qt,1,,qt,12)Σm=112ΣnS(m)pnt+1,mqnt,mΣm=112ΣnS(m)pnt,mqnt,m(22.13)
PP(pt,1,,pt,12;pt+1,1,,pt+1,12;qt+1,1,,qt+1,12)Σm=112ΣnS(m)pnt+1,mqnt+1,mΣm=112ΣnS(m)pnt,mqnt+1,m(22.14)
PF(pt,1,,pt,12;pt+1,1,,pt+1,12;qt,1,,qt,12;qt+1,1,,qt+1,12)PL(pt,1,,pt,12;pt+1,1,,pt+1,12;qt,1,,qt,12)×PP(pt,1,,pt,12;pt+1,1,,pt+1,12;qt+1,1,,qt+1,12)(22.15)

22.37 The above formulae can be rewritten in price relative and monthly expenditure share form as follows:

PL(pt,1,,pt,12;pt+1,1,,pt+1,12;σ1tst,1,,σ12tst,12)Σm=112ΣnS(m)σmtsnt,m(pnt+1,m/pnt,m)=Σm=112σmtPL(pnt,m,pnt+1,m,st,m)(22.16)
PP(pt,1,,pt,12;pt+1,1,,pt+1,12;σ1t+1st+1,1,,σ12t+1st+1,12)[Σm=112ΣnS(m)σmt+1snt+1,m(pnt+1,m/pnt,m)1]1[Σm=112σmt+1ΣnS(m)snt+1,m(pnt+1,m/pnt,m)1]1[Σm=112σmt+1[PP(pt,m,pt+1,m,st+1,m)]1]1(22.17)

where the expenditure share for month m in year t is defined as:

PF(pt,1,,pt,12;pt+1,1,,pt+1,12;σ1tst,1,,σ12tst,12;σ1t+1st+1,1,….,σ12t+1st+1,12)Σm=112ΣnS(m)σmtsnt,m(pnt+1,m/pnt,m)Σm=112ΣnS(m)σmt+1snt+1,m(pnt+1,m/pnt,m)1(22.18)
=Σm=112σmt[PL(pt,m,pnt+1,m,st,m)]Σm=112σmt+1[PL(pt,m,pnt+1,m,st+1,m)1]σntΣnS(m)pnt,mqnt,mΣi=112ΣjS(i)pjt,iqjt,im=1,2,,12;t=0,1,,T(22.19)

and the year-over-year monthly Laspeyres and Paasche (chain link) price indices PL(pt,m, pt+1,m, st,m) and PP(pt,m, pt+1,m, st,m) are defined by equations (22.4) and (22.5), respectively. As usual, the annual chain link Fisher index PF defined by equation (22.18), which compares the prices in every month of year t with the corresponding prices in year t+ 1, is the geometric mean of the annual chain link Laspeyres and Paasche indices, PL and PP, defined by equations (22.16) and (22.17). The last equations in (22.16), (22.17) and (22.18) show that these annual indices can be defined as (monthly) share-weighted averages of the year-over-year monthly chain link Laspeyres and Paasche indices, PL(pt,m, pt+1,m, st,m) and PP(pt,m, pt+1,m, st,m) defined by equations (22.4) and (22.5). Hence once the year-over-year monthly indices defined above have been calculated numerically, it is easy to calculate the corresponding annual indices.

22.38 Fixed base counterparts to the formulae defined by equations (22.16) to (22.18) can readily be defined: simply replace the data pertaining to period t by the corresponding data pertaining to the base period 0.

22.39 The annual fixed base Laspeyres, Paasche and Fisher indices, as calculated using the data from the artificial data set tabled in paragraphs 22.14 and 22.15, are listed in Table 22.14, which shows that by 1973, the annual fixed base Laspeyres index exceeds its Paasche counterpart by 4.5 per cent. Note that each series increases steadily.

22.40 The annual fixed base Laspeyres, Paasche and Fisher indices can be approximated by replacing any current shares by the corresponding base year shares. The resulting annual approximate fixed base Laspeyres, Paasche and Fisher indices are listed in Table 22.15. Also listed in the last column of Table 22.15 is the fixed base geometric Laspeyres annual index, PGL. This is the weighted geometric mean counterpart to the fixed base Laspeyres index, which is equal to a base period weighted arithmetic average of the long-term price relatives; see Chapter 19. It can be shown that PGL approximates the approximate fixed base Fisher index, PAF, to the second order around a point where all the long-term price relatives are equal to unity.14 It can be seen that the entries for the Laspeyres price indices are exactly the same in Tables 22.14 and 22.15. This is as it should be, because the fixed base Laspeyres price index uses only expenditure shares from the base year 1970: hence the approximate fixed base Laspeyres index is equal to the true fixed base Laspeyres index. Comparing the columns labelled PP and PF in Table 22.14 with the columns PAP and PAF in Table 22.15 shows that the approximate Paasche and approximate Fisher indices are quite close to the corresponding annual Paasche and Fisher indices. Hence, for the artificial data set, the true annual fixed base Fisher index can be very closely approximated by the corresponding approximate Fisher index, PAF (or the geometric Laspeyres index, PGL), which, of course, can be computed using the same information set that is normally available to statistical agencies.

