Change in house price structure with time and housing price index-Centered around the approach to the problem of structural changeChihiro SHIMIZU* Hideoki TAKATSUJI** Hiroya ONO** Kiyohiko G. NISHIMURA***

Abstract
The purpose of this study is to estimate the hedonic price index of secondhand condominiums while
taking into account seasonal sample selection bias and structural changes, using the 23 wards ofTokyo as a subject. When housing price indices are estimated using a hedonic price model, theproblem of temporal sample selection bias in addition to changes in the housing market structure
should be considered. We propose an overlapping-period hedonic model (OPHM: this model was
proposed by Ono, et al first), which can accommodate seasonal sample selection bias and structuralchanges. In addition, we estimate housing price indices for the 23 wards of Tokyo from 1986
through 2006, and demonstrate biases in price indices because of differences in the functions usedin the models. Results of the estimation using the OPHM demonstrate that the structure of the
housing market changes with time, and these changes occur continuously with time. It is also
demonstrated that structurally restricted indices that do not account for structural changes involve alarge time lag compared with indices that do account for structural changes during periods withsignificant price fluctuations. This study proposes a method of estimating hedonic housing priceindices under the conditions of successively added data and structural changes.
Key Words: Hedonic housing price index, seasonal sample selection bias, (un)restricted hedonic
model, overlapping-period hedonic model

JEL Code: C43 - Index Numbers and Aggregation, R31 - Housing Supply and Markets.

===================================* Visiting Associate Professor, Center for Spatial Information Science (CSIS), University of Tokyo
Associate Professor, The International School of Economics and Business Administration, Reitaku
University
2-1-1 Hikarigaoka, Kashiwa-Shi, Chiba, 277-8686 Japan
Tel. +81-(0)4-7173-3439, Fax. +81-(0)4-7173-1100
e-mail: cshimizu@reitaku-u.ac.jp** Professor , The International School of Economics and Business Administration, Reitaku University
2-1-1 Hikarigaoka, Kashiwa-Shi, Chiba, 277-8686 Japan***Member of the Policy Board, Bank of Japan
2-2-1 Hongoku-cho, Chuo-Ku, Tokyo 103-8660, Japan

1.Objectives of the study
The specifications and facilities of each house are different from each other in varying degrees, so there
are no two houses of identical quality. Even when the specifications and facilities are identical, if the ageof the building differs, the degree of deterioration differs accordingly, so that the houses are not identical.
In other words, houses have “particularity with few equivalents”. In addition to such a problem, thequality of houses (in particular, condominiums) changes with time owing to fairly rapid technologicalprogress. Such characteristics are particularly evident in the housing market of Japan compared with theUnited States and other countries (Shimizu, Nishimura and Asami, 2004).
There are two approaches in constructing a housing price index that takes into account issues resulting
from the above particularity with few equivalents and changes in quality: they are hedonic price modeland the repeat-sales method. In the current study, we use hedonic price model to estimate the priceindices of the secondhand condominium market in the 23 wards of Tokyo.
When the repeat-sales method and hedonic price model are compared, the following problems are noted
for the respective methods.
In the repeat-sales method, the following two problems are noted: (i) a sample selection bias issue, thatis, houses that are repeatedly on sale have different characteristics from the houses traded in the market(so-called lemons) (Clapp and Giaccotto, 1992), and (ii) accommodating qualitative changes andstructural changes because of the assumption that there are no changes in the property characteristics andtheir parameters during the transaction period in the repeat-sales method (Case and Shiller, 1987, 1989;
Clapp and Giaccotto, 1992, 1998, 1999; Goodman and Thibodeau, 1998; Case, Pollakowski and Wachter,1991).
Furthermore, the estimation of housing price indices by hedonic price model has the following two
major problems: (iii) the occurrence of bias in housing price indices because of the difficulty incollecting all the variables required for the estimation of functions as well as because of the presence ofunobservable factors such as environmental variables (see Case and Quigley, 1991; Clapp, 2003), and(iv) a structural change issue, that is, it is necessary to accommodate changes in the house price structure,1

because the housing market is examined over a long period of time (Case, Pollakowski and Wachter,
1991; Clapp, Giaccotto and Tirtiroglu, 1991; Clapp and Giaccotto, 1992, 1998; Shimizu and Nishimura,
2006,2007). However, while the two approaches have problems in terms of estimation, it can also be
shown that as the analysis period increases, the difference between the hedonic price index and the priceindex evaluated by the repeat-sales method decreases (Clapp and Giaccotto, 1998, 1999).
With regard to problem (i) of the repeat-sales method, because not all the transaction data are collected
and estimated, the problem of sample selection bias also exists for hedonic price model , although its
level is relatively low. Problems (ii) and (iv) occur in both the repeat-sales method and hedonic pricemodel when housing price indices over a long period of time are estimated (i.e., there is a problem ofcoping with structural changes).
Regarding problem (iii), problems relating to the control of unobservable environmental variables can be
avoided in the repeat-sales method (Case and Quigley, 1991; Case and Shiller, 1987, 1989; Thibodeau,1997). Furthermore, the calculation procedure is simpler, and hence the calculation load is smaller in therepeat-sales method than in hedonic price model . Therefore, it has been considered that, at a glance, therepeat-sales method is more practical (Bourassa, Hoesli and Sun, 2006).
However, because the fluidity of housing markets is considerably lower (i.e., the market is thinner) inJapan than in the United States and other countries, and because institutional restrictions strongly
suppress reselling within a short period, in accordance with the law based on the National Land Use Plan,problems with repeat-sales sample selection bias unique to Japan will still occur. If the repeat-sales
method is to be applied in Japan, such a sample selection bias will be an extremely large problem, and in
addition to that, the estimation of housing prices with high renewal frequency is impossible because of
the small number of samples, and the estimation of indices in a limited area is difficult; consequently, therepeat-sales method is not a very practical method.
Under such circumstances, the importance of estimating hedonic housing price indices with highaccuracy while solving the above problems involved in hedonic price model is extremely high in Japan.
Therefore, in this study, we focus on the greatest problem involved in hedonic price model, which relates2

to changes in the market structure. Problem (iii) will be discussed in another report (according to Clapp(2003), unobservable variables are handled by adding coordinate data).
We start with the estimation of a structurally restricted hedonic model (hereafter, also referred to as theRHM) under the assumption of no changes in the market structure and a structurally unrestricted hedonic
model (hereafter, also referred to as the URHM) under the assumption that the structure changes in eachperiod (Case, Pollakowski and Wachter, 1991; Clapp, Giaccotto and Tirtiroglu, 1991).
In Section 2, the structures of the structurally restricted hedonic housing price index (hereafter, also
referred to as the RHI) estimated using the restricted hedonic model and of the structurally unrestricted
hedonic housing price index (hereafter, also referred to as the URHI) estimated using the unrestricted
hedonic model are explained together with the repeat-sales index. This is aimed at clarifying thecharacteristics of hedonic price model in terms of estimation, by means of comparison with the repeatsales method. Then, we propose a new housing price index taking into account structural changes andseasonal sample selection bias; the overlapping-period hedonic housing index (hereafter, also referred toas the OPHI). Data are explained in Section 3, and RHI, URHI and OPHI are estimated for thesecondhand condominium market in the 23 wards of Tokyo, and the estimated housing price indices areevaluated in Section 4.
The results show temporal changes in the house price structure in the secondhand condominium market,
and in particular, regarding the floor space, the sign is reversed during some periods. Therefore, it isnecessary to estimate the price index taking into account structural changes. In addition, because housingtransactions in Japan tend to be concentrated in periods when large numbers of people move, the numberof samples varies considerably with the season; accordingly, seasonal sample selection bias should beconsidered. To accommodate changes in the market structure and the problem of seasonal sample
selection bias, it has been demonstrated that OPHM with an overlapped estimate period τ of 12 months
is effective.

32.Changes in market structure and housing price indices2.1.Structurally restricted hedonic housing price index (RHI) and structurally unrestricted
hedonic housing price index (URHI)
Regarding the estimation of quality-adjusted housing price indices, the hedonic estimation method andthe repeat-sales estimation method can be used. The price indices of the hedonic estimation method
include the structurally restricted price index and the structurally unrestricted price index. We summarize
these estimation methods, and clarify the characteristics of the hedonic price indices in terms of
estimation in comparison with the repeat-sales method.

2.1.1.Structurally restricted hedonic housing price index: RHI
Assume that we have data for the house price and residential property characteristics, which are pooled
for all the periods t = 1, 2, …, T, and that the number of data samples in each period is nt. A house priceestimation model that can be used to obtain a structurally restricted price index is given as follows.
K

τ

k =1

s =1

ln Pit = ∑ β k X ikt + ∑ δ s Ds + ε it(1)

t = 1, 2, …, T.
i = 1, 2, …, nt (designates ith data among the nt data samples in period t).
Pit = price of house i in period t (designates ith data among the data in period t, instead ofdesignating the same house i over each of the t periods).

βk = parameter of residential property characteristic k.
Xikt = value of property characteristic k of house i in period t.

δs = parameter of the time dummy variable in period s.
Ds: when s = 1, this takes a constant value of 1 (constant term). When 2 ≤ s ≤ T, this is a time
dummy variable, and it takes a value of 1 when s = t and a value of 0 otherwise.

εit = random disturbance term.

