Short-run momentum, long-run mean reversion and excess volatility: An elementary housing model

Economics Letters 176 (2019) 43–46

Contents lists available at ScienceDirect

Economics Letters journal homepage: www.elsevier.com/locate/ecolet

Short-run momentum, long-run mean reversion and excess volatility: An elementary housing model ∗

Noemi Schmitt , Frank Westerhoff University of Bamberg, Department of Economics, Feldkirchenstrasse 21, 96045 Bamberg, Germany

article

info

Article history: Received 16 November 2018 Accepted 8 December 2018 Available online 19 December 2018

a b s t r a c t We propose an elementary housing model that replicates the key properties of housing bubbles, namely short-run momentum, long-run mean reversion, and excess volatility. We analytically proof that such dynamics can only emerge if homebuyers place sufficient weight on extrapolative expectations. © 2018 Elsevier B.V. All rights reserved.

JEL classification: D84 G12 R21 Keywords: Housing markets Bubbles and crashes Extrapolative expectations

1. Introduction The U.S. housing bubble between 1996 and 2012 was extreme, but hardly unique. Glaeser and Nathanson (2015) show that history is replete with dramatic real estate cycles. They stress that extrapolative expectations, as reported by Case and Shiller (2003), are major drivers of house prices that require consideration to improve our understanding of housing dynamics. However, any model that deviates from the assumption of fully rational agents faces the wilderness of the bounded rationality problem. Glaeser (2013, p. 4) explicitly warns that ‘‘economic models will lose all discipline if they treat investors as blank slates that irrationally absorb any foolish notion that they hear’’, but also suggests that an ‘‘alternative approach is to assume that such beliefs are limited by sensible models of asset valuation’’. Similarly, Hommes (2013) argues that agents rely on a set of plausible rules to form expectations and select among them based on their past performance. We thus propose an elementary housing model to explore the impact of homebuyers’ expectation formation and their learning behavior regarding house price dynamics. The underlying housing model follows Poterba (1984), while homebuyers’ expectations are modeled according to Brock and Hommes (1997). Simulations reveal that our model’s boom–bust dynamics displays short-run momentum, long-run mean reversion, and excess volatility. Glaeser (2013) states that these are the key ingredients forming the typical shape of housing bubbles, and that they need to be better explained ∗ Corresponding author. E-mail address: [email protected] (N. Schmitt). https://doi.org/10.1016/j.econlet.2018.12.013 0165-1765/© 2018 Elsevier B.V. All rights reserved.

by housing models. Most importantly, we analytically demonstrate that such dynamics can only arise if homebuyers rely heavily on extrapolative expectations. 2. An elementary housing model We combine Poterba’s (1984) user cost framework, which includes explicit relations between house prices, the rent level and the housing stock, with the heuristic switching approach by Brock and Hommes (1997), which captures the formation of homebuyers’ expectations and their learning behavior. Poterba (1984) assumes that there is a single price for housing that is paid by all homebuyers. House price Pt , derived from an intertemporal no-arbitrage condition, equals the discounted value of the next period’s expected house price Et [Pt +1 ] plus (hypothetical) rent payments Rt , paid at the end of the period. Formally, this results in E [Pt +1 ]+Rt Pt = t 1+ , where r > 0 is the risk-free interest rate and 0 < r +δ δ < 1 represents the housing depreciation rate. The fundamental house price – the steady state at which expectations are realized ∗ – corresponds to the discounted value of future rents F = rR+δ , where r + δ reflects user costs of house ownership. See Glaeser and Nathanson (2015) for an appraisal of this framework. Following Brock and Hommes (1997), homebuyers select among different types of expectation rules, depending on the evolutionary fitness of these rules. For analytical tractability, we concentrate on two representative rules. Extrapolative expectations are formalized by EtE [Pt +1 ] = Pt −1 + c(Pt −1 − Pt −2 ), where c > 0 indicates homebuyers’ extrapolation strength. Regressive expectations are expressed by EtR [Pt +1 ] = Pt −1 + f (F − Pt −1 ),

