Chapter 6 – Calculation of the Consumer Price Index

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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 1

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 indexes.

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 691 elementary product classes in each of the 19 geographical strata of the CPI.Note 2 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 3

6.6 Not all elementary indices are derived directly from observed prices. At the Canada level, 76% of elementary indices, by basket weight, are derived directly from observed prices within their product class and geography. Table 6.1 shows that 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 4

Table 6.1
Proportion of elementary aggregates, by basket weight, estimated with direct price observation
Table summary
This table displays the results of Proportion of elementary aggregates. The information is grouped by Geography (appearing as row headers), Proportion (%) (appearing as column headers).
Geography Proportion (%)
Canada 75.8
Newfoundland and Labrador 87.8
Prince Edward Island 87.8
Nova Scotia 87.8
New Brunswick 86.7
Quebec 79.0
Ontario 75.2
Manitoba 86.9
Saskatchewan 87.4
Alberta 86.0
British Columbia 76.2
Yellowknife 80.5
Whitehorse 81.7
Iqaluit 67.8

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 5

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

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' licenses 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.Note 7

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 t1:t = i=1 n ( p i t p i t1 ) 1 / n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamysa8aadaqhaaWcbaWdbiaadQeacaGGSaGaamyyaaWdaeaapeGa aiiDaiabgkHiTiaaigdacaGG6aGaamiDaaaakiabg2da9maarahaba WaaeWaaeaadaWcaaqaaiaadchadaqhaaWcbaGaamyAaaqaaiaadsha aaaakeaacaWGWbWaa0baaSqaaiaadMgaaeaacaWG0bGaeyOeI0IaaG ymaaaaaaaakiaawIcacaGLPaaaaSqaaiaadMgacqGH9aqpcaaIXaaa baGaamOBaaqdcqGHpis1aOWdamaaCaaaleqabaWaaWbaaWqabeaape GaaGymaaaaliaac+capaWaaSbaaWqaa8qacaWGUbaapaqabaaaaaaa @5195@

(6.1)

Where:

I J,a t1:t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa aaleaacaWGkbGaaiilaiaadggaaeaacaGG0bGaeyOeI0IaaGymaiaa cQdacaWG0baaaaaa@3DAD@ is the implicitly weighted Jevons price index for elementary aggregate a between period t-1 and period t;
n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36EA@ is the number of POs i in elementary aggregate a; and
p i t p i t1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGWbWaa0baaSqaaiaadMgaaeaacaWG0baaaaGcbaGaamiCamaaDaaa leaacaWGPbaabaGaamiDaiabgkHiTiaaigdaaaaaaaaa@3DCA@ is the price relative for PO i between period t-1and 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 t1:t = i=1 n ( p i t ) 1 n i=1 n ( p i t1 ) 1 n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa aaleaacaWGkbGaaiilaiaadggaaeaacaGG0bGaeyOeI0IaaGymaiaa cQdacaWG0baaaOGaeyypa0ZaaSaaaeaadaqeWbqaaiaacIcacaWGWb Waa0baaSqaaiaadMgaaeaacaWG0baaaOGaaiykamaaCaaaleqabaWa aSGaaeaacaaIXaaabaGaamOBaaaaaaaabaGaamyAaiabg2da9iaaig daaeaacaWGUbaaniabg+GivdaakeaadaqeWbqaaiaacIcacaWGWbWa a0baaSqaaiaadMgaaeaacaWG0bGaeyOeI0IaaGymaaaakiaacMcada ahaaWcbeqaamaaliaabaGaaGymaaqaaiaad6gaaaaaaaqaaiaadMga cqGH9aqpcaaIXaaabaGaamOBaaqdcqGHpis1aaaaaaa@58C1@

(6.2)