This model is called the structurally restricted hedonic model (RHM) because it assumes that the4regression coefficient βk of the house-price determining factor Xikt is constant throughout all the periods.
From this, the RHI is obtained as follows. The estimated price P̂t of a house with residential property
characteristic values {Xk} (k = 1, 2, …, K) in period t (t = 1, 2, …, T) is given as follows.
K

ln Pˆt = ∑ βˆk X k + δˆ1 + δˆt
k =1

(2)

K

ln Pˆ1 = ∑ βˆk X k + δˆ1
k =1

(3)

Here, β̂k , δˆ1 , δˆt are estimated values of the parameters. Accordingly, the housing price index Pˆt / Pˆ1 inperiod t, where the house price in period t = 1 is used as the reference, is obtained as follows.

ln(Pˆt / Pˆ1 ) = δˆt(4)

In addition, the change in the price index from period t−1 to period t can be expressed as follows.

ln(Pˆt / Pˆt −1 ) = δˆt − δˆt −1(5)

In this case, the price index is obtained under the assumption of specific residential property
characteristic values {Xk}; however, as we can see in the above process, price indices are expressedusing only time dummy variables without involving residential property characteristic values in the RHI.

2.1.2.Structurally unrestricted hedonic housing price index: URHI
Using similar data to those described above, a house price estimation model that can be used to obtain a
URHI is given as follows.

5

K

ln Pit = ∑ β kt X ikt + δ t + ε it
k =1

(6)

Here, no time dummy variables are used, and instead, the parameter βkt and a constant term δt related toresidential property characteristics are assumed to change in each period. Namely, because the model
does not assume the restriction of constant parameters, it is called the URHM. From equation (6), the
URHI is obtained as follows. The estimated price P̂t of a house with residential property characteristic
values {Xk} (k = 1, 2, …, K) in period t (t = 1, 2, …, T) is given as follows.
K

ln Pˆt = ∑ βˆkt X k + δˆt
k =1

(7)

K

ln Pˆ1 = ∑ βˆk1 X k + δˆ1
k =1

(8)

Therefore, the housing price index Pˆt / Pˆ1 in period t, where the house price in period t = 1 is used as thereference, is obtained as follows.
Kln( Pˆt / Pˆ1 ) = ∑ ( βˆkt − βˆk1 )X k + (δˆt − δˆ1 )
k =1

(9)

In addition, the change in the price index from period t−1 to period t can be expressed as follows.
Kln(Pˆt / Pˆt −1 ) = ∑ ( βˆkt − βˆk ,t −1 )X k + (δˆt − δˆt −1 )
k =1

(10)

Thus, for the URHI, price indices are obtained for specific residential property characteristics. Whenspecific residential property characteristics change, the price index changes accordingly.
62.1.3. Repeat-sales housing price index
Next, the repeat-sales method is summarized. For house h, it is assumed that its price is determined by
the residential property characteristics and the time point of the transaction. It is also assumed that theresidential property characteristics do not change with time, and the strength of their effect on priceformation also does not change. Thus, the house price model in this case can be expressed as follows.
K

T

k =1

s =1

ln Pht = ∑ β k X hk + ∑ δ s Ds + ε ht

(11)

Pht is the price of house h in period t. Here, we assume that house h appears repeatedly in different
periods. Xhk is the value of property characteristic k of house h, which does not change with time.
Accordingly, we assume that parameter βk of Xhk also does not change with time. Ds is a time dummy
variable and equals 1 when s = t (the period of the transaction) and 0 otherwise. Here, we assume that D1= 1 (a constant term). δs is the parameter of the time dummy variable. It is assumed that house h is
subject to a transaction twice, in periods t1 and t2, during the estimation period of t = 1, 2, …, T. The
house prices in the periods of transaction can be expressed, using the above model, as follows.
K

ln Pht1 = ∑ β k X hk + δ1 + δ t1 + ε ht1

(12)

k =1

K

ln Pht2 = ∑ β k X hk + δ1 + δ t 2 + ε ht2

(13)

k =1

From equations (12) and (13), the price change Pht 2 / Pht1 is given as follows.

(ln( Pht 2 / Pht1 ) = δ t 2 − δ t1 + ε ht 2 − ε ht1

)

(14)

Therefore, in this model, the price change is determined from the difference in the two time points of the7transactions irrespective of residential property characteristics. We now formulate the model to estimatechanges in the house price with respect to data collected for various transactions of houses at various
time points. We obtain:Tln(Pht 2 / Pht1 ) = ∑ δ s Ds + μh ,

(15)

s =1

where μh is a random disturbance term and Ds is a time dummy variable, which takes a value of 1 at thesecond transaction (t = t2), −1 at the first transaction (t = t1), and 0 in other periods. In addition,
parameter δs of each time dummy variable estimated using this model represents the price index of eachperiod. This represents the typical repeat-sales model.
On the basis of the above summary of the indices and under the assumption that the market structurechanges, the structurally restricted hedonic price index is problematic in that it has the restriction thatparameters for Xki are identical throughout all the periods. In the repeat-sales method, the same
assumption that parameters for Xki remain constant is adopted, and in addition, the very strong
assumption that there are no changes in the property characteristics during the transaction period isadopted as well. The latter assumption is too strong for Japan. In practice, the values of houses change
because of the extension and reconstruction, or renovation of buildings, and because of the developmentof physical damage accompanying the increasing age of buildings. In Japan, large-scale renovation is
performed every couple of years, and the life of houses is short compared with that in Europe and theUnited States; therefore, house prices significantly decrease as the age of buildings increases. Moreover,because of the weak restrictions on city planning in Japan, it is extremely unrealistic to assume that there
are no changes in the city environment or in building property characteristics.
In the URHM , although the restriction on parameters for Xki can be eliminated, an assumption that theparameters including error terms are independent in each period is necessary. However, in a real market,
it cannot be expected that the structure always changes randomly. In addition, a new issue arises. Whenhousing price indices are set with a period of less than one year, i.e., quarterly or monthly, a problem of8seasonal sample selection bias occurs. For example, in Japan there are periods with a large number oftransactions, from January to March when many people move, and periods with a small number oftransactions, such as July and August, and these seasonal changes in the number of transactions may
affect and bias the price index. Thus, although the URHM can possibly accommodate structural changes,
it disconnects the estimation from the continuity of the market conditions and generates the problem of
seasonal sample selection bias.

2.2.Overlapping-period hedonic housing price index: OPHIThe URHI assumes that the market structure changes successively. Such a structural change of themarket occurs as a result of various external shocks; it is considered that there is, in reality, a certain
adjustment period before such a change penetrates into the market. Accordingly, regression coefficientsshould be regarded as changing successively rather than instantaneously. However, generally, theestimation of a model with structural changes is performed by dividing observation data into several
periods with break points, then using the divided data of each period (for example, Ono et al,2004;
Shimizu and Nishimura, 2006; Shimizu and Nishimura, 2007). Namely, the continuity of the observation
data is disconnected at the break points. Therefore, it is rather difficult to use such an estimation method,under the assumption of the occurrence of successive structural changes, to determine regression
coefficients allowing for successive changes. Instead, it may be more natural and desirable to estimate
regression coefficients on the basis of a process of successive change by taking a certain period length τas the estimation period, and by shifting this period, similar to the process of obtaining moving averages.
This process can be formulated as follows.
Assuming that we have pooled data over the periods 1, 2, …T. With respect to some of these periods, i.e.,a period length τ, we assume the following basic model.
K

τ

k =1

s =1

ln Pit = ∑ β k X ikt + ∑ δ s Ds + ε it

,

(16)
9

where t = 1, 2, …, τ (taking part of the entire pooled data consisting of 1, 2, …T periods, namely,
taking a certain period length τ, the periods within this range are numbered from 1 to τ).
i = 1, 2, …, nt (ith datum among nt items of data in period t).
Pit: price of house i in period t.

βk: parameter of residential property characteristic k; βk is assumed not to change within the
period length τ.
Xkit: value of property characteristic k of house i in period t.

δs: parameter of the time dummy variable in period s.
Ds: when s = 1, this takes a constant value of 1 (constant term). When 2 ≤ s ≤ τ, this is a time
dummy variable and takes a value of 1 when s = t and a value of 0 otherwise.

εit: random disturbance term.

In addition, we express a period with length τ starting from period r among periods 1, 2, …T, as [r, r + τ− 1]. Then, our estimation method is obtained by applying the above basic model to periods [1, τ], [2, τ+ 1], …, [r, r + τ − 1], …,[T − τ + 1, T] successively. From this, successive changes of the market
structure can be reflected in changes in the parameters. We call this model the OPHM, and the period
lengthτ, the overlapped estimate period length.
OPHM is a RHM with respect to a certain period lengthτ. Accordingly, the parameter of the time dummy
variable represents the price index of each period with the starting period of length τ as the reference.
Thus, price indices can be obtained directly from the basic model within the period lengthτ. With the
OPHM, models for all the periods are estimated by successively shifting the period length τ by one
period. Here, the problem remains of how to connect the price indices obtained by the estimation in eachperiod length τ to construct the price index for all periods. Our method is as follows.
We designate the housing price index throughout all the periods as qr. This represents the price index of
period r among periods 1, 2, …T. We designate the reference period as period 1, and assume that q1 = 0.
We also designate a parameter of the time dummy variable obtained by applying the basic model to thedata for the period length τ starting from period r among periods 1, 2, …T, i.e., [r, r + τ − 1], as10δˆ1( r ) , δˆ2( r ) ,L, δˆτ( r ) by explicitly expressing the period r.
The procedure to obtain the housing price index qr is as follows.

(Set 1)The basic model is applied to the first [1, τ] period to obtain the parameter of the time dummy variable.

