Assessing the Existence of a Housing Bubble Using an Error Correction Model

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Assessing the existence of a bubble in the
housing market using an Error Correction
Model1

Cyrus Laurent
Master of Science internship thesis

Master of Science in Economics
University of Lausanne, Unil
Supervisor: Philippe Bacchetta

Traineeship
European Commission, JRC,
Euro Area Macroeconomics modeling unit
Supervisor: Marco Ratto

Expert: Ivan Jaccard

European Central Bank

Monetary Policy Research Division

January 27, 2010
I would like to thank Paolo Paruolo for his useful comments and suggestions

1
The work is of the responsibility of the author, in no way does it engage the responsibility of the university, the
European Commission nor of the supervisors.

Assessing the existence of a bubble in the housing market using an ECM Laurent Cyrus

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Abstract

Detecting the existence of the housing bubble in the early 2000’s would have permitted policy makers to act in
order to deflate it slowly. Arguments in favor of taking action to deflate housing prices have to be strongly
justified given the potential negative impact of a non-required intervention. This implies that the methodology
used to assess the existence of a bubble must produce reliable results. We examine what conclusions can be
made from an Error Correction (EC) model through a study of the recent bubble in the US housing market. We
compare the results from two models anterior to the burst of the bubble which have different conclusions, one
from McCarthy & Peach (2004) concluding that there was no bubble and one from Soerensen(2006)
concluding that there was a bubble. We try to identify the key differences which could have led to this
conclusion. Then, we try to construct a model similar to the one from McCarthy and Peach (2004), in order to
have another comparison point and we identify some key issues in the construction of such model. Finally, we
assess the validity of such models by comparing them with a model without fundamentals. The results
obtained lead us to conclude that this methodology is not able to give clear conclusion about the existence of a
bubble. However, we try to point out the key issues that one would face when using this methodology.

Résumé

Détecter l’existence d’une bulle immobilière au début des années 2000 aurait permit aux dirigeants politiques
d’intervenir de manière à la déflater lentement. Les arguments en faveur d’une action pour déflater les prix
immobiliers doivent ètre solidement justifiés étant donné l’impact potentiel qu’aurait une intervention inutile.
Ceci implique qu’une méthodologie utilisée pour évaluer l’existence d’une bulle doit produire des résultats
fiables. Nous examinons les conclusions pouvant ètre formulées à partir d’un modèle à Erreures Corrigées à
travers l’étude de la bulle récente sur le marché immobilier américain. Nous examinons les résultats de deux
modèles antérieurs à l’éclatement de la bulle ayant différentes conclusions, l’un de McCarthy et Peach (2004)
concluant qu’il n’y avait pas de bulle et l’autre de Soerensen (2006) concluant à l’existence d’une bulle. Nous
essayons d’identifier les différences clés qui pourraient avoir mener à cette conclusion. Puis, nous essayons de
construire un modèle similaire à celui de McCarthy et Peach (2004), de façon à avoir another point de
comparaisons et d’identifier les problèmes relatifs à la construction de tel modèles. Enfin, nous examinons la
valilité de tels modèles en les comparant avec un modèle sans fondamentaux. Les résultats obtenus nous
pousse à conclure que cette méthodologie ne peut pas donner de conclusions claires quant à l’existence d’une
bulle. Toutefois, nous essayons de souligner les points important à considérer lors de l’utilisation de cette
méthodologie.

Keywords: ECM; Housing market; Bubble

JEL classification: C51; C52; R31; E22


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CONTENTS
1. INTRODUCTION ........................................................................................................................... 4

2. LITERATURE REVIEW................................................................................................................. 7

2.1 The economic foundations ....................................................................................................... 7

2.2 Econometric Issues in the methodology .................................................................................... 8

2.3 Soerensen (2006)’s and McCarthy & Peach (2004)’s models ................................................... 10

3. THE EMPIRICAL APPROACH.................................................................................................... 11

3.1 Motivations ............................................................................................................................ 11

3.2 Economic theory .................................................................................................................... 12

3.3 Empirical analysis .................................................................................................................. 15

3.3.1 The model ...................................................................................................................... 15

3.3.2 Long run identification ................................................................................................... 20

3.3.3 EC representation ........................................................................................................... 23

3.3.4 Interpretation of the results ............................................................................................. 25

4. CONCLUSION ............................................................................................................................. 30

REFERENCES...................................................................................................................................... 32

ECONOMETRIC SOFTWARE......................................................................................................... 35

APPENDICES ...................................................................................................................................... 36

A- Data ....................................................................................................................................... 36

B- STATISTICAL RESULTS..................................................................................................... 39

C- NORMALITY GRAPH ......................................................................................................... 40

D- CONSTANCY GRAPH ......................................................................................................... 42

E- IDENTIFICATION OF THE LONG RUN RELATIONS........................................................... 44

F- COINTEGRATION RELATIONS............................................................................................. 46

G- VECM IDENTIFICATION RESULTS .................................................................................. 46

H- AR IDENTIFICATION RESULTS ....................................................................................... 48


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1. INTRODUCTION

Detecting the existence of the housing bubble in the early 2000’s would have permitted policy
makers to act in order to deflate it slowly. Arguments in favor of taking action to deflate housing
prices have to be strongly justified given the potential negative impact of a non-required
intervention. Policy makers have to balance a trade off between two types of error, intervening
on the market where it was not necessary or letting a bubble inflates without making any
intervention. In order to reduce the possibility of policy decision mistakes in cases such as these,
it is necessary to have reliable tools to assess the existence of a bubble. In this paper, we examine
what conclusions can be made from an Error Correction (EC) model through a study of the
recent bubble in the US housing market. We compare our model and its results with the results
from two previously conducted studies. One from McCarthy and Peach which reached the
conclusion that there was no bubble at the end of 2003 and the other from Soerensen which
concluded that there was some evidences of a bubble during the same period. These two papers
offer an example of the work that could be obtained from a real-time analysis.

One method used to assess whether an asset is overvalued is to consider deviations from their
long-run value of some ratios of the housing prices. The information given by these measures is
often sufficient to provide a fairly good idea of the evolution of the value of the asset in the long-
run. These ratios are expected to have fairly stationary values in the long run, so if an
observation of a significant deviation from the long-run value is seen then it is probable that a
correction will occur in the near future. In the housing market, the conventional metrics used for
this purpose are the prices-to-income ratio, the prices-to-rents ratio and the growth rate of real
housing prices. They are all intended to be relatively stationary, however, in 2000, this was not
the case. At this time, these ratios were indicating some important and persisting deviations in
housing prices from the value implied by the long-run value of the ratios while at the same time,
the economic situation was quite unusual due to the recent burst of the Dotcom bubble. Given
these economic conditions, it was difficult to assess whether the rapid increase in housing prices
was part of a cycle and could be justified by the value of the fundamentals or if the increase was
primarily due to speculation. The burst of the Dotcom bubble changed the way that investors
were attributing risk to the different assets in which they can invest, and the housing sector


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appeared to be relatively less risky than the stock market which had just crashed. This led to an
increase in housing demand and an increase in housing prices. Home owners saw the value of
their houses rapidly increasing, creating a wealth effect on the economy. This change in the
saving behavior of the household softened the crisis as mentioned by Shiller (2004). At the same
time, the GDP growth rate increased due to an increase in productivity. In such a context,
distinguishing the housing prices changes driven by the fundamentals from the speculation part
was quite challenging.

