## Electronic Supplementary Material

Article Title: Are there rational bubbles in REITs? New evidence from a complex systems approach

Journal Name: Journal of Real Estate Finance and Economics

Author Names: Maximilian Brauers, Matthias Thomas, Joachim Zietz

**Corresponding Author**: Maximilian Brauers, EBS Universität für Wirtschaft und Recht, Germany; email:

**Micro-level derivation of the hazard rate**

Market participants influence each other from one period to another in their investment decisions. The decision of one agent to buy or sell (+ or-) within the next period depends on the observable behavior of the agents within her network and an idiosyncratic signal she alone receives,

sit+1=sign Ki j=1nsj+ σiεi, / (1)where sit+1 is the decision of an agent whether to buy, hold or sell within the next period described as a signum function; Ki is the coupling strength between traders of a network and can be heterogeneous across pairs of neighbors; n is the set of traders who influence traderi. Sj is the current state of trader j; σi is the agent’s tendency towards idiosyncratic behavior; εi~N0,1 is an individually received signal.

The purpose of this equation is to formalize the decision process of an individual investor. In non-bubble regimes disorder rules, i.e. agents disagree on whether to buy or sell. Prices lead to market clearance by matching demand and supply. But as Zhou and Anderson (2011) show, herding in REIT markets is time varying. When herding and imitation is on the rise, the average of the Ki (K) rises towards a critical level Kc and order appears in the market. The system becomes increasingly susceptible to react in a coordinated way to any new signal. It is increasingly likely that many market participants make the same decision, e.g. they sell and thus trigger a coordinated sell-off. The susceptibility of the system and thus the hazard rate of an endogenous collapse grows as K reaches its critical level Kc.

In the simplest scenario, all agents (si) have the same number of neighbors in their network and all networks have the same size, i.e. all agents have the same influence on the market. Then the hazard rate for an endogenous collapse is given by Onsager’s solution (Onsager 1944) for the two dimensional Ising model (Ising 1925) of phase transitions:

hK≈B'Kc-K-∝'. / (2)The hazard rate of an endogenous collapse depends on the evolution of the imitation strength (K) over time. The evolution of K cannot be observed directly in real time, but assuming that K evolves linearly over time, the hazard rate can be expressed as a function of time (t) instead of K. Thus, the critical point (tc), i.e. the most likely point in time for a collapse, can be estimated

Kc-K≈c*tc-t,where c is a constant, / (3)

ht=Btc-t-∝, / (4)

where t denotes time and where tc is the critical time, i.e. the most likely time for a collapse. The exponent α depends inversely on the average number of traders within a network, α=ε-1-1. Therefore, the exponent must be between 0 and 1.

A hierarchical model of agents in networks reflects their unequal market power and abandons the unrealistic assumption that all agents have the same influence on each other. Based on the solution of Derrida et al. (1983) for a hierarchical model, we can describe the hazard rate under the more realistic assumption of hierarchical structures in asset markets by letting the power law exponent be a complex exponent. The solution for markets with hierarchical structures, i.e. differing influence of agents on each other and thus on the market, is given by a power law augmented by log-periodic oscillations,

ht= B0'tc-t-γ+B1''tc-t-γcosωlntc-t+ψ. / (5)As derived in section 2, the solution for the logarithm of the price trajectory during a rational bubble regime is given by

lnptn=lnpt0+κt0tnhtdt. / (6)Plugging (5) into (6), as derived in section 2, leads to the equation for the price trajectory as a function of time before an endogenous collapse,

lnpt=lnpt0+κβB0'tc-tβ+B1''tc-tβcosωlntc-t+ψ, / (7)with β=1-γ. Simplifying (7) gives the log-periodic power-law (LPPL) function of the price trajectory as a function of time t before an endogenous crash as provided in section 2,

lnp(t)=A+Btc-tβ+Ctc-tβcosωlntc-t+Φ. / (8)The log-periodicity of the LPPL model is captured by the third term. A function is defined as being log-periodic when it is periodic in the log scale of its input variable. The input variable of the LPPL model is time t, or more precisely tc-t, i.e., the time from the beginning of the buildup of the instability phase or bubble until its critical point or crash (Sornette et al. 1996).