44

N. Schmitt and F. Westerhoff / Economics Letters 176 (2019) 43–46

where 0 < f < 1 denotes homebuyers’ expected mean reversion speed. Forecasts are conditional on information available at period t − 1. The fitness of the rules depends on squared prediction errors. Hence, AEt = −g(Pt −1 − EtE−2 [Pt −1 ])2 and ARt = −g(Pt −1 − EtR−2 [Pt −1 ])2 − K , with g > 0. While extrapolative expectations are free, regressive expectations require the computation of F , incurring information costs K ≥ 0. The weights of extrapolative and regressive expectations follow the discrete choice expressions WtE =

Exp[hAEt ] Exp[hAEt ]+Exp[hAR t]

and WtR =

Exp[hAR t] Exp[hAEt ]+Exp[hAR t]

. Accordingly,

homebuyers tend to choose the rule with the higher fitness. Furthermore, as values of the intensity of choice parameter h ≥ 0 increase, more homebuyers opt for the rule that yields better predictions. Homebuyers’ average house price expectations are defined by Et [Pt +1 ] = WtE EtE [Pt +1 ] + WtR EtR [Pt +1 ]. Hommes (2013) provides an excellent review of this approach. Finally, we assume that the rent level depends negatively on the existing housing stock, i.e. Rt = α − β Ht , with α , β > 0, and that the housing stock evolves as Ht = It + (1 − δ )Ht −1 , where new housing construction depends positively on past house prices, i.e. It = γ Pt −1 , with γ > 0. New housing construction is consistent with assumptions that builders face a one-period production lag, maximize expected profits subject to a quadratic cost function Ct = 21γ It2 , and form naive expectations. Therefore, a lower value of γ implies higher building costs and a more sluggish housing supply. 3. Analytical and numerical results We explore our model through a combination of mathematical analysis and numerical simulations. The following proposition states our main analytical results: Proposition 1. The model’s unique steady state, given by P ∗ = γ F = βγ +δαδ(r +δ ) and H ∗ = δ P ∗ , is locally asymptotically stable iff

f <

(1−m ) δ (1 − m∗ ) + 2(1+2cr βγ < +δ )−c(1−m∗ ) R∗ E∗ = Wt − Wt = −Tanh[ hK ]. 2

4+2r −δ (r +δ )−βγ +c(1−m∗ )(2−δ ) and c (1+m∗ )(1−0.5δ ) 2(2δ+r) ∗ m∗ ) , where m 1−δ

∗

f (1 + + Moreover, violation of the first (second) inequality is associated with a flip (Neimark–Sacker) bifurcation.

Note that the fundamental house price does only depend on real parameters. For instance, a reduction in the interest rate causes an increase of F . Moreover, we are particularly interested in which conditions may contribute to endogenous oscillations.1 First of all, straightforward computations reveal that the Neimark–Sacker condition is always fulfilled for c = 0, representing a model with naive versus regressive expectations. Put differently, endogenous cycles can only occur if homebuyers form extrapolative expectations. Second, if either information costs K or the intensity of choice h are very high (low), implying that all (half of) homebuyers form extrapolative expectations, i.e. m∗ = −1 (m∗ = 0), then the Neimark–Sacker condition will be violated at the latest when c ≥ 1 + r + δ (c ≥ 2 + 2r + 2δ ), suggesting that an increase in K or h destabilizes the dynamics. To simulate our model, we assume α = 20, β = 0.1, γ = 0.05, δ = 0.05, r = 0.05, f = 0.75, g = 1, h = 5 and K = 0.8. Homebuyers’ extrapolation strength serves as a sensitivity parameter. Accordingly, the fundamental house price P ∗ = F = 100 undergoes a Neimark–Sacker bifurcation at c ≈ 1.0748. Such an extrapolation strength is not unrealistic: Case and Shiller (2003) report that homebuyers held very optimistic beliefs during the recent U.S. housing boom, predicting home prices to rise by 1 Economically, the flip bifurcation, generating a period-two cycle, is less interesting. However, for 0 < f < 1, this bifurcation may be observed if βγ becomes sufficiently large.

