Using methods originating from statistical physics we model bubbles in English house prices. It is found that there was a nationwide housing bubble 2002-2007. Typically prices were 30-40% over-valued and fell around 20%. London is atypical in that the level of over-pricing was lower, only around 20%, and experienced a drop in prices of only around 15%. There is some suggestion of contagious effects, with the bubble in London affecting prices in Yorkshire and the North.financial crashes; super-exponential growth; illusion of certainty; contagion; housing-bubble; Enlish house prices

Fry, J. M.

English

Research Papers in Economics

Bubbles and contagion in English house
prices
John M. Fry
Economics, School of Social Sciences,
The University of Manchester, Manchester, UK ¤
October 2009
Abstract
Using methods originating from statistical physics we model bubbles in English
house prices. It is found that there was a nationwide housing bubble 2002-2007.
Typically prices were 30-40% over-valued and fell around 20%. London is atypical
in that the level of over-pricing was lower, only around 20%, and experienced a drop
in prices of only around 15%. There is some suggestion of contagious e®ects, with
the bubble in London a®ecting prices in Yorkshire and the North.
Keywords: ¯nancial crashes, super-exponential growth, illusion of certainty, contagion,
housing-bubble, English house prices.
1 Introduction
Housing bubbles are of obvious topical interest given the credit crunch of 2007-8
(Parkinson et al. (2009)). The simple truth is that housing matters. Housing is typically
the major asset in household portfolios and can have major implications for the economy
as a whole. As we have seen, the banking sector is particularly exposed to to the housing
market (Hott and Monnin (2008)), and house price crashes tend to have a stronger impact
on the economy than stock market crashes such as the 1987 stock market crash (Black
¤Email: frymaths@googlemail.com
1et al. (2006), Helbling and Terrones (2003)). Symmetrically, however, housing o®ers
potentially greater rewards in that the wealth e®ects for housing assets are typically
greater than those for ¯nancial assets (Case et al. (2005)). As well as economy-wide
issues there are also a number of pertinent housing renewal/social policy implications of
the recent crisis in the UK (Ferrari (2007), Parkinson et al. (2009)).
Housing markets are particularly susceptible to bubbles. A succession of booms and
slumps have been documented in worldwide housing markets; in the UK (Parkinson et
al. (2009), Hott and Monnin (2008), Black et al. (2006)) and the rest of the world
including the USA, Japan and Switzerland (Hott and Monnin (2008)). Housing bubbles
have been widely studied, see e.g. Black et al. (2006), Hott and Monnin (2008) and a
host of references therein. The comment is made in Black et al. (2006) that since much of
the housing market is based on consumption rather than investment, subsequent market
ine±ciencies mean that housing markets are prone to bubbles and speculative behaviour.
In this paper we apply our theoretical model to English house prices { a subject with
obvious socio-economic implications. Our approach allows for univariate and multivariate
models and leads to simple answers to questions such as \Are there bubbles?" and \What
is the apparent level of over-pricing?". In particular, we apply the model in Fry (2009)
to English house prices over the years 2002-2007. For additional background on the
modelling work see the papers by Feigenbaum and Sornette and co-workers cited in the
references, especially Johansen et al. (2000).
The layout of this document is as follows. Section 2 provides the main analysis. Section
3 concludes. Section 4 provides a self-contained mathematical appendix.
2 Analysis
The analysis in this paper is as follows. We model quarterly house price data for ten
English regions obtained from the nationwide website1 over the years 2002-2007. Our aim
in this section is purely data-analytic and a self-contained mathematical Appendix can
be found at the end of this paper. Our analysis splits into two parts. Firstly, we use a
univariate model where we model data for each region individually. We test for bubbles,
and provide estimates of fundamental values and for the level of over-pricing. Further, we
are able to demonstrate that prices eventually converge towards fundamental levels out of
sample. Secondly, we use a bivariate model which allows us to model data for two regions
simultaneously. This gives us an alternative test for bubbles and in addition allows us to
1[http://www.nationwide.co.uk/hpi/historical.htm]
2model contagion. See Section 2.2.
