Analysis of the real estate market in Las
Vegas: Bubble, seasonal patterns, and
prediction of the CSW indexes 1
Wei-Xing Zhou a, Didier Sornette b,∗
aSchool of Business, School of Science, and Research Center of Systems
Engineering, East China University of Science and Technology, Shanghai 200237,
China
bD-MTEC, ETH Zurich, CH-8032 Zurich, Switzerland
Abstract
We analyze 27 house price indexes of Las Vegas from Jun. 1983 to Mar. 2005,
corresponding to 27 different zip codes. These analyses confirm the existence of a
real-estate bubble, defined as a price acceleration faster than exponential, which is
found however to be confined to a rather limited time interval in the recent past
from approximately 2003 to mid-2004 and has progressively transformed into a
more normal growth rate comparable to pre-bubble levels in 2005. There has been
no bubble till 2002 except for a medium-sized surge in 1990. In addition, we have
identified a strong yearly periodicity which provides a good potential for fine-tuned
prediction from month to month. A monthly monitoring using a model that we have
developed could confirm, by testing the intra-year structure, if indeed the market
has returned to “normal” or if more turbulence is expected ahead. We predict the
evolution of the indexes one year ahead, which is validated with new data up to Sep.
2006. The present analysis demonstrates the existence of very significant variations
at the local scale, in the sense that the bubble in Las Vegas seems to have preceded
the more global USA bubble and has ended approximately two years earlier (mid
2004 for Las Vegas compared with mid-2006 for the whole of the USA).
Key words: Econophysics, Real estate market, Periodicity, Power law, Prediction
∗ Corresponding author. Address: KPL F 38.2, Kreuzplatz 5, ETH Zurich, CH-8032
Zurich, Switzerland. Phone: +41 44 632 89 17, Fax: +41 44 632 19 14.
Email addresses: wxzhou@ecust.edu.cn (Wei-Xing Zhou), sornette@ethz.ch
(Didier Sornette).
URL: http://www.er.ethz.ch/ (Didier Sornette).
1 We are grateful to Signature Homes (Las Vegas) for providing us the data. This
work was partially supported by the NSFC (Grant 70501011), the Fok Ying Tong
Preprint submitted to International Journal of Forecasting 23 October 2018
http://arxiv.org/abs/0704.0589v1
1 Introduction
Zhou and Sornette (2003) analyzed the deflated quarterly average sales prices
p(t) from December 1992 to December 2002 of new houses sold in all the states
in the USA and by regions (northeast, midwest, south and west) and found
that, while there was undoubtedly a strong growth rate, there was no evidence
of a bubble in the latest six years (as qualified by a super-exponential growth).
Then, Zhou and Sornette (2006) analyzed the quarterly average sale prices of
new houses sold in the USA as a whole, in the northeast, midwest, south,
and west of the USA, in each of the 50 states and the District of Columbia
of the USA up to the first quarter of 2005, to determine whether they have
grown faster-than-exponential (which is taken as the diagnostic of a bubble).
Zhou and Sornette (2006) found that 22 states (mostly Northeast and West)
exhibit clear-cut signatures of a fast growing bubble. From the analysis of the
S&P 500 Home Index, they concluded that the turning point of the bubble
would probably occur around mid-2006. The specific statement found at the
bottom of page 306 of Ref.[Zhou and Sornette (2006)] is: “We observe a good
stability of the predicted tc ≈ mid-2006 for the two LPPL models (2) and (3).
The spread of tc is larger for the second-order LPPL fits but brackets mid-
2006. As mentioned before, the power-law fits are not reliable. We conclude
that the turning point of the bubble will probably occur around mid-2006.”
It should be stressed that these studies departed from most other reports by
analysts and consulting firms on real estate prices in that Zhou and Sornette
(2003, 2006) did not characterize the housing market as overpriced in 2003. It is
only in 2004-2005 that they confirmed that the signatures of an unsustainable
bubble path has been revealed.
Let us briefly analyze how this prediction has fared. The upper panel of Figure
1 shows the quarterly house price indexes (HPIs) in the 21 states and in the
District of Columbia (DC) from 1994 to the fourth quarter of 2006 released
by the OFHEO. It is evident that the growth in most of these 22 HPIs has
slowed down or even stopped during the year of 2006. When we look at the
S&P Case-Shiller Home Indexes of the 20 major US cities, as illustrated in the
lower panel of Figure 1, we observe that the majority of the S&P/CSIs had
a maximum denoted by a solid dot in the middle of 2006, validating the pre-
diction of Zhou and Sornette (2006). Specifically, the times of the maxima are
respectively 2006/06/01, 2006/09/01, 2005/11/01, 2006/05/01, 2006/08/01,
2006/05/01, 2006/12/01, 2006/07/01, 2006/08/01, 2006/09/01, 2005/09/01,
2005/12/01, 2006/09/01, 2006/09/01, 2006/08/01, 2006/06/01, 2006/07/01,
2006/09/01, 2006/08/01, 2006/12/01, 2006/06/01, and 2006/07/01 for the 20
cities shown in the legend of the lower panel. The only two cities with a max-
Education Foundation (Grant 101086), and the Alfred Kastler Foundation which
supported W.-X. Zhou for a visiting position in France.
imum occurring later towards the end of 2006 (2006/12/01) are Miami and
Seattle. However their growth rates decreased remarkably in 2006 as shown
in the figure. Furthermore, the S&P/CS Home Price Composite-10 reached
its historical high 226.29 on 2006/06/01 and the Composite-20 culminated to
206.53 on 2006/07/01, again confirming remarkably well the validity of the
forecast of Zhou and Sornette (2006).
1994 1996 1998 2000 2002 2004 2006 2008 2010
2000 2001 2002 2003 2004 2005 2006 2007
Phoenix − AZ
Los Angeles
San Diego
San Francisco
Denver
Washington
Miami
Tampa − FL
Atlanta − GA
Chicago
Boston
Detroit − MI
Minneapolis − MN
Charlotte − NC
Las Vegas
New York
Cleveland − OH
Portland − OR
Dallas − TX
Seattle − WA
Fig. 1. Evaluation of the prediction of Zhou and Sornette (2006) that “the turning
point of the bubble will probably occur around mid-2006” using the OFHEO HPI
data (upper panel) and the S&P CSI data (lower panel).
In this note, we provide a more regional study of the diagnostic of bubbles
and the prediction of their demise. Specifically, we analyze the Case-Shiller-
Weiss (CSW) Zip Code Indexes of 27 different Las Vegas regions calculated
with a monthly rate from June-1983 to March-2005. The CSW Indexes are
based on the so-called repeat sales methods which directly measure house
price appreciations. The key to these data is that they are observations of
multiple transactions on the same property, repeated over many properties
and then pooled in an index. Prices from different time periods are combined
to create “matched pairs,” providing a direct measure of price changes for a
given property over a known period of time. Bailey et al. (1963) proposed the
basic repeat sales method over four decades ago, but only after the work by
Case and Shiller (1987, 1989, 1990) did the idea receive significant attention
in the housing research community.
Studying the Las Vegas database is particular suitable since Las Vegas belongs
to a state which was identified by Zhou and Sornette (2006) as one of the
22 states with a fast growing bubble in 2005. With access to 27 different
CSW Zip Code Indexes of Las Vegas, we are able to obtain more reliable and
fine-grained measures, which both confirm and extend the previous analyses
of Zhou and Sornette (2003, 2006). The next section recalls the conceptual
background underlying our empirical approach. Then, section 3 analyzes the
regional CSW indexes for Las Vegas, showing that there is a regime shift
separated by a bubble around year 2004. Section 4 identifies and then analyzes
the yearly periodicity and intra-year pattern detected in the growth rate of
the regional CSW indexes. Section 5 offers a preliminary forecast based on the
periodicity analyses in Sec. 4. Section 6 concludes.
2 Conceptual background of our empirical analysis
2.1 Humans as social animals and herding
Humans are perhaps the most social mammals and they shape their envi-
ronment to their personal and social needs. This statement is based on a
growing body of research at the frontier between new disciplines called neuro-
economics, evolutionary psychology, cognitive science, and behavioral finance.
This body of evidence emphasizes the very human nature of humans with
its biases and limitations, opposed to the previously prevailing view of ratio-
nal economic agents optimizing their decisions based on unlimited access to
information and to computation resources.
Here, we focus on an empirical question (the existence and detection of real-
estate bubbles) which, we hypothesize, is a footprint of perhaps the most
robust trait of humans and the most visible imprint in our social affairs: im-
itation and herding. Imitation has been documented in psychology and in
neuro-sciences as one of the most evolved cognitive process, requiring a de-
veloped cortex and sophisticated processing abilities. In short, we learn our
basics and how to adapt mostly by imitation all along our life. It seems that
imitation has evolved as an evolutionary advantageous trait, and may even
have promoted the development of our anomalously large brain (compared
with other mammals). It is actually “rational” to imitate when lacking suffi-
cient time, energy and information to take a decision based only on private
information and processing, that is..., most of the time. Imitation, in obvious
or subtle forms, is a pervasive activity of humans. In the modern business,
economic and financial worlds, the tendency for humans to imitate leads in
its strongest form to herding and to crowd effects.
Based on a theory of cooperative herding and imitation, we have shown that
imitation leads to positive feedbacks, that is, an action leads to consequences
which themselves reinforce the action and so on, leading to virtuous or vicious
circles. We have formalized these ideas in a general mathematical theory which
has led to observable signature of herding, in the form of so-called log-periodic
power law acceleration of prices. A power law acceleration of prices reflects
the positive feedback mechanism. When present, log-periodicity takes into
account the competition between positive feedback (self-fulfilling sentiment),
negative feedbacks (contrariant behavior and fundamental/value analysis) and
inertia (everything takes time to adjust). Sornette (2003) presented a general
introduction, a synthesis and examples of applications.
2.2 Definition and mechanism for bubbles
The term “bubble” is widely used but rarely clearly defined. Following Case and Shiller
(2003), the term “bubble” refers to a situation in which excessive public ex-
pectations of future price increases cause prices to be temporarily elevated.
During a housing price bubble, homebuyers think that a home that they would
normally consider too expensive for them is now an acceptable purchase be-
cause they will be compensated by significant further price increases. They
will not need to save as much as they otherwise might, because they expect
the increased value of their home to do the saving for them. First-time home-
buyers may also worry during a housing bubble that if they do not buy now,
they will not be able to afford a home later. Furthermore, the expectation of
large price increases may have a strong impact on demand if people think that
home prices are very unlikely to fall, and certainly not likely to fall for long,
so that there is little perceived risk associated with an investment in a home.
What is the origin of bubbles? In a nutshell, speculative bubbles are caused
by “precipitating factors” that change public opinion about markets or that
have an immediate impact on demand, and by “amplification mechanisms”
that take the form of price-to-price feedback, as stressed by Shiller (2000). A
number of fundamental factors can influence price movements in housing mar-
kets. On the demand side, demographics, income growth, employment growth,
changes in financing mechanisms or interest rates, as well as changes in loca-
tion characteristics such as accessibility, schools, or crime, to name a few, have
been shown to have effects. On the supply side, attention has been paid to
construction costs, the age of the housing stock, and the industrial organiza-
tion of the housing market. The elasticity of supply has been shown to be a
key factor in the cyclical behavior of home prices. The cyclical process that
we observed in the 1980s in those cities experiencing boom-and-bust cycles
was caused by the general economic expansion, best proxied by employment
gains, which drove demand up. In the short run, those increases in demand
encountered an inelastic supply of housing and developable land, inventories
of for-sale properties shrank, and vacancy declined. As a consequence, prices
accelerated. This provided an amplification mechanism as it led buyers to
anticipate further gains, and the bubble was born. Once prices overshoot or
supply catches up, inventories begin to rise, time on the market increases,
vacancy rises, and price increases slow down, eventually encountering down-
ward stickiness. The predominant story about home prices is always the prices
themselves (see Shiller, 2000; Sornette, 2003); the feedback from initial price
increases to further price increases is a mechanism that amplifies the effects of
the precipitating factors. If prices are going up rapidly, there is much word-of-
mouth communication, a hallmark of a bubble. The word of mouth can spread
optimistic stories and thus help cause an overreaction to other stories, such as
stories about employment. The amplification can also work on the downside
as well. Price decreases will generate publicity for negative stories about the
city, but downward stickiness is encountered initially.
