Measuring value change for used automobiles within the Canadian Client Value Index


Key messages


  • Introducing used car costs within the Client Value Index (CPI) is a part of Statistics Canada’s dedication to present essentially the most well timed, dependable and correct knowledge which displays the expertise of Canadians.
  • As a part of Statistics Canada’s rigorous and ongoing efforts to take care of the standard and relevance of the CPI, this technical paper explains the proposed timing, knowledge and methodology for together with the costs of used automobiles within the CPI’s buy of passenger car index.
  • Statistics Canada has recognized a dependable knowledge supply for the costs and traits of used automobiles, and the upcoming annual basket replace in June will incorporate this new supply of knowledge in its calculation of the CPI. The CPI beforehand accounted for used car costs by together with a weight for used automobiles and utilizing new car costs as a proxy.
  • We’ll proceed to observe costs for used automobiles and leverage extra new knowledge sources for the acquisition of passenger automobiles index. This may make sure the CPI stays an correct, strong and related technique of measuring inflation.


The Client Value Index (CPI) measures the change in the price of a hard and fast basket of shopper items and providers over time. To precisely mirror traits available in the market and shopper conduct, Statistics Canada periodically updates the strategies and sources utilized to numerous parts of the CPI.

The acquisition of passenger automobiles index within the CPI measures the common change over time within the costs of passenger automobiles. It includes 6.21% of the 2020 CPI basket. The burden of the acquisition of passenger automobiles index includes family expenditures on new automobiles, plus web family expendituresWord 1 on used automobiles, which alone make up between one quarter and one third of the 6.21% weight share of the acquisition of passenger automobiles index.Word 2 At present, Statistics Canada makes use of new car costs to estimate the whole lot of the acquisition of passenger automobiles index, successfully utilizing new car costs as a proxy for used car costs.

Amid the COVID-19 pandemic, a divergence in value actions for brand spanking new and used vehicles was noticed in a number of nations, significantly the US. Provide chain disruptions, notably for the semiconductor chips utilized in numerous parts of newly manufactured automobiles, and pandemic-related plant closures proceed to affect the manufacture of latest automobiles, resulting in decreased inventories. With fewer new vehicles and vans out there for buy and prolonged delays for supply of latest automobiles bought, customers have sought out used vehicles, driving up demand. On the similar time, fewer customers are buying and selling of their used fashions, making a provide scarcity within the used car market. These shifting market dynamics have, consequently, resulted in steeper value will increase for used automobiles than for brand spanking new automobiles. This divergence within the value actions signifies that new car costs not function an efficient proxy for used car costs within the Canadian CPI. Statistics Canada recommends to introduce enhancements to the calculation of the acquisition of passenger automobiles index by together with used car costs. The enhancement can be applied with the CPI basket replace on June 22, 2022. On the similar time, used passenger automobiles can be added to the CPI basket as a broadcast mixture.

Enhancements to the index

To be able to higher measure value change for passenger automobiles, enhancements can be made to the index together with:

  • the creation of two new elementary aggregates for the acquisition of latest passenger automobiles and the acquisition of used passenger automobiles as parts of a single buy of passenger automobiles index
  • using a dependable knowledge supply for used car costs and traits
  • the introduction of applicable modelling to calculate a used automobiles index that accounts for high quality change and depreciation over time

The transaction knowledge used to cost used automobiles will come from JD Energy, offering entry to costs and traits of automobiles (used and new) bought by households, from dealerships. The month-to-month transaction knowledge is obtained as an mixture such that every make and mannequin of auto has a single value,Word 3 classic age, odometer studying, and the pattern transaction rely. The worth, classic age and odometer studying are averages which can be calculated utilizing weights primarily based on car registrations to make sure their representativeness. Hedonic modelling of auto costs is already executed by Statistics Canada within the deflation of used motorcar costs within the nationwide accounts, although the mannequin isn’t relevant to the wants of the CPI. The CPI will use an analogous hedonic mannequin, with the principle variations involving adjustments to the specification, weighting, durations of curiosity, and segmentation. A hedonic method is employed as a result of used automobiles of the identical mannequin sort might differ in observable traits, resembling utilization or classic, which means that direct value comparisons of the identical mannequin sort over time might result in biased estimates. This hedonic method capabilities as a measure of change in mixture car mannequin costs with high quality changesWord 4 for aggregates of vintage-age and utilization.

Development of month-to-month value kin

The CPI measures pure value change, guaranteeing that value comparisons are revamped time for like merchandise, explicitly accounting for variations in observable high quality traits. Utilizing transaction knowledge signifies that a given mannequin of used car, as a result of its depreciation, might have various high quality between durations. Due to this fact, in an effort to management for high quality change and estimate pure value change, a hedonic time dummy is employed alongside a rolling five-month window.Word 5

The logarithm of value is modeled as a perform of the logarithm of vintage-age,Word 6 and logarithm of odometer studying of automobiles, in addition to mannequin fastened results and a dummy variable for every of the final 4 months of the window. Formally:




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  • car fashions are weighted in line with estimated expenditures on them through the window, and these weights are constructed individually for every CPI strata


The regression specification is much like the methodology employed within the measurement of used automotive value actions in New Zealand.Word 7 Whereas the observable traits of a given car are usually not explicitly managed for, there’s comparatively little variation inside fashions (primarily coming from completely different trims), in comparison with throughout fashions. Moreover, the inclusion of express traits would require the acquisition and processing of extra knowledge every interval, which was deemed unfeasible underneath the present constraints of CPI manufacturing. For these causes, using mannequin fastened results has been employed.Word 8 The above specification was discovered to offer adjusted R-squares that tended to vary inside the low .90s (largely inside .90 to .94) for some lessons, and the excessive .90s (largely inside .95 to .98) for others.Word 9

Separate regressions are run for every CPI geography and sophistication of auto. The change in time dummy coefficient from




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fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqeaaa@375E@




is a distinction operator.

Additional particulars on the derivation of a month-to-month value relative from the hedonic time dummy mannequin are given beneath, first by discussing the weighting inside the regression mannequin, then by developing the relative from the estimated regression coefficients.