δˆ1(1) ,δˆ2(1) ,L,δˆτ(1)Using these parameters, we define the price index qr (r = 1, 2, …,τ) for the [1, τ] period as follows.

q1 = 0q2 = δˆ2(1)q3 = δˆ3(1)

(17)

･･･

qτ = δˆτ(1)(Set 2)
To obtain the next price index qτ+1 under the assumption that those up to qτ have been determined as
described above, an estimated amount, which is considered to be the change from qτ to qτ+1, is added toqτ. We consider that this estimated amount is based on the following parameters:δˆ1( 2) , δˆ2( 2) ,L, δˆτ( 2) ,(18)

which are determined by the estimation using the basic model for the next [2, τ + 1] period, as follows.
11δτ( 2) − δτ(−21)(19)

Accordingly, qτ+1 is defined as follows.

qτ +1 = qτ + (δτ( 2) − δτ( −21) )(20)(Set 3)
Similarly, to obtain the next price index qτ+r−1 under the assumption that those up to qτ+r−2 have been
determined, an estimated amount, which is considered to be the change from qτ+r−2 to qτ+r−1 is added toqτ+r−2. Accordingly, on the basis of the parameters determined by the estimation of the basic model forthe [r, τ + r − 1] period,δˆ1( r ) , δˆ2( r ) ,L, δˆτ( r ) ,(21)

the estimated price index is defined as follows.

qτ +τ −1 = qτ +τ − 2 + (δτ( r ) − δτ( −r )1 )(22)

Thus, we can obtain the price indices by OPHM for all periods.
Here, we note one point regarding the estimation of the basic model (equation (16)). With respect to therandom disturbance term εit, when we assume:

Var (ε it ) = σ t2

,

(23)

it has been confirmed by our previous analysis that:

σ i2 ≠ σ 2j(i ≠ j )(24)

12

holds. Namely, heterogeneity of the variance is observed. Therefore, the basic model (equation (16)) isreestimated using feasible generalized least squares (FGLS). That is, σˆ t is obtained from the residual
2

upon estimation using the basic model (equation (16)), and then the parameter is estimated using thefollowing.
K

τ

k =1

s =1

(ln Pit ) / σˆ t = ∑ β k ( X ikt / σˆ t ) + ∑ δ s ( Ds / σˆ t ) + (ε it / σˆ t )(25)
2.3.Setup of estimation model
In this study, the secondhand condominium market in the 23 wards of Tokyo is used as the analyticalsubject. The basic equation of the RHM is as follows.

log RP / FS = a0 + ∑ a1h log X h + ∑ a 2 i log Z i + ∑ a3・
j LD jhij+ ∑ a4・k RDk + ∑ a 5・l TDl +ε
k

l

(26)

RP: Resale price of condominium (yen)Xh: Main variables
FS: Floor space (square meters)
Age: Age of building (months)
TS: Time to the nearest station (minutes)
TT: Travel time to central business district
Zi: Other variables
BS: Balcony space (square meters)
NU: Number of units
BC: Other building characteristics
13RT: Market reservation time (weeks)
LDj: Location (ward) dummy (j = 0 … J)
RDk: Railway line dummy (k = 0 … J)
TDl: Time dummy (l = 0 … K)
Residential property characteristics as explanatory variables include floor space (FS), time to nearest
station (TS), travel time (TT) to the central business district (CBD), age of building (Age), area of
balcony (BS) and other building property characteristics (Zh), as well as a railway dummy variable (RDj)and location (ward) dummy variable (LDk) as location factors. In addition, we have a time dummy
variable (TDk). The regression coefficient of this time dummy variable (a12k) represents the secondhand
condominium price index. In the RHM, the estimation is performed using this equation with pooled datain the periods t = 1 ··· T.
In contrast, the basic equation of the URHM is given by equation (27) below; this is the equation of the
RHM from which the time dummy variable is excluded. Estimation is performed for each period using
the data of that period (period t). Setting the period length to one month, a model is estimated for all
periods. Then, using the estimated models and the assumption of houses having the same quality, theprice of the houses in each period is estimated and its change-over time is examined.

log RPt / FS t = a 0 + ∑ a1th log X th + ∑ a 2t i log Z ti + ∑ a 3 ・
j LD j + ∑ a 2tk・RD tk +ε
h

i

j

k

t = 1LLLT(27)

Using the OPHM, the RHM of equation (26) is adopted for the estimation period lengthτ.

3.Data3.1.Secondhand condominium price data
The subject of the analysis is the 23 wards of Tokyo metropolitan area (621 square kilometers), and the14

analysis period is approximately 20 years between January 1986 and September 2006.
As the main information source, we used the prices of secondhand condominiums published in
Residential Information Weekly (or Shukan Jyutaku Joho in Japanese) published by RECRUIT, Co. Thismagazine provides information on the quality and asking price of listed properties on a weekly basis, andincludes the historical price data of individual properties from the time they are placed in the magazine
for the first time until they are removed because of sale or other reasons. There are three items of
information regarding the price: i) the initial asking price (first offer price) upon appearance in themarket, ii) the price upon removal from the magazine (estimated purchase price: first bid price), and iii)
the transaction price, which are collected as a sample. The first asking price represents the seller’s
desired price rather than the market value. In contrast, some of the transaction prices may partly reflect
the specific situations of individual transactions such as the desire for a quick sale or hasty purchase.
Therefore, among the information published in Residential Information Weekly, we decided to use theprice when the listing of the house is removed from the magazine upon the conclusion of the contract as
the explanatory variable in the model. The price at the time of removal from the magazine is the first bid
price offered by a prospective buyer, such a bid is offered through the process in which several
particulars of quality and price are disclosed to the market via the magazine, and the price is decreased
until the buyer responds to that information. Thus, the price we used indicates the upper range ofpossible bid prices and can be regarded as a competitive market price that is relatively free fromindividual specific conditions associated with transactions.

3.2.Data regarding house quality
In the condominium market, while condominiums with light-gauge steel structures are also included,transactions are mainly for condominiums with reinforced concrete (RC) structures or steel-reinforced
concrete (SRC) structures; therefore, we used condominiums with RC or SRC structures as the subjectof our study. A list of data analyzed is shown in table 1.
The transport accessibility of each condominium location is represented by the TT to the CBD and the15TS. The former is measured in the following way. First, we defined the CBD. The Tokyo Metropolitan
area is composed of the 23 wards of Tokyo as its center, with a dense railway network developed therein.
We designated seven terminal stations, which include six on the Yamanote Line, Tokyo, Shinagawa,Shibuya, Shinjuku, Ikebukuro, and Ueno, as well as Otemachi as the central station of the Tokyo Metro(Teito Rapid Transit Authority). Then, we investigated average travel times during the day from eachstation to the seven terminal stations, and set the minimum value as the TT to the CBD for that station.
By considering travel times to multiple terminal stations, the timesaving effect of a new railway
development on the whole transport network can be embedded into the model. When a new line isdeveloped or a new station is constructed, or when a timetable is changed, this index will change.
Therefore, travel times are renewed once every half year (April and October).
Regarding TS, different means of transportation are available. There are three transportation means: onfoot, by bus and by car. However, because a very dense transportation network is established in the 23
wards of Tokyo, and many condominiums are built in transportation-convenient areas, condominiums
only within walking distance or bus-transportation distance are included in the analysis data. Therefore,
any difference in the transportation distance between the former and latter is controlled for using the bus
dummy variable (BD). In addition, the walking time (in minutes) is recorded when the condominium is
within walking distance, and the walking time from the condominium to the bus stop and the on-board
time from the bus stop to the nearest station (in minutes) are recorded in the case of condominiums in a
bus-transportation area. The TS is defined as; (walking time to nearest station) + (walking time to the busstop) + (on-board time from the bus stop to the nearest station). Then, with respect to the bustransportation area, the cross term of the constant dummy variable with the TS is incorporated in the BD.
Details of the model are described below.
The sale price of each property is also affected by the fluidity and thickness of the market. The timespent until the contract is concluded is considered to be affected by the period and location and by thelevel of activity of transactions in the market. We explain such market factors using variables such as
market reservation time (RT). This is the time period between the date when a house is placed on the16market by a seller, and the time period when a buyer appears. Properties with a long RT are regarded as
those having a price higher than the equilibrium price, or those in a thin market. Conversely, properties
with a short RT are regarded as those in a market with high fluidity or those having a price close to or
lower than the equilibrium price. For our purposes, the RT is defined as the time period from when theproperty is listed in the magazine for the first time until it is removed from the magazine.
We also identified some quantitative measures representing building property characteristics. They are
FS, Age, BS and the number of units (NU). Age is the time period from its construction to the time period
when the contract is concluded. NU is regarded as a proxy variable for the grade of the entire
condominium and the quality of the common space. In addition, we created other dummy variables: theground-floor dummy variable is used because the price of a ground-floor property is expected to belower than that of higher-floor properties, and the highest-floor dummy variable is used because theprice is expected to again be higher on the top floor. In terms of the direction of properties, such aswhether their opening parts are south facing or not, we define a south-facing dummy variable (SD).
Furthermore, we define a ferroconcrete structure dummy variable (FD) to incorporate the difference instructural strength.
The above variables are all related to the location or building characteristics of condominiums. It isreasonable to assume that other regional factors may also affect the house price. Therefore, we set anadministrative-area dummy variable, LD, to reflect differences in the quality of public services, as wellas the “Ji-Gurai” or area rank. Furthermore, the railway dummy variable, RD, represents the pricestructure for condominiums along railway lines, because most Japanese residential developments havebeen carried out along railway lines.
Finally, the time dummy variable (TD) is used to control for differences in the prices between different
time periods.The observation data consist of 211,179 samples collected between January 1986 and
September 2006.

17

Table 1. List of analyzed data.

SymbolsFS

Variables

Contents

Floor space/ square meters Floor space.