Using the prices-to-rent ratio and the prices-to-income ratio, a number of papers failed to predict
the existence of a bubble, for example Case and Shiller (2003), or Himmelberg, Mayer and Sinai
(2005). They justified the deviations from the long term value of the ratios mainly by the low-
interest rate policy implemented at this time. By comparing housing prices index and rents,
Learner (2002), and Hatzius (2004) pointed to some evidences of overvaluation of the housing
prices. And Krainer and Weil (2005), using the same methodology found similar conclusions.

Due to the apparent lack of link between these ratios and the other variables driving the housing
market, economic forecasters face considerable difficulties in making reliable conclusions
regarding the over-valuation of housing prices. A misunderstanding of the relation between
housing prices and their fundamentals is partly responsible of the wrong assessment of the
situation. A model which relates housing prices to the values for all the fundamentals would give
a better analysis of such situation by including all the relevant information in a joint analysis.
When the housing prices are above the equilibrium value determined by the model, there is a
positive deviation and if this deviation remains positive for a given period of time, then it is a
bubble. This method of defining a bubble is compatible with the commonly used definition from
Stiglitz : "the basic intuition is straightforward: if the reason that the price is high today is
because investors believe that the selling price will be high tomorrow-when "fundamental"
factors do not seem to justify such a price-then a bubble exists." (Stiglitz 1990, p 13).

This paper investigates the quality a a methodology based on the Error Correction Model (ECM).
The model is used to decompose the movements in housing prices into short and long-run
determinants. The long relation is assumed to define the fundamental value of housing prices
toward which they will converge in the long run. The use of an ECM allows avoiding spurious
regression problems and estimating more precisely the fundamental relation(s) of housing prices


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than what would be obtained using classical regressions. McCarthy and Peach used their model
to analyse the impact of monetary policy on the housing market in a first paper published in 2002
and then used it as a tool to assess the valuation of housing prices in M&P2 (2004). The model
from M&P failed to predict an overvaluation of the housing prices. It is difficult to determine
whether this is due to their methodology of assessing the existence of a bubble or to its
implementation. This paper tries to identify some clues of the reasons which could explain it.

The model from Soerensen (2006) is based on the same methodology and is presented as an
illustration that one could obtain correct conclusion using an EC framework, and that the results
from McCarthy and Peach are not representative of the methodology. We discuss some possible
reasons for the good results obtained by this model. We show that the theorical foundations of
the model used by M&P are commonly used in the literature and that despite some issues related
to their implementation, they seem not to be the cause of their failure. The construction of an EC
model using the same theorical foundations is used following this statement. Despite some strong
evidences of overvaluation, the construction of our model is subject to some issues and doesn’t
allow us to obtain satisfying results. We are not able to estimate stable long-run relations which
would prevent these results to be reliable in a real time analysis. Moreover, we show that the
quality of the fundamentals choice may be difficult to justify through the comparison of the
ECM predicted prices and the one from a model without fundamentals.

Our review of the methodology doesn’t identify clearly the reason that could explained its
failure. The estimation of a non-observed relation is difficult and subject to many statistical
issues related to the construction of the model. This leads us to recommend a transparent and
detailed presentation of the model in order to increase their reliability and to consider with
caution the conclusion issue from this methodology.

2
We denote McCarthy and Peach by M&P in the remaining part of the paper.


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2. LITERATURE REVIEW

2.1 The economic foundations

The model developed in this paper is based on M&P (2002) which is itself based on one of the
references of the empirical housing literature from Poterba (1984). This model from Poterba has
emerged as being an important reference of the housing sector, it is chosen here to give more
strength to the argument that the choice of fundamentals is complete. A condition which must be
fulfilled in order to produce reliable conclusions. This simple model is composed of two main
equations, a demand equation (Eq (1)) and a supply equation (Eq (5)). Using a small asset-
pricing model of the housing rents market, the model integrates the interest rate through the
notion of user cost first introduced by Jorgenson (1963). The user cost represents the opportunity
cost of living in a house instead of renting it. It has as advantage that it relates the rent-to-price
ratio to the interest rate and therefore, links monetary policy to the housing market3. This was
perfectly suited to the analysis of the situation at the beginning of the 2000’s, when the impact of
the policy of low interest rate was often given as the reason for the fast growth of the housing
prices. It also integrates the ratio of consumption to the housing stock which allows
consideration of an important factor of the dynamic of the housing market, the wealth effect
generated by changes in housing prices as recently documented in Guo (2009). These two terms
are related to the ratio of the housing prices to consumption prices in the demand equation. One
can mention as an important omission on this side of the market a liquidity variable, for
measuring the quantity of credits available for investment. This would be important to model the
liquidity constraint effect and the consumption dynamics. As mentioned by Soerensen (2006),
credits availability plays an important role in the short run dynamics of the housing market. This
variable would have been really useful to explain the dynamic of the housing market after the
crash of the Dotcom bubble. However, it is difficult to find a proper measure to use in an
empirical analysis of the housing sector. And this variable is left out of many of the empirical
studies of the housing market.

The main features of the supply side of the housing market are explained in Topel and Rosen
(1988) or Poterba (1984). They find that the supply curve is inelastic, interest rate is a demand

3
see Mishkin (2007) for a complete explanation of the importance of the monetary policy in the housing market


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shifter and it is hard to estimate the link between prices and houses supplied due to the
sluggishness of adjustment in the supply. These features led to a consensus that the housing
supply was accurately modeled using the Tobin’s Q theory (1954) as exemplified by Summers
(1981), Poterba (1991), Abraham and Hendershoot (1996), Rosen and Topel (1986), and Barrot
(2001). M&P (2002) assumed in their model that the fundamental prices are defined only by the
demand side of the market following the consensus about the inelasticity of the supply side.

Some additional elements can be added to the model specification depending on the scope of the
study. For example, Mankiw and Weil (1989) and Madsen (2008) introduce demographic factor.
Barrot (2001) introduces the debt to income ratio in a similar model for the Swedish market,
Berger-Thomson and Ellis (2004) estimate a model for different countries to analyze the effect of
institutional factor, Kenny (2004) identifies the impact of monetary policy for Ireland, and
Poterba (1984) analyzes the impact of taxes changes. The variables used in the model here are
often presented as being sufficient for the determination of the fundamentals but many other
elements may be added as suggested by Soerensen (2006). This model from Poterba (1984) is
also mentioned or used in Besson and Boissinot (2005) and Soerensen (2006) and prove to be
trustable for people using this methodology.

