CSIS Discussion Paper No.125 (University of Tokyo)

Separating the Age Eﬀect from a Repeat Sales Index:
Land and Structure Decomposition
SK Wong∗ , KW Chau† , K Karato‡ , C Shimizu§
December 10, 2013

Summary
Since real estate is heterogeneous and infrequently traded, the repeat sales model has
become a popular method to estimate a real estate price index. However, the model
fails to adjust for depreciation, as age and time between sales have an exact linear
relationship. This paper proposes a new method to estimate an age-adjusted repeat
sales index by decomposing property value into land and structure components. As
depreciation is more relevant to the structure than land, the property’s depreciation rateshould depend on the relative size of land and structure. The larger the land component,
the lower is the depreciation rate of the property. Based on housing transactions datafrom Hong Kong and Tokyo, we ﬁnd that Hong Kong has a higher depreciation rate
(assuming a ﬁxed structure-to-property value ratio), while the resulting age adjustment
is larger in Tokyo because its structure component has grown larger from the ﬁrst to
second sales.

1

Introduction
A price index aims to capture the price change of products free from any variations inquantity or quality. When it comes to real estate, the core problem is that it is heterogeneousand infrequently traded. Mean or median price indices are simple to compute, but properties
sold in one period may diﬀer from those in another period. To overcome this problem, tworegression-based approaches are used to construct a constant-quality real estate price index(Shimizu et al. (2010)).
One is the hedonic price model, which speciﬁes the property attributes to be controlled
and uses time dummies to capture price changes over time. This method is often employedwhen data on property attributes are readily available, although omission of unobserved
attributes or unique features of a property could lead to estimation bias. The other approachis the repeat sales model advanced by Bailey, Muth and Nourse (1963). It controls for∗ The University of Hong Kong
† The University of Hong Kong
‡ University of Toyama§ Correspondence: Chihiro Shimizu, Reitaku University & The University of British Columbia, Kashiwa,
Chiba 277-8686, Japan. E-mail: cshimizu@reitaku-u.ac.jp.

1quality variations, including the uniqueness of each property, by tracking the price changeof properties that have been sold twice. This method is most useful when repeat sales areabundant.1
The repeat sales model is sometimes challenged for making a fundamentally incorrect
assumption: quality could change over time even for the same property. For example, a
repeat sales index may wrongly capture the price increase of a property due to the additionof a bedroom. But this is not the failure of the repeat sales model. If the change is known,
the repeat sales model can be easily modiﬁed to account for it (Bailey, Muth and Nourse,
1963, p.935). If the change is unknown, both the hedonic and repeat sales models would
suﬀer from the same problem. The real issue concerning the repeat sales model is that it isincapable of controlling for depreciation:“Unfortunately, a depreciation adjustment cannot be readily estimated along with the priceindex using our regression method. . . Assuming that properties depreciate at a constant rate
per unit time, . . . the x matrix [regressors] is singular. . . In applying our model, therefore,
additional information would be needed in order to adjust the price index for depreciation.”(Bailey, Muth and Nourse, 1963, p.936).
Unlike any other quality change, the increase in age of a property is always identical to thetime elapsed between two sales. Including both age and time diﬀerences in the repeat sales
regression gives rise to perfect collinearity. Not only are the age and time eﬀects inseparable,
but estimation also becomes impossible. Therefore, in many applications, the depreciation
problem is simply ignored, resulting in a repeat sales index that is biased downward.
Following Bailey, Muth and Nourse’s (1963) suggestion, several attempts were made to ﬁnd
the “additional information” needed to solve the identiﬁcation problem. Palmquist (1980)
proposed a two-stage method: ﬁrst obtain an independent estimate of the deprecation ratefrom the hedonic price model and then add it to back to the repeat sales index. Case andQuigley (1991), Hill, Knight, and Sirmans (1997), and Englund, Quigley and Redfearn (1998)
oﬀered a similar idea but combined the hedonic and repeat sales regressions into a hybrid
model for joint estimation. In both cases, cross-sectional diﬀerences in property age serve as
the additional information to identify the depreciation rate. Chau, Wong, and Yiu (2005)
found another instrument to separate the age and time eﬀects. They treated depreciation
as a discounted cash ﬂow problem and showed that for leasehold properties, the age eﬀect
varies inversely with real interest rates. They also proved that simply converting the age
variable into dummies does not work. For instance, Cannaday, Munneke, and Yang (2005)
had to drop two age dummies arbitrarily in order to avoid perfect collinearity, although ahigh degree of collinearity still remains.2
This paper proposes a new method to estimate an age-adjusted repeat sales index by
1 Diewert(2007) summarized advantages and disadvantages in Repeat Sales measure and Hedonic measure. And, Shimizu et al. (2010) compared Hedonic Index, Traditional Repeat Sales Index, Case-Shiller
type Repeat Sales Index, Age adjusted Repeat Sales Index and new type Hedonic Index or Rolling Window
Hedonic Index using Tokyo data..
2 Shimizu, Nishimura and Watanabe(2010) and Karato, Movshuk and Shimizu (2010) also proposed new
estimation method to separate the age and time eﬀects. Karato, Movshuk and Shimizu (2010) tried toseparate age eﬀect , time eﬀect and cohort eﬀect using semi -parametric approach; Generalized Additive
Model or GAM.