almost 15 percent per year over ten years. Anufriev and Hommes (2012) document extrapolation parameters of c = 1.3. In Panel (a) of Fig. 1, we set c = 0.95. Although the housing market is stable, an initial one percent shock causes temporary cycles with decreasing amplitude. Panel (b) is based on c = 1.1. Since the Neimark–Sacker bifurcation boundary has been crossed, the housing market displays endogenous fluctuations in the form of a limit cycle. Assuming c = 1.5, as in Panel (c), the model produces complex (chaotic) dynamics, characterized by short-run momentum, long-run mean reversion, and excess volatility. While the housing market shows limited fluctuations for some time, a major bubble emerges around period t = 60. According to Glaeser and Nathanson (2015), these empirical regularities are the defining properties of actual housing bubbles, and are difficult to explain using standard models. Panel (d) presents a bifurcation diagram in which house prices are plotted against homebuyers’ extrapolation strength. Clearly, a more aggressive extrapolative behavior is associated with more pronounced fluctuations.2 The model functions as follows. During an upswing, extrapolative expectations deliver better predictions than regressive expectations, gain significant weight, and add momentum to the bubble. When the bubble bursts, regressive expectations outperform extrapolative expectations, initiating a mean reversion period. As the market enters a stable period, prediction errors of both rules decrease. Since regressive expectations are costly, extrapolative expectations experience a revival, triggering the next bubble. What causes the bubble to burst? Here, Glaeser and Nathanson (2015) stress the role of the supply side. Within our model, higher house prices induce more housing construction; a higher housing stock depresses rent levels, making the housing market less attractive and forcing the housing bubble to collapse. Glaeser and Nathanson (2015) conclude that ignoring the impact of the supply side of the housing market is a pervasive error made by homebuyers. Nevertheless, homebuyers are not irrational. Glaeser (2013, p. 38) remarks that, during the recent U.S. housing boom, homebuyers’ ‘‘optimistic expectations have been justifiable based on recent experience’’. It is exactly this kind of learning behavior that drives the dynamics of our model. 4. Conclusions Housing markets exhibit large booms and busts, often with dire economic consequences. Unfortunately, the economics of housing is still in its infancy. Here, we present an elementary housing model in which homebuyers rely on a mix of extrapolative and regressive expectations to forecast house prices. To counteract the wilderness of the bounded rationality problem, homebuyers are capable of learning, and they select expectation rules that have produced lower prediction errors in the past. We analytically demonstrate that endogenous house price cycles are only triggered if extrapolative expectations are heavy-handed. Such dynamics display shortrun momentum, long-run mean reversion, and excess volatility — the defining features of actual housing bubbles. Appendix Using the difference in fractions mt = WtR − WtE = Tanh[ 2h (ARt − AEt )] and the auxiliary variables Xt = Pt −1 , Yt = Xt −1 and Zt = Yt −1 , 2 The Neimark–Sacker bifurcation is accompanied by a Chenciner bifurcation. Endogenous fluctuations with a significant amplitude thus emerge once P ∗ becomes unstable. Furthermore, P ∗ is locally stable, and coexists with an attracting limit cycle between c ≈ 1.0672 and c ≈ 1.0748, allowing interesting attractor-switching phenomena.

N. Schmitt and F. Westerhoff / Economics Letters 176 (2019) 43–46

45

Fig. 1. Panels (a)–(c) depict house price dynamics for increasing extrapolation strength. Panel (d) shows a bifurcation diagram in which house prices are plotted against the extrapolation parameter. The parameter setting is given in Section 3. f (1+m )−2+2βγ δ − 1 + c(m −1)+2(1 , a2 = +r +δ ) ∗ ∗ ∗ c(m −1)(δ−2)+(f (1+m )−2)(δ−1) c(m −1)(δ−1) ∗ , a3 = − 2(1+r +δ ) and m = Tanh 2(1+r +δ ) [− hK ] . Since two eigenvalues are zero, the stability of the steady 2