2.2 Econometric Issues in the methodology

The quality of the statistical properties of the model should be of main concerned as they may
change the conclusion from the model. A model proving to have good statistical properties
would be more reliable and of higher value for policy-making.

The implementation of our theory is subject to many problems. One of the main problems is the
lack of proper data. Demers (2005) for example found that using a perpetual inventory equation
(Eq (4)) to compute the housing stock may lead to find a root of order two in the system,
invaliding the properties of model. In his study of the Canadian housing market, Demers found
that if the depreciation rate is small enough4, the housing stock will be by construction integrated
of order two. To solve this problem, he mentioned that one could extend his work by using a

4
The relation can be rewritten as ∆ . If the average value of ⁄ is not too different from on
the sample considered, one may not observed a too strong I(2) behavior I the model, and ignore it.


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polynomial cointegration for stock-flow model as suggested by Haldrup (1994) and Tom
Engsted and Niels Haldrup (1997). Using our dataset, we find some evidences of I(2)5 roots
under many different assumptions and sample size using a rank test. Some I(2) roots in the
dataset would prevent one from imposing the theorical constraints used in M&P’s papers as the
residuals left after the estimation of the “long-run” relations would be non-stationary. This
wouldn’t invalidate their interpretation as long-run relations6 and some properties of the model.
No attempt has been done to build an I(2) model of the housing market to my knowledge. The
I(2) model is more difficult to interpret than an I(1) model, and also much more difficult to build.
This problem could be solved in a few cases when a reduction of the I(2) model to an I(1) can be
implemented. This is what is called the nominal-to-real transformation (Kongsted (2005)). After
a few trials, it seems that this is not suited in our case; the possible transformations were rejected
by the data on the sample where some signs of I(2) roots are present. After many trials to deal
with this problem, we chose a sample such that the I(2) properties of the model were minimized.

In many studies, this problem is not mentioned, and it is difficult to know whether the possibility
that the dataset contains some I(2) roots has been taken into consideration. As mentioned in
Juselius (1999), the prices index are often found to have an I(2) root. The number of possible
sources of I(2) is important in such dataset7, not checking whether it is effectively driven by
some I(2) roots may be misleading. It wouldn’t invalidate the prediction properties of the model
and would still allow the model to be used in its complete form. However, it would impact the
reliability of the results found as the model would features a low-mean reversion in the
cointegration relations and loose its interpretation into economics terms as it would be possible
to qualify the cointegrations relations of long-run relations. The model developed in Soerensen
(2006) is less subject to I(2) problems. The use of annual data changes the statistical properties
due to the reduction in the frequency. The long swings in the value of the variables under
quarterly frequency look rather like simple fluctuations of an I(1) stochastic variable at an annual
frequency. This choice of frequency gives to the model better statistical properties, and leads to
the estimation of more stable fundamental relations than what can be obtained from a quarterly
analysis.

5
I(2) is used as qualitative expression for “Integrated of order two”
6
The interpretation of the “long-run” relations has to be modified as explained in Juselius K. (2004).
7
Prices index are often found to be I(2) as mentioned in Juselius K. (1999)


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The studies of the housing market are often conducted with a careless consideration for the
statistical properties of the model. The number of cointegration relations is sometimes chosen
with few attentions for the statistical results due to their fragility as in Bessone and Boissinot
(2005). This kind of choice is done following an a priori about the number of relations to be
found in the data but it may create a bias in the whole model and its results.

2.3 Soerensen (2006)’s and McCarthy & Peach (2004)’s models

As pointed out in Dirk J. (2009), a review of the literature forecasting the crisis, the findings of
Soerensen (2006) were noteworthy as they were among the few ones predicting a crisis. In his
empirical analysis of the housing market in different countries, Soerensen defined a bubble in a
similar way as M&P, suggesting that this kind of approach use to assess the existence of a bubble
may have some potential. The fundamental relation is defined as being the long-run relation
estimated in the two steps procedure of the construction of an ECM. A positive difference
between the observed value and the equilibrium prices defined by the model is a deviation from
the prices implied by the fundamentals. Therefore, if the deviation persists, it is a bubble8. One
drawback of this methodology is that the fundamental relation must be properly defined. The
original aim of Soerensen was to use a dataset similar to the one of M&P, but due to the
difficulty of acquiring the data on his sample, he renounced and used a subset of the original one.
Our two benchmark models have been constructed using a dataset ranging until the end of 2003,
Soerensen’s model using annual data and M&P using quarterly data. To compensate his choice
of a lower frequency, Soerensen use a dataset starting in 1913. One drawback of using a long
sample as Soerensen is that the fundamental relation may have been subject to structural
changes, and that the estimated relation may not be accurate to assess the valuation of housing
prices in 2004. On the other hand, using a shorter sample makes the final relations more sensitive
to the choice of its beginning and to the influence of outliers and structural break.

One problem in M&P’s model is that some choices are assumed based on the theory, but these
choices may not be verified in the data. A more objective analysis of the data would analyse all
the possible fundamentals available, as done in Soerensen (2006). The choice to use different

8
A precise definition should specify the duration and the size of the deviations required in order to claim about the
existence of a bubble. We left this element for future analysis.


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variables to describe the short run and the long run or the choice of using only the estimate of the
demand as fundamental prices is choices which can and should be tested. The discussion of the
construction of their model is rather short, and it is difficult to assess the point which may have
lead to the failure of their model. Finally, a last important methodological difference which could
explain the different conclusions is that M&P used a multivariate procedure for the estimation,
and then assumed that they can consider only one of their two equations to assess housing prices
valuation. Such problems illustrate an important point raised by the Dahlem report9 (2009). In
examining why the economists failed to warn about the impending crisis, it mentioned an
obvious problem of communication about the limitations of their models. The model from M&P
is rather employed as an illustration of an idea rather than an objective analysis of the US
housing market. The limitations of their model are not discussed. On this aspect the paper from
Soerensen, even if some other points could be criticized, gives a much more transparent
description of the construction of his model, and can be considered in this respect as a better
approach of modeling.

3. THE EMPIRICAL APPROACH

3.1 Motivations

The methodology presented here is frequently used to assess the value of an asset, see for
example Bessone and Boissiniot (2005), Antipa and Lecat (2009), Francke, Vujic and Vos
(2009), Wong (2001) and Shen, Chi-Man Hui Liu (2005) for the housing market or Miller and
Ratti (2009), Gautier (2001) for the stock market or Adland (2006) for the shipping industry.

The benchmark paper for the empirical approach is the paper from M&P (2004) which describes
an Error Correction (EC) model for the housing market. This approach gives a good description
of the variables and is based on economical foundations. Approaches similar to the one of M&P
are commonly used to study the housing market or to assess the value of assets. It defines an

9
The Dahlem report makes reference to the paper issued from the discussions in the Dahlem workshop : Colander, D. & Föllmer,
H. &Haas, A. & Goldberg, M. & Juselius, K. & Kirman, E. & Lux, T. & Sloth, B. (2009) “The Financial Crisis and the Systemic
Failure of Academic Economics,” Kiel Working papers 1489, Kiel Institute for the World Economy


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equilibrium in which some deviations can occur. The long-run equilibrium is defined in the
model by the cointegration relations. They are identified so as to be in agreement with the same
economic theory. The short-run dynamic is defined by the lags of the variables which are
considered as relevant for the definition of the dynamic of the variable.