To examine the meaning of ω w split equation (8) into its two component parts, the power law and the log-periodicity. The log-periodicity can be regarded as a complex exponent. This term describes the real part of a complex number in its trigonometric representation. For a detailed explanation of the link between log-periodicity and complex exponents, see Sornette (1998) and Newman et al. (1995). Here, we take a closer look at the cosine function of the third term of the LPPL model:

cosωlntc-t+Φ=r. / (9)Recall that the cosine function is a periodic function. Its radian output r lies between -1 and 1. We select an arbitrary value, which we call λ for now, for the observable input tc-t that maximizes the function such that r=1. We further reduce the problem by omitting the phase constant Φ[1]. The resulting equation is given as:

cosωlnλ=1. / (10)Rearranging (10) one obtains:

ωlnλ=2π / (11)This relation results in an identity for λ and ω:

λ≡e2π/ω / (12)Because we chose λ such that r equals 1, any integer multiple of λ will result in r=1 for the log scale of tc-t. This relation translates into a geometric series on the linear scale. Any integer power of λ (λp) will lead to a local maximum, i.e., r=1.[2] Therefore, λ is the preferred scaling factor. The log-periodicity of the model reflects the existence of a preferred scaling factor in the price trajectory during a bubble regime on a linear scale.

This fact alone might not be surprising because the micro-level derivation of the model is based on the solution of a hierarchical network that exhibits a preferred scaling factor. The preferred scaling factor is essentially a symmetry property of the underlying assumed hierarchical structure. The preferred scaling factor directly reflects how one hierarchical level is linked to another (Johansen et al. 1999). The LPPL model assumes the existence of a hierarchical structure that reflects a built-in market organization (Vanderwalle et al. 1999). The organizational market structure accounts for the differing market powers of the individual participants, e.g., institutional investors have a higher market power than private investors. This phenomenon is depicted in hierarchical lattices, such as the diamond lattice (Appendix Figure 1). The more influential market participants have more neighbors, whereas private investors are depicted as participants with a few neighbors, because they only have a limited influence.

Appendix Fig. 1 Diamond lattice reflecting hierarchical networksNotes: Magnifications that multiply the number of links by any power of 4 (4n) leave the lattice invariant. The system is only self-similar under discrete scale invariance. Here, the preferred scaling factor is λ=4.

The initial LPPL model is derived from the solution of Derrida et al. (1983) for the diamond lattice (Appendix Figure 1). However, as shown in equation (12), the estimated parameter ω measures the value of the preferred scaling factor λ. Therefore, we can derive empirical values of the preferred scaling factor λ from the estimates of ω, i.e. ω reflects the degree of hierarchy in the underlying market structure.

References

Derrida, B., De Seze, L., & Itzykson, C. (1983). Fractal structure of zeros in hierarchical models. Journal of Statistical Physics, 33(3), 559-569.

Ising, E. (1925). Beitrag zur Theorie des Ferromagnetismus. *Zeitschrift für Physik (Germany), 31*(1), 253-258.

Johansen, A., Sornette, D., & Ledoit, O. (1999). Predicting financial crashes using discrete scale invariance. Journal of Risk, 1, 5-32.

Newman, W., Turcotte, D., & Gabrielov, A. (1995). Log-periodic behavior of a hierarchical failure model with applications to precursory seismic activation. Physical Review E, 52, 4827–4835.

Onsager, L. (1944). Crystal statistics. I. A two-dimensional model with an order-disorder transition. Physical Review, 65, 117-149.

Sornette, D., Johansen, A., & Bouchard, J.-P. (1996). Stock market crashes, precursors and replicas. Journal Physica I (France), 6, 167-175.

Sornette, D. (1998). Discrete scale invariance and complex dimensions. Physics Reports, 297(5), 239-270.

Vanderwalle, N., Ausloos, M., Boveroux, P., & Minguet, A. (1999). Visualizing the log-periodic pattern before crashes. The European Physical Journal B, 9, 355-359.

Zhou, J., & Anderson, R. I. (2011). An empirical investigation of herding behavior in the U.S. REIT market. *Journal of Real Estate Finance and Economics*, doi: 10.1007/s11146-011-9352-x.

1

[1] The phase constant controls for units when the LPPL model is estimated with empirical data. The meaning of the phase constants lies in the fact that they make the estimated parameter values comparable over different time series. This is important for our purposes when we compare the findings over different samples.

[2] The same holds true if we chose λ so as to minimize the cosine function.