CSIS Discussion Paper No.156

Property Price Index Theory and Estimation:
A Survey∗
Chihiro Shimizu† and Koji Karato‡
November 28, 2018
Abstract
Property has the particularity of being a non-homogeneous good, and based on this,
it is necessary to perform quality adjustment when estimating property price indexes.
Various methods of quality adjustment have been proposed and applied, such as thehedonic method often used in price statistics and, due to the fact that the informationthat can be used in estimation is limited, the repeat sales price method, methods using
property appraisal price information, and so forth. However, since there is a lack oftheoretical knowledge and data restrictions, it is no exaggeration to say that it is diﬃcult
to evaluate their practical application in the present situation. Therefore, focusing on
the hedonic method that has been proposed as a quality adjustment method for property
price indexes, in addition to repeat sales price indexes and indexes employing property
appraisal prices, this paper aimed to outline the underlying econometric theory andclarify the advantages and disadvantages of the respective estimation methods.
Key Words:
Hedonic price index; Repeat sales price index; Age eﬀect; Hybrid method; Property
appraisal price method; SPAR
Journal of Economic Literature Classiﬁcation Numbers:
C2, C23, C43, D12, E31, R21.

∗ The authors thank Erwin Diewert, David Geltner and Marc Francke for helpful discussions.

The ﬁrst
author gratefully acknowledges the ﬁnancial support of Nomura Foundation of Japan.
† Correspondence: Nihon University, Setagaya Campus, 34-1 Shimouma 3 chome, Setagaya-ku, Tokyo
154-8513,Japan. Professor, Nihon University, shimizu.chihiro@nihon-u.ac.jp & Research Aﬃliate, MITCenter for Real Estate, cshimizu@mit.edu.
‡ Professor, Faculty of Economics, University of Toyama. kkarato@eco.u-toyama.ac.jp.

11 Introduction
The formation and collapse of property bubbles has a profound impact on the economic
administration of many leading nations. The property bubble that began around the mid1980s in Japan has been called the 20th century’s biggest bubble. In its aftermath, the country
faced a period of long-term economic stagnation dubbed the “lost decade.” Many countries
had similar experiences with this kind of problem – for example, Sweden’s economic crisis in
the 1990s and the global ﬁnancial crisis and economic stagnation caused by the formation andcollapse of the U.S.-centered property bubble in the early 21st century.
In light of this, it was pointed out that the “information gap” which existed between policymaking authorities and the property (including housing) and ﬁnancial markets was a problem.
In 2009, the IMF proposed the creation of a housing price index to the G20 in order to ﬁll in thisinformation gap, and the proposal was adopted. Furthermore, in 2011, it was suggested thatthe next economic crisis would be caused by land – i.e., proﬁt-generating property (commercial
property) – and it was decided to create a commercial property index as well. But how should
these property price indexes be created?
Property standards and facilities vary to a greater or lesser extent for each building, so there
is no such thing as identical properties. Even if one assumed that the standards and facilities
were the same, the process by which quality deteriorated would diﬀer by building age, so
the buildings would become non-homogeneous over time. In other words, property has theparticularity of being a non-homogeneous good. In addition to this problem, the developmentof building technology is relatively fast, so quality changes over time. That is, not only doesa building’s functionality decline over time, but it becomes economically obsolescent withthe advance of technology. As well, in cases where the surrounding environment changes
signiﬁcantly through redevelopment and the like, location characteristics such as transport
accessibility of the city center also change.
When attempting to capture temporal ﬂuctuations in property prices while dealing with the
problems caused by property being a non-homogeneous good and changes in quality, it isnecessary to perform quality adjustment.
In order to address these problems, there are quite a few points that can be adapted fromexisting index theory, as typiﬁed by consumer price statistics. For example, with regard tochanges in quality accompanying technological development, the quality adjustment method
known as the hedonic approach is used in private vehicle price statistics and the like. It would
therefore be natural to also consider quality adjustment with the hedonic method for property
price indexes, since this enables consistency with other types of economic statistics.
However, when it comes to methods of quality adjustment for property price indexes, if one
looks at the Residential Property Price Indices Handbook published by EuroStat in 2013, it
present a variety of methods along with their advantages and disadvantages: a) Stratiﬁcation
or Mix Adjustment Methods，b) Hedonic Regression Methods，c) Repeat Sales Methods, andd) Appraisal-Based Methods. This is because, in reality, multiple methods have been appliedin the estimation of property price indexes.
Why have approaches other than the hedonic method been applied in practice?
The ﬁrst reason is the diﬃculty of quality adjustment. As explained previously, the reason
for performing quality adjustment of property is that it is a good for which no homogeneity
exists, so it is strongly heterogeneous. In such a case, in addition to the problems relating
to quality changes faced in consumer price statistics and the like, one must also address said2

heterogeneity. In other words, quality adjustment involves a high degree of diﬃculty.
The second reason is the lack of usable price information at the micro level when estimating
property price indexes. If attempting to apply the hedonic method, not only transaction
price, transaction time, and land/building size but also location-related information such as
the time to the city center and detailed information related to the building age and features are
required. When there is no such information, the price index must be estimated with limited
data. With the repeat sales method, quality adjustment is possible with just the transaction
price and transaction time, so it has the advantage of minimizing the information needed withrespect to property-related variables. That being the case, when attempting to measure pricechanges when information is limited, creating an index using only data for properties that aretransacted repeatedly is consistent with the general thinking behind price index estimation
methods.
However, unlike other goods and service markets, the property transaction market is extremely
thin. Thus, when attempting, for example, to create a monthly price index, one may easily
anticipate that many problems will occur, since – unlike markets where goods and services ofidentical quality are transacted frequently – this is a market in which property with identicalcharacteristics is transacted only once every few years.
Third, in actual property transaction practice, it is not uncommon for property appraisal
prices to be used. Not only is property strongly heterogeneous but there are few transactions –depending on the region and usage, there are even markets where almost no transactions exist.
In light of this, when trying to determine prices, using prices based on property appraisals is
a valid approach.
Thus, attempting to estimate property price indexes involves many diﬃculties. Compared tothe housing market, where there is a relatively large number of transactions and the quality
gap is small (i.e., it is more homogeneous), these diﬃculties mean that for commercial property(oﬃces, retail facilities, hotels, logistics facilities, hospitals, farmland, etc.): 1) there will be
more heterogeneity and a greater lack of information, 2) there will be a reduction in repeat
sales samples, and 3) there is a greater probability of property appraisal prices being used.
For markets for which property price information is relatively easy to obtain, the aim of thispaper is to outline the characteristics of the hedonic method and repeat sales method that
may be used in creating property price indexes, as well as estimation methods using property
appraisal prices. Speciﬁcally, focusing on the hedonic method and repeat sales method, it
will provide a comprehensive survey relating to quality adjustment methods when estimating property price indexes and clarify the characteristics of the various estimation methods.
What’s more, drawing on this outline, it will present a view of how property price indexes
should be created, from the perspective of estimation method theory.

2 The Hedonic Price Method
2.1 The Hedonic Approach
The hedonic approach is a technique established theoretically by Rosen (1974)[55]. Speciﬁcally,it treats a given product’s price as an aggregate (bundle of attributes) of the values of the
product’s various attributes (characteristics) and estimates the various attribute prices using
regression analysis. For many products circulating on the market, even when their intended
use is the same, considerable diﬀerentiation exists based on their performance, functions,etc. Diﬀerences in attributes are reﬂected in the product’s production costs. One could also
say that consumer evaluations of the product’s speciﬁc performance and functions are also3

reﬂected in the price determined by the market. However, the attributes themselves are notnecessarily bought and sold on the market. With the hedonic approach, by regressing productprice on variables representing attribute quality and quantity, it is possible to measure theshadow price of non-market goods based on the estimated coeﬃcient value.
Lancaster (1966)[44] has conducted theoretical analysis of consumer behavior based on theassumption that consumer utility depends not on the product itself but on the various features,functions, etc., that comprise the product. The product’s market price is thought to be
determined based on supply and demand in relation to its various characteristics. However,
the market with respect to these characteristics is not necessarily explicit but may be hiddenin the background of product price determination. Lancaster’s aim was to explicitly treat such
underlying mechanisms and analyze consumer behavior in diﬀerentiated goods markets.
Rigorously examining the relationship between diﬀerentiated product prices and consumer
behavior is essential in preparing price indexes. For example, in the case of digital consumer
electronics, passenger vehicles, housing, etc., even if the price is the same, quality will improveand functions increase as time passes. With the Laspeyres method, since a market basket isﬁxed at a baseline point in time, price indexes based on this method ignore changes in quality
and functionality. Using the hedonic approach helps estimate the performance ratio between
new and old products.
Rosen (1974)[55]’s price analysis of diﬀerentiated goods is a study that theoretically clariﬁes
the manner in which product prices comprised by bundles of attributes are generated on themarket. The study rigorously examines the relationship between the product supplier oﬀerfunction, product demander bid function, and hedonic market price function, and characterizes
the market price of products based on consumer and producer behavior. When this hedonic
market price function is used, it is possible to obtain the acceptable payment amount for
product attributes.
Section 2.2 below outlines Rosen (1974)[55]’s hedonic approach theory, while Section 2.3 addresses issues relating to the estimation of hedonic market price functions. Following Diewert,
Heravi and Silver (2007)[27], Section 2.4 summarizes diﬀerences based on a hedonic dummyindex and hedonic imputed index. Section 2.5 explains the characteristics of a producer
price-related quality-adjusted hedonic index. Section 2.6 summarizes the characteristics ofthe hedonic price method.

2.2 Hedonic Approach Theory
2.2.1 The Bid Function
Following Rosen (1974)[55]’s method, we will demonstrate the theoretical basis of the hedonic
approach, using real estate as an example. The value of characteristic k comprising realestate shall be expressed as zk (k = 1, 2, . . . , K). Real estate characteristics represent size,building structure, kitchen, bathroom, accessibility of transportation, natural environment,social environment, and so forth. According to Rosen, the relationship between real estatemarket price p and characteristic value z1 , . . . , zk , . . . , zK may be expressed with the following
hedonic price function h:
p = h(z1 , . . . , zk , . . . , zK )(1)
The main objective of Rosen’s analysis is to clarify how (1) is determined by the market.
Given market price function (1), consumers select the optimal combination of real estate4

characteristics. The issue of utility maximization may be formulated as follows:
max U (x, z)(2)s.t. I = x + h(z)

(3)

x,z

Here, U (·) is a well-behaved, strictly concave function, x is composite goods including goods
and services other than real estate, z = (z1 , . . . , zk , . . . , zK ) is the real estate characteristic
vector, and I is income. The composite goods price is standardized as 1. Based on theparameters of this optimization issue step, Uk /Ux = hk (z) is established. Note that Uk =∂U (x,z)∂U (x,z)∂h(z)，and hk (z) = ∂zk . In other words, this shows that the marginal∂zk ， Ux =∂xutility of the real estate characteristic measured using the marginal utility of income is equal
to the marginal contribution value of the attribute in market prices.
It is possible to determine the market price function using the bid function. Based on a given
utility level u and income I, if the bid oﬀered by a housing demander for real estate possessing
characteristic z is taken as θ, then based on (2), this may be written as U (I − θ, z) = u.
If one solves this for θ, the amount that a consumer is able to spend on housing with respect tovarious combinations of characteristic z may be expressed as the bid function θ(z; I, u), given
the utility level and income. In order to raise³(lower) the utility level´ u, the bid for housing∂θ(z;I,u)−1
with characteristic z must decrease (increase)= −Ux < 0 . Therefore, this shows∂uthat θ, when it reaches utility level u, is the maximum price that may be paid for housingwith characteristic z.
Based on (2), (3), and the bid function θ(z; I, u), one may write that U (I − θ(z; I, u), z) = u.
If this formula is partially diﬀerentiated for zk and 0 is included, the following is obtained:−Ux∂θ(z; I, u)+ Uk = 0∂zk

When the utility is maximized at the level of u∗ , since Uk /Ux = hk (z ∗ ) for the optimal
combination of characteristics z ∗ , the following two equations are deﬁnitely established:∂θ(z ∗ ; I, u∗ )= hk (z ∗ )∂zkθ(z ∗ ; I, u∗ ) = h(z ∗ )