22.41 Using the artificial data set in Tables 22.1 and 22.2, the annual chained Laspeyres, Paasche and Fisher indices can readily be calculated, using the formulae (22.16) to (22.18) for the chain links. The resulting indices are listed in Table 22.16, which shows that the use of chained indices has substantially narrowed the gap between the Paasche and Laspeyres indices. The difference between the chained annual Laspeyres and Paasche indices in 1973 is only 1.5 per cent (1.3994 versus 1.3791), whereas from Table 22.14, the difference between the fixed base annual Laspeyres and Paasche indices in 1973 is 4.5 per cent (1.4144 versus 1.3536). Thus the use of chained annual indices has substantially reduced the substitution (or representativity) bias of the Laspeyres and Paasche indices. Comparing Tables 22.14 and 22.16, it can be seen that for this particular artificial data set, the annual fixed base Fisher indices are very close to their annual chained Fisher counterparts. The annual chained Fisher indices should, however, normally be regarded as the more desirable target index to approximate, since this index will normally give better results if prices and expenditure shares are changing substantially over time.15

Table 22.14

Annual fixed base Laspeyres, Paasche and Fisher price indices

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Table 22.15

Annual approximate fixed base Laspeyres, Paasche, Fisher and geometric Laspeyres indices

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22.42 Obviously, the current year weights, st,mn and σtm and st+1,mn and σt+1m, which appear in the chain link formulae (22.16) to (22.18), can be approximated by the corresponding base year weights, s0,mn and σ0m. This leads to the annual approximate chained Laspeyres, Paasche and Fisher indices listed in Table 22.17.

22.43 Comparing the entries in Tables 22.16 and 22.17 shows that the approximate chained annual Laspeyres, Paasche and Fisher indices are extremely close to the corresponding true chained annual Laspeyres, Paasche and Fisher indices. Hence, for the artificial data set, the true annual chained Fisher index can be very closely approximated by the corresponding approximate Fisher index, which can be computed using the same information set that is normally available to statistical agencies.

22.44 The approach to computing annual indices outlined in this section, which essentially involves taking monthly expenditure share-weighted averages of the 12 year-over-year monthly indices, should be contrasted with the approach that simply takes the arithmetic mean of the 12 monthly indices. The problem with the latter approach is that months where expenditures are below the average (e.g., February) are given the same weight in the unweighted annual average as months where expenditures are above the average (e.g., December).

Rolling year annual indices

22.45 In the previous section, the price and quantity data pertaining to the 12 months of a calendar year were compared to the 12 months of a base calendar year. There is, however, no need to restrict attention to calendar-year comparisons: any 12 consecutive months of price and quantity data could be compared to the price and quantity data of the base year, provided that the January data in the non-calendar year are compared to the January data of the base year, the February data of the non-calendar year are compared to the February data of the base year, and so on, up to the December data of the non-calendar year being compared to the December data of the base year.16 Alterman, Diewert and Feenstra (1999, p. 70) called the resulting indices rolling year or moving year indices.17

Table 22.16

Annual chained Laspeyres, Paasche and Fisher price indices

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Table 22.17

Annual approximate chained Laspeyres, Paasche and Fisher price indices

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22.46 In order to theoretically justify the rolling year indices from the viewpoint of the economic approach to index number theory, some restrictions on preferences are required. The details of these assumptions can be found in Diewert (1996b, pp. 32-34; 1999a, pp. 56-61).