We use the Johansen maximum likelihood estimation to estimate the long-run relations in a
Cointegrated Vector Autoregressive (CVAR) model. This method is used to deal with the
problem of spurious regression between I(1) variables, and allow to interpret the system in terms
of long run and short run dynamics. The CVAR model can be rewritten in an EC form. This
representation incorporated the influence of the deviations from the long-run relations in the
variables called EC terms. They stand for the difference between the observed values and the
values predicted by the long-run relations and impact the variables of the system and therefore
the system is driven back to the dynamic equilibrium10. M&P use the CVAR model to deal with
the problems of estimation of the long-run relations only. The short run coefficients are
estimated through an univariate procedure. In order to have a more objective description of the
data, we choose to estimate a CVAR model in its Vector Error Correction representation rather
than using an univariate approach; this choices takes into account the correlation between the
different variables and the different steps of the construction of the model can be tested.

With this model, we illustrate that an approach based on a similar theory and framework as M&P
could indicate some evidences of a bubble. We use this estimation to compare these results with
the one of an AutoRegressive (AR) process of order one. We show that the reliability of our EC
model is quite low and that a model without fundamentals like an AR(1) could find results which
are qualitatively similar to a model with fundamentals. From these results, we conclude that
when using this methodology,one has to be really carefull in the formulation of his conclusions.

3.2 Economic theory

Using simple ratios to evaluate the possible overvaluation of housing prices is confusing as the
link between the information of these different elements is not explicitly defined. A model of

10
A complete description of the CVAR model in terms of pushing and pulling forces can be fond in Johansen
(1996)


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demand-supply can be used to link the different fundamentals in a unique framework. The real
housing prices, the ratio of consumption to housing stock, the real rents, the ratio of housing
prices to construction and the growth rate of the housing stock are variables which are supposed
to be stationary to some extend. These variables are related in these foundations through to
equations, deviations in one of them can persist as long as it would be compensated by a
deviation in one or several other variables. This formulation of the metrics in this model is less
restrictive than using them independently, and should be less likely to indicate some evidences of
a bubble. We present briefly here the theorical foundations used by M&P and our model. They
include a supply and a demand equation in addition to the notion of user cost.

On the demand side (Eq. 1), the ratio of housing prices ( ) to consumption prices ( ) is a
function of the ratio of consumption ( ) to housing wealth ( ), and of the user cost ( ). The
first term integrates that housing consumption should depend on its affordability.

, (1)

The permanent-income theory from Friedman (1957) states that the proportion of the permanent
income of households11 used for consumption should be stationary in the long run, because
consumption decisions are based on expected total lifetime income. Permanent deviations from
the ratio should be due to change in relative prices. We expect a positive coefficient as a
permanent increase in the ratio of consumption to the housing stock should be enhanced by a
change in the relative prices.

The user cost relates interest rate to the rent-to-price ratio through an arbitrage condition. It links
the housing market to the interest rate variable, and is used to estimate the impact of the
monetary policy on the housing market. The coefficient of this variable represents the degree of
substitution between renting and buying a house. Different approximations of the user cost are
employed in the literature. The one used in M&P (2004) is defined by equation (2):

1 (2)

11
Permanent income is usually proxy by consumption of non-durable and services as in M&P. We use the general
definition of consumption which as been found to give more stability.


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Where is the expected inflation of real housing prices, is the housing
prices inflation and is the consumption prices inflation. is the real12 rental price, is the
real housing price, is the income tax rate, is the real depreciation rate including other cost
(reparation and maintenance cost). We expect a positive sign related to the user cost13. The last
term in the equation (1), stands for the deviations from the estimated demand equation and is
supposed to be stationary.

The housing stock is defined in our data according to the perpetual inventory equation (Eq. 4).

1 (4)

This equation is often used in the housing market empirical literature (e.g. Barot (2001),
Kenny(1999)). It defines the value of the housing stock which is also used as a proxy for the
quantity of housing unit provided14.

The equation (5) is the supply equation, and is based on the Tobin’s Q theory:

, (5) .

The Q-theory represents the value of one extra unit of housing relative to its replacement cost. If
the Tobin’s Q is above one, the marginal value of one extra unit of housing is above its marginal
cost, and investment increase. It is often assumed that the shadow cost of construction is equal to
the construction cost (Summers (1981), Poterba (1991), Abraham and Hendershott (1996)). In
this analysis, due to the lack of data on the land prices, we allow for to be different from -1 as
the construction cost is not properly defined. The coefficient of the growth rate of the housing
stock ⁄ is expected to be positive, and the trend is corresponding to the quarterly
depreciation rate so we expect to be negative and equal to a few percentage points.

12
Real means here that the prices index is normalized using the consumer index prices
13
Our version of the user cost is a simplified one. Due to some data problem, we compute it using the approximation
presented in the appendices.
14
The formulation of the model implies that there is no vacancy


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3.3 Empirical analysis

3.3.1 The model

This quarterly analysis is based on the following data set: is the real consumption per capita,
is the consumer prices index, is the construction price index, is the housing
price index, is the real housing investment per capita, is the user cost of housing and ,
15
is the housing stock per capita. The dataset used has been transformed using log.

The initial representation of the data is the following:

∆X Γ ∆X Γ ∆X ΠX Φ t = 1,…,T (6)

where the variables aredefinedasfollow: Dt is a vector of deterministic components. Γi, Π and Φ
are matrix of coefficients. The residuals εt are independent and identically distributed, with zero
mean and positive definite covariance matrix.

, ′

, , , , , , ′ is the vector of endogenous variables,

. ′

Under the assumption that our variables are integrated of order 1, the matrix Π is statistically not
full rank and a linear combination of the rows or columns can be found to approximate another
one such that this matrix can be decomposed as: Π (7)
where and are p×r full rank matrix, p ≤ r . are the cointegration vectors. They determine
the linear combination of the variables which are stationary and interpreted as being the long run
relations. are the vector of parameters measuring the speed of adjustment of the system toward
its long-run equilibrium defined by the long-run relations. Γ ∆X Γ ∆X are the lags of the
variables in differences, they define the short run effect of the variables.

We proceed in an iterative way to determine the optimal number of lags, the rank, the
deterministic components, and the exogeneity assumption imposed on our model. Because the
15
Variables definition and sources are given in the Appendix A


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choice of each assumption will influence the results of the tests for the other assumptions, they
have to be determined jointly.