(4)
(5)

(4) and (5) show that when the optimal characteristics are selected, the slope of the bidfunction and the slope of the market price function are consistent and the bid and market
price are also equal. In other words, based on the optimal characteristic value, the bid function
and market price function are contiguous.
When consumer incomes and preferences vary, the bid function also varies. However, since thebid function and market price function must be contiguous in market equilibrium, the market
price function is an envelope of the bid function for all consumers, with their various incomesand preferences.
2.2.2 The Oﬀer Function
It is also possible to deﬁne the price oﬀer function for real estate suppliers and theorize therelationship with the market price function from the issue of proﬁt maximization. For a given
level of technology, the oﬀer function is the minimum price oﬀered when a given proﬁt isreached. When a company selects the optimal characteristics and produces real estate, the5slope of the oﬀer function and the slope of the market price (per unit of real estate) function
will be consistent based on proﬁt maximization behavior, and the oﬀer price and market pricewill also be consistent. Therefore, based on the optimal characteristic value, the oﬀer function
and market price function are contiguous. Since heterogeneity exists in real estate producers’
technology, oﬀer prices also vary in accordance with this. Since the oﬀer price and market
price need to be consistent in equilibrium, the market price function is an envelope of the oﬀerfunction for various companies.
Based on the above, the hedonic market price function is an envelope of both the bid function
for an inﬁnite number of real estate demanders and the oﬀer function for an inﬁnite numberof real estate suppliers. As well, in the case of there being one supplier company, the bidfunction is equal to the marginal cost if one additional unit of real estate is produced (or the
average cost per unit of real estate). As a result, the market price function is equal to the
supplier’s marginal cost.
2.2.3 Willingness to Pay
If the bid function is used, it is possible to obtain consumers’ willingness to pay with respect
to changes in attributes. For z ∗ , let us now assume that p∗ = θ(z ∗ ; I, u∗ ) = h(z ∗ ). When real∗∗∗estate K’s characteristic zK
is increased to (zK), the demander’s willingness to pay (W T P )may be deﬁned with the following formula:¡¢∗∗∗W T P ≡ θ z1∗ , . . . , zK−1
, zK
; I, u∗ − p∗(6)
In other words, when the characteristic value changes incrementally, the willingness to pay isthe additional value that may be paid for real estate without changing the utility level. Since
the utility function U is∂2
U 2 Ukk − 2Ux Uk Uxk + Uk2 Uxx<0θ(z ∗ ; I, u∗ ) = x
2
∂zk
Ux3
when it is a strict concave function (the Hessian matrix is a negative deﬁnite matrix), the bidfunction is a concave function. (4) and (5) are established based on the optimal characteristic
value combination z ∗ , and given that the bid function is a concave function, one can derive
from
¡
¢
¡
¢
∗
∗∗
∗
∗∗
θ z1∗ , . . . , zK−1
, zK
; I, u∗ < h z1∗ , . . . , zK−1
, zK
= p′
That

p′ − p∗ > W T P(7)

In other words, caution is required with regard to the limit value of market price function characteristics, since as long as demanders are not homogeneous, it is possible that the willingness
to pay is overestimated. However, if it is assumed that changes in characteristic values will
be suﬃciently small, the market price function limit value may be used as an approximationof the willingness to pay.

2.3 Hedonic Market Price Function Estimation
2.3.1 Function Types
In order to accurately measure willingness to pay, estimation of the bid function is required,
but in general an approximation is used by estimating the hedonic market price function (1).
6

When estimating the hedonic market price function, the function type is an issue. Given that
simple estimation is possible, models such as double logarithms, semi-logs, and line shapes areoften used.
When real estate price at multiple points in time is observed as data, the hedonic market priceat time t for property n may be described with the following formula:tynt = αt + z t′n γ + εn(n = 1, 2, . . . , N (t); t = 0, 1, . . . , T )(8)Here, ynt is the housing price logarithm (ln ptn ) or exact numeric value (ptn ), αt is the unknownttt, . . . , znk)′ is the explanatory variable (characteristic) vector
time eﬀect, z tn = (1, zn1, . . . , znK
including a constant term, γ = (γ0 , γ1 , . . . , γk , . . . , γK ) is the coeﬃcient vector, and εtn is theerror term. As an example, a semi-log model including the time eﬀect may be written as:
ynt = ln ptn = αt + γ0 +

K
X

t
+ εtn
γk znk(9)

k=1

In this model, the estimation value of coeﬃcient γ shows the eﬀect of the characteristic value
with respect to real estate price, and if a dummy variable is used for each point in time, estimation may be made estimated based on the method of least squares. To avoid multicollinearityand distinguish all parameters, it is necessary to perform some kind of standardization for αtand γ0 . Typically, at the observation starting point t = 0, it is considered that α0 = 0, and adummy variable for each point in time is used with respect to t = 1, 2, . . . , T .
Since the function type of the hedonic market price function h cannot be speciﬁed in theoreticalterms, it must be selected with a statistical technique. Even if speciﬁed in a double logarithmic
model or semi-log model, the form is not necessarily the ideal one. Studies from the 1980s
onward, such as Linneman (1980)[45], have performed non-linear estimation using Box-Coxtransformation. In this case, the left side of (9) can be rewritten as follows:λ p − 1 λ ̸= 0
λ
y=
(10)

 ln pλ=0Here, λ is an unknown parameter. Halvorson and Polakowski (1981)[36] tested various function forms by applying Box-Cox transformation to a ﬂexible function form using a two-step
approximation formula including a cross-term between explanatory variables. In response to
their paper, Cassel and Mendelsohn (1985)[11] increased the explanatory power by including
multiple cross-terms between variables, but pointed out that there is a reduction in the reliability of the coeﬃcient estimation value due to multicollinearity and that interpretation of themarginal eﬀect of hedonic characteristic values becomes more diﬃcult. Cropper, Deck, andMcConnell (1988)[13] performed statistical tests based on a translog form and Diewert-typeutility function (Barten (1964)[3], Diewert (1971)[19], Diewert (1973)[20]), showing that if observational errors are included in the variables, a linear model or linear Box-Cox transformation
model is superior to quadratic form Box-Cox transformation when it comes to formulation.
There is also research that has proposed using a non-parametric method or semi-parametricmethod instead of a parametric function form to formulate the hedonic price function. With
these approaches, attribute prices are inferred directly from the data without specifying afunction form in advance (Knight et al. 1993[40], Anglin and Gencay (1996)[1], Pace 1995[53]).
However, it has also been pointed out that, as with parametric analysis techniques, these do7

not free one from data-related problems (multicollinearity). In tests relating to the selectionof parametric versus non-parametric models, Anglin and Gencay (1996)[1] have shown that itis relatively easy to dismiss parametric models. It is not that the parametric model variable
structure is weak; rather, this result was demonstrated even for parametric models that passed
a number of standard tests for model selection. Using a more ﬂexible Generalized Additive
Model (GAM), Pace (1998)[52] estimated a semi-parametric-type hedonic price function anddemonstrated that it was superior to all types of parametric model. Since GAM itself is anestablished statistical technique, this ﬁnding shows that the incorporation of a non-parametricmethod in the hedonic approach is extremely eﬀective.
2.3.2 The Problem of Distinguishing the Marginal Bid Value Function
If characteristic values have a signiﬁcant eﬀect on market prices, the willingness to pay willcause divergence between the hedonic market price function and bid value function, so it isnecessary to estimate the bid value function or bid value marginal eﬀect. As a method of
estimating the bid value function, Rosen (1974)[55] has proposed a method that regresses themarket price function marginal eﬀect on characteristic values and other exogenous variables.
ĥk = Dk (z, A)(11)ĥk = Sk (z, B)(12)Here, ĥk is the marginal eﬀect for hedonic market price function characteristic k, D(·) andS(·) are the characteristic’s demand and supply functions, and A and B are vectors showing
the real estate demander and supplier type, respectively (based on income, manufacturing
technology, etc.). Since the marginal eﬀect is the shadow price of the characteristic value, (11)and (12) are supply and demand simultaneous equations using inverse demand (bid value)
and inverse supply (oﬀer price), and supply and demand are distinguished using A and B as
instrumental variables.
Following Rosen’s model, Witte, Sumka, and Erekson (1979)[59] estimated simultaneous equations for three characteristics covering multiple housing markets. However, as Brown and
Rosen (1982)[6] have pointed out, it is not possible to properly distinguish between characteristic value supply and demand with estimation based on this method. Since the market
price function marginal eﬀect ĥk estimated in the ﬁrst step is derived from h(z), one may
consider that the characteristic price shown with the marginal eﬀect is also a function of z.
Demand for z depends on the various characteristic prices, and there is a correlation between
characteristic prices and characteristic demand equation errors. In other words, it is possiblethat the eﬀect of characteristic prices on characteristic demand is estimated with a bias.
This problem of distinguishing the bid value function and oﬀer function has been considered by
Diamond and Smith (1985)[18] and Mendelsohn (1985)[49]. First, with regard to estimationof the ﬁrst step hedonic market price function, it is pointed out that, apart from characteristic
vectors, there is a need for exogenous variables not included in either the bid value function
or oﬀer function as well as for a characteristic value exponential term. Then, in the second
step, a marginal bid value function simultaneous equation system is estimated simultaneously
using exogenous variables solely to meet the distinction conditions. Sheppard (1999)[56] hasdiscussed the distinction problem in greater detail. Ekeland, Heckman, and Nesheim (2004)[29]and Heckman, Matzkin, and Nesheim (2010)[39] proposed a distinction method for hedonic
price estimation using a non-parametric approach.

8

2.4 Price Index Estimation Based on the Hedonic Approach
2.4.1 Time Dummy Hedonic Regression
The hedonic approach is a useful technique when creating quality-adjusted price indexes.
There are two representative types of hedonic price index: (i) time dummy hedonic indexesand (ii) imputed hedonic indexes. Following Diewert, Heravi and Silver (2007)[27], we discuss
diﬀerences between the two types of price index below.
In (8), taking the observation period as two points in time, (t = 0, 1), one may assume thefollowing estimation model that regresses logarithmic price on an explanatory variable vector
with the time dummy and constant term excluded:
ynt ≡ ln ptn = αt +KXtγk znk+ εtn(n = 1, 2, . . . , N (t); t = 0, 1)

(13)

k=1

Here, αt shows the average level of the product’s quality-constant price for each period, andthe overall scale of logarithmic price changes from time 0 to time 1 is α1 − α0 .
Let us take 1t as an N (t) dimension vector comprising everything from 1 and 0t as an N (t)dimension vector comprising everything from 0. As well, let us take y 0 and y 1 as the N (0)and N (1) dimension vectors for the time 0 and time 1 logarithmic prices respectively, Z0 andZ1 as the N (t) × K explanatory variable matrices for time 0 and time 1 respectively, and ε0and ε1 as the N (0), N (1) dimension error vectors for time 0 and time 1 respectively. If werepresent (13) as matrices for time 0 and time 1, they may be written as follows:
y 0 = 10 α0 + 00 α1 + Z0 γ + ε0(14)

y 1 = 01 α0 + 11 α1 + Z1 γ + ε1

(15)Here, if we take αt∗ , γ ∗ as estimators based on the method of least squares, one can formulate
the following using the estimators and the realized value et of the least squares residual error:
y 0 = 10 α0∗ + 00 α1∗ + Z0 γ ∗ + e0(16)

y 1 = 01 α0∗ + 11 α1∗ + Z1 γ ∗ + e1

(17)

£
¤′
¤′
£
For (16) and (17), if we deﬁne y = y 0′ y 1′ （(N (0) + N (1)) × 1 vector)， e· = e0′ e1′ ¸£¤′
1 00 Z0（(N (0)+N (1))×1 vector), ϕ∗ = α0∗ α1∗ γ ∗′ （(2+K)×1 vector), and X = 0
01 11 Z1（(N (0) + N (1)) × (2 + K) matrix), then (16) and (17) may be rewritten as follows:y = Xϕ∗ + e(18)Here, since X and the residual error e are orthogonal, we can obtain:
X′ e = X′ (y − Xϕ∗ ) = 02+K

(19)