2

decomposing property value into land and structure components.3 The key idea is that
depreciation is more relevant to the structure than land. Since a property is the sum ofboth components, its depreciation rate depends on the relative size of land and structure.
The larger the land component, the lower is the depreciation rate of the property. Section
2 derives the relationship between the depreciation rate and a structure-to-property priceratio, and shows how the ratio can be used as additional information to separate the age
eﬀect from a repeat sales index. Moreover, similar to Chau, Wong, and Yiu (2005), the age
eﬀect is allowed to be non-linear by adopting a ﬂexible functional form.
The proposed method is then used to estimate the repeat sales indices and depreciation
rates for the housing markets in Hong Kong and Tokyo. While both are densely developed
cities with high property prices, we expect them to have diﬀerent deprecation patterns
because the maintenance of condominiums (in particular the common areas) is a functionof the institutions governing the rights and duties of unit owners. Speciﬁcally, buildings inJapan have better upkeep in order to withstand earthquakes and fulﬁll the relevant legal(Building Standards Law) and insurance requirements. Section 3 will present the data andestimation results. Section 4 will further check the robustness of the estimates using a hybrid
and hedonic model. The last section is the conclusion.

2

An age-adjusted repeat sales model

Suppose property value (P ) is the sum of land value (L) and structure value (S).4 The
value of a new property is:
P (0) = L + S(0)(1)

where the number in brackets represents the building age. Aging reduces the value of thestructure value but not the land. Therefore, L is not a function of building age.
After A periods, the property reaches age A. Assuming a stable economy where L andS(0) remain unchanged, the property value will decline solely due to aging:
P (A) = L + S(0) × (1 − δ A)

(2)

where δ is the depreciation rate of the structure. Equation 2 can be easily generalized toallow for a non-linear depreciation pattern. One way is to replace A with a more ﬂexible
function g(A):P (A) = L + S(0) × [1 − δ g(A)](3)

3 Diewert and de Haan (2011) and Diewert and Shimizu (2013) proposed a new hedonic model to decompose the Land and Structure component.
4 Thorsnes (1997; 101) assumed that a related supply side model held instead of equation (2). He assumed
that housing was produced by a CES production function H(L, K) = [αLρ + βK ρ ]1/ρ where K is structuret)quantity and ρ ̸= 0; α > 0; β > 0 and α + β = 1. He assumed that property value Vnt is equal to pt H(Ltn , Kn
where pt ,ρ , α and β are parameters to be estimated. However, Diewert’s builder’s model assumes that the
production functions that produce structure space and that produce land are independent of each other.

3

Take g(A) = Aλ as an example. δ represents the initial depreciation rate of the structure.
The depreciation rate would rise with age if λ>1, decrease with age if 0 < λ < 1, and remain
at δ if λ = 1.
According to Equation 3, as long as land value is not zero, the property would depreciate
at a rate lower than the structure. Speciﬁcally, the depreciation rate of P depends not only
on δ but also on the ratio of new structure value to new property value:(P (A) − P (0))/(P (0)) = −S(0)/P (0) δ g(A)(4)

Equation 4 has two important implications. First, given the same δ (e.g. building technology) for all structures, properties in a high land value area would depreciate more slowly
than those in a low land value area. Second, even if the structure depreciates at a constantrate, the property’s depreciation rate is likely to be time-varying because structure and land
values do not move at the same pace.5 The property would depreciate more (less) whenconstruction costs increase (decrease) faster than land costs.
The second implication provides us with a new perspective to resolve the perfect collinearity between age and time in the repeat sales method. This new perspective is diﬀerent fromthe previous attempts that aimed to break down the linear relationship between age andtime by introducing “error” to the age variable. For instance, Cannaday, Munneke, andYang (2005) converted age into a set of dummy variables and dropped two such dummies toavoid perfect collinearity with time by making an arbitrary assumption that the depreciation
rate is the same for certain ages. Our approach to the problem relies on the use of external
information to disentangle the eﬀects of age and time. Chau, Wong, and Yiu (2005) also
followed this approach to derive the age eﬀect as a function of real interest rates, but theirmodel is more applicable to leasehold interests. This paper proposes a more general framework to separate the age and time eﬀects based on the relative size of land and structure.
The ratio of new structure value to new property value is the external information we relyupon.
To motivate our age-adjusted repeat sales model, we can start with a hedonic price equation and supplement it with the age term from Equation 4:
lnPit = Xi β + αt − δ Rt g(Ait ) + ϵit(5)where Pit is the price of property i at time t; Xi is a vector of property attributes excluding
building age; β is a vector of the implicit price of the attributes; αt is the property priceindex at time t; and ϵit is a random error.
The third term on the right captures the age eﬀect, where Rt is the ratio of construction
cost to new property price at time t. The smaller the Rt , the larger the non-depreciable
land component, hence the smaller the age eﬀect on property prices. This generalizes thestandard hedonic model, which assumes Rt = 1.
5 In general, the inelastic supply of land makes land value more volatile and more sensitive to economic
shocks than building value.