22.47 The problems involved in constructing rolling year indices for the artificial data set are now considered. For both fixed base and chained rolling year indices, the first 13 index number calculations are the same. For the year that ends with the data for December of 1970, the index is set equal to 1 for the Laspeyres, Paasche and Fisher moving year indices. The base year data are the 44 non-zero price and quantity observations for the calendar year 1970. When the data for January 1971 become available, the three non-zero price and quantity entries for January of calendar year 1970 are dropped and replaced by the corresponding entries for January 1971. The data for the remaining months of the comparison year remain the same; i.e., for February to December of the comparison year, the data for the rolling year are set equal to the corresponding entries for February to December 1970. Thus the Laspeyres, Paasche or Fisher rolling year index value for January 1971 compares the prices and quantities of January 1971 with the corresponding prices and quantities of January 1970. For the remaining months of this first moving year, the prices and quantities of February to December 1970 are simply compared with exactly the same prices and quantities of February to December 1970. When the data for February 1971 become available, the three non-zero price and quantity entries for February for the last rolling year (which are equal to the three non-zero price and quantity entries for February 1970) are dropped and replaced by the corresponding entries for February 1971. The resulting data become the price and quantity data for the second rolling year. The Laspeyres, Paasche or Fisher rolling year index value for February 1971 compares the prices and quantities of January and February 1971 with the corresponding prices and quantities of January and February 1970. For the remaining months of this first moving year, the prices and quantities of March to December 1970 are compared with exactly the same prices and quantities of March to December 1970. This process of exchanging the price and quantity data of the current month in 1971 with the corresponding data of the same month in the base year 1970 in order to form the price and quantity data for the latest rolling year continues until December 1971 is reached, when the current rolling year becomes the calendar year 1971. Thus the Laspeyres, Paasche and Fisher rolling year indices for December 1971 are equal to the corresponding fixed base (or chained) annual Laspeyres, Paasche and Fisher indices for 1971, listed in Tables 22.14 or 22.16.

22.48 Once the first 13 entries for the rolling year indices have been defined as indicated above, the remaining fixed base rolling year Laspeyres, Paasche and Fisher indices are constructed by taking the price and quantity data of the last 12 months and rearranging the data so that the January data in the rolling year are compared to the January data in the base year, the February data in the rolling year are compared to the February data in the base year, and so on, up to the December data in the rolling year being compared to the December data in the base year. The resulting fixed base rolling year Laspeyres, Paasche and Fisher indices for the artificial data set are listed in Table 22.18.

22.49 Once the first 13 entries for the fixed base rolling year indices have been defined as indicated above, the remaining chained rolling year Laspeyres, Paasche and Fisher indices are constructed by taking the price and quantity data of the last 12 months and comparing these data to the corresponding data of the rolling year of the 12 months preceding the current rolling year. The resulting chained rolling year Laspeyres, Paasche and Fisher indices for the artificial data set are listed in the last three columns of Table 22.18. Note that the first 13 entries of the fixed base Laspeyres, Paasche and Fisher indices are equal to the corresponding entries for the chained Laspeyres, Paasche and Fisher indices. It will also be noted that the entries for December (month 12) of 1970, 1971, 1972 and 1973 for the fixed base rolling year Laspeyres, Paasche and Fisher indices are equal to the corresponding fixed base annual Laspeyres, Paasche and Fisher indices listed in Table 22.14. Similarly, the entries in Table 22.18 for December (month 12) of 1970, 1971, 1972 and 1973 for the chained rolling year Laspeyres, Paasche and Fisher indices are equal to the corresponding chained annual Laspeyres, Paasche and Fisher indices listed in Table 22.16.

22.50 Table 22.18 shows that the rolling year indices are very smooth and free from seasonal fluctuations. For the fixed base indices, each entry can be viewed as a seasonally adjusted annual consumer price index that compares the data of the 12 consecutive months that end with the year and month indicated with the corresponding price and quantity data of the 12 months in the base year, 1970. Thus rolling year indices offer statistical agencies an objective and reproducible method of seasonal adjustment that can compete with existing time series methods of seasonal adjustment.18

22.51 Table 22.18 shows that the use of chained indices has substantially narrowed the gap between the fixed base moving year Paasche and Laspeyres indices. The difference between the rolling year chained Laspeyres and Paasche indices in December 1973 is only 1.5 per cent (1.3994 versus 1.3791), whereas the difference between the rolling year fixed base Laspeyres and Paasche indices in December 1973 is 4.5 per cent (1.4144 versus 1.3536). Thus, the use of chained indices has substantially reduced the substitution (or represent at iv-ity) bias of the Laspeyres and Paasche indices. As in the previous section, the chained Fisher rolling year index is regarded as the target seasonally adjusted annual index when seasonal commodities are in the scope of the CPI. This type of index is also a suitable index for central banks to use for inflation targeting purposes.19 The six series in Table 22.18 are charted in Figure 22.1. The fixed base Laspeyres index is the highest one, followed by the chained Laspeyres, the two Fisher indices (which are virtually indistinguishable), and the chained Paasche. Finally, the fixed base Paasche is the lowest index. An increase in the slope of each graph can clearly be seen for the last eight months, reflecting the increase in the month-to-month inflation rates that was built into the data for the last 12 months of the data set.20

22.52 As in the previous section, the current year weights, st,mn and σtm and st+1,mn and σt+1m, which appear in the chain link formulae (22.16) to (22.18) or in the corresponding fixed base formulae, can be approximated by the corresponding base year weights, sn0,m and =0m. This leads to the annual approximate fixed base and chained rolling year Laspeyres, Paasche and Fisher indices listed in Table 22.19.