In other words, we can obtain 1′0 e0 = 0, 1′1 e1 = 0 and Z0′ e0 + Z1′ e1 = 0K .
Therefore, using the residual errors for (16) and (17):1′0 y 0 = N (0)α0∗ + 1′0 Z0 γ ∗(20)

1′1 y 1 = N (1)α1∗ + 1′1 Z1 γ ∗

(21)

9

If we work out α0∗ and α1∗ from this, we can obtain:¡ 0¢
y − Z0 γ ∗
N (0)
N (0)
N (0)
¢
¡ 1
′
′ 1
′ 1 ∗
11 y − Z1 γ ∗11 y
11 Z γ
∗
α1 =
−
=
N (1)
N (1)
N (1)
1′
1′ y 01′ Z0 γ ∗α0∗ = 0
− 0
= 0

(22)
(23)

(22) and (23) show the quality-constant logarithmic price level. 1′t y t /N (t) shows the arithmetic mean of the logarithmic price for time t = 0, 1 and 1′t Zt /N (t) shows the arithmetic
mean of the characteristic vector for time t = 0, 1. In other words, α0∗ is equal to the result
obtained by subtracting the average value of all characteristic values from the average value ofthe logarithmic price (arithmetic average of the quality-adjusted logarithmic price). Based on
the above, the hedonic time dummy estimation value based on the logarithmic price change
from time 0 to time 1 is the following diﬀerential:
LPHD = α1∗ − α0∗(24)The explanatory variable matrices for (18) are expressed as follows:
¸
· 0¸
·
Z
10 00
, Z=
V=
Z1
01 11
Here, V is a (N (0)+N (1))×2£ matrix¤ and Z is a (N (0)+N (1))×K matrix. If the explanatory
variable is rewritten as X = V Z , since the residual error vector is e = y − Vα∗ − Zγ ∗ ,the least squares estimator
· ∗¸
α
∗
(25)
α = 0∗ = (V′ V)−1 V′ (y − Zγ ∗ )α1can be obtained from ∂e′ e/∂α∗ = 0. Based on (25), the residual error may be rewritten as
e = M (y − Zγ ∗ ). Here,
¸
· 0¸
·
0
M
I0 − 10 1′0 /N (0)
′
−1 ′
,
M=
=
I
−
V(V
V)
V
=
0
I1 − 11 1′1 /N (1)M1
I is a (N (0)+N (1))×(N (0)+N (1)) identity matrix, and It is a N (t)×N (t) identity matrix. If′
we deﬁne y ∗ = My and Z∗ = MZ, the error sum of squares is e′ e = (y ∗ − Z∗ γ ∗ ) (y ∗ − Z∗ γ ∗ ),so the least squares estimator for γ can be obtained as follows:−1γ ∗ = (Z∗′ Z∗ )¡¢−1 ¡ 0∗′ 0∗¢
Z∗′ y ∗ = Z0∗′ Z0∗ + Z1∗′ Z1∗
Z y + Z1∗′ y 1∗

(26)

If we ﬁrst calculate γ ∗ from (26) and then plug it into (25) (or (22) or (23)), the time eﬀect
estimator α∗ can be obtained.
2.4.2

Imputed Hedonic Indexes

Instead of performing estimation one time for two periods by pooling data, it is also possible
to estimate the characteristic price parameter for each period. Taking η t as the N (t) × 1 error
term vector, the regression model for time t = 0 and time t = 1 may be written as follows:y t = 1t βt + Zt γ t + η t
10

(27)

Here, it is assumed that the characteristic price parameters γ 0 , γ 1 vary depending on theobservation period. If one includes βt∗ and γ t∗ as least squares estimators, the following
formula may be established using the least squares residual error vector ut :y t = 1t βt∗ + Zt γ t∗ + ut£
Based on the nature of the residual error, 1t
obtained:

(28)

¤′
£
¤′
Zt ut = 0 0′K , so the following can be

1′0 y 0 = N (0)β0∗ + 1′0 Z0 γ 0∗(29)

1′1 y 1 = N (1)β1∗ + 1′1 Z1 γ 1∗

(30)

Therefore, the estimator for (29) and the time eﬀect based on (29) can be obtained by solving
these for β0∗ , β1∗ .
¡¢1′0 y 0 − Z0 γ 0∗
1′0 y 01′0 Z0 γ 0∗
∗
β0 =
−
=
(31)
N (0)
N (0)
N (0)
¡
¢
1′1 y 1 − Z1 γ 1∗
1′1 y 11′1 Z1 γ 1∗
∗
β1 =
−
=
(32)
N (1)
N (1)
N (1)
The estimation value of the hedonic time dummy based on the logarithmic price change fromtime 0 to time 1 may be obtained using the diﬀerential LPHD = α1∗ − α0∗ . However, since it isassumed that the parameters γ 0 , γ 1 for quality adjustment based on the formulation of (27)vary across periods for the two times, it is not possible to simply deﬁne the logarithmic pricechange from the diﬀerential of β0∗ , β1∗ . Therefore, Laspeyres type and Paasche type measuresof price change in imputed hedonic model are shown as follows :µ¶ µ¶1′1 Z1 γ 1∗
1′1 Z1 γ 0∗∗∗∗[Laspeyres] φL = β1 +− β0 +
(33)
N (1)
N (1)
µ
¶ µ
¶
1′0 Z0 γ 1∗
1′0 Z0 γ 0∗∗∗∗[Paasche]φP = β1 +− β0 +
(34)
N (0)
N (0)
where characteristic value change of a Laspeyres type measure is calculated as the arithmeticalmean of Z1 , and that of a Paasche type measure is calculated as an arithmetical mean of Z0 .
For both the diﬀerential φ∗L and φ∗P , adjustment with characteristic price is asymmetrical.
Therefore, using the median value of the two diﬀerentials, the hedonic imputed estimation
value based on the logarithmic price change from time 0 to time 1 is written as follows:
1
1 ∗φL + φ∗P
2 © 2
©
¡
¡
¢ª
¢ª
1′0 y 0 − Z0 12 γ 0∗ + 21 γ 1∗
1′1 y 1 − Z1 12 γ 0∗ + 12 γ 1∗
−
=
N (1)
N (0)

LPHI =

(35)

t t∗
Here,
see
¡ one can¢ that quality adjustment of price is performed not with Z γ but with1 1∗
t 1 0∗
Z 2 γ + 2 γ . If the sample sizes for the two times are identical and the characteristics andcharacteristic prices are constant over time, there is no diﬀerence between the two techniques.

112.4.3 Diﬀerences Between Time Dummy Indexes and Imputed Indexes
In order to look at the diﬀerences between LPHD (24) and LPHI (35), the diﬀerential of thetwo may be expressed as follows:
µ ′ 1
¶µ
¶
11 Z
1′0 Z01 0∗ 1 1∗∗
LPHD − LPHI =−γ + γ −γ(36)
N (1) N (0)22
In other words, if the average of the characteristic prices is equivalent for each time and if thepooled hedonic regression model characteristic price is equivalent to the hedonic characteristic
price median value estimated for each time, (24) and (35) are fully equivalent.
Based on (31) and (32), the β0 , β1 least squares estimator regressed on each observation periodis:¢¡βt∗ = 1′t y − Zt γ t∗ /N (t)Using this, the least squares residual error may be written as follows:
ut = Mt y t − Mt Zt γ t∗(37)Here, Mt = I − 1t 1′t /N (t). If we deﬁne y t∗ = Mt y t and Zt∗ = Mt Zt , the estimated characteristic price vector is as follows:¡¢−1 t∗′ t∗
Z y(38)
γ t∗ = Zt∗′ Zt∗¡ 0∗′ 0∗¢Here, if we multiply Z Z + Z1∗′ Z1∗ by both sides of (26), the characteristic price using
pooled data in (26) becomes:¡ 0∗′ 0∗¢¡¢Z Z + Z1∗′ Z1∗ γ ∗ = Z0∗′ y 0∗ + Z1∗′ y 1∗ = Z0∗′ Z0∗ γ 0∗ + Z1∗′ Z1∗ γ 1∗
(39)Note that, based on (38), (Z0∗′ Z0∗ )γ 0∗ = Z0∗′ y 0∗ and (Z1∗′ Z1∗ )γ 1∗ = Z1∗′ y 1∗ . If γ 0∗ and
γ 1∗ are equivalent, (39) shows that γ ∗¡ is necessarily the ¢shared ¡characteristic vector¢ of these.
1 0∗
1 1∗
∗
0∗′ 0∗
1∗′ 1∗
If we multiply the right side of (36) 2 γ + 2 γ − γ by 2 Z Z + Z Z
from theright side, we obtain the following:µ¶¡ 0∗′ 0∗
¢ 1 0∗ 1 1∗
1∗′ 1∗∗
2 Z Z +Z Z
γ + γ −γ22¡¢0∗′ 0∗ 0∗0∗′ 0∗ 1∗= Z Z γ + Z Z γ + Z1∗′ Z1∗ γ 0∗ + Z1∗′ Z1∗ γ 1∗ − 2 Z0∗′ Z0∗ + Z1∗′ Z1∗ γ ∗= Z0∗′ Z0∗ γ 0∗ + Z0∗′ Z0∗ γ 1∗ + Z1∗′ Z1∗ γ 0∗ + Z1∗′ Z1∗ γ 1∗ − 2Z0∗′ Z0∗ γ 0∗ − 2Z1∗′ Z1∗ γ 1∗
= −Z0∗′ Z0∗ γ 0∗ + Z0∗′ Z0∗ γ 1∗ + Z1∗′ Z1∗ γ 0∗ − Z1∗′ Z1∗ γ 1∗¡¢¡¢
= − Z1∗′ Z1∗ − Z0∗′ Z0∗ γ 1∗ − γ 0∗
In other words,µ¶¢¡¢¢−1 ¡ 1∗′ 1∗
1 0∗ 1 1∗
1¡∗× Z Z − Z0∗′ Z0∗ γ 1∗ − γ 0∗ (40)γ + γ −γ
= − Z0∗′ Z0∗ + Z1∗′ Z1∗2
2
2
If we plug (40) into (36), the diﬀerential of the time dummy index and imputed index shownwith the logarithmic price diﬀerence may be rewritten using the following formula:
LPHD − LPHIµ¶¢¡¢¢−1 ¡ 1∗′ 1∗
1 1′1 Z11′0 Z0 ¡ 0∗′ 0∗=−
× Z Z − Z0∗′ Z0∗ γ 1∗ − γ 0∗
Z Z + Z1∗′ Z1∗−
2 N (1) N (0)
12

(41)

Based on the above, when any of the following conditions are met, the two logarithmic pricediﬀerentials based on the hedonic time dummy and hedonic imputation method are identical:
1′ Z1