4

To allow for a non-linear depreciation pattern of the structure, the age function, g(Ait ),adopts the Box-Cox transformation as shown in Equation 6.6 If λ = 1, the structure
depreciates at a constant rate δ. If λ > 1, the structure depreciates faster as its age increases.
If λ < 1, the structure depreciates more slowly as its age increases. In all cases, δ is expected
to be positive.
(g(Ait ) =Aλit − 1λ)(6)Given that property i is sold twice at time s and t (where t > s) and there is no change inproperty attributes between the sales, our age-adjusted repeat sales model can be derived
from the (t − s)th diﬀerence of Equation 5:
(
ln

Pit
Pis

)
= (αt − αs ) − δ [Rt g(Ait ) − Rs g(Ais )] + (ϵit − ϵis )(7)

A key advantage of the repeat sales model is that it is less vulnerable to omitted variablebias – as long as the unobserved attributes do not change between sales, they will be cancelled
out and do not aﬀect estimation of the change in price index, αt − αs . Assuming ϵit − ϵisis normally distributed, the parameters in Equation 7 can be estimated by the maximum
likelihood method. We call our new model in Equation 7 the “Age-R model”, which spells
out the key feature that building age is interacted with the structure-to-property value ratio.
The traditional repeat sales model without the age term (called the BMN model) will also
be estimated and compared against the Age-R model.
Again, if the Age-R model is correct, δ should be positive. Even if the structure depreciates
at a constant rate (i.e. λ = 1), the Age-R model is still free from perfect collinearity because
the structure-to-property value ratio is unlikely to be ﬁxed over time, especially upon thearrival of economic shocks. A time-varying structure-to-property value ratio indeed gives us
a new angle to interpret the age eﬀect. Consider a property of age 10 in 2012. For simplicity,let λ = 1, δ = 2%, and R2012 =50%. Other things being equal, in 2013, aging by one year
should depreciate the structure by 2% and the property by 1%. However, the structure-toproperty value ratio may change. If R2013 is 60%, the property would actually depreciateby 1.2% – the greater depreciation was aquired as if the property had reached age 13 in one
year. By contrast, if R2013 is 40%, the property would have looked as young as age 9, as theproperty would only depreciate by 0.8%. Therefore, the structure-to-property value ratio
eﬀectively breaks the linear relationship between age and time diﬀerences between sales,making it possible to separate the age eﬀect from a repeat-sales index.

6 Box-Cox transformation requires that the age variable is strictly positive.

5

Table 1: Descriptive statistics of the repeat sales data in Hong Kong and Tokyo
Hong Kong IslandstMean
Std.Dev.
Minimum
Maximum
N=190,890

nd

st

nd

Price at 1 sale
(HK$ million)

Price at 2 sale
(HK$ million)

Age at 1 sale(quarters)

Age at 2 sale(quarters)

4.02675.32570.101
184.8

4.72996.9459
0.101
338

63.32
43.11
1
230

81.72
43.41
2
246

Tokyo
st

Mean
Std.Dev.
Minimum
Maximum
N=36,212

nd

st

nd

Price at 1 sale
¥10,000

Price at 2 sale
¥10,000

Age at 1 sale(quarters)

Age at 2 sale(quarters)

3,998.36
3,180.38
34
80,000.00

3,402.43
2,582.23
968,000.00

61.45
34.73
1
192

76.78
36.86
2
201

3

Estimated results of the age-adjusted repeat sales model

3.1

Data

Housing sales in Hong Kong and Tokyo are used to estimate the repeat sales property
price indices for the two cities. The time period runs from 1993Q1 to 2012Q2. 190,890
pairs of repeat sales in Hong Kong Island were collected from the EPRC database, and36,212 pairs in the special 23 wards of Tokyo from a weekly magazine Shukan Jutaku Joho
(Residential Information Weekly) published by Recruit Co., Ltd., which is one of the largest
vendors of residential lettings information in Japan.7 Most of them are sales of condo units.
The average sale price in Hong Kong is HK$4-5 million (US$600,000), whereas the average
price in Tokyo is 30-40 million (US$400,000). The average building age in the repeat sales
sample is similar in both places: about 15 years old in the ﬁrst sale and 20 years old in thesecond sale. The descriptive statistics of their sale prices and building ages are shown in
Table 1.
An important variable for the age-R model is Rt, the ratio of construction cost to newproperty price. The average construction cost of a new residential building in Hong Kong is
obtained from a major quantity surveying consultancy Rider Levett Bucknall (RLB), which
publishes the RLB Hong Kong Cost Report every quarter. In Japan, the construction cost
is obtained from the Ministry of Land, Infrastructure, Transportation and Tourism (MLIT).
Estimating the price of new property is problematic. New properties constitute only asmall part of the existing stock and their prices can vary widely depending on their locationand developer. This means the use of new property prices could introduce a lot of noise to Rt .
7 Recruit Co., Ltd. provided us with information on contract prices for about 24 percent of all listings.
Using this information, we were able to conﬁrm that prices in the ﬁnal week were almost always identicalto the contract prices; see Shimizu et al. (2012).

