1/26Research Department Complex Methods
Wilko BoltDe Nederlandsche Bank
Maria DemertzisDe Nederlandsche Bank
Cees Diks University of Amsterdam, CeNDEF
Marco van der LeijUniversity of Amsterdam, CeNDEF
November 2011
Complex Methods in EconomicsAn Example of Behavioural Heterogeneity in House Prices
Complex Methods in EconomicsAn Example of Behavioural Heterogeneity in House Prices
Research Department
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Why Complexity?
Our current understanding of the world The concept of equilibrium is given and
unique All deviations are small and temporary Everybody is the same Little to no interaction between
agentsWhole=sum of its parts
The world is more complex
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A Way to think about Complexity
1. Critical Transitions
2. Heterogeneous Agents Models (H.A.Ms)
3. Networks
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Example of Critical Transition: Birth of the Sahara
relatively moist area until about 6,000 years ago gradual change in solar radiation, due to subtle variation in Earth’s orbit abrupt shift in climate and vegetation cover over Saharah
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Detecting Critical Transitions
Scheffer, Bascompte, Brock et al (Nature, Sept. 2009) Slow recovery from perturbations Memory of the system increases
Moving window estimation
Critical slowdown prior to regime shift characterised by
Increasing variance, and Increasing autocorrelation
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SP500: 1987 Crash
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Catastrophic Bifurcations
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The housing market
1rf
t t t t t t t t tH Pr P Pq P
Imputed rents
1 11 1 1 1 1
,
1
rft t t t t t t t t t
rft tt t t t t
t
Q P r Pq P R
P QR r r r
P
Actual rents -returns on housing
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Heterogeneous beliefs and the housing market (1)
, 1 , , 1 ,h t t h t h t t h tE R z aVar R z
Agent’s demand zh,t determined by maximising risk-adjusted expected future excess returns, Rt+1zh,t:
This gives the demand for agent h:
, 1 1,
/ 1h t t t th t
E P Q P rz
aV
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Heterogeneous beliefs and the housing market (2)
Aggregation over 2 types of agents, market clearing
2
, , 1 1
1 2 11
/ 1, , 1h t h t t t t
h
n E P Q P rS n n n
aV
Leads to the price equation:
2
, , 1 11
1 t h t h t t th
r P n E P Q
aVS where
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Heterogeneous beliefs and the housing market (3)
1 11
1t t t t
t t
r P E P Q
gP Q
r g
Under rational expectations on the first conditional moment:
2
, , 11
1 1
1t h t h t th
rX n E X
g
Define Xt as the ratio of price to fundamental price:1t
t
P
P
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Beliefs: Two types of agents
Beliefs: 1, 1 1 1 2, 1 2 1 1 2, ,t t t t t tE X X E X X
Performance (realised profits):
, 1 1 2 , 2
1 2 1 3 2.
h t t t h t
t t t t
X X z
cnst X X X X
Fractions determined by logistic switching model:
1, 1
1, 1 2, 1
1 2 1 2 3
1,
2, 1,
1
11
t
t t
t t t
t
X X X
t t
en
e e
en n
Estimation via nonlinear OLS
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Stability condition: a simulated exampleThe =0 dynamics is locally stable if:
1 2 12
We assume =1.05, =500
1 20.94, 1.14 1 20.96, 1.16
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The US housing market
Actual and estimated fundamental prices
Their difference (in logs)
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Parameter Estimates - US
Y
Y
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Estimated time-dependent fractions -US
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Fancharts US: House price deviationsUSA - BHM insample forecasts
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
1970
.25
1972
.25
1974
.25
1976
.25
1978
.25
1980
.25
1982
.25
1984
.25
1986
.25
1988
.25
1990
.25
1992
.25
1994
.25
1996
.25
1998
.25
2000
.25
2002
.25
2004
.25
2006
.25
2008
.25
2010
.25
X 95% Confidence Bands 85% Confidence Bands median forecast
USA -AR in sample forecasts
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
1970
.25
1971
.50
1972
.75
1974
.00
1975
.25
1976