1′ Z0

• The average value of each characteristic is equivalent for the two times: N1(1) = N0(0)• The characteristic value variance/covariance matrices are equivalent for the two times:
Z1∗′ Z1∗ = Z0∗′ Z0∗• The quality-adjusted prices obtained with the hedonic price estimation method for eachtime are identical: γ 1∗ = γ 0∗
2.4.4 Summary of Hedonic Dummy Indexes and Hedonic Imputed Indexes
As shown above, we have identiﬁed factors that show the diﬀerences between a hedonic dummyindex and hedonic imputed index. In the regression equations, if it is possible to use information for two times and formulate the indexes with identical function forms, taking the(geometric) average of the two is perhaps a viable method when the two approaches show different results. However, rather than doing this, using either one index or the other is preferable
for various reasons.
A major issue of concern when using the hedonic time dummy (HD) method is that it has thefollowing restriction: the characteristic price is ﬁxed over time. However, the null hypothesisthat the characteristic variable parameter is ﬁxed throughout the observation period has infact been dismissed by a number of papers. In contrast to this, the hedonic imputed index(HI) method is inherently more ﬂexible than the time dummy model, which is a signiﬁcantadvantage.
In Section 2.4.3, we showed that the diﬀerence between the two approaches depends on thefollowing three variable factors:• The characteristic average value• The variance/covariance matrix of the characteristic value• The estimated hedonic characteristic price
What’s more, multiplication of the diﬀerence between the two periods produces the ultimate
diﬀerence. Therefore, the stability of the characteristic price parameter alone is not necessarily
a problem. For example, even if the parameter is unstable, its instability will be alleviated byslight changes in other characteristics, and the same may be true for the price index.
Due to the nature of the HD method, it uses independent variables observed for the two times,
and it is restricted such that the characteristic price parameters are the same for the two times,
and regression analysis ends up being executed one time only. In this sense, it may be said
that the HD method is not ﬂexible due to the presence of these restrictions. Why, then, arethese restrictions imposed? Presumably, the reasons include the following:• To not lose a degree of freedom.
• To provide an unambiguous estimation value for the overall price change from time 0
to time 1.
• To minimize the eﬀect of abnormal values in conditions where there is a small degreeof freedom.
In contrast to this, the HI method allows for diachronic changes in characteristic prices andformulation is more ﬂexible. However:• A degree of freedom is lost.
• The estimation value for the overall price change in the two times is diﬃcult to repro13

duce.
Due to these and other issues, analysis costs increase. The latter of the two issues pointed out
above may in fact not be all that serious, because Laspeyres- and Paasche-type estimation
values for price changes are well established in index theory. Bearing these points in mind,
the HI method may be considered the preferred method as long as the degree of freedom is
not extremely restricted.
In light of the above, “the rolling window hedonic method ” that merges the hedonic dummy
method and hedonic imputed method has been proposed. Market structural changes occur asa result of various exogenous shocks, but it is thought that a certain adjustment period exists
until shocks are absorbed by the market and changes are realized. Therefore, the regression
coeﬃcient likewise does not change instantaneously but should instead be viewed as changing
sequentially.
However, if estimating a model where the data is divided into various periods and observation
data for each period is used (as with the HI method), the links to prior and subsequent data
are severed. As a result, under the assumption that structural changes occur sequentially, thismethod ends up making it more diﬃcult to capture price changes within the sequential change
process. Instead, as a more natural approach, a method of estimating price indexes within thesequential change process by taking an estimation period of a certain duration and estimating
the model while moving this period – as if obtaining a moving average – may be preferable.
A method that has been proposed based on this idea is “the rolling window hedonic method ”.
This approach is employed in the estimation of housing price indexes in Ireland and Japan.

2.5 Hedonic Production Price Index Measurement and Quality Adjustment
2.5.1 The Producer Revenue Maximization Problem
In this section, we will explain the characteristics of quality-adjusted hedonic indexes for producer prices, based on Diewert (2002)[24]. The Konus-type price index proposed in that paper
is deﬁned using a revenue function that is a value function of the revenue maximization problem based on company technology and resource constraints. The revenue function is derived
from characteristic values constituting the product price, production technology, production
factors, and product.
We shall deﬁne the hedonic price (producer’s willingness to pay) based on the characteristic
vector z as:
Πt (z) = ρt f t (z)(42)
Here, ρt is the price showing the value of all characteristic values used for the product attime t and f t (z) shows the cardinal utility separable from the utility function. In (42), it isassumed that the utility function is equivalent for the two times.
f0 = f1(43)

Given the hedonic price (42), the producer performs revenue maximization. First, we shalldeﬁne the production function F as follows:
q = F t (z, v)(44)

Here, q is the production volume and v is the production factor vector. For a given level of14

production technology, the following revenue-maximizing value function may be obtained:©
ªR(ρs f s , F t , Z t , v) ≡ max ρs f s (z)q : q = F t (z, v); z ∈ Z tq,z©
ª= max ρs f s (z)F t (z, v); z ∈ Z t(45)zHere, Z t shows the feasible set of characteristic values.
When the characteristics and input factors for time t are taken as z t , v t , the corresponding
production volume is:q t = F t (z t , v t )(46)
Therefore, the maximized revenue function for time t may be written as follows:©
ªR(ρt f t , F t , Z t , v t ) ≡ max ρt f t (z)q : q = F t (z, v t ); z ∈ Z tq,z= ρt f t (z t )q t ;2.5.2

t = 0, 1(47)

Konus-Type Hedonic Production Price Indexes

Using the maximized revenue function (47), the Konus-type hedonic product price index between time 0 and time 1 is deﬁned as follows:
P (ρ0 f 0 , ρ1 f 1 , F t , Z t , v) =R(ρ1 f 1 , F t , Z t , v)
R(ρ0 f 0 , F t , Z t , v)

(48)

1 1
The diﬀerences between
revenue functions
are caused
hedonic prices© 1 1the twoª© 1 0 by theª ρ f and
t
t
0 0
t
t
t
t
ρ f . Since maxz ρ f (z)F (z, v ); z ∈ Z = maxz ρ f (z)F (z, v ); z ∈ Z based on
hypothesis (43), (48) may be rewritten as follows:

P (ρ0 f 0 , ρ1 f 1 , F t , Z t , v) =ρ1
ρ1 R(ρ0 f 0 , F t , Z t , v)=
ρ0
ρ0 R(ρ0 f 0 , F t , Z t , v)

(49)

In estimation of the hedonic price, if we assume that the utility of the characteristic portionis diachronically constant, the Konus-type product price index may be estimated very easily.
Let us consider general cases that do not meet hypothesis (43). Taking the price index in(49) as our base, we can deﬁne an observable hedonic Laspeyres production price index andPaasche production price index with the following inequalities, using:
P (ρ0 f 0 , ρ1 f 1 , F 0 , Z 0 , v 0 ) =R(ρ1 f 1 , F 0 , Z 0 , v 0 )ρ1 f 1 (z 0 )≥= PHL
ρ0 f 0 (z 0 )R(ρ0 f 0 , F 0 , Z 0 , v 0 )(50)

P (ρ0 f 0 , ρ1 f 1 , F 1 , Z 1 , v 1 ) =ρ1 f 1 (z 1 )R(ρ1 f 1 , F 1 , Z 1 , v 1 )≤= PHP
ρ0 f 0 (z 1 )R(ρ0 f 0 , F 1 , Z 1 , v 1 )(51)Here, P (ρ0 f 0 , ρ1 f 1 , F 0 , Z 0 , v 0 ) and P (ρ0 f 0 , ρ1 f 1 , F 1 , Z 1 , v 1 ) are theoretical production priceindexes that cannot be observed. (50) shows that the theoretical production price index
P (ρ0 f 0 , ρ1 f 1 , F 0 , Z 0 , v 0 ) has the observable Laspeyres production price index PHL as its lower
limit, and (51) shows that the theoretical production price index P (ρ0 f 0 , ρ1 f 1 , F 1 , Z 1 , v 1 ) has
the observable Paasche production price index PHP as its upper limit.
By using these convex combination equations (weighted averages) instead of F 0 , Z 0 , v 0 or
F 1 , Z 1 , v 1 constituting the production price index, it is possible to deﬁne the range that can15be covered by the theoretical Laspeyres production price index and Paasche production priceindex. If the scalar λ ∈ [0, 1] is used, the convex combinations for F t , Z t , v t for period t = 0, 1
may be written as follows.
Z(λ) = (1 − λ)Z 0 + λZ 1v(λ) = (1 − λ)v 0 + λv 1
F (λ) = (1 − λ)F 0 (z, v(λ)) + λF 1 (z, v(λ))
Therefore, the hedonic production price function may be written as:©
ªmaxz ρ1 f 1 (z)F (λ); z ∈ Z(λ)
R(ρ1 f 1 , F (λ), Z(λ), v(λ))
P (λ) ==
R(ρ0 f 0 , F (λ), Z(λ), v(λ))
maxz {ρ0 f 0 (z)F (λ); z ∈ Z(λ)}(52)

When λ = 0，since P (λ) signiﬁes that P (ρ0 f 0 , ρ1 f 1 , F 0 , Z 0 , v 0 )，the following may be derived
from inequality (50):
ρ1 f 1 (z 0 )P (0) ≥ PHL = 0 0 0(53)
ρ f (z )As well, when λ = 1, since P (λ) signiﬁes that P (ρ0 f 0 , ρ1 f 1 , F 1 , Z 1 , v 1 ), the following may bederived from inequality (51):ρ1 f 1 (z 1 )P (1) ≤ PHP = 0 0 1(54)ρ f (z )By using Diewert’s proof (1983; 1060-1061)[22], if P (λ) is a continuous function for [0, 1], it ispossible to show that λ∗ exists, whereby
0 ≤ λ∗ ≤ 1 and PHL ≤ P (λ∗ ) ≤ PHP

PHP ≤ P (λ∗ ) ≤ PHL .

In other words, one can see that the theoretical hedonic production price index for the periodt = 0, 1, when considered via P (λ∗ ) described above, exists between the observable Laspeyres
production price index and Paasche production price index. Note that to obtain this result,one must assume the continuity of λ in the hedonic model price functions ρ1 f 1 (z 0 ), ρ0 f 0 (z 0 ) in
the numerator and denominator of Formula (52), the production functions F 0 (z, v), F 1 (z, v),
and the feasible characteristic value sets Z 0 , Z 1 . The suﬃcient conditions for continuity are:
• The production functions F 0 (z, v), F 1 (z, v) are positive and continuous for z and v.
• The hedonic model price functions f 0 (z), f 1 (z) are positive and continuous for z.
• ρ0 , ρ1 are positive.
• Sets Z 0 , Z 1 are convex sets, bounded, and closed.
Based on the above, one can see that the boundary range for the theoretical price index isdetermined by the observable price index. In order to obtain the best value for approximating
the theoretical index, it is natural to take the adjusted average of the two boundary values. If
the adjusted average function for the Laspeyres and Paasche production price indexes is written as m(PHL , PHP ), we can conﬁrm, based on Diewert’s argument (1997; 138)[23], that m()
must be the geometric average. In other words, the best candidate in terms of approximating
the theoretical production price index is the following observable Fisher hedonic production
price index, using (50) and (51):
1/2

PHF = (PHL PHP )

ρ1
= 0
ρ

µ

16

f 1 (z 0 )f 0 (z 0 )¶1/2 µf 1 (z 1 )f 0 (z 1 )¶1/2

If the hypothesis f 0 = f 1 is fulﬁlled by the hedonic model price function being the same forthe two times, then this can be transformed into PHF = ρ1 /ρ0 . As well, if the respective
observable prices are deﬁned as
P 0 = ρ0 f 0 (z 0 ) and P 1 = ρ1 f 1 (z 1 )(55)

the Laspeyres and Paasche production price indexes can be shown as quality-adjusted pricecomparisons:ρ1 f 1 (z 0 )P 1 /f 1 (z 1 )=ρ0 f 0 (z 0 )P 0 /f 1 (z 0 )ρ1 f 1 (z 1 )P 1 /f 0 (z 1 )PHP = 0 0 1 = 0 0 0
ρ f (z )P /f (z )PHL =Therefore, the Fisher hedonic production price index may be written as follows:µ 1 1 1 ¶1/2 µ 1 0 1 ¶1/2
P /f (z )P /f (z )1/2
PHF = (PHL PHP )=P 0 /f 1 (z 0 )P 0 /f 0 (z 0 )

(56)
(57)

(58)

In other words, the Fisher hedonic production price index may be obtained from the geometricaverage of the two quality-adjusted price indexes obtained by estimating the hedonic regression
model. The hedonic approach is useful not just for quality adjustment of the product user
price but also for quality adjustment of the product supplier price. In this chapter, in order todeﬁne a product price index assuming competitive company production activities, we used arevenue function (total willingness to pay) maximized based on Konus. If the cardinal utility
function for the characteristic portion is the same at the two points in time, the theoreticalproduction price index may be shown by comparison with the observable product price.
In addition, in general cases, based on certain restrictions, we showed that the theoreticalproduction price index is present in the range that forms the boundary values of the observable
Laspeyres and Paasche production price indexes.

