Predictability of Downturns in Housing Markets : A Complex Systems Approach

1. Predictability of Downturns in Housing Markets:A Complex Systems Approach Maximilian Brauers Wiesbaden, 15 June 2012

2. Inflation-Adjusted Regional US House Price Indices Introduction Is it possible to predict crashes in housing markets such as in 2007?

3. 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

4. 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. ofTheoryandappliedFinancerejected 2nd round RFS, Sornette & Zhou (2006), International Journal of Forecast (With a review on the link betweenherdingandstatisticalphysicsmodels). • Further independent tests on Stock Markets: : Vandewalle et al., (1998) PhysicaB; 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

5. Exogen Introduction Downturns Endogen Foto-Fläche

6. Theoretical Background 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!)

7. 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 DISORDERAND ORDER

8. 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 ⟶Kc⟹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

9. Theoretical Background • The Hazard Rate of a Crash: • In the simplest scenario for aggregate K evolves linearly with time. Assuming that each • trader has four neighbors arranged in a regular two dimensional grid, then the • susceptibility of the system near the critical value, Kc, can be shown to be given • by 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.

10. Theoretical Background • The Hazard Rate of a Crash dependent on t: • Instead of *Kcwe take tc≡ the most probable time of the crash: • withα ≡ (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

11. 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

12. Theoretical Background • Synopsis / Intuitive summary of the Model: • Mathematical Formalization: • Power Law Function • Hierarchical Structures • Discrete Scale Invariance • Economic Theory: • Rational Expectations • No Arbitrage Condition • Herding Behavior • Positive Feedback 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.

13. Methodology • Objective function: • Slave linear parameters in OLS first order condition ⟹ reduces 7 parameter fit to only 4 • Choose tc and ω and let them run on a grid ⟹ leaves only 2 parameters left for estimation • Estimate z and Φ with an algorithm ⟹ avoiding local minima

14. Methodology • Behavior of Objective Function:

15. Methodology • Original Parameter Restrictions

16. Methodology • 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 stationarityin 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.

17. 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

18. Data Housing Market Crashes in US CensusDivisions

19. Results • Windows of Best Fits

20. Results • Out-of-Sample Predictions of Downturns

21. Results