New approach for estimating the rent component of the Consumer Price Index

by Roobina Keshishbanoosy and Lance Taylor

 

Release date: February 27, 2019

The Consumer Price Index (CPI) measures the change of prices of consumer goods and services over time. To accurately reflect trends in the market and in consumer behaviour, Statistics Canada periodically reviews and updates the methods applied to various components of the CPI.

The release of the January 2019 CPI (published on February 27th, 2019) marks the implementation of methodological changes for the calculation of the rent index.Note 1

The rent index represents around 6.24 % of the 2017 CPI basket and is part of Shelter, one of the major CPI components.

This document describes the new methodology in estimating the rent index.

A new hedonic model for estimating the rent index

The new methodology based on a hedonic model uses data collected from the Labor Force Survey (LFS) to estimate the rent index.Note 2 The LFS uses a rotating panel multi-stage stratified sample. Households in the sample are surveyed for a period of 6 months (one-sixth of the sample is replaced every month). The dwellings are followed (not the households), so the tenants might change during the survey period.

Following a review of different methods, a characteristics approach hedonic index is proposed to replace the current methodology that is based on the matched-model approach. A hedonic model is estimated using monthly cross-sections of LFS data at the national level. The lowest geographical level indices are constructed using average characteristics as quantities and estimated coefficients as prices, while the higher level indices use weighted averages of lower level estimated expenditures.

The hedonic model is a log-linear regression in which the explanatory variables include observed dwelling characteristics, such as the number of bedrooms, as well as locational characteristics captured by postal codes.

The regression specification is as follows:

y * = β 0 + β 1 services+ β 2 age+ β 3 bedrooms+ β 4 dwelling+ β 5 FSA+ϵ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG5bWdamaaCaaaleqabaWdbiaacQcaaaGccqGH9aqpcqaHYoGy paWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaey4kaSIaeqOSdi2dam aaBaaaleaapeGaaGymaaWdaeqaaOWdbiaadohacaWGLbGaamOCaiaa dAhacaWGPbGaam4yaiaadwgacaWGZbGaey4kaSIaeqOSdi2damaaBa aaleaapeGaaGOmaaWdaeqaaOWdbiaadggacaWGNbGaamyzaiabgUca Riabek7aI9aadaWgaaWcbaWdbiaaiodaa8aabeaak8qacaWGIbGaam yzaiaadsgacaWGYbGaam4Baiaad+gacaWGTbGaam4CaiabgUcaRiab ek7aI9aadaWgaaWcbaWdbiaaisdaa8aabeaak8qacaWGKbGaam4Dai aadwgacaWGSbGaamiBaiaadMgacaWGUbGaam4zaiabgUcaRiabek7a I9aadaWgaaWcbaWdbiaaiwdaa8aabeaak8qacaWGgbGaam4uaiaadg eacqGHRaWktuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb ciab=v=aYdaa@7700@

where:

y * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG5bWdamaaCaaaleqabaWdbiaabQcaaaaaaa@380E@ is the log of observed rent.

All explanatory variables (characteristics) are dummy variables such that:
services MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGZbGaamyzaiaadkhacaWG2bGaamyAaiaadogacaWGLbGaam4C aaaa@3DA3@ represents whether the rent cost includes furniture, a washing machine, a refrigerator, cable or heat, age MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGHbGaam4zaiaadwgaaaa@38D3@ represents the age of building, bedrooms MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGIbGaamyzaiaadsgacaWGYbGaam4Baiaad+gacaWGTbGaam4C aaaa@3D9A@ represents the number of bedrooms, dwelling MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGKbGaam4DaiaadwgacaWGSbGaamiBaiaadMgacaWGUbGaam4z aaaa@3D95@ represents the type of the building, and FSA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGgbGaam4uaiaadgeaaaa@3880@ is a vector of dummies defined from the first three digits of the postal codes.Note 3

The intercept represents the baseline observation, which is a studio in a high rise apartment that is over 40 years old with no included services, in FSA C1A, which is the most sampled FSA in the LFS data.

Before running the regression model, the subject-matter experts analyze the data to flag and remedy some of the possible errors and other anomalies in the data. After that, for each period, a regression is run twice for all observations across Canada, with the first regression being used to calculate the Cook’s distance of each observation to filter out outliers. The optimal filtering threshold for the Cook’s distance has been chosen based on rigorous tests.

Overall, based on all monthly cross-sectional data available, the model tests and estimation results are quite satisfactory with an average adjusted R-squared of around 0.65, and the coefficients for the major dwelling characteristics are mostly significant and stable over time with the expected signs. In addition, the regression results for the locational characteristics support the claim that conurbation matters.