Where:

i=1 n ( p i t ) 1 n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaebCaeaaca GGOaGaamiCamaaDaaaleaacaWGPbaabaGaamiDaaaakiaacMcadaah aaWcbeqaamaaliaabaGaaGymaaqaaiaad6gaaaaaaaqaaiaadMgacq GH9aqpcaaIXaaabaGaamOBaaqdcqGHpis1aaaa@4217@ is the geometric mean price for all POs i for elementary aggregate a in period t; and
i=1 n ( p i t1 ) 1 n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaebCaeaaca GGOaGaamiCamaaDaaaleaacaWGPbaabaGaamiDaiabgkHiTiaaigda aaGccaGGPaWaaWbaaSqabeaadaWccaqaaiaaigdaaeaacaWGUbaaaa aaaeaacaWGPbGaeyypa0JaaGymaaqaaiaad6gaa0Gaey4dIunaaaa@43BF@ 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 8 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 9 Secondly, elementary price indices that are calculated as geometric means 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 691 elementary product indices there are several departures from the standard approach.Note 10 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 formulas 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 11

I D,a t1:t = i=1 n 1 n p i t i=1 n 1 n p i t1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa aaleaacaWGebGaaiilaiaadggaaeaacaWG0bGaeyOeI0IaaGymaiaa cQdacaWG0baaaOGaeyypa0ZaaSaaaeaadaaeWbqaamaalaaabaGaaG ymaaqaaiaad6gaaaGaamiCamaaDaaaleaacaWGPbaabaGaamiDaaaa aeaacaWGPbGaeyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoaaOqaam aaqahabaWaaSaaaeaacaaIXaaabaGaamOBaaaacaWGWbWaa0baaSqa aiaadMgaaeaacaWG0bGaeyOeI0IaaGymaaaaaeaacaWGPbGaeyypa0 JaaGymaaqaaiaad6gaa0GaeyyeIuoaaaaaaa@55BB@

(6.3)

Where:

I D,a t1:t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa aaleaacaWGebGaaiilaiaadggaaeaacaGG0bGaeyOeI0IaaGymaiaa cQdacaWG0baaaaaa@3DA7@ is the Dutot price index for elementary aggregate a between period t-1 and period t;
n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36EA@ is the number of POs i in elementary aggregate a;
i=1 n 1 n p i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaada WcaaqaaiaaigdaaeaacaWGUbaaaiaadchadaqhaaWcbaGaamyAaaqa aiaadshaaaaabaGaamyAaiabg2da9iaaigdaaeaacaWGUbaaniabgg HiLdaaaa@4097@ is the arithmetic mean price for all POs i for elementary aggregate a in period t; and
i=1 n 1 n p i t1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaada WcaaqaaiaaigdaaeaacaWGUbaaaiaadchadaqhaaWcbaGaamyAaaqa aiaadshacqGHsislcaaIXaaaaaqaaiaadMgacqGH9aqpcaaIXaaaba GaamOBaaqdcqGHris5aaaa@423F@ 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 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 t1:t = i=1 n ( p i t ) w i / i=1 n w i i=1 n ( p i t1 ) w i / i=1 n w i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa aaleaacaWGxbGaamOsaiaacYcacaWGHbaabaGaaiiDaiabgkHiTiaa igdacaGG6aGaamiDaaaakiabg2da9maalaaabaWaaebCaeaadaqada qaaiaadchadaqhaaWcbaGaamyAaaqaaiaadshaaaaakiaawIcacaGL PaaadaahaaWcbeqaamaalyaabaGaam4DamaaBaaameaacaWGPbaabe aaaSqaamaaqahabaGaam4DamaaBaaameaacaWGPbaabeaaaeaacaWG PbGaeyypa0JaaGymaaqaaiaad6gaa4GaeyyeIuoaaaaaaaWcbaGaam yAaiabg2da9iaaigdaaeaacaWGUbaaniabg+GivdaakeaadaqeWbqa amaabmaabaGaamiCamaaDaaaleaacaWGPbaabaGaamiDaiabgkHiTi aaigdaaaaakiaawIcacaGLPaaadaahaaWcbeqaamaalyaabaGaam4D amaaBaaameaacaWGPbaabeaaaSqaamaaqahabaGaam4DamaaBaaame aacaWGPbaabeaaaeaacaWGPbGaeyypa0JaaGymaaqaaiaad6gaa4Ga eyyeIuoaaaaaaaWcbaGaamyAaiabg2da9iaaigdaaeaacaWGUbaani abg+Givdaaaaaa@6AE5@

