Predictability of Downturns in Housing Markets: A Complex Systems Approach

Maximilian BrauersWiesbaden, 15 June 2012

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Inflation-Adjusted Regional US House Price Indices

Introduction

Is it possible to predict crashes in housing markets such as in 2007?

1991.0000001991.0833331991.1666671991.2500001991.3333331991.4166671991.5000001991.5833331991.6666671991.7500001991.8333331991.9166671992.0000001992.0833331992.1666671992.2500001992.3333331992.4166671992.5000001992.5833331992.6666671992.7500001992.8333331992.9166671993.0000001993.0833331993.1666671993.2500001993.3333331993.4166671993.5000001993.5833331993.6666671993.7500001993.8333331993.9166671994.0000001994.0833331994.1666671994.2500001994.3333331994.4166671994.5000001994.5833331994.6666671994.7500001994.8333331994.9166671995.0000001995.0833331995.1666671995.2500001995.3333331995.4166671995.5000001995.5833331995.6666671995.7500001995.8333331995.9166671996.0000001996.0833331996.1666671996.2500001996.3333331996.4166671996.5000001996.5833331996.6666671996.7500001996.8333331996.9166671997.0000001997.0833331997.1666671997.2500001997.3333331997.4166671997.5000001997.5833331997.6666671997.7500001997.8333331997.9166671998.0000001998.0833331998.1666671998.2500001998.3333331998.4166671998.5000001998.5833331998.6666671998.7500001998.8333331998.9166671999.0000001999.0833331999.1666671999.2500001999.3333331999.4166671999.5000001999.5833331999.6666671999.7500001999.8333331999.9166672000.0000002000.0833332000.1666672000.2500002000.3333332000.4166672000.5000002000.5833332000.6666672000.7500002000.8333332000.9166672001.0000002001.0833332001.1666672001.2500002001.3333332001.4166672001.5000002001.5833332001.6666672001.7500002001.8333332001.9166672002.0000002002.0833332002.1666672002.2500002002.3333332002.4166672002.5000002002.5833332002.6666672002.7500002002.8333332002.9166672003.0000002003.0833332003.1666672003.2500002003.3333332003.4166672003.5000002003.5833332003.6666672003.7500002003.8333332003.9166672004.0000002004.0833332004.1666672004.2500002004.3333332004.4166672004.5000002004.5833332004.6666672004.7500002004.8333332004.9166672005.0000002005.0833332005.1666672005.2500002005.3333332005.4166672005.5000002005.5833332005.6666672005.7500002005.8333332005.9166672006.0000002006.0833332006.1666672006.2500002006.3333332006.4166672006.5000002006.5833332006.6666672006.7500002006.8333332006.9166672007.0000002007.0833332007.1666672007.2500002007.3333332007.4166672007.5000002007.5833332007.6666672007.7500002007.8333332007.9166672008.0000002008.0833332008.1666672008.2500002008.3333332008.4166672008.5000002008.5833332008.6666672008.7500002008.8333332008.9166672009.0000002009.0833332009.1666672009.2500002009.3333332009.4166672009.5000002009.5833332009.6666672009.7500002009.8333332009.9166672010.0000002010.0833332010.1666672010.2500002010.3333332010.4166672010.5000002010.5833332010.6666672010.7500002010.8333332010.9166672011.0000002011.0833332011.1666672011.2500000

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East North Central East South Central

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HPI

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Is it possible to predict crashes in housing markets such as in 2007?

A model originating in the field of statistical physics is claiming to be able to predict such downturns in financial markets (Johansen and Sornette, 2010).

We test the model’s validity, predictive power and its success rate on 20 years of housing price data for nine regional sub-markets of the U.S. housing market. We propose a new model restriction to remedy estimation issues due to the low frequency of housing price data. This restriction constitutes a new test for exponential price growth against a power law growth in low frequency datasets.

Introduction

First proposal: for a connection between crashes in FTS and critical points was made by Sornette et al. (1996) applied to Oct. 1987 Crash in Physika France.

Confirmed independently: by Feigenbaum & Freund (1996) for 1929/1987 on arXiv.org (Cornell University), Sornette & Johansen (1997) Physika A, Johansen Sornette, (1999) Risk, Johansen, Ledoit, Sornette, (2000) Int J. of Theory and applied Finance rejected 2nd round RFS, Sornette & Zhou (2006), International Journal of Forecast (With a review on the link between herding and statistical physics models).

Further independent tests on Stock Markets: : Vandewalle et al., (1998) Physica B; Feigenbaum and Freund, (1998), Intern. J. of Modern Ph.; Gluzman and Yukalov (1998) arXiv.org; Laloux et al., 1998, Euro Physics Letter; Bree (2010), DP.

Noise and Estimation Issues: Feigenbaum (2001) QF; Bothmer et al. (2003) Physika A; Chang & Feigenbaum (2006) QF; Lin et al. (2009) WPS; Gazola et al. (2008) The European Physical Journal B, (Computational Issues: Liberatori, 2010 QF).

Recent Summary on state of the art: Zhi, Sornette et al 2009 QF auf arXIV.org

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Introduction

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Foto-Fläche

Exogen Endogen

Downturns

Introduction

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Rational Endogenous Bubble

C(t) is not deterministic but following stochastic path.

⟹ The movement can be captured by the Log Periodic Power Law Model.(Which itself is a deterministic function describing a hazard rate!)