Unit
2

m

AGE

Age of Building:Number of years sinceconstruction

Period between the date when the data is deleted
from the magazine and the date of construction of
the building.

month

TS

Time to nearest station

Time distance to the nearest station (Time by Walkor Bus).

minuteTT

Travel Time to central
business district

Minimum of railway riding time in daytime to
Terminal 7 stations in 2005*.

minuteBS

Balcony space/ square
metersBalcony space.

m

NU

Number of units

Total units of the condominium.

unit

RT

Market reservation time

Period between the date when the data appear in
the magazine for the first time and the date of being
deleted.

week

FDFirst floor dummy

HF

Highest floor dummy

SDSouth-facing dummy

The property is on the ground floor 1,
on other floors 0.
The property is on the top floor 1,
on the other floors 0.
Fenestrae facing south 1，other directions 0.

Steel reinforced concrete frame structure 1,
other structure 0.
j th administrative district 1,
LDj (j=0,…,J)
Location (Ward) dummyother district 0.
k th railway line 1,
RDk (k=0,…,K)
Railway line dummyother railway line 0.
l th month 1,
TDl (l=0,…,L)Time dummy (monthly)other month 0.
*Terminal Staion : Tokyo,Shinagawa,Shibuya,Shinjuku,Ikebukuro,Ueno, and Ootemachi
FD

Ferroconcrete dummy

2

(0,1)
(0,1)
(0,1)
(0,1)
(0,1)
(0,1)
(0,1)

18

3.3.Statistical distribution of secondhand condominium price data
Table 2 shows descriptive statistics of the major variables. The average resale price of a condominium is39.04 million yen, the minimum value is 8.50 million yen, and the maximum value is 195.00 million yen,with a fairly large standard deviation of 23.48 million yen. The data include a wide range ofcondominiums from studio-apartment-class small properties to the so-called 100-million-yen-class large
properties. The average unit price is approximately 0.7 million yen/m2 with a right-skewed distribution.
Regarding the FS, the minimum value is 16.00 m2, the maximum value is 134.99 m2, and the average is
56.57 m2, including all condominiums from single-person households to large-family condominiums.
Regarding the Age, the average value is 165 months (13.75 years), with a maximum value of 413 months(34.42 years). Because the history of condominiums in Japan is short, it is expected that this index will
increase over time.
Regarding the TS, we only observed the distribution of data on the time axis; there are properties with aminimum value of 0 minutes that are located in front of a station. The maximum value is 32 minutes andthe average value is 7.60 minutes. On average, while many properties are conveniently located, some are
beyond walking distance. This indicates that, in general, convenience is emphasized in the constructionof condominiums because of the required characteristics of condominiums.
Regarding the TT to the CBD, the average is 15 minutes with a maximum value of 30 minutes,
indicating that most condominiums are concentrated in areas of high convenience.

19

Table 2. Summary of statistical values of secondhand condominium price data.

Variables

AverageStandard
Deviation

Minimum

Maximum

RP : Resale Price of Condominimum(10,000 Yen)

3,904.66

2,348.54

850.0019,500.00

FS : Floor space (㎡)

56.57

19.40

16.00

134.99RP /FS

70.93

36.78

24.00

270.90Age :Age of Building(months)

165.74

91.98

5.00

413.00TS : Time to the nearest station:(minutes)

7.60

4.27

0.00

32.00

TT : Travel Time to Central
Business District (minutes)

15.32

5.30

0.00

30.00

NU : The Number of Units

100.03131.05

10.00

1149.00

RT : Market reservation time (week)

11.58

10.62

1.00

64.00

1986/01-2006/09

n=211,179

4.Estimation results
4.1.Estimation of RHI
The estimated RHI for the 23 wards of Tokyo is as follows.
log RP / FS = 4.631 + 0.0126・log FS − 0.189・log Age − 0.078・log TS − 0.117・log TT + 0.019・log NU(498.23)(+10.81)(−337.38)(−99.69)(−36.21)(40.90)− 0.276・BD + 0.058・( BD × log WT ) − 0.026・FF + 0.018・HF − 0.097 FD + 0.0093・SD(−13.140) (6.970)(−19.210)(8.000) (−10.150) (10.790)+ β 1h ∑ LD h + β 2i ∑ RDi + β 3 j ∑ TD j
h

i

j

20

Adjusted R-square: 0.837
Number of observations: 211,178

Because the coefficient of determination adjusted for the degrees of freedom is 0.837, the estimatedmodel has a fairly high explanatory power (refer to Table 3 for details).
Because the data was pooled for sales between 1986 and 2006, we corrected the time point by forcibly
introducing the TD, so that the structure of the secondhand condominium price was estimated using
property characteristics specific to condominiums and the RD. Among the property characteristics
specific to condominiums, FS, BS and NU have positive values, and Age, TS, and TT to the CBD are
estimated with negative values.
First, regarding FS, the unit price was shown to increase with increasing floor space. A similar tendency
was observed for BS and NU. This indicates that consumers show a strong preference for the floor spaceof each property as well as the floor space of the entire condominium.
As Age increases, we expect not only functional deterioration but also economic deterioration because of
the improvement of facilities in newer condominiums. The results obtained showed that as TS and TT to
the CBD increase, the convenience decreases because of the greater distance from populated areas,
resulting in a decrease in the price.
Furthermore, the level of public service differs for each administrative ward, and there are broaddifferences in the residential environment depending on administrative cities and wards or railway line
areas, which cannot be taken into consideration in our estimated function; therefore, these differenceswere estimated using the dummy variables.

21

Table 3. Estimation results of the RHM: 23 wards of Tokyo.
Method of Estimation
OLS

Dependent Variable
RP : Resale Price of Condominiums (in log)
Independent Variables
Property Characteristics (in log) Coefficientt-value

Railway/Subway Line Dummy

498.230
10.810-337.380-99.690-36.21040.90032.530LDk (k=0,…,K)
Yamanote
Ginza
Marunouchi
HibiyaTozai
ChiyodaYurakucho
HanzomonToeiAsakusa
ToeiShinjuku
Keihinkyukou
Keihintohoku
IkegamiTokyutoyoko
Setagaya
OdakyuInogashira
keio
Chuo
Seibushinjuku
Seibuikebukuro
Toubutojou
SaikyoTakasakiToubuisezaki
JoubanSoubu
Time DummyTDl (l=0,…,L)
yes(see Figure)

Constant
FS : Floor space
Age : Age of building
TS : Time to the nearest station
TT : Travel Time to CBD
NU : Number of units
RT : Market reservation time

4.631
0.013
-0.190-0.078-0.0400.019
0.014

Property Characteristics (dummy variables)

Coefficientt-value-0.2760.059-0.0260.018-0.0100.009-13.140
6.970-19.2108.000-10.150
10.790Coefficientt-value

0.625
0.347
0.552
0.407
0.356
0.047
-0.030
0.315
0.443
0.233
0.407
0.583
0.284
0.248
0.243
0.092
-0.0640.007
0.146-0.171-0.144-0.080110.740
82.770154.730
115.62095.06010.080-8.97086.020109.280
62.930115.890155.950
65.62060.99061.22017.620-14.5801.960
37.100-43.080-38.390-21.890

BD : Bus Dummy
TS × BD
FF : First Floor Dummy
HF :Highest floor dummy
FD :Ferroconcrete dummy
SD :South-facing dummy
Location (Ward) Dummy
LDj (j=0,…,J)
ChiyodaChuo
minatoShinjuku
Bunkyo
Taito
Koto
ShinagawaMeguroOta
SetagayaShibuyaNakanoSuginami
Toshima
Kita
Arakawa
Itabashi
NerimaAdachi
Katsushika
Edogawa

Coefficientt-value

0.033
0.158
0.056
0.085
0.040
0.067
0.053
-0.029-0.265-0.338-0.214-0.265-0.0890.036-0.091-0.0250.076
0.032-0.045-0.0530.040-0.1260.065-0.063-0.073-0.111-0.122

4.236
11.460
5.556
9.039
4.727
7.858
3.609
-2.621-2.420-10.244-15.225-6.692-6.782
1.712-5.465-1.6495.800
2.361-1.621-4.0242.455-10.4165.680-5.655-2.694-7.410-5.927

Coefficientt-value

Adjusted R square= 0.837
Number of Observations= 211,179

22

4.2.Estimation of URHI
Next, we estimated the URHM. In accordance with the definition of equation (27), we divided the data
into t periods (here, monthly) and estimated the structure of house prices. Regarding the price index, we
estimated the prices of secondhand condominiums in each period by substituting the specific residential
property characteristics common to all periods into the explanatory variables, and obtained thestructurally unrestricted hedonic housing price indices relative to the reference period based on theestimated prices.
Table 4 shows the estimated regression coefficients of the major variables, and Fig. 1 shows changes in
the number of samples and the coefficient of determination adjusted for the degrees of freedom.
The adjusted coefficient of determination decreased from 1986 through 1995, then increased from 1996.
However, on the whole, it maintained an average of around 0.75, showing fairly good results.
The number of samples was approximately 500 per month from 1986 through 1989, which thenincreased significantly to an average value of 844. However, there is more than a threefold difference
depending on the month. In each year, transactions are concentrated from January to March, which is theend of the fiscal year, when there are large movements of people in Japan, and the number oftransactions significantly decreases around July and August, thus showing seasonal changes. However,there is no apparent correlation between the number of samples and adjusted coefficient of determination.
Next, we focused on the regression coefficients of the estimated model. Table 5 shows descriptive
statistical values for the regression coefficients over 250 periods. Figs 2–6 show changes in the
regression coefficients with time. All of the regression coefficients show major fluctuations in eachperiod or every several periods. However, the fluctuations show a certain tendency, although not a
gradual change with time. In addition, we can see that all the variables are around (higher or lower than)
the regression coefficients estimated using the RHM.
Using the data in Table 5, we calculated the coefficients of variance (standard deviation/average): FS =2.428, Age = –0.179, TS = –0.232, TT to CBD = –0.779. In other words, FS shows the largest change,
including a change of the sign (+/–), followed by TT to the CBD, TS, and Age.
23

Table 4. Estimation results of the URHM : the 23 wards of Tokyo: 1986/01–2006/09.