2.6 Characteristics, Advantages, and Disadvantages of the Hedonic Method
Rosen (1974)[55] developed a market equilibrium theory for diﬀerentiated products. This
study rigorously examined the relationship between the structures of the product supplieroﬀer function, product demander bid value function, and hedonic market price function, andcharacterized product market price based on consumer and producer behavior.
If the bid value function is used, it is possible to obtain the consumer’s willingness to pay withrespect to changes in characteristics. In market equilibrium, not only are the market priceand bid value consistent, but the slope of the hedonic function and bid value function are
also consistent. Since the bid value function is a concave function, the willingness to pay withrespect to incremental changes in characteristics is smaller than the change in the market price.
In other words, caution is required with respect to the market price function characteristic limit
value, since it is possible that the willingness to pay will be overestimated as long as demanders
are not homogeneous. However, if one assumes that changes in characteristic values will be
suﬃciently small, the market price function limit value may be used as an approximation of
the willingness to pay. Therefore, the market price function is generally estimated in existing
research.
If the hedonic approach is used, it is possible to measure changes in quality-adjusted price using
samples at another point in time. The simplest and most widely used method is to estimate
17the time eﬀect for the hedonic function based on the characteristic price being constant andusing a time dummy with pooled data. Hedonic imputed indexes estimate a hedonic function
for each observation time, allowing for changes in characteristic prices, and measure pricechanges using a Laspeyres-type scale or Paasche-type scale. Hedonic dummy indexes andhedonic imputed indexes produce diﬀerent results, but this is not caused solely by diﬀerences
based on whether or not characteristic prices are constant for the two times. Diﬀerences
between the two indexes occur if the average value for each characteristic varies for the twotimes or if the characteristic variance /covariance matrices vary for the two periods.
The hedonic approach is useful not just for quality adjustment of the product user price but
also for quality adjustment of the product supplier price. It is possible to deﬁne a producer
price index using a revenue function (total willingness to pay) maximized based on Konus. In
this case, if the cardinal utility function for the characteristic portion is the same at the twopoints in time, the theoretical production price index may be shown by comparison with theobservable product price. In general cases, based on certain restrictions, it has been shownthat the theoretical production price index is present in the range that forms the boundary
values of the observable Laspeyres and Paasche production price indexes.
Thus, one can see that price indexes with what is broadly called the “hedonic method” varyconsiderably depending on the approach employed in estimation.
The advantages and disadvantages of the hedonic method in the estimation of property priceindexes are outlined below. The following may be considered as advantages:• As well as having a basis in economic theory and index theory, the theoretical biases of
the hedonic method are clear.
• Compared to other approaches, it is possible to use all transaction price data, so it may
be considered the most eﬃcient approach.
• Since it makes it possible to control for the many characteristics of property, it enables
the sorting of data into specialized indexes by purpose/region.
• Since it is already used in the estimation of consumer price statistics and the like, it ispossible to be consistent with other economic statistics.
Disadvantages include:• Since it is necessary to collect many property-related characteristics, informationgathering costs are high.
• In cases where it is not possible to collect important characteristics for determining
property prices, one faces the problem of omitted variable bias.
• Calculated indexes vary depending on the function form used. In other words, there isa low level of reproducibility.
• In cases of strong heterogeneity, it may not be possible to control for quality.
• Since the underlying economic theory and statistical procedures are complicated, theorganizations creating the indexes require specialized skills, and explaining the indexes
to users is diﬃcult.

3 Repeat Sales Method
3.1 Standard Repeat Sales Price Index
3.1.1 Regression for Repeat Sales

18

Apart from the hedonic method, the most used approach is the repeat sales method elaboratedby Bailey, Muth, and Nourse (1963)[2] and Case and Shiller (1987)[9], (1989)[10]. With therepeat sales method, since use of the data generation process in the hedonic price regression
model is assumed, some of the problems occurring with the hedonic method are inherited.
However, since the same product is compared, underestimation bias is eliminated if there is
no change in the characteristics or characteristic prices. Given that the estimation method
is straightforward, it has the beneﬁts of being a technique with high reproducibility andestimation eﬃciency.
With either method, there is a bias that exists due to the estimation technique. Since the purpose of a price index is to observe price data over an extended time, as the observation period
becomes longer, “aggregation bias” is to be expected, due to changes in the characteristics
and characteristic prices of identical properties.
In particular, the fact that it is not possible to separate eﬀects common to the market as awhole (time eﬀects) that are factors in the housing market supply-demand balance from eﬀects
related to changes in individual housing, especially deterioration (age eﬀects), is an extremely
important issue when using the repeat sales method. If the eﬀects of housing deterioration
are ignored, it is to be expected that repeat sales price indexes will have a strong downwardbias.
As well, since only properties transacted multiple times are selected for use, the sample size
shrinks and there is also concern that selection bias occurs in the samples. Moreover, while itis strongly assumed that there will be no changes in property quality during the period when
repeat transactions are conducted, it is easy to predict that property deterioration, investment
in renovations, or changes to the surrounding environment will occur, so the assumption is
not consistent with the reality.
Below, we provide an overview of the repeat sales method and describe what kinds of problems
occur with it. Section 3.2 explains the analysis structure and price index characteristics of thestandard repeat sales method. Section 3.3 and Section 3.4 present the problems of aggregation
bias and sample selection bias with the standard repeat sales method, along with methods ofresolving them.
When housing prices at multiple points in time are observed as data, the hedonic market priceof property n at time t may be described in the form of the following regression model:ttynt ≡ ln ptn = αt + z t′n γ + εn(n = 1, 2, . . . , N (t); t = 0, 1, . . . , T )(59)

Here, ynt is the housing price logarithm (ln ptn )，αt is the unknown time eﬀect at time t，ttt, . . . , znk, . . . , znK)′ is the explanatory variable (characteristic) vector including az tn = (1, zn2ttt ′constant term, γ = (γ1 , γ2 , . . . , γkt , . . . , γK
) is the unknown coeﬃcient vector, and εtn is thetis theerror term. γ1 is the constant term coeﬃcient for the overall model and γ2t , . . . , γK
characteristic marginal eﬀect (characteristic quality-adjusted parameter).
Let us take it that property n is transacted on the market twice, at time s and time t (t > s).
In this case, for example, the logarithmic price diﬀerential for the two times in (59) may be
written as follows:
¡
¢
t
s′ s
Yn ≡ ln ptn − ln psn = (αt − αs ) + z t′
+ vn
(60)
n γ − zn γHere, vn is the diﬀerential of the error terms for the respective times (εtn − εsn ). In otherwords, the price rate of change (logarithmic diﬀerential) may be treated as data occurring
based on diﬀerences in the time eﬀect, changes in characteristic values (characteristic qualities
and quantities), and errors.
19

The repeat sales method of Bailey, Muth, and Nourse (1963)[2] and Case and Shiller (1987)[9],(1989)[10] reformulated (60) above by implicitly establishing the following hypotheses:
Hypothesis 1. All characteristics are constant over time.
Hypothesis 2. All characteristic parameters are constant over time.
In other words, hypothesis 1 signiﬁes that z n = z tn = z sn and hypothesis 2 that γ = γ t = γ s .
If one takes property n as being transacted for the ﬁrst time at time s and for the second timeat time t, the hedonic regression formula (59) may be rewritten as follows for time s and timet, respectively, using time dummy variables based on hypothesis 1 and 2:′

yns = dn α + z ′n γ + εsn′

ynt = dn α + z ′n γ + εtn
′

1

(61)
(62)
′

T

Note that dn = (dn , . . . , dn ) is the time dummy variable for the ﬁrst transaction and dn =1T

(dn , . . . , dn ) is the time dummy variable for the second transaction, and they are deﬁned asfollows:
½
½
u
u
1 u=s
1 u=t
dn =
, dn =
0 u ̸= s
0 u ̸= t
As well, α = (α1 , . . . , αs , . . . , αt , . . . , αT ) is the time eﬀect vector. Since there is a linear
relationship between the constant term z 1 for the overall model and the dummy variable, the0time eﬀect for time 0 is standardized here as α0 = 0. Therefore, the time dummy variables dn
0

and dn for time 0 are omitted.
The diﬀerential of the time dummy variables for the ﬁrst and second hedonic regression equations is deﬁned with the following T × 1 vector.
D n = dn − dnNote that:u = t (2nd transaction)
 1−1 u = s (1st transaction)
Dnu =

0
other cases(63)(n = 1, 2, . . . , N (t); u = 1, . . . , s, . . . , t, . . . , T )Y = (Y1 , . . . , Yn , . . . , YN )′ and D = (D 1 , . . . , D n , . . . , D N )′ are included, the matrix representation repeat sales regression model may be deﬁned as follows:
Y = Dα + v(64)The least squares estimator for (64) is α̂ = (D′ D)−1 D′ Y . The theoretical value (logarithmic
price diﬀerential) of a random property transacted for the ﬁrst time at time s and the second
time at time t is:µ t¶pnŶ = ln= α̂t − α̂spsnTherefore, taking time s as the baseline, the price index at time t (price comparison) is
exp(α̂t − α̂s ). Since the time dummy variable for time 0 was omitted and α0 = 0 included inorder to avoid multicollinearity, exp(α̂t ) is the price index taking time 0 as the baseline. The“BMN” price index presented in Bailey, Muth, and Nourse (1963)[2] is:
I BM N = {exp(α̂t ), exp(α̂1 ), . . . , exp(α̂T )}
20

(65)

3.1.2 Random Walk Error Term
The following is assumed with respect to the error terms in (61) and (62).
¤£E(εtn ) = 0, E (εtn )2 = σ 2 n = 1, . . . , N.
E(εsm εtn ) = 0m, n = 1, . . . , N ; s, t = 1, . . . , T ; n ̸= m, t ̸= s(66)(66) shows that for the respective hedonic regression equations, the error terms are homogeneously variant and there is no serial correlation. In this case, since the error term for (6),which is the diﬀerential of (3) and (4), is£¤E(vn ) = 0, E (vn )2 = 2σ 2 n = 1, . . . , N.
(67)
E(vn vm ) = 0m, n = 1, . . . , N ; n ̸= m(67) fulﬁlls the conditions of being homogeneously variant and having no serial correlation.
Bailey, Muth, and Nourse (1963)[2]’s price index is also estimated based on this type ofassumption.
With regard to this, Case and Shiller (1987)[9], (1989)[10] presented a repeat sales regression
model which assumes that as the interval between transactions becomes larger, the variancein noise associated with housing-speciﬁc structural factors becomes greater, and logarithmic
price changes are not homogeneously variant. In this paper, the error term with respect tologarithmic price ﬂuctuation is hypothesized with the following formulas including a randomwalk:
εtn = htn + νnt , νnt ∼ i.i.d. N (0, σν2 ) n = 1, . . . , N ; t = 1, . . . , T(68)

htn = hnt−1 + ηnt , ηnt ∼ i.i.d. N (0, ση2 ) n = 1, . . . , N ; t = 1, . . . , T(69)

Here, the left-side ﬁrst term in (68) is the random walk shown in (69), and the second itemin (68) is assumes white noise νnt , ηnt :sE(νnt νm
) = 0,sE(νnt ηm
) = 0,t s
E(ηn ηm ) = 0,n ̸= m = 1, . . . , N ; t ̸= s = 1, . . . , T
n = 1, . . . , N ; t = 0, . . . , Tn ̸= m = 1, . . . , N ; t ̸= s = 0, . . . , T(70)(64) is the repeat sales regression model error term. Here, based on
vn = εtn − εsn = (htn − hsn ) + (νnt − νns ),the following can be obtained:
E(vn ) = 0(71)

E(vn 2 ) = 2σν2 + (t − s)ση2(72)

In this case, one can see that if the transaction interval t − s becomes greater, the repeat sales
regression model error variance also increases (heterogeneous variance).
With regard to this heterogeneous variance, Case and Shiller (1987)[9], (1989)[10] proposed athree-step estimation method, the Weighted Repeat Sales (WRS) method.

21

1. (64) is estimated in the same way as when a BMN price index is obtained, the logarithmic price diﬀerential is regressed on the time dummy diﬀerential, and the least squares
residual error vbn is obtained.
2. In order to estimate σν2 , ση2 in (72), the error value of squares vbn2 is regressed on theconstant term and transaction interval An = t − s. (bvn2 = a + bAn + errorn )2
a + bbAn and its reciprocal square root
3. Taking the theoretical value in Step 2 as bvb = bn

2
1/bvbn as the weight, the weighted method of least squares is implemented for (64).

If the weight in Step 3 (N × N diagonal matrix) is deﬁned as


2
0
1/b
vb1


..
,
ω
b=
.