The regression mannequin is estimated utilizing the weighted least squares technique, the place the burden of remark



i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@




at time



t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36F0@




is constructed as follows:

  • take the noticed pattern expenditure on a mannequin in every interval, so




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    =
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    MathType@MTEF@5@5@+=
    feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
    hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
    4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
    vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
    fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaDa
    aaleaacaWGPbGaaiilaiaadshaaeaacaWGTbaaaOGaeyypa0Jaamiv
    aiaadoeadaqhaaWcbaGaamyAaiaacYcacaWG0baabaGaamyBaaaaki
    abgwSixlaadchadaqhaaWcbaGaamyAaiaacYcacaWG0baabaGaamyB
    aaaaaaa@47FD@







    • T

      C

      i
      ,
      t

      m


      MathType@MTEF@5@5@+=
      feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
      hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
      4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
      vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
      fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaiaado
      eadaqhaaWcbaGaamyAaiaacYcacaWG0baabaGaamyBaaaaaaa@3B4E@




      is the pattern transaction rely of



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      MathType@MTEF@5@5@+=
      feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
      hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
      4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
      vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
      fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E4@




      throughout



      t
      MathType@MTEF@5@5@+=
      feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
      hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
      4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
      vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
      fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36F0@








    • p

      i
      ,
      t

      m


      MathType@MTEF@5@5@+=
      feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
      hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
      4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
      vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
      fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDa
      aaleaacaWGPbGaaiilaiaadshaaeaacaWGTbaaaaaa@3AA2@




      is the worth of



      i
      MathType@MTEF@5@5@+=
      feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
      hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
      4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
      vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
      fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E4@




      throughout



      t
      MathType@MTEF@5@5@+=
      feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
      hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
      4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
      vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
      fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36F0@


  • cut up the mannequin’s complete noticed expenditure within the window equally throughout durationsWord 10 in window, so





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    =
    0

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    m
    )



    MathType@MTEF@5@5@+=
    feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
    hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
    4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
    vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
    fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyzayaara
    Waa0baaSqaaiaadMgacaGGSaGaam4Daaqaaiaad2gacaGGSaGaam4y
    aiaad2gaaaGccqGH9aqpdaWcaaqaaiabggHiLpaaDaaaleaacaWG0b
    Gaeyypa0JaaGimaaqaaiaadsfaaaGccaWGLbWaa0baaSqaaiaadMga
    caGGSaGaamiDaaqaaiaad2gaaaaakeaacaWGUbGaaiikaiaad2gaca
    GGPaaaaaaa@4BB1@







    • n
      (
      m
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      MathType@MTEF@5@5@+=
      feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
      hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
      4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
      vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
      fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaacI
      cacaWGTbGaaiykaaaa@3935@




      is the variety of months within the window that the car mannequin was noticed

    • a mannequin


      m
      MathType@MTEF@5@5@+=
      feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
      hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
      4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
      vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
      fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaaaa@36E8@




      exists solely inside a given class-make




      c
      m

      MathType@MTEF@5@5@+=
      feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
      hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
      4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
      vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
      fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiaad2
      gaaaa@37D0@



  • take remark


    i
    MathType@MTEF@5@5@+=
    feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
    hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
    4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
    vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
    fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@



    ’s share of the expenditures on the class-make throughout



    t
    MathType@MTEF@5@5@+=
    feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
    hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
    4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
    vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
    fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36F0@




    (i.e. in every interval of the window, a class-make’s expenditures are distributed primarily based on the window’s sampled expenditures of fashions), so





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    =




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    MathType@MTEF@5@5@+=
    feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
    hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
    4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
    vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
    fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaDa
    aaleaacaWGPbGaaiilaiaadshacaGGSaGaam4Daaqaaiaad2gacaGG
    SaGaam4yaiaad2gacaGGSaGaaiOlaiaac6cacaGGUaaaaOGaeyypa0
    ZaaSaaaeaaceWGLbGbaebadaqhaaWcbaGaamyAaiaacYcacaWG3baa
    baGaamyBaiaacYcacaWGJbGaamyBaaaaaOqaaiabfo6atnaaBaaale
    aacaWGPbGaeyicI4Saam4uamaaBaaameaacaWG0bGaaiilaiaadoga
    caWGTbaabeaaaSqabaGcceWGLbGbaebadaqhaaWcbaGaamyAaiaacY
    cacaWG3baabaGaamyBaiaacYcacaWGJbGaamyBaaaaaaaaaa@5A14@









    • S

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      MathType@MTEF@5@5@+=
      feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
      hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
      4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
      vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
      fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa
      aaleaacaWG0bGaaiilaiaadogacaWGTbaabeaaaaa@3A7E@





      is the pattern set of automobiles in class-make




      c
      m

      MathType@MTEF@5@5@+=
      feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
      hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
      4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
      vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
      fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiaad2
      gaaaa@37D1@




      equivalent to interval



      t
      MathType@MTEF@5@5@+=
      feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
      hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
      4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
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      fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36F0@



  • the expenditure related to remark


    i
    MathType@MTEF@5@5@+=
    feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
    hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
    4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
    vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
    fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E4@




    throughout



    t
    MathType@MTEF@5@5@+=
    feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
    hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
    4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
    vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
    fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36F0@




    is then the share of class-make expenditures occasions the class-make’s earlier window value up to date expenditures, so





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    =
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    MathType@MTEF@5@5@+=
    feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
    hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
    4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
    vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
    fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaDa
    aaleaacaWGPbGaaiilaiaadshacaGGSaGaam4Daaqaaiaad2gacaGG
    SaGaam4yaiaad2gacaGGSaGaaiOlaiaac6cacaGGUaaaaOGaeyypa0
    JaamiuaiaadcfacaWGwbWaa0baaSqaaiaadsfacqGHsislcaaIXaGa
    aiilaiaadwhacaWGZbGaamyzaiaadsgaaeaacaWGJbGaamyBaiaacY
    cacaGGUaGaaiOlaiaac6caaaGccqGHflY1caWGZbWaa0baaSqaaiaa
    dMgacaGGSaGaamiDaiaacYcacaWG3baabaGaamyBaiaacYcacaWGJb
    GaamyBaiaacYcacaGGUaGaaiOlaiaac6caaaaaaa@5EE9@