6

Table 2: Maximum likelihood estimates of the age eﬀect
Hong Kong Islandδ
0.1029*λ0.5547*
Log-likelihood-19,185.82
Note: * significant at the 1% levelTokyo
0.1035*0.3212*
11,097.23

We propose using the average property price in the entire housing market instead. Although
the average property price is a biased estimate of the new property price due to aging, thebias is likely to be stable as new properties will be added to and old properties removed fromthe existing stock. With a large sample of transactions in the existing stock, the averageage of transacted properties could remain more or less the same over time. Therefore, any
underestimation from the use of all transacted properties could be compensated by a smaller
δ in Equation 7. The model for estimating the age eﬀect on property (rather than structure)is unaﬀected by this, but δ should be interpreted as a lower-bound estimate of the structure’s
depreciation rate.
The ratio of construction cost to average property price is shown in Figure 1. As expected,
the properties in Hong Kong and Tokyo have a larger portion for land than structure. On
average, the structure component only comprises 24% of the property value in Hong Kongand 37% in Tokyo. Note that these are upper-bound estimates because of the use of average
prices of all transacted properties instead of just new property prices.

3.2

Results of Age-R

The maximum likelihood estimates of the parameters δ and λ in Equation 7 are reported
in Table 2, whereas the price index results will be reported separately later. With Box-Coxtransformation, the marginal eﬀect of age on log price is:
E[lnP (A)] = −δ Rt A(λ−1) f or A > 0(8)

As expected, δ is positive and signiﬁcant at the 1% level, conﬁrming that the age eﬀect
on property price is negative. Moreover, the magnitude of δ in Hong Kong is almost thesame as that in Tokyo. At ﬁrst sight, according to Equation 8, this can be interpreted asthat the structure shares a similar initial depreciation rate in both cities. But as mentioned
before, the magnitude of δ could be biased downward due to the use of average price of alltransacted properties in estimating Rt .8 So, δ alone cannot tell if the two cities really havea similar initial depreciation rate at the structure level; rather, δ has to be combined with
Rt in order to compare the depreciation rates at the property level.
While the magnitude of δ appears to be large, it is only the initial depreciation rate. Since
λ is smaller than 1 and signiﬁcant, the depreciation pattern is conﬁrmed to be non-linear:
8 The downward bias in the δ estimate reinforces our prediction that the true δ should be positive.

7

the structure depreciates more slowly as its age increases. We can examine the depreciation
pattern of property by considering δ and λ together. Assuming a property is worth $100 at
age 1 and the ratio of construction cost to property price stays at its average level (Rt ), theexpected price of property at age A is:()
EP (A) = exp ln P (1) − δRt(9)Figure 2 plots the depreciation pattern calculated from Equation 9. Properties in Tokyo
depreciate faster than those in Hong Kong initially, but the depreciation rate in Hong Kong
picks up soon and the rates reach a “break-even” at about Age 20. Over 50 years, property
value has depreciated by 29% in Hong Kong and 26% in Tokyo – by that time, the property
value is mostly derived from the land. The cumulative depreciation can be converted toan average annual depreciation rate of 0.58% and 0.52%, respectively. Bear in mind thatthese estimates are based on a constant structure-to-property value ratio. If the structurecomponent gets larger (smaller), the depreciation rate will go up (down) accordingly.
The higher average depreciation rate in Hong Kong could be caused by under-maintenanceof old buildings. In Hong Kong, the multiple ownership system for condominiums has createda lot of problemd in maintaining and managing the common areas (e.g. external walls,
lifts, staircases, and lobbies). Court cases on building management issues and accidents
due to building neglect are not uncommon. Legislation on mandatory building inspection
and maintenance did not exist until recently. In contrast, buildings in Japan are generally
better maintained because they have to withstand earthquakes and fulﬁll the relevant legal(Building Standard Law) and insurance requirements.
The coeﬃcients of the time dummies from the Age-R model in Equation 7 provide us withthe age-adjusted price indices for Hong Kong and Tokyo. As shown in Figures 3 and 4, ourage-R indices are above the BMN indices in both Hong Kong and Tokyo, suggesting thatproperty returns are higher after age adjustments. This result is intuitive, as the return
captured by the BMN index is after depreciation.
Descriptive statistics of the indices are reported in Table 3. In Hong Kong, the mean
return is 1.63% per quarter based on the age-R index and 1.56% based on the BMN index.
The diﬀerence is 0.07% per quarter. In Tokyo, the mean return is -0.67% per quarter basedon the age-R index and -0.97% based on the BMN index. The diﬀerence is 0.30% perquarter. The age adjustment is larger for Tokyo than Hong Kong because Tokyo has alarger component of structure relative to land. Moreover, the structure-to-property value
ratio in Tokyo is, on average, larger in the second sales than the ﬁrst sales, whereas the ratio
in Hong Kong is smaller in the second sales. This implies a bigger age adjustment for the
Tokyo housing market.