TimeConstant

FS :
Floor
space

198601198602198603198604
198605198606198607
198608198609198610198611
198612

4.402
4.508
4.464
4.413
4.669
4.343
3.930
4.401
4.526
4.250
4.310
4.822

0.007
0.066
-0.032
0.051
0.012
0.057
0.083
0.034
0.069
0.043
-0.001
0.077

TS :
TT :
RT :
NU :
Age : Time toTravelMarket BD : Bus WT × Number of Adjusted
Observation
Numberthe
Age ofTime toreservatio DummyBD
R squaresof units
building nearest
CBD
n timestation-0.143 -0.100 -0.048 -0.011-0.0101.333 -0.495
416
0.761
-0.144 -0.089 -0.099 -0.010-0.021 -0.323
0.068
528
0.776
-0.110 -0.070 -0.046 -0.007-0.022 -0.994
0.480
489
0.823
-0.161 -0.106 -0.029
0.006-0.0121.160 -0.489
455
0.824
-0.155 -0.096 -0.095 -0.002-0.0340.722 -0.268
605
0.727
-0.133 -0.132 -0.025
0.014-0.020 -0.912
0.268
446
0.751
-0.120 -0.116 -0.054
0.002-0.046 -0.470
0.168
430
0.785
-0.128 -0.113 -0.009
0.021-0.0580.078 -0.095
564
0.791
-0.146 -0.146 -0.018
0.008-0.020 -0.308
0.084
394
0.838
-0.114 -0.092 -0.025
0.014-0.0240.177 -0.122
560
0.826
-0.109 -0.121
0.053
0.010
-0.0310.036 -0.080
340
0.866
-0.154 -0.133 -0.142 -0.006-0.032 -1.199
0.425
342
0.896

199001

5.831-0.114-0.154-0.084-0.067

0.022

0.005

-0.1980.025

857

199501

4.820

0.090

-0.208-0.070-0.048

0.011

0.044

-0.203

0.064

1,109

0.641

200001

4.402

0.071

-0.209-0.036-0.035

0.021

0.005

-0.373

0.125

778

0.697

200501
200601
200602
200603
200604
200605200606200607200608200609

4.548
4.303
4.484
4.584
4.441
4.583
4.807
4.530
4.742
4.566

0.035
0.084
0.115
0.056
0.067
0.060
0.033
0.063
0.024
0.020

-0.208-0.191-0.201-0.201-0.182-0.186-0.208-0.187-0.198-0.198-0.057-0.093-0.106-0.087-0.091-0.080-0.072-0.092-0.079-0.072-0.015-0.011-0.040-0.024-0.024-0.036-0.0220.001-0.015
0.012

0.018
0.023
-0.002
0.025
0.012
0.015
0.000
0.007
0.011
0.018

0.009
-0.002-0.010-0.007-0.011-0.014-0.018-0.006-0.011-0.005-0.752-0.576-0.382-0.670-0.242-0.224-0.248-0.039
0.324
0.089

0.294
0.188
0.125
0.207
0.051
0.049
0.039
-0.043-0.135-0.070

702
650
768
1,015
826
966
776
819
901
900

0.757
0.809
0.766
0.785
0.775
0.774
0.775
0.770
0.784
0.766

0.763

24

0.9

0.8
0.75
0.7

2,000

0.65

adjusted R square

0.85

1,500

1,000

500

0

1986/06
1986/12
1987/06
1987/121988/06
1988/12
1989/06
1989/12
1990/06
1990/12
1991/06
1991/12
1992/06
1992/12
1993/06
1993/12
1994/06
1994/12
1995/06
1995/12
1996/06
1996/12
1997/06
1997/12
1998/06
1998/12
1999/06
1999/12
2000/06
2000/12
2001/06
2001/12
2002/06
2002/12
2003/06
2003/12
2004/06
2004/12
2005/06
2005/12
2006/06

Number of observation

0.6

Figure1.Estimation accuracy of the URHM: between 1986/01 and 2006/09.
Table 5. Statistical values of major regression coefficients (URHM).

Principal Independent RHI :1986.01 Variables
2006.09

NRHI :Summary statistics of estimated parameter
AverageStandard
deviation

Skewness

Kurtosis

FS :Floor space/square
meters

0.013

0.033

0.081

-0.758-0.627

Age :Age of building-0.190-0.185

0.033

0.474

0.110

WT :Distance tonearest station-0.078-0.0820.019-0.6400.799

TT :Travel Time tocentral business
district-0.040-0.041

0.032-0.3200.136

Adjusted-R Square

0.837

0.741

0.054

0.190

-0.379Number of Samples

211,179844.720282.977

0.369

0.123

1986.01 - 2006.09:Monthly ,Number of Mode=25025

1986/06
1986/12
1987/06
1987/121988/06
1988/12
1989/06
1989/12
1990/06
1990/12
1991/06
1991/12
1992/06
1992/12
1993/06
1993/12
1994/06
1994/12
1995/06
1995/12
1996/06
1996/12
1997/06
1997/12
1998/06
1998/12
1999/06
1999/12
2000/06
2000/12
2001/06
2001/12
2002/06
2002/12
2003/06
2003/12
2004/06
2004/12
2005/06
2005/12
2006/06

Coefficient

1986/06
1986/12
1987/06
1987/121988/06
1988/12
1989/06
1989/12
1990/06
1990/12
1991/06
1991/12
1992/06
1992/12
1993/06
1993/12
1994/06
1994/12
1995/06
1995/12
1996/06
1996/12
1997/06
1997/12
1998/06
1998/12
1999/06
1999/12
2000/06
2000/12
2001/06
2001/12
2002/06
2002/12
2003/06
2003/12
2004/06
2004/12
2005/06
2005/12
2006/06

Coefficient
6.5

6

5.5

0.05
5
RHI
NRHI

4.5

4

3.5

Fig2. Time profile of regression coefficient of the URHM, constant term cnst: 1986/01–
2006/09.
0.2

0.15

0.1
NRHI

RHI

0

-0.05

-0.1

-0.15

-0.2

Fig3. Time profile of regression coefficient of the URHM, floor space FS: 1986/01–
2006/09.

26

1986/06
1986/12
1987/06
1987/121988/06
1988/12
1989/06
1989/12
1990/06
1990/12
1991/06
1991/12
1992/06
1992/12
1993/06
1993/12
1994/06
1994/12
1995/06
1995/12
1996/06
1996/12
1997/06
1997/12
1998/06
1998/12
1999/06
1999/12
2000/06
2000/12
2001/06
2001/12
2002/06
2002/12
2003/06
2003/12
2004/06
2004/12
2005/06
2005/12
2006/06

Coefficient

1986/06
1986/12
1987/06
1987/121988/06
1988/12
1989/06
1989/12
1990/06
1990/12
1991/06
1991/12
1992/06
1992/12
1993/06
1993/12
1994/06
1994/12
1995/06
1995/12
1996/06
1996/12
1997/06
1997/12
1998/06
1998/12
1999/06
1999/12
2000/06
2000/12
2001/06
2001/12
2002/06
2002/12
2003/06
2003/12
2004/06
2004/12
2005/06
2005/12
2006/06

Coefficient
-0.05

-0.1

-0.15
NRHI

-0.2
RHI

-0.25

-0.3

Fig4. Time profile of regression coefficient of the URHM, age of building Age:
1986/01–2006/09.

-0.02

-0.04

-0.06
RHI

-0.08

-0.1
NRHI

-0.12

-0.14

-0.16

Fig5. Time profile of regression coefficient of the URHM, time to nearest station TS:
1986/01–2006/09.

27

0.06
0.04
0.02
0

Coefficient

-0.02

RHI

-0.04
-0.06
-0.08
-0.1

NRHI

-0.12
-0.14

1986/06
1986/12
1987/06
1987/121988/06
1988/12
1989/06
1989/12
1990/06
1990/12
1991/06
1991/12
1992/06
1992/12
1993/06
1993/12
1994/06
1994/12
1995/06
1995/12
1996/06
1996/12
1997/06
1997/12
1998/06
1998/12
1999/06
1999/12
2000/06
2000/12
2001/06
2001/12
2002/06
2002/12
2003/06
2003/12
2004/06
2004/12
2005/06
2005/12
2006/06-0.16

Fig6. Time profile of regression coefficient of the URHM, travel time to CBD TT:
1986/01–2006/09.
Thus, structural changes are expected to occur as a background to changes in the regression coefficients.
For example, with respect to FS, the regression coefficient is estimated as positive in the basic model,
while it is estimated as negative from 1987 to the end of 1995. This indicates that the structureapparently differs for this period (1987–1995) compared with other periods.
Meanwhile, large changes occur within short periods, the range of which exceeds that of the long-termchanges. For example, when we look at the changes in regression coefficients of the TS and the TT to the
CBD shown in Figs 5 and 6, we can see the tendency of the coefficient line to increase throughout theperiod. For Age (Fig. 4), the trend is a decrease until the mid 1990s, followed by an increase thereafter.
While there are such changes observed over all periods, there are major fluctuations within short periods
beyond the range of such changes (beyond the difference in the regression coefficient between the
beginning and ending periods). These changes in the short periods cannot be considered to be structuralchanges.
On the basis of these findings, we consider that part of the change in the regression coefficients over28

time is caused by structural changes. It is also expected that some bias in the observation data group ineach period may have a certain effect.