2.3 Was there a bubble? Status of the argument based on the ratio of cost of
owning versus cost of renting
In recent years, there has been increasing debates on whether there was a real
estate bubble or not in the United States of America. Case and Shiller (2003),
Shiller (2006) and Smith and Smith (2006) argued that the house prices over
the period 2000-2005 were not abnormal as they reflected only the convergence
of the prices to their fundamentals from below. In contrast, Zhou and Sornette
(2006) and Roehner (2006) have suggested that there was a bubble, which be-
came identifiable only after 2003, that is, after the work of Zhou and Sornette
(2003).
In this context, it is instructive to comment on the study by Himmelberg et al.
(2005), from the Federal Reserve Bank of New York , as it reflects the never
ending debate between tenants of the fundamental valuation explanation and
those invoking speculative bubbles. We are resolutely part of the second group.
Himmelberg et al. (2005) constructed measures of the annual cost of single-
family housing for 46 metropolitan areas in the United States over the last 25
years and compared them with local rents and incomes as a way of judging the
level of housing prices. In a nutshell, they claimed in 2005 that conventional
metrics like the growth rate of house prices, the price-to-rent ratio, and the
price-to-income ratio can be misleading and lead to incorrect conclusions on
the existence of the real-estate bubble. Their measure showed that, during
the 1980s, houses looked most overvalued in many of the same cities that
subsequently experienced the largest house price declines. But they found that
from the trough of 1995 to 2004, the cost of owning rose somewhat relative
to the cost of renting, but not, in most cities, to levels that made houses look
overvalued.
The rosy conclusion of Himmelberg et al. (2005), that 2004-2005 prices were
justifiable and that there was no risk of deflation as no bubble was present, is
based on a particularly curious comparison between cost of owning and cost of
renting, as noticed by Jorion (2005), in a letter to the Wall Street Journal. In-
deed, they candidly revealed however that their “cost of owning” calculations
imply an “expected appreciation on the property” coefficient. The value for
this factor is no doubt derived from figures for appreciation as currently ob-
served on the housing market, meaning they regarded the current appreciation
level as a reasonable assumption for what would indeed happen next – which
is precisely what our analyses and that of others question. In other words, the
authors had unwittingly hard-wired into their model the assertion that there
was no housing bubble; little wonder then that this is also what they felt au-
thorized to conclude. The circularity of their reasoning is particularly obvious
in an illustration they gave for San Francisco where for more than 60 years the
price-to-rent ratio has exceeded the national average, which, so they claimed,
“does not necessarily make owning there more expensive than renting.” The
reason why is that “high financing costs are offset by above-average expected
capital gains.” Translated, this means that as long as there is a bubble, prices
will go up and investing in a house remain a profitable operation. This trivial
statement is hollow; the real question is whether the trend that is observed
now remains sustainable.
In addition to this criticism put forward by Jorion (2005), there are other
reasons to doubt the validity of the conclusion of Himmelberg et al. (2005). In
the own words of Himmelberg et al. (2005), “the ratio of the cost of owning to
the cost of renting is especially sensitive to the real long-term interest rates.”
They are right in their rosy conclusion... as long as the long-term interest rates
remain exceptionally low. It is particularly surprising that their estimation of
the ratio of the cost of owning to the cost of renting was based on the most
recent rates over the preceding year of their analysis (2004), while the price of a
house is a long-term investment: what will be the long-term rates in 10, 20, 30,
or 50 years? Another problem is that their analysis was “mono-dimensional”:
they proposed that everything depends only on the ratio of the cost of owning
to the cost of renting. But they missed the interest rates as an independent
variable. As a consequence, it is not reasonable to compare the 1980s and the
present time, as the long-term interest rates had nothing in common. Another
problem with their analysis is that they assumed “equilibrium,” while people
are sensitive to the history-dependent path followed by the prices. In other
words, people are sensitive to the way prices reach a certain level, if there is
an acceleration that can self-fuel itself for a while, while Himmelberg et al.
(2005) discussed only the mono-dimensional level of the price, and not how it
got there. We think that this general error made by “equilibrium” economists
constitutes a fundamental flaw which fails to capture the real nature of the
organization of human societies and their decision process. In the sequel, we
actually focus our attention on signatures of price trajectories that highlight
the importance of history dependence for prediction.
This discussion is reminiscent of the proposition by Mauboussin and Hiler
(1999), offered close to the peak of the Internet and new technology bubble
that culminated in 2000, that better business models, the network effect, first-
to-scale advantages, and real options effect could account rationally for the
high prices of dot.com and other New Economy companies. These interest-
ing views expounded in early 1999 were in synchrony with the bull market
of 1999 and preceding years. They participated in the general optimistic view
and added to the strength of the herd. Later, after the collapse of the bubble,
these explanations seem less attractive. This did not escape the then U.S. Fed-
eral Reserve chairman Alan Greenspan (1997), who said : “Is it possible that
there is something fundamentally new about this current period that would
warrant such complacency? Yes, it is possible. Markets may have become more
efficient, competition is more global, and information technology has doubt-
less enhanced the stability of business operations. But, regrettably, history is
strewn with visions of such new eras that, in the end, have proven to be a
mirage. In short, history counsels caution.”
3 Regime shift in the CSW Zip Code Indexes of Las Vegas
3.1 Description of the data
We now turn to the analysis of the CSW indexes of 27 different Las Vegas zip
regions obtained with a monthly rate. The 27 monthly CSW data sets start
from June-1983 and end in March-2005. Figure 2 shows the price trajectories
of all the 27 CSW indexes. Visual inspection shows (i) a very similar behavior
of all the different zip codes and (ii) a sudden increase of the indexes since
Mid-2003. Let us now analyze this data quantitatively.
3.2 Power law fits
The simplest mathematical equation capturing the positive feedback effect
and herding is the power law formula (see Broekstra et al., 2005, for a simple
introduction in a similar context)
I(t) = A+B|tc − t|
m , (1)
1980 1985 1990 1995 2000 2005 2010
Fig. 2. Time evolution of the Case-Shiller-Weiss (CSW) Zip Code Indexes of 27 Las
Vegas zip regions from June-1983 to March-2005.
where B < 0 and 0 < m < 1 or B > 0 and m < 1. Others cases do not
qualify as a power law acceleration. For B < 0 and 0 < m < 1 or B > 0
and m < 0, the trajectory of I(t) described by (1) expresses the existence of
an accelerating bubble, which is faster than exponential. This is taken as one
hallmark of the existence of a bubble.
Notice also that this formula expresses the existence of a singularity at time tc,
which should be interpreted as a change of regime (the mathematical singu-
larity does not exist in reality and is rounded off by so-called finite-size effects
and the appearance of a large susceptibility to other mechanisms). This criti-
cal time tc must be interpreted as the end of the bubble and the time where
the regime is transiting to another state through a crash or simply a plateau
or a slowly moving correction.
We have fitted each of the 27 individual CSW indexes using the pure power
law model (1). The data used for fitting is from Dec-1995 to Jun-2005. We do
not show the results as the signature of a power law growth is not evident,
essentially because the acceleration is only over a rather short period of time
from approximately 2002 to 2004. As a consequence, power law fits give unre-
liable critical time tc too much in the future (like 2008 and beyond). We have
thus redone the fits of the 27 CSW indexes over a shorter time interval from
Aug-2001 to Jun-2005. A typical example is shown in Fig 3. All other 26 CSW
are very similar, with some variations of the parameters, but the message is
the same: while there is a clear faster-than-exponential acceleration over most
of the time interval, the price trajectory has clearly transitioned into another
regime in the latter part of the time interval considered here. The transition
occurred smoothly from mid-2004 to mid-2005 (the end of the time period
analyzed here).
It is important to recognize that the power law regime is expected only rela-
2001 2002 2003 2004 2005 2006
 = 2012.94;   m = −12;   χ = 0.007679
Fig. 3. Typical evolution of a CSW index from Aug-2001 to Jun-2005 and its fit by
a power law, showing both the faster-than-exponential growth up to mid-2004 and
the smooth transition to a much slower growth at later times. The root-mean-square
χ of the residuals of the fit as well at tc and m are given inside the figure.
tively close to the critical time tc, while other behaviors are expected far from
tc. The simplest model is to consider that, far from tc, the price follows an
exponential growth with an approximately constant growth rate µ:
I(t) = a+ beµt . (2)
A fuller description is thus to consider that formula (2) holds from the begin-
ning of the time series up to a cross-over time t∗, beyond which expression (1)
takes over. Any given price trajectory should thus be fitted by (2) from some
initial time tstart to time t
∗ and then by (1) from t∗ to the end of the time
series. Technically, t∗ is known from the parameters a, b, µ, A,B, tc, m by the
condition of continuity of I(t) at t = t∗, that is, both formulas give the same
value at t = t∗. We can further determine one of the parameters a, b or µ by
imposing a condition of differentiability at t∗, that is, the first time-derivative
of I(t) is continuous at t∗. This approach is known in numerical analysis as
“asymptotic matching” (see Bender and Orszag, 1978).
A simplified description of such a cross-over between a standard exponential
growth and the power law super-exponential acceleration is obtained by using
a more compact formulation
I(t) = A+B tanh[(tc − t)/τ ]
m , (3)
where tanh denotes the hyperbolic tangent function. This expression derives
from a study of the transition from the non-critical to critical regime in rup-
ture processes (of which bubbles and their terminal singularity belong to)
conducted by Sornette and Andersen (1998). This expression has the virtue
of providing automatically a smooth transition between the exponential be-
havior (2) and the pure power law (1), since tanh[(tc − t)/τ ] ≈ (tc − t)/τ for
tc− t < τ and tanh[(tc− t)/τ ] ≈ 1− 2e
2(t−tc)/τ for tc− t > τ . In this later case
tc − t > τ , expression (3) becomes of the form (2) with m = 1 and
a=A+B , (4)
b=−2Be−2tc/τ , (5)
µ=1/τ . (6)
In contrast, for tc − t < τ , expression (3) becomes of the form (1) with the
correspondence B/τm → B. Expression (3) has only five free parameters, in
contrast with the model involving the cross-over from (2) to (1) which has 7
free parameters (a, b, µ, A,B, tc, m) while t
∗ is determined by the asymptotic
matching). The pure power law formula (1) has 4 parameters while the ex-
ponential law (2) has just 3 parameters. The problem with expression (3) is
that it does not recover a pure exponential growth even for tc − t > τ , when
m 6= 1. Thus, expression (3) is limited in fully describing a possible cross-over
from a standard mild exponential growth and an super-exponential power law
acceleration. Our tests (not shown) find that a fit with model (3) retrieve the
pure power law model (1) with the same critical time tc and exponent m and
the same root-mean-square residual r.m.s. (the fit adjusts the parameter τ to a
very large value, ensuring that the fit is always in the regime tc−t ≪ τ so that
the hyperbolic tangential model reduces to the pure power law model). Thus,
contrary to our initial hopes, this approach does not provide any additional
insight.
Inspired by these tests, we could propose the following modified model
I(t) = a+ beµt(tc − t)
m . (7)
It has 5 adjustable parameters, like model (3), but it seems more flexible to
describe the looked-for cross-over: for large tc− t, the power law term (tc− t)
changes slowly, especially for 0 < m < 1 as is expected here; for small tc−t, the
power law term changes a lot while the exponential term is basically constant.
But, this model is correct for a critical point only if m < 0 so that b > 0;
otherwise, if 0 < m < 1, b < 0 and for tc− t large, the exponential term which
dominate does not describe a growth but an exponentially accelerating decay.
For 0 < m < 1, we thus need a different formulation. We propose
I(t) = a+ beµt + c(tc − t)
m . (8)
We have fitted this formula to the data over the four periods 1983 - Oct. 2004,
1991 - Oct. 2004, 1983 - Mar. 2005, 1991 - Mar. 2005 and, while the fits are
reasonable, the critical time tc is found to overshoot to 2007-2008, which is a
typical signature that the model is not predictive.
In conclusion of this first preliminary study, the presence of a bubble (faster-
than-exponential growth) is confirmed but the determination of the end of
this phase is for the moment unreliable.