2
1/b
vbN

0

the weighted repeat sales regression model may be written as follows:Y ∗ = D∗ α + v ∗(73)

Note that Y ∗ = ω
b Y , D∗ = ω
b D, and v ∗ = ω
b v. Therefore, a workable generalized least squares
estimator is:b W LS = (D′ ωαb′ωb D)−1 D′ ω
b′ω
bY
(74)
Based on (16), Case and Shiller’s WRS price index is:
I W RS = {exp(0), exp(eα1 ), . . . , exp(e
αT )}

(75)

Hill, Knight, and Sirmans (1997)[42] and , and Knight, Hill and Sirmans (1999)[43] deﬁne thehedonic price index error term in cases where the serial correlation autoregressive parameteris 1 as:2εtn = εnt−1 + νnt , νnt ∼ i.i.d. N (0, σν,n) n = 1, . . . , N ; t = 0, . . . , T
Here, νnt assumes unknown heterogeneous variance. Therefore, the repeat sales regression
model error variance is:2
E[(εtn − εsn )2 ] = (t − s)σν,nWhen the transaction interval is taken as An = t − s, then2vn ≡ εtn − εsn ∼ N (0, An σν,n),and the weighted repeat sales regression model may be written as follows:Y ∗∗ = D∗∗ α + v ∗∗Note that Y ∗∗ = ωY , D∗∗ = ωD, v ∗∗ = ωv, and
 √
1/ A1

..

.

√

ω=
1/ An


0
22

(76)

0

..

.

√
1/ AN










3.2 Aggregation Bias
3.2.1 Bias Due to Omission of the Age Eﬀect
With the repeat sales method, aggregation bias is an important issue that has been pointed
out in much research. Aggregation bias is a problem relating to the two hypotheses in repeat
sales regression model (60) (hypothesis 1: all characteristics are constant over time; hypothesis
2: all characteristic parameters are constant over time). For example, if there are changes
in characteristic values due to deterioration/obsolescence of real estate capital, renovations
and maintenance, changes in the surrounding geographic environment, etc., such hypotheses
are not valid. As well, Dombrow, Knight, and Sirmans (1997)[41] tested whether or not
characteristic parameters change with the observation period (hypothesis 2). In order toestimate a stable price index, an observation period of suﬃcient length is necessary, but thelonger the observation period, the more liable these kinds of structural changes are to occur.
The biggest problem is the downward eﬀect on real estate prices due to deterioration pointed
out by Bailey et al. (1963)[2], Palmquist (1979)[54], and others. Below, we will show that it
is not possible to separate the time eﬀect and age eﬀect in standard repeat sales price indexes
and discuss methods for simultaneously estimating the time eﬀect and age eﬀect.
As the transaction interval becomes greater, real estate depreciates and the market value
declines. This is explained by McMillen (2003)[47]’s succinct model. The real estate at time
t is deﬁned as pt = Qt H t . Here, pt = exp(αt + γx x) is the price per unit of ﬂoor space, which
varies based on the location (distance from city center) x, αt is the time eﬀect, and γx < 0.
H t is the real estate ﬂoor space, which is produced using land L and capital K t (a linear
homogeneous Cobb-Douglas production function is hypothesized: H t = L1−ξ K ξ ). Since realestate materials deteriorate over time, this is deﬁned as K t = K[0] exp(cτ ). Here, K[0] is thereal estate capital when the building age is 0, c < 0 is the capital decrease rate per period,and τ is the building age. Based on the above, the logarithm of the real estate price at time
t may be written as follows:
ln pt = ln Qt + ln H t = αt + γx x + (1 − ξ) ln L + ξ ln K[0] + θτNote that θ = ξc < 0. Therefore, the logarithmic price diﬀerential for property n transacted
twice at time s and time t is as follows:
Yn = ln pt − ln ps = αt − αs + θAn + νn(77)Here, An = t − s > 0 ，and vn is the error term diﬀerential. If taking housing capital
deterioration into account, it is necessary to consider not only the time eﬀect diﬀerence forthe logarithmic price diﬀerential but also the age eﬀect diﬀerence.
The time eﬀect estimator with the BMN-type repeat sales method is α̂ = (D′ D)−1 D′ Y , but
if the real data generation process is (77), then based onα̂ = α + (D′ D)−1 D′ (Aθ + v),
one obtains

E(α̂) − α = (D′ D)−1 D′ Aθ

(78)

and as long as it is θ < 0, the BMN-type repeat sales method time eﬀect has a bias. In otherwords, an age eﬀect is included in the BMN-type time eﬀect.
23

However, it is not possible to simultaneously estimate the time eﬀect and age eﬀect in (77)in order to distinguish them. If (63) is used, the linear relationship shown below is formed
between the age diﬀerence for property n and the time dummy diﬀerence (Cannaday, Munneke,
and Yang, 2005[50]):
An = t − s = D ′n u =

T
X
u=1

Dnu u = Dn1 · 1 + · · · + Dns · s + · · · + Dnt · t + · · · + DnT · T
|{z}
|{z}
|{z}
|{z}
=−1

=0

=1

=0

Therefore, the repeat sales regression model with the age eﬀect is
Y = Dα + Aθ + v = D(α + uθ) + v(79)

and it is not possible to distinguish between α and θ. Based on the above, there is a diﬃcult
problem with the BMN-type repeat sales method: if the eﬀect of deterioration is ignored, abias occurs in the price index, and if the eﬀect of deterioration is considered, it cannot be
estimated due to multicollinearity.
3.2.2 Age Eﬀect Estimation
The problem of not being able to distinguish the time eﬀect and age eﬀect with the standard
repeat sales method has been pointed out by Bailey, Muth, and Nourse (1963)[2]，Palmquist(1979)[54], Hill, Knight and Sirmans (1997)[42], Hill, Sirmans and Knight (1999)[43], Chau,
Wong, and Yiu (2005)[51], Cannaday, Munneke, and Yang (2005)[50], and others. It is common for the age eﬀect to be ignored in the estimation of repeat sales price indexes.
However, estimating the age eﬀect with the repeat sales method is a very signiﬁcant issue.
First, in economic accounting, as typiﬁed by the SNA, estimation of housing stocks is performed, but no consistent method has been established for measuring the depreciation rate.
In general, the hedonic method is used for this kind of quality adjustment, but a lot of housing
characteristic information is needed to estimate the hedonic function. When collecting thisinformation is diﬃcult, it becomes necessary to perform estimation using another approach. If
measurement of the depreciation rate is possible within the framework of repeat sales method
estimation, it could be applied in many countries. An additional issue is the elimination of thebiases inherent in housing price indexes. There is a strong possibility that biases due to being
unable to eliminate the age eﬀect with the repeat sales method will be a serious problem,
especially in Asian countries like Hong Kong and Japan where high depreciation rates may be
expected.
In an attempt to estimate (21), Palmquist (1979)[54] estimated θ independently of the repeat
sales regression equation, then adjusted the time eﬀect to satisfy (21). In order to estimate
the age eﬀect, Cannaday, Munneke, and Yang (2005)[50] proposed a multivariate repeat sales
model that incorporates a dummy variable for building age instead of a continuous term
as in (77). In addition to this, a method has been proposed that performs estimation by
disrupting the linear relationship between the time dummy variable and age. Chau, Wong,and Yiu (2005)[51] distinguished the time eﬀect and age eﬀect by hypothesizing non-linearity
in the age eﬀect (Box-Cox transformation), and Hill, Knight, and Sirmans (1997)[42] did soby reﬁning Case and Quigley (1991)[8]’s hybrid method (joint hedonic and repeat sales model
estimation).
Cannaday, Munneke, and Yang (2005)[50] proposed the following model using an age dummy
variable:
Yn = D ′n β + B ′n θ + vn , n = 1, . . . , N(80)24

Here, β = (β1 , . . . , βt , . . . , βT )′ is the unknown time eﬀect parameter, and B n =(Bn1 , . . . , Bnj , . . . , BnJ−1 )′ is the dummy variable corresponding to building age, deﬁned
as follows:j=τ
Building age at second transaction time 1
−1 j = τ − (t − s) Building age at ﬁrst transaction time
Bnj =

0
Other
As well, θ = (θ1 , . . . , θj , . . . , θJ−1 )′ is the age dummy coeﬃcient vector. For the building
age dummy, in the case of new construction (j = 0), Bn0 is removed, and in the case of the
maximum building age value in the sample (j = J), BnJ is removed. By dropping the time
dummy for time 0 Dn0 and removing Bn0 and BnJ , it is possible to avoid multicollinearity.
According to Cannaday et al. (2005)[50], the reason for this is that since, in general, building
age [0, J] has a broader range than the observation period [0, T ] in the data used, it may beconsidered that the degree of freedom lowering eﬀect is less if the two building age dummies
are dropped. Dropping the ﬁrst and last building age dummies is equivalent to assuming thatthe price change rate is 0 in this sample range. Now, let us take the average value of the pricechange rate per year of building age as θ̄. However, since in general it may be considered thatθ̄ < 0, assuming that the price change rate is 0 for the interval between the ﬁrst and second
transactions t − s means overestimating the price change rate average value θ̄, which leads asa result to underestimating the time eﬀect. Therefore, for the time eﬀect, it is necessary to
perform upward correction for −θ̄(t − s) > 0 only, and for the age eﬀect, it is necessary to
perform downward correction for θ̄(t − s) < 0 only.
Taking the above into account, if we estimate β̂, θ̂ from (80) with the method of least squares,
using time s as the baseline year and time t as the comparison year, we can deﬁne an ageadjusted multivariate repeat sales price index (AAMRS price index) as follows when thebuilding age for the baseline year is j:ih´³
exp β̂t − θ̄(t − s) + θj+(t−s) + θ̄(t − s)
AAM RS´³(81)= exp β̂t − β̂s + θj+(t−s) − θjIs,t,j=exp β̂s + θj
Here, for the period t − s, the price change rate based on the time eﬀect is β̂t − β̂s ，and theprice change rate based on the age eﬀect is θj+(t−s) − θj . θ̄ is estimated asln pt = const. + θt τnt + errorn ,n = 1, 2, . . . , N (t)for the sample in each period,
P and the average value of the change rate in relation to building
age is obtained from θ̄ = t θt /T . Here, τnt is the building age of property n at time t (81)adjusts the price change based on deterioration in addition to the time eﬀect price change.
A price index in which building age is controlled as a constant (age-constant multivariate
repeat sales price index, or ACMRS price index) can be obtained from the following:´³
AAM RS(82)
Is,t,j= exp β̂t − β̂s − θ̄tIn this paper, this will be referred to as a “pure time index.”
When the two price indexes were estimated using four cities (Cleveland, Ohio; Miami, Florida;
San Francisco, California; and Champaign, Illinois), the results varied for the AAMRS price25

index when the initial building age value was set as j = 1 and j = 45. Compared to the orthodox Case-Shiller price index, the price increase rate was smaller for the recently constructedbuilding (j = 1) and the price increase rate was higher for the older building (j = 45). In
other words, one can see that the price index is important even in terms of what should be thebaseline building age level. As well, the ACMRS price index, which separates the age eﬀect
included in the orthodox Case-Shiller price index from the time eﬀect and holds the age eﬀect
constant, showed that, as predicted with (20), three of the cities advanced at a higher level
than with the Case-Shiller price index, and the further one gets from the baseline year, thegreater the divergence becomes.
Hill, Knight, and Sirmans (1997)[42] distinguished the time eﬀect and age eﬀect by reﬁning
Case and Quigley (1991)[8]’s hybrid method (hedonic and repeat sales method joint model
estimation). The hedonic regression model is recognized to be deﬁned as follows:ynt ≡ ln ptn = z ′n γ + τnt θ + d′n β + εtn(n = 1, 2, . . . , N (t); t = 0, 1, . . . , T )(83)

Here, ynt is the logarithm of price ptn of property n at time t, z n is the characteristic vector,
τnt is the building age of property n at time t, dn is the time dummy variable, (γ, θ, β) is theunknown parameter that should be estimated, and εtn is the error term. θ shows the age eﬀect
and β the time eﬀect.
From sample n = 1, . . . , N (t), let us take NR as a property that is transacted twice. Whenthe building age of property n at the ﬁrst transaction point s is τ − (t − s) and the building
age at the second transaction point t is τ , the repeat sales regression model may be written
as follows:
Yn = An θ + D ′n β + vn (n = 1, 2, . . . , NR )(84)
Here, An = τn − {τn − (t − s)} is the diﬀerential of the building age at time s and time t. If
all samples N + NR for (83) and (84) are pooled, the following regression model is obtained:
 
