# The Canadian Consumer Price Index Reference Paper

Chapter 8 – Weights and Basket Updates

## Meaning and Construction of the Consumer Price Index Weights

**8.1 **The Consumer Price Index (CPI) basket weights are expenditures derived
primarily from the Survey of
Household
Spending (SHS)^{Note } or from the national/provincial Household Final Consumption Expenditure (HFCE)^{Note } series for a given reference year.^{Note } The basket weights are actually hybrid expenditures, meaning that the prices and
quantities of the expenditures come from different periods. Hybrid expenditure weights are essential to the fixed basket
concept of the CPI.^{Note }

**8.2 **The CPI weights
are constructed from aggregate household expenditures.
This type of weighting, known as plutocratic, implies that each household contributes to the total weight of an
elementary aggregate proportionally to their respective spending.^{Note }

**8.3 **The HFCE data is used to derive the basket weights for higher level commodity classes, and SHS data used to derive the proportions for the elementary aggregates by mapping the HFCE and SHS estimates to the product and
geographical classifications of the CPI. However, the SHS sometimes does not provide sufficient detail and thus
basket weights are in some instances constructed from
alternative sources.

**8.4 **The basket weights for the homeowners’ replacement cost and mortgage
interest cost elementary indices are two examples for which supplementary data are
required to construct the weight.^{Note } Additionally, alternative data sources which include other Statistics Canada
surveys, administrative data, and scanner data received from retailers are used to break down aggregate expenditures further for product classes in which the SHS does not provide sufficient detail.^{Note }

**8.5 **Supplementary data are also used to confront specific HFCE or SHS
expenditure estimates which may be suspected of bias. For example, expenditures for alcohol and
tobacco are often thought to be under-reported in household expenditure surveys, as the survey
estimates are typically lower than reported in retail sales and government excise tax revenue data.^{Note }

**8.6 **At the time of a basket update, Statistics Canada also uses the
Bortkiewicz-Szulc decomposition to evaluate expenditures used as basket weights.^{Note } This method compares relative
changes in quantities with the
corresponding
relative changes in prices in order to assess the reliability of the expenditure
weights.

**8.7 **Assessing the quality of expenditure data also helps Statistics
Canada determine the number of basic classes in the CPI
(that is, the levels in the product and geographical classifications at which
the quantity weights are fixed for the duration of a
basket).^{Note }

**8.8 **Basic classes are determined based on the availability and quality
of the consumer expenditure data as well as the stability of the distribution of
spending within elementary aggregates. For example, if the distribution of consumer spending within a given
elementary aggregate changes frequently, then it may be advantageous to allow the quantities in the
expenditure weight to be updated when new information on consumer spending is available. In such a
case Statistics Canada will designate the basic class to be
the one above the elementary
aggregates where quantities may be updated during the life of the basket.

**8.9 **The practice of changing the quantities below the basic class
level between basket updates provides benefits in that it allows for new information on
consumer spending to be incorporated into the CPI in a timely manner.

## Updating the Consumer Price Index Basket

**8.10 **The process of updating the CPI basket is to make the weights
assigned to elementary aggregates representative of current consumer spending
patterns. In the past, the basket for the CPI was updated every four to five years^{Note } using new expenditure data from
the most recent SHS. Starting with the 2011 basket update, the CPI weights were updated biennially until the 2021 basket update, from which point the basket update is performed on an annual basis. While there is no rule as to how often a CPI basket should be updated, there is general agreement among
CPI compilers that more frequent basket updates are preferred.^{Note }

**8.11 **In addition to updating and assuring the
quality of the weights, the exercise of a basket update also provides an opportunity to review and update other
aspects of the indices which may include:

**8.11.1**Changing the product and/or geographical classifications to be more representative.**8.11.2**Reviewing and updating the sample of representative products (RP) and outlets.**8.11.3**Updating weights below the elementary aggregate level.**8.11.4**Reviewing methods and concepts for the elementary indices.**8.11.5**Updating documentation and products for dissemination.

**8.12 **The final stage of a basket update is to chain-link the new
fixed-quantity basket to the old fixed-quantity basket in order to produce
indices that are a continuous time series. For this reason, the CPI is referred
to as a chain of fixed-basket indices.