3.3 Dependence of the growth rate on the index value
The monthly growth rate g(t) of a given CSW index at time t is defined by
g(t) = ln[p(t)/p(t− 1)] , (9)
where p(t) is the price of that CSW index at time t. Figure 4 shows the
evolution of the growth rates of the 27 CSW indexes from June-1983 to March-
2005. While there are some variations, all 27 CSW indexes follow practically
the same pattern. We clearly observe a large peak of growth over the period
2003-2005. Notice that this recent peak is much larger and coherent than the
previous one ending in 1991, which was followed by a price stabilization and
even a price drop in certain cases. This figure stresses that the acceleration in
growth rate is a very localized event which occurred essentially in 2003-2004
and the subsequent growth rate has leveled off to pre-bubble times. We can
conclude that there has been no bubble from 1990 to 2002, approximately,
then a short-lived bubble until mid-2004 followed by a smoothed transition
back to normal.
1980 1985 1990 1995 2000 2005 2010
−0.04
−0.02
Fig. 4. Evolution of the growth rates of the 27 regional CSW indexes from June-1983
to March-2005.
Fig. 5 plots the price growth rate g(t) versus the price p(t) itself for the 27
CSW indexes. A linear regression of the data points on Fig. 5, shown as the
red straight line, gives a correlation coefficient of 0.494. If we perform lin-
ear regression for each index, then we find an average correlation coefficient
0.503 ± 0.036, confirming the robustness of this estimation of the correlation
between growth rate and price level. The obtained relation between g and p
obtained from this correlation analysis is captured by the following mathe-
matical regression
g = 0.00922×
− 0.00747 . (10)
In words, if p is large, then g is large on average, which confirms the concept
of a positive feedback of price on its further growth. The continuous time limit
of g(t) defined by (9) is
g(t) =
d ln p
. (11)
This last equation together with (10), that we write as g(t) = αp − β (with
α = 0.00922/100 and β = 0.00747), implies the following ordinary differential
equation
= αp2 − βp , (12)
which indeed gives a power law acceleration p(t) ∼ 1/(tc − t) asymptotically
close to the critical time tc. Note that this critical time is determined by
the initial conditions, and is called in mathematics a movable singularity. We
conclude from this first analysis that the rough linear growth of the growth rate
confirms the existence of a bubble growing faster than exponential according
to an approximate power law. But of course, the exponent of this power law
is poorly constrained, in particular from the fact that the growth rate g(t)
exhibits significant variability and furthermore nonlinearity, as can be seen in
Fig. 5.
50 100 150 200 250 300 350 400 450
−0.04
−0.02
Jul. 1983 − Sep. 2003
Oct. 2003 − Sep. 2004
Oct. 2004 − Mar. 2005
Fig. 5. Dependence on the data price p for all CSW indexes of its growth rate g.
The overall correlation coefficient is 0.494. The red line is the linear fit of the data
points.
It is useful to refine this analysis by separating the whole time interval into
three distinct intervals. The corresponding plot of the growth rate g as a
function of price is shown in Fig. 5 with different symbols: period 1 is Jul.
1983 to Sept. 2003, period 2 is from Oct. 2003 to Sept. 2004, and period 3 is
from Oct. 2004 to Mar. 2005. An anomaly can be clearly outlined, associated
with the red dots which correspond to the anomalous peak in the growth rate
in the period from Oct. 2003 to Sept. 2004. Notice also that the most recent
time interval from Oct. 2004 to Mar. 2005 shows practically the same behavior
as the first period before 2003. In other words, when removing the data in red
for the period from Oct. 2003 to Sept. 2004, the growth rate g(t) is practically
independent of p, which qualifies the normal regime. We can thus conclude that
this so-called “phase-portrait” of the growth rate versus price has identified
clearly an anomalous time interval associated with extremely fast accelerating
prices followed by a more recent period where the price growth has resumed
a more normal regime.
4 Yearly periodicity and intra-year structure
4.1 Yearly periodicity from superposed year analysis and spectral analysis
In Fig. 4, the time dependence of the monthly growth rate exhibits a clear sea-
sonality (or periodicity), which appear visually to be predominantly a yearly
phenomenon. This visual observation is made quantitative by performing a
spectral Fourier analysis. The power spectrum of a typical CSW index is shown
in Fig. 6 (all CSW indexes show the same power spectrum). Since the unit of
time used here is one year, the frequency f is in unit of 1/year. A periodic
behavior with period one year should translate into a peak at f = 1 plus all
its harmonics f = 2, 3, 4, · · · , which is indeed observed in Fig. 6. Note also
that the spectrum has large peaks at f = 4 and f = 8 among the harmonics of
f = 1, which indicates a weak periodicity with period of one quarter. This is
consistent with Fig. 7, where four oscillations in the averaged monthly growth
rates can be observed.
0 1 2 3 4 5 6 7 8 9 10 11 12
Fig. 6. Spectrum analysis to confirm the strong periodicity in g(t).
Note that the power spectrum itself is periodic with a period of 12, which is
the sampling frequency, equal to the double of the Nyquist frequency. There
are also many peaks in the low-frequency region (larger that one-year time
scale) close to f = 0, which are associated with the time scales of the global
trends produced by the big peaks in g(t) around year 2004 as well as around
1990.
To further explore this seasonal variability of the price growth rates, we cal-
culate the averages of the growth rates for given months, where the average is
performed over all years. Consider for instance the month of January: we look
up the growth rate for all the data over all years for the month of January
and take the average. We do the same for each successive months. The result
is shown in Figure 7 for two time periods, which gives the average growth rate
〈g〉 for different months of the year. The red dash line and circles give the
resultant 〈g〉 for all the data and the black dash line and triangles give the
standard deviation σg for all data (which is a measure of the variability from
year to year and from zip code to zip code around the average). The difference
between the two time periods is precisely the time interval from June 2003 to
March 2005: this period is responsible for a significant increase of the average
growth rate (compare the red dashed line (filled circles) with the red continu-
ous line (open circles)) and an even larger increase of the variability (compare
the black continuous line (filled triangles) with the dashed black line (open
triangles)), again confirming the evidence of an anomalous behavior in that
period. In 2005, it appears that the growth rate relaxed back to the normal
level (according to the historical record).
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Month
Fig. 7. Monthly average growth rate (circles) and its standard deviation (triangles)
as a function of the month within the year. Dash: results obtained over all 27
indexes over the period from Jun. 1983 to Mar. 2005; Solid: results obtained over
all 27 indexes over the period from Jun. 1983 to May 2003.
4.2 Yearly periodicity and intra-year structure with a scale and translation
modulated model
Inspired by these results, we propose the following quantitative model. Con-
sider a time t in units of month. We write t = 12T +m, where T is the year
and m is the month within that year and thus goes from 1 (January) to 12
(December). For instance, t = 26 corresponds to T = 2 and m = 2 (Febru-
ary), while t = 38 corresponds to T = 3 and again the same month m = 2
(February) within the year. We propose to model the intra-year structure of
the growth rate g(t) together with possible yearly variations by the following
expression
g(t = 12T +m) = f(T )h(m) + j(T ) . (13)
In words, the growth rate has an intra-year structure h(m) modulated from
year to year in amplitude by f(T ) up to a possible overall translation j(T )
which can also vary from year to year. We can expect f(T ) and j(T ) to be
approximately constant for most years, except around 1990 and 2004 for which
we should see an anomaly in either or both of them, since these two periods
had bubbles. Note that this model (13) gives an exact yearly periodicity if
f(T ) and j(T ) are constant. A non-constant f(T ) describes an amplitude
modulation of the yearly periodicity. In particular, we expect a strong peak
around T = 2004. With this model, we can focus on predicting f(T ) and j(T )
only, because we have removed the complex intra-year structure.
We have thus fitted the model (13) to three subsets of the whole available
time series for the growth rate g(t) and also to the whole set taken globally, in
order to test for the robustness of the model. For this, we use the cost function
Tmax∑
[g(t = 12T +m)− f(T )h(m)− j(T )]2 (14)
which is minimized with respect to the 12 unknown variables h(1), ..., h(12)
and the 2 × Tmax variables [f(1), j(1)], ..., f(Tmax), j(Tmax). There are 12Tmax
terms in the sum and 12 + 2 × Tmax unknown variables. This shows that the
system is well-constrained as soon as Tmax ≥ 2. For instance for Tmax = 20, we
have 52 unknown variables to fit and 240 terms in the sum to constrain the
Figure 8 illustrates the result of the fit of model (13) to the growth rate over
the whole time interval from 1985 to 2005. As expected, we can observe a clear
peak in the amplitude f(T ) corresponding to the year 2004, while there is not
appreciable peak around 1990. This means that the recent bubble appears
significantly stronger than any other episodes in the last 20 years and dwarfs
them. The anomalous nature of the recent bubble is reinforced by the existence
of a peak in j(T ) for the same year 2004, showing that both the amplitude and
translation components of the growth rates has been completely anomalous
in 2004. The middle graph of the top panel of figure 8 shows the intra-year
pattern captured by the model, which is in remarkable agreement with the
pattern shown in figure 7: one can observe a peak in March, May, August and
December, the largest peak being in May. The bottom panel of figure 8 shows
visually how well (or badly) the model fits the actual data. The quality of
the fit is excellent, except in 2004-2005. In other words, we clearly identify
a very anomalous or exceptional behavior in 2004-2005, again providing a
confirmation that something exceptional or anomalous has occurred during
that period.
1985 1990 1995 2000 2005
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
−0.025
−0.02
−0.015
−0.01
1985 1990 1995 2000 2005
1980 1985 1990 1995 2000 2005 2010
−0.03
−0.02
−0.01
Fig. 8. Upper panels: three graphs showing the three functions f(T ), h(m) and
j(T ) fitted on the growth rate over the whole time interval from 1985 to 2005.
Lower panel: Comparison between the growth rate data (empty blue circles) and
the model (13) (red line).
Figure 9 is the same as figure 8 for the period from 1985 to 1990. One can
clearly here observe a peak in the scaling amplitude f(T ) at T =1988 and in
the translation term j(T ) at T =1986, suggesting that the first bubble of the
1985-2005 period occurred over a relatively large time period 1985-1990, with
two successive contributions. The intra-year structure h(m) has also its peaks
on March, May, August and December, but this intra-year structure is weaker
than for other sub-periods. The lower panel of figure 9 shows that the model
captures very well the overall trend as well as the intra-year structure. The
main discrepancies are in the amplitude of the large peaks and valleys, which
are not fully predicted.
1985 1985.5 1986 1986.5 1987 1987.5 1988 1988.5 1989 1989.5 1990
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
0.185
0.195
0.205
1985 1985.5 1986 1986.5 1987 1987.5 1988 1988.5 1989 1989.5 1990
−0.29
−0.28
−0.27
1984 1985 1986 1987 1988 1989 1990 1991
−0.03
−0.02
−0.01
Fig. 9. Same as figure 8 for the period from 1985 to 1990.
Figure 10 is the same as figure 8 for the period from 1991 to 2000. One can
clearly here observe a peak in the scaling amplitude f(T ) at T =1995 and
in the translation term j(T ) at T =1994. This thus identifies a small bubble
in the mid-1990s. The intra-year structure h(m) has also its peaks on March,
May, August and December, with very large amplitudes. The lower panel of
figure 10 shows a truly excellent fit.
1991 1992 1993 1994 1995 1996 1997 1998 1999 2000
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
−0.12
−0.115
−0.11
−0.105
1991 1992 1993 1994 1995 1996 1997 1998 1999 2000
0.155
0.165
1990 1992 1994 1996 1998 2000 2002
−0.01
−0.005
0.005
0.015
0.025
Fig. 10. Same as figure 8 for the period from 1991 to 2000.
Figure 11 is the same as figure 8 for the period from 2001 to 2005. One can
clearly here observe a peak in the scaling amplitude f(T ) at T =2004 and
in the translation term j(T ) also at T =2004. This thus clearly identifies the
bubble as peaking in 2004. The intra-year structure h(m) has also its peaks
on March, May, August and December, with very large amplitudes and very
good agreement with the other three figures. The lower panel of figure 11
shows an excellent fit up to the early 2003 and then a rather large discrepancy
starting early 2003 all the way to the last data point approaching mid-2005. In
particular, note that the intra-year structure is washed out by the anomalous
growth rate culminating in mid-2004. Symmetrically, the intra-year structure
is also absent in the fast decay of the growth rate back to normal. We do
not have enough data to ascertain if the growth rate has resumed its normal
intra-year pattern. We believe that this is a very important diagnostic to
characterize abnormal behavior and this could be a very useful variable to
monitor on a monthly basis.
2001 2001.5 2002 2002.5 2003 2003.5 2004 2004.5 2005
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
−0.025
−0.02
−0.015
−0.01
2001 2001.5 2002 2002.5 2003 2003.5 2004 2004.5 2005
2000 2001 2002 2003 2004 2005 2006
−0.01
Fig. 11. Same as figure 8 for the period from 2001 to 2005.