µ ¶
¶ γ
µ ¶ µ
ε
z τ d  
y
θ +
(85)
=
v
0 A D
Y
β
Case and Quigley (1991)[8] estimated this with the generalized least squares (GLS) method,
assuming the error terms for (8) and (9). However, Hill, Knight, and Sirmans (1997)[42]
perform the estimation using the maximum-likelihood method, assuming the AR1 process
εtn = ρεt−1+ νnt for the hedonic regression model error term. Here, ρ is the autoregressiven
coeﬃcient, and a time-homogeneous error term is assumed. If ρ = 1, the model is the random
walk error term in Case and Shiller, but the parameters must be tested. A distinctive featureof this approach is that, by pooling a hedonic regression model and repeat sales regression
model, it disrupts the linear relationship between A and D and makes it possible to estimate
the age eﬀect θ.
Hill, Knight, and Sirmans (1997)[42] report that the age eﬀect when hedonic regression model(83) is estimated using OLS or GLS and the age eﬀect of the pooled joint model (85) are roughly
the same value, the error term’s autocorrelation coeﬃcient is signiﬁcant, and ρ̂ = 0.54. In thecase of the repeat sales price index when (84) is estimated alone, since a negative age eﬀect isincluded, it is estimated at a lower value than the hedonic price index when (83) is estimatedalone. The price index based on the joint model (85) maximum-likelihood method progresses
with a somewhat higher value than the hedonic price index. As well, it is shown that theestimator based on the serial correlation obtained with the maximum-likelihood method is
also eﬃcient in Monte Carlo testing.
26

For the repeat sales method, estimation is not workable unless some kind of hypothesis isincluded for the error term’s heterogeneous variance based on factors other than age, but
Hill, Sirmans, and Knight (1999)[43] also performed Monte Carlo testing for the repeat sales
regression model, assuming various heterogeneous variances and serial correlations, and showedthat a model with an error term that assumes time-homogeneous serial correlation provides
results that are preferable to a random walk.

3.3 Sample Selection Bias
3.3.1 Selection Bias Elimination
With a repeat sales index, since the price index is estimated using samples that are transacted
repeatedly, it has been pointed out that sample selection bias likely exists. For example,
Shimizu, Nishimura, and Watanabe (2010)[57] used a hedonic index and repeat sales index
to analyze the diﬀerence between the two indexes. The ﬁndings they obtained showed thatwhen a hedonic index was estimated using only repeat sales samples, the extent to which priceﬂuctuations lagged behind market turning points became greater as the number of repeat sales
increased. They concluded that this suggests the existence of a structural sampling bias inrepeat sales samples.
As a hypothesis for explaining whether or not housing will appear on the market as a good to
be exchanged, the following condition may be considered: the seller’s oﬀer price must exceed
his or her reservation price. Gatzlaﬀ and Haurin (1997)[31], (1998)[32] veriﬁed that, if changes
in the housing market’s economic conditions inﬂuence the determination of oﬀer prices andreservation prices, there is a possibility that housing samples that are actually sold are notrandom samples. In other words, actual observed transaction prices depend on the stochastic
process that generates oﬀer prices and reservation prices. With that in mind, selection bias
is eliminated by applying a two-stage estimation method (Heckit method) based on Heckman(1979)[37].
Since the transaction prices at the ﬁrst and second sale times are observed as a paired data-set
only if the seller’s oﬀer price exceeds his or her reservation price, the use of selected samples
in analysis cannot be avoided. In Gatzlaﬀ and Haurin (1997)[31], correction of selection bias
is performed by applying the Heckit method, using the simplest repeat sales regression model
proposed by Bailey et al. (1963)[2] as a base. In order for a property to be sold on themarket, the seller’s oﬀer price has to exceed his or her reservation price, and a transaction
price is observed only when that happens. Therefore, based on the fact that the conditional
expected value for the closed hedonic price error distribution is not 0, selection bias occurs in
the hedonic price.
In the hedonic regression model presented by Gatzlaﬀ and Haurin (1998)[32], selection bias
elimination is performed using the method below. Taking the seller’s reservation (logarithmic)
price as yntR and oﬀer (logarithmic) price as yntO , the following hedonic regression model may
be written:
yntR = z ′n γ R + d′n α + εtRn ,n = 1, . . . , N ; t = 0, 1, . . . , T(86)yntO = z ′n γ O + d′n α + εtOnn = 1, . . . , N ; t = 0, 1, . . . , T

(87)

tO
Here, the average of the error terms εtRn , εn is 0, and the variance/covariance matrix is:¶µ
σRR σRO(88)
Σ=σRO σOO

27

The actual transacted price ynt is observed only when the oﬀer price exceeds the reservation
price. In other words:(
yntO
if yntO − yntR ≥ 0
ynt =
(89)
unobserved if yntO − yntR < 0
Therefore, the transaction price expected value is:
¡
¢
tO
tR
E(ynt ) = z ′n γ + d′n α + E εtO
n | yn − yn ≥ 0(90)

Since the error term expected value is not 0 and selection bias occurs, Heckman (1979)[37]’s
approach is used to correct this. That is, probit estimation is performed for the selection
function that determines whether or not housing is put up for sale in the ﬁrst step, and OLS
estimation using an inverse Mills ratio is performed in the second step.
Gatzlaﬀ and Haurin (1997)[31] expanded the above model to the repeat sales regression
method. Here, for time s, let us take Sns∗ as a latent variable that represents the choiceof whether or not to put housing on sale, where the selection mechanism may be written withthe following regression equation:s
Sns∗ = W s′n π + ϕn(91)

Here, W sn is the characteristic vector including the seller’s individual characteristics, housing
characteristics, geographic environment, etc., π is the unknown parameter, and ϕsn is the error
term. The latent variable Sns∗ cannot actually be observed. For the ﬁrst transaction, since
the price is observed only when the oﬀer price surpasses the reservation price – i.e., whenyn1O − yn1R ≥ 0 – this is taken as Sn1 = 1. For the second transaction to be observed, theﬁrst transaction must actually occur. Therefore, the binary variable expressing this may bedeﬁned as follows:
if yn1O − yn1R ≥ 0 and if yn2O − yn2R ≥ 0 10
if yn1O − yn1R ≥ 0 and if yn2O − yn2R < 0
Sn2 =
(92)


1O
1R
unobserved if yn − yn < 0
Therefore, the ﬁrst and second prices yn1 , yn2 are observed if Sn2 = 1 (yn1O − yn1R ≥ 0 and
yn2O − yn2R ≥ 0) and cannot be observed in other cases. Taking the error terms of the selection
function that determines the ﬁrst and second sales as ϕ1n , ϕ2n and the error terms of the hedonic
regression models as ε1n , ε2n , the variance/covariance matrix may be deﬁned as follows:
1 σ12 σ13 σ14
σ12
1 σ23 σ24 

(93)
Σ=
σ13 σ23 σ33 σ34 
σ14 σ24 σ34 σ44
Since both the ﬁrst and second transaction prices yn1 , yn2 are only observed when Sn1 = 1
and Sn2 = 1 are established simultaneously, the hedonic regression¡ 1 1model error¢ term2
expectedvalues,
based
on
the
conditional
expectedvalues
E
y
|
S
=
1
and
S
and
n
n
n
¡ 2 1
¢
2
E yn | Sn = 1 and Sn , are:¡¢E ε1n | Sn1 = 1 and Sn2 = σ13 λ1n + σ23 λ2n¡¢
E ε2n | Sn1 = 1 and Sn2 = σ14 λ1n + σ24 λ2n28Here, λ1n , λ2n are inverse Mills ratios. Therefore, the repeat sales regression model corrected
for sample selection bias may be written as follows:
Yn = D ′n α + (σ14 − σ13 )λ1n + (σ24 − σ23 )λ2n + ηn ,n = 1, . . . , N(94)

Analysis by Gatzlaﬀ and Haurin (1997)[31] using Miami housing market data showed that astandard repeat sales price index has an upward bias compared to a price index estimatedwith (36).
3.3.2

Matching Estimation

Properties transacted multiple times are limited even for the housing market as a whole, andwith the standard repeat sales method, data transacted only once is not used at all. Therefore,
the problem of selection bias discussed in the previous section occurs, and the reduction insample size also leads to a decrease in estimation eﬃciency.
McMillen (2012)[48] proposes a price index estimation method using a matching approachto handle this problem. The time eﬀect α in the hedonic regression model corresponds totreatment eﬀects in policy evaluation. As a simple example, the hedonic regression model inthe case of the two times t = 0, 1 is written as follows:
ynt = α0 + α1 d1n + z ′n γ + εtn= α0 + (α1 − α0 )d1n + z ′n γ + εtn

n = 1, . . . , N (t); t = 0, 1

(95)Here, d1n is a time dummy, where if t = 1, then d1n = 1; otherwise, d1n = 0. Taking thebaseline point as t = 0, it is possible to measure the price change rate from d1n ’s coeﬃcientestimation value. In the case of the repeat sales method, the price change rate can be obtainedby regressing yn2 − yn1 on d1n . One can see that the repeat sales method price change rate
observes the diﬀerence between a representative “post-treatment” value and representative“pre-treatment” value.
Apart from these methods, it is possible to measure the extent to which price changes occur
with the average for the whole sample, using the project evaluation method. In other words,
it is necessary to obtain the average treatment eﬀect (ATE):
N (1)AT E =1 X 1
d E(yn1 − yn0 )N (1) n=1 nThis shows that the average price change expected value for property transacted at the baseline
time t = 0 and re-sold at time t = 1 is equivalent to AT E. Or, it shows the average diﬀerencein the pre-treatment and post-treatment values. To measure the price index with respect to
t = 1, 2, . . . , T , it is necessary to obtain the following:
N (t)

1 X t
AT E =d E(ynt − yn0 ),N (t) n=1 n

t = 1, 2, . . . , T(96)

To approximate the ATE as the “treatment group average treatment eﬀect,” the data for allobservation times must be randomly sampled data. In the case of data used with the repeat
sales method, treatment group data may only be observed at the points when properties are
actually sold. Therefore, it is necessary to match data corresponding to the baseline year t = 0
control group to t = 1, 2, . . . , T at each point in time. To perform matching, we ﬁrst obtain29

a propensity score after performing logit regression of the time dummy on the characteristic
variable used in the hedonic regression model. Next, we create the treatment group matchingdata for each time based on kernel matching (Heckman, Ichimura, and Todd, 1998[38]), andﬁnally measure the price index by calculating (38).
McMillen (2012)[48] estimated a price index using quarterly data from 1993 to 2008 (approximately 60 quarters) for single-family housing in Chicago. The data size is approximately
169,000 samples, of which 52,000 are repeat sales data transacted at least twice. The baseline
point is Q1 1993, and the control group data consists of 1,651 samples. Due to the nature of
repeat sales data, there are few samples at the initial observation time and even fewer samples
at the ﬁnal observation time. The number of matched samples, however, is roughly the sameat each point. The total number of matched samples for the period from Q1 1993 to Q4 2008
is 102,000, which exceeds the repeat sales data. When price indexes were estimated based on
the hedonic approach using the initial 169,000 samples and 102,000 matched samples, there
was almost no diﬀerence between the two. In other words, when it comes to the matching
estimator, this shows that a hedonic approach-based estimator is extremely robust.
The matching estimator clearly diﬀers from the simple price change average. The price index
based on the average value for each period using all 169,000 samples is easily inﬂuenced byvalues that are outliers from the distribution. In the case of Chicago, since the variation in
the 2005 data is greater (in particular, the left side is ﬂat) than that of the 1995 data, it isshown that the price index based on the average value for each time is pulled downward.
The repeat sales method of estimating price indexes could be called an extreme version ofmatching method-based estimation. Matching housing transaction data that is not necessarily
identical but is similar has a number of advantages compared to price index estimation witheither the repeat sales method or hedonic method. Compared to the standard repeat sales
method, the general matching method dramatically increases the sample size and the likelihoodof obtaining more eﬃcient estimation values. For example, even in cases where there are few
samples (small region, short observation period, etc.), it may make it possible to create a priceindex. This means that matching-based estimation could be a useful price index estimation
method.