8

Table 3: Descriptive statistics of the BMN and Age-R indicesHong Kong Island

Tokyo

Mean return per quarter

1.63%-0.67%

Return volatility per quarter

5.26%

1.71%

1.56%
7.05%

-0.97%
2.18%

0.07%

0.30%

－
－

-0.53%
2.21%－－-0.52%
2.06%－-0.14%－-0.15%

Age-R

BMN
Mean return per quarter
Return volatility per quarter
Age adjustment in property return(per quarter)
HybridMean return per quarter
Return volatility per quarter
Hedonic
Mean return per quarter
Return volatility per quarter
Diffremces in property return with Hybrid(per quarter)
Diffremces in property return with Hedonic(per quarter)

4

Comparison with other traditional models

4.1

Hybrid Model and Hedonic Model

Here, we will compare the new Age-R index with indices based on the hedonic and hybrid
models. As discussed in Introduction, the hedonic and hybrid models allow us to use a larger
dataset comprising of both single and repeat sales, but they are more vulnerable to omitted
variable bias. Therefore, the purpose of this section is to check whether the three models –based on diﬀerent assumptions and samples – will give similar estimates or not. If diﬀerent
methods share similar results, convergent validity is achieved. But it is not our purpose tojudge which model is right or wrong.
In the Tokyo data, there is suﬃcient attribute data (characteristics) about condominiums
to estimate a hedonic function. Thus, the hybrid method that was proposed to modify therepeat sales method proposed by Case and Quigley (1991) may also be applied. Here, toappraise the new Age-R proposed in the preceding section, we decided to compare the house
price indices estimated by the hedonic method and the hybrid method using Tokyo data.
First, we will examine the hedonic model and hybrid model.
Hill, Knight, and Sirmans (1997) distinguished the time eﬀect and age eﬀect by reﬁning
Case and Quigley’s (1991) hybrid method (hedonic and repeat sales method joint model
estimation). The hedonic regression model is deﬁned as:

9′

yit = ln Pit + Xi β + δAit + αt + ϵit(10)where Pit is the price of property i at time t; Xi is a vector of property attributes excluding
building age; β is a vector of the implicit price of the attributes; is age eﬀect; is i th property
age at time t; αt is the ﬁxed time eﬀect at time t; and ϵit is a random error.
From sample i = 1, ..., N , let us take as a property that is transacted twice. The repeat
sales regression model may be written as follows:
Yi = yit − yis = lnPit= τi δ + αt − αs + υi
Pis(i = 1, 2, ..., NR )(11)

Here,τi = Ait − Ais is the diﬀerential of the building age at time s and time t. If all
samples for (10) and (11) are pooled, the following regression model is obtained:
(

y
Y

)

(
=

X

A

d

0

τ

D

)


(
)
β
ϵ


 δ +
υα(12)where d and D are the matrices of time dummy variables. A distinctive feature of thisapproach is that, by pooling a hedonic regression model and repeat sales regression model,
the linear relationship between τi = Ait − Ais and D is disrupted and makes it possible toestimate simultaneously the age eﬀect δ and time eﬀect α.
The relationship between log price and the age term are linear in the hedonic equation of
the hybrid model. To compare the age eﬀects with the Box Cox transformed Age-R model,
we estimate the hedonic regression model with the Box Cox transformed age term as follows:′

ln Pit = Xi β + δ

4.2

Aλit − 1+ αt + ϵit
λ(13)

Estimated results of Hybrid and Hedonic Model

Let us compare the data collected for the hedonic estimate in Tokyo wards with the data
used for estimation of repeat sales price (Table 4).
Although the number of repeat sales samples in the estimation period was 36,212, hedonic
samples increased to 375,374, or more than about 10 times. That is, by the repeat sales
method, only 1/10 of the transactions that occurred from 1993 to the 2nd quarter of 2012
could be utilized. If such a repeat sales sample is a typical random sample, it will besatisfactory, but if there is a bias in the sampling, there will be a bias in the index that is
estimated.
First, looking at the average for prices, whereas this is 36,370,000 yen in the data for thehedonic model, in the repeat sales sample, the price of the ﬁrst transaction is 39,980,000
yen, and that of the second is 34,020,000 yen. In other words, the average for the hedonic
sample is between the average for the ﬁrst transaction and the average for the second transaction. Looking at the number of post-construction years (A), whereas the average for the10Table 4: Descriptive statistics of the repeat sales data and hedonic data
Hedonic Sample

Mean

Std. Dev.