The four figures 8-11 validate model (13): in particular, they show the very
robust intra-year structure with peaks in March, May, August and December.
One possible contribution to this quarterly periodicity comes from the con-
struction of the CSI: the monthly indexes use a three-month moving average
algorithm. Home sales pairs are accumulated in rolling three-month periods,
on which the repeat sales methodology is applied. The index point for each
reporting month is based on sales pairs found for that month and the pre-
ceding two months. For example, the December 2005 index point is based
on repeat sales data for October, November and December of 2005. This av-
eraging methodology is used to offset delays that can occur in the flow of
sales price data from county deed recorders and to keep sample sizes large
enough to create meaningful price change averages. A three month rolling
window construction corresponds in general to a convolution of the bare price
with a kernel which possesses a three month periodicity (or size). The Fourier
transform of the convolution is the product of Fourier transforms. Thus the
spectrum of the signal should contain the peaks of the Fourier spectrum of
the kernel, which by construction contains a peak at three months. However,
our synthetic tests (not shown) suggest that this effect is by far too small to
explain the strong amplitude of the observed quarterly periodicity. It would
be important to understanding why such intra-year structure develops: is it
the result of a natural intra-day organization of buyers’ behaviors associated
with taxes/ income constraints or a problem of reporting or perhaps the effect
of other calendar regularities? Or is it the result of patterns coming from the
supply part of the equation, namely home-builders, developers, and perhaps
in the time modulation of the rates of allocated permits? Answering these
questions is important to determine how much emphasis one should give to
these results. But if indeed the intra-day structure is a genuine non-artificial
phenomenon, we believe that it offers a remarkable opportunity for monitoring
in real time the normal versus abnormal evolution of the market and also for
developing forecasts on a month time horizon.
4.3 Intra-year pattern from signs of growth rate increments
The existence of a strong and robust intra-year structure in the price growth
rate can be further demonstrated by studying the sign of g(t + 1) − g(t). A
positive (negative) sign mean that the growth rate tends to increase (decrease)
from one month to the next.
Based on the seasonality of the growth rate, we are able to answer the following
question: given the current growth rate g(t), will the growth rate increase or
decrease at time t+1? This amounts to asking what is the sign of g(t+1)−g(t)?
Technically, we construct the (unconditional) number of times the sign of the
increment g(t + 1) − g(t) is positive or negative irrespective of what is g(t).
From Fig. 4, we obtain a sequence of signs: −−+−+−−+−−++. For each
month, we calculate the percentage of positive and negative signs, respectively.
The second and the third rows of Table 1 gives the percentage of positive and
negative signs for each month. The third and fourth rows gives the signs and
the associated percentages.
For instance, the table says that the “probability” of the sign of g(t = Feb)−
g(t = Jan) being “-” is about 92.1%. If we know g(t = Jan), we can say that it
is very probable that the growth rate of February will be less than this January
value. Thus, this table has predictive power in the sense that the probabilities
to predict the signs are much higher than the value of 75% obtained under the
null hypothesis that g(t) is a white noise process (see Sornette and Andersen,
2000). This table is another way to rephrase and expand on our preceding
analysis on the yearly periodicity by identifying a very strong and robust
intra-year structure.
Table 1
Analysis of the signs of g(t + 1) − g(t). The second and the third rows gives the
percentage of positive and negative signs for each month. The third and fourth rows
give the sign for each month that dominates and the associated percentages.
Mon Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
+% 7.91 17.2 88.0 5.64 97.7 8.47 8.47 91.4 6.57 8.92 84.2 82.2
-% 92.1 82.8 12.0 94.4 2.29 91.5 91.5 8.59 93.4 91.1 15.8 17.8
sign - - + - + - - + - - + +
% 92.1 82.8 88.0 94.4 97.7 91.5 91.5 91.4 93.4 91.1 84.2 82.2
Since our initial analysis performed in the summer of 2005 which used data
up to March 2005, new data for the 27 CSW indexes has become available
which covers the interval from Apr. 2005 to Sept. 2006. It is very interesting
to check if the sign of the growth variations obtained in Table 1 using the data
until March 2005 still applies to the new data. The realized signs of the newly
available months are calculated and the sequence of signs is the following: -
(Apr. 2005, 27 CSW indexes out of 27), + (May. 2005, 27 out of 27), - (Jun.
2005, 27 out of 27), - (Jul. 2005, 27 out of 27), + (Aug. 2005, 27 out of 27), -
(Sep. 2005, 27 out of 27), - (Oct. 2005, 27 out of 27), + (Nov. 2005, 27 out of
27), + (Dec. 2005, 21 out of 27), - (Jan. 2006, 27 out of 27), - (Feb. 2006, 27 out
of 27), + (Mar. 2006, 27 out of 27), - (Apr. 2006, 27 out of 27), + (May. 2006,
27 out of 27), - (Jun. 2006, 27 out of 27), - (Jul. 2006, 27 out of 27), + (Aug.
2006, 27 out of 27), and - (Sep. 2006, 27 out of 27). Thus, table 1 predicts
exactly the signs of the growth rate variations of all 27 CSW indexes for all
months except for Dec. 2005 for which there are 6 errors: table 1 predicts that
the growth rate variation from Dec. 2005 to Jan. 2006 should be +, which is
correct for 21 CSW indexes out of 27, corresponding to a success ratio of 77%
(close to the white noise case). This score is slightly lower than the previously
estimated probability of 82.2% for the month of December, which is the lowest
among all months. Overall, the success rate is remarkably high, adding further
evidence that the Las Vegas property market has returned to a more normal
phase (no bubble from April 2005 to Sept. 2006).
5 Predicting the monthly growth rate
Conditional of the evidence that the anomalous faster than exponential growth
has ended, let us attempt to predict the future evolution of the CSW indexes
based only on the strong seasonality of the growth rate. Figure 12 presents
the predictions one year ahead for the 27 regional CSW indexes. Two different
prediction schemes are used. The RED lines are based on the average growth
rate obtained from all 27 indexes, while the MAGENTA lines are based on the
average growth rate obtained from the individual index under investigation.
There is not discernable difference.
A similar prediction of the Clark County (Las Vegas MSA) indexes (NVC003Q
and NVC003C) has also been made using the average growth rates obtained
from all 27 regional indexes. Since these two indexes are only available from
July-2000 to March-2005, we do not have enough data to calculate the average
growth rates using the indexes themselves. The results are shown in Fig. 13.
2002 2003 2004 2005 2006 2007
Fig. 12. Predicting regional CSW indexes one year ahead. Red lines: Prediction
using average growth rate obtained from all 27 indexes; Magenta lines: Prediction
using average growth rate obtained from the individual index under investigation.
The two kinds of prediction are almost undistinguishable.
2000 2001 2002 2003 2004 2005 2006 2007
NVC003C: Raw data
NVC003C: Prediction
NVC003Q: Raw data
NVC003Q: Prediction
Fig. 13. Predicting Clark County (Las Vegas MSA) indexes (NVC003Q and
NVC003C) one year ahead.
6 Conclusion
We have analyzed 27 house price indexes of Las Vegas from Jun. 1983 to
Mar. 2005, corresponding to 27 different zip codes. These analyses confirm
the existence of a real-estate bubble, defined as a price acceleration faster
than exponential. This bubble is found however to be confined to a rather
limited time interval in the recent past from approximately 2003 to mid-2004
and has progressively transformed into a more normal growth rate in 2005.
The data up to mid-2005 suggests that the current growth rate has now come
back to pre-bubble levels. We conclude that there has been no bubble from
1990 to 2002 except for a medium-sized surge in 1995, then a short-lived but
very strong bubble until mid-2004 which has been followed by a smoothed
transition back to what appears to be normal. It thus seems that, while the
strength of the real-estate bubble has been very strong over the period 2003-
2004, the price appreciation rate has returned basically to normal.
In addition, we have identified a strong yearly periodicity which provides a
good potential for fine-tuned prediction from month to month. As the intra-
year structure is likely a genuine non-artificial phenomenon, it offers a re-
markable opportunity for monitoring in real time the normal versus abnormal
evolution of the market and also for developing forecasts on a monthly time
horizon. In particular, a monthly monitoring using a model that we have de-
veloped here could confirm, by testing the intra-year structure, if indeed the
market has returned to “normal” or if more turbulence is expected ahead.
In addition, it would provide a real-time observatory of upsurges and other
anomalous behavior at the monthly scale. This requires additional technical
developments and tests beyond this report.
Compared with previous analysis of Zhou and Sornette (2003, 2006) at the
scale of states and whole regions (northeast, midwest, south and west), the
present analysis demonstrates the existence of very significant variations at the
local scale, in the sense that the bubble in Las Vegas seems to have preceded
the more global USA bubble and has ended approximately two years earlier
(mid 2004 for Las Vegas compared with mid-2006 for the whole of the USA).
References
Bailey, M., Muth, R., Nourse, H., 1963. A regression method for real estate
price index construction. Journal of the American Statistical Association
58, 933–942.
Bender, C., Orszag, S. A., 1978. Advanced Mathematical Methods for Scien-
tists and Engineers. McGraw-Hill, New York.
Broekstra, G., Sornette, D., Zhou, W.-X., 2005. Bubble, critical zone and the
crash of Royal Ahold. Physica A 346, 529–560.
Case, K. E., Shiller, R. J., 1987. Prices of single-family homes since 1970: New
indexes for four cities. New England Economic Review Sep/Oct, 45–56.
Case, K. E., Shiller, R. J., 1989. The efficiency of the market for single-family
homes. American Economic Review 79, 125–137.
Case, K. E., Shiller, R. J., 1990. Forecasting prices and excess returns in the
housing market. AREUEA J. 18, 253–273.
Case, K. E., Shiller, R. J., 2003. Is there a bubble in the housing market.
Brookings Papers on Economic Activity (2), 299–362.
Greenspan, A., 1997. Federal ReserveÕs semiannual monetary policy report,
before the Committee on Banking, Housing, and Urban Affairs, U.S. Senate,
February 26.
Himmelberg, C., Mayer, C., Sinai, T., September 2005. Assessing high house
prices: Bubbles, fundamentals, and misperceptions. Tech. Rep. Staff Report
no. 218, Federal Reserve Bank of New York.
Jorion, P., 2005. Is housing market surge really sustainable? The Wall Street
Journal September 22, A17.
Mauboussin, M. J., Hiler, R., 1999. Rational Exuberance? Equity research
report of Credit Suisse First Boston, January 26.
Roehner, B. M., 2006. Real estate price peaks: A comparative overview. Evo-
lutionary and Institutional Economics Review in press, physics/0605133.
Shiller, R. J., 2000. Irrational Exuberance. Princeton University Press, New
York.
Shiller, R. J., 2006. Long-term perspectives on the current boom in home
prices. Economists’ Voice 3(4), Art. 4.
Smith, M. H., Smith, G., 2006. Bubble, bubble, where’s the housing bubble?
Brookings Papers on Economic Activity (1), 1–67.
Sornette, D., 2003. Why Stock Markets Crash: Critical Events in Complex
Financial Systems. Princeton University Press, Princeton.
Sornette, D., Andersen, J.-V., 1998. Scaling with respect to disorder in time-
to-failure. European Physical Journal B 1, 353–357.
Sornette, D., Andersen, J.-V., 2000. Increments of uncorrelated time series
can be predicted with a universal 75% probability of success. International
Journal of Modern Physics C 11, 713–720.
Zhou, W.-X., Sornette, D., 2003. 2000-2003 real estate bubble in the UK but
not in the USA. Physica A 329, 249–263.
Zhou, W.-X., Sornette, D., 2006. Is there a real-estate bubble in the US?
Physica A 361, 297–308.
Biographies:
Wei-Xing Zhou is a Professor of Finance at the School of Business in the East
China University of Science and Technology. He received his PhD in Chemical
Engineering from the East China University of Science and Technology in
2001. His current research interest focuses on the modeling and prediction of
catastrophic events in complex systems.
Didier Sornette holds the Chair of Entrepreneurial Risks at the Depart-
ment of Management, Technology and Economics of ETH Zurich. He received
his PhD in Statistical Physics from the University of Nice, France. His current
research focuses on the modeling and prediction of catastrophic events in com-
plex systems, with applications to finance, economics, seismology, geophysics
and biology.
