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62, US
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rices
roper
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down
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not g
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price may be a biased indicator of the real market condi-
tions. The degree of such bias is dependent on the fre-
quency of delisting. In hot market where sales are quick
1051-1377/$ - see front matter � 2011 Elsevier Inc. All rights reserved.
⇑ Corresponding author.
E-mail addresses: pcheng@fau.edu (P. Cheng), llin@cobilan.msstate.
edu (Z. Lin), yingchun.Liu@fsa.ulaval.ca (Y. Liu).
1 Obviously, the implicit assumption is that all listed properties are
reasonably priced. In practice, this is ensured to a large extent by brokers
who presumably tend to refuse to list any properties with unrealistic price
expectations in the first place, especially in cold market. Therefore,
properties listed and then delisted represent true (shadow) supply, while
properties whose owners have unrealistic high price expectations are not
considered as true supply.
Journal of Housing Economics 20 (2011) 152–157
Contents lists available at ScienceDirect
Journal of Housing Economics
journal homepage: www.elsevier .com/locate / jhec
doi:10.1016/j.jhe.2011.02.002
houses are part of the true supply (or shadow supply) in
the market, given demand, the true equilibrium price (or
the shadow price) may be significantly lower than the ob-
served market prices based on the sold houses. Intuitively,
if all the houses listed for sale must remain on the market
to be sold, competition among sellers will likely be inten-
sified, thus the eventual prices on average will be even
lower, and the expected time-on-market (the shadow
time-on-market) will be longer than those observed only
from successful sales. In other words, the observed market
typically based on only transaction prices, they are likely
to suffer from such a bias. Simply put, transaction-based
housing indices tend to overstate the true equilibrium price
and understate the price decline from previous periods, thus
making the housing market appear to be in better condition
than it really is during down cycles.
This problem is intuitively well understood. But how
can we correct such pricing bias and find out the true mar-
ket condition? Theoretical and empirical solutions to this
problem have significant policy implications because the
Keywords:
Property delisting
Housing cycle
House prices
A wide-spread phenomenon in a
estate market (such as the recent U
that houses listed for sale often do
tended period of time-on-market.
them are pulled off the market with
This is known as property delistin
cycle of the real
using market) is
et sold after ex-
tually, many of
cceptable offers.
nce the delisted
and delisting is rare, the bias is negligible. But in cold mar-
ket, like in the recent years, the bias can be severe in the
face of rampant property delisting.1 An important implica-
tion of such bias is that it may cause biased housing price
indices. Since major housing market indices, such as the
widely watched S&P/Case-Shiller Home Price Indices, are
JEL classifications:
G11
G12
makers to form less biased views of the true state of the housing market, especially during
the down cycles.
� 2011 Elsevier Inc. All rights reserved.
Property delisting, housing cycle an
Ping Cheng a, Zhenguo Lin b,⇑, Yingchun Liu c
aDepartment of Finance, College of Business, Florida Atlantic University, 777
bDepartment of Finance and Economics, Mississippi State University, MS 397
cDepartment of Finance, Insurance and Real Estate, Laval University, Quebec,
a r t i c l e i n f o
Article history:
Received 27 July 2010
Available online 19 February 2011
a b s t r a c t
This paper provides
vable transaction p
numbers of listed p
to identify and corre
ricing bias
Road, Boca Raton, FL 33431, USA
A
a G1V 0A6
mal analysis on a well-known issue of the housing market – obser-
are a biased indication of the true market condition if significant
ties are delisted without sale. We provide a closed-form formula
ch pricing bias. The model can help market participants and policy
degree of the index bias, besides causing potential mis-
judgment among market participants, is likely to be most
severe during the worst market times, when accurate
assessment of the true market condition is most needed
for appropriate policy responses. The purpose of this study,
therefore, is to seek general understanding of the impact of
property delisting on property price, and to develop an
for such a successful bid to arrive (i.e., time-on-market)
should follow an exponential distribution with parameter
of gk.6 This has recently been empirically confirmed by the
findings of Bond et al. (2007), in which UK data are used
A
tP
*
tp O
Observable
B Unobservable
tP
4 In fact, our essential results would hold under a wide variety of more
complex distribution function assumptions.
5 In certain circumstances, it is well recognized that potential buyers can
get into a ‘‘bidding war’’ in which they bid a price above the asking price;
however, this happens rarely and only when the market is exceptionally
‘‘hot’’ or a property is dramatically underpriced. Based on the data from the
National Association of Realtors, Green and Vandell (1998) find that such a
situation occurs in about five percent of transactions.
6 Statistics principles state that the exponential distribution occurs
naturally when describing the lengths of the inter-arrival times in a
homogeneous Poisson process.
