The Canadian Consumer Price Index Reference Paper
Chapter 6 – Calculation of the Consumer Price Index

6.1 The Consumer Price Index (CPI) is calculated in two stages, termed the lower level and the upper level.

6.2 At the lower level of calculation, price change is estimated for elementary aggregates. These are found at the lowest level in the product and geographical classifications of the CPI and are most often calculated using a Jevons (geometric mean) index number formula. Elementary aggregates consist of similar groups of products in a geographical stratum.^Note 

6.3 At the upper level, an asymmetrically-weighted fixed-basket Lowe price index formula (Laspeyres-type) is used to combine elementary aggregates in order to obtain upper level aggregate indices.

6.4 This chapter will discuss the two-stage calculation of the CPI, first explaining the computation of elementary indices at the lower level. While the chapter will focus on the standard method for computing indices, some non-standard methods used in the CPI will also be discussed. Then the chapter will explain the method used to aggregate elementary price indices to the upper level.

Calculation of Elementary Indices (lower level)

6.5 At the lower level, elementary price indices are calculated for about 645 elementary product classes in each of the 19 geographical strata of the CPI.^Note  Elementary indices can be understood as the building blocks of the CPI and represent the lowest level of the fixed-basket index hierarchy. Estimation of price change at this level is usually done via the standard approach for elementary price index calculation. Exceptions are made for special cases addressed later in this chapter.^Note 

6.6 Not all elementary indices are derived directly from observed prices. At the Canada level, about 75% of elementary indices, by basket weight, are derived directly from observed prices within their product class and geography. The proportion of elementary indices estimated with direct price observation varies across geography. The remaining portion of elementary indices is imputed, either from another closely related product class, or from the same product class in another geographic stratum.^Note 

6.7 Most of the elementary aggregates that are not calculated using observed prices are catch-all product classes; as such, they represent more marginal and diverse varieties of products which do not fit neatly into any of the other elementary product classes. Typically these catch-all product classes would also be significantly more expensive to estimate via direct price observation. Their price change is usually estimated by imputing the price movement from another elementary price index for which prices are observed.

6.8 While it would appear ideal that all elementary price indices be calculated using observed prices within their product class, this is not always necessary. Since the goal of the CPI is to measure price change, and not absolute price levels, sampling strategies are developed to reflect which Product Offers (POs) are the most important to capture directly, and which others may be suitably estimated via imputation.^Note 

6.9 The CPI follows the matched-model approach for calculating elementary price indices whereby identical (unchanging quantity and quality) POs are followed through time. However, it is not always possible to follow the same products across time, as new goods and services are constantly emerging and old ones disappearing. When an identical PO cannot be collected in a subsequent period, a replacement PO must be observed. This chapter will not discuss situations where POs are replaced.^Note 

6.10 Examples where the calculation of elementary price indices is a relatively simple matter are the few elementary aggregates for which there is one product having a single price. These product classes typically have goods or services for which prices are determined by a level of government, such as drivers’ licences or passport fees. In such cases, the ratio of one month’s price over the previous month is the best estimate of price change. However, for the majority of elementary product classes reality is more complex, mainly because of the availability of many competing and continuously changing product types.

6.11 In the majority of cases, elementary price indices are based on a sample of prices for one or more goods or services belonging to the elementary product class. The sampled POs receive equal weighting in this elementary calculation, because consumer expenditure weighting information is usually not available at this level.

6.12 The following section describes the standard approach for calculating elementary price indices. The chapter will then go on to discuss several of the elementary price indices for which estimation methods differ from the standard approach either because of the complex nature of estimating price change for the goods and services within the elementary product class or because additional information is available that can be used to produce an improved elementary price index.

The Standard Approach for Calculating Elementary Price Indices

6.13 The standard approach refers to the most commonly used method of combining prices, in order to estimate price change for elementary aggregates in the CPI. Typically consumer expenditure patterns below the elementary aggregate level are not known and therefore the implicitly weighted geometric mean, known as the Jevons formula (6.1), is used to calculate an average price relative from the sample of the collected POs. This means the price relative of each collected PO is assigned equal importance in the calculation. The Jevons formula has been used by Statistics Canada since 1995 as its primary formula for the calculation of elementary price indices in the CPI.

$$I_{J,a}^{t-1:t}={{\displaystyle \prod _{i=1}^n\left(\frac{p_i^t}{p_i^{t-1}}\right)}}^{{}^1{/}_n}\text{\hspace{1em}\hspace{1em}\hspace{1em}(6.1)}$$

where:

$I_{J,a}^{t-1:t}$ is the implicitly weighted Jevons price index for elementary aggregate $a$ between period $t-1$ and period $t$ ;

$n$ is the number of POs $i$ in elementary aggregate $a$ ; and

$\frac{p_i^t}{p_i^{t-1}}$ is the price relative for PO $i$ between period $t-1$ and period $t$.