(6.4)

Where:

I WJ,a t1:t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa aaleaacaWGxbGaamOsaiaacYcacaWGHbaabaGaaiiDaiabgkHiTiaa igdacaGG6aGaamiDaaaaaaa@3E89@ is the explicitly weighted Jevons price index for elementary aggregate a between period t-1 and period t; n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36EA@ is the number of collected POs i in elementary aggregate a;
i=1 n ( p i t ) w i / i=1 n w i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaebCaeaada qadaqaaiaadchadaqhaaWcbaGaamyAaaqaaiaadshaaaaakiaawIca caGLPaaadaahaaWcbeqaamaalyaabaGaam4DamaaBaaameaacaWGPb aabeaaaSqaamaaqahabaGaam4DamaaBaaameaacaWGPbaabeaaaeaa caWGPbGaeyypa0JaaGymaaqaaiaad6gaa4GaeyyeIuoaaaaaaaWcba GaamyAaiabg2da9iaaigdaaeaacaWGUbaaniabg+Givdaaaa@4ABB@ is the explicitly weighted geometric mean price for all POs i in elementary aggregate a in period t;
i=1 n ( p i t1 ) w i / i=1 n w i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaebCaeaada qadaqaaiaadchadaqhaaWcbaGaamyAaaqaaiaadshacqGHsislcaaI XaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaadaWcgaqaaiaadEhada WgaaadbaGaamyAaaqabaaaleaadaaeWbqaaiaadEhadaWgaaadbaGa amyAaaqabaaabaGaamyAaiabg2da9iaaigdaaeaacaWGUbaaoiabgg HiLdaaaaaaaSqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGH pis1aaaa@4C63@ is the explicitly weighted geometric mean price for all POs i for elementary aggregate a in period t-1; and
w i / i=1 n w i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWbaaSqabe aadaahaaadbeqaamaalyaabaGaam4DamaaBaaabaGaamyAaaqabaaa baWaaabCaeaacaWG3bWaaSbaaeaacaWGPbaabeaaaeaacaWGPbGaey ypa0JaaGymaaqaaiaad6gaaiabggHiLdaaaaaaaaaaaa@4049@ 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 12

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 fares 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 13

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 14

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 = i=1 n p i t q i 0 i=1 n p i 0 q i 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa aaleaacaWGmbGaaiilaiaadgeaaeaacaaIWaGaaiOoaiaadshaaaGc cqGH9aqpdaWcaaqaamaaqahabaGaamiCamaaDaaaleaacaWGPbaaba GaamiDaaaakiaadghadaqhaaWcbaGaamyAaaqaaiaaicdaaaaabaGa amyAaiabg2da9iaaigdaaeaacaWGUbaaniabggHiLdaakeaadaaeWb qaaiaadchadaqhaaWcbaGaamyAaaqaaiaaicdaaaGccaWGXbWaa0ba aSqaaiaadMgaaeaacaaIWaaaaaqaaiaadMgacqGH9aqpcaaIXaaaba GaamOBaaqdcqGHris5aaaaaaa@5402@

(6.5)

Where:

I L,A 0:t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa aaleaacaWGmbGaaiilaiaadgeaaeaacaaIWaGaaiOoaiaadshaaaaa aa@3BA9@ is the Laspeyres price index of aggregate class A between period 0 and t;
n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36EA@ is the number of elementary aggregates i in the aggregate class A;
p i t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDa aaleaacaWGPbaabaGaamiDaaaaaaa@38FF@ is the price of elementary aggregate i, in time t;
p i 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDa aaleaacaWGPbaabaGaaGimaaaaaaa@38C0@ is the price of elementary aggregate i, in time 0; and
q i 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCamaaDa aaleaacaWGPbaabaGaaGimaaaaaaa@38C1@ 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 consumedNote 15 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 = i=1 n p i t q i b i=1 n p i 0 q i b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa aaleaacaWGmbGaam4BaiaacYcacaWGbbaabaGaaGimaiaacQdacaWG 0baaaOGaeyypa0ZaaSaaaeaadaaeWbqaaiaadchadaqhaaWcbaGaam yAaaqaaiaadshaaaGccaWGXbWaa0baaSqaaiaadMgaaeaacaWGIbaa aaqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aaGcba WaaabCaeaacaWGWbWaa0baaSqaaiaadMgaaeaacaaIWaaaaOGaamyC amaaDaaaleaacaWGPbaabaGaamOyaaaaaeaacaWGPbGaeyypa0JaaG ymaaqaaiaad6gaa0GaeyyeIuoaaaaaaa@5550@

(6.6)

Where:

I Lo,A 0:t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa aaleaacaWGmbGaam4BaiaacYcacaWGbbaabaGaaGimaiaacQdacaWG 0baaaaaa@3C9D@ is the Lowe price index of aggregate class A between period 0 and t;
n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36EA@          is the number of elementary aggregates i in the aggregate class A;
p i t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDa aaleaacaWGPbaabaGaamiDaaaaaaa@38FF@   is the price of elementary aggregate i, in time t;
p i 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDa aaleaacaWGPbaabaGaaGimaaaaaaa@38C0@ is the price of elementary aggregate i, in time 0; and
q i b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCamaaDa aaleaacaWGPbaabaGaamOyaaaaaaa@38EE@ is the quantity weight of elementary aggregate i, in the weight reference period b,
with b≤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 = i=1 n ( p i t / p i 0 ) s i 0b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa aaleaacaWGmbGaam4BaiaacYcacaWGbbaabaGaaGimaiaacQdacaWG 0baaaOGaeyypa0ZaaabCaeaacaGGOaGaamiCamaaDaaaleaacaWGPb aabaGaamiDaaaakiaac+cacaWGWbWaa0baaSqaaiaadMgaaeaacaaI WaaaaOGaaiykaiaadohadaqhaaWcbaGaamyAaaqaaiaaicdacaWGIb aaaaqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aaaa @4F2D@

(6.7)

Where:

p i t / p i 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca WGWbWaa0baaSqaaiaadMgaaeaacaWG0baaaaGcbaGaamiCamaaDaaa leaacaWGPbaabaGaaGimaaaaaaaaaa@3BEA@ is the price index of elementary aggregate (i) between period 0 and t, and;
s i 0b p i 0 q i b i=1 n p i 0 q i b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaDa aaleaacaWGPbaabaGaaGimaiaadkgaaaGccqGHHjIUdaWcaaqaaiaa dchadaqhaaWcbaGaamyAaaqaaiaaicdaaaGccaWGXbWaa0baaSqaai aadMgaaeaacaWGIbaaaaGcbaWaaabCaeaacaWGWbWaa0baaSqaaiaa dMgaaeaacaaIWaaaaOGaamyCamaaDaaaleaacaWGPbaabaGaamOyaa aaaeaacaWGPbGaeyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoaaaaa aa@4D08@

(6.8)