## Chain-linking Indices Across Baskets

**8.13 **Published
consumer price indices are calculated as a chain of fixed-basket indices. This
means that a sequence of fixed-basket indices have been
chained together to create a continuous time series. This type of chaining is not to be confused with the calculation of monthly
chained indices^{Note } but rather refers to the process of chaining indices across baskets. This is necessary to
avoid having breaks in an index when a
basket update is performed.

**8.14 **Chain-linking indices across baskets takes
place at the time of a basket update. In order to chain indices across baskets, hybrid expenditure weights for the old and new
baskets must be expressed at the prices of a common period. This common period is called
the link month.

**8.15 **Link month
weights are obtained by price-updating the original expenditure weights
to obtain the hybrid expenditures expressed at prices of the link
month.

**8.16 **Since the basket reference period *b* of the CPI is a *full year*, a process called weight adjustment is
necessary to obtain *monthly
*hybrid expenditures for the link month. Monthly hybrid expenditures for the
link month are calculated in two steps.

**8.17 **First, the
annual expenditures for the basket reference year *b* are divided by the average price change for the basket reference year. This calculation provides a monthly
expenditure, called the initial value, for the month preceding the basket reference year *b*. This first step implicitly assumes that the quantities of the basket are constant for each month of the
basket reference year.

**8.18 **In the
second step, the initial values are price updated to the link month in order to
express the value of the fixed quantities of the
basket at the prices of the link month.^{Note }
Once the link month hybrid expenditures for the new basket are obtained,
aggregate indices can be calculated using the new basket.

**8.19 **In the month
following the basket link month, price indices calculated using the new basket
are multiplied by the index levels previously
published for the old basket.

**8.20 **Chain-linking
of indices is done separately for each basic class.^{Note } Currently the CPI is published
with an index reference period of 2002=100. In 2002 the CPI was based on
the 1996 basket. Between the 1996 basket and June 2022, there have been ten basket updates with the following link months:

- 2001 basket linked in December 2002;
- 2001 revised basket linked in June 2004;
- 2005 basket linked in April 2007;
- 2009 basket linked in April 2011;
- 2011 basket linked in January 2013;
- 2013 basket linked in December 2014;
- 2015 basket linked in December 2016;
- 2017 basket linked in December 2018;
- 2020 basket linked in May 2021; and
- 2021 basket linked in April 2022.

**8.21 **For example, following the introduction of the 2017 basket until the next basket, any chain-linked index with an index reference period
of 2002=100 is a chain of nine fixed baskets (8.1).

$$\begin{array}{l}{I}_{chained}^{2002:t}={I}_{2017}^{Dec2018:t}\times {I}_{2015}^{Dec2016:Dec2018}\times {I}_{2013}^{Dec2014:Dec2016}\times {I}_{2011}^{Jan2013:Dec2014}\times {I}_{2009}^{Apr2011:Jan2013}\times {I}_{2005}^{Apr2007:Apr2011}\times \\ {I}_{2001r}^{Jun2004:Apr2007}\times {I}_{2001}^{Dec2002:Jun2004}\times {I}_{1996}^{2002:Dec2002\text{\hspace{1em}\hspace{1em}\hspace{1em}(8.1)}}\end{array}$$

where:

${I}_{chained}^{2002:t}$ is a chained index for the price observation period ** t**
with a price reference period equal to 2002;

${I}_{2017}^{Dec2018:t}$ is an index for the price observation period *t* with December 2018 as the price reference
period,
calculated using the 2017 basket;

${I}_{2015}^{Dec2016:Dec2018}$ is an index for December 2018 with December 2016 as the price reference period, calculated using the 2015 basket;

${I}_{2013}^{Dec2014:Dec2016}$ is an index for December 2016 with December 2014 as the price reference period, calculated using the 2013 basket;

${I}_{2011}^{Jan2013:Dec2014}$ is an index for December 2014 with January 2013 as the price reference period, calculated using the 2011 basket;

${I}_{2009}^{Apr2011:Jan2013}$ is an index for January 2013 with April 2011 as the price reference period, calculated using the 2009 basket;