    • P
      P

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      MathType@MTEF@5@5@+=
      feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
      hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
      4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
      vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
      fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaadc
      facaWGwbWaa0baaSqaaiaadsfacqGHsislcaaIXaGaaiilaiaadwha
      caWGZbGaamyzaiaadsgaaeaacaWGJbGaamyBaiaacYcacaGGUaGaai
      Olaiaac6caaaaaaa@443F@




      is the earlier interval value up to date expenditures on the used car class-make




      c
      m

      MathType@MTEF@5@5@+=
      feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
      hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
      4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
      vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
      fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiaad2
      gaaaa@37D1@



  • the burden used within the regression mannequin is then that expenditure as a share of the interval’s expenditures, divided by the variety of durations within the window, so



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    =



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    ,





    (
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    +
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    a
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    s





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    ,






    MathType@MTEF@5@5@+=
    feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
    hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
    4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
    vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
    fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaadE
    gacaWG0bWaa0baaSqaaiaadMgacaGGSaGaam4DaiaacYcacaWG0baa
    baGaam4yaiaadYgacaWGHbGaam4CaiaadohacaGGSaGaaiOlaiaac6
    cacaGGUaaaaOGaeyypa0ZaaSaaaeaacaWGLbWaa0baaSqaaiaadMga
    caGGSaGaamiDaiaacYcacaWG3baabaGaamyBaiaacYcacaWGJbGaam
    yBaiaacYcacaGGUaGaaiOlaiaac6caaaaakeaacaGGOaGaamivaiab
    gUcaRiaaigdacaGGPaGaeyyXICTaeu4Odm1aaSbaaSqaaiaadMgacq
    GHiiIZcaWGtbWaaSbaaWqaaiaadshacaGGSaGaam4yaiaadYgacaWG
    HbGaam4CaiaadohaaeqaaaWcbeaakiaadwgadaqhaaWcbaGaamyAai
    aacYcacaWG0bGaaiilaiaadEhaaeaacaWGTbGaaiilaiaadogacaWG
    TbGaaiilaiaac6cacaGGUaGaaiOlaaaaaaaaaa@6ECC@



In abstract, regression mannequin weights have been constructed such that:

  • for every interval that it’s noticed within the window, a given used car mannequin had a relentless absolute expenditure
  • in every interval, a class-make had the identical absolute expenditure it did in some other interval of the window during which it had a sale recorded within the pattern
  • the class-make share might differ by interval, however solely proportionally, as they solely change if a class-make had no observations in that interval of the window
  • every interval has an equal share of the burden within the regression mannequin, i.e.



    w
    g

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    ,
    t


    c
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    s
    s
    ,







    i


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The next discusses the development of the month-to-month value relative from the regression mannequin. The method is much like the time-product dummy index mentioned by de Haan and Hendriks (2013) and de Haan and Krsinich (2018).

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, its imputed value for interval



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A geometrical imply of imputed costs from the weighted least sq. estimates for interval



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to



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4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaaaa@36D0@




is taken, we receive:











i



p
^


i
,
T


c
l
a
s
s
,
w
,





w
g

t
w

c
l
a
s
s
,




¯



w
g

t

i
,
w
,
T


c
l
a
s
s
,
….















i



p
^


i
,
t


c
l
a
s
s
,
w
,





w
g

t
w

c
l
a
s
s
,




¯



w
g

t

i
,
w
,
t


c
l
a
s
s
,
….










=

e


Δ
t



δ
^

T

c
l
a
s
s
,
w
,



+


β
^

1

c
l
a
s
s
,
w
,




Δ
t





l
n
O
d
o
m
e
t
e
r

¯


T

+


β
^

2

c
l
a
s
s
,
w
,




Δ
t





l
n
A
g
e

¯


T

+

Σ

m
=
1

M



γ
^

m

c
l
a
s
s
,
w
,




Δ
t



D
¯

T

m
o
d
e
l





MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaada
qeaaqaamaaBaaaleaacaWGPbaabeaakiqadchagaqcamaaDaaaleaa
caWGPbGaaiilaiaadsfaaeaacaWGJbGaamiBaiaadggacaWGZbGaam
4CaiaacYcacaWG3bGaaiilaiaac6cacaGGUaGaaiOlamaaxacabaWa
a0aaaeaacaWG3bGaam4zaiaadshadaqhaaadbaGaam4Daaqaaiaado
gacaWGSbGaamyyaiaadohacaWGZbGaaiilaiaac6cacaGGUaGaaiOl
aaaaaaaabeqaaiaadEhacaWGNbGaamiDamaaDaaabaGaamyAaiaacY
cacaWG3bGaaiilaiaadsfaaeaacaWGJbGaamiBaiaadggacaWGZbGa
am4CaiaacYcacaGGUaGaaiOlaiaac6cacaGGUaaaaaaaaaaaleqabe
qdcqGHpis1aaGcbaWaaebaaeaadaWgaaWcbaGaamyAaaqabaGcceWG
WbGbaKaadaqhaaWcbaGaamyAaiaacYcacaWG0baabaGaam4yaiaadY
gacaWGHbGaam4CaiaadohacaGGSaGaam4DaiaacYcacaGGUaGaaiOl
aiaac6cadaWfGaqaamaanaaabaGaam4DaiaadEgacaWG0bWaa0baaW
qaaiaadEhaaeaacaWGJbGaamiBaiaadggacaWGZbGaam4CaiaacYca
caGGUaGaaiOlaiaac6caaaaaaaqabeaacaWG3bGaam4zaiaadshada
qhaaqaaiaadMgacaGGSaGaam4DaiaacYcacaWG0baabaGaam4yaiaa
dYgacaWGHbGaam4CaiaadohacaGGSaGaaiOlaiaac6cacaGGUaGaai
OlaaaaaaaaaaWcbeqab0Gaey4dIunaaaGccqGH9aqpcaWGLbWaaWba
aSqabeaacqGHuoardaahaaadbeqaaiaadshaaaWccuaH0oazgaqcam
aaDaaameaacaWGubaabaGaam4yaiaadYgacaWGHbGaam4Caiaadoha
caGGSaGaam4DaiaacYcacaGGUaGaaiOlaiaac6caaaWccqGHRaWkcu
aHYoGygaqcamaaDaaameaacaaIXaaabaGaam4yaiaadYgacaWGHbGa
am4CaiaadohacaGGSaGaam4DaiaacYcacaGGUaGaaiOlaiaac6caaa
WccqGHuoardaahaaadbeqaaiaadshaaaWcdaqdaaqaaiaadYgacaWG
UbGaam4taiaadsgacaWGVbGaamyBaiaadwgacaWG0bGaamOCaiaadw
gaaaWaaSbaaWqaaiaadsfaaeqaaSGaey4kaSIafqOSdiMbaKaadaqh
aaadbaGaaGOmaaqaaiaadogacaWGSbGaamyyaiaadohacaWGZbGaai
ilaiaadEhacaGGSaGaaiOlaiaac6cacaGGUaaaaSGaeyiLdq0aaWba
aWqabeaacaWG0baaaSWaa0aaaeaacaWGSbGaamOBaiaadgeacaWGNb
GaamyzaaaadaWgaaadbaGaamivaaqabaWccqGHRaWkcqqHJoWudaqh
aaadbaGaamyBaiabg2da9iaaigdaaeaacaWGnbaaaSGafq4SdCMbaK
aadaqhaaadbaGaamyBaaqaaiaadogacaWGSbGaamyyaiaadohacaWG
ZbGaaiilaiaadEhacaGGSaGaaiOlaiaac6cacaGGUaaaaSGaeyiLdq
0aaWbaaWqabeaacaWG0baaaSGabmirayaaraWaa0baaWqaaiaadsfa
aeaacaWGTbGaam4BaiaadsgacaWGLbGaamiBaaaaaaaaaa@E676@