We use an unrestricted constant in our model. Based on the inspection of the data, it seems well
suited to have a trend restricted to the cointegration relations. This allows linear, but no quadratic
trends in the variables. This means that we can have trend-stationary variables or trend-stationary
cointegrating relations. From a visual inspection of the graph of the data in appendices, it seems
that a break in the trend will be required to take into account the change in the growth rate of
most of the variable. We identify it as being the results from the oil prices shock and the crash in
the housing market which occurred at this time. These events impacted the US economy as a
whole and are found to be statistically significant using a t-test.

Another structural change occurred at the beginning of the 2000’s due to the burst of the Dotcom
bubble. Using the data after 2000 may be misleading, as the period is subject to important
instability and the number of observations which could be gained from using M&P sample is
very low. The instability of the cointegration relations leads us to avoid using the exact same
sample as M&P. As a consequence, we restrict ourselves to a sample ranging until 2000 in the
estimation of the model in order to have good statistical properties. The estimation of the
unrestricted model features long swings in the cointegration relations. Using some seasonal
dummies doesn’t help to solve this problem. M&P use four lags in order to capture the unstable
dynamic of the system but this has been found not to be helpful in solving the problem in our
case. The low mean reversion of the cointegration relations at this step suggests that the
cointegration properties of the model are poor. In order to estimate cointegration relations that
would be robust to the choice of the sample, one has to find an important number of mean
reversions, indicating that the system is pushed or pulled toward the equilibrium defined by the
model.

In order to increase the quality of the cointegration relations of the model, we assume that some
of the variables are exogenous. Such an assumption is strong. The variables with this property
generate a stochastic trend which, in the long-run, drives the dynamic of the system but these
variables are not impacted by the system itself. Consumption and the user cost are found to be
statistically weakly exogenous. We imposed them as being strictly exogenous on the system.


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is also found statistically as being weakly exogenous. However, we decide not to impose
it as exogenous in the system as it doesn’t help to improve the model.

It can be seen in the graphs of the differences of the variables (appendices A: figures 5 to 11) that
some of the variables could be integrated of order two. In order to construct an I(1) CVAR
model, one has to carefully check the order of integration of the variables. The long swings
which can be found in the cointegration relations are often a sign of I(2) roots in the model. We
start the rank determination by checking that the model is not I(2) using a rank test.

Table 3.1 Models corresponding to the statistics presented in table 3.216
p-r r Models
5 0 0,0 0,1 0,2 0,3 0,4 0,5 = 0

4 1 1,0 1,1 1,2 1,3 1,4 1

3 2 2,0 2,1 2,2 2,3 = 2

2 3 3,0 3,1 3,2 = 3

1 4 4,0 4,1 = 4
S2 5 4 3 2 1 0

The table is constructed in the following way: a statistic is computed for each model under all the
possible numbers of I(1) and I(2) roots for each rank choice. p-r is the number of common
trends, s2 the number of I(2) roots and r the rank. These models are partially nested in the way
presented in the table 3.1 implying that the numbers of I(1) and I(2) roots/trends have to be
determined jointly. The models on the same row have all the same rank and are all nested, the
models with the higher number of I(2) roots are the most restricted. The models with no I(2)
roots are also all nested implying that it is possible to compare the models between different
rows. A model with a smaller rank number r is more restricted. This allows us to determine the
correct model starting from the top-left corner and proceeding row-wise until the first acceptance
16
Interpretation of the I(2) rank test can be found in Nielsen and Rahbek (2007), and a formal description of the I(2)
model in Johansen (1995).


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level. All the previous models are rejected, and all the models with a significant probability
cannot be rejected. The model which should be chosen is the more global one. The number of
I(1) trend in our model is determined using the relation p-r=s1+ s2 where s1 is the number of I(1)
roots. Despite the swings which can be observed in the difference of the variables in appendix,
the I(2) rank test in table 3.2 suggests that an I(1) model with two I(1) roots is correct.17

Table 3.2 Test of the 2 rank indices in the unconstrained VAR
p-r r 5 4 3 2 1 0
5 0 610.716 416.541 322.537 273.757 242.187 227.397
(0.000) (0.000) (0.000) (0.000) (0.000) (0.000)
4 1 298.392 216.302 175.654 146.192 142.307
(0.000) (0.000) (0.000) (0.000) (0.000)
3 2 160.173 116.042 90.812 88.260
(0.000) (0.000) (0.000) (0.000)
2 3 76.377 53.323 47.493
(0.000) (0.000) (0.002)
1 4 26.817 18.753
(0.014) (0.081)

The results from the I(2) rank test are not very robust. The power of the test is generally very low
for I(2) or near I(2) data. This test is use to investigate the presence of I(2) but it shouldn’t be
used as a substitute for the I(1) rank test. Due to the low power of the rank tests, the choice of the
rank should be done using a much informaton as possible. Table 3.3 presents the results of the
I(1) rank test (Nielsen&Rahbek (2004)). This table gives the statistics for different rank number
of nested model. The models are nested in the following way: H 0 H 1
H , where p is the number of variable in the system. The choice of the rank is determined
by a top-to-bottom inspection of the table. We stop at the first significant probability which
indicates that the hypothesis of having a rank is not rejected. Then the rank is accepted.
The I(1) rank test suggests four cointegration relations in accordance with the results given by
the I(2) rank test.

17
For the purpose of assessing the value of housing prices, some I(2)ness in the model wouldn’t have any significant
impact on our conclusion as it doesn’t impact the prediction properties of the model.


of 48

The unrestricted cointegration relations found are approximatively stationary. This confirms the
number of cointegration relations determined by the rank tests.

The overall statistical properties of the residuals are good according to all the criteria18 and the
properties required to make correct inference are satisfied19. The skewness is small and the
kurtosis close to three. There is no significant autoregressive conditional heteroskedasticity effect
or important autocorrelation which implies that the quality of the estimates should be good. The
size and the number of large residuals are reasonably small enough to have good statistical
properties. There are no outliers that we should correct for, the kurtosis of the consumption
prices index is slightly above three, but still remains reasonable.

Table 3.3 Simulated rank test
p-r r Eig.Value Trace Frac95 P-Value
5 0 0.659 227.397 119.146 0.000
4 1 0.495 142.307 90.184 0.000
3 2 0.403 88.260 63.123 0.000
2 3 0.305 47.493 40.280 0.018
1 4 0.211 18.753 20.861 0.129

The table 3.4 indicates that housing prices have the biggest standard error. This is due for a large
part to the asset properties of the housing stock which is subject to speculations of investors. The
correlation of the residual is important between investment and housing stock. Using a
multivariate estimation of the model allows us to take into consideration the correlation in the
residuals. The other correlation coefficients are significant and will make the identification of the
short-run structure less precise but won’t affect the results of the estimation of the long-run
equations.