Aggregation Methodology

Quantity and Expenditure Aggregations

In order to build indices, it is necessary to construct quantity and expenditure aggregates. To do so, for each characteristic the average of its dummy variable is calculated by Census Metropolitan Area (CMA). As a simple example, if there are 100 observations in a CMA and 40 of them are 2 bedrooms, then the quantity of that characteristic is 0.4. In practice, a weighted average is used where the weights are the number of renters in each LFS stratum. In short, the quantity x j char,cma MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG4bWdamaaDaaaleaapeGaamOAaaWdaeaapeGaam4yaiaadIga caWGHbGaamOCaiaacYcacaWGJbGaamyBaiaadggaaaaaaa@3F90@ of characteristic char MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGJbGaamiAaiaadggacaWGYbaaaa@39C9@ in CMA cma MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGJbGaamyBaiaadggaaaa@38D7@ during period j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGQbaaaa@3706@ is:

x j char,cma = n=1 N cma w n strat,j x n,j char,cma,strat MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG4bWdamaaDaaaleaapeGaamOAaaWdaeaapeGaam4yaiaadIga caWGHbGaamOCaiaacYcacaWGJbGaamyBaiaadggaaaGccqGH9aqpda GfWbqabSWdaeaapeGaamOBaiabg2da9iaaigdaa8aabaWdbiaad6ea paWaaSbaaWqaa8qacaWGJbGaamyBaiaadggaa8aabeaaa0qaa8qacq GHris5aaGccaWG3bWdamaaDaaaleaapeGaamOBaaWdaeaapeGaam4C aiaadshacaWGYbGaamyyaiaadshacaGGSaGaamOAaaaakiaadIhapa Waa0baaSqaa8qacaWGUbGaaiilaiaadQgaa8aabaWdbiaadogacaWG ObGaamyyaiaadkhacaGGSaGaam4yaiaad2gacaWGHbGaaiilaiaado hacaWG0bGaamOCaiaadggacaWG0baaaaaa@6340@

where strat MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGZbGaamiDaiaadkhacaWGHbGaamiDaaaa@3ADE@ is the LFS stratum of the observation and strat MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGZbGaamiDaiaadkhacaWGHbGaamiDaaaa@3ADE@ belongs to the CMA cma, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGJbGaamyBaiaadggacaGGSaaaaa@3986@ N cma MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGobWdamaaBaaaleaapeGaam4yaiaad2gacaWGHbaapaqabaaa aa@3A03@ is the number of observations in the CMA cma, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGJbGaamyBaiaadggacaGGSaaaaa@3986@ and w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG3baaaa@3712@ is the weight.

The estimated average rent expenditures at the CMA level are calculated as the exponent of the fitted values from inserting the average characteristics by CMA back into the model. Formally, the estimated average rent expenditure  p q i,j cma MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaW2bqaaaaa aaaaWdbiaacckacaWGWbGaamyCa8aadaqhaaWcbaWdbiaadMgacaGG SaGaamOAaaWdaeaapeGaam4yaiaad2gacaWGHbaaaaaa@3F54@ in a CMA cma MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaW2bcaWGJb GaamyBaiaadggaaaa@392D@ using period i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaW2bcaWGPb aaaa@375B@ prices B i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaW2bqaaaaa aaaaWdbiaadkeapaWaaSbaaSqaa8qacaWGPbaapaqabaaaaa@389C@ and period j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaW2bcaWGQb aaaa@375C@ quantities isNote 4 p q i,j cma = e B i X j cma e σ i 2 ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaW2bqaaaaa aaaaWdbiaadchacaWGXbWdamaaDaaaleaapeGaamyAaiaacYcacaWG Qbaapaqaa8qacaWGJbGaamyBaiaadggaaaGccqGH9aqpcaWGLbWdam aaCaaaleqabaWdbiaadkeapaWaaSbaaWqaa8qacaWGPbaapaqabaWc peGaamiwa8aadaqhaaadbaWdbiaadQgaa8aabaWdbiaadogacaWGTb GaamyyaaaaaaGccaWGLbWdamaaCaaaleqabaWdbmaalaaapaqaamaa HaaabaWdbiabeo8aZ9aadaqhaaadbaWdbiaadMgaa8aabaWdbiaaik daaaaal8aacaGLcmaaaeaapeGaaGOmaaaaaaaaaa@4E89@ where B i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGcbWdamaaBaaaleaapeGaamyAaaWdaeqaaaaa@3826@ is the vector of coefficient estimates from period i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaW2bqaaaaa aaaaWdbiaadMgaaaa@377B@ and X j cma MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaW2bqaaaaa aaaaWdbiaadIfapaWaa0baaSqaa8qacaWGQbaapaqaa8qacaWGJbGa amyBaiaadggaaaaaaa@3B84@ is the vector of quantities for all characteristics in CMA cma MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaW2bqaaaaa aaaaWdbiaadogacaWGTbGaamyyaaaa@394D@ during period j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaW2bqaaaaa aaaaWdbiaadQgaaaa@377C@ .