	Introduction
	Conceptual background of our empirical analysis
	Humans as social animals and herding
	Definition and mechanism for bubbles
	Was there a bubble? Status of the argument based on the ratio of cost of owning versus cost of renting
	Regime shift in the CSW Zip Code Indexes of Las Vegas
	Description of the data
	Power law fits
	Dependence of the growth rate on the index value
	Yearly periodicity and intra-year structure
	Yearly periodicity from superposed year analysis and spectral analysis
	Yearly periodicity and intra-year structure with a scale and translation modulated model
	Intra-year pattern from signs of growth rate increments
	Predicting the monthly growth rate
	Conclusion

  We analyze 27 house price indexes of Las Vegas from Jun. 1983 to Mar. 2005,
corresponding to 27 different zip codes. These analyses confirm the existence
of a real-estate bubble, defined as a price acceleration faster than
exponential, which is found however to be confined to a rather limited time
interval in the recent past from approximately 2003 to mid-2004 and has
progressively transformed into a more normal growth rate comparable to
pre-bubble levels in 2005. There has been no bubble till 2002 except for a
medium-sized surge in 1990. In addition, we have identified a strong yearly
periodicity which provides a good potential for fine-tuned prediction from
month to month. A monthly monitoring using a model that we have developed could
confirm, by testing the intra-year structure, if indeed the market has returned
to ``normal'' or if more turbulence is expected ahead. We predict the evolution
of the indexes one year ahead, which is validated with new data up to Sep.
2006. The present analysis demonstrates the existence of very significant
variations at the local scale, in the sense that the bubble in Las Vegas seems
to have preceded the more global USA bubble and has ended approximately two
years earlier (mid 2004 for Las Vegas compared with mid-2006 for the whole of
the USA).

Introduction
Zhou and Sornette (2003) analyzed the deflated quarterly average sales prices
p(t) from December 1992 to December 2002 of new houses sold in all the states
in the USA and by regions (northeast, midwest, south and west) and found
that, while there was undoubtedly a strong growth rate, there was no evidence
of a bubble in the latest six years (as qualified by a super-exponential growth).
Then, Zhou and Sornette (2006) analyzed the quarterly average sale prices of
new houses sold in the USA as a whole, in the northeast, midwest, south,
and west of the USA, in each of the 50 states and the District of Columbia
of the USA up to the first quarter of 2005, to determine whether they have
grown faster-than-exponential (which is taken as the diagnostic of a bubble).
Zhou and Sornette (2006) found that 22 states (mostly Northeast and West)
exhibit clear-cut signatures of a fast growing bubble. From the analysis of the
S&P 500 Home Index, they concluded that the turning point of the bubble
would probably occur around mid-2006. The specific statement found at the
bottom of page 306 of Ref.[Zhou and Sornette (2006)] is: “We observe a good
stability of the predicted tc ≈ mid-2006 for the two LPPL models (2) and (3).
The spread of tc is larger for the second-order LPPL fits but brackets mid-
2006. As mentioned before, the power-law fits are not reliable. We conclude
that the turning point of the bubble will probably occur around mid-2006.”
It should be stressed that these studies departed from most other reports by
analysts and consulting firms on real estate prices in that Zhou and Sornette
(2003, 2006) did not characterize the housing market as overpriced in 2003. It is
only in 2004-2005 that they confirmed that the signatures of an unsustainable
bubble path has been revealed.
Let us briefly analyze how this prediction has fared. The upper panel of Figure
1 shows the quarterly house price indexes (HPIs) in the 21 states and in the
District of Columbia (DC) from 1994 to the fourth quarter of 2006 released
by the OFHEO. It is evident that the growth in most of these 22 HPIs has
slowed down or even stopped during the year of 2006. When we look at the
S&P Case-Shiller Home Indexes of the 20 major US cities, as illustrated in the
lower panel of Figure 1, we observe that the majority of the S&P/CSIs had
a maximum denoted by a solid dot in the middle of 2006, validating the pre-
diction of Zhou and Sornette (2006). Specifically, the times of the maxima are
respectively 2006/06/01, 2006/09/01, 2005/11/01, 2006/05/01, 2006/08/01,
2006/05/01, 2006/12/01, 2006/07/01, 2006/08/01, 2006/09/01, 2005/09/01,
2005/12/01, 2006/09/01, 2006/09/01, 2006/08/01, 2006/06/01, 2006/07/01,
2006/09/01, 2006/08/01, 2006/12/01, 2006/06/01, and 2006/07/01 for the 20
cities shown in the legend of the lower panel. The only two cities with a max-
Education Foundation (Grant 101086), and the Alfred Kastler Foundation which
supported W.-X. Zhou for a visiting position in France.
imum occurring later towards the end of 2006 (2006/12/01) are Miami and
Seattle. However their growth rates decreased remarkably in 2006 as shown
in the figure. Furthermore, the S&P/CS Home Price Composite-10 reached
its historical high 226.29 on 2006/06/01 and the Composite-20 culminated to
206.53 on 2006/07/01, again confirming remarkably well the validity of the
forecast of Zhou and Sornette (2006).
1994 1996 1998 2000 2002 2004 2006 2008 2010
2000 2001 2002 2003 2004 2005 2006 2007
Phoenix − AZ
Los Angeles
San Diego
San Francisco
Denver
Washington
Miami
Tampa − FL
Atlanta − GA
Chicago
Boston
Detroit − MI
Minneapolis − MN
Charlotte − NC
Las Vegas
New York
Cleveland − OH
Portland − OR
Dallas − TX
Seattle − WA
Fig. 1. Evaluation of the prediction of Zhou and Sornette (2006) that “the turning
point of the bubble will probably occur around mid-2006” using the OFHEO HPI
data (upper panel) and the S&P CSI data (lower panel).
In this note, we provide a more regional study of the diagnostic of bubbles
and the prediction of their demise. Specifically, we analyze the Case-Shiller-
Weiss (CSW) Zip Code Indexes of 27 different Las Vegas regions calculated
with a monthly rate from June-1983 to March-2005. The CSW Indexes are
based on the so-called repeat sales methods which directly measure house
price appreciations. The key to these data is that they are observations of
multiple transactions on the same property, repeated over many properties
and then pooled in an index. Prices from different time periods are combined
to create “matched pairs,” providing a direct measure of price changes for a
given property over a known period of time. Bailey et al. (1963) proposed the
basic repeat sales method over four decades ago, but only after the work by
Case and Shiller (1987, 1989, 1990) did the idea receive significant attention
in the housing research community.
Studying the Las Vegas database is particular suitable since Las Vegas belongs
to a state which was identified by Zhou and Sornette (2006) as one of the
22 states with a fast growing bubble in 2005. With access to 27 different
CSW Zip Code Indexes of Las Vegas, we are able to obtain more reliable and
fine-grained measures, which both confirm and extend the previous analyses
of Zhou and Sornette (2003, 2006). The next section recalls the conceptual
background underlying our empirical approach. Then, section 3 analyzes the
regional CSW indexes for Las Vegas, showing that there is a regime shift
separated by a bubble around year 2004. Section 4 identifies and then analyzes
the yearly periodicity and intra-year pattern detected in the growth rate of
the regional CSW indexes. Section 5 offers a preliminary forecast based on the
periodicity analyses in Sec. 4. Section 6 concludes.
2 Conceptual background of our empirical analysis
2.1 Humans as social animals and herding
Humans are perhaps the most social mammals and they shape their envi-
ronment to their personal and social needs. This statement is based on a
growing body of research at the frontier between new disciplines called neuro-
economics, evolutionary psychology, cognitive science, and behavioral finance.
This body of evidence emphasizes the very human nature of humans with
its biases and limitations, opposed to the previously prevailing view of ratio-
nal economic agents optimizing their decisions based on unlimited access to
information and to computation resources.
Here, we focus on an empirical question (the existence and detection of real-
estate bubbles) which, we hypothesize, is a footprint of perhaps the most
robust trait of humans and the most visible imprint in our social affairs: im-
itation and herding. Imitation has been documented in psychology and in
neuro-sciences as one of the most evolved cognitive process, requiring a de-
veloped cortex and sophisticated processing abilities. In short, we learn our
basics and how to adapt mostly by imitation all along our life. It seems that
imitation has evolved as an evolutionary advantageous trait, and may even
have promoted the development of our anomalously large brain (compared
with other mammals). It is actually “rational” to imitate when lacking suffi-
cient time, energy and information to take a decision based only on private
information and processing, that is..., most of the time. Imitation, in obvious
or subtle forms, is a pervasive activity of humans. In the modern business,
economic and financial worlds, the tendency for humans to imitate leads in
its strongest form to herding and to crowd effects.
Based on a theory of cooperative herding and imitation, we have shown that
imitation leads to positive feedbacks, that is, an action leads to consequences
which themselves reinforce the action and so on, leading to virtuous or vicious
circles. We have formalized these ideas in a general mathematical theory which
has led to observable signature of herding, in the form of so-called log-periodic
power law acceleration of prices. A power law acceleration of prices reflects
the positive feedback mechanism. When present, log-periodicity takes into
account the competition between positive feedback (self-fulfilling sentiment),
negative feedbacks (contrariant behavior and fundamental/value analysis) and
inertia (everything takes time to adjust). Sornette (2003) presented a general
introduction, a synthesis and examples of applications.
2.2 Definition and mechanism for bubbles
The term “bubble” is widely used but rarely clearly defined. Following Case and Shiller
(2003), the term “bubble” refers to a situation in which excessive public ex-
pectations of future price increases cause prices to be temporarily elevated.
During a housing price bubble, homebuyers think that a home that they would
normally consider too expensive for them is now an acceptable purchase be-
cause they will be compensated by significant further price increases. They
will not need to save as much as they otherwise might, because they expect
the increased value of their home to do the saving for them. First-time home-
buyers may also worry during a housing bubble that if they do not buy now,
they will not be able to afford a home later. Furthermore, the expectation of
large price increases may have a strong impact on demand if people think that
home prices are very unlikely to fall, and certainly not likely to fall for long,
so that there is little perceived risk associated with an investment in a home.
What is the origin of bubbles? In a nutshell, speculative bubbles are caused
by “precipitating factors” that change public opinion about markets or that
have an immediate impact on demand, and by “amplification mechanisms”
that take the form of price-to-price feedback, as stressed by Shiller (2000). A
number of fundamental factors can influence price movements in housing mar-
kets. On the demand side, demographics, income growth, employment growth,
changes in financing mechanisms or interest rates, as well as changes in loca-
tion characteristics such as accessibility, schools, or crime, to name a few, have
been shown to have effects. On the supply side, attention has been paid to
construction costs, the age of the housing stock, and the industrial organiza-
tion of the housing market. The elasticity of supply has been shown to be a
key factor in the cyclical behavior of home prices. The cyclical process that
we observed in the 1980s in those cities experiencing boom-and-bust cycles
was caused by the general economic expansion, best proxied by employment
gains, which drove demand up. In the short run, those increases in demand
encountered an inelastic supply of housing and developable land, inventories
of for-sale properties shrank, and vacancy declined. As a consequence, prices
accelerated. This provided an amplification mechanism as it led buyers to
anticipate further gains, and the bubble was born. Once prices overshoot or
supply catches up, inventories begin to rise, time on the market increases,
vacancy rises, and price increases slow down, eventually encountering down-
ward stickiness. The predominant story about home prices is always the prices
themselves (see Shiller, 2000; Sornette, 2003); the feedback from initial price
increases to further price increases is a mechanism that amplifies the effects of
the precipitating factors. If prices are going up rapidly, there is much word-of-
mouth communication, a hallmark of a bubble. The word of mouth can spread
optimistic stories and thus help cause an overreaction to other stories, such as
stories about employment. The amplification can also work on the downside
as well. Price decreases will generate publicity for negative stories about the
city, but downward stickiness is encountered initially.
2.3 Was there a bubble? Status of the argument based on the ratio of cost of
owning versus cost of renting
In recent years, there has been increasing debates on whether there was a real
estate bubble or not in the United States of America. Case and Shiller (2003),
Shiller (2006) and Smith and Smith (2006) argued that the house prices over
the period 2000-2005 were not abnormal as they reflected only the convergence
of the prices to their fundamentals from below. In contrast, Zhou and Sornette
(2006) and Roehner (2006) have suggested that there was a bubble, which be-
came identifiable only after 2003, that is, after the work of Zhou and Sornette
(2003).
In this context, it is instructive to comment on the study by Himmelberg et al.