2.1 Univariate analysis
The results obtained using the univariate bubble model (equation (2) in the Appendix)
are shown in Table 1. We have strong evidence for bubbles in each of the individual price
series. The results for London appear outlying, with the estimated bubble component
(equation (7)) comprising only 20% of prices compared to 30-40% for the rest of England
and a fall in prices (maximum-to-minimum before subsequent price rises) of 15% compared
to 20% for much of the rest of England. These estimates of over-pricing compare with
similar estimates of between 12-25% in Black et al. (2006) and 28-53% in Hott and
Monnin (2008).
This picture of speculative bubbles is reinforced once we take into account of the
estimates of fundamental value in equation (6). A graph of observed prices and estimated
fundamental price in the years 2002-2007 by region is shown in Figure 1. In all cases
prices appear way in excess of fundamental levels. In Table 2 we compare estimates of
fundamental value, calculated only using data from 2002-2007, with historically observed
prices over the years 2008-2009. The suggestion is that prices have converged towards
fundamental values by the ¯rst quarter of 2009.
Region p-value Estimated speculative % Drop
component
North 0.001 0.375 0.164
Yorkshire 0.009 0.358 0.185
North West 0.002 0.276 0.179
East Midlands 0.000 0.374 0.193
West Midlands 0.000 0.381 0.175
East Anglia 0.000 0.300 0.216
South East 0.000 0.337 0.200
Outer Met 0.001 0.247 0.191
London 0.007 0.203 0.156
South West 0.000 0.362 0.181
Table 1: p-values for null hypothesis of no bubble, estimated speculative component and
percentage drop in prices by English region.
35
0
0
0
0
1
0
0
0
0
0
1
5
0
0
0
0
2
0
0
0
0
0
2
5
0
0
0
0
3
0
0
0
0
0
Time
P
r
i
c
e
2003 2004 2005 2006 2007 2008
Key
 
 
 
 
London
South West
West Midlands
North
Figure 1: Observed historical prices (wavy lines) and estimated fundamental price
(straight lines) by region
Date Actual price Estimated fundamental price 95% C. I
2008 Q1 179,363 122,326 100,583-144,068
2008 Q2 174,514 123,602 100,478-146,725
2008 Q3 165,188 124,891 100,338-149,444
2008 Q4 156,828 126,194 100,162-152,226
2009 Q1 149,709 127,511 99,948-155,073
2009 Q2 154,066 128,841 99,695-157,986
2009 Q3 130,185 99,402-160,968
2009 Q4 131,543 99,067-164,018
Table 2: Uk fundamental house prices estimated out-of-sample using data from 2002-2007
only.
2.2 Bivariate bubbles and contagion
Assessing contagion is a delicate theoretical and empirical issue in economics. A
distinction needs to be made between genuine contagion and simple co-dependence, with 4much of the literature failing to make an adequate distinction between the two (Forbes
and Rigobon (2002)). Asset prices are assumed to exhibit non-zero correlations in normal
times. Contagion occurs when there is a genuine change in correlation structure brought
about by speci¯c events or crises. Anything else is simply co-dependence.
In this paper we model contagion as occurring if, under the bubble model given by
equation (9), region X is more informative about prices in region Y than Y is about
X in a sense to be made precise in the Appendix in Section 4.3. Our analysis in this
subsection splits into two parts. Firstly, our aim is to investigate a putative north-south
divide. We examine the e®ect of London upon prices in the Northern and Midlands
regions, see e.g. Parkinson et al. (2009) Chapter 3. We use equations (8-10) to test for
bubbles. Further, we test the additional null hypothesis of no contagion (see equation
(11) in Section 4.3). Secondly, we repeat the analysis along geographical lines for each of
the Northern, Midlands and South Eastern regions of England.
North-South divide. The results are shown in Table 3 and are suggestive of bubbles in
each case, although the p-value of 0.07 for Yorkshire-London is inconclusive in isolation.