3.4 Characteristics, Advantages, and Disadvantages of Repeat Sales Indexes
This chapter has presented an overview of the repeat sales method and discussed it, with afocus on what kind of problems occur. Since the repeat sales method involves price comparisonof the same property, if there is no change in characteristics or characteristic prices, theproblem of underestimation bias that occurs with the hedonic method is eliminated. As well,since the estimation method is simple, it has the beneﬁts of being an approach with highreproducibility and estimation eﬃciency.
In order to create a more stable price index, it is necessary to observe price data over an
extended period. However, when the observation period becomes longer, aggregation bias
occurs due to changes in characteristics and characteristic values for the same property.
Since the price of housing changes due to deterioration and investment in renovations (housing
age eﬀect), an age eﬀect is included in the time eﬀect in the standard repeat sales method.
However, since a perfectly linear relationship exists between the time dummy and the variableindicating the transaction interval, it is not possible to distinguish the time eﬀect and age eﬀect
in the standard method. What have been proposed to date are methods that intentionallydisrupt the linear relationship between the time dummy and transaction interval variable andmethods that extrapolate the price index using exogenous data.
30

As well, since only property transacted multiple times is selected for use with the repeat sales
method, the sample size is reduced and the occurrence of selection bias in the sample is aconcern. If changes in housing market economic conditions inﬂuence the determination of theoﬀer price and reservation price, since the transactions at the times of the ﬁrst and second
sales are observed as a paired data-set only when the seller’s oﬀer price exceeds the reservation
price, the use of selected samples in analysis cannot be avoided. In this case, the traditional
method of correcting bias by estimating a selection function has been proposed.
Properties transacted multiple times are limited even for the housing market as a whole, andwith the standard repeat sales method, data transacted only once is not used at all. It has
been shown that the matching method is useful in improving this point. Using data transacted
only once as re-sold data based on the matching method increases the sample size dramaticallyand the likelihood of obtained more eﬃcient estimation values. This may, for example, makeit possible to create a price index even in cases where there are few samples (small region,
short observation period, etc.).
To summarize the above points, the advantages include:• Since the index is created by comparing prices of repeatedly transacted properties at
diﬀerent points in time, there is no need for information relating to the property characteristics.
• The problem of omitted variable bias that occurs with the hedonic method is avoided.
• The estimation method is simple and there is a high level of reproducibility.
• Even in the case of strong heterogeneous property, the probability of estimating index
is high.
• Due to simple concept, it is easy to explain to users.
The disadvantages include:• Since the price index is estimated using only information for properties transacted atleast twice (information for properties transacted only once is discarded), this method
is ineﬃcient. As a result, its use is diﬃcult in countries or regions where liquidity islow, and it often becomes diﬃcult to estimate indexes restricted to certain regions orproperty uses.
• Since the depreciation that accompanies the aging of the building between the twotransaction times is ignored, there is a downward bias if this is not controlled for.
• If investment in renovations is made between the two transaction times, there is an
upward bias if this is not controlled for.
• Depending on the database composition, it may be cost-intensive to identify transactions
involving the same property (there are quite a few countries where it is diﬃcult to
identify transactions involving the same property).
• It is impossible to create separate indexes for land and buildings.
• When new transaction price information is generated, the data – including even pastseries – changes, so it is not possible to produce deﬁnite values.

4 Price Indexes Based on Property Appraisal Prices
4.1 Property Appraisal Price Indexes
If the property market has few transactions (i.e., it is thin) and property is strongly heterogeneous, price surveys are conducted by property appraisal experts. In addition, in the many31

countries with property taxes, there are quite a few that use property assessment values forthe purpose of tax assessment.
Moreover, in recent years, with the dramatic growth in the property investment market, it
has become possible to obtain property appraisal prices that are periodically surveyed for the
purpose of measuring the performance of investment properties. In light of this, eﬀorts havebeen made to create property prices indexes using property appraisal prices.
In particular, when attempting to capture the movements of markets that are strongly heterogeneous with few transactions, using property appraisal prices may be a valuable methodof capturing changes.
However, it has been pointed out that there are valuation error, lagging, and smoothingproblems surrounding property appraisal prices. The ﬁrst problem occurs because property
appraisal prices are determined based on the judgment of property appraisal experts, so there
is a certain degree of error in the price determination absolute value. The second problem
occurs because the information property appraisers are able to use in price determination ispast information, so there is a certain lag in price determination. The ﬁnal problem, which isrelated to the ﬁrst and second problems, occurs because not only is there a strong possibilityof misjudging market turning points, but changes also undergo smoothing, so price changes
occur only gradually.
Furthermore, property appraisal systems diﬀer by country, so there are cases where the deﬁnition of the price obtained by property appraisers also varies. Furthermore, assessed values
for the purpose of tax assessment diﬀer from normal property appraisal values, and since they
are assessed values, there is an even stronger possibility that they do not properly capture
market changes.
While property appraisal price information is a valuable information source for markets withfew transactions, and the possibility of creating an index using this information exists, suﬃcient
care must be taken with regard to its biases.

4.2 Hedonic Method Based on Pooling of Property Appraisal Prices and Transaction Prices
When attempting to estimate hedonic price indexes using transaction price information, one
faces cases where index estimation is diﬃcult due to a lack of such information. In addition,
as mentioned previously, since valuation error and smoothing problems exist with property
price information, it is known to have certain biases. In order to control for these biases andcompensate for insuﬃcient transaction price information, the following has been proposed:
attempting to estimate price indexes by pooling property appraisal price information andtransaction price information, then using the hedonic method.
Since assessed values obtained for tax purposes incorporate various factors that are likely
included in transaction price information, they are helpful in explaining transaction prices.
The regression equation using assessed values is written as follows:
ln ptn = αt + ς ln atn + εtn(97)Here, ptn is the transaction price and atn is the assessed value. Note that there is no guaranteethat the true property market value can be assessed correctly, so there is always an error in
the assessed value. For example, taking the true property market value as Vnt , the assessedvalue atn may be an observation value accompanied by a probability error, as follows:
ln atn = ln Vnt + ηnt
32

(98)

In other words, the assessed value is the true property market value with the probability error
ηnt added to it. If the assessed value in (99) is taken as a proxy for the true property market
value, since ln Vnt = ln atn − ηnt based on (98), (99) may be rewritten as follows:¡¢¡¢
ln ptn = αt + ς ln atn − ηnt + εtn = αt + ς ln atn + εtn − ςηnt
(99)
Since the explanatory variable ln atn is clearly correlated to the error term, the coeﬃcient’s
least-squares estimator has a bias.
This estimation method is currently being researched and developed by the European Central
Bank with the aim of applying it in practice.

4.3 The SPAR Method
Since there is also a lack of price information for properties transacted multiple times whenestimating repeat sales indexes, one may be faced with the problem of being unable to estimate
the price index. In light of this, along with the method of artiﬁcially increasing the number of
repeat sales using the previously mentioned matching method, an estimation method known
as the SPAR (sale price appraisal ratio) method, which obtains the ﬁrst transaction price withthe property appraisal price, has been proposed and applied in practice.
The sale price of property n at comparison point t is taken as ptn (n = 1, 2, . . . , N (t)). In
addition, the appraisal price of said property at the baseline point 0 is taken as a0n (n =1, 2, . . . , N (0)). In this case, the sale-appraisal price ratio is ptn /a0n . If all quantities are
standardized as 1, the appraisal price-based arithmetic average price index may be deﬁned asfollows:
PN (t) t
µ t¶
N (t)
X
pn
0
0t
n=1 pn(100)=
wn (t)
PAP = PN (t)
0
0
a
an
n
n=1
n=1

Here, wn0 (t) in the second.Pformula on the right side of (100) is the weight based on the appraisal
N (t) 000
price, and wn (t) = an
n=1 an . Since this weight is deﬁned by the quantity (standardized

as 1) of sample N (t) at the comparison point, wn0 (t) is the expenditure weight calculated withthe baseline point price and comparison point quantity. Therefore, since (100) is the weighted
average based on wn0 (t) in the sale-appraisal price ratio ptn /a0n , one can see that it is a Paaschetype index. Note that in general, the baseline point sample size N (0) and comparison point
sample size N (t) are not equivalent.
The problem with (100) is that it takes the appraisal value as the baseline point price. Since
the sale price is not used, the price index is not 1 at the baseline point. The arithmetic
average sales price appraisal ratio method (arithmetic method) index overcomes this problem
by dividing by the baseline point sale-appraisal price ratio, as follows:
PN (t) t0t
n=1 pnPSP=
PN (t) 0AR
n=1 anÃP

N (0) 0n=1 pnPN (0) 0
n=1 an!−1

PN (t)

t
n=1 pn /N (t)
= PN
(0) 0
n=1 pn /N (0)ÃP

N (0) 0n=1 an /N (0)PN (t) 0
n=1 an /N (t)

!
(101)

(101) is a reciprocal multiplication of the sale price arithmetic average ratio and appraisal
price arithmetic average ratio. The reciprocal of the appraisal price arithmetic average plays
a role in adjusting structural changes that occur from the baseline point to the comparison
point.

33

4.4 Characteristics, Advantages, and Disadvantages of Property Appraisal Price
Indexes
Property appraisal price information is, needless to say, an extremely important source of
information in the estimation of property price indexes. In particular, in regions where there
are few transactions and markets which are strongly heterogeneous, such as the logistics facility, hotel, or hospital markets, there are quite a few cases where one has to rely on property
appraisal price information.
In light of this, not only are there price indexes that make direct use of property appraisal
prices, but many inventive approaches have also been developed, such as methods like the
SPAR method that correct the repeat sales method by using property appraisal prices andmethods that perform estimation by combining property appraisal prices and transaction
prices in hedonic method estimation. Their respective advantages and disadvantages are
outlined below. First, the SPAR method’s advantages include:• It preserves the advantages of the repeat sales method.
• Since it enables the use of more information than the repeat sales method, it is highly
eﬃcient.
• Since it is a method based on traditional index theory, it is easy to understand, and theestimation method is simple, it has a high level of reproducibility.
Its disadvantages include:• It inherits the disadvantages of the repeat sales method.
• Since the initial transaction is obtained with the property appraisal price, it is aﬀectedby the valuation error and smoothing problems with property appraisal prices.
• The quality adjustment issues surrounding property price indexes.
In the case of estimation using property appraisal price information and transaction priceinformation with the hedonic method, while it artiﬁcially increases the number of samples andincreases the eﬃciency when estimating the hedonic function, numerous problems remain interms of estimation theory, such as how to set the probability that transactions will occur.

5 How Should Property Price Indexes Be Estimated?
How should property price indexes be estimated?
When estimating a property price index, the estimation method varies considerably based on
the limitations of available information. If no such limitations exist, the hedonic method hasan advantage when one considers the underlying economic theory, the consistency with othertypes of economic statistics, its application in the System of National Accounts, and so forth.
However, in reality, while it may be viable for the housing market, where the transaction
quantity is relatively large and quality is relatively high, or even for the oﬃce market when it
comes to commercial property, in markets that are strongly heterogeneous, there are quite afew cases where it is diﬃcult to apply the hedonic method.
The repeat sales method is eﬀective when there is a suﬃcient quantity of transactions, even if
the market is strongly heterogeneous. However, in markets where the number of transactions
is limited, application of the repeat sales method is also diﬃcult.
In such cases, it may be possible to create indexes using property appraisal price information.
34

There are various possibilities, such as the SPAR method, estimation based on the hedonic
method using property appraisal prices and transaction prices, and appraisal price indexes
that use property appraisal prices as is.
However, when there are few transactions, one faces the problem of how property appraisers
determine property appraisal prices and, in light of that, how reliable the determined property
appraisal prices are.
Furthermore, in cases where there is a lack of property transaction price information, there isfurther scope to consider creating indexes using property revenue information and so on.
Going forward, in an attempt to properly capture property market trends, it is likely thatmultiple indexes will be created by combining various sources of information with appropriate
estimation methods for those sources.

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