we can summarize the model by the five-dimensional map

⎧ m +1 1−mt (Pt −1 +c(Pt −1 −Xt −1 ))+ t2 (Pt −1 +f (F −Pt −1 ))+α−β Ht ⎪ Pt = 2 ⎪ 1+r +δ ⎪ ⎪ ⎪ ⎨Xt = Pt −1 S : Yt = Xt −1 ⎪ ⎪ ⎪ Z = Yt −1 ⎪ ⎪ ⎩ t Ht = γ Pt −1 + (1 − δ )Ht −1

where a1

,

where h mt = Tanh[ (−g(Pt −1 − (Yt −1 + f (F − Yt −1 ))))2 − K 2 + g(Pt −1 − (Yt −1 + c(Yt −1 − Zt −1 )))2 ]. Straightforward computations reveal that the dynamical system admits a unique steady state S ∗ = (P ∗ , X ∗ , Y ∗ , Z ∗ , H ∗ ) = γ α−β H ∗ (P ∗ , P ∗ , P ∗ , P ∗ , δ P ∗ ), where P ∗ = βγ +δαδ(r +δ ) . As P ∗ = F = r +δ , ∗ S is called the fundamental steady state. From the Jacobian of the fundamental steady state ∗

1+r +δ

⎜ ⎜ =⎜ ⎝

β (1−δ ) c(1−m ) − 2(1 0 0 − 1+r +δ +r +δ )

1 0 0

γ

we obtain the characteristic polynomial

λ2 (λ3 − a1 λ2 − a2 λ − a3 ) = 0,

∗

0 1 0 0

0 0 1 0

0 0 0 0

0 0 0

1−δ

⎞ ⎟ ⎟ ⎟, ⎠

∗

state can be explored by using a set of stability conditions for the remaining third-degree characteristic polynomial. We follow Lines (2007) and use (i) 1 + a1 + a2 + a3 > 0, (ii) 1 − a1 + a2 − a3 > 0, (iii) 1 − a2 + a1 a3 − a23 > 0 and (iv) 3 + a1 − a2 − 3a3 > 0, which correspond to the (i) fold, (ii) flip and (iii) Neimark–Sacker bifurcation, respectively. Note that no parameter constellation satisfies the fourth condition as an equality while conditions (i), (ii) and (iii) are satisfied simultaneously. It can be shown that condition (i) is always satisfied, while inequalities (ii) and (iii) are not. These can be rewritten as f <

4 + 2r − δ (r + δ ) − βγ + c(1 − m∗ )(2 − δ ) (1 + m∗ )(1 − 0.5δ )

and c δ (1 − m∗ ) +

J(S )

⎛ 0.5(1+c)(1−m∗ )+0.5(1−f )(1+m∗ )−βγ

∗

=

2c βγ (1 − m∗ ) 2(1 + r + δ ) − c(1 − m∗ )

< f (1 + m∗ ) +

2(2δ + r) 1−δ

.

References Anufriev, M., Hommes, C., 2012. Evolutionary selection of individual expectations and aggregate outcomes in asset pricing experiments. Am. Econ. J. Microecon. 4, 35–64. Brock, W., Hommes, C., 1997. A rational route to randomness. Econometrica 65, 1059–1095. Case, K., Shiller, R., 2003. Is there a bubble in the housing market? Brookings Papers Econ. Activity 2, 299–362.

46

N. Schmitt and F. Westerhoff / Economics Letters 176 (2019) 43–46

Glaeser, E., 2013. A nation of gamblers: Real estate speculation and American history. Amer. Econ. Rev. 103, 1–42. Glaeser, E., Nathanson, C., 2015. Housing bubbles. In: Duranton, G., Henderson, V., Strange, W. (Eds.), Handbook of Regional and Urban Economics, vol. 5B. NorthHolland, Amsterdam, pp. 701–751.

Hommes, C., 2013. Behavioral Rationality and Heterogeneous Expectations in Complex Economic Systems. Cambridge University Press, Cambridge. Lines, M., 2007. Practical tools for identifying dynamics in discrete systems. In: Working paper. Department of Statistics, University of Udine. Poterba, J., 1984. Tax subsidies to owner-occupied housing: An asset market approach. Q. J. Econ. 99, 729–752.