6.14 The Jevons formula (6.1) can also be calculated by taking the ratio of the implicitly weighted geometric mean prices of the observed POs in the two periods being compared (6.2).

$$I_{J,a}^{t-1:t}=\frac{{\displaystyle \prod _{i=1}^n{(p_i^t)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}}}{{\displaystyle \prod _{i=1}^n{(p_i^{t-1})}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}}}\text{\hspace{1em}\hspace{1em}\hspace{1em}(6.2)}$$

where:

${\displaystyle \prod _{i=1}^n{(p_i^t)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}}$ is the geometric mean price for all POs $i$ for elementary aggregate $a$ in period $t$ ; and

${\displaystyle \prod _{i=1}^n{(p_i^{t-1})}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}}$ is the geometric mean price for all POs $i$ for elementary aggregate $a$ in period $t-1$.

6.15 The Jevons formula was adopted because it has advantages over the previously used Dutot formula.^Note  Firstly, the geometric mean of price relatives (Jevons) is less influenced by extreme prices than is the ratio of arithmetic mean prices (Dutot). The resulting elementary price indices are less volatile.^Note  Secondly, elementary price indices that are calculated as geometric mean of price relatives (Jevons) can be interpreted in two ways; first, as an average of price changes (6.1) and second as a change in average prices (6.2). The first interpretation, which is only applicable to the Jevons formula, is convenient for explaining the composition of aggregate price changes.

Other Methods for Calculating Elementary Price Indices

6.16 Among the elementary product indices there are several departures from the standard approach. Exceptions to the standard approach are usually made because more complete information is available on the universe of transactions within the elementary aggregate.

6.17 Post-1995, arithmetic formulae were retained for the calculation of a few elementary price indices (Rent, Passenger vehicle insurance premiums and Tuition fees). What sets these elementary aggregates apart is that the sampled POs are drawn from a population frame and there is confidence that the sample sufficiently represents the universe of consumer expenditures for these product classes. Furthermore, the contractual nature of the expenditures in these product classes means that it is likely that product substitution will not take place over the period of price comparison. The unweighted arithmetic formula used in the Canadian CPI is the Dutot (6.3).^Note 

$$I_{D,a}^{t-1:t}=\frac{{\displaystyle \sum _{i=1}^n\frac{1}{n}{p}_i^t}}{{\displaystyle \sum _{i=1}^n\frac{1}{n}{p}_i^{t-1}}}\text{\hspace{1em}\hspace{1em}\hspace{1em}(6.3)}$$

where:

$I_{D,a}^{t-1:t}$ is the Dutot price index for elementary aggregate $a$ between period $t-1$ and period $t$ ;

$n$ is the number of POs $i$  in elementary aggregate $a$ ;

${\displaystyle \sum _{i=1}^n\frac{1}{n}{p}_i^t}$ is the arithmetic mean price for all POs $i$  for elementary aggregate $a$ in period $t$ ; and

${\displaystyle \sum _{i=1}^n\frac{1}{n}{p}_i^{t-1}}$ is the arithmetic mean price for all POs $i$  for elementary aggregate $a$  in period $t-1$ .

6.18 An explicitly weighted Jevons formula (6.4) is used in a few special cases where more detailed expenditure information is available below the elementary aggregate level. Examples where an explicitly weighted Jevons formula is used are the indices for postal fees, newspapers and magazines, urban transit and parking rates.

$$I_{WJ,a}^{t-1:t}=\frac{{\displaystyle \prod _{i=1}^n{\left(p_i^t\right)}^{w_i/{\displaystyle \sum _{i=1}^n{w}_i}}}}{{\displaystyle \prod _{i=1}^n{\left(p_i^{t-1}\right)}^{w_i/{\displaystyle \sum _{i=1}^n{w}_i}}}}\text{\hspace{1em}\hspace{1em}\hspace{1em}(6.4)}$$

where:

$I_{WJ,a}^{t-1:t}$ is the explicitly weighted Jevons price index for elementary aggregate $a$ between period $t-1$ and period $t$ ; $n$  is the number of collected POs $i$  in elementary aggregate $a$ ;

${\displaystyle \prod _{i=1}^n{\left(p_i^t\right)}^{w_i/{\displaystyle \sum _{i=1}^n{w}_i}}}$ is the explicitly weighted geometric mean price for all POs $i$ in elementary aggregate $a$  in period $t$ ;

${\displaystyle \prod _{i=1}^n{\left(p_i^{t-1}\right)}^{w_i/{\displaystyle \sum _{i=1}^n{w}_i}}}$ is the explicitly weighted geometric mean price for all POs $i$  for elementary aggregate $a$  in period $t-1$ ; and

${}^{{}^{w_i/{\displaystyle \sum _{i=1}^n{w}_i}}}$ is the weight of PO $i$  as a proportion of the aggregate weight for all POs.