P. Cheng et al. / Journal of Housing Economics 20 (2011) 152–157 153
analytical framework that enables us to determine the sha-
dow price and the shadow time-on-market with observed
transaction and delisting information. Using empirical
data, we demonstrate that, when there is rampant prop-
erty delisting (as reported in the media),2 the true market
condition is likely to be significantly worse than the major
housing indices suggest.
1. A model of the real estate transaction process
In this section, we examine the matching process of the
real estate transaction. This process is characterized by
sequential search and bargaining. During the search pro-
cess, a seller receives offers over time from a stream of
buyers whose offer prices and timing of arrival are stochas-
tic in nature. The buyers make bids based on the informa-
tion acquired from their search. Each time a buyer makes
an offer; the seller evaluates the costs and benefits of wait-
ing for a potentially better offer, and decides whether to
accept the current bid. If the bid is rejected, the search con-
tinues. A common stopping rule for this process is to as-
sume that the seller will accept the first bid above the
reservation price, and to reject all bids below it.3 The reser-
vation price is affected by the seller’s holding costs due to
his/her unique personal and financial constraints. Generally
speaking, sellers who are more constrained tend to set lower
reservation prices in order to increase the probability of sale
within a planned or expected time frame. To formally model
such a process, we make the following assumptions:
a) The distribution of buyers’ arrivals. We assume the
buyers’ stochastic arrival follows a Poisson process
with rate k. This assumption is widely adopted by
previous studies including Sirmans et al. (1995),
Arnold (1999), Glower et al. (1998), Miceli (1989),
Cheng et al. (2008), among others. It is also widely
used in standard search models in labor economics.
b) The distribution of bidding prices. Consider a seller
who places a property on the market at time 0 and
sells it at time t. As shown in Fig. 1, assume the dis-
tribution of bidding prices is uniformly distributed
over ½P; P� with density function f(Pbid), where P
and P are the minimum and maximum bid prices,
respectively, and p�t is the seller’s reservation price.
By the stopping rule, a seller only accepts an offer
that is at least as high as the reservation price. That
2 See, for example, The New York Times report: ‘‘Home Sales Put Shares In
a Tailspin’’, Jul 25, 2008, Page 1, By Michael M Grynbaum and Eric Dash
3 Early studies in labor economics literature often rely on this assump-
tion (e.g. Stigler (1962), Whipple (1973) and Barron (1975)), and it has been
extensively applied to the real estate market since the 1980s (e.g. Yinger
(1981), Read (1988), Quan and Quigley (1991), Yavas (1992), Arnold (1999),
Lin and Vandell (2007), Lin and Liu (2008)).
is, the distribution of transaction prices is a trun-
cated distribution of bidding prices. The degree of
the truncation depends on the reservation price.
The higher the reservation price, the higher the
likely transaction price, but the smaller the range
of transaction price variations.
The assumption of bidding prices being uniformly dis-
tributed is another widely adopted assumption in early
studies including Read (1988), Yavas (1992), Sirmans
et al. (1995), Arnold (1999), and Cheng et al. (2008). For
technical simplicity, we adopt the same assumption.4 Since
no buyer will normally bids above the listing price, P can be
regarded as the listing price.5 In addition, we should point
out that the underlying value of the property is closely re-
lated to both holding period and market conditions. To sim-
plify notation, we intentionally omit the subscripts denoting
market state and holding period. Hence, the buyers’ bid price
is distributed as
f ðPbidÞ ¼
1
ðP�PÞ ; P
bid 2 ½P; P�
0; otherwise
(
ð1Þ
Given the two assumptions above, it is obvious that
there are two random processes in the determination of
the probability of a real estate sale. The first is the potential
buyers’ stochastic arrival, which is assumed to follow a
Poisson distribution with constant rate k. The second is
the probability of the offering price being above the seller’s
reservation price, P�p
�
P�P ; given k. Let g ¼
P�p�
P�P , the arrival rate
of a successful bid is then the joint probability of the two
stochastic processes, i.e. gk. Therefore, the time it takes
t
Fig. 1. Real estate bidding prices and transaction prices.
TDL ¼ gkð1� dÞ ð3Þ
transactions (T) rather than the shadow time-on-market
(TDL), the sellers have to compete harder. Under the
assumption of homogeneous property and market partici-
pants, the implication is that sellers will have to lower
154 P. Cheng et al. / Journal of Housing Economics 20 (2011) 152–157
Eqs. (2) and (3) suggest
TDL ¼ T1� d ð4Þ
Since T is observable from transactions, the true (or sha-
dow) time-on-market TDL can be computed given delisting
percentage d. Also, since (0 6 d 6 1), Eq. (4) suggests that,
in the presence of delisting, the shadow TOM is always
longer than what can be observed solely from the sold
properties, and that a higher percentage of delisting (often
occurs in weak real estate market) implies a bigger differ-
ence between the two.
