# 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 over 500 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. 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 (PO) 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 equally 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:

$\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

$\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 (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 $a$ ;

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

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

**6.18 **An explicitly weighted geometric mean 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 geometric mean 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 $a$ ;

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

$\prod _{i=1}^{n}{\left({p}_{i}^{t-1}\right)}^{{w}_{i}/{\displaystyle \sum _{i=1}^{n}{w}_{i}}}$ 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 of the weighted geometric mean 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 (6.5).

$${I}_{U,a}^{t-1:t}=\frac{\left(\frac{{\displaystyle \sum _{i=1}^{n}{q}_{i}^{t}{p}_{i}^{t}}}{{\displaystyle \sum _{i=1}^{n}{q}_{i}^{t}}}\right)}{\left(\frac{{\displaystyle \sum _{i=1}^{n}{q}_{i}^{t-1}{p}_{i}^{t-1}}}{{\displaystyle \sum _{i=1}^{n}{q}_{i}^{t-1}}}\right)}\text{\hspace{1em}\hspace{1em}\hspace{1em}(6.5)}$$

where:

$\frac{{\displaystyle \sum _{i=1}^{n}{q}_{i}^{t}{p}_{i}^{t}}}{{\displaystyle \sum _{i=1}^{n}{q}_{i}^{t}}}$
is the quantity-weighted average price in period *t* using quantities from period *t*

$\frac{{\displaystyle \sum _{i=1}^{n}{q}_{i}^{t-1}{p}_{i}^{t-1}}}{{\displaystyle \sum _{i=1}^{n}{q}_{i}^{t-1}}}$
is the quantity-weighted average price in period *t*-1 using quantities from period *t*-1.

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.

**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 or the Canadian System of National Accounts’ (CSNA) Household Final Consumption Expenditure (HFCE) data 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 *$b$*. In the case of the CPI, $b$ 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.

$${I}_{Lo,A}^{0:t}={\displaystyle \sum _{i=1}^{n}\left(\frac{{p}_{i}^{t}}{{p}_{i}^{0}}\right){s}_{i}^{0b}}\text{\hspace{1em}\hspace{1em}\hspace{1em}(6.7)}$$

where:

$\frac{{p}_{i}^{t}}{{p}_{i}^{0}}$ 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 $\frac{{p}_{i}^{0}}{{p}_{i}^{b}}$. 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 }

## Contributions to Price Change

**6.36 ** A fixed-basket composite price index for a given aggregate ${I}_{A}^{0:t}$
is made up of price indices ${I}_{i}^{0:t}$
and weights ${w}_{i}^{0}$
for the sub-aggregates that are contained in the given aggregate.^{Note } Therefore it is possible to explain a given aggregate’s price change (month-over-month or 12-month) in terms of the influence exerted by its particular sub-aggregates. Analyses of this kind are referred to as contributions to percentage change. Contributions explain how many percentage points of the aggregate’s percentage change come from a given sub-aggregate. For example, the gasoline index (a sub-aggregate) contributed 0.5 percentage points to the 1.0 percent change in the all-items CPI.

**6.37 ** The influence exerted by a given sub-aggregate on a composite price change depends on both its price change and on its importance in the basket, as measured by its weight. Calculating contributions to composite price change across chained baskets requires additional steps.^{Note }

**6.38 **Any composite price index that relates to one fixed basket can be written as a weighted arithmetic average of the corresponding indices for all its constituent sub-aggregates. In other words, the aggregate index ${I}_{A}^{0:t}$
that expresses the change in prices between period $0$ and $t$ is a weighted mean of all the indices ${I}_{i}^{0:t}$ expressing the change in prices during the same period for all its constituting sub-aggregates.

$${I}_{A}^{0:t}={\displaystyle \sum _{i=1}^{n}{I}_{i}^{0:t}\times {w}_{i}^{0b}\text{\hspace{1em}\hspace{1em}\hspace{1em}(6.13)}}$$

where

${w}_{i}^{0b}\equiv \frac{{p}_{i}^{0}{q}_{i}^{b}}{{\displaystyle \sum _{i=1}^{n}{p}_{i}^{0}{q}_{i}^{b}}}$ is the hybrid expenditure share,^{Note }

${p}_{i}^{0}$ is the price for sub-aggregate $i$ in period $0$;

${q}_{i}^{b}$ is the quantity for sub-aggregate $i$ in period $b$, and;

$n$ is the number of sub-aggregates in the aggregate $A$.

**6.39 ** Using (2.1), it is possible to decompose the monthly price change of the aggregate index between $t-1$ and $t$ in terms of the monthly change of its sub-aggregates.^{Note } By construction, the sum of all the sub-aggregates’ monthly price changes will be equal to the monthly price change of the aggregate.

$$\left(\frac{{I}_{A}^{0:t}}{{I}_{A}^{0:t-1}}-1\right)=\frac{{I}_{A}^{0:t}-{I}_{A}^{0:t-1}}{{I}_{A}^{0:t-1}}=\frac{{\displaystyle \sum _{i\text{\hspace{0.17em}}\mathrm{=}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1}^{n}}\left({I}_{i}^{0:t}-{I}_{i}^{0:t-1}\right){w}_{i}^{0b}}{{I}_{A}^{0:t-1}}\text{\hspace{1em}\hspace{1em}\hspace{1em}(6.14)}$$

where:

$\frac{\left({I}_{i}^{0:t}-{I}_{i}^{0:t-1}\right){w}_{i}^{0b}}{{I}_{A}^{0:t-1}}$ represents the contribution of each sub-aggregate $i$ to the aggregate $A$.

**6.40 ** The share of the basket weight ${w}_{i}^{0b}$ of the sub-aggregate index $i$, together with the size and direction of its price change will determine the size and direction of its contribution to the percentage change in the aggregate index $A$
. An increase/decrease in a sub-aggregate index will most often translate into an upward/downward contribution to the aggregate index percentage change.^{Note } The sum of the contributions of all sub-aggregates of the all-items CPI is equal to its overall rate of change (1-month or 12-month).

**6.41 ** The difference in contributions gives the impact of a sub-aggregate on the difference in the percentage change of its aggregate index. This is commonly referred to as acceleration or deceleration and is obtained by subtracting the contribution in period $t-1$ from the contribution in period $t$
. For example, assuming that the gasoline index contributed 0.5 percentage points in period
$t-1$
to the 1.0 percent change in the all-items CPI and in period
$t$
contributed 0.7 percentage points to the 1.4 percent change in the all-items CPI, it can be interpreted that the gasoline index contributed 0.2 percentage points (0.7 – 0.5) to the 0.4 percentage point acceleration (1.4 – 1.0) of the all-items CPI between periods
$t-1$ and $t$.

**6.42 ** The analysis provided by Statistics Canada in the various release items for the CPI is based on an understanding of the contributions of sub-aggregate indices to the 1-month or 12-month percentage change in the all-items CPI or another aggregate index.

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