Min

Max

Price (\10,000)

3,637.92

2,684.75

185

112,000

Age (quarters)

68.45

41.93

0

352

Price at 1st sale (\10,000)

3,998.36

3,180.38

34

80,000Price at 2nd sale (\10,000) 3,402.43

2,582.23

968,000Age at 1st sale (quarters)

61.45

34.73

1

192

Age at 2nd sale (quarters)

76.78

36.86

2

201

N=375,374
Repeat Sales Sample

N=36,212

hedonic sample is the 68th quarter, in the repeat sales sample, the average number of postconstruction years for the ﬁrst transaction was the 61st quarter, and for the second, it was
the 76th quarter. Thus, similarly to prices, the average number of post-construction years
is between the average for the ﬁrst transaction and the average for the second transactions.
This means that there is no signiﬁcant bias in the repeat sales sample.
Using the hedonic database thus constructed, we estimated the hedonic function together
with the hybrid technique.
Here, comparing the aggregates for the hybrid model and the hedonic model based on
Table 5, and comparing estimates, the coeﬃcient for ﬂoor space is 1.098 for the hybrid
method and 1.091 for the hedonic method, so there is not much diﬀerence. As for distanceto the nearest station, this is -0.0071 by the hybrid method, and -0.009 by the hedonic
method, whereas for times to the CBD (Center of the Business District), i.e., the time toTokyo Station, it is -0.0257 for the hybrid method and -0.006 for the hedonic method.

4.3

Comparison Age adjusted RS with traditional indexes

Figure 5 shows the estimated price index computed from the hybrid model and hedonic
model, compared with BMN repeat sales price index and Age-R. The hybrid price index
and hedonic price index almost overlap; the new repeat sales price index (Age-R) follows asimilar trend but is less volatile. In particular, the BMN repeat sales price index is strongly
biased downwards, and by analogy, there is a depreciation bias that was clear from thisseries of studies. On the other hand, the Age-R proposed in this research, as comparedto the BMN repeat sales price index, is largely shifted so that it approximates the hedonic
price index and hybrid price index. As a result, the aggregation problem due to depreciation
bias is largely improved. This result is extremely signiﬁcant, although not as much as in the
hybrid method.
However, even if we adjust for depreciation, a certain discrepancy remains between AgeR and the hedonic/hybrid price index. This discrepancy may be due to the fact that the11Table 5: Estimated results of Hedonic and Hybrid
VariableCoef.

Age (δ)Box Cox (λ)
Log (Floor Space)
Distance to Nearest Station
Distance to Tokyo Sta.
Building Construction Dummy
Ward DummyTime Dummies
const.
Number of obs.
R-squared
Adjusted R-squared
S.E. of regression
Log likelihoodt -value-0.0046－
1.1021-0.0071-0.0257
Yes
Yes
4.860

-219.809－
568.018-42.0458-56.8187

416.450

108,969
0.998
0.998
0.189
26915.9

Coef.

t -value-0.0192
0.63531.0890-0.0089-0.0058
Yes
Yes
Yes
5.041

-72.135
182.6801330.660-121.360-41.792

659.798

375,374
0.889
0.889
0.182
106200.0

samples were diﬀerent or the choice of variables in the hedonic equation. In other words,
although depreciation can be controlled for in both the Age-R and hedonic/hybrid models,the potential sample selection bias inherent in the repeat sales price method and the potential
omitted variable bias in the hedonic-based method cannot be resolved.
Moreover, if we compare the magnitudes of the depreciation bias and sample selection
bias, the bias due to depreciation is much larger. In other words, the Age-R evidently
resolves the major part of the bias inherent in the repeat sales price method.
Next, Figure 6 compares the age indexes using aggregate parameters that correspond to
the number of post-construction years (A) estimated in the hybrid model, hedonic model,
and the Age-R proposed in this research. Figure 7 and Table 6 show the marginal eﬀects.