Theoretical Background

Derivation of c(t):Micro-level:The general opinion in a network of single traders is given by: The optimal choice is given by:

ε N(0; 1) ≡ idiosyncratic signal∼σ ≡ tendency towards idiosyncratic behavior K ≡ measure for the strength of imitation. s ≡ selling (-1) or buying (+1)

⟹ THE FIGHT BETWWEEN DISORDER AND ORDER

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𝑠𝑖 (𝑡+1 )=𝑠𝑖𝑔𝑛(∑𝑗=1

𝑛∈𝑵𝐾 𝑗 𝑠 𝑗+𝜎 𝜀𝑖)

Theoretical Background

Theoretical Background

Macro-Level:

K ≡ strength of imitation, average over Ki’s

Kc ≡ critical size of K, i.e. point with highest probability for a crash

K < Kc ⟹ disorder rules on the market, i.e. agents do not agree with each other

K K⟶ c ⟹ order starts appearing, i.e. agents do agree with each other. At this point Kc, the system is extremely unstable and sensitive to any new

information.

⟹ THE HAZARD RATE OF A CRASH DEPENDS ON K

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Theoretical Background

The Hazard Rate of a Crash:In the simplest scenario for aggregate K evolves linearly with time. Assuming that

eachtrader has four neighbors arranged in a regular two dimensional grid, then thesusceptibility of the system near the critical value, Kc, can be shown to be givenby the approximation:

⟹ The Crash Hazard Rate quantifies the probability that a large group of investors sells/buys simultaneously. The mechanism for this lies in imitation behavior, herding and (un)willing cooperation in networks (friends, colleagues, family). When order wins, the bubble ends, the crash happens.

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Theoretical Background

The Hazard Rate of a Crash dependent on t:

Instead of *Kc we take tc ≡ the most probable time of the crash:

with α ≡ (ε-1)-1 , with ε ≡ number of traders in one network.

The Way the hazard rate evolves depends on the assumed market structure represented by a lattice.

Hierarchies in a market are described by a diamond structure. The solution for the hazard rate is than extended by log periodicity:

*We cannot observe K directly at least in real time, maybe one could argue that ‘s been done by Case & Shiller (2003) for the US-Housing Market

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Theoretical Background

The Log Periodic Power Law Model:

⟹ Back in (1) and simplifying, gives for the price:

A,B,C ≡ linear parameters, with

≡ exponent of power law (some say up to 2, but no super-exponential growth?)

≡ angular frequency of oscillations ≡ phase constant

≡ critical point in time, i.e. most probable time of crash

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Theoretical Background

Mathematical Formalization:

• Power Law Function• Hierarchical Structures• Discrete Scale Invariance

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Economic Theory:

• Rational Expectations• No Arbitrage Condition

• Herding Behavior• Positive Feedback

Synopsis / Intuitive summary of the Model:

Imitation leads to herding and herding to positive feedback. The positive feedback can be detected as super-exponential growth in the price, i.e. a self reinforcing trend.

This trend is corrected by log periodic oscillations due to the market structure. i.e. the way and strength of individual participants on each other.

Methodology

Objective function:

1. Slave linear parameters in OLS first order condition reduces 7 parameter fit to ⟹only 4

2. Choose tc and ω and let them run on a grid leaves only 2 parameters left for ⟹estimation

3. Estimate z and Φ with an algorithm avoiding local minima⟹

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Methodology

Behavior of Objective Function:

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Methodology

Original Parameter Restrictions

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Parameter Range Reason for Restriction

𝐵< 0 Theoretical restriction that ensures a positive growth rate for bullish bubbles

0 < ȁ�𝐵ȁ�− ȁ�𝐶ȁ� Theoretical restriction for a positive hazard rate

0 < 𝑧< 1 Theoretical restriction that ensures power law growth, reflecting a self-enforcing trend

RMSE< 0.008 Empirical implication for an acceptable goodness of fit

New Restriction

• Our restriction constitutes a null hypothesis for testing against the LPPL model in low frequency price series.

• We take the first log differences in the house price series and test with the KPSS test without trend for stationarity.

• If we cannot reject stationarity in the first log differences, we cannot reject the null hypothesis of exponential growth in the price trajectory and, therefore, we must reject the power law and LPPL fit.

• We choose the KPSS test as it offers the highest power for testing against I(0) within a time series.

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Methodology

Data

HPI provided by the Federal Housing Agency:

• Weighted repeat sales index in order to qualify as constant quality index. Purchase only.

• Obtained from repeat mortgage transaction securitized by Fannie Mae or Freddie Mac since January 1975

• As of December 1995 there were over 6.9 million repeat transactions in the national sample

• Inflation-adjusted

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Data

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Housing Market Crashes in US Census Divisions

Results

Windows of Best Fits

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Panel A: Three Crash Census Divisions Pacific Mountain South Atlantic Crash size - 36.4 % - 29.89 % - 21.57 %

Turning point 2/2006 10/2006 02/2007

Estimated turning point 4/2006 11/2006 3/2007

Starting time of best fit 2/2000 4/1998 10/2001

RMSE 0.0070 0.0067 0.0066

Results

Out-of-Sample Predictions of Downturns

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Census division 𝑡𝑐 𝑡𝑐𝑑𝑖𝑠𝑡തതതതതത 𝜔ഥ zത Pacific 2/2006 -0.143

(0.185) 2.129

(0.198) 0.926

(0.106)

Mountain 10/2006 -0.278 (0.146)

2.093 (0.12)

0.862 (0.105)

South Atlantic 2/2007 -0.021 (0.086)

2.083 (0.133)

0.369 (0.491)

Results

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Success Rate of Crash Predictions Using the Additional Restriction Before

Crash Not before Crash

Total Success Rate

Crash Prediction 23 14 37 62.16 %

No Crash Prediction 25 424 449

Success Rate of Crash Predictions Using the Standard Restrictions Before

Crash Not before Crash

Total Success Rate

Crash Prediction 26 146 172 15.11 %

No Crash Prediction 22 292 314