From the price perspective, if all the properties were to
be sold within the same time frame as those observed
to investigate a number of possible assumptions about the
distribution of times to sale, such as the normal, chi-square,
gamma and Weibull distributions. Bond et al. (2007) finds
that the exponential distribution explains the data better
than the others. For exponentially distributed time-on-mar-
ket, the mean, or the expected TOM of the sale, is simply
T ¼ EðTOMÞ ¼ 1
gk
ð2Þ
It is important to notice that, given the bidding price distri-
bution, g is determined by the seller’s reservation price p⁄,
which in turn is affected by the seller’s constraint. In other
words, g can be considered as an indicator of the seller’s
constraint.
2. Correction of pricing bias in the homogeneous
market
Based on the model above, we now study the effect of
property delisting on real estate price/return and volatility.
We begin with a simple setting of a homogeneous real es-
tate market, in which we assume there is only one type of
properties and all market participants have homogeneous
expectations. This is a useful simplification if one agrees
with that the heterogeneous real estate market can be con-
sidered as to consist of many relatively homogeneous seg-
ments. In this section we focus on the within-segment
analysis. In the later section, we will extend the model to
inter-segments analysis in the context of heterogeneous
housing market.
As previously discussed, if the supposedly delisted
properties were to remain on the market until a successful
sale, the observed average time-on-market of all sold prop-
erties would be longer. This notion can be demonstrated
formally. Suppose that there is a pool of housing supply
in the market, let d (0 6 d 6 1) denote the percentage of
properties to be delisted in the pool, so the percentage of
the properties remaining on the market is (1 � d). The
probability of a successful sale now can be viewed as the
joint probability of the buyer’s arrival (k), the offering price
being above the reservation price (g), and the house’s
remaining on the market (1 � d). Similar to Eq. (2), the ex-
pected TOM upon a successful sale should be,
1
their reservation price to P⁄ < p⁄ in order to increase their
chance of successful sale to g� ¼ P�P�
P�P . From Eq. (2) and
(3), let TDL ¼ 1g�kð1�dÞ ¼ T ¼ 1gk, we can conclude that (g⁄)
must satisfy:
g� ¼ g
1� d ð5Þ
It is obvious that, since g < g⁄, g indicated by the suc-
cessful sales will overstate the reservation price (thus the
expected sales price) if more properties are likely to be
delisted (i.e. d > 0).7 The degree of the overestimation is pos-
itively related to d – from negligible in hot market when all
properties are sold quickly to very high in cold market when
a significant number of properties are pulled off the market.
Now we move to examine the impact of delisting on
real estate return and risk. As illustrated in Fig. 1, each bid-
ding price ~Pbid received by the seller implies a total return
~Rbid over a holding period s, which can be expressed as
~Rbid ¼
~Pbid � P0
P0
ð6Þ
where ~Pbid is uniformly distributed over ½P; P� and P0 is the
original purchase price.8 Similarly, the total return upon
sale for a seller with reservation price p⁄ or g can be ex-
pressed as
~Rg ¼
~Pg � P0
P0
; ð7Þ
where ~Pg is uniformly distributed over ½p�; P�.From Eq. (6),
the expected total return of bidding prices is
uM;s ¼ E
~Pbid � P0
P0
" #
¼ ðP þ PÞ
2P0
� 1 ð8Þ
And the volatility (or risk) of such return is
r2M;s ¼ Var
~Pbid � P0
P0
" #
¼ ðP � PÞ
2
12P20
; ð9Þ
where (uM;s;r2M;s) characterize the distribution of bidding
prices from potential buyers (or buyers’ valuation). Simi-
larly, the expected total return and volatility upon a suc-
cessful sale in the absence of delisting can be computed as,
us ¼ E
~Pg � P0
P0
~P P p�
��� i
"
¼ p
� þ P
2P0
� 1 ð10Þ
r2s ¼ Var
~Pg � P0
P0
~P P p�
��� i
"
¼ ðP � p
�Þ2
12P20
ð11Þ
Since g ¼ P�p�
P�P , we have
1
g
� 1 ¼ p
� � P
P � p� ð12Þ
7 This can also be seen by noting g� ¼ P�P�
P�P and g ¼
P�p�
P�P , where P
⁄ < p⁄.