Table 22.18

Rolling year Laspeyres, Paasche and Fisher price indices

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Figure 22.1
Figure 22.1

Rolling year fixed base and chained Laspeyres, Paasche and Fisher indices

22.53 Comparing the indices in Tables 22.18 and 22.19, it can be seen that the approximate rolling year fixed base and chained Laspeyres, Paasche and Fisher indices listed in Table 22.19 are very close to their true rolling year counterparts listed in Table 22.18. In particular, the approximate chain rolling year Fisher index (which can be computed using just base year expenditure share information, along with current information on prices) is very close to the preferred target index, the rolling year chained Fisher index. In December 1973, these two indices differ by only 0.014 per cent (1.3894/ 1.3892= 1.00014). The indices in Table 22.19 are charted in Figure 22.2. It can be seen that Figures 22.1 and 22.2 are very similar; in particular, the Fisher fixed base and chained indices are virtually identical in both figures.

Table 22.19

Rolling year approximate Laspeyres, Paasche and Fisher price indices

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Figure 22.2
Figure 22.2

Rolling year approximate fixed base and chained Laspeyres, Paasche and Fisher indices

22.54 From the above tables, it can be seen that year-over-year monthly indices and their generalizations to rolling year indices perform very well using the modified Turvey data set; like is compared to like, and the existence of seasonal commodities does not lead to erratic fluctuations in the indices. The only drawback to the use of these indices is that it seems that they cannot give any information on short-term, month-to-month fluctuations in prices. This is most evident if seasonal baskets are totally different for each month since in this case, there is no possibility of comparing prices on a month-to-month basis. In the following section, it is shown how a current period year-over-year monthly index can be used to predict a rolling year index that is centred on the current month.

Predicting a rolling year index using a current period year-over-year monthly index

22.55 It might be conjectured that under a regime where the long-run trend in prices is smooth, changes in the year-over-year inflation rate for a particular month compared to the previous month could give valuable information about the long-run trend in price inflation. For the modified Turvey data set, this conjecture turns out to be true, as seen below.

22.56 The basic idea is illustrated using the fixed base Laspeyres rolling year indices listed in Table 22.18 and the year-over-year monthly fixed base Laspeyres indices listed in Table 22.3. In Table 22.18, the fixed base Laspeyres rolling year entry for December of 1971 compares the 12 months of price and quantity data pertaining to 1971 with the corresponding prices and quantities pertaining to 1970. This index number, PL, is the first entry in Table 22.20. Thus the PLRY column of Table 22.20 shows the fixed base rolling year Laspeyres index, taken from Table 22.18, starting at December 1971 and carrying through to December 1973, which is 24 observations in all. Looking at the first entry of this column, it can be seen that the index is a weighted average of year-over-year price relatives over all 12 months in 1970 and 1971. Thus this index is an average of year-over-year monthly price changes, centred between June and July of the two years for which prices are being compared. Hence, an approximation to this annual index could be obtained by taking the arithmetic average of the June and July year-over-year monthly indices pertaining to the years 1970 and 1971 (see the entries for months 6 and 7 for the year 1971 in Table 22.3, 1.0844 and 1.1103).21 The next rolling year fixed base Laspeyres index corresponds to the January 1972 entry in Table 22.18. An approximation to this rolling year index, PARY, could be obtained by taking the arithmetic average of the July and August year-over-year monthly indices pertaining to the years 1970 and 1971 (see the entries for months 7 and 8 for the year 1971 in Table 22.3, 1.1103 and 1.0783). These arithmetic averages of the two year-over-year monthly indices that are in the middle of the corresponding rolling year are listed in the PARY column of Table 22.20. From Table 22.20, it can be seen that the PARY column does not approximate the PLRY column particularly well, since the approximate indices in the PARY column are seen to have some pronounced seasonal fluctuations, whereas the rolling year indices in the PLRY column are free from seasonal fluctuations.

22.57 Some seasonal adjustment factors (SAF) are listed in Table 22.20. For the first 12 observations, the entries in the SAF column are simply the ratios of the entries in the PLRY column, divided by the corresponding entries in the PARY column; i.e., for the first 12 observations, the seasonal adjustment factors are simply the ratio of the rolling year indices starting at December 1971, divided by the arithmetic average of the two year-over-year monthly indices that are in the middle of the corresponding rolling year.22 The initial 12 seasonal adjustment factors are then just repeated for the remaining entries for the SAF column.

Table 22.20

Rolling year fixed base Laspeyres and seasonally adjusted approximate rolling year price indices

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