The place





Δ
t


MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdq0aaW
baaSqabeaacaWG0baaaaaa@3884@






x

T


MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8
qacaWG4bWdamaaBaaaleaapeGaamivaaWdaeqaaaaa@3846@





is a distinction operator in




x

MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8
qacaWG4bWdamaaBaaaleaapeGaamivaaWdaeqaaaaa@3846@



from



t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36F0@




to



T
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaaaa@36D0@



.

Rearrange to get (be aware the swapping of subscripts on adjustments in pattern means):











i



p
^


i
,
T


c
l
a
s
s
,
w
,





w
g

t
w

c
l
a
s
s
,




¯



w
g

t

i
,
w
,
T


c
l
a
s
s
,
….















i



p
^


i
,
t


c
l
a
s
s
,
w
,





w
g

t
w

c
l
a
s
s
,




¯



w
g

t

i
,
w
,
t


c
l
a
s
s
,
….












e



β
^

1

c
l
a
s
s
,
w
,




Δ
T





l
n
O
d
o
m
e
t
e
r

¯


t

+


β
^

2

c
l
a
s
s
,
w
,




Δ
T





l
n
A
g
e

¯


t

+

Σ

m
=
1

M



γ
^

m

c
l
a
s
s
,
w
,




Δ
T



D
¯

t

m
o
d
e
l




=

e


Δ
t



δ
^

T

c
l
a
s
s
,
w
,






MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaada
qeaaqaamaaBaaaleaacaWGPbaabeaakiqadchagaqcamaaDaaaleaa
caWGPbGaaiilaiaadsfaaeaacaWGJbGaamiBaiaadggacaWGZbGaam
4CaiaacYcacaWG3bGaaiilaiaac6cacaGGUaGaaiOlamaaxacabaWa
a0aaaeaacaWG3bGaam4zaiaadshadaqhaaadbaGaam4Daaqaaiaado
gacaWGSbGaamyyaiaadohacaWGZbGaaiilaiaac6cacaGGUaGaaiOl
aaaaaaaabeqaaiaadEhacaWGNbGaamiDamaaDaaabaGaamyAaiaacY
cacaWG3bGaaiilaiaadsfaaeaacaWGJbGaamiBaiaadggacaWGZbGa
am4CaiaacYcacaGGUaGaaiOlaiaac6cacaGGUaaaaaaaaaaaleqabe
qdcqGHpis1aaGcbaWaaebaaeaadaWgaaWcbaGaamyAaaqabaGcceWG
WbGbaKaadaqhaaWcbaGaamyAaiaacYcacaWG0baabaGaam4yaiaadY
gacaWGHbGaam4CaiaadohacaGGSaGaam4DaiaacYcacaGGUaGaaiOl
aiaac6cadaWfGaqaamaanaaabaGaam4DaiaadEgacaWG0bWaa0baaW
qaaiaadEhaaeaacaWGJbGaamiBaiaadggacaWGZbGaam4CaiaacYca
caGGUaGaaiOlaiaac6caaaaaaaqabeaacaWG3bGaam4zaiaadshada
qhaaqaaiaadMgacaGGSaGaam4DaiaacYcacaWG0baabaGaam4yaiaa
dYgacaWGHbGaam4CaiaadohacaGGSaGaaiOlaiaac6cacaGGUaGaai
OlaaaaaaaaaaWcbeqab0Gaey4dIunaaaGccqGHflY1caWGLbWaaWba
aSqabeaacuaHYoGygaqcamaaDaaameaacaaIXaaabaGaam4yaiaadY
gacaWGHbGaam4CaiaadohacaGGSaGaam4DaiaacYcacaGGUaGaaiOl
aiaac6caaaWccqGHuoardaahaaadbeqaaiaadsfaaaWcdaqdaaqaai
aadYgacaWGUbGaam4taiaadsgacaWGVbGaamyBaiaadwgacaWG0bGa
amOCaiaadwgaaaWaaSbaaWqaaiaadshaaeqaaSGaey4kaSIafqOSdi
MbaKaadaqhaaadbaGaaGOmaaqaaiaadogacaWGSbGaamyyaiaadoha
caWGZbGaaiilaiaadEhacaGGSaGaaiOlaiaac6cacaGGUaaaaSGaey
iLdq0aaWbaaWqabeaacaWGubaaaSWaa0aaaeaacaWGSbGaamOBaiaa
dgeacaWGNbGaamyzaaaadaWgaaadbaGaamiDaaqabaWccqGHRaWkcq
qHJoWudaqhaaadbaGaamyBaiabg2da9iaaigdaaeaacaWGnbaaaSGa
fq4SdCMbaKaadaqhaaadbaGaamyBaaqaaiaadogacaWGSbGaamyyai
aadohacaWGZbGaaiilaiaadEhacaGGSaGaaiOlaiaac6cacaGGUaaa
aSGaeyiLdq0aaWbaaWqabeaacaWGubaaaSGabmirayaaraWaa0baaW
qaaiaadshaaeaacaWGTbGaam4BaiaadsgacaWGLbGaamiBaaaaaaGc
cqGH9aqpcaWGLbWaaWbaaSqabeaacqGHuoardaahaaadbeqaaiaads
haaaWccuaH0oazgaqcamaaDaaameaacaWGubaabaGaam4yaiaadYga
caWGHbGaam4CaiaadohacaGGSaGaam4DaiaacYcacaGGUaGaaiOlai
aac6caaaaaaaaa@E8F4@