6.27 The expenditure shares s i 0b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaDa aaleaacaWGPbaabaGaaGimaiaadkgaaaaaaa@39AA@  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 expenditures (6.8) are obtained by updating the original expenditure weights p i b q i b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDa aaleaacaWGPbaabaGaamOyaaaakiaadghadaqhaaWcbaGaamyAaaqa aiaadkgaaaaaaa@3BEF@  (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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDa aaleaacaWGPbaabaGaaGimaaaakiaac+cacaGGWbWaa0baaSqaaiaa dMgaaeaacaWGIbaaaaaa@3C73@ . This process is often referred to as price-updating and thus hybrid expenditure weights are frequently termed price-updated weights.Note 16 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 order in which 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 transitivityNote 17, whereby the ratio of two Lowe indices using the same set of basket reference quantities q b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCamaaCa aaleqabaGaamOyaaaaaaa@3801@  is also a Lowe index (6.9).Note 18 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 0:t I Lo,A 0:t1 )= i=1 n p i t q i b i=1 n p i 0 q i b i=1 n p i t1 q i b i=1 n p i 0 q i b = i=1 n p i t q i b i=1 n p i t1 q i b I Lo,A t1:t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaadMeadaqhaaWcbaGaamitaiaad+gacaGGSaGaamyqaaqa aiaaicdacaGG6aGaamiDaaaaaOqaaiaadMeadaqhaaWcbaGaamitai aad+gacaGGSaGaamyqaaqaaiaaicdacaGG6aGaamiDaiabgkHiTiaa igdaaaaaaaGccaGLOaGaayzkaaGaeyypa0ZaaSaaaeaadaWcaaqaam aaqahabaGaamiCamaaDaaaleaacaWGPbaabaGaamiDaaaakiaadgha daqhaaWcbaGaamyAaaqaaiaadkgaaaaabaGaamyAaiabg2da9iaaig daaeaacaWGUbaaniabggHiLdaakeaadaaeWbqaaiaadchadaqhaaWc baGaamyAaaqaaiaaicdaaaGccaWGXbWaa0baaSqaaiaadMgaaeaaca WGIbaaaaqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5 aaaaaOqaamaalaaabaWaaabCaeaacaWGWbWaa0baaSqaaiaadMgaae aacaWG0bGaeyOeI0IaaGymaaaakiaadghadaqhaaWcbaGaamyAaaqa aiaadkgaaaaabaGaamyAaiabg2da9iaaigdaaeaacaWGUbaaniabgg HiLdaakeaadaaeWbqaaiaadchadaqhaaWcbaGaamyAaaqaaiaaicda aaGccaWGXbWaa0baaSqaaiaadMgaaeaacaWGIbaaaaqaaiaadMgacq GH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aaaaaaGccqGH9aqpdaWc aaqaamaaqahabaGaamiCamaaDaaaleaacaWGPbaabaGaamiDaaaaki aadghadaqhaaWcbaGaamyAaaqaaiaadkgaaaaabaGaamyAaiabg2da 9iaaigdaaeaacaWGUbaaniabggHiLdaakeaadaaeWbqaaiaadchada qhaaWcbaGaamyAaaqaaiaadshacqGHsislcaaIXaaaaOGaamyCamaa DaaaleaacaWGPbaabaGaamOyaaaaaeaacaWGPbGaeyypa0JaaGymaa qaaiaad6gaa0GaeyyeIuoaaaGccaWGjbWaa0baaSqaaiaadYeacaWG VbGaaiilaiaadgeaaeaacaWG0bGaeyOeI0IaaGymaiaacQdacaWG0b aaaaaa@9BD9@

(6.9)

Where:

I Lo,A 0:t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa aaleaacaWGmbGaam4BaiaacYcacaWGbbaabaGaaGimaiaacQdacaWG 0baaaaaa@3C9E@ A between period 0 and period t;
I Lo,A 0:t1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa aaleaacaWGmbGaam4BaiaacYcacaWGbbaabaGaaGimaiaacQdacaWG 0bGaeyOeI0IaaGymaaaaaaa@3E46@ A between period 0 and period t-1; and
I Lo,A t1:t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa aaleaacaWGmbGaam4BaiaacYcacaWGbbaabaGaamiDaiabgkHiTiaa igdacaGG6aGaamiDaaaaaaa@3E85@  is the short-term Lowe index for aggregate class A between period t-1 and period t.