8 To simplify notation, we intentionally omit the subscript denoting the
holding period in prices.
ffi
Application of the formulas in Eq. (21) and (22) requires
knowledge of the percentage of delisting (d). Empirically, d
narios and shows the impact of delisting on the average
P. Cheng et al. / Journal of Housing Economics 20 (2011) 152–157 155
Given
ðP þ PÞ
2P0
� 1 ¼ ðP þ p
�Þ
2P0
� 1�
ffiffiffi
3
p p� � P
P � p�
� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðP � p�Þ2
12P20
s
ð13Þ
and
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðP � PÞ2
12P20
s
¼ 1þ p
� � P
P � p�
� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðP � p�Þ2
12P20
s
ð14Þ
using Eqs. (8)–(12), we can therefore rewrite Eqs. (13) and
(14) as
uM;s ¼ us �
ffiffiffi
3
p
ð1
g
� 1Þrs ð15Þ
rM;s ¼ 1grs ð16Þ
Eqs. (15) and (16) illustrate how the buyers’ valuation
can be inferred from the observed transaction if there are
no properties to be delisted. At the presence of delisting,
in order to achieve the observed TOM in Eq. (2), the seller
needs to set lower reservation prices P⁄ where g⁄ should
satisfy Eq. (5) and P⁄ < p⁄. Suppose uDL;s and rDL;s are the ex-
pected return and volatility when g⁄ satisfies Eq. (5). The
buyer’s valuation of the same property (uM;s;r2M;s) can be
inferred in the similar fashion:
uM;s ¼ uDL;s �
ffiffiffi
3
p
ð 1
g�
� 1ÞrDL;s ð17Þ
rM;s ¼ 1g� rDL;s ð18Þ
From Eqs. (15)–(18), we can have
us �
ffiffiffi
3
p
ð1
g
� 1Þrs ¼ uDL;s �
ffiffiffi
3
p
ð 1
g�
� 1ÞrDL;s ð19Þ
1
g
rs ¼ 1g� rDL;s ð20Þ
Simplifying Eqs. (19) and (20) by using Eq. (5) yields the
relationship between the observed transactions indicated
by (us, rs) and the shadow return and volatility described
by (uDL,s, rDL,s):
uDL;s ¼ us �
ffiffiffi
3
p d
1� drs ð21Þ
rDL;s ¼ 11� drs ð22Þ
Note that if d = 0, that is, no property is delisted, there
will be no pricing bias and the observed return and volatil-
ity are in fact unbiased estimates of the expected return
and volatility. But if d > 0, it can be easily shown that,
uDL;s � us < 0 and @ðuDL;s � usÞ
@d
< 0 ð23Þ
rDL;s � rs > 0 and @ðrDL;s � rsÞ
@d
> 0 ð24Þ
Eqs. (23) and (24) suggest that, in the presence of delist-
ing, the observed return based on transacted properties is
upward biased, while the observed volatility (risk) is
downward biased. Such biases are greater if a larger per-
centage of properties on the market are delisted.
index return and volatility during down market periods.
The shadow return and volatility are computed based
on Eqs. (21) and (22) for each assumed delisting percent-
age. As shown above, if the shadow return and volatility
indicate the true market conditions, it is clear that the ob-
served return significantly overstates the true market re-
turn and understates its volatility when delisting is not
negligible (i.e. 5% or more). If one accepts that property
delisting is widespread in down markets such as the one
we are currently experiencing, the shadow return and
9 In comparison, during the up market periods, the average annual (year-
over-year) index return and volatility are 5.63% and 3.61%, respectively.
for a particular time period (i.e. a quarter) can be estimated
from the total number of properties listed for sale at the
beginning of the quarter and then pulled off the market
without sale during that quarter. For residential properties,
this information, though not often reported, can be ex-
tracted from the Multiple Listing Service (MLS). This study
highlights the usefulness of the delisting information and
calls for systematic tracking and recording delisting per-
centage over time, since this information is so critical to
understand the true housing market conditions.
3. Numerical example
Now we present a numerical example to demonstrate
the application of Eqs. (21) and (22) to provide some prac-
tical sense as to the magnitudes of the pricing biases due to
property delisting. To begin with, it is necessary to stress
that the formulas can be applied to any relevant data in
specific markets, as long as the delisting information is
available. For convenience, we use the data from one con-
venient source: the OFHEO Repeat-transaction Home Price
Index for the period from 1975 to 2008. We use the OFHEO
index to compute the quarterly US home price growth for
the period of analysis.
Fig. 2 presents a historic perspective of the US housing
market via the OFHEO Home price indices. During the past
33 years or so, the housing market exhibits clear cyclical
patterns. In nominal terms, the housing market never
experienced negative year-over-year growth rate until
very recently. In real terms (inflation-adjusted), negative
growth rate happened during the early 1980s, the early
1990s, and, of course, the most recent two years of 2007
and 2008. During these down market periods, it is ex-
pected that sales are slow and time-on-market is relatively
long. Property delisting, therefore, is more likely during
these periods. From the index data, we can observe the
average annual (year-over-year) index return and standard
deviation (volatility) over these down market periods
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