Table 6: Average depreciation rate
Age adjusted RS Age adjusted RS
(HK)
(TKO)

Hedonic(TKO)

Hybrid(TKO)over 50 yrs-0.58%-0.52%-1.14%-1.18%

over 40 yrs-0.65%-0.59%-1.29%-1.27%

0-10 yrs-1.09%-1.22%-2.36%-1.51%

10-20 yrs

-0.64%-0.53%-1.47%-1.51%

20-30 yrs

-0.51%-0.38%-1.23%-1.51%

30-40 yrs

-0.44%-0.30%-1.09%-1.51%

40-50 yrs

-0.40%-0.26%-1.00%-1.51%

12

As can be seen from Figure 1, the proportion accounted for by house construction costs
in Japan was about 20% in 1991 during the country’s economic bubble years, but recently
it increased to about 35%. Considering that the average between 1991-2012 is also about
33%, it may be said that the average for 2012 is around the average level for the last 20
years. In other words, considering there is no depreciation of land, regardless of how muchhouse prices have depreciated, the maximum value of this depreciation is 35%, and 65% of
the price has been maintained.
Looking at Figure 6 from this assumption, the result estimated from the hybrid model andthe hedonic model is that 65% of house prices are factored in 15 years post-construction,and they will be at the 40% level at 50 years post-construction. This means, since land
suﬀers no depreciation, that the value of property is estimated to be negative. On the other
hand, looking at the result estimated from Age-R, the value at 50 years post-construction is70%, so that although it depreciates close to 65% that includes land values, a small property
value remains.
From this result, it is clear that if we calculate the rate of depreciation by the hedonic or
hybrid method taking land and property together, it will be largely over-estimated.
Comparing the marginal eﬀects for depreciation (Figure 7, Table 6), in the 50 year-average,
the rates of depreciation estimated from the hedonic method and hybrid method are about
twice the rate of appreciation estimated from the new Age-R method proposed here, andthe discrepancy becomes larger, the larger the number of post-construction years.
In section 3, we suggested that property returns are higher after age adjustments.
In addition to this analysis, we compared age R with hybrid and hedonic indexes in Tokyo.
The mean return of Age R is smaller than 0.14 and 0.15 % than that of Hybrid and Hedonic
indexes(Table 3). Age R is overcoming depreciation problem in estimating Repeat Sales
regression, while the returns from the hedonic and hybrid indexes are higher probably due
to the over-adjustment of depreciation as discussed above.

5

Conclusion

Since real estate is heterogeneous and infrequently traded, the repeat sales model has
become a popular method to estimate a property price indexes. However, the model fails toadjust for depreciation, as age and time between sales have an exact linear relationship. This
paper proposes a new method to estimate an age-adjusted repeat sales index by decomposing
property value into land and structure components. As depreciation is more relevant to thestructure than land, the property’s depreciation rate should depend on the relative size ofland and structure. The larger the land component, the lower the depreciation rate of theproperty. Based on housing transactions data from Hong Kong and Tokyo, Hong Kong
has a higher depreciation rate (assuming a ﬁxed structure-to-property value ratio), while
the resulting age adjustment is larger in Tokyo because its structure component has grownlarger from the ﬁrst to second sales.
To evaluate the Age-R proposed here, we input the Tokyo data into both the hybrid model13

and the hedonic model, and then compared the estimated price indexes. This comparison
revealed that newly proposed Age-R cancels out the inherent depreciation problem of therepeat sales model. Although the sample selection bias of the repeat sales model is expected
to remain, the magnitude of this problem was shown to be far smaller than that of the bias
resulting from depreciation. We also compared the estimated depreciation rate here withthe depreciation rate estimated by using the hedonic model and hybrid model. Housing
consists of land and structure, but the depreciation that occurs only applies to the structure.
When the hedonic model and hybrid model are used to estimate the depreciation rate of the
combined price of land and structure, the study revealed that as the structure value becomes
negative over a speciﬁed period of time, depreciation cuts into the land value. This was nota realistic result. Essentially, this result reiterates the importance of separating land andstructure when attempting to calculate the depreciation of housing, just as was proposed by
Diewert and Shimizu (2013). Nevertheless, in regions where the repeat sales model, and not
the hedonic model, can be applied, the use of the Age-R is considered to be a valid method
for calculating an appropriate depreciation rate.
In the “Handbook of Residential Property Indexes” that Eurostat began to distributein 2012,9 the hedonic model is recommended for estimating price indexes. However, whenattempting to estimate the real estate price index for the purpose of oﬃcial statistics, there
will be many countries where the hedonic method cannot be applied due to existing restrictions on data. This is because a variety of data about the characteristics of housing must
be collected in order to apply the hedonic method. In these countries, the depreciation bias
will be a major problem when using the repeat sales model to estimate the price index. This
fact is not only noted in the Eurostat handbook, but has also been made clear in this paper.
In conclusion, we believe that the new Age-R proposed in this paper is a valid means forsolving the problem of depreciation bias.