We test for contagion using a nonlinear t-test based on the delta method, Bingham and
Fry (2010) Chapter 7, and restrict to a one-sided test since we know apriori that London
is much more likely to exert a causal in°uence on the Northern and Midlands regions
than vice-versa. The results give no evidence of contagion in the majority of cases but
some contagious e®ects are apparent with London prices in°uencing those in the North
and Yorkshire.
Geographical contagion. The results are shown in Table 4 and give conclusive evidence
of a speculative bubble in each of the pairwise comparisons made. No evidence of
contagion is found, suggesting that in each case the neighbouring regions have a roughly
equal impact upon each other.
Region No bubble No contagion
p-value (One-sided) p-value (One-sided)
London-North 0.006 0.058 (¢)
London-North West 0.023 0.210
London-Yorkshire 0.071 (¢) 0.065 (¢)
London-East Mids 0.000 0.242
London-West Mids 0.001 0.281
London-East Anglia 0.001 0.443
Table 3: p-values for null hypotheses of no bubble and of no contagion London versus
Northern and Midlands regions.
5Area Regions No bubble No contagion
p-value (One-sided) p-value (Two-sided)
\North" North-Yorkshire 0.005 0.899
North-North West 0.002 0.269
Yorkshire-North West 0.006 0.408
\Midlands" East Mids-West Mids 0.000 0.906
East Mids-East Anglia 0.000 0.573
West Mids-East Anglia 0.000 0.633
\South East" London-South East 0.000 0.612
London-Outer Met 0.001 0.922
South East-Outer Met 0.000 0.617
Table 4: p-values for null hypotheses of no bubble and of no contagion by geographical
location.
3 Conclusions
We analysed English house prices over the period 2002-2007 based on the model for asset
price bubbles in Fry (2009). Using both univariate and bivariate models strong evidence
for bubbles was found, with bubbles estimated to contribute 30-40% of observed prices.
This ¯gure compares reasonably to estimates of 12-25% in Black et al. (2006) and of
28-53% in Hott and Monnin (2008). Historical price falls were slightly lower than our
estimates of over-pricing and were typically in the range of 20-30%. Out of sample, prices
appear to converge towards estimated fundamental prices and away from the previous
speculative highs. The results for London were slightly atypical in that both the estimated
bubble component and the historical fall in prices were less than those experienced in the
rest of England. There was some suggestion that prices in London had a contagious
e®ect, causing more rapid price rises in Yorkshire and the North of England than would
ordinarily be expected.
4 Mathematical Appendix
4.1 Univariate bubble model (Table 1)
In this subsection we give a brief overview of the model in Fry (2009) which forms the
basis of the analysis here. Let Xt denote the house price at time t, t = 1;:::;n. We
model purely random or non-bubble behaviour as
dXt = rdt + ¾dWt: (1)
6In contrast, our model for a bubble is
dXt =
µ
r +
·¯t¯¡1
®¯ + t¯
¶
dt +
r
¾2 ¡
·2¯t¯¡1
®¯ + t¯ dWt: (2)
The interpretation of (2) is that a representative investor is compensated by the risk of