4.3.Estimation of OPHI
4.3.1.Estimation of model and changes in price structure
Next, an OPHM is estimated on the basis of the procedure described in Section2.2. In the OPHM, toabsorb changes during a short period, the presence of which became apparent in the estimation of the
URHM, data over a certain period are pooled for the estimation. Using such an estimation method, we
expect to absorb changes during a short period. However, upon estimation of the OPHM, setting an
overlapped estimation period length (τ) is the key.
In the estimation of the URHI, it has been clarified that there is a seasonal characteristic of changes inthe thickness (more precisely, the number of transactions) of the housing market, and that the number oftransactions increases at the end of the fiscal year between January and March, and the number decreases
in July and August. When we attempt to absorb such a seasonally fluctuating characteristic in the market,
the estimation period should be set to exceed one year. When the period is set to be longer, parameters
are expected to be stable. However, it becomes difficult to accurately represent changes in the market
structure. In this study, on the basis of such assumptions, we examine the effects of varying τ between 12
months and 36 months on the price index and on the regression coefficients that represent the priceforming structure of the major variables.
First, estimation results using τ of 12 months are shown in Table 6 and Fig. 7. The coefficient ofdetermination decreases from 1986 through 1995, similar to that in the URHM; then it increases fromthe second half of 1996. On the whole, the coefficient of determination maintains an average ofapproximately 0.75, indicating a good result similar to that of the URHM.
Next, we focused on the regression coefficients of the estimated model. Table 7 shows descriptive
statistical values of the regression coefficients for 238 periods. Figs 8–12 show temporal changes in the
regression coefficients.
29

When we look at the changes in the regression coefficients over time, the wide fluctuations observed in
the URHM are eliminated and smooth changes are shown, making it easier to clarify overall tendencies.
The absolute values of the regression coefficients of the TS and TT to the CBD tend to decrease withtime. In other words, elasticity in terms of distance is decreasing. Regarding Age, quasi-periodic changes
in the regression coefficients are observed. The regression coefficient of FS is estimated to be negativefrom 1989 through 1995, then positive thereafter. It showed a stable value at around 0.1 for several years
after 1996, but since then, the elasticity has been gradually decreasing. In summary, we can see that
consumer preference is shifting its emphasis from the location to the amount of FS and Age in recent
years.
The coefficients of variation were obtained as follows (Table 7): FS = 2.424 (2.428 in the URHM), Age= −0.163 (−0.179 in the URHM), TS = −0.178 (−0.232 in the URHM), TT to CBD = −0.554 (−0.779 in
the URHM). On the whole, the values are smaller than those in the URHM. However, the variations of
the coefficients of FS and Age are not significantly different from those of the URHM, indicating that the
regression coefficients changed not because of temporal changes caused by sample bias, but because oflarge structural changes. The coefficients of variance of TS and TT to the CBD were significantly lower
than those in the URHM. This is speculated to be because of large temporal changes in the regression
coefficients caused by sample bias existing in a unit of time.
Next, in Figs 8–12, we observed changes in the regression coefficients when τ was varied between 12
months and 36 months. Compared with the estimation results of the URHM, with regard to the temporalchanges in the regression coefficients of each variable, we can see the presence of a time lag in thechanges in regression coefficients as τ becomes longer. This tendency is commonly observed in all thevariables. Such a time lag is also expected to affect the price indices.

30

0.9

0.8
0.75
0.7

16,0000.65

14,00012,00010,000
8,000
6,000
4,000
2,000
0

1986/06
1986/12
1987/06
1987/121988/06
1988/12
1989/06
1989/12
1990/06
1990/12
1991/06
1991/12
1992/06
1992/12
1993/06
1993/12
1994/06
1994/12
1995/06
1995/12
1996/06
1996/12
1997/06
1997/12
1998/06
1998/12
1999/06
1999/12
2000/06
2000/12
2001/06
2001/12
2002/06
2002/12
2003/06
2003/12
2004/06
2004/12
2005/06
2005/12
2006/06

Number of observations

adjusted R square

0.85

Fig7. Estimation accuracy of the OPHM: between 1986/01 and 2006/09.
Table6. Estimation results of the OPHM (τ = 12): 1986/01–2006/09.

TimeConstant

FS :
Floor
space

198612198701198702198703198704
198705
198706198707
198708198709198710198711
198712

4.232
4.176
4.133
4.067
4.038
3.994
4.015
4.053
4.104
4.139
4.183
4.293
4.315

0.041
0.055
0.058
0.068
0.075
0.085
0.089
0.087
0.089
0.091
0.091
0.093
0.092

TS :
Age : Time toAge ofthebuilding nearest
station-0.129 -0.108-0.129 -0.112-0.126 -0.117-0.126 -0.120-0.123 -0.122-0.120 -0.124-0.122 -0.119-0.121 -0.118-0.120 -0.115-0.120 -0.111-0.119 -0.111-0.123 -0.108-0.121 -0.109

199001
199501200001

5.522
5.191
4.244

-0.0830.030
0.105-0.154-0.209-0.201-0.092-0.074-0.059

TT :
RT :
NU :
TravelMarket BD : Bus WT ×
NumberTime toreservati Dummy
BDof units
CBD
on timeNumber of
Adjusted
Observation
R squares-0.046-0.047-0.043-0.045-0.050-0.048-0.050-0.050-0.057-0.059-0.059-0.066-0.066

0.002
0.003
0.007
0.008
0.011
0.014
0.013
0.013
0.012
0.012
0.012
0.010
0.012

-0.028-0.028-0.025-0.022-0.021-0.019-0.021-0.021-0.018-0.020-0.020-0.016-0.009-0.156-0.106-0.075-0.023-0.088-0.133-0.095-0.053-0.049-0.084-0.162-0.231-0.2570.035
0.010-0.009-0.034-0.006-0.005-0.019-0.037-0.041-0.031
0.002
0.024
0.033

5,497
5,425
5,446
5,383
5,492
5,316
5,268
5,372
5,083
4,986
4,888
4,863
4,792

0.785
0.796
0.812
0.824
0.835
0.849
0.854
0.856
0.859
0.857
0.852
0.846
0.839

-0.074-0.047-0.008

0.020
0.019
0.021

0.007
0.030
0.008

-0.275-0.360-0.307

0.049
0.105
0.084

12,36014,903
10,033

0.788
0.681
0.710

200501

4.327

0.069

-0.185-0.076-0.020

0.023

0.000

-0.172

0.027

8,131

0.755

200601
200602
200603
200604
200605200606200607200608200609

4.454
4.443
4.450
4.439
4.438
4.446
4.449
4.485
4.494

0.057
0.062
0.063
0.064
0.069
0.069
0.070
0.064
0.060

-0.189-0.191-0.191-0.190-0.189-0.189-0.189-0.190-0.192-0.075-0.076-0.079-0.080-0.080-0.082-0.083-0.084-0.083-0.026-0.025-0.024-0.023-0.024-0.022-0.018-0.018-0.015

0.018
0.017
0.017
0.017
0.016
0.014
0.014
0.014
0.015

-0.006-0.006-0.006-0.007-0.007-0.008-0.008-0.007-0.008-0.404-0.461-0.544-0.486-0.486-0.444-0.409-0.347-0.249

0.117
0.141
0.172
0.149
0.148
0.130
0.118
0.098
0.061

9,684
9,622
9,506
9,617
9,844
9,699
9,726
9,837
9,920

0.776
0.777
0.780
0.778
0.777
0.778
0.777
0.778
0.778

31

Table7. Statistical values of major regression coefficients (τ = 12).
OPHM :Summary statistics of estimated parameter

Principal Independent RHI :1986.01 Variables
2006.09

AverageStandard
deviation

Skewness

Kurtosis

FS :Floor space/square
meters

0.013

0.033

0.079

-0.880-0.770Age :Age of building-0.190-0.185

0.030

0.545

-0.392

TS :Time to nearest
station-0.078-0.0820.015-0.9840.214

TT :Travel Time tocentral business
district-0.040-0.0420.023-0.518-1.084

Adjusted-R Square

0.837

0.738

0.049

0.237

-0.745

Number of Samples

211,179

10,178.252

2,709.339-0.198-0.467

1986.12 - 2006.09:Monthly ,Number of Mode=238

6.5

6Coefficient

5.5
12months
5

36months
24months

4.5

4

1987/05
1987/11
1988/05
1988/11
1989/05
1989/11
1990/05
1990/11
1991/05
1991/11
1992/05
1992/11
1993/05
1993/11
1994/05
1994/11
1995/05
1995/11
1996/05
1996/11
1997/05
1997/11
1998/05
1998/11
1999/05
1999/11
2000/05
2000/11
2001/05
2001/11
2002/05
2002/11
2003/05
2003/11
2004/05
2004/11
2005/05
2005/11
2006/05

3.5

Fig. 8. Time profile of regression coefficient of the OPHM, constant term cnst:
1986/01–2006/09.

32

1987/05
1987/11
1988/05
1988/11
1989/05
1989/11
1990/05
1990/11
1991/05
1991/11
1992/05
1992/11
1993/05
1993/11
1994/05
1994/11
1995/05
1995/11
1996/05
1996/11
1997/05
1997/11
1998/05
1998/11
1999/05
1999/11
2000/05
2000/11
2001/05
2001/11
2002/05
2002/11
2003/05
2003/11
2004/05
2004/11
2005/05
2005/11
2006/05

Coefficient

1987/05
1987/11
1988/05
1988/11
1989/05
1989/11
1990/05
1990/11
1991/05
1991/11
1992/05
1992/11
1993/05
1993/11
1994/05
1994/11
1995/05
1995/11
1996/05
1996/11
1997/05
1997/11
1998/05
1998/11
1999/05
1999/11
2000/05
2000/11
2001/05
2001/11
2002/05
2002/11
2003/05
2003/11
2004/05
2004/11
2005/05
2005/11
2006/05

Coefficient
0.12

0.08
0.1

0.06

0.04

0.02

-0.02
0

-0.04

-0.12

-0.16

-0.22
12months

-0.06

-0.08
36months

-0.1

-0.14
24months

Fig9. Time profile of regression coefficient of the OPHM, floor space FS: 1986/01–
2006/09.
-0.1

-0.12

-0.14

24months-0.18
36months-0.2

12months

-0.24

-0.26

Fig10. Time profile of regression coefficient of the OPHM, age of building Age:
1986/01–2006/09.

