Prices Analytical Series
Measuring price change for used vehicles in the Canadian Consumer Price Index

• Introducing used vehicle prices in the Consumer Price Index (CPI) is part of Statistics Canada’s commitment to provide the most timely, reliable and accurate data which reflects the experience of Canadians.
• As part of Statistics Canada’s rigorous and ongoing efforts to maintain the quality and relevance of the CPI, this technical paper explains the proposed timing, data and methodology for including the prices of used vehicles in the CPI’s purchase of passenger vehicle index.
• Statistics Canada has identified a reliable data source for the prices and characteristics of used vehicles, and the upcoming annual basket update in June will incorporate this new source of data in its calculation of the CPI. The CPI previously accounted for used vehicle prices by including a weight for used vehicles and using new vehicle prices as a proxy.
• We will continue to monitor prices for used vehicles and leverage additional new data sources for the purchase of passenger vehicles index. This will ensure the CPI remains an accurate, robust and relevant means of measuring inflation.

The logarithm of price is modeled as a function of the logarithm of vintage-age,^Note 6 and logarithm of odometer reading of vehicles, as well as model fixed effects and a dummy variable for each of the last four months of the window. Formally:

$ln{p}_{i,}^{class,w,\mathrm{...}}={\beta }_0^{class,w,\mathrm{...}}+{\beta }_1^{class,w,\mathrm{...}}lnOdomete{r}_i+{\beta }_2^{class,w,\mathrm{...}}lnAg{e}_i+{\sum }_{m=1}^M{\gamma }_m^{class,w,\mathrm{...}}{D}_i^{model}(\text{m})$ $+{\sum }_{t=1}^T{\delta }_t^{class,w,\mathrm{...}}{D}_i^{period}(\text{t})+{\epsilon }_i$

Where:

• observation $i$ is the average set of characteristics (price, odometer, vintage-age) for a class-make-model $m$ sold in a given month $t$
• these observations are reported nationally, though prices have provincial taxes applied to them
• $D_i^{model}(\text{m})$ is 1 if the model of observation $i$ is equal to $m$ , and zero otherwise
• $D_i^{period}(\text{t})$ is 1 if the sales month of observation $i$ is equal to $t$ , and zero otherwise
• the regression window $w$ is an interval consisting of a current period and $T=4$ periods back into the past, e.g. if the current period was January, it would be a 5 month interval of September $(t=0)$ through January $(t=4=T)$
• vehicle models are weighted according to estimated expenditures on them during the window, and these weights are constructed separately for each CPI strata

Further details on the derivation of a monthly price relative from the hedonic time dummy model are given below, first by discussing the weighting within the regression model, then by constructing the relative from the estimated regression coefficients.

The regression model is estimated using the weighted least squares method, where the weight of observation $i$ at time $t$ is constructed as follows:

• take the observed sample expenditure on a model in each period, so $e_{i,t}^m=T{C}_{i,t}^m\cdot p_{i,t}^m$
  • $T{C}_{i,t}^m$ is the sample transaction count of $i$ during $t$
  • $p_{i,t}^m$ is the price of $i$ during $t$
• split the model’s total observed expenditure in the window equally across periods^Note 10 in window, so ${\overline{e}}_{i,w}^{m,cm}=\frac{{\sum }_{t=0}^T{e}_{i,t}^m}{n(m)}$
  • $n(m)$ is the number of months in the window that the vehicle model was observed
  • a model $m$ exists solely within a given class-make $cm$
• take observation $i$ ’s share of the expenditures on the class-make during $t$ (i.e. in each period of the window, a class-make’s expenditures are distributed based on the window’s sampled expenditures of models), so $s_{i,t,w}^{m,cm,\mathrm{...}}=\frac{{\overline{e}}_{i,w}^{m,cm}}{{\Sigma }_{i\in S_{t,cm}}{\overline{e}}_{i,w}^{m,cm}}$
  • $S_{t,cm}$ is the sample set of vehicles in class-make $cm$ corresponding to period $t$
• the expenditure associated with observation $i$ during $t$ is then the share of class-make expenditures times the class-make’s previous window price updated expenditures, so $e_{i,t,w}^{m,cm,\mathrm{...}}=PP{V}_{T-1,used}^{cm,\mathrm{...}}\cdot s_{i,t,w}^{m,cm,\mathrm{...}}$
  • $PP{V}_{T-1,used}^{cm,\mathrm{...}}$ is the previous period price updated expenditures on the used vehicle class-make $cm$
• the weight used in the regression model is then that expenditure as a share of the period’s expenditures, divided by the number of periods in the window, so $wg{t}_{i,w,t}^{class,\mathrm{...}}=\frac{e_{i,t,w}^{m,cm,\mathrm{...}}}{(T+1)\cdot {\Sigma }_{i\in S_{t,class}}{e}_{i,t,w}^{m,cm,\mathrm{...}}}$