a crash associated with the bubble by an increase in growth (·¯t¯¡1)=(®¯ + t¯)dt term
and a decrease in volatility (¡·2¯t¯¡1)=(®¯ + t¯) term. If we de¯ne ¢Xt = Xt+1 ¡ Xt
the log-likelihood function under equation (1) is
l(µ) = ¡
1
2
n¡1 X
t=1
log(2¼¾
2) ¡
1
2¾2
n¡1 X
t=1
(¢Xt ¡ r)
2 : (3)
Under (2) l(µ) is given by
¡
1
2
n¡1 X
t=1
log
µ
2¼
µ
¾
2 ¡ ·
2 ln
µ
®¯ + (t + 1)¯
®¯ + t¯
¶¶¶
¡
³
¢Xt ¡ r ¡ ·ln
³
®¯+(t+1)¯
®¯+t¯
´´2
¾2 ¡ ·2 ln
³
®¯+(t+1)¯
®¯+t¯
´ : (4)
We test for the presence of bubbles by calculating the likelihood ratio statistic (¯nd a
maximising set of parameters then subtract (3) from (4) and double) and perform a
one-sided test against the mixture distribution
1
2
Â
2
2 +
1
2
Â
2
3; (5)
high values indicating a bubble. The distribution in (5) is obtained by randomly sampling
from Â2
2 with probability 0.5 and from Â2
3 with probability 0.5. As discussed in Fry (2009)
under model (2) the mean price is
PB(t) = E(P(t)) = E(e
X(t)) = P(1)e
³
r+ ¾2
2
´
(t¡1)
µ
®¯ + t¯
®¯ + 1
¶·¡ 1
2·2
:
Under the fundamental model (1) the mean price is P(1)expf(r + ¾2=2)(t ¡ 1)g, and we
estimate fundamental values by
PF(t) = P(1)e
³
r+ ¾2
2
´
(t¡1): (6)
The estimated speculative bubble component is given by
1 ¡
0
@ n ¡ 1
R n
1
³
PB(t)
PF(t)
´
dt
1
A = 1 ¡
(n ¡ 1)
¡
®¯ + 1
¢·¡ 1
2·2
R n
1 (®¯ + t¯)
·¡ 1
2·2
dt
; (7)
7which can be calculated numerically.
4.2 Bivariate bubble model (Tables 3-4)
In the previous subsection we described the price of one asset. Here, we model the joint
behaviour of two assets Zt = (Xt;Yt)T. Our fundamental model is
dZt = rdt + §
1
2dWt; (8)
where r is a 2£1 vector and § is a 2£2 covariance matrix (see Fry (2009)) for further
details. Our model for bubbles becomes
dZt =
µ
r +
·¯t¯¡1
®¯ + t¯
¶
dt +
r
¾2 ¡
··T¯t¯¡1
®¯ + t¯ dWt: (9)
For the model (8) the likelihood equation is
l(µ) = ¡(n ¡ 1)log(2¼) ¡
(n ¡ 1)
2
log(¾
2
X¾
2
Y ¡ ¾
2
XY ) ¡
1
2
n¡1 X
t=1
(¢Xt ¡ rX)2
³
¾2
X ¡
¾2
XY
¾2
Y
´
+
n¡1 X
t=1
(¢Xt ¡ rX)(¢Yt ¡ rY)
µ
¾XY
¾2
X¾2
Y ¡ ¾2
XY
¶
¡
1
2
n¡1 X
t=1
(¢Xt ¡ rX)2
³
¾2
Y ¡
¾2
XY
¾2
X
´:
Under the bubble model (9) the likelihood equation is
l(µ) = ¡(n ¡ 1)log(2¼) ¡
(n ¡ 1)
2
log(¾
2
X;t¾
2
Y;t ¡ ¾
2
XY;t) ¡
1
2
n¡1 X
t=1
(¢Xt ¡ ¹X;t)2
³
¾2
X;t ¡
¾2
XY;t
¾2
Y;t
´
+
n¡1 X
t=1
(¢Xt ¡ ¹X;t)(¢Yt ¡ ¹Y;t)
Ã
¾XY;t
¾2
X;t¾2
Y;t ¡ ¾2
XY;t
!
¡
1
2
n¡1 X
t=1
(¢Yt ¡ ¹Y;t)2
³
¾2
Y;t ¡
¾2
XY;t
¾2
X;t
´;
where
¾
2
X;t = ¾
2
X ¡ ·
2
X ln
µ
®¯ + (t + 1)¯
®¯ + t¯
¶
¾
2
Y;t = ¾
2
Y ¡ ·
2
Y ln
µ
®¯ + (t + 1)¯
®¯ + t¯
¶
¾
2
XY;t = ¾
2
XY ¡ ·X·Y ln
µ
®¯ + (t + 1)¯
®¯ + t¯
¶
¹X;t = r + ·X ln
µ
®¯ + (t + 1)¯
®¯ + t¯
¶
8¹Y;t = r + ·Y ln
µ
®¯ + (t + 1)¯
®¯ + t¯
¶
:
We can again use the di®erence in the log-likelihood to test for bubbles. As before,
calculate the likelihood ratio statistic and perform a one-sided test against the mixture
distribution
1
4
Â
2
2 +
1
2
Â
2
3 +
1
4
Â
2
4; (10)
with high values indicating a bubble. The distribution in (10) is obtained by randomly
sampling from Â2
2 with probability 0.25, from Â2
3 with probability 0.5, and from Â2
4 with
probability 0.25.