(2005), from the Federal Reserve Bank of New York , as it reflects the never
ending debate between tenants of the fundamental valuation explanation and
those invoking speculative bubbles. We are resolutely part of the second group.
Himmelberg et al. (2005) constructed measures of the annual cost of single-
family housing for 46 metropolitan areas in the United States over the last 25
years and compared them with local rents and incomes as a way of judging the
level of housing prices. In a nutshell, they claimed in 2005 that conventional
metrics like the growth rate of house prices, the price-to-rent ratio, and the
price-to-income ratio can be misleading and lead to incorrect conclusions on
the existence of the real-estate bubble. Their measure showed that, during
the 1980s, houses looked most overvalued in many of the same cities that
subsequently experienced the largest house price declines. But they found that
from the trough of 1995 to 2004, the cost of owning rose somewhat relative
to the cost of renting, but not, in most cities, to levels that made houses look
overvalued.
The rosy conclusion of Himmelberg et al. (2005), that 2004-2005 prices were
justifiable and that there was no risk of deflation as no bubble was present, is
based on a particularly curious comparison between cost of owning and cost of
renting, as noticed by Jorion (2005), in a letter to the Wall Street Journal. In-
deed, they candidly revealed however that their “cost of owning” calculations
imply an “expected appreciation on the property” coefficient. The value for
this factor is no doubt derived from figures for appreciation as currently ob-
served on the housing market, meaning they regarded the current appreciation
level as a reasonable assumption for what would indeed happen next – which
is precisely what our analyses and that of others question. In other words, the
authors had unwittingly hard-wired into their model the assertion that there
was no housing bubble; little wonder then that this is also what they felt au-
thorized to conclude. The circularity of their reasoning is particularly obvious
in an illustration they gave for San Francisco where for more than 60 years the
price-to-rent ratio has exceeded the national average, which, so they claimed,
“does not necessarily make owning there more expensive than renting.” The
reason why is that “high financing costs are offset by above-average expected
capital gains.” Translated, this means that as long as there is a bubble, prices
will go up and investing in a house remain a profitable operation. This trivial
statement is hollow; the real question is whether the trend that is observed
now remains sustainable.
In addition to this criticism put forward by Jorion (2005), there are other
reasons to doubt the validity of the conclusion of Himmelberg et al. (2005). In
the own words of Himmelberg et al. (2005), “the ratio of the cost of owning to
the cost of renting is especially sensitive to the real long-term interest rates.”
They are right in their rosy conclusion... as long as the long-term interest rates
remain exceptionally low. It is particularly surprising that their estimation of
the ratio of the cost of owning to the cost of renting was based on the most
recent rates over the preceding year of their analysis (2004), while the price of a
house is a long-term investment: what will be the long-term rates in 10, 20, 30,
or 50 years? Another problem is that their analysis was “mono-dimensional”:
they proposed that everything depends only on the ratio of the cost of owning
to the cost of renting. But they missed the interest rates as an independent
variable. As a consequence, it is not reasonable to compare the 1980s and the
present time, as the long-term interest rates had nothing in common. Another
problem with their analysis is that they assumed “equilibrium,” while people
are sensitive to the history-dependent path followed by the prices. In other
words, people are sensitive to the way prices reach a certain level, if there is
an acceleration that can self-fuel itself for a while, while Himmelberg et al.
(2005) discussed only the mono-dimensional level of the price, and not how it
got there. We think that this general error made by “equilibrium” economists
constitutes a fundamental flaw which fails to capture the real nature of the
organization of human societies and their decision process. In the sequel, we
actually focus our attention on signatures of price trajectories that highlight
the importance of history dependence for prediction.
This discussion is reminiscent of the proposition by Mauboussin and Hiler
(1999), offered close to the peak of the Internet and new technology bubble
that culminated in 2000, that better business models, the network effect, first-
to-scale advantages, and real options effect could account rationally for the
high prices of dot.com and other New Economy companies. These interest-
ing views expounded in early 1999 were in synchrony with the bull market
of 1999 and preceding years. They participated in the general optimistic view
and added to the strength of the herd. Later, after the collapse of the bubble,
these explanations seem less attractive. This did not escape the then U.S. Fed-
eral Reserve chairman Alan Greenspan (1997), who said : “Is it possible that
there is something fundamentally new about this current period that would
warrant such complacency? Yes, it is possible. Markets may have become more
efficient, competition is more global, and information technology has doubt-
less enhanced the stability of business operations. But, regrettably, history is
strewn with visions of such new eras that, in the end, have proven to be a
mirage. In short, history counsels caution.”
3 Regime shift in the CSW Zip Code Indexes of Las Vegas
3.1 Description of the data
We now turn to the analysis of the CSW indexes of 27 different Las Vegas zip
regions obtained with a monthly rate. The 27 monthly CSW data sets start
from June-1983 and end in March-2005. Figure 2 shows the price trajectories
of all the 27 CSW indexes. Visual inspection shows (i) a very similar behavior
of all the different zip codes and (ii) a sudden increase of the indexes since
Mid-2003. Let us now analyze this data quantitatively.
3.2 Power law fits
The simplest mathematical equation capturing the positive feedback effect
and herding is the power law formula (see Broekstra et al., 2005, for a simple
introduction in a similar context)
I(t) = A+B|tc − t|
m , (1)
1980 1985 1990 1995 2000 2005 2010
Fig. 2. Time evolution of the Case-Shiller-Weiss (CSW) Zip Code Indexes of 27 Las
Vegas zip regions from June-1983 to March-2005.
where B < 0 and 0 < m < 1 or B > 0 and m < 1. Others cases do not
qualify as a power law acceleration. For B < 0 and 0 < m < 1 or B > 0
and m < 0, the trajectory of I(t) described by (1) expresses the existence of
an accelerating bubble, which is faster than exponential. This is taken as one
hallmark of the existence of a bubble.
Notice also that this formula expresses the existence of a singularity at time tc,
which should be interpreted as a change of regime (the mathematical singu-
larity does not exist in reality and is rounded off by so-called finite-size effects
and the appearance of a large susceptibility to other mechanisms). This criti-
cal time tc must be interpreted as the end of the bubble and the time where
the regime is transiting to another state through a crash or simply a plateau
or a slowly moving correction.
We have fitted each of the 27 individual CSW indexes using the pure power
law model (1). The data used for fitting is from Dec-1995 to Jun-2005. We do
not show the results as the signature of a power law growth is not evident,
essentially because the acceleration is only over a rather short period of time
from approximately 2002 to 2004. As a consequence, power law fits give unre-
liable critical time tc too much in the future (like 2008 and beyond). We have
thus redone the fits of the 27 CSW indexes over a shorter time interval from
Aug-2001 to Jun-2005. A typical example is shown in Fig 3. All other 26 CSW
are very similar, with some variations of the parameters, but the message is
the same: while there is a clear faster-than-exponential acceleration over most
of the time interval, the price trajectory has clearly transitioned into another
regime in the latter part of the time interval considered here. The transition
occurred smoothly from mid-2004 to mid-2005 (the end of the time period
analyzed here).
It is important to recognize that the power law regime is expected only rela-
2001 2002 2003 2004 2005 2006
 = 2012.94;   m = −12;   χ = 0.007679
Fig. 3. Typical evolution of a CSW index from Aug-2001 to Jun-2005 and its fit by
a power law, showing both the faster-than-exponential growth up to mid-2004 and
the smooth transition to a much slower growth at later times. The root-mean-square
χ of the residuals of the fit as well at tc and m are given inside the figure.
tively close to the critical time tc, while other behaviors are expected far from
tc. The simplest model is to consider that, far from tc, the price follows an
exponential growth with an approximately constant growth rate µ:
I(t) = a+ beµt . (2)
A fuller description is thus to consider that formula (2) holds from the begin-
ning of the time series up to a cross-over time t∗, beyond which expression (1)
takes over. Any given price trajectory should thus be fitted by (2) from some
initial time tstart to time t
∗ and then by (1) from t∗ to the end of the time
series. Technically, t∗ is known from the parameters a, b, µ, A,B, tc, m by the
condition of continuity of I(t) at t = t∗, that is, both formulas give the same
value at t = t∗. We can further determine one of the parameters a, b or µ by
imposing a condition of differentiability at t∗, that is, the first time-derivative
of I(t) is continuous at t∗. This approach is known in numerical analysis as
“asymptotic matching” (see Bender and Orszag, 1978).
A simplified description of such a cross-over between a standard exponential
growth and the power law super-exponential acceleration is obtained by using
a more compact formulation
I(t) = A+B tanh[(tc − t)/τ ]
m , (3)
where tanh denotes the hyperbolic tangent function. This expression derives
from a study of the transition from the non-critical to critical regime in rup-
ture processes (of which bubbles and their terminal singularity belong to)
conducted by Sornette and Andersen (1998). This expression has the virtue
of providing automatically a smooth transition between the exponential be-
havior (2) and the pure power law (1), since tanh[(tc − t)/τ ] ≈ (tc − t)/τ for
tc− t < τ and tanh[(tc− t)/τ ] ≈ 1− 2e
2(t−tc)/τ for tc− t > τ . In this later case
tc − t > τ , expression (3) becomes of the form (2) with m = 1 and
a=A+B , (4)
b=−2Be−2tc/τ , (5)
µ=1/τ . (6)
In contrast, for tc − t < τ , expression (3) becomes of the form (1) with the
correspondence B/τm → B. Expression (3) has only five free parameters, in
contrast with the model involving the cross-over from (2) to (1) which has 7
free parameters (a, b, µ, A,B, tc, m) while t
∗ is determined by the asymptotic
matching). The pure power law formula (1) has 4 parameters while the ex-
ponential law (2) has just 3 parameters. The problem with expression (3) is
that it does not recover a pure exponential growth even for tc − t > τ , when
m 6= 1. Thus, expression (3) is limited in fully describing a possible cross-over
from a standard mild exponential growth and an super-exponential power law
acceleration. Our tests (not shown) find that a fit with model (3) retrieve the
pure power law model (1) with the same critical time tc and exponent m and
the same root-mean-square residual r.m.s. (the fit adjusts the parameter τ to a
very large value, ensuring that the fit is always in the regime tc−t ≪ τ so that
the hyperbolic tangential model reduces to the pure power law model). Thus,
contrary to our initial hopes, this approach does not provide any additional
insight.
Inspired by these tests, we could propose the following modified model
I(t) = a+ beµt(tc − t)
m . (7)
It has 5 adjustable parameters, like model (3), but it seems more flexible to
describe the looked-for cross-over: for large tc− t, the power law term (tc− t)
changes slowly, especially for 0 < m < 1 as is expected here; for small tc−t, the
power law term changes a lot while the exponential term is basically constant.
But, this model is correct for a critical point only if m < 0 so that b > 0;
otherwise, if 0 < m < 1, b < 0 and for tc− t large, the exponential term which
dominate does not describe a growth but an exponentially accelerating decay.
For 0 < m < 1, we thus need a different formulation. We propose
I(t) = a+ beµt + c(tc − t)
m . (8)
We have fitted this formula to the data over the four periods 1983 - Oct. 2004,
1991 - Oct. 2004, 1983 - Mar. 2005, 1991 - Mar. 2005 and, while the fits are
reasonable, the critical time tc is found to overshoot to 2007-2008, which is a
typical signature that the model is not predictive.
In conclusion of this first preliminary study, the presence of a bubble (faster-
than-exponential growth) is confirmed but the determination of the end of
this phase is for the moment unreliable.
3.3 Dependence of the growth rate on the index value
The monthly growth rate g(t) of a given CSW index at time t is defined by
g(t) = ln[p(t)/p(t− 1)] , (9)
where p(t) is the price of that CSW index at time t. Figure 4 shows the
evolution of the growth rates of the 27 CSW indexes from June-1983 to March-
2005. While there are some variations, all 27 CSW indexes follow practically
the same pattern. We clearly observe a large peak of growth over the period
2003-2005. Notice that this recent peak is much larger and coherent than the
previous one ending in 1991, which was followed by a price stabilization and
even a price drop in certain cases. This figure stresses that the acceleration in
growth rate is a very localized event which occurred essentially in 2003-2004
and the subsequent growth rate has leveled off to pre-bubble times. We can
conclude that there has been no bubble from 1990 to 2002, approximately,
then a short-lived bubble until mid-2004 followed by a smoothed transition
back to normal.
1980 1985 1990 1995 2000 2005 2010
−0.04
−0.02
Fig. 4. Evolution of the growth rates of the 27 regional CSW indexes from June-1983
to March-2005.