CPI Strata Expenditure and Index Aggregation

To build the final indices at the CPI strata level, the average cost of rent is calculated as the mean of the CMA level expenditures, using the estimated count of renters in each CMA as weights. That is:

p q i,j CS = cma=1 N CS w cma p q i,j  cma MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaW2bqaaaaa aaaaWdbiaadchacaWGXbWdamaaDaaaleaapeGaamyAaiaacYcacaWG Qbaapaqaa8qacaWGdbGaam4uaaaakiabg2da9maawahabeWcpaqaa8 qacaWGJbGaamyBaiaadggacqGH9aqpcaaIXaaapaqaa8qacaWGobWd amaaBaaameaapeGaam4qaiaadofaa8aabeaaa0qaa8qacqGHris5aa GccaWG3bWdamaaCaaaleqabaWdbiaadogacaWGTbGaamyyaaaakiaa dchacaWGXbWdamaaDaaaleaapeGaamyAaiaacYcacaWGQbGaaiiOaa WdaeaapeGaam4yaiaad2gacaWGHbaaaaaa@54E9@

where w cma MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaW2bqaaaaa aaaaWdbiaadEhapaWaaWbaaSqabeaapeGaam4yaiaad2gacaWGHbaa aaaa@3A95@ is the CMA cma MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGJbGaamyBaiaadggaaaa@38D7@ 's share of renters in the CPI stratum CS MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaW2bqaaaaa aaaaWdbiaadoeacaWGtbaaaa@382D@ , and i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaW2bqaaaaa aaaaWdbiaadMgaaaa@377B@ and j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaW2bqaaaaa aaaaWdbiaadQgaaaa@377C@ are the periods of the prices and quantities used, each taking the values 1 (current period) or 0 (previous period). Finally, N CS MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaW2bqaaaaa aaaaWdbiaad6eapaWaaSbaaSqaa8qacaWGdbGaam4uaaWdaeqaaaaa @395A@ is the number of CMAs in the CPI strata CS MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaW2bqaaaaa aaaaWdbiaadoeacaWGtbaaaa@382D@ .

The final price index I p CS MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaW2bqaaaaa aaaaWdbiaadMeapaWaa0baaSqaa8qacaWGWbaapaqaa8qacaWGdbGa am4uaaaaaaa@3A5B@ for prices going from the previous period to current period in a given CPI strata CS MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaW2bqaaaaa aaaaWdbiaadoeacaWGtbaaaa@382D@ is:

I p CS =  ( p q 1,0 CS p q 0,0 CS    p q 1,1 CS p q 0,1 CS ) 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGjbWdamaaDaaaleaapeGaamiCaaWdaeaapeGaam4qaiaadofa aaGccqGH9aqpcaGGGcWaaeWaa8aabaWdbmaalaaapaqaa8qacaWGWb GaamyCa8aadaqhaaWcbaWdbiaaigdacaGGSaGaaGimaaWdaeaapeGa am4qaiaadofaaaaak8aabaWdbiaadchacaWGXbWdamaaDaaaleaape GaaGimaiaacYcacaaIWaaapaqaa8qacaWGdbGaam4uaaaaaaGccaGG GcGaaiiOamaalaaapaqaa8qacaWGWbGaamyCa8aadaqhaaWcbaWdbi aaigdacaGGSaGaaGymaaWdaeaapeGaam4qaiaadofaaaaak8aabaWd biaadchacaWGXbWdamaaDaaaleaapeGaaGimaiaacYcacaaIXaaapa qaa8qacaWGdbGaam4uaaaaaaaakiaawIcacaGLPaaapaWaaWbaaSqa beaapeWaaSaaa8aabaWdbiaaigdaa8aabaWdbiaaikdaaaaaaaaa@5B49@

As of the release of the January 2019 CPI (published on February 27th, 2019) this new index replaces the previous matched model rent index.

References

Hoffmann, J and Kurtz, C., 2003, “Rent Indices for Housing in West Germany”, European Central Bank Working Paper Series, Working Paper no. 116

Pakes, Ariel, 2003, “A Reconsideration of Hedonic Price Indexes with an Application to PC’s”, The American Economic Review, Vol, 93, no., pp.1578-1596

Soumare, Amadou, 2017, “Shelter in the Canadian CPI: An overview”, Price Analytical Series, 62F0014M, Statistics Canada.

Statistics Canada. (2015a). “The Canadian Consumer Price Index Reference Paper: Catalogue no. 62-553-X.

Note


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