Because the weight of an remark is zero if it didn’t exist in a given interval,









i



p
^


i
,
t


c
l
a
s
s
,
w
,





w
g

t
w

c
l
a
s
s
,




¯



w
g

t

i
,
w
,
t


c
l
a
s
s
,
….








=






i


S
t





p
^


i


S

t



t


c
l
a
s
s
,
w
,





w
g

t
w

c
l
a
s
s
,




¯



w
g

t

i


S
t

,
w
,
t


c
l
a
s
s
,
….









MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaebaaeaada
WgaaWcbaGaamyAaaqabaGcceWGWbGbaKaadaqhaaWcbaGaamyAaiaa
cYcacaWG0baabaGaam4yaiaadYgacaWGHbGaam4CaiaadohacaGGSa
Gaam4DaiaacYcacaGGUaGaaiOlaiaac6cadaWfGaqaamaanaaabaGa
am4DaiaadEgacaWG0bWaa0baaWqaaiaadEhaaeaacaWGJbGaamiBai
aadggacaWGZbGaam4CaiaacYcacaGGUaGaaiOlaiaac6caaaaaaaqa
beaacaWG3bGaam4zaiaadshadaqhaaqaaiaadMgacaGGSaGaam4Dai
aacYcacaWG0baabaGaam4yaiaadYgacaWGHbGaam4CaiaadohacaGG
SaGaaiOlaiaac6cacaGGUaGaaiOlaaaaaaaaaaWcbeqab0Gaey4dIu
nakiabg2da9maaraaabaWaaSbaaSqaaiaadMgacqGHiiIZcaWGtbWa
aSbaaWqaaiaadshaaeqaaaWcbeaakiqadchagaqcamaaDaaaleaaca
WGPbGaeyicI4Saam4uamaaBaaameaacaWG0bGaai4jaaqabaWccaWG
0baabaGaam4yaiaadYgacaWGHbGaam4CaiaadohacaGGSaGaam4Dai
aacYcacaGGUaGaaiOlaiaac6cadaWfGaqaamaanaaabaGaam4Daiaa
dEgacaWG0bWaa0baaWqaaiaadEhaaeaacaWGJbGaamiBaiaadggaca
WGZbGaam4CaiaacYcacaGGUaGaaiOlaiaac6caaaaaaaqabeaacaWG
3bGaam4zaiaadshadaqhaaqaaiaadMgacqGHiiIZcaWGtbWaaSbaae
aacaWG0baabeaacaGGSaGaam4DaiaacYcacaWG0baabaGaam4yaiaa
dYgacaWGHbGaam4CaiaadohacaGGSaGaaiOlaiaac6cacaGGUaGaai
OlaaaaaaaaaaWcbeqab0Gaey4dIunaaaa@97AB@



. Because the time dummies trigger WLS residuals to sum to zero in every interval of the regression window,









i



p
^


i
,
t


c
l
a
s
s
,
w
,





w
g

t
w

c
l
a
s
s
,




¯



w
g

t

i
,
w
,
t


c
l
a
s
s
,









=





i


p

i
,
t


c
l
a
s
s
,
w
,





w
g

t
w

c
l
a
s
s
,




¯



w
g

t

i
,
w
,
t


c
l
a
s
s
,










MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaebaaeaada
WgaaWcbaGaamyAaaqabaGcceWGWbGbaKaadaqhaaWcbaGaamyAaiaa
cYcacaWG0baabaGaam4yaiaadYgacaWGHbGaam4CaiaadohacaGGSa
Gaam4DaiaacYcacaGGUaGaaiOlaiaac6cadaWfGaqaamaanaaabaGa
am4DaiaadEgacaWG0bWaa0baaWqaaiaadEhaaeaacaWGJbGaamiBai
aadggacaWGZbGaam4CaiaacYcacaGGUaGaaiOlaiaac6caaaaaaaqa
beaacaWG3bGaam4zaiaadshadaqhaaqaaiaadMgacaGGSaGaam4Dai
aacYcacaWG0baabaGaam4yaiaadYgacaWGHbGaam4CaiaadohacaGG
SaGaaiOlaiaac6cacaGGUaaaaaaaaaaaleqabeqdcqGHpis1aOGaey
ypa0ZaaebaaeaadaWgaaWcbaGaamyAaaqabaGccaWGWbWaa0baaSqa
aiaadMgacaGGSaGaamiDaaqaaiaadogacaWGSbGaamyyaiaadohaca
WGZbGaaiilaiaadEhacaGGSaGaaiOlaiaac6cacaGGUaWaaCbiaeaa
daqdaaqaaiaadEhacaWGNbGaamiDamaaDaaameaacaWG3baabaGaam
4yaiaadYgacaWGHbGaam4CaiaadohacaGGSaGaaiOlaiaac6cacaGG
UaaaaaaaaeqabaGaam4DaiaadEgacaWG0bWaa0baaeaacaWGPbGaai
ilaiaadEhacaGGSaGaamiDaaqaaiaadogacaWGSbGaamyyaiaadoha
caWGZbGaaiilaiaac6cacaGGUaGaaiOlaaaaaaaaaaWcbeqab0Gaey
4dIunaaaa@8BAD@



. This makes the ultimate equation equal to











i


p

i
,
T


c
l
a
s
s
,
w
,





w
g

t
w

c
l
a
s
s
,




¯



w
g

t

i
,
w
,
t


c
l
a
s
s
,
….