.50
1977
.75
1979
.00
1980
.25
1981
.50
1982
.75
1984
.00
1985
.25
1986
.50
1987
.75
1989
.00
1990
.25
1991
.50
1992
.75
1994
.00
1995
.25
1996
.50
1997
.75
1999
.00
2000
.25
2001
.50
2002
.75
2004
.00
2005
.25
2006
.50
2007
.75
2009
.00
2010
.25
X 95% Confidence Bands 85% Confidence Bands median forecast
USA- BHM out of sample forecasts
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
1970
.25
1971
.50
1972
.75
1974
.00
1975
.25
1976
.50
1977
.75
1979
.00
1980
.25
1981
.50
1982
.75
1984
.00
1985
.25
1986
.50
1987
.75
1989
.00
1990
.25
1991
.50
1992
.75
1994
.00
1995
.25
1996
.50
1997
.75
1999
.00
2000
.25
2001
.50
2002
.75
2004
.00
2005
.25
2006
.50
2007
.75
2009
.00
2010
.25
2011
.50
2012
.75
2014
.00
2015
.25
X
X 95% Confidence Bands 85% Confidence Bands median forecast
USA - AR out of sample
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
1970
.25
1971
.50
1972
.75
1974
.00
1975
.25
1976
.50
1977
.75
1979
.00
1980
.25
1981
.50
1982
.75
1984
.00
1985
.25
1986
.50
1987
.75
1989
.00
1990
.25
1991
.50
1992
.75
1994
.00
1995
.25
1996
.50
1997
.75
1999
.00
2000
.25
2001
.50
2002
.75
2004
.00
2005
.25
2006
.50
2007
.75
2009
.00
2010
.25
2011
.50
2012
.75
2014
.00
2015
.25
X
X 95% Confidence Bands 85% Confidence Bands median forecast
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Bifurcation Results – US:
Bifurcation diagram (with and without noise)
Slowly varying can induce critical transitions But noise overwhelms the dynamics (early transitions and/or repetitive jumps between two stochastic attractors
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Bifurcation Results – US: Y
Pitchfork bifurcation
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The NL housing market
Actual and estimated fundamental prices
Their difference (in logs)
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Parameter Estimates - NL
Y
Y Y
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Estimated time-dependent fractions -NL
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Fancharts NL: House price deviations
NL - BHM insample forecasts
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1970
.25
1972
.25
1974
.25
1976
.25
1978
.25
1980
.25
1982
.25
1984
.25
1986
.25
1988
.25
1990
.25
1992
.25
1994
.25
1996
.25
1998
.25
2000
.25
2002
.25
2004
.25
2006
.25
2008
.25
2010
.25
X 95% Confidence Bands 85% Confidence Bands median forecast
NL- BHM out of sample forecasts
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1970
.25
1971
.50
1972
.75
1974
.00
1975
.25
1976
.50
1977
.75
1979
.00
1980
.25
1981
.50
1982
.75
1984
.00
1985
.25
1986
.50
1987
.75
1989
.00
1990
.25
1991
.50
1992
.75
1994
.00
1995
.25
1996
.50
1997
.75
1999
.00
2000
.25
2001
.50
2002
.75
2004
.00
2005
.25
2006
.50
2007
.75
2009
.00
2010
.25
2011
.50
2012
.75
2014
.00
2015
.25
X
X 95% Confidence Bands 85% Confidence Bands median forecast
NL - AR insample
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1970
.25
1971
.50
1972
.75
1974
.00
1975
.25
1976
.50
1977
.75
1979
.00
1980
.25
1981
.50
1982
.75
1984
.00
1985
.25
1986
.50
1987
.75
1989
.00
1990
.25
1991
.50
1992
.75
1994
.00
1995
.25
1996
.50
1997
.75
1999
.00
2000
.25
2001
.50
2002
.75
2004
.00
2005
.25
2006
.50
2007
.75
2009
.00
2010
.25
X 95% Confidence Bands 85% Confidence Bands median forecast
NL - AR out of sample
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1970
.25
1971
.50
1972
.75
1974
.00
1975
.25
1976
.50
1977
.75
1979
.00
1980
.25
1981
.50
1982
.75
1984
.00
1985
.25
1986
.50
1987
.75
1989
.00
1990
.25
1991
.50
1992
.75
1994
.00
1995
.25
1996
.50
1997
.75
1999
.00
2000
.25
2001
.50
2002
.75
2004
.00
2005
.25
2006
.50
2007
.75
2009
.00
2010
.25
2011
.50
2012
.75
2014
.00
2015
.25
X
X 95% Confidence Bands 85% Confidence Bands median forecast
24/26Research Department Complex Methods
Bifurcation Results – NL:
25/26Research Department Complex Methods
Bifurcation Results – NL: Y
Value of fixed at 1.01Pitchfork bifurcation
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What have we learned?
Univariate model Data justifies multiplicity of equilibria (estimated) Visualisation of herding Models predict very different
Multivariate model Use institutional factors to estimate fundamental
prices Multiple countries