In summary, regression model weights have been constructed such that:

• for each period that it is observed in the window, a given used vehicle model had a constant absolute expenditure
• in each period, a class-make had the same absolute expenditure it did in any other period of the window in which it had a sale recorded in the sample
• the class-make share may vary by period, but only proportionally, as they only change if a class-make had no observations in that period of the window
• each period has an equal share of the weight in the regression model, i.e. $wg{t}_{w,t}^{class,\mathrm{...}}\equiv {\displaystyle {\sum }_{i}wg{t}_{i,t,w}^{class,\mathrm{...}}={\displaystyle {\sum }_{i}wg{t}_{i,T,w}^{class,\mathrm{...}}}}$ for all $t$

The following discusses the construction of the monthly price relative from the regression model. The approach is similar to the time-product dummy index discussed by de Haan and Hendriks (2013) and de Haan and Krsinich (2018).

For observation $i$ , its imputed price for period $t$ under the regression model would be:

${\widehat{p}}_{i,t}^{class,w,\mathrm{...}}=E(p_{i,t}^{class,w,\mathrm{...}})=e^{{\widehat{\delta }}_t^{class,w,\mathrm{...}}+{\widehat{\beta }}_0^{class,w,\mathrm{...}}+{\widehat{\beta }}_1^{class,w,\mathrm{...}}lnOdomete{r}_i+{\widehat{\beta }}_2^{class,w,\mathrm{...}}lnAg{e}_i+{\Sigma }_{m=1}^M{\widehat{\gamma }}_m^{class,w,\mathrm{...}}{D}_i^{model}}$

A geometric mean of imputed prices from the weighted least square estimates for period $t$ is then:

${\displaystyle \prod }_i{\widehat{p}}_{i,t}^{class,w,\mathrm{...}}{}^{\frac{wg{t}_{i,w,t}^{class,\mathrm{...}}}{{\Sigma }_{i}wg{t}_{i,w,t}^{class,\mathrm{...}}}}$

$=e^{{\widehat{\delta }}_t^{class,w,\mathrm{...}}+{\widehat{\beta }}_0^{class,w,\mathrm{...}}+{\widehat{\beta }}_1^{class,w,\mathrm{...}}\frac{{\Sigma }_{i}wg{t}_{i,t,w}^{class,\mathrm{...}}\cdot lnOdomete{r}_{i,t}}{{\Sigma }_{i}wg{t}_{i,w,t}^{class,\mathrm{...}}}+{\widehat{\beta }}_2^{class,w,\mathrm{...}}\frac{{\Sigma }_{i}wg{t}_{i,w,t}^{class,\mathrm{...}}\cdot lnAg{e}_{i,t}}{{\Sigma }_{i}wg{t}_{i,w,t}^{class,\mathrm{...}}}+{\Sigma }_{m=1}^M{\widehat{\gamma }}_m^{class,w,\mathrm{...}}\frac{{\Sigma }_{i}wg{t}_{i,w,t}^{class,\mathrm{...}}\cdot D_{i,t}^{model}}{{\Sigma }_{i}wg{t}_{i,w,t}^{class,\mathrm{...}}}}$

i.e.