i


p

i
,
t


c
l
a
s
s
,
w
,





w
g

t
w

c
l
a
s
s
,




¯



w
g

t

i
,
w
,
T


c
l
a
s
s
,
….












e



β
^

1

c
l
a
s
s
,
w
,




Δ
T





l
n
O
d
o
m
e
t
e
r

¯


t

+


β
^

2

c
l
a
s
s
,
w
,




Δ
T





l
n
A
g
e

¯


t

+

Σ

m
=
1

M



γ
^

m

c
l
a
s
s
,
w
,




Δ
T



D
¯

t

m
o
d
e
l




=

e


Δ
t



δ
^

T

c
l
a
s
s
,
w
,






MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaada
qeaaqaamaaBaaaleaacaWGPbaabeaakiaadchadaqhaaWcbaGaamyA
aiaacYcacaWGubaabaGaam4yaiaadYgacaWGHbGaam4Caiaadohaca
GGSaGaam4DaiaacYcacaGGUaGaaiOlaiaac6cadaWfGaqaamaanaaa
baGaam4DaiaadEgacaWG0bWaa0baaWqaaiaadEhaaeaacaWGJbGaam
iBaiaadggacaWGZbGaam4CaiaacYcacaGGUaGaaiOlaiaac6caaaaa
aaqabeaacaWG3bGaam4zaiaadshadaqhaaqaaiaadMgacaGGSaGaam
4DaiaacYcacaWG0baabaGaam4yaiaadYgacaWGHbGaam4Caiaadoha
caGGSaGaaiOlaiaac6cacaGGUaGaaiOlaaaaaaaaaaWcbeqab0Gaey
4dIunaaOqaamaaraaabaWaaSbaaSqaaiaadMgaaeqaaOGaamiCamaa
DaaaleaacaWGPbGaaiilaiaadshaaeaacaWGJbGaamiBaiaadggaca
WGZbGaam4CaiaacYcacaWG3bGaaiilaiaac6cacaGGUaGaaiOlamaa
xacabaWaa0aaaeaacaWG3bGaam4zaiaadshadaqhaaadbaGaam4Daa
qaaiaadogacaWGSbGaamyyaiaadohacaWGZbGaaiilaiaac6cacaGG
UaGaaiOlaaaaaaaabeqaaiaadEhacaWGNbGaamiDamaaDaaabaGaam
yAaiaacYcacaWG3bGaaiilaiaadsfaaeaacaWGJbGaamiBaiaadgga
caWGZbGaam4CaiaacYcacaGGUaGaaiOlaiaac6cacaGGUaaaaaaaaa
aaleqabeqdcqGHpis1aaaakiabgwSixlaadwgadaahaaWcbeqaaiqb
ek7aIzaajaWaa0baaWqaaiaaigdaaeaacaWGJbGaamiBaiaadggaca
WGZbGaam4CaiaacYcacaWG3bGaaiilaiaac6cacaGGUaGaaiOlaaaa
liabgs5aenaaCaaameqabaGaamivaaaalmaanaaabaGaamiBaiaad6
gacaWGpbGaamizaiaad+gacaWGTbGaamyzaiaadshacaWGYbGaamyz
aaaadaWgaaadbaGaamiDaaqabaWccqGHRaWkcuaHYoGygaqcamaaDa
aameaacaaIYaaabaGaam4yaiaadYgacaWGHbGaam4CaiaadohacaGG
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adbeqaaiaadsfaaaWcdaqdaaqaaiaadYgacaWGUbGaamyqaiaadEga
caWGLbaaamaaBaaameaacaWG0baabeaaliabgUcaRiabfo6atnaaDa
aameaacaWGTbGaeyypa0JaaGymaaqaaiaad2eaaaWccuaHZoWzgaqc
amaaDaaameaacaWGTbaabaGaam4yaiaadYgacaWGHbGaam4Caiaado
hacaGGSaGaam4DaiaacYcacaGGUaGaaiOlaiaac6caaaWccqGHuoar
daahaaadbeqaaiaadsfaaaWcceWGebGbaebadaqhaaadbaGaamiDaa
qaaiaad2gacaWGVbGaamizaiaadwgacaWGSbaaaaaakiabg2da9iaa
dwgadaahaaWcbeqaaiabgs5aenaaCaaameqabaGaamiDaaaaliqbes
7aKzaajaWaa0baaWqaaiaadsfaaeaacaWGJbGaamiBaiaadggacaWG
ZbGaam4CaiaacYcacaWG3bGaaiilaiaac6cacaGGUaGaaiOlaaaaaa
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That is an interpretation of the hedonic time dummy mannequin which lets us consider the change in time dummy coefficients as some measure of change in common costs that’s quality-adjusted to mirror adjustments within the pattern technique of automobiles traits.Word 11 Since we’re estimating value change from




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to



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Aggregation of month-to-month value kin

The month-to-month value kin constructed for every class are used alongside the class-make expenditures to roll-up as much as an mixture used car value motion, after which to an total buy of used passenger automobiles value motion by price-updating and summing expenditures.

The category-make value kin come from the time dummy coefficients, i.e.,









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vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
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aiilaiaac6cacaGGUaGaaiOlaaaakiabg2da9iaadwgadaahaaWcbe
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, and they’re used to cost replace a class-make expenditure, i.e.,




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4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
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, the place




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4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
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fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaadc
facaWGwbWaa0baaSqaaiaadshacaGGSaGaamyDaiaadohacaWGLbGa
amizaaqaaiaadogacaWGSbGaamyyaiaadohacaWGZbGaaiilaiaad2
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refers back to the used motorcar expenditures for a given class and make in interval



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vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36EF@



.

Total price-updated used automobiles expenditures are the sum throughout class-makes, so




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vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaadc
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amizaaqaaiaacYcacaGGUaGaaiOlaiaac6caaaGccqGH9aqpcqqHJo
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eu4Odm1aaSbaaSqaaiaad2gacaWGHbGaam4AaiaadwgaaeqaaOGaam
iuaiaadcfacaWGwbWaa0baaSqaaiaadshacaGGSaGaamyDaiaadoha
caWGLbGaamizaaqaaiaadogacaWGSbGaamyyaiaadohacaWGZbGaai
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. The general used automobiles value motion is then simply the sum of present interval price-updated class-make expenditures over the earlier interval’s corresponding sum, i.e.