6.32 The transitive 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 = [ i=1 n p i 1 q i b i=1 n p i 0 q i b ] I Lo,A 0:1 × [ i=1 n p i 2 q i b i=1 n p i 1 q i b ] I Lo,A 1:2 ×....× [ i=1 n p i t2 q i b i=1 n p i t3 q i b ] I Lo,A t3:t2 × [ i=1 n p i t1 q i b i=1 n p i t2 q i b ] I Lo,A t2:t1 I Lo,A 0:t1 = i=1 n p i t1 q i b i=1 n p i 0 q i b × [ i=1 n p i t q i b i=1 n p i t1 q i b ] I Lo,A t1:t = i=1 n ( p i t p i t1 ) s i t1b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa aaleaacaWGmbGaam4BaiaacYcacaWGbbaabaGaaGimaiaacQdacaWG 0baaaOGaeyypa0ZaaGbaaeaadaagaaqaamaadmaabaWaaSaaaeaada aeWbqaaiaadchadaqhaaWcbaGaamyAaaqaaiaaigdaaaGccaWGXbWa a0baaSqaaiaadMgaaeaacaWGIbaaaaqaaiaadMgacqGH9aqpcaaIXa aabaGaamOBaaqdcqGHris5aaGcbaWaaabCaeaacaWGWbWaa0baaSqa aiaadMgaaeaacaaIWaaaaOGaamyCamaaDaaaleaacaWGPbaabaGaam OyaaaaaeaacaWGPbGaeyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoa aaaakiaawUfacaGLDbaaaSqaaiaadMeadaqhaaadbaGaamitaiaad+ gacaGGSaGaamyqaaqaaiaaicdacaGG6aGaaGymaaaaaOGaayjo+dGa ey41aq7aaGbaaeaadaWadaqaamaalaaabaWaaabCaeaacaWGWbWaa0 baaSqaaiaadMgaaeaacaaIYaaaaOGaamyCamaaDaaaleaacaWGPbaa baGaamOyaaaaaeaacaWGPbGaeyypa0JaaGymaaqaaiaad6gaa0Gaey yeIuoaaOqaamaaqahabaGaamiCamaaDaaaleaacaWGPbaabaGaaGym aaaakiaadghadaqhaaWcbaGaamyAaaqaaiaadkgaaaaabaGaamyAai abg2da9iaaigdaaeaacaWGUbaaniabggHiLdaaaaGccaGLBbGaayzx aaaaleaacaWGjbWaa0baaWqaaiaadYeacaWGVbGaaiilaiaadgeaae aacaaIXaGaaiOoaiaaikdaaaaakiaawIJ=aiabgEna0kaac6cacaGG UaGaaiOlaiaac6cacqGHxdaTdaagaaqaamaadmaabaWaaSaaaeaada aeWbqaaiaadchadaqhaaWcbaGaamyAaaqaaiaadshacqGHsislcaaI YaaaaOGaamyCamaaDaaaleaacaWGPbaabaGaamOyaaaaaeaacaWGPb Gaeyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoaaOqaamaaqahabaGa amiCamaaDaaaleaacaWGPbaabaGaamiDaiabgkHiTiaaiodaaaGcca WGXbWaa0baaSqaaiaadMgaaeaacaWGIbaaaaqaaiaadMgacqGH9aqp