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

2/26Research Department Complex Methods

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

3/26Research Department Complex Methods

A Way to think about Complexity

1. Critical Transitions

2. Heterogeneous Agents Models (H.A.Ms)

3. Networks

4/26Research Department Complex Methods

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

5/26Research Department Complex Methods

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

6/26Research Department Complex Methods

SP500: 1987 Crash

7/26Research Department Complex Methods

Catastrophic Bifurcations

8/26Research Department Complex Methods

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

9/26Research Department Complex Methods

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

10/26Research Department Complex Methods

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

11/26Research Department Complex Methods

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

12/26Research Department Complex Methods

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

13/26Research Department Complex Methods

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

14/26Research Department Complex Methods

The US housing market

Actual and estimated fundamental prices

Their difference (in logs)

15/26Research Department Complex Methods

Parameter Estimates - US

Y

Y

16/26Research Department Complex Methods

Estimated time-dependent fractions -US

17/26Research Department Complex Methods

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

18/26Research Department Complex Methods

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

19/26Research Department Complex Methods

Bifurcation Results – US: Y

Pitchfork bifurcation

20/26Research Department Complex Methods

The NL housing market

Actual and estimated fundamental prices

Their difference (in logs)

21/26Research Department Complex Methods

Parameter Estimates - NL

Y

Y Y

22/26Research Department Complex Methods

Estimated time-dependent fractions -NL

23/26Research Department Complex Methods

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

26/26Research Department Complex Methods

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