### Publications

### The Canadian Consumer Price Index Reference Paper

# Chapter 8 – Weights and Basket Updates

- Meaning and Construction of the Consumer Price Index Weights
- Updating the Consumer Price Index Basket
- Chain-linking Indices Across Baskets
- Contributions to Index Percentage Change Across Baskets
- Rebasing an Index

## 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) for a given reference year.^{Note 1} 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 2}

**8.2** Generally speaking, the SHS is designed to provide information on spending by private households that is detailed enough for, and consistent with the CPI scope and definitions. 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 3}

**8.3** For the most part the SHS is used to derive the weights for the elementary aggregates by concording the 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 Replacement cost and Mortgage interest cost elementary indices are two examples in which supplementary data are required to construct the weight.^{Note 4} Additionally, alternative data sources which include other Statistics Canada surveys, administrative data and scanner data from retailers are used to break down aggregate expenditures further for product classes in which the SHS does not provide sufficient detail.

**8.5** Supplementary data are also used to confront specific 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 5}

**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 6} 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 7}

**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 is 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 8} using new expenditure data from the most recent SHS. Starting with the 2011 basket update, the CPI weights are updated biennially. 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 9}

**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 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 10} 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 11} 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 12} Currently the CPI is published with an index reference period of 2002=100. In 2002 the CPI was based on the 1996 basket. Since the 1996 basket there have been five 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; and
- 2011 basket linked in January 2013.

**8.21** Following the introduction of the 2011 basket, any chain-linked index with an index reference period of 2002=100 is a chain of six fixed-baskets (8.1).

$${I}_{chained}^{2002:t}={I}_{2011}^{Jan2013:t}\times {I}_{2009}^{Apr2011:Jan2013}\times {I}_{2005}^{Apr2007:Apr2011}\times {I}_{2001r}^{Jun2004:Apr2007}\times {I}_{2001}^{Dec2002:Jun2004}\times {I}_{1996}^{2002:Dec2002}$$

(8.1)

Where:

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

${I}_{2011}^{Jan2013:t}$
is an index for the price observation period *t* 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;

${I}_{2001r}^{Apr2007:Apr2011}$ 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;

${I}_{2001}^{Dec2002:Jun2004}$ 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 13}

**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).

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

(8.2)

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 ${I}^{g:t}$
to express it for a new index reference period *g*, 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 index reference period *h* equal to 100.

$${I}^{h:t}=\frac{{I}^{g:t}}{{I}^{g:h}}\times 100$$

(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${I}^{2002:t}$
. These indices have been converted into the following two new index reference periods: January 2012=100 and 2012=100. 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 changes 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.