caaIXaaabaGaamOBaaqdcqGHris5aaaaaOGaay5waiaaw2faaaWcba GaamysamaaDaaameaacaWGmbGaam4BaiaacYcacaWGbbaabaGaaiiD aiabgkHiTiaaiodacaGG6aGaaiiDaiabgkHiTiaaikdaaaaakiaawI J=aiabgEna0oaayaaabaWaamWaaeaadaWcaaqaamaaqahabaGaamiC amaaDaaaleaacaWGPbaabaGaamiDaiabgkHiTiaaigdaaaGccaWGXb Waa0baaSqaaiaadMgaaeaacaWGIbaaaaqaaiaadMgacqGH9aqpcaaI XaaabaGaamOBaaqdcqGHris5aaGcbaWaaabCaeaacaWGWbWaa0baaS qaaiaadMgaaeaacaWG0bGaeyOeI0IaaGOmaaaakiaadghadaqhaaWc baGaamyAaaqaaiaadkgaaaaabaGaamyAaiabg2da9iaaigdaaeaaca WGUbaaniabggHiLdaaaaGccaGLBbGaayzxaaaaleaacaWGjbWaa0ba aWqaaiaadYeacaWGVbGaaiilaiaadgeaaeaacaGG0bGaeyOeI0IaaG OmaiaacQdacaGG0bGaeyOeI0IaaGymaaaaaOGaayjo+daaleaacaWG jbWaa0baaWqaaiaadYeacaWGVbGaaiilaiaadgeaaeaacaaIWaGaai OoaiaadshacqGHsislcaaIXaaaaSGaeyypa0ZaaSaaaeaadaaeWbqa aiaadchadaqhaaadbaGaamyAaaqaaiaadshacqGHsislcaaIXaaaaS GaamyCamaaDaaameaacaWGPbaabaGaamOyaaaaaeaacaWGPbGaeyyp a0JaaGymaaqaaiaad6gaa4GaeyyeIuoaaSqaamaaqahabaGaamiCam aaDaaameaacaWGPbaabaGaaGimaaaaliaadghadaqhaaadbaGaamyA aaqaaiaadkgaaaaabaGaamyAaiabg2da9iaaigdaaeaacaWGUbaaoi abggHiLdaaaaGccaGL44pacqGHxdaTdaagaaqaamaadmaabaWaaSaa aeaadaaeWbqaaiaadchadaqhaaWcbaGaamyAaaqaaiaadshaaaGcca WGXbWaa0baaSqaaiaadMgaaeaacaWGIbaaaaqaaiaadMgacqGH9aqp caaIXaaabaGaamOBaaqdcqGHris5aaGcbaWaaabCaeaacaWGWbWaa0 baaSqaaiaadMgaaeaacaWG0bGaeyOeI0IaaGymaaaakiaadghadaqh aaWcbaGaamyAaaqaaiaadkgaaaaabaGaamyAaiabg2da9iaaigdaae aacaWGUbaaniabggHiLdaaaaGccaGLBbGaayzxaaaaleaacaWGjbWa a0baaWqaaiaadYeacaWGVbGaaiilaiaadgeaaeaacaWG0bGaeyOeI0 IaaGymaiaacQdacaWG0baaaSGaeyypa0ZaaabCaeaadaqadaqaamaa laaabaGaamiCamaaDaaameaacaWGPbaabaGaamiDaaaaaSqaaiaadc hadaqhaaadbaGaamyAaaqaaiaadshacqGHsislcaaIXaaaaaaaaSGa ayjkaiaawMcaaiaadohadaqhaaadbaGaamyAaaqaaiaadshacqGHsi slcaaIXaGaamOyaaaaaeaacaWGPbGaeyypa0JaaGymaaqaaiaad6ga a4GaeyyeIuoaaOGaayjo+daaaa@420C@