Table 3.5 Lags length selection test
Model Lags T Regr Log-Lik SC H-Q
VAR(5) 5 76 41 2458.126 -53.006 -56.780
VAR(4) 4 76 34 2395.497 -53.352 -56.482
VAR(3) 3 76 27 2346.937 -54.069 -56.554
VAR(2) 2 76 20 2317.447 -55.287 -57.128
VAR(1) 1 76 13 2244.530 -55.363 -56.559

18
The complete results of this test can be found in the appendix
19
In our particular case, the choice of this assumption is very important as the failure in some particular assumptions
(see Nielsen and Rahbek 2000 for the relative importance of the failure of these assumptions) would have important
consequences on the appropriate choice of rank


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The table 3.5 presents the results from the lag length selection tests. The number of lag suggested
by the Hannan Queen test is 2 and by the Schwartz criterion is 1. The choice of only one lag is
likely to fail to catch the dynamic of the system of variables, and choosing a higher number of
lags would increase too much the number of parameters to estimate without improving
significantly the fit.

3.3.2 Long run identification

The model has been estimated on the sample 1981:1 to 1999:4 as it has been found to have better
statistical properties. Our identification of the economic foundations is accepted only when we
restricted ourselves to this sample. Consumption and the user cost are imposed as exogenous. A
structural break has been modeled through a permanent dummy on the period ranging from
1990:01 to 1999:4.Using longer periods would probably lead to find a model subject to a bias
due to the instability after 2000, and lead to find less obvious evidences of bubbling growth. By
choosing such sample, we assume that the fundamentals are well defined on this period after
modeling the 90’s periods with a structural break, and that the fundamental equations defined by
our model can still be applied on the periods after the Dotcom bubble.

Our final statistical representation of the data is the following:

∆X Γ ∆X Γ ∆X αβ′ X Φ t = 1,…,T (8)

where the variables are defined as previously. 9001 and 9001 are the restricted
deterministic component corresponding to the inclusion of the break in the trend.

, , , 9001, ′

, , , , ′,

9001 , . ′

The β structure is identified with a relatively low acceptance probability of the overidentifying
restriction of 0.079. The estimated Tobin’s Q equation for the supply side is:

1.009 0.327 0.006 , (9)


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The coefficient of the real construction price index is lower than what we would expect but it has
the correct sign. The trend can be interpreted as being the depreciation of the housing stock. It
implies an annual depreciation rate of 2.4% in agreement with what we would expect. The
estimates found are of the expected sign and they are all significant20.

The estimated housing demand is:

0.756 0.312 0.009 , (10)21

We assumed a unitary elasticity between housing wealth and consumption as in M&P in order to
find estimates of the correct value. All signs are of the correct sign and are significant.

In addition to these two equations predicted by the economic theory, we have two other
equations describing the dynamic equilibrium of the system. The first relation (Eq. 11)
approximates the housing stock as a broken stochastic trend.

0.015 0.01 90 , (11)

This restriction is accepted due to the fact that the variation of the housing stock is very small
compared to its absolute value. The graph of the first differences of the housing stock shows
clearly a change in its growth rate at the beginning of the 90’s.

And the second relation is an empirical relation approximating the long-run relation between
investments and housing stock.

3.2 0.002 , (12)

The graphs of constancy22 presented in appendix are used to detect some possible instability in
the model. They are the result of the computation of some elements on different sample. The low
fluctuations of these elements in the R-form suggest that the long-run estimates are more robust

20
The complete results of the estimation can be found in the appendix.
21
Equation (10) has been restricted following M&P so as to have ratio of consumption to housing wealth equal to
the unity. Consumption has been found weakly exogenous and the exogeneity has been imposed on the system. It
seems that our system doesn’t integrate a long-run impact of the collateral constraint. The weakly exogeneity results
indicate that consumption is not affected by the other variables in the long run. Due to the large number of omitted
factors influencing the evolution of consumption, it was difficult to relax this assumption without creating an
important instability of the system.
22
For an explanation about the way to interpret these graphs, see “the cointegrated VAR model” from K. Juselius


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than the short run, but their stability seem not very good. The important differences between the
R and the X form indicate that the short-run estimates are more sensitive to the sample used for
their estimation. The stability of the coefficients of the long run relation(s) as well as the
acceptance probability of the identifying restrictions are important indicators of the stability of
the fundamental relation(s) defined.

Equations (9) and (10) are consistent with the economic theory. All the signs found are correct
and the coefficients are all significant23. Equations (10) and (11) are additional relations more
difficult to relate to the economic theory. We are not able to reproduce perfectly the results from
M&P (2004). Our data don’t contain the same information as they are defined differently, and
we may have interpreted the results of some tests differently.

However, our model reflects perfectly the information contained in our dataset, and offer a more
transparent description of the information contain in the data which ensures a higher degree of
reliability. The p-value of 0.079 found may seem small at first sight for the reader compared to
the one found in M&P. This may be due to some data problems such as the construction of the
housing variable or the user cost but it can be also due to the differences in the hypothesis
imposed on the model.

The cointegration relations presented in appendix F features some important non-stationarity.
They can be due to some I(2) features of our model not detected by the I(2) rank test or to the
difficulty to catch the long swings of the housing market by our model. The non-stationarity
becomes more important in the periods after 2000. The slow mean-revertion of the cointegration
relations makes it difficult to interpret them as being long-run relations. A large number of mean
reversions of the cointegration relation(s) increasing the evidences that some corrections occur in
the market toward the relation found. The number of mean reversion in M&P (2004) is relatively
low, this could explain why the model failed to give the good results.

Another argument that can be given is that the fundamental relation has been affected by non-
modeled structural break and that the true relation may have changed. For example, one could
consider the possibility that the proportion of wealth invested in the housing market increases
relatively to the other asset markets due to a change in the relative risk premium attributed to the
housing market after the burst of the Dotcom bubble. This question would need to be assessed. It
23
The complete table of the results can be found in the appendices.


of 48

is left out without any answer as the aim of this analysis is not to assess the existence of a
bubble, but to point the important issues related to the construction of such model. A structural
break in our model, it does not have an impact on the supply and demand equation directly but it
has an impact on the demand equation through the estimation of the short-run parameters and of
the other cointegration relations.

A third arguments can be used to explain their failure. Our four long-run relations describe
jointly the long-run equilibrium of the model. A shock can create a deviation from one or several
of these relations but in the long-run, the system is pushed back to the equilibrium and these
deviations do not persist. Deviations from the cointegration relations are the error correction
(EC) term and are noted as , . The EC terms have to be considered jointly. The nonstationarity
in one cointegration relation can be balanced by the nonstationarity in one other relation
implying that they compensate which would implied that the system is still in equilibrium. By
construction, all the cointegration relations are linked as it is indicated by the equation (7). The
assumption that the fundamental prices is driven by the demand side as it is used by M&P
simplify the analysis, but may lead even a correct model, to produce misleading conclusions.