6.19 The weights used in the calculation do not have to relate to the period of price comparison, however in each comparison period they are fixed. The weights are obtained from administrative records or other data sources. These cases can be seen as improvements on the standard approach because rather than giving implicit equal importance to each price relative (6.1) they make use of additional information about the relative importance, or size, of each group of transactions.

6.20 In cases where there are different product types available within one elementary aggregate, but each product type is homogeneous, a unit value index is a preferred method for calculating elementary price indices. A unit value index is simply the quantity-weighted average transaction price for all products within an elementary aggregate in one period, divided by the quantity-weighted average transaction price in the previous period. The rationale for using a unit value calculation must be based on a reasonable assumption that the changes in these average prices do not reflect a change in quality over time. Otherwise the index could be prone to bias.^Note 

6.21 The CPI uses a unit value calculation for the spectator entertainment index, which includes prices for stadium sports seating and live staged performances. The assumption behind this index is that if the stadium or theatre is full in each of the two periods being compared, there is likely to be no change in the overall quality, even though seats may be valued differently. In effect, the price of all seats in the stadium or theatre is used rather than a few individual seats. A similar approach is used to calculate the Air transportation index.

6.22 A unit value calculation is also used in the property taxes elementary price index. A sample of properties is drawn so that the average annual property tax paid in a given municipality can be calculated. These calculated average annual taxes are then multiplied by the total stock of dwellings in each municipality in order to obtain the average annual property tax paid in each CPI geographical stratum. No attempt is made to control for differences in the quality of services that homeowners receive in exchange for their tax payments from one municipality to another. Additionally, there is no treatment to control for changes in the quality of municipal services from one period to another. Accounting for these differences is impractical as there are no data available which associate specific municipal services to proportions of property taxes paid.^Note 

Calculation of the Consumer Price Index Above Elementary Indices (upper level)

6.23 The calculation of the CPI at the upper level is relatively straightforward compared to the lower level. It involves aggregating calculated elementary price indices by applying an asymmetrically weighted arithmetic fixed-basket formula in order to obtain aggregate indices which culminate in the All-items CPI.^Note 

6.24 The Laspeyres formula (6.5) is a basic method for calculating price indices and is consistent with the CPI’s fixed basket concept. It expresses the change in the cost between period $0$ and period $t$ of buying a fixed basket of products, by aggregating the prices of the products in the basket using quantities consumed from the price reference period $0$ as weights.

$$I_{L,A}^{0:t}=\frac{{\displaystyle \sum _{i=1}^n{p}_i^t{q}_i^0}}{{\displaystyle \sum _{i=1}^n{p}_i^0{q}_i^0}}\text{\hspace{1em}\hspace{1em}\hspace{1em}(6.5)}$$

where:

$I_{L,A}^{0:t}$ is the Laspeyres price index of aggregate class $A$  between period $0$  and $t$ ;

$n$ is the number of elementary aggregates $i$ in the aggregate class $A$ ;

$p_i^t$ is the price of elementary aggregate $i$ , in time $t$ ;

$p_i^0$ is the price of elementary aggregate $i$ , in time $0$ ; and

$q_i^0$ is the quantity weight of elementary aggregate $i$ , in the price reference period $0$ .

6.25 In practice, the Laspeyres index is not commonly used to calculate the CPI because it requires information on the quantities consumed^Note  in the price reference period $0$ and these data are not available in a timely manner. This has to do with the fact that household expenditure surveys are typically produced with a lag. Therefore, since Statistics Canada aims to produce a CPI that is timely, in that it measures changes in prices for recent periods, the Laspeyres formula must be altered to use quantities from a period preceding the price reference period $0$. This transformation is the Lowe formula (6.6), a more general form of a Laspeyres index because the quantities come from a chosen weight reference period $b$  which precedes the price reference period $0$.

$$I_{Lo,A}^{0:t}=\frac{{\displaystyle \sum _{i=1}^n{p}_i^t{q}_i^b}}{{\displaystyle \sum _{i=1}^n{p}_i^0{q}_i^b}}\text{\hspace{1em}\hspace{1em}\hspace{1em}(6.6)}$$

where:

$I_{Lo,A}^{0:t}$ is the Lowe price index of aggregate class $A$  between period $0$  and $t$ ;

$n$ is the number of elementary aggregates $i$  in the aggregate class $A$ ;

$p_i^t$ is the price of elementary aggregate $i$ , in time $t$ ;

$p_i^0$ is the price of elementary aggregate $i$ , in time $0$ ; and

$q_i^b$ is the quantity weight of elementary aggregate $i$ , in the weight reference period $b$ , with $b\le 0<t$ .