${I}_{2005}^{Apr2007:Apr2011}$ is an index for April 2011 with April 2007 as the price reference period, calculated using the 2005 basket;

${I}_{2001r}^{Jun2004:Apr2007}$ is an index for April 2007 with June 2004 as
the price reference period, calculated
using the
2001 revised basket; ^{Note }

${I}_{2001}^{Dec2002:Jun2004}$ is an index for June 2004 with December 2002 as the price reference period, calculated using the 2001 basket;

${I}_{1996}^{2002:Dec2002}$ is an index for December 2002 with 2002 as the price reference period, calculated using the 1996 basket.

## Contributions to Index Percentage Change Across Baskets

**8.22 **The calculation of contributions to
percentage change must be modified when the 12-month percentage change of an index spans two baskets, that
is, when a basket update was performed between the two periods of comparison (period *t* and period *t*-12). This
is because indices chained across baskets are computed using more than one fixed basket. Hence there can
be no single expression of the importance (weight) of each sub-aggregate.^{Note }

**8.23 **The 12-month contribution to change for a composite price index that is
chained across two baskets
$\left(\frac{{I}_{A}^{0:t}}{{I}_{A}^{0:t-12}}-1\right)$ is calculated in two parts. The
first relates to the old basket and the second to the new basket. Unchained indices must be used to derive
contributions across baskets (8.2).

$$\begin{array}{l}\left(\frac{{I}_{A}^{0:t}}{{I}_{A}^{0:t-12}}-1\right)=\underset{\text{oldbasketcontributions}}{\underbrace{\left[{\displaystyle \sum _{i}\left(\frac{{I}_{i}^{0:link}}{{I}_{i}^{0:t-12}}-1\right)\times {w}_{i}^{t-12\_old}}\right]}}+\left[\underset{\text{newbasketcontributions}}{\underbrace{{\displaystyle \sum _{i}\left(\frac{{I}_{i}^{link:t}}{{I}_{i}^{link:link}}-1\right)\times {w}_{i}^{link\_new}}}}\times {I}_{A}^{t-12:link}\right]\\ \\ \text{with}{I}_{i}^{link:link}=100& \text{\hspace{1em}\hspace{1em}\hspace{1em}(8.2)}\end{array}$$

where:

${w}_{i}^{t-12\_old}$ is the weight of component $i$ according to the old basket valued at the $t-12$ period price;

${w}_{i}^{link\_new}$ is the weight of component $i$ according to the new basket valued at the link month period price; and

${I}_{A}^{t-12:link}$ is the aggregate index in the link month with price reference period $t-12$.

**8.24 **When calculating
contributions to 12-month percentage change on an index that spans across two
baskets, it is possible that the summed old basket
contributions and summed new basket contributions have opposite signs (+/-). The resulting contribution to the 12-month percentage
change in the aggregate index could
therefore
have the opposite sign of the corresponding 12-month percentage change in the
index. In other words, a given sub-aggregate can have a
positive 12-month contribution to its aggregate while posting a negative 12-month price change and
vice-versa.

## Rebasing an Index

**8.25 **As discussed in Chapter 2, the index reference period or index
base period is the period in which the index is set to equal 100. For the CPI, the index base
period is usually a calendar year expressed as “index year=100”. Currently the index base period for the CPI
is 2002=100. However, the index reference period of the CPI is changed periodically to coincide with the
index base period of other major economic indicators produced
by Statistics Canada. The
process of changing the index base period is known as rebasing.

**8.26 **There are many reasons why users may need CPI series with index
base periods other than those used in the published CPI. For example, they might
need a series whose index reference period corresponds to the starting period of a particular wage or
payment contract, so they can easily calculate the adjustments to be made. Those interested in comparing consumer
price changes between countries might need a CPI series on an index reference period that corresponds
with the index base period of another country. The need to change the index base period of CPI series may also
result from the technical requirements of an index computation procedure, such as chain-linking
across baskets.