3.3.3 EC representation

M&P assume that some additional variables are required to describe the short-run dynamics to
their Error Correction Model (EC). This choice of using some variables for the short-run only
can be justified by the willingness to impose the intensively used theorical foundations from
Poterba. However, including these variables in the long-run estimation and testing whether they
were significant or not is econometrically more correct. Their choice may not be able to give an
appropriate representation of the local data generating process defined in the general-to-specific
methodology24. This methodology ensures that the resulting ECM would be congruent with the
description of the data given by the original CVAR model. The general-to-specific approach
favors the empirical quality of the model over its economical foundations. The drawback of such
approach is that it is likely to reject the theorical foundations that were expected. However, such
choice can be justified as the loss of the reliability grant by the economical foundations may be

24
a complete description of the methodology can be find in Campos, Ericsson Hendry (2005)


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compensated by the gain in reliability due to the improvement in the statistical properties of the
model.

In this approach, the reduction of the model is tested until a reasonably good identification is
reached. We use a likelihood ratio test to assess the validity of the elimination of each non-
significant coefficient. We discriminate between the elimination of the coefficients toward the
“best”25 possible identification using the significance of the coefficients.

M&P assumed that the housing market is driven only by the demand. This assumption is in
accordance with the idea that the supply is inelastic but may not be able to give a correct
description of the housing prices dynamic as it was previously mentioned. They chose to assess
housing prices valuation using only the EC term from the demand side. Following the general-to-
specific methodology, this assumption is initially relaxed, by estimating a more general form in
which all the possible EC terms are included. Then, the significance of each of the EC terms is
tested in order to select which one could be removed from the EC equation without a significant
loss of information.

We identify a Vector Error Correction Model (VECM) from our dataset using the sample
ranging until 1999:4. From the constancy graph introduced previously, we saw that the short-run
estimates were subject to an important instability depending on the sample used. Using data
after 2000 would create even more instability in the estimates. In a “real time” modelization of
the housing sector, one should try to construct some model using different samples, in order to
assess how the short-run estimates26 impacted the conclusion given by the model. Using a
sample until 2003:3 as M&P would create a bias in the estimates due to the important growth of
the housing prices at the beginning of the 2000’s therefore may not reflect the fundamental
relation precisely. The model of Soerensen is less subject to this critique. By using annual data,
the number of observation impacted by the burst of the Dotcom bubble is very low, and so their
influence on the parameters estimates smaller. This way of dealing with the parameters
instability is the more objective one, it avoids that through the choice of the sample, one may
also create a bias in the estimate that is not reflecting the true fundamental relation.

25
The identification given here is not unique. Best is used in term of precision of the short-run parameter estimates.
26
We found that our long run estimates are less subject to instability, but this question may have to be addressed. For
example, the long-run estimates found by M&P in the paper from 2002 and the one from 2004 are quite different.
An additional restriction is required in the paper of 2004 in order to identify the same theorical foundations.


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Our model is not able to give clear evidences of the existence of a bubble based on the
observations of the long run relations only. We assess the quality of the fundamentals used by
comparing the bubble produce using the complete ECM with the one produce by a model
without fundamentals. This way of processing doesn’t allow considering the long-run relation
only. Due to the important number of cointegration relations use to describe the long-run
dynamic of the system, using only the demand equation would produce misleading results as it
wouldn’t describe properly the equilibrium of the model. The stability of the whole model may
be comparable to the one found for the demand equation only.

To avoid the over-parameterization of the model, we use only one lag for the short-run dynamic.
The equation identified is the following27 :

Δ 0.1153Δ 0.5956Δ 0.031469 1 0.0470415 2 0.111011

where 1 , 2 stand respectively for the error correction term of the equations (9)
and (10). The two other correction terms have been found to be statistically non-significant.
Housing prices are found to be driven in the short-run by the demand factors which are
consumption and the user cost. Supply side fundamentals are not found to have a direct impact
on housing prices in the short-run, but it is found to have an indirect impact through its
corresponding EC term, in opposition to what is assumed by M&P.

3.3.4 Interpretation of the results

It is difficult to formulate clear conclusions from the error correction graph presented in the
appendix F. Even if one can observe some persisting positive deviations from the cointegration
relations for demand and supply, it is hard to conclude whether they are significant or not. And
this is even more difficult when one had to the analysis the two remaining cointegration relations
as it should be.

In order to illustrate the issues of determining whether deviations is significant or not, we
construct a simple AR(1) model, as benchmark for the results found by the EC model. The

27
The complete indentified VECM model is given in the appendices.


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AR(1) model is not using any fundamentals contrarily to the EC model but it should be able to
catch the speculation process through an autoregressive expectation mechanism.

An AR(1) model doesn’t integrate the impact of the fundamentals on the housing prices. It is
well-suited to describe a process with a constant mean and some persistence in the dynamics
such as inflation. The estimated AR(1) with constant term is the following :

Δ 0.3536Δ 0.007

The graph 3.1 presents the one-step-ahead housing inflation prediction from the EC model and
the AR model. The predicted prices defined by both models are rather similar. This observation
shows that a model with fundamentals, the EC model, and a model without fundamentals, the
AR model, can produce results which are qualitatively comparable. They both suggest that the
housing prices were deviating from the equilibrium prices. And produce graphs giving more
evidences that the housing prices were bubbling.

Graph 3.1 Observed and computed housing inflation


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The strong similarities of the deviations in the two models suggest that it is difficult to determine
the amplitude of the deviations needed in order to state that the housing prices are creating a
bubble. In a real-time analysis, in light of these results, one could argue to someone presenting
evidences of a bubble using this methodology that his model is not doing better than being
enable to explain the growth of housing prices. The omission of some fundamentals responsible
for the important growth in housing prices may lead to the detection of a bubble even if it doesn’t
exist. The question about whether the fundamentals used in this kind of model produces a proper
description of the housing prices inflation dynamics is difficult to address. Soerensen also
reaches this conclusion in his analysis. A complete analysis of the housing market would need to
assess clearly the relevance of all fundamentals, to determine whether some have been omitted.

Graph 3.2 Bubble processes

The graph 3.2 presents the observed inflation minus the equilibrium prices defined by the one-
step-ahead prediction for both models. It represents the bubble dynamic according to definition
implied by our framework for the AR model and the ECM. It is difficult to distinguish some
important differences in the two bubbles process. One has to be careful in choosing the
fundamentals. The omission of some important factors or the inclusion of some non-relevant


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factors may lead to inaccurate results. This graph shows that without any criteria to assess the
difference between the two curves, it is not possible to conclude with certainty whether the
deviations are due to a failure of the model to catch the change in growth rate or whether it is due
to a growth not explained by the fundamentals.

Graph 3.3 Observed and fundamental real housing prices

In the graph 3.3, we assume that the housing inflation price index is equal to the fundamental
prices in the first observation. The position of the equilibrium prices curve is often chosen in a
way that the average difference between the observed prices and the fundamental prices is equal
to zero, reflecting the assumption that the deviations from the fundamental relation should be
stationary. The graph 3.3 makes clearer the differences between the AR and the EC model than
the graph 3.2. There is an important difference in the 90’s between the prices implied by the two
models. This is due to the fact that the crash in the housing prices in this period is modeled in
the ECM with a structural break. This illustrates one of the critical point of this methodology,
about the importance of structural break in the definition of the fundamentals relation. One can
think that some events impacts the relation between housing prices and fundamentals. For
example, one could think that the burst of the Dotcom bubble changed the proportion of wealth


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that was invested in the housing sector due to a change in the relative risk aversion of investors.
M&P don’t mention any possible structural break, but it is important to consider them in order to
produce a model with good statistical properties and stable cointegration relations. The influence
of the growth due to expectations during the bubbling periods is partly taken out of the value of
the coefficients related to the fundamental through their use.