Fig. 5 plots the price growth rate g(t) versus the price p(t) itself for the 27
CSW indexes. A linear regression of the data points on Fig. 5, shown as the
red straight line, gives a correlation coefficient of 0.494. If we perform lin-
ear regression for each index, then we find an average correlation coefficient
0.503 ± 0.036, confirming the robustness of this estimation of the correlation
between growth rate and price level. The obtained relation between g and p
obtained from this correlation analysis is captured by the following mathe-
matical regression
g = 0.00922×
− 0.00747 . (10)
In words, if p is large, then g is large on average, which confirms the concept
of a positive feedback of price on its further growth. The continuous time limit
of g(t) defined by (9) is
g(t) =
d ln p
. (11)
This last equation together with (10), that we write as g(t) = αp − β (with
α = 0.00922/100 and β = 0.00747), implies the following ordinary differential
equation
= αp2 − βp , (12)
which indeed gives a power law acceleration p(t) ∼ 1/(tc − t) asymptotically
close to the critical time tc. Note that this critical time is determined by
the initial conditions, and is called in mathematics a movable singularity. We
conclude from this first analysis that the rough linear growth of the growth rate
confirms the existence of a bubble growing faster than exponential according
to an approximate power law. But of course, the exponent of this power law
is poorly constrained, in particular from the fact that the growth rate g(t)
exhibits significant variability and furthermore nonlinearity, as can be seen in
Fig. 5.
50 100 150 200 250 300 350 400 450
−0.04
−0.02
Jul. 1983 − Sep. 2003
Oct. 2003 − Sep. 2004
Oct. 2004 − Mar. 2005
Fig. 5. Dependence on the data price p for all CSW indexes of its growth rate g.
The overall correlation coefficient is 0.494. The red line is the linear fit of the data
points.
It is useful to refine this analysis by separating the whole time interval into
three distinct intervals. The corresponding plot of the growth rate g as a
function of price is shown in Fig. 5 with different symbols: period 1 is Jul.
1983 to Sept. 2003, period 2 is from Oct. 2003 to Sept. 2004, and period 3 is
from Oct. 2004 to Mar. 2005. An anomaly can be clearly outlined, associated
with the red dots which correspond to the anomalous peak in the growth rate
in the period from Oct. 2003 to Sept. 2004. Notice also that the most recent
time interval from Oct. 2004 to Mar. 2005 shows practically the same behavior
as the first period before 2003. In other words, when removing the data in red
for the period from Oct. 2003 to Sept. 2004, the growth rate g(t) is practically
independent of p, which qualifies the normal regime. We can thus conclude that
this so-called “phase-portrait” of the growth rate versus price has identified
clearly an anomalous time interval associated with extremely fast accelerating
prices followed by a more recent period where the price growth has resumed
a more normal regime.
4 Yearly periodicity and intra-year structure
4.1 Yearly periodicity from superposed year analysis and spectral analysis
In Fig. 4, the time dependence of the monthly growth rate exhibits a clear sea-
sonality (or periodicity), which appear visually to be predominantly a yearly
phenomenon. This visual observation is made quantitative by performing a
spectral Fourier analysis. The power spectrum of a typical CSW index is shown
in Fig. 6 (all CSW indexes show the same power spectrum). Since the unit of
time used here is one year, the frequency f is in unit of 1/year. A periodic
behavior with period one year should translate into a peak at f = 1 plus all
its harmonics f = 2, 3, 4, · · · , which is indeed observed in Fig. 6. Note also
that the spectrum has large peaks at f = 4 and f = 8 among the harmonics of
f = 1, which indicates a weak periodicity with period of one quarter. This is
consistent with Fig. 7, where four oscillations in the averaged monthly growth
rates can be observed.
0 1 2 3 4 5 6 7 8 9 10 11 12
Fig. 6. Spectrum analysis to confirm the strong periodicity in g(t).
Note that the power spectrum itself is periodic with a period of 12, which is
the sampling frequency, equal to the double of the Nyquist frequency. There
are also many peaks in the low-frequency region (larger that one-year time
scale) close to f = 0, which are associated with the time scales of the global
trends produced by the big peaks in g(t) around year 2004 as well as around
1990.
To further explore this seasonal variability of the price growth rates, we cal-
culate the averages of the growth rates for given months, where the average is
performed over all years. Consider for instance the month of January: we look
up the growth rate for all the data over all years for the month of January
and take the average. We do the same for each successive months. The result
is shown in Figure 7 for two time periods, which gives the average growth rate
〈g〉 for different months of the year. The red dash line and circles give the
resultant 〈g〉 for all the data and the black dash line and triangles give the
standard deviation σg for all data (which is a measure of the variability from
year to year and from zip code to zip code around the average). The difference
between the two time periods is precisely the time interval from June 2003 to
March 2005: this period is responsible for a significant increase of the average
growth rate (compare the red dashed line (filled circles) with the red continu-
ous line (open circles)) and an even larger increase of the variability (compare
the black continuous line (filled triangles) with the dashed black line (open
triangles)), again confirming the evidence of an anomalous behavior in that
period. In 2005, it appears that the growth rate relaxed back to the normal
level (according to the historical record).
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Month
Fig. 7. Monthly average growth rate (circles) and its standard deviation (triangles)
as a function of the month within the year. Dash: results obtained over all 27
indexes over the period from Jun. 1983 to Mar. 2005; Solid: results obtained over
all 27 indexes over the period from Jun. 1983 to May 2003.
4.2 Yearly periodicity and intra-year structure with a scale and translation
modulated model
Inspired by these results, we propose the following quantitative model. Con-
sider a time t in units of month. We write t = 12T +m, where T is the year
and m is the month within that year and thus goes from 1 (January) to 12
(December). For instance, t = 26 corresponds to T = 2 and m = 2 (Febru-
ary), while t = 38 corresponds to T = 3 and again the same month m = 2
(February) within the year. We propose to model the intra-year structure of
the growth rate g(t) together with possible yearly variations by the following
expression
g(t = 12T +m) = f(T )h(m) + j(T ) . (13)
In words, the growth rate has an intra-year structure h(m) modulated from
year to year in amplitude by f(T ) up to a possible overall translation j(T )
which can also vary from year to year. We can expect f(T ) and j(T ) to be
approximately constant for most years, except around 1990 and 2004 for which
we should see an anomaly in either or both of them, since these two periods
had bubbles. Note that this model (13) gives an exact yearly periodicity if
f(T ) and j(T ) are constant. A non-constant f(T ) describes an amplitude
modulation of the yearly periodicity. In particular, we expect a strong peak
around T = 2004. With this model, we can focus on predicting f(T ) and j(T )
only, because we have removed the complex intra-year structure.
We have thus fitted the model (13) to three subsets of the whole available
time series for the growth rate g(t) and also to the whole set taken globally, in
order to test for the robustness of the model. For this, we use the cost function
Tmax∑
[g(t = 12T +m)− f(T )h(m)− j(T )]2 (14)
which is minimized with respect to the 12 unknown variables h(1), ..., h(12)
and the 2 × Tmax variables [f(1), j(1)], ..., f(Tmax), j(Tmax). There are 12Tmax
terms in the sum and 12 + 2 × Tmax unknown variables. This shows that the
system is well-constrained as soon as Tmax ≥ 2. For instance for Tmax = 20, we
have 52 unknown variables to fit and 240 terms in the sum to constrain the
Figure 8 illustrates the result of the fit of model (13) to the growth rate over
the whole time interval from 1985 to 2005. As expected, we can observe a clear
peak in the amplitude f(T ) corresponding to the year 2004, while there is not
appreciable peak around 1990. This means that the recent bubble appears
significantly stronger than any other episodes in the last 20 years and dwarfs
them. The anomalous nature of the recent bubble is reinforced by the existence
of a peak in j(T ) for the same year 2004, showing that both the amplitude and
translation components of the growth rates has been completely anomalous
in 2004. The middle graph of the top panel of figure 8 shows the intra-year
pattern captured by the model, which is in remarkable agreement with the
pattern shown in figure 7: one can observe a peak in March, May, August and
December, the largest peak being in May. The bottom panel of figure 8 shows
visually how well (or badly) the model fits the actual data. The quality of
the fit is excellent, except in 2004-2005. In other words, we clearly identify
a very anomalous or exceptional behavior in 2004-2005, again providing a
confirmation that something exceptional or anomalous has occurred during
that period.
1985 1990 1995 2000 2005
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
−0.025
−0.02
−0.015
−0.01
1985 1990 1995 2000 2005
1980 1985 1990 1995 2000 2005 2010
−0.03
−0.02
−0.01
Fig. 8. Upper panels: three graphs showing the three functions f(T ), h(m) and
j(T ) fitted on the growth rate over the whole time interval from 1985 to 2005.
Lower panel: Comparison between the growth rate data (empty blue circles) and
the model (13) (red line).
Figure 9 is the same as figure 8 for the period from 1985 to 1990. One can
clearly here observe a peak in the scaling amplitude f(T ) at T =1988 and in
the translation term j(T ) at T =1986, suggesting that the first bubble of the
1985-2005 period occurred over a relatively large time period 1985-1990, with
two successive contributions. The intra-year structure h(m) has also its peaks
on March, May, August and December, but this intra-year structure is weaker
than for other sub-periods. The lower panel of figure 9 shows that the model
captures very well the overall trend as well as the intra-year structure. The
main discrepancies are in the amplitude of the large peaks and valleys, which
are not fully predicted.
1985 1985.5 1986 1986.5 1987 1987.5 1988 1988.5 1989 1989.5 1990
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
0.185
0.195
0.205
1985 1985.5 1986 1986.5 1987 1987.5 1988 1988.5 1989 1989.5 1990
−0.29
−0.28
−0.27
1984 1985 1986 1987 1988 1989 1990 1991
−0.03
−0.02
−0.01
Fig. 9. Same as figure 8 for the period from 1985 to 1990.
Figure 10 is the same as figure 8 for the period from 1991 to 2000. One can
clearly here observe a peak in the scaling amplitude f(T ) at T =1995 and
in the translation term j(T ) at T =1994. This thus identifies a small bubble
in the mid-1990s. The intra-year structure h(m) has also its peaks on March,
May, August and December, with very large amplitudes. The lower panel of
figure 10 shows a truly excellent fit.
1991 1992 1993 1994 1995 1996 1997 1998 1999 2000
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
−0.12
−0.115
−0.11
−0.105
1991 1992 1993 1994 1995 1996 1997 1998 1999 2000
0.155
0.165
1990 1992 1994 1996 1998 2000 2002
−0.01
−0.005
0.005
0.015
0.025
Fig. 10. Same as figure 8 for the period from 1991 to 2000.
Figure 11 is the same as figure 8 for the period from 2001 to 2005. One can
clearly here observe a peak in the scaling amplitude f(T ) at T =2004 and
in the translation term j(T ) also at T =2004. This thus clearly identifies the
bubble as peaking in 2004. The intra-year structure h(m) has also its peaks
on March, May, August and December, with very large amplitudes and very
good agreement with the other three figures. The lower panel of figure 11
shows an excellent fit up to the early 2003 and then a rather large discrepancy
starting early 2003 all the way to the last data point approaching mid-2005. In
particular, note that the intra-year structure is washed out by the anomalous
growth rate culminating in mid-2004. Symmetrically, the intra-year structure
is also absent in the fast decay of the growth rate back to normal. We do
not have enough data to ascertain if the growth rate has resumed its normal
intra-year pattern. We believe that this is a very important diagnostic to
characterize abnormal behavior and this could be a very useful variable to
monitor on a monthly basis.
2001 2001.5 2002 2002.5 2003 2003.5 2004 2004.5 2005
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
−0.025
−0.02
−0.015
−0.01
2001 2001.5 2002 2002.5 2003 2003.5 2004 2004.5 2005
2000 2001 2002 2003 2004 2005 2006
−0.01
Fig. 11. Same as figure 8 for the period from 2001 to 2005.
The four figures 8-11 validate model (13): in particular, they show the very
robust intra-year structure with peaks in March, May, August and December.