(6.10)

Where:

I Lo,A 0:t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa aaleaacaWGmbGaam4BaiaacYcacaWGbbaabaGaaGimaiaacQdacaWG 0baaaaaa@3C9D@ is the long-term Lowe index for aggregate class A between period 0 and t;
I Lo,A t1:t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa aaleaacaWGmbGaam4BaiaacYcacaWGbbaabaGaamiDaiabgkHiTiaa igdacaGG6aGaamiDaaaaaaa@3E84@ is the monthly short-term Lowe index for aggregate A; and
s i t1b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaDa aaleaacaWGPbaabaGaamiDaiabgkHiTiaaigdacaWGIbaaaaaa@3B91@ 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 t1b p i t1 q i b i=1 n p i t1 q i b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaDa aaleaacaWGPbaabaGaamiDaiabgkHiTiaaigdacaWGIbaaaOGaeyyy IO7aaSaaaeaacaWGWbWaa0baaSqaaiaadMgaaeaacaWG0bGaeyOeI0 IaaGymaaaakiaadghadaqhaaWcbaGaamyAaaqaaiaadkgaaaaakeaa daaeWbqaaiaadchadaqhaaWcbaGaamyAaaqaaiaadshacqGHsislca aIXaaaaOGaamyCamaaDaaaleaacaWGPbaabaGaamOyaaaaaeaacaWG PbGaeyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoaaaaaaa@52BD@

(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 (6.12), where monthly price relatives ( p i t p i t1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaadchadaqhaaWcbaGaamyAaaqaaiaadshaaaaakeaacaWG WbWaa0baaSqaaiaadMgaaeaacaWG0bGaeyOeI0IaaGymaaaaaaaaki aawIcacaGLPaaaaaa@3F5D@  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 = i=1 n ( p i t p i t1 )( p i t1 q i b ) i=1 n ( p i 0 q i b ) = i=1 n ( p i t p i t1 )( p i t1 p i 0 ) ( p i 0 q i b ) i=1 n ( p i 0 q i b ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabbeaacaWGjb Waa0baaSqaaiaadYeacaWGVbGaaiilaiaadgeaaeaacaaIWaGaaiOo aiaadshaaaGccqGH9aqpdaWcaaqaamaaqahabaWaaeWaaeaadaWcaa qaaiaadchadaqhaaWcbaGaamyAaaqaaiaadshaaaaakeaacaWGWbWa a0baaSqaaiaadMgaaeaacaWG0bGaeyOeI0IaaGymaaaaaaaakiaawI cacaGLPaaadaqadaqaaiaadchadaqhaaWcbaGaamyAaaqaaiaadsha cqGHsislcaaIXaaaaOGaamyCamaaDaaaleaacaWGPbaabaGaamOyaa aaaOGaayjkaiaawMcaaaWcbaGaamyAaiabg2da9iaaigdaaeaacaWG UbaaniabggHiLdaakeaadaaeWbqaamaabmaabaGaamiCamaaDaaale aacaWGPbaabaGaaGimaaaakiaadghadaqhaaWcbaGaamyAaaqaaiaa dkgaaaaakiaawIcacaGLPaaaaSqaaiaadMgacqGH9aqpcaaIXaaaba GaamOBaaqdcqGHris5aaaaaOqaaiabg2da9maalaaabaWaaabCaeaa daqadaqaamaalaaabaGaamiCamaaDaaaleaacaWGPbaabaGaamiDaa aaaOqaaiaadchadaqhaaWcbaGaamyAaaqaaiaadshacqGHsislcaaI XaaaaaaaaOGaayjkaiaawMcaamaabmaabaWaaSaaaeaacaWGWbWaa0 baaSqaaiaadMgaaeaacaWG0bGaeyOeI0IaaGymaaaaaOqaaiaadcha daqhaaWcbaGaamyAaaqaaiaaicdaaaaaaaGccaGLOaGaayzkaaaale aacaWGPbGaeyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoakmaabmaa baGaamiCamaaDaaaleaacaWGPbaabaGaaGimaaaakiaadghadaqhaa WcbaGaamyAaaqaaiaadkgaaaaakiaawIcacaGLPaaaaeaadaaeWbqa amaabmaabaGaamiCamaaDaaaleaacaWGPbaabaGaaGimaaaakiaadg hadaqhaaWcbaGaamyAaaqaaiaadkgaaaaakiaawIcacaGLPaaaaSqa aiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aaaaaaaa@91E4@

(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 19

Notes

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