The graphs 3.1 and 3.3 show clearly that the housing prices were growing faster than what was
implied by the models. By the end of 2003, some evidences of housing overvaluation could have
been observed from our EC model. But to conclude whether this observation was significant
enough or not to conclude to the existence of a bubble is difficult to determine. Such analysis
would probably be not sufficient to conclude to the existence of a bubble in the context where
the two papers presented here have been published. The choice of a methodology that define a
bubble as being the error produce by a fundamentals model seems to be not satisfying.


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4. CONCLUSION

The conclusions that could be drawn from this methodology are not robust. Defining a bubble as
being the error of a model requires the assumption that the model constructed is correct and
complete. Such assumption is strong and difficult, if possible, to demonstrate.

The problem in M&P’s model is not only that it gave the wrong conclusion but also and mainly
that the lack of details in the construction of their model which doesn’t allow us to identify
clearly its failures. The critics about Soerensen methodology are more difficult to formulate as
the conclusion reached was the correct one. However, his choice of the fundamentals used seems
to be very poor and it is likely not to represent the true fundamental relation. Moreover, the
questions of the stability of the relation and of the degree of significance of the deviations cannot
be addressed.

The important question for policy-makers is the degree of reliability that can be given to the
conclusions of the model. The quality of the statistical properties of the model is an important
issue. The comparison between the results produced by the ECM and the AR model suggest that
it is difficult to distinguish between a lack of fit of the model and a bubble. This methodology
relies mainly on the assumption that the estimated relation is properly defined. It seems that such
analysis cannot be judged as sufficient by itself to obtain the reliability sufficient in order to
make any important choices. Such model is subject to important instability due to the numerous
variables close to be integrated of order two. The CVAR model and its EC form is a good
representative of the model which can be used in what is called the general-to-specific approach.
This approach offers a framework in which every step of the analysis can be tested rather than
assumed. Its correct implementation would insure more reliability to the model than many of the
studies found in the literature which are imposing restrictions based on theorical a priori only.

In order to produce more acceptable results, some issues have to receive careful attention. A
good analysis should try to assess the impact of as many variables as possible to ensure that the
definition of the fundamentals used is complete. In addition to this, many other issues have to be
considered: the stability of the coefficients, the possible existence of non-modeled event(s)
affecting the fundamental relation(s), and the choice of the sampling period. The modeler should
carry out a sensitivity analysis of the impact of the different choices made upon the conclusions


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of the model. Transparency in the construction of the model is necessary in order to give some
reliability to the conclusions from the model. The estimation of a relation relating a non-
observed variables such as the equilibrium prices to its fundamentals is a difficult exercise.

Two of the recommendations from the Dahlem report (2009) are that economists should
communicate the limitations of their models and that the methodology used to construct models
should be more data-driven. This paper illustrates these recommendations.

This methodology is also subject to two important limitations which point toward new directions
of research. It doesn’t model the expectations process which is of main importance to assess the
existence of speculation, some expectations mechanism could be integrated in a Dynamic
Stochastic General Equilibrium model. And this methodology use of macroeconomics
datawhereas the housing market features important heterogeneities due to geographic
differentiation. In order to be able to identify local bubbles which may not be identifiable to the
aggregate level one could consider a dynamic panel data approach.


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ECONOMETRIC SOFTWARE
Estimation of the Cointegrated VAR model has been realized using WINRATS 6.2/CATS
version 2 (Dennis et al. 2005.)

Estimation of the ECM has been done using PcGive version 12 and Matlab 7.8.


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APPENDICES

A- Data
User cost is often found to be negative; this implies that it is not possible to do a log-transformation of the
variables. We avoid this problem by using the 30 years mortgage rate as interest rate. The value of this
interest rate is sensibly higher but its fluctuations are very close to the fluctuations are quite similar, and it
doesn’t impact the results. We construct the user cost as being the the nominal interest rate and set the
quarterly depreciation rate at one percent. The expected house price expectations is approximated using a
12 quarter Moving average of past prices inflation.The user cost has been constructed using the mortgage
rate as the nominal interest rate and set the quarterly depreciation rate to 1% and we approximate house
price expectations with a 12 quarter moving average of past house price inflation rates.

The housing stock has been constructed using the perpetual inventory equation. The quarterly
depreciation rate chosen is of 0.0625%, and the initial value of the housing stock has been chosen such
that the variable constructed was able to reproduce some features of the housing stock observed in the
annual data. The housing prices index is the OFHEO index, this is the most commonly used in the
literature, and is used in M&P too.

The data are quarterly and obtained from the BEA, they are seasonally adjusted. Investment housing stock
and consumption are real and per capita. Prices index are transformed taking log.


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B- STATISTICAL RESULTS

RESIDUAL ANALYSIS:

Tests for Autocorrelation
Ljung-Box(19): (430) = 969.273 [0.000]
LM(1): (7) = 41.732 [0.019]
LM(2): (7) = 38.263 [0.044]
Test for Normality: (10) = 10.554 [0.393]

Test for ARCH:
LM(1): (225) = 255.867 [0.077]
LM(2): (450) = 446.981 [0.531]

Mean Std.Dev Skewness Kurtosis Maximum Minimum
∆ ln -0.000 0.004 0.271 3.045 0.013 -0.008
∆ ln -0.000 0.002 0.145 2.886 0.006 -0.006
∆ ln -0.000 0.017 0.077 3.130 0.044 -0.038
∆ ln 0.000 0.001 0.547 3.782 0.004 -0.004
∆ ln -0.000 0.001 0.052 3.046 0.003 -0.003

ARCH(2) Normality R-Squared
∆ ln 1.792 [0.408] 1.208 [0.547] 0.601
∆ ln 1.468 [0.480] 0.380 [0.827] 0.862
∆ ln 0.220 [0.896] 0.819 [0.664] 0.827
∆ ln 0.486 [0.784] 4.706 [0.095] 0.914
∆ ln 6.672 [0.036] 0.521 [0.771] 0.996

Residual correlation coefficients and standard error
Correlation ∆pHouse ∆ ∆ ∆ ∆
∆ 1.000

∆ -0.365 1.000

∆ -0.231 -0.027 1.000

∆ -0.149 0.325 0.081 1.000

∆ -0.347 0.226 0.714 0.448 1.000

Standard error 0.0038690 0.0021339 0.0169321 0.0013858 0.0010940


Assessing the Existence of a Housing Bubble Using an Error Correction Model

Assessing the Existence of a Housing Bubble Using an Error Correction Model

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