33

1987/05
1987/11
1988/05
1988/11
1989/05
1989/11
1990/05
1990/11
1991/05
1991/11
1992/05
1992/11
1993/05
1993/11
1994/05
1994/11
1995/05
1995/11
1996/05
1996/11
1997/05
1997/11
1998/05
1998/11
1999/05
1999/11
2000/05
2000/11
2001/05
2001/11
2002/05
2002/11
2003/05
2003/11
2004/05
2004/11
2005/05
2005/11
2006/05

Coefficient

1987/05
1987/11
1988/05
1988/11
1989/05
1989/11
1990/05
1990/11
1991/05
1991/11
1992/05
1992/11
1993/05
1993/11
1994/05
1994/11
1995/05
1995/11
1996/05
1996/11
1997/05
1997/11
1998/05
1998/11
1999/05
1999/11
2000/05
2000/11
2001/05
2001/11
2002/05
2002/11
2003/05
2003/11
2004/05
2004/11
2005/05
2005/11
2006/05

Coefficient
-0.05

-0.06

-0.07

-0.08
12months

-0.09

-0.06
24months
36months

-0.1

-0.11

-0.12

-0.13

Fig11. Time profile of regression coefficient of the OPHM, time to nearest station TS:
1986/01–2006/09.

0

-0.02

-0.04

12months
24months-0.08
36months-0.1

Fig12. Time profile of regression coefficient of the OPHM, travel time to CBD TT:
1986/01–2006/09.

34

4.3.2.Evaluation of indices by different τ
We set τ in order to avoid the effects of specific bias in the observation data when the data were divided
monthly; the bias, if present, is reflected in the regression coefficients and may generate effects that are
difficult to differentiate from structural changes. The background of the bias is not clear at this stage ofthe study, but from the monthly samples, seasonal changes in the timing of when properties are put on
the market were confirmed. Therefore, if we wish to avoid the seasonal fluctuation characteristic, at least
12-month periods are necessary. Namely, the data division should accommodate the bias resulting fromseasons with or without a large movement of people. However, when τ was extended, the presence of atime lag relative to changes in the market was also observed.
Therefore, we varied τ between 12 months and 36 months, and observed the changes in the price indices,
as shown in Fig. 13.

907036months
60

50

40

12months

30

20
1987/05
1987/11
1988/05
1988/11
1989/05
1989/11
1990/05
1990/11
1991/05
1991/11
1992/05
1992/11
1993/05
1993/11
1994/05
1994/11
1995/05
1995/11
1996/05
1996/11
1997/05
1997/11
1998/05
1998/11
1999/05
1999/11
2000/05
2000/11
2001/05
2001/11
2002/05
2002/11
2003/05
2003/11
2004/05
2004/11
2005/05
2005/11
2006/05

Index=yen per square meter80

Fig13. Time profile of OPHI: 1986/01–2006/09.

35

We can see that the time lag in the regression coefficients also affects the price indices. When wecompare changes in the price indices (Fig. 13) with changes in the constant term (Fig. 8), we can see thata time lag is generated in the coefficient of the constant term (Fig. 8) as the connection period length
increases. In contrast, for the price index (Fig. 13), as the connection period length increases, theincrease in price occurs earlier. In other words, as the connection period length increases, while the pricechange excluding the effects of regression coefficients is delayed, the increase in the price index occurs
at an earlier period than the actual time of the price increase because the effect of the price increase in
the future is incorporated in advance. This effect is particularly strong in the bubble-economy years of1986 and after, when the price increase was particularly rapid.
These findings indicate that τ should be short for the estimation of the price index. Then, because it isnecessary to overlap at least 12 months to exclude the seasonal change in the number of transactions, theoptimal τ is determined to be 12 months.

4.4.Comparison between RHI/ URHI and OPHI
In the analysis described in the previous section, the OPHI with τ = 12 months was shown to be the mostaccurate in representing the market trends. Here, we compare the RHI, the URHI, and the OPHI for τ =12 months (Fig. 14).
When these indices are compared, large fluctuations in the URHI are noted (Fig. 14). The magnitude ofthese fluctuations seems to be different from our actual experience of price changes, because we did notexperience any large increase or decrease in the prices of condominiums with specific qualities during
the periods when the indices showed a large increase or decrease. In particular, because the fluidity of
the condominium market is low, instantaneous changes in the price are not expected.

36

3

2.5

Ristrict

Index:2000.01=1.0

2

1.5
N-Ristrict1

OPHM12

1986/06
1986/12
1987/06
1987/121988/06
1988/12
1989/06
1989/12
1990/06
1990/12
1991/06
1991/12
1992/06
1992/12
1993/06
1993/12
1994/06
1994/12
1995/06
1995/12
1996/06
1996/12
1997/06
1997/12
1998/06
1998/12
1999/06
1999/12
2000/06
2000/12
2001/06
2001/12
2002/06
2002/12
2003/06
2003/12
2004/06
2004/12
2005/06
2005/12
2006/06

0.5

Fig14. Comparison of indices between RHI,URHI and OPHI(τ=12): 1986/01–2006/10.
The magnitude of fluctuation in an index is not a priori a point of evaluation of the quality of the index.
However, such excessive fluctuations cannot be justified. In particular, while large changes wereobserved in the regression coefficient, it is difficult to expect that consumers in the market randomly andsignificantly changed their preferences; therefore, we speculate that the regression coefficient largely
changed because of bias in the data, and that, as a result, fluctuations in the price index occurred.
When we compare the RHI with the OPHI, there are large differences between the two from 1986
through 1990. When we further compare the URHI with the above two indices, the URHI showed
random fluctuations centered around the OPHI. While the URHI enables us to represent market changes
most sensitively, it fluctuates considerably because of the large bias in samples. Considering thesecharacteristics, rather than the RHI, the URHI can represent changes in the market structure more
appropriately and is expected to be more accurate.
The results of the above analysis demonstrated that among the RHI, URHI, and OPHI (τ = 12 months),
the OPHI (τ = 12 months) is superior, because the RHM is not able to accurately estimate the market
trends under the circumstance of large changes in the price level and the price structure during the years
37from 1986 through 1990, and because the URHI fluctuates considerably because of the seasonal
fluctuation of transactions as well as other biases.

5.Conclusions
In the series of analyses described above, we focused on temporal changes in the house (condominium)
price structure, and examined the effects of structural change and seasonal sample bias on price indices.
As a result of the analyses, we clarified the following.
First, when using the model of URHI, which takes into account changes in the market structure, theregression coefficients widely fluctuate in each period or every couple of periods. Although thisfluctuation is observed within short periods, a specific trend is observed in the long term. The fluctuationis speculated to be because of not only the occurrence of structural changes in the condominium market,
but also the occurrence of bias in the transaction samples in each period. More precisely, we find that
there is a seasonal fluctuation in the condominium market, which leads to active or sluggish transactions
depending on the time of year, and that such changes affect the regression coefficients.
To eliminate the bias existing in transaction samples in each period, which are mainly exhibited as theseasonal fluctuation, and to eliminate changes in the market structure, we proposed OPHM. Here, toeliminate the bias in the number of transactions and in samples affected by the seasonal change in thenumber of transactions, we set τ from 12 months to 36 months for the estimation. The results depicting
the changes in major regression coefficients over time showed gradual changes instead of the wide
fluctuations observed in the URHM, thus, it became easier to understand the trend.
Temporal changes in the regression coefficients revealed by OPHM showed that the absolute values of
the regression coefficients of TS and TT to the CBD have tended to gradually decrease in recent years,
indicating that the elasticity in terms of distance has become smaller. There were slight periodic changes
in the regression coefficient of Age, and the regression coefficient of FS was estimated to be negativefrom 1989 through 1995, then positive thereafter. These results demonstrate that consumer preference ismoving towards the amount of FS and Age, rather than location.
38

When τ between 12 months and 36 months were compared, the existence of a time lag in the regression
coefficients was observed. This time lag in the regression coefficients from 1986 during the bubbleeconomy years when prices were rapidly increasing also resulted in a large time lag in the price index.
From these findings, τ should be short. In addition, considering the aim of resolving the bias because of
seasonal changes in the number of transactions, the most suitable τ was determined to be 12 months.
The comparison among RHI, URHI and OPHI (τ = 12 months) showed that there are large fluctuationsin URHI, and that the magnitude of the fluctuations differs significantly from that in actual prices.
The comparison between URHI and OPHI showed the presence of large differences between the two
indices from 1986 through 1990. When URHI is added to this comparison, we can see random changes
in the URHI centered around the values of the OPHI. From the results of the comparison of these three
price indices, it was speculated that, because there were large changes in the price structure from 1986
through 1990, RHI was unable to accommodate such changes, so that large differences between URHI
and URHI/OPHI (τ = 12 months) were observed.
The results of the above series of analyses indicated the superiority of the estimation by OPHI (τ = 12months) in the secondhand condominium market in Japan, when structural changes in the market are to
be accommodated.