6.26 The Lowe index can also be expressed as the weighted sum of elementary price indices (6.7) with the weights expressed as expenditure shares.

$$I_{Lo,A}^{0:t}={\displaystyle \sum _{i=1}^n(p_i^t/p_i^0)s_i^{0b}}\text{\hspace{1em}\hspace{1em}\hspace{1em}(6.7)}$$

where:

$p_i^t/p_i^0$ is the price index of elementary aggregate ( $i$ ) between periods $0$  and $t$ , and;

$$s_i^{0b}\equiv \frac{p_i^0{q}_i^b}{{\displaystyle \sum _{i=1}^n{p}_i^0{q}_i^b}}\text{\hspace{1em}\hspace{1em}\hspace{1em}(6.8)}$$

6.27 The expenditure shares $s_i^{0b}$ in the Lowe formula (6.7) are hybrid expenditures because the prices and quantities (that equal the expenditures when multiplied) are from different periods, $0$ and $b$.

6.28 Hybrid expenditure shares (6.8) are obtained by updating the original expenditure weights $p_i^b{q}_i^b$ (observed in the weight reference period $b$ ) to reflect the prices of the price reference period $0$ using the price relatives $p_i^0/p_i^b$. This process is often referred to as price-updating and thus hybrid expenditure weights are frequently termed price-updated weights.^Note  The use of price-updated or hybrid expenditure weights is essential to the fixed-quantity basket concept of the CPI.

6.29 Because the weights used in the calculation of the CPI are obtained from consumer expenditure data with a weight reference period that precedes the price reference period $0$, the Lowe index formula is the practical option for computing a timely CPI.

6.30 Notwithstanding this practical advantage, the Lowe formula also has many desirable properties. One is its consistency in aggregation. This means that no matter in which order the elementary price indices are aggregated (for example first by geographical stratum and then by product class, or the reverse) the aggregate index results are the same.

6.31 Another desirable property of the Lowe formula is its transitivity^Note  , whereby the ratio of two Lowe indices using the same set of basket reference quantities $q^b$ is also a Lowe index (6.9).^Note  This property is useful because it enables index compilers to calculate short-term price movements. For example, price change between period $t-1$ and period $t$ can be estimated by taking the ratio of two long-term Lowe price indices, one comparing periods $0$ and $t-1$ and the other comparing periods $0$ and $t$.

$$I_{Lo,A}^{t-1:t}=\frac{{\displaystyle \sum _{i=1}^n{p}_i^t{q}_i^b}}{{\displaystyle \sum _{i=1}^n{p}_i^{t-1}{q}_i^b}}=\frac{\frac{{\displaystyle \sum _{i=1}^n{p}_i^t{q}_i^b}}{{\displaystyle \sum _{i=1}^n{p}_i^0{q}_i^b}}}{\frac{{\displaystyle \sum _{i=1}^n{p}_i^{t-1}{q}_i^b}}{{\displaystyle \sum _{i=1}^n{p}_i^0{q}_i^b}}}=\left(\frac{I_{Lo,A}^{0:t}}{I_{Lo,A}^{0:t-1}}\right)\text{\hspace{1em}\hspace{1em}\hspace{1em}(6.9)}$$

where:

$I_{Lo,A}^{t-1:t}$ is the short-term Lowe index for aggregate $A$  between period $t-1$ and period;

$I_{Lo,A}^{0:t}$ is the long-term Lowe index for aggregate $A$  between period $0$  and period $t$ , and;

$I_{Lo,A}^{0:t-1}$ is the long-term Lowe index for aggregate $A$  between period $0$  and period $t-1$.