References[1] Bailey, M. J., R. F. Muth, and H. O. Nourse. (1963). A Regression Model for Real
Estate Price Index Construction, Journal of the American Statistical Association 58,
933-942.
[2] Case, B and J. M. Quigley (1991), The dynamics of real estate prices, Review of Economics and Statistics 22, 50-58.
[3] Chau, K.W., S. K. Wong, and C. Y. Yiu (2005). Adjusting for Non-Linear Age Eﬀects in
the Repeat Sales Index, Journal of Real Estate Finance and Economics 31:2, 137-153.
[4] Diewert, W. E (2007), The Paris OECD-IMF Workshop on Real Estate Price Indexes:
Conclusions and Future Directions. Discussion Paper 07-01, Department of Economics,
University of British Columbia, Vancouver, Canada, V6T1Z1.

9 RPPI Handbook is available from the following website: http://epp.eurostat.ec.europa.eu/portal/page/portal/hicp/methodology/h14

[5] Diewert, W.E., J. de Haan and R. Hendriks (2011). “Hedonic Regressions and the Decomposition of a House Price index into Land and Structure Components,” Discussion
Paper 11-01, Department of Economics, University of British Columbia, Vancouver,
Canada, V6T1Z1. Forthcoming in Econometric Reviews.
[6] Diewert, W. E. and C. Shimizu (2013). “A Conceptual Framework for Commercial
Property Price Indexes,” Discussion Paper 13-11, Vancouver School of Economics, University of British Columbia.
[7] Englund, P., J. M. Quigley, and C. L. Redfearn (1998). Improved Price Indexes for Real
Estate: Measuring the Course of Swedish Housing Prices, Journal of Urban Economics
44, 171-196.
[8] Gatzlaﬀ, D. H. and D. R. Haurin (1997). ”Sample Selection Bias and Repeat-Sales
Index Estimates,” Journal of Real Estate Finance and Economics, 14 (1), 33-50.
[9] Gatzlaﬀ, D. H. and D. R. Haurin (1998). “Sample Selection and Biases in Local House
Value Indices,” Journal of Urban Economics, 43(2), 199-222.
[10] Hill, R. C., J. R. Knight and C. F. Sirmans (1993) ”Estimation of Hedonic Housing
Price Models Using non Sample Information: A Montecarlo Study,” Journal of Urban
Economics, 34(3), 319–346.
[11] Karato, K, O. Movshuk and C. Shimizu (2010). “Semiparametric Estimation of Time,Age and Cohort Eﬀects in A Hedonic Model of House Prices,” Faculty of Economics,
University of Toyama, Working Paper No. 25[12] Palmquist, R. B (1980). Alternative Techniques for Developing Real Estate Price Indexes, Review of Economics and Statistics 66, 394-404.
[13] Shimizu, C., K. G. Nishimura and T. Watanabe (2010). “House Prices in Tokyo A Comparison of Repeat-sales and Hedonic Measures-,” Journal of Economics andStatistics, 230 (6), 792-813[14] Shimizu, C., K.G. Nishimura and T. Watanabe (2012). “House Prices from Magazines,
Realtors, and the Land Registry,” Property Market and Financial Stability, BIS Papers
No.64, Bank of International Settlements, March 2012, 29-38.
[15] Thorsnes, P. (1997). “Consistent Estimates of the Elasticity of Substitution between
Land and Non-Land Inputs in the Production of Housing,” Journal of Urban Economics
42, 98-108.

15

50%
45%
40%
35%
30%
25%
20%
15%
10%
5%
0%

Hong Kong

Tokyo

Figure 1: Ratio of construction cost to average property price

120
HK
110

Tokyo

100

Age Index

90
80
70
60
50
40
0

5

10

15

20

25
Age (years)

30

35

40

45

Note: A constant structure-to-property value ratio is assumed

Figure 2: Property depreciation pattern in Hong Kong (HK) and Tokyo
i

3.5
3
2.5
2
1.5
1
0.5
0

BMN

Age-R

Figure 3: Hong Kong Island’s repeat sales indices

1.2

1

0.8

0.6

0.4

0.2

0

BMN

Age-R

BMN is the traditional repeat sales index; Age-R is the age-adjusted repeat sales index basedon Equation 7

Figure 4: Tokyo’s repeat sales indices
ii

1.1

1.0

0.9

BMN

Age-R

HybridHedonic

0.8

0.7

0.6

0.5

0.4

Figure 5: Comparison of BMN, Age-R, Hedonic and Hybrid property priceindexes in Tokyo

iii

120
Age-R .
Hybrid .
Hedonic .

100

Age Index

80

60

40

20

0
0

5

10

15

20

25
Age (year)

30

35

40

45

Figure 6: Comparison of Age-R, Hedonic and Hybrid Age indexes

0.0%
-0.2%
-0.4%

Marginal Effect

-0.6%
-0.8%
Age-R(HK) .
-1.0%
Age-R(Tokyo) .
-1.2%
Hybrid(Tokyo) .
-1.4%
Hedonic(Tokyo) .
-1.6%
-1.8%
-2.0%
0

5

10

15

20

25
Age (year)

30

35

40

Figure 7: Marginal effect of depreciation rateiv

45

Appendix: Histogram of Age of Building in Hong Kong and Tokyov