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4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
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laaabaGaamiuaiaadcfacaWGwbWaa0baaSqaaiaadshacaGGSaGaam
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.


Areas for future enchancment

Statistics Canada is dedicated to knowledge accuracy, high quality and timeliness in measuring value change and producing a CPI that displays the expertise of Canadians. Statistics Canada is conscious of some limitations of the above method, primarily associated to the granularity of the out there knowledge. Every of those limitations is attributable to constraints in entry to detailed knowledge. Nevertheless, Statistics Canada is actively working to deal with these limitations:

  • Statistics Canada is within the strategy of buying extra granular knowledge on transacted automobiles in an effort to account for added traits and results resembling car trims within the high quality adjustment course of.
  • At present, there’s a one month lag within the value knowledge. Statistics Canada is working to enhance the timeliness of knowledge entry and processing, in an effort to produce essentially the most present estimates of month-to-month value change.

Information

Utilizing the strategies outlined above, value actions have been derived for used automobiles (Desk 1). Desk 1 comprises the decomposed value actions for brand spanking new and used passenger automobiles, in addition to a derived buy of passenger automobiles index primarily based on the proposed method.













Desk 1

New and used passenger automobiles, 12-month change, Canada

Desk abstract

This desk shows the outcomes of New and used passenger automobiles. The data is grouped by Reference Month (showing as row headers), Buy of latest passenger automobiles

(equal to the revealed buy of passenger automobiles index )

, Buy of used passenger automobiles (calculated utilizing proposed method) and Buy of passenger automobiles

(calculated utilizing proposed method, if launched to the CPI in June 2021), calculated utilizing % items of measure (showing as column headers).
Reference Month Buy of latest passenger automobiles

(equal to the revealed buy of passenger automobiles index )
Buy of used passenger automobiles
(calculated utilizing proposed method)
Buy of passenger automobilesDesk 1 Word 1

(calculated utilizing proposed method, if launched to the CPI in June 2021)
%
December 2021 +7.2 +18.3 +11.2
January 2022 +5.2 +19.7 +9.2
February 2022 +4.7 +20.6 +8.8
March 2022 +7.0 +24.5 +11.7

Used vs. new automobiles

Inside evaluation signifies that value change for used automobiles has, till lately, tracked new car value change to the extent that new automobiles served as an appropriate long run proxy. Costs of used automobiles started to diverge from these of latest automobiles within the fall of 2020 amid the COVID-19 pandemic.

Information desk for Chart 1





































Chart 1
New and used automobiles, Canada, January 2020 to March 2022Word 1


Desk abstract

This desk shows the outcomes of Information desk for Chart 1 New automobiles and Used automobiles, calculated utilizing 12-month share change items of measure (showing as column headers).

New automobiles Used automobiles
12-month share change
2020
January 2.27 1.03
February 2.24 2.78
March 1.03 0.73
April 1.88 0.68
Might 1.98 -0.49
June 2.82 0.20
July 3.29 2.80
August 2.19 3.62
September 2.69 3.36
October 2.94 3.33
November 2.01 5.30
December 2.45 5.62
2021
January 2.87 7.11
February 2.79 8.13
March 3.49 8.24
April 3.39 10.20
Might 4.94 11.73
June 4.10 14.44
July 5.52 12.61
August 7.13 13.82
September 7.22 13.57
October 6.13 15.28
November 6.03 14.98
December 7.21 18.32
2022
January 5.20 19.71
February 4.70 20.64
March 7.20 24.50



The introduction of used car costs with the 2021 CPI basket will safe towards future divergences in pattern from new car costs.

Comparability of used car costs in Canada and the US

Whereas comparable traits within the passenger car market, the place progress in used car costs is at the moment outpacing progress in new car costs, have been noticed in each nations, Canadian customers haven’t seen value will increase of the magnitude of these noticed in the US.

There are key market variations between the 2 nations. Given the completely different sizes and scopes of vehicle manufacturing in Canada and the US, value actions might differ between the 2 nations for particular person fashions. Not all used automobiles have proven the identical value actions prior to now 12 months, with some lessons of auto rising in value considerably greater than others. Pattern composition, which is, in flip, influenced by what class of automobiles customers are shopping for in Canada in contrast with the US, could also be contributing to the divergence in costs between the 2 nations. There may be additional potential for pattern composition results on the lowest degree of element due to variations by way of out there fashions in every nation.

Whereas each Statistics Canada and the Bureau of Labor Statistics (BLS) use a web family expenditures method to calculating used car weights, the weights are markedly completely different within the two nations. Passenger automobiles comprise 9.29% of the US CPI basket of products and providers, in contrast with 6.21% in Canada. Of that weight, used automobiles make up 4.14% of the basket in the US, in contrast with 1.84% in Canada’s 2020 CPI basket. These variations can also contribute to a special pre-pandemic seasonal sample in Canada in contrast with the US.

Chart 2 Used vehicles in Canada and the United States, 12-month price change, January 2020 to March 2022

Information desk for Chart 2





































Chart 1
New and used automobiles, Canada, January 2020 to March 2022

Desk abstract

This desk shows the outcomes of Information desk for Chart 2 United States and Canada, calculated utilizing 12-month share change items of measure (showing as column headers).