6.32 The transitivity property of the Lowe formula also enables index compilers to calculate long-term price change by chaining together short-term price indices. For example, a Lowe index comparing prices in period $t$ to prices in the price reference period $0$ is obtained by multiplying the Lowe index comparing period t to period $t-1$ by the Lowe index comparing period $t-1$ with the price reference period $0$ (6.10). The product of monthly chained indices provides identical results to an index that directly compares prices in period $t$ to prices in the price reference period $0$.

$$I_{Lo,A}^{0:t}=\underset{I_{Lo,A}^{0:t-1}=\frac{{\displaystyle \sum _{i=1}^n{p}_i^{t-1}{q}_i^b}}{{\displaystyle \sum _{i=1}^n{p}_i^0{q}_i^b}}}{\underbrace{\underset{I_{Lo,A}^{0:1}}{\underbrace{\left[\frac{{\displaystyle \sum _{i=1}^n{p}_i^1{q}_i^b}}{{\displaystyle \sum _{i=1}^n{p}_i^0{q}_i^b}}\right]}}\times \underset{I_{Lo,A}^{1:2}}{\underbrace{\left[\frac{{\displaystyle \sum _{i=1}^n{p}_i^2{q}_i^b}}{{\displaystyle \sum _{i=1}^n{p}_i^1{q}_i^b}}\right]}}\times \mathrm{....}\times \underset{I_{Lo,A}^{t-3:t-2}}{\underbrace{\left[\frac{{\displaystyle \sum _{i=1}^n{p}_i^{t-2}{q}_i^b}}{{\displaystyle \sum _{i=1}^n{p}_i^{t-3}{q}_i^b}}\right]}}\times \underset{I_{Lo,A}^{t-2:t-1}}{\underbrace{\left[\frac{{\displaystyle \sum _{i=1}^n{p}_i^{t-1}{q}_i^b}}{{\displaystyle \sum _{i=1}^n{p}_i^{t-2}{q}_i^b}}\right]}}}}\times \underset{I_{Lo,A}^{t-1:t}={\displaystyle \sum _{i=1}^n\left(\frac{p_i^t}{p_i^{t-1}}\right)s_i^{t-1b}}}{\underbrace{\left[\frac{{\displaystyle \sum _{i=1}^n{p}_i^t{q}_i^b}}{{\displaystyle \sum _{i=1}^n{p}_i^{t-1}{q}_i^b}}\right]}}\text{\hspace{1em}\hspace{1em}\hspace{1em}(6.10)}$$

where:

$I_{Lo,A}^{0:t}$ is the long-term Lowe index for aggregate class $A$  between period $0$  and $t$;

$I_{Lo,A}^{t-1:t}$ is the monthly short-term Lowe index for aggregate $A$; and

$s_i^{t-1b}$ is the hybrid expenditure share of elementary aggregate $i$ , with quantities from the basket reference period $b$  expressed at period $t-1$ prices, derived as (6.11).

$$s_i^{t-1b}\equiv \frac{p_i^{t-1}{q}_i^b}{{\displaystyle \sum _{i=1}^n{p}_i^{t-1}{q}_i^b}}\text{\hspace{1em}\hspace{1em}\hspace{1em}(6.11)}$$

6.33 In any given period $t$ the hybrid expenditure shares price-updated to period $t-1$ are used to aggregate elementary price indices. Since hybrid expenditure weights are an estimate of the value of purchasing the quantities from the weight reference period b expressed in period $t-1$ prices, they do not reflect changes in consumer purchasing patterns. These are necessary in order to maintain the fixed quantity concept of the Lowe formula.

6.34 In the ongoing practice of compiling the CPI, hybrid expenditure shares (6.11) are not explicitly calculated. Instead, the equivalent Lowe formula is used (6.12), where monthly price relatives $\left(\frac{p_i^t}{p_i^{t-1}}\right)$ multiplied by hybrid expenditure weights expressed at period $t-1$ prices are compared to the hybrid expenditures expressed at period $0$ prices in order to obtain price change between period $0$  and $t$ .

$$I_{Lo,A}^{0:t}=\frac{{\displaystyle \sum _{i=1}^n\left(\frac{p_i^t}{p_i^{t-1}}\right)\left(p_i^{t-1}{q}_i^b\right)}}{{\displaystyle \sum _{i=1}^n\left(p_i^0{q}_i^b\right)}}=\frac{{\displaystyle \sum _{i=1}^n\left(\frac{p_i^t}{p_i^{t-1}}\right)\left(\frac{p_i^{t-1}}{p_i^0}\right)}\left(p_i^0{q}_i^b\right)}{{\displaystyle \sum _{i=1}^n\left(p_i^0{q}_i^b\right)}}\text{\hspace{1em}\hspace{1em}\hspace{1em}(6.12)}$$

6.35 Despite all the practical advantages of using the Lowe formula for calculating the upper level of the CPI, it is an asymmetrically weighted price index, meaning that the weights used to aggregate elementary price indices refer to a period preceding the price reference month. For this reason the Lowe formula does not represent the current spending patterns of consumers and therefore is subject to substitution bias.^Note