**8.27 **The
rebasing of an index (that is, its conversion from one index reference period
to another) is an arithmetic operation that does not affect the rate of price
change measured by the series between any two periods. To rebase an index ${I}^{g:t}$
and express it in terms of a new index
reference period $h$
, all values in the
index time series are divided by a constant. This constant ${I}^{g:h}$
is an index for price observation period $h$
(which will be the new index reference period)
with the initial index reference period $g$
. The calculated results are then multiplied by 100 in order to obtain the new rebased index,
with the index for the reference period $h$
equal to 100.

$${I}^{h:t}=\frac{{I}^{g:t}}{{I}^{g:h}}\times 100\text{\hspace{1em}\hspace{1em}\hspace{1em}(8.3)}$$

where:

${I}^{h:t}$ is the index for a price observation period $t$ with the new index reference period $h$ ;

${I}^{g:t}$ is the index for a price observation period $t$ with the initial index reference period $g$ ; and

${I}^{g:h}$ is the index for price observation period $h$ with the initial index reference period $g$ .

**8.28 **As an example take the all-items CPI for Canada published with an
index reference period 2002=100. An extract of this series is shown in Table 8.1
in the column headed
${I}^{2002:t}$
. This index
series has been converted into two new index series with
index reference periods January 2012=100 and 2012=100 respectively. They are presented in Table 8.1 in the columns headed ${I}^{Jan2012:t}$
and ${I}^{2012:t}$
.

**8.29 **To calculate ${I}^{Jan2012:t}$
using the original index ${I}^{2002:t}$
the series
is divided by the constant ${I}^{2002:Jan2012}$
. To calculate ${I}^{2012:t}$
using the
original index ${I}^{2002:t}$
the series is divided by the constant ${I}^{2002:2012}$
.

Index price observation period t |
${I}^{2002:t}$ | ${I}^{Jan2012:t}$ | ${I}^{2012:t}$ |
---|---|---|---|

Jan-12 | 120.7 | 100.0= $\frac{{I}^{2002:Jan2012}}{{I}^{2002:Jan2012}}\times 100=\frac{120.7}{120.7}\times 100$ |
99.2= $\frac{{I}^{2002:Jan2012}}{{I}^{2012:Jan2012}}\times 100=\frac{120.7}{121.7}\times 100$ |

Feb-12 | 121.2 | 100.4 | 99.6 |

Mar-12 | 121.7 | 100.8 | 100.0 |

Apr-12 | 122.2 | 101.2 | 100.4 |

May-12 | 122.1 | 101.2 | 100.3 |

Jun-12 | 121.6 | 100.7 | 99.9 |

Jul-12 | 121.5 | 100.7 | 99.9 |

Aug-12 | 121.8 | 100.9 | 100.1 |

Sep-12 | 122.0 | 101.1 | 100.3 |

Oct-12 | 122.2 | 101.2 | 100.4 |

Nov-12 | 121.9 | 101.0 | 100.2 |

Dec-12 | 121.2 | 100.4 | 99.6 |

2012 average | 121.7 | 100.9 | 100.0 |

Jan-13 | 121.3 | 100.5 | 99.7 |

Feb-13 | 122.7 | 101.7 | 100.8 |

Mar-13 | 122.9 | 101.8 | 101.0 |

Apr-13 | 122.7 | 101.7 | 100.8 |

May-13 | 123.0 | 101.9 | 101.1 |

Jun-13 | 123.0 | 101.9 = $\frac{{I}^{2002:Jun2013}}{{I}^{2002:Jan2012}}\times 100=\frac{123}{120.7}\times 100$ |
101.1= $\frac{{I}^{2002:Jun2013}}{{I}^{2012:Jan2012}}\times 100=\frac{123}{121.7}\times 100$ |

Source: Statistics Canada, CANSIM Table 326-0020. |

**8.30 **Since all indices in any given
column of Table 8.1 are derived from original indices with an index reference period 2002=100, divided by a constant, the
rate of price change in all the rebased series is the same as in the original series. Small differences in
percentage change may result due to rounding when average index values are calculated. It should be noted,
however, that differences between index levels, sometimes referred to as differences in index points, vary with
the change of the index reference period. Therefore, users who would like to use the CPI for purposes of
indexation are advised to use the rate of price change
(the percentage change between
index values) rather than using the difference in index points.

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