Abstract
In this paper, we examine changes in the time series properties of three widely used housing market indicators (real house prices, pricetoincome ratios, and pricetorent ratios) for a large set of countries to detect episodes of explosive dynamics. Dating such episodes of exuberance in housing markets provides a timeline as well as empirical content to the narrative connecting housing exuberance to the global 2008 −09 recession. For our empirical analysis, we employ two recursive univariate unit root tests recently developed by Phillips and Yu (International Economic Review 52(1):201–226, 2011) and Phillips et al. (2015). We also propose a novel extension of the test developed by Phillips et al. (2015) to a panel setting in order to exploit the large crosssectional dimension of our international dataset. Statistically significant periods of exuberance are found in most countries. Moreover, we find strong evidence of the emergence of an unprecedented period of exuberance in the early 2000s that eventually collapsed around 2006 −07, preceding the 2008 −09 global recession. We examine whether macro and financial variables help to predict (insample) episodes of exuberance in housing markets. Longterm interest rates, credit growth and global economic conditions are found to be among the best predictors. We conclude that global factors (partly) explain the synchronization of exuberance episodes that we detect in the data in the 2000s.
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Notes
 1.
We complement the Dallas Fed dataset with housing rents from the OECD for 16 of the 22 countries for which there is consistent data over the same sample period (Girouard et al. 2006).
 2.
Mildly explosive behavior is modeled by an autoregressive process with a root that exceeds unity, but remains within the vicinity of one. This represents a small departure from martingale behavior, but is consistent with the submartingale property often used in the rational bubbles literature (see Section “The Univariate SADF and GSADF Procedures” for further details). Phillips and Magdalinos (2007a, b) and Magdalinos (2012) provide a large sample asymptotic theory for this class of processes that enables econometric inference in this case, unlike for purely explosive processes.
 3.
The SADF and GSADF tests better detect mildly explosive behavior in time series data than standard methods such as unit root/cointegration tests (Diba and Grossman 1988), but also variance bound tests (LeRoy 1981; Shiller 1981), specification tests (West 1987), and Chow and CUSUMtype tests (Homm and Breitung 2012). A few studies have implemented these techniques in the context of housing markets (Phillips and Yu 2011; Yiu et al. 2013), but only on domestic and not across international markets.
 4.
The existing empirical evidence points out that house prices may temporarily deviate from fundamentals (e.g., the time series and crosssection evidence in Clayton 1996; Hwang and Quigley 2006; Mikhed and Zemcik 2009a; Capozza et al. 2004; Adams and Fuss 2010) and the importance of the bank lending channel in housing (e.g., Mian and Sufi 2009; Pavlov and Wachter 2011; Berkovec et al. 2012).
 5.
 6.
We implicitly use this demand equation for rental housing to relate house prices to personal disposable income. In doing so, however, the definition of fundamentals has to be augmented with a particular specification of the rental housing demand.
 7.
Loglinear approximations are also commonly used but may be less relevant with nonstationary data where sample means do not converge to population constants (Campbell and Shiller 1988; Campbell et al. 1997). Further discussion on these approximations can be found in Lee and Phillips (2011). In this paper, we work with levels. Using logs does not qualitatively alter the results.
 8.
For the standard dividend discount model in which the payoff stream \(\left \{ X_{t}\right \}_{t=1}^{\infty }\) grows at a constant rate, see Gordon and Shapiro (1956). Blanchard and Watson (1982) and Campbell et al. (1997) examine more general processes for \(\left \{ X_{t}\right \}_{t=1}^{\infty }\).
 9.
For a discussion of a more general solution with loglinear approximation methods, see Engsted et al. (2012).
 10.
 11.
We note that, apart from income and rent, there are other fundamental drivers of housing prices, such as the cost of foregone interest, the cost of property taxes and maintenance costs (see, e.g., the discussion in Himmelberg et al. 2005). Lack of consistent and comparable data across countries for fundamental factors like this remains a limitation for applied research in housing.
 12.
The recursive representation of the discount rate is equivalent to the following alternative characterization,
 13.
For a discussion on the characteristics of the volatility process in house prices with data from the International House Price Database see Mack and MartínezGarcía (2012). These authors provide empirical evidence of an increase in house price volatility that is consistent with the stylized implications of declining discount rates laid out here.
 14.
We can also show that the persistence term \(\frac {\theta _{t1}}{\theta _{t2}}\) in the house price equation is bounded below by g and above by \(g^{k^{\prime }k}\) over the period from t up to \(t+k^{\prime }\).
 15.
For the numerical example, we set ρ=0.02, g=1.0002774397, \(k^{\prime }k=70\) and \(\sigma _{\epsilon }^{2}=0.01\).
 16.
Evans (1991) shows using simulation methods that standard unit root and cointegration tests cannot reject the null of no explosive behavior when such periodically collapsing episodes are present in the data. Price increases during the boom followed by a decline during the correction phase make it look like a meanreverting (stationary) process. Intuitively, this is the reason why many nonrecursive unit root tests wrongly suggest that processes that incorporate periodically collapsing boombust episodes are stationary—as indicated by Evans (1991).
 17.
These approaches are used to test a permanent change in persistence from a random walk to an explosive process. As a consequence, they perform well only in cases where the series becomes explosive but never bursts insample.
 18.
The Backward SADF (BSADF) statistic relates to the GSADF statistic as follows,
 19.
Exploring alternative minimum window sizes can be computationally demanding since for each r _{0} new critical values must be computed.
 20.
The choice of a fixed lag length is appealing because it allows us to employ a recursive least squares approach, which substantially reduces the computational cost of estimation.
 21.
Using asymptotic critical values doesn’t qualitatively change our results. Asymptotic values are provided in Phillips et al. (2015).
 22.
We are grateful to an anonymous referee for motivating this extension.
 23.
National house price indices aggregate across dwelling types and diverse locations within a country which may impact the performance of the econometric tests described in the previous section. In order to examine the effect of aggregation on the properties of the SADF and GSADF tests, we have conducted a large simulation experiment based on the S&P/CaseShiller 10City Composite Home Price Index and its constituent series. The results of the simulation experiment (which are available upon request from the authors) illustrate that aggregating lowers the power of both the SADF and GSADF tests. The effect is much larger for the SADF test than for the GSADF test, which gives us another reason to prefer the latter in our econometric strategy.
 24.
The variability of E X U _{ t } within a country is limited because the GSADF methodology does not detect many episodes of exuberance. An advantage of the pooled probit model is that, by incorporating the full variability across countries, it increases the number of episodes leading to more tightly identified results.
 25.
While the StoneGeary reduces to the CobbDouglas utility function whenever the parameters 𝜃 _{ H } and 𝜃 _{ C } are both set equal to zero, the specification permits both the rental rate elasticity and the income elasticity to vary with both rental rates and income—unlike the CobbDouglas where both elasticities are constant or the constant elasticity of substitution utility function for which the income elasticity is constant.
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Acknowledgments
We would like to thank María Teresa Martínez García and Itamar Caspi for providing helpful assistance and suggestions. We acknowledge the support of the Federal Reserve Bank of Dallas. All remaining errors are ours alone. The views expressed in this paper are those of the authors and do not necessarily reflect the views of the Federal Reserve Bank of Dallas or the Federal Reserve System.
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Additional information
The International House Price Database can be accessed online at http://www.dallasfed.org/institute/houseprice/index.cfm. An earlier version of the paper circulated under the title “Monitoring Housing Markets for Periods of Exuberance. An Application of the Phillips et al. (2012, 2013) GSADF Test on the Dallas Fed International House Price Database.”
Appendix A: Demand Equation for Rental Housing
Appendix A: Demand Equation for Rental Housing
Consider the maximization of the StoneGeary utility function with housing units rented, H _{ t }, and consumption of other goods, C _{ t }, i.e.,^{Footnote 25}
subject to the intratemporal budget constraint,
where the price of the consumption good is normalized to 1. X _{ t }≡x _{ t } H _{ t } is the housing rents—rental expenditures—paid and x _{ t } the rental rate per unit rented, Y _{ t } refers to disposable income, and α, 𝜃 _{ H } and 𝜃 _{ C } are preference parameters.
From firstorder conditions, the StoneGeary utility function subject to the standard intratemporal budget constraint gives a linear expenditure system where the demand for rental housing takes the following form:
or in expenditure terms,
Under the assumption that in equilibrium the units rented are constant (i.e., H _{ t }=H) and normalized to 1, the demand equation that determines housing rents in Eq. 39 reduces to an affine transformation of disposable income (Y _{ t }), i.e.,
where \(\delta \equiv \frac {\alpha }{1\left (1\alpha \right ) \theta _{H}}\) and \(\theta _{F}\equiv \frac {\alpha }{1\left (1\alpha \right ) \theta _{H}} \theta _{C}\).
B The Panel GSADF Test
The bootstrap procedure consists of the following steps:

1.
For each country, impose the null hypothesis of a unit root and fit the restricted ADF regression equation,
$${\Delta} y_{i,t}=a_{i,r_{1},r_{2}}+ \sum\nolimits_{j=1}^{k} \psi_{i,r_{1},r_{2}}^{j}{\Delta} y_{i,tj}+\epsilon_{i,t}, $$to obtain coefficient estimates (\(\widehat {a}_{i,r_{1},r_{2}}\), and \(\widehat {\psi }_{i,r_{1},r_{2}}^{j}\) for \(j=1,\dots ,k\)) and residuals (\(\widehat {\epsilon }_{i}\)).

2.
Create a residual matrix with typical element \(\widehat {\epsilon }_{t,i}\).

3.
In order to preserve the covariance structure of the error term, generate bootstrap residuals, \(\epsilon _{i,t}^{b}\), by sampling with replacement draws from the residual matrix.

4.
Use the bootstrap residuals and the estimated coefficients to recursively generate bootstrap samples for first differences,
$${\Delta} y_{i,t}^{b}=\widehat{a}_{i,r_{1},r_{2}}+ \sum\nolimits_{j=1}^{k} \widehat{\psi}_{i,r_{1},r_{2}}^{j}{\Delta} y_{i,tj}^{b}+\epsilon_{i,t}^{b}, $$and for levels,
$$y_{i,t}^{b}= {\displaystyle\sum\nolimits_{p=1}^{t}} {\Delta} y_{i,p}^{b}. $$ 
5.
Compute the sequence of panel BSADF statistics and the panel GSADF statistic for \(y_{i,t}^{b}\).

6.
Repeat steps (3) to (5) a large number of times to obtain the empirical distribution of the test statistics under the null of a unit root.
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Pavlidis, E., Yusupova, A., Paya, I. et al. Episodes of Exuberance in Housing Markets: In Search of the Smoking Gun. J Real Estate Finan Econ 53, 419–449 (2016). https://doi.org/10.1007/s1114601595312
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Keywords
 House prices
 Mildly explosive time series
 Sup ADF test
 Generalized sup ADF test
 Panel GSADF
 Probit model
JEL Classification:
 C22
 G12
 R30
 R31