39

Appendix. Statistical nature of RHI and URHI

Characteristics of RHI and URHI
The characteristics of price indices generated by the two methods are summarized. Some of thecharacteristics, which are basic principles of regression analysis, will also be noted in relation to thehedonic housing price index.
The principles set in this paper are modified as shown below, using different notations. First, we attempt
to obtain a price index using only two periods, i.e., period 0 and period 1, with period 0 as the reference
point. The price data in each period are expressed as y0(n0 × 1), y1(n1 × 1), and the explanatory variabledata are expressed as X0(n0 × K), X1(n1 × K). Here, the notation of (n × m) represents the size of thematrix, i.e., number of rows × number of columns. n0 and n1 are the number of items of observation datain each period. k is the number of explanatory variables (including the constant term). The RHM is given
below, using the pooled observation data for periods 0 and 1 as follows.

~y = X~β + u
~
,

(28)

where:⎛y ⎞ ~ ⎛0~y = ⎜⎜ 0 ⎟⎟, X = ⎜⎜⎝1
⎝ y1 ⎠

X0 ⎞⎛ a ⎞ ~ ⎛u0 ⎞⎟⎟, β = ⎜⎜ 1 ⎟⎟, u= ⎜⎜ ⎟⎟
X1 ⎠⎝b⎠⎝ u1 ⎠.

0 is a column vector in which all elements are zero.
1 is a column vector in which all elements are one.
The first columns of both X0 and X1 are vector 1 because of the constant terms.

~

The first column (0 ' , 1' ) of X is a time dummy variable of period 1.
a1 is the regression coefficient of the time dummy variable of period 1.

40

b(K × 1) is a regression coefficient vector corresponding to the explanatory variables (including theconstant term) from which the time dummy variable is excluded.

β( K + 1 × 1) is a regression coefficient vector comprising a and b(K × 1).
1

Both u0(n0 × 1) and u1(n1 × 1) are random disturbance vectors.
In contrast, the URHM is given by the equations below, using the observation data for each period.

y 0 = X 0 β 0 + u0(29)

y1 = X 1 β1 + u1

(30)

Characteristic 1:

The sum of the errors between the observed value and the estimated value ofthe price for each period is zero.

~ ~

~

~

~ ~ ~

~

y, ~
In the case of the RHM, from βˆ = ( X ' X ) −1 X ' ~y is estimated as yˆ = Xβˆ = X ( X ' X ) −1 X ' ~y and the
~ ~ ~

~

~

y − yˆ = ( I − X ( X ' X ) −1 X ' ) ~y . By multiplying both sides of this equation by X 'error is given as uˆ = ~~~from the left, we derive X ' uˆ = 0 . The first column vector of the data matrix X is the time dummy
variable for period 1 and the second column vector is 1 (because it corresponds to the constant term).

~
Then, X ' uˆ = 0 indicates that the sum of the estimation errors for period 0 = 0 and the sum of theestimation errors for period 1 = 0. The same result can be derived for the URHM .

~Characteristic 2: The estimated equation yˆ = Xβˆ of the RHM passes through the center of gravity ofthe observation data of each period.

~

From the estimated equation yˆ = Xβˆ , we obtain:yˆ 0 = X 0 bˆ , uˆ 0 = ~y 0 − yˆ 0 → ~y 0 − uˆ 0 = X 0 bˆ

(31)

and

41

yˆ 1 = aˆ1 + X 1bˆ , uˆ 1 = ~y1 − yˆ 1 → ~y1 − uˆ1 = aˆ1 + X 1bˆ .

(32)

If both sides are multiplied by 1' (vector with all the elements equal to 1) and the results are divided by
the number of items of observation data of each period, with consideration given to characteristic 1, wethen obtain:

y 0 = x 0 ' bˆ(33)

y1 = aˆ1 + x1 ' bˆ .

(34)

and

We find y 0 and y1 to be the average prices and x 0 and x1 to be the averages of the explanatory
variables in the two periods. Similarly, in the URHM , we obtain:

y 0 = x 0 ' βˆ 0(35)y1 = x1 ' βˆ1 .

(36)

and

Characteristic 3: If the averages of the explanatory variables of period 0 and period 1 are equal( x 0 = x1 ), then the RHI and the URHI of the houses having this average value of quality are equal.
When x 0 = x1 = x , the estimated house prices with the quality value of x are as below, fromcharacteristic 2.

y 0 = x ' bˆ = x ' βˆ 0(37)y1 = aˆ1 + x ' bˆ = x ' βˆ1

(38)
42

This means that the estimated house prices using both models (RHM and URHM ) for houses with thequality value of x are the average house prices in each period. Therefore, characteristic 3 holds.

The relationship with the URHI
In addition to the RHM of equation (1) and the URHM based on equations (9) and (10), we add another
RHM without time dummy variables.

~y = Xβ + u
*
*

(39)

Here,

~y = ⎛⎜ y 0 ⎞⎟ X = ⎛⎜ X 0 ⎞⎟ u = ⎛⎜ u*0 ⎞⎟⎜y ⎟⎜ X ⎟ * ⎜u ⎟⎝ 1 ⎠,⎝ 1⎠,⎝ *1 ⎠
holds. β * represents the regression coefficient vector, which contains a constant term but does notcontain time dummy variables.
Characteristic 4: If the variance of the random disturbance term of each period is the same, theregression coefficient β̂ * of the RHM without the time dummy variable is the weighted average of theregression coefficients βˆ 0 , βˆ1 of the URHM using the inverse matrix of the respective variance–
covariance matrix of the regression coefficient.
The estimated value of β * of the model based on equation (10) is as follows.

βˆ * = ( X ' X ) −1 X ' ~y= ( X 0 ' X 0 + X 1 ' X 1 ) −1 ( X 0 ' y 0 + X 1 ' y1 )(40)

Meanwhile, the estimated values of β 0 , β1 of the URHM based on equation (7) and (8) are as follows.

βˆ 0 = ( X 0 ' X 0 ) −1 X 0 ' y 0

(41)
43

βˆ1 = ( X 1 ' X 1 ) −1 X 1 ' y1

(42)

Assuming that u0～N (0, σ 2 I 0 ) and u1～N (0 , σ 2 I 1 ) , namely, the variance of the random disturbanceterm in each period is the same, we obtain:

Var ( βˆ 0 ) = σ 2 ( X 0 ' X 0 ) −1 （ = V0 ,）(43)

Var ( βˆ1 ) = σ 2 ( X 1 ' X 1 ) −1 （ = V1 ）.

(44)

Equation (28) can then be rewritten using equations (29), (30), (31) and (32) as follows.

βˆ * = (σ 2V0−1 + σ 2V1−1 ) −1 (σ 2V0−1 βˆ 0 + σ 2V1−1 βˆ1 )= (V −1 + V −1 ) −1 (V −1 βˆ + V −1 βˆ )
0

1

0

0

1

1

(45)

Namely, the regression coefficient β̂ * of the RHM is obtained by calculating the weighted average ofthe regression coefficients βˆ 0 , βˆ 1 of the URHM using the inverse matrix of the respective variance–
covariance matrix of the regression coefficient.
Actually, also in the case of the RHM represented by equation (26), which includes time dummy
variables, the regression coefficients other than the constant term and time dummy variables are theweighted averages of the corresponding regression coefficients of the URHM using the inverse matrixof the respective variance–covariance matrix. This is simply because the RHM represented by equation(26) is obtained by shifting the regression plane of the RHM without the time dummy variables,
represented by equation (28), parallel as it passes through the center of gravity of the observation data
for each period (Characteristic 2). Consequently, the regression coefficients of equations (26) and (28)are the same except for the constant term and time dummy variables.
Characteristic 5: The RHI and the URHI are identical for those houses whose property characteristics
are defined by calculating the weighted averages of the explanatory variables for each period using the44

variance–covariance matrix of the regression coefficients βˆ 0 , βˆ 1 of the URHM.
As is evident from the method of deriving structurally restricted indices described in Section 2.1, theRHI for period 1 generated using the RHM based on equation (26) is represented as a time dummy
variable â1 . Let us consider a case where the URHI and restricted price index have the same value.
Namely, the URHI of houses with quality m for period 1 can be described as follows.

yˆ1 − yˆ 0 = m ' βˆ1 − m ' βˆ 0 = m ' ( βˆ1 − βˆ 0 )(46)

We attempt to solve this equation to find the value of m for which equation (46) equals â1 . This helps us
understand under which conditions the two indices, i.e., the RHI and URHI, show similar movements.
Provided that y 0 and y1 are the mean prices, and x 0 and x1 are the averages of the explanatory
variables in each period, the RHI â1 is expressed as follows.

aˆ1 = ( y1 − x1 ' βˆ * ) − ( y 0 − x 0 ' βˆ * )= ( y − y ) − ( x '− x ' ) βˆ
1

0

1

0

*

(47)

−1
−1 −1
−1
−1
This equation can be transformed by βˆ * = (V0 + V1 ) (V0 βˆ 0 + V1 βˆ1 ) , which is derived fromthe analysis of equation (27), as follows.

aˆ1 = ( x 0 'W0 + x1 'W1 )( βˆ1 − βˆ 0 )(48)

We know that:

W0 = V0 (V0 + V1 ) −1(49)

W1 = V1 (V0 + V1 ) −1

(50)

45

hold. Therefore, we can obtain the value of m for which m ' ( βˆ1 − βˆ 0 ) = aˆ1 as follows.

m ' = x 0 'W0 + x1 'W1(51)

This shows that m is obtained by a weighted averaging of the averages of the explanatory variables foreach period using the variance–covariance matrix of the regression coefficients βˆ 0 , βˆ1 of the URHM .
This once more demonstrates the points described in Characteristic 3, because m = x holds
when x 0 = x1 = x .

46

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