One possible contribution to this quarterly periodicity comes from the con-
struction of the CSI: the monthly indexes use a three-month moving average
algorithm. Home sales pairs are accumulated in rolling three-month periods,
on which the repeat sales methodology is applied. The index point for each
reporting month is based on sales pairs found for that month and the pre-
ceding two months. For example, the December 2005 index point is based
on repeat sales data for October, November and December of 2005. This av-
eraging methodology is used to offset delays that can occur in the flow of
sales price data from county deed recorders and to keep sample sizes large
enough to create meaningful price change averages. A three month rolling
window construction corresponds in general to a convolution of the bare price
with a kernel which possesses a three month periodicity (or size). The Fourier
transform of the convolution is the product of Fourier transforms. Thus the
spectrum of the signal should contain the peaks of the Fourier spectrum of
the kernel, which by construction contains a peak at three months. However,
our synthetic tests (not shown) suggest that this effect is by far too small to
explain the strong amplitude of the observed quarterly periodicity. It would
be important to understanding why such intra-year structure develops: is it
the result of a natural intra-day organization of buyers’ behaviors associated
with taxes/ income constraints or a problem of reporting or perhaps the effect
of other calendar regularities? Or is it the result of patterns coming from the
supply part of the equation, namely home-builders, developers, and perhaps
in the time modulation of the rates of allocated permits? Answering these
questions is important to determine how much emphasis one should give to
these results. But if indeed the intra-day structure is a genuine non-artificial
phenomenon, we believe that it offers a remarkable opportunity for monitoring
in real time the normal versus abnormal evolution of the market and also for
developing forecasts on a month time horizon.
4.3 Intra-year pattern from signs of growth rate increments
The existence of a strong and robust intra-year structure in the price growth
rate can be further demonstrated by studying the sign of g(t + 1) − g(t). A
positive (negative) sign mean that the growth rate tends to increase (decrease)
from one month to the next.
Based on the seasonality of the growth rate, we are able to answer the following
question: given the current growth rate g(t), will the growth rate increase or
decrease at time t+1? This amounts to asking what is the sign of g(t+1)−g(t)?
Technically, we construct the (unconditional) number of times the sign of the
increment g(t + 1) − g(t) is positive or negative irrespective of what is g(t).
From Fig. 4, we obtain a sequence of signs: −−+−+−−+−−++. For each
month, we calculate the percentage of positive and negative signs, respectively.
The second and the third rows of Table 1 gives the percentage of positive and
negative signs for each month. The third and fourth rows gives the signs and
the associated percentages.
For instance, the table says that the “probability” of the sign of g(t = Feb)−
g(t = Jan) being “-” is about 92.1%. If we know g(t = Jan), we can say that it
is very probable that the growth rate of February will be less than this January
value. Thus, this table has predictive power in the sense that the probabilities
to predict the signs are much higher than the value of 75% obtained under the
null hypothesis that g(t) is a white noise process (see Sornette and Andersen,
2000). This table is another way to rephrase and expand on our preceding
analysis on the yearly periodicity by identifying a very strong and robust
intra-year structure.
Table 1
Analysis of the signs of g(t + 1) − g(t). The second and the third rows gives the
percentage of positive and negative signs for each month. The third and fourth rows
give the sign for each month that dominates and the associated percentages.
Mon Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
+% 7.91 17.2 88.0 5.64 97.7 8.47 8.47 91.4 6.57 8.92 84.2 82.2
-% 92.1 82.8 12.0 94.4 2.29 91.5 91.5 8.59 93.4 91.1 15.8 17.8
sign - - + - + - - + - - + +
% 92.1 82.8 88.0 94.4 97.7 91.5 91.5 91.4 93.4 91.1 84.2 82.2
Since our initial analysis performed in the summer of 2005 which used data
up to March 2005, new data for the 27 CSW indexes has become available
which covers the interval from Apr. 2005 to Sept. 2006. It is very interesting
to check if the sign of the growth variations obtained in Table 1 using the data
until March 2005 still applies to the new data. The realized signs of the newly
available months are calculated and the sequence of signs is the following: -
(Apr. 2005, 27 CSW indexes out of 27), + (May. 2005, 27 out of 27), - (Jun.
2005, 27 out of 27), - (Jul. 2005, 27 out of 27), + (Aug. 2005, 27 out of 27), -
(Sep. 2005, 27 out of 27), - (Oct. 2005, 27 out of 27), + (Nov. 2005, 27 out of
27), + (Dec. 2005, 21 out of 27), - (Jan. 2006, 27 out of 27), - (Feb. 2006, 27 out
of 27), + (Mar. 2006, 27 out of 27), - (Apr. 2006, 27 out of 27), + (May. 2006,
27 out of 27), - (Jun. 2006, 27 out of 27), - (Jul. 2006, 27 out of 27), + (Aug.
2006, 27 out of 27), and - (Sep. 2006, 27 out of 27). Thus, table 1 predicts
exactly the signs of the growth rate variations of all 27 CSW indexes for all
months except for Dec. 2005 for which there are 6 errors: table 1 predicts that
the growth rate variation from Dec. 2005 to Jan. 2006 should be +, which is
correct for 21 CSW indexes out of 27, corresponding to a success ratio of 77%
(close to the white noise case). This score is slightly lower than the previously
estimated probability of 82.2% for the month of December, which is the lowest
among all months. Overall, the success rate is remarkably high, adding further
evidence that the Las Vegas property market has returned to a more normal
phase (no bubble from April 2005 to Sept. 2006).
5 Predicting the monthly growth rate
Conditional of the evidence that the anomalous faster than exponential growth
has ended, let us attempt to predict the future evolution of the CSW indexes
based only on the strong seasonality of the growth rate. Figure 12 presents
the predictions one year ahead for the 27 regional CSW indexes. Two different
prediction schemes are used. The RED lines are based on the average growth
rate obtained from all 27 indexes, while the MAGENTA lines are based on the
average growth rate obtained from the individual index under investigation.
There is not discernable difference.
A similar prediction of the Clark County (Las Vegas MSA) indexes (NVC003Q
and NVC003C) has also been made using the average growth rates obtained
from all 27 regional indexes. Since these two indexes are only available from
July-2000 to March-2005, we do not have enough data to calculate the average
growth rates using the indexes themselves. The results are shown in Fig. 13.
2002 2003 2004 2005 2006 2007
Fig. 12. Predicting regional CSW indexes one year ahead. Red lines: Prediction
using average growth rate obtained from all 27 indexes; Magenta lines: Prediction
using average growth rate obtained from the individual index under investigation.
The two kinds of prediction are almost undistinguishable.
2000 2001 2002 2003 2004 2005 2006 2007
NVC003C: Raw data
NVC003C: Prediction
NVC003Q: Raw data
NVC003Q: Prediction
Fig. 13. Predicting Clark County (Las Vegas MSA) indexes (NVC003Q and
NVC003C) one year ahead.
6 Conclusion
We have analyzed 27 house price indexes of Las Vegas from Jun. 1983 to
Mar. 2005, corresponding to 27 different zip codes. These analyses confirm
the existence of a real-estate bubble, defined as a price acceleration faster
than exponential. This bubble is found however to be confined to a rather
limited time interval in the recent past from approximately 2003 to mid-2004
and has progressively transformed into a more normal growth rate in 2005.
The data up to mid-2005 suggests that the current growth rate has now come
back to pre-bubble levels. We conclude that there has been no bubble from
1990 to 2002 except for a medium-sized surge in 1995, then a short-lived but
very strong bubble until mid-2004 which has been followed by a smoothed
transition back to what appears to be normal. It thus seems that, while the
strength of the real-estate bubble has been very strong over the period 2003-
2004, the price appreciation rate has returned basically to normal.
In addition, we have identified a strong yearly periodicity which provides a
good potential for fine-tuned prediction from month to month. As the intra-
year structure is likely a genuine non-artificial phenomenon, it offers a re-
markable opportunity for monitoring in real time the normal versus abnormal
evolution of the market and also for developing forecasts on a monthly time
horizon. In particular, a monthly monitoring using a model that we have de-
veloped here could confirm, by testing the intra-year structure, if indeed the
market has returned to “normal” or if more turbulence is expected ahead.
In addition, it would provide a real-time observatory of upsurges and other
anomalous behavior at the monthly scale. This requires additional technical
developments and tests beyond this report.
Compared with previous analysis of Zhou and Sornette (2003, 2006) at the
scale of states and whole regions (northeast, midwest, south and west), the
present analysis demonstrates the existence of very significant variations at the
local scale, in the sense that the bubble in Las Vegas seems to have preceded
the more global USA bubble and has ended approximately two years earlier
(mid 2004 for Las Vegas compared with mid-2006 for the whole of the USA).
References
Bailey, M., Muth, R., Nourse, H., 1963. A regression method for real estate
price index construction. Journal of the American Statistical Association
58, 933–942.
Bender, C., Orszag, S. A., 1978. Advanced Mathematical Methods for Scien-
tists and Engineers. McGraw-Hill, New York.
Broekstra, G., Sornette, D., Zhou, W.-X., 2005. Bubble, critical zone and the
crash of Royal Ahold. Physica A 346, 529–560.
Case, K. E., Shiller, R. J., 1987. Prices of single-family homes since 1970: New
indexes for four cities. New England Economic Review Sep/Oct, 45–56.
Case, K. E., Shiller, R. J., 1989. The efficiency of the market for single-family
homes. American Economic Review 79, 125–137.
Case, K. E., Shiller, R. J., 1990. Forecasting prices and excess returns in the
housing market. AREUEA J. 18, 253–273.
Case, K. E., Shiller, R. J., 2003. Is there a bubble in the housing market.
Brookings Papers on Economic Activity (2), 299–362.
Greenspan, A., 1997. Federal ReserveÕs semiannual monetary policy report,
before the Committee on Banking, Housing, and Urban Affairs, U.S. Senate,
February 26.
Himmelberg, C., Mayer, C., Sinai, T., September 2005. Assessing high house
prices: Bubbles, fundamentals, and misperceptions. Tech. Rep. Staff Report
no. 218, Federal Reserve Bank of New York.
Jorion, P., 2005. Is housing market surge really sustainable? The Wall Street
Journal September 22, A17.
Mauboussin, M. J., Hiler, R., 1999. Rational Exuberance? Equity research
report of Credit Suisse First Boston, January 26.
Roehner, B. M., 2006. Real estate price peaks: A comparative overview. Evo-
lutionary and Institutional Economics Review in press, physics/0605133.
Shiller, R. J., 2000. Irrational Exuberance. Princeton University Press, New
York.
Shiller, R. J., 2006. Long-term perspectives on the current boom in home
prices. Economists’ Voice 3(4), Art. 4.
Smith, M. H., Smith, G., 2006. Bubble, bubble, where’s the housing bubble?
Brookings Papers on Economic Activity (1), 1–67.
Sornette, D., 2003. Why Stock Markets Crash: Critical Events in Complex
Financial Systems. Princeton University Press, Princeton.
Sornette, D., Andersen, J.-V., 1998. Scaling with respect to disorder in time-
to-failure. European Physical Journal B 1, 353–357.
Sornette, D., Andersen, J.-V., 2000. Increments of uncorrelated time series
can be predicted with a universal 75% probability of success. International
Journal of Modern Physics C 11, 713–720.
Zhou, W.-X., Sornette, D., 2003. 2000-2003 real estate bubble in the UK but
not in the USA. Physica A 329, 249–263.
Zhou, W.-X., Sornette, D., 2006. Is there a real-estate bubble in the US?
Physica A 361, 297–308.
Biographies:
Wei-Xing Zhou is a Professor of Finance at the School of Business in the East
China University of Science and Technology. He received his PhD in Chemical
Engineering from the East China University of Science and Technology in
2001. His current research interest focuses on the modeling and prediction of
catastrophic events in complex systems.
Didier Sornette holds the Chair of Entrepreneurial Risks at the Depart-
ment of Management, Technology and Economics of ETH Zurich. He received
his PhD in Statistical Physics from the University of Nice, France. His current
research focuses on the modeling and prediction of catastrophic events in com-
plex systems, with applications to finance, economics, seismology, geophysics
and biology.
	Introduction
	Conceptual background of our empirical analysis
	Humans as social animals and herding
	Definition and mechanism for bubbles
	Was there a bubble? Status of the argument based on the ratio of cost of owning versus cost of renting
	Regime shift in the CSW Zip Code Indexes of Las Vegas
	Description of the data
	Power law fits
	Dependence of the growth rate on the index value
	Yearly periodicity and intra-year structure
	Yearly periodicity from superposed year analysis and spectral analysis
	Yearly periodicity and intra-year structure with a scale and translation modulated model
	Intra-year pattern from signs of growth rate increments
	Predicting the monthly growth rate
	Conclusion