${\displaystyle \prod _i{\widehat{p}}_{i,t}^{class,w,\mathrm{...}\stackrel{wg{t}_{i,w,t}^{class,\mathrm{...}}}{\overline{{\Sigma }_{i}wg{t}_{i,w,t}^{class,\mathrm{...}}}}}}=e^{{\widehat{\delta }}_t^{class,w,\mathrm{...}}+{\widehat{\beta }}_0^{class,w,\mathrm{...}}+{\widehat{\beta }}_1^{class,w,\mathrm{...}}{\overline{lnOdometer}}_t+{\widehat{\beta }}_2^{class,w,\mathrm{...}}{\overline{lnAge}}_t+{\Sigma }_{m=1}^M{\widehat{\gamma }}_m^{class,w,\mathrm{...}}{\overline{D}}_t^{model}}$

Where ${\overline{lnOdometer}}_t$ is the sample mean of $lnOdometer$ in $t$ , and the same is applied to other characteristics. If the ratio of geomeans from $t$ to $T$ is taken, we obtain:

$\frac{{\displaystyle \prod {}_i{\widehat{p}}_{i,T}^{class,w,\mathrm{...}\stackrel{wg{t}_{i,w,T}^{class,\mathrm{....}}}{\overline{wg{t}_w^{class,\mathrm{...}}}}}}}{{\displaystyle \prod {}_i{\widehat{p}}_{i,t}^{class,w,\mathrm{...}\stackrel{wg{t}_{i,w,t}^{class,\mathrm{....}}}{\overline{wg{t}_w^{class,\mathrm{...}}}}}}}=e^{{\Delta }^t{\widehat{\delta }}_T^{class,w,\mathrm{...}}+{\widehat{\beta }}_1^{class,w,\mathrm{...}}{\Delta }^t{\overline{lnOdometer}}_T+{\widehat{\beta }}_2^{class,w,\mathrm{...}}{\Delta }^t{\overline{lnAge}}_T+{\Sigma }_{m=1}^M{\widehat{\gamma }}_m^{class,w,\mathrm{...}}{\Delta }^t{\overline{D}}_T^{model}}$

Where ${\Delta }^t$ $x_T$ is a difference operator in $x$ from $t$ to $T$ .

Rearrange to get (note the swapping of subscripts on changes in sample means):

$\frac{{\displaystyle \prod {}_i{\widehat{p}}_{i,T}^{class,w,\mathrm{...}\stackrel{wg{t}_{i,w,T}^{class,\mathrm{....}}}{\overline{wg{t}_w^{class,\mathrm{...}}}}}}}{{\displaystyle \prod {}_i{\widehat{p}}_{i,t}^{class,w,\mathrm{...}\stackrel{wg{t}_{i,w,t}^{class,\mathrm{....}}}{\overline{wg{t}_w^{class,\mathrm{...}}}}}}}\cdot e^{{\widehat{\beta }}_1^{class,w,\mathrm{...}}{\Delta }^T{\overline{lnOdometer}}_t+{\widehat{\beta }}_2^{class,w,\mathrm{...}}{\Delta }^T{\overline{lnAge}}_t+{\Sigma }_{m=1}^M{\widehat{\gamma }}_m^{class,w,\mathrm{...}}{\Delta }^T{\overline{D}}_t^{model}}=e^{{\Delta }^t{\widehat{\delta }}_T^{class,w,\mathrm{...}}}$