United States Canada
12-month share change
2020
January -2.00 1.03
February -1.20 2.78
March 0.10 0.73
April -0.80 0.68
Might -0.30 -0.49
June -2.70 0.20
July -0.90 2.80
August 4.00 3.62
September 10.30 3.36
October 11.50 3.33
November 10.80 5.30
December 10.00 5.62
2021
January 10.00 7.11
February 9.40 8.13
March 9.40 8.24
April 21.00 10.20
Might 29.80 11.73
June 45.30 14.44
July 41.60 12.61
August 31.90 13.82
September 24.40 13.57
October 26.40 15.28
November 31.40 14.98
December 37.30 18.32
2022
January 40.51 19.71
February 41.15 20.64
March 35.30 24.50



Latest market situations are possible additionally at play. Between Canada and the US, there have been vital variations within the scope and period of public well being measures launched to restrict the unfold of COVID-19, in addition to the financial helps supplied. Whereas periodic stimulus cheques have been despatched to People, the Canadian authorities supplied extra constant, focused helps to those that had misplaced employment on account of the pandemic. Notably, the most important spike in used automobiles costs in the US occurred between April and June 2021, which coincided with the third stimulus cost, tax refund seasonWord 12 and an finish to public well being measures in lots of jurisdictions. An equal motion was not noticed in Canada, which remained underneath some type of lockdown in a lot of the nation till July 2021. Lockdown insurance policies themselves might have additionally performed a task in shifting demand: as costs for used automobiles surged in the US through the spring of 2021, Canadians, who have been re-entering lockdown measures in a number of provinces, decreased their mobility charges to a better extent than their American counterparts.Word 13

There are additionally two variations within the methodological approaches utilized by the 2 nations:

  • Statistics Canada makes use of a hedonic mannequin, whereas the US BLSWord 14 makes use of choice price adjustment primarily based on data from automotive dealerships for high quality adjustment;
  • Completely different value knowledge sources are used, with Statistics Canada utilizing transaction knowledge from level of sale and the BLS utilizing evaluation valuation knowledge from an business information.

Impression on headline CPI

An analytical sequence was calculated to evaluate the affect of introducing used car costs on the headline CPI. Given the burden of used automobiles (1.84%) within the 2020 CPI basket, if used car costs had been launched with the June 2021 CPI, coinciding with the final basket replace, the headline CPI for March 2022 is estimated to have been 0.2 share factors larger, in contrast with the revealed CPI (+6.7%).

2021 CPI basket

The introduction of the 2021 CPI basket will mark the implementation of the above enhancements to the calculation of the acquisition of passenger automobiles index and the introduction of used car costs to the CPI. Right now, the used automobiles index can be added to the CPI classification as a broadcast mixture:

  • Transportation

    • Personal transportation

      • Buy, leasing and rental of passenger automobiles

        • Buy and leasing of passenger automobiles

          • Buy of passenger automobiles

            • Buy of vehicles (2013=100)Word 15
            • Buy of vans, vans and sport utility automobiles (2013=100)Word 15
            • Buy of latest passenger automobiles (2022-04=100)Word 16
            • Buy of used passenger automobiles (2022-04=100)Word 16

As a result of the CPI is a non-revisable index, used car costs are proposed to be launched with the Might 2022 month-to-month value change with no degree adjustment for historic value adjustments. This method is according to the best way different merchandise have been included within the CPI resembling mobile providers, digital units and hashish. This method follows worldwide finest practices in addition to the Client Value Index Handbook (Chapter 7) and suggestions by Statistics Canada’s Value Measurement Advisory Committee. Though the sort of ‘catch-up’ adjustment would account extra totally for the affect of the current will increase in Canadian used car costs within the CPI, it could be problematic for indexation and escalation of contracts that took impact prior to now.

In abstract

As of the introduction of the 2021 CPI basket, a brand new method for measuring value change in used automobiles is really helpful to exchange the earlier technique of measuring used automobiles value change by proxy.

Statistics Canada continues to work with value consultants, nationwide statistical organizations and different companions to make sure knowledge and strategies used within the calculation of the CPI are aligned with worldwide requirements and finest practices. The company is continuous to observe costs for used automobiles and buying new knowledge sources for the measurement of the acquisition of passenger automobiles index ensures the continued accuracy and relevance of the CPI.

For added data or to offer feedback on the proposed enhancement, customers might contact the Client Costs Division at statcan.cpddisseminationunit-dpcunitedediffusion.statcan@canada.ca.

References

Akay, E., Bolukbasi, O., & Bekar, E. (2018). Sturdy and resistant estimations of hedonic costs for second hand vehicles: an utility to the Istanbul automotive market. Worldwide Journal of Economics and Monetary Points 8(1), 39-47.

Bode, B., & van Dalen, J. (2001, April 2-6). High quality-corrected value indexes of latest passenger vehicles within the Netherlands, 1990-1999 [Presentation]. Worldwide Working Group on Value Indices, Canberra, Australia.

Cheng, J. (2015, Might 20-22). High quality adjustment of second-hand motorcar – utility of hedonic method in Hong Kong’s shopper value index [Presentation]. Ottawa Group on Value Indices, Ottawa, Canada.

de Haan, J., & Hendriks, R. (2013, November 28-29). On-line knowledge, fastened results and the development of high-frequency value indexes [Presentation]. Financial Measurement Group Workshop, Sydney, Australia.

de Haan, J., & Krsinich, F. (2018). Time dummy hedonic and quality-adjusted unit worth indexes: do they actually differ?, Evaluation of Revenue and Wealth 64(4), 757-770.

Krsinich, F. (2014). High quality adjustment within the New Zealand Shoppers value index. In S. Forbes & A. Victorio,The New Zealand CPI at 100. Historical past and Interpretation. Victoria College Press.

Larsen, M. (2011, March 25). Experimental use of hedonics for brand spanking new vehicles within the Danish HICP [Presentation]. Ottawa Group on Value Indices, Ottawa, Canada.

Nielsen, M. (2018, Might 7-9). High quality adjustment strategies when calculating CPI [Presentation]. Assembly of the Group of Consultants on Client Value Indices, Geneva, Switzerland.

Reis, H. & Silva, J. (2002). Hedonic value indexes for brand spanking new passenger vehicles in Portugal (1997-2001). Financial Bulletin and Monetary Stability Report Articles and Banco de Portugal Financial Research, Financial institution of Portugal, Economics and Analysis Division.

Requena-Silvente, F., & Walker, J. (2006). Calculating hedonic value indices with unobserved product attributes: an utility to the UK automotive market. Economica 73(291), 509-532.

Tomat, G. (2002). Sturdy items, value indexes and high quality change: an utility to vehicle costs in Italy, 1988-1998. European Central Financial institution Working Paper.

Varela-Irimia, X. (2014). Age results, unobserved traits and hedonic value indexes: the Spanish automotive market within the Nineties. SERIEs: Journal of the Spanish Financial Affiliation 5(4), 419-455.



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