4.3 Contagion from Xt to Yt
The condition for contagion from Xt to Yt is
·X < ·Y: (11)
We give an explanation for this interpretation below. Under the bubble model (9) the
conditional variance of YtjXt is
(V ar)(YtjXt) =
¡
1 ¡ ½
2
t
¢
¾
2
Y;t
where
½
2
t =
¾XY(t ¡ 1) ¡ ·X·Y ln
³
®¯+t¯
®¯+1
´
r³
¾2
X(t ¡ 1) ¡ ·2
X ln
³
®¯+t¯
®¯+1
´´³
¾2
Y(t ¡ 1) ¡ ·2
Y ln
³
®¯+t¯
®¯+1
´´;
¾
2
Y;t = ¾
2
Y(t ¡ 1) ¡ ·
2
Y ln
µ
®¯ + t¯
®¯ + 1
¶
:
Symmetrically,
(V ar)(XtjYt) =
¡
1 ¡ ½
2
t
¢
¾
2
X;t;
¾
2
X;t = ¾
2
X(t ¡ 1) ¡ ·
2
X ln
µ
®¯ + t¯
®¯ + 1
¶
:
Knowing Xt reduces the amount of uncertainty in Yt by the amount
¾
2
Y;t ¡ (1 ¡ ½
2
t)¾
2
Y;t = ½
2
t¾
2
Y;t:
9Similarly, knowledge of Yt reduces uncertainty in Xt by the amount
¾
2
X;t ¡ (1 ¡ ½
2
t)¾
2
X;t = ½
2
t¾
2
X;t:
We say that contagion occurs from Xt to Yt if ½2
t¾2
Y;t > ½2
t¾2
X;t, i.e. if Xt is more informative
about Yt than Yt is about Xt, since knowledge of Xt produces the greater reduction in the
conditional variance.
In the bubble models (2) and (9), crashes occur when there is a phase transition from
stochastic to deterministic behaviour in prices (Yeomans (1992), Sornette (2004)), i.e.
by a decrease in the volatility function (see equation (2) and Fry (2009)). Moreover,
in (2) we must have that ¾2 ¡ ·2¯t¯¡1=(®¯ + t¯)¸0 with ¾2 ¡ ·2¯t¯¡1=(®¯ + t¯) = 0
corresponding to the situation where randomness completely disappears and prices are
purely deterministic. These considerations imply the constraints
¾
2
X =
·2
X(¯ ¡ 1)
1¡ 1
¯
®
;
¾
2
Y =
·2
Y(¯ ¡ 1)
1¡ 1
¯
®
:
Contagion from Xt to Yt occurs if
¾
2
X;t < ¾
2
Y;t;
¾
2
X(t ¡ 1) ¡ ·
2
X ln
µ
®¯ + t¯
®¯ + 1
¶
< ¾
2
Y(t ¡ 1) ¡ ·
2
Y ln
µ
®¯ + t¯
®¯ + 1
¶
;
·
2
X
"
(¯ ¡ 1)
1¡ 1
¯(t ¡ 1)
®
¡ ln
µ
®¯ + t¯
®¯ + 1
¶#
< ·
2
Y
"
(¯ ¡ 1)
1¡ 1
¯(t ¡ 1)
®
¡ ln
µ
®¯ + t¯
®¯ + 1
¶#
;
i.e. ·X < ·Y.
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12

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