Since the weight of an observation is zero if it didn’t exist in a given period, ${\displaystyle \prod {}_i{\widehat{p}}_{i,t}^{class,w,\mathrm{...}\stackrel{wg{t}_{i,w,t}^{class,\mathrm{....}}}{\overline{wg{t}_w^{class,\mathrm{...}}}}}}={\displaystyle \prod {}_{i\in S_t}{\widehat{p}}_{i\in S_{t\text{'}}t}^{class,w,\mathrm{...}\stackrel{wg{t}_{i\in S_t,w,t}^{class,\mathrm{....}}}{\overline{wg{t}_w^{class,\mathrm{...}}}}}}$ . Since the time dummies cause WLS residuals to sum to zero in each period of the regression window, ${\displaystyle \prod {}_i{\widehat{p}}_{i,t}^{class,w,\mathrm{...}\stackrel{wg{t}_{i,w,t}^{class,\mathrm{...}}}{\overline{wg{t}_w^{class,\mathrm{...}}}}}}={\displaystyle \prod {}_i{p}_{i,t}^{class,w,\mathrm{...}\stackrel{wg{t}_{i,w,t}^{class,\mathrm{...}}}{\overline{wg{t}_w^{class,\mathrm{...}}}}}}$ . This makes the final equation equivalent to $\frac{{\displaystyle \prod {}_i{p}_{i,T}^{class,w,\mathrm{...}\stackrel{wg{t}_{i,w,t}^{class,\mathrm{....}}}{\overline{wg{t}_w^{class,\mathrm{...}}}}}}}{{\displaystyle \prod {}_i{p}_{i,t}^{class,w,\mathrm{...}\stackrel{wg{t}_{i,w,T}^{class,\mathrm{....}}}{\overline{wg{t}_w^{class,\mathrm{...}}}}}}}\cdot e^{{\widehat{\beta }}_1^{class,w,\mathrm{...}}{\Delta }^T{\overline{lnOdometer}}_t+{\widehat{\beta }}_2^{class,w,\mathrm{...}}{\Delta }^T{\overline{lnAge}}_t+{\Sigma }_{m=1}^M{\widehat{\gamma }}_m^{class,w,\mathrm{...}}{\Delta }^T{\overline{D}}_t^{model}}=e^{{\Delta }^t{\widehat{\delta }}_T^{class,w,\mathrm{...}}}$

This is an interpretation of the hedonic time dummy model which lets us think of the change in time dummy coefficients as some measure of change in average prices that is quality-adjusted to reflect changes in the sample means of vehicles characteristics.^Note 11 Since we are estimating price change from $T-1$ to $T$ , the price relative is defined as $e^{\Delta {\widehat{\delta }}_T^{class,w,\mathrm{...}}}$ .

The class-make price relatives come from the time dummy coefficients, i.e., ${\frac{p_t}{p_{t-1}}}^{class,make,\mathrm{...}}=e^{\Delta {\delta }_t^{class,\mathrm{...}}}={\frac{p_t}{p_{t-1}}}^{class,\mathrm{...}}$ , and they are used to price update a class-make expenditure, i.e., $PP{V}_{t,used}^{class,make,\mathrm{...}}={\frac{p_t}{p_{t-1}}}^{class,\mathrm{...}}\cdot PP{V}_{t-1,used}^{class,make,\mathrm{...}}$ , where $PP{V}_{t,used}^{class,make,\mathrm{...}}$ refers to the used motor vehicle expenditures for a given class and make in period $t$ .

Overall price-updated used vehicles expenditures are the sum across class-makes, so $PP{V}_{t,used}^{,\mathrm{...}}={\Sigma }_{class}{\Sigma }_{make}PP{V}_{t,used}^{class,make,\mathrm{...}}$ . The overall used vehicles price movement is then just the sum of current period price-updated class-make expenditures over the previous period’s corresponding sum, i.e. ${\frac{p_t}{p_{t-1}}}^{used,\mathrm{...}}=\frac{PP{V}_{t,used}^{\mathrm{...}}}{PP{V}_{t-1,used}^{\mathrm{...}}}$ .