Date-stamping US housing market explosivity

In this paper the authors set out to date-stamp periods of US housing price explosivity for the period 1830–2013. They make use of several robust techniques that allow them to identify such periods by determining when prices start to exhibit explosivity with respect to its past behaviour and when it recedes to long term stable prices. The first technique used is the Generalized supADF (GSADF) test procedure developed by Phillips et al. (Testing for multiple bubbles: Historical episodes of exuberance and collapse in the S&P 500, 2011a), which allows the recursive identification of multiple periods of price explosivity. The second approach makes use of Robinson’s test statistic (Efficient tests of nonstationary hypotheses, 1994), comparing the null of a unit root process against the alternative of specified orders of fractional integration. The analysis date-stamps several periods of US house price explosivity, allowing the authors to contextualize its historic relevance. JEL C22 G15 G14


Introduction
In the wake of the recent global financial crisis, it became pertinently clear that bubbles in core asset markets can cause tremendous real economic consequences if abruptly corrected. It is becoming harder to argue that we can simply dismiss the need to intervene during such episodes and simply "mop-up" following corrections (as the previous Fed chairman once argued). This follows as the previously held belief that financial markets have become sufficiently self-stabilizing fails to hold at times, particularly in markets where price corrections do not happen as smoothly.
The steep rise and subsequent fall of US house prices in the late 2000s have been the subject of much debate over the last few years. This follows largely from its role as underlying asset class to many of the derivative instruments that contributed to the financial crisis of 2008.
Property markets and residential houses, in particular, constitute a key asset class to the portfolio of most households worldwide. Abrupt movements of house prices, therefore, have a very real impact on households' abilities to consume and save. This in turn significantly impacts the economy's production and job creation capacity. As such, policies that curb unstable and bubble-like expansions in prices of houses in the economy could be considered a core policy objective, as sharp and sudden corrections in such prices could dramatically impact general price stability in the economy.
Since Shiller (1980) introduced the idea that prices of assets could deviate significantly from their underlying fundamentals (however defined), a large literature has emerged that aims to explain, document and even suggest preventions for asset price bubble formations. Although some efficient market proponents dismiss such notions, most accept that high transaction costs and limits to short selling could indeed lead to prices diverging from fundamental levels. As noted in Glaeser, Gyourko, and Saiz (2008), e.g., such market failures that hamper the ability of markets to correct price inefficiencies is particularly applicable to housing markets, where transaction costs are very high and short selling exceptionally difficult. This implies periods of price inefficiencies, and in particular periods of bubble-like behaviour, could feasibly exist with relatively little scope for arbitrage.
Our aim in this paper is to identify periods of bubble-like house price expansions over the last two centuries for the US market. This will serve to put the most recent bubble episode into historical perspective, and shed light on past price trends. We defer from making policy recommendations on curbing such behaviour, instead focusing on defining historical periods of US house price explosivity.
The key research question is how to tell when rapidly rising house prices constitute a bubble. Case and Shiller (2003) defines a housing bubble as being driven by home buyers who are willing to pay inflated prices for houses due to their expectations that houses will keep experiencing unrealistic appreciation in the future. This notion might be based on high expected returns, with the "dividend" portion of holding the asset being the value of residing in the residence (or the rental income earned), and the capital gain the expected rising price of the home. In fact, both can be expected to experience periods of rapidly rising prices in the short run, which can fuel the demand for homebuyers and mortgage originators alike, as the value of the underlying asset rises. But, as seen in the US market in 2007, external factors might lead to costly corrections with very real economic impacts felt across income divides.
Indeed, house prices may also experience such costly corrections as a result of deteriorating macroeconomic factors, even though it might not have experienced a rapid increase before. It might also experience a gradual downward correction with little or no noticeable real costs. Our objective is not to estimate the costs or consequences of these periods of explosive price buildups, but merely to document and contextualize their historic occurrences. We also do not attempt to distinguish between the type of bubble which occurred (albeit "irrational exuberance" or "credit-boom" driven), as we believe this falls outside the scope of our paper.
Our paper's contribution to the literature is to add estimates of past house-price bubbles that have not yet been applied to this asset class. We make use of two robust and efficient techniques that allow us to date-stamp periods of explosivity of these measures. The first technique that we will use is the Generalized sup ADF (GSADF) test procedure developed by Phillips, Shi, and Yu (2013), which is a recursive right-tailed unit root testing procedure that allows the identification of multiple periods of price explosivity. The second approach makes use of Robinson (1994)'s test of unit roots against the alternative of specified orders of fractional integration. We use the approach developed by Balcilar, Ozdemir, and Cakan (2015), which extends Robinson (1994)'s test statistic, to allow the identification of multiple periods of deviations from unit root behaviour in the presence of multiple endogenously determined structural breaks at unknown dates. This approach also provides the added benefit of testing a broader range of persistence than that which is measured using the unit root alternative in the first test.
Using these techniques, we identify several periods of explosivity for real US house prices that can be used in future studies.
Our paper is then structured as follows: section 2 discusses literature relevant to our study. Thereafter, section 4 outlines the methodologies used to identify periods of explosivity of US house prices, while section 3 describes the data used in the study. Section 5 discusses our findings, while section 6 concludes our study.

House Price Bubble Literature
Our study is not the first to consider the historical underlying price trends of housing markets. Jordà, Schularick, and Taylor (2015) focuses on a similarly long-dated sample of house prices for 17 countries dating back to 1870. In their study, they focus on identifying periods of credit-driven house price bubbles and the economic consequences of such events. 2 Our first challenge is identifying a fundamental level for house prices from which to deviate in order to define a bubble period. As we face a lack of historical data on measures that have previously been used to define fundamental house prices (including rental prices, construction costs and gross margins to home builders, as suggested by, among others, Himmelberg, Mayer, and Sinai (2005) and Glaeser et al. (2008)) we use another broad measure to define a level to which prices converge. 3 Our premise is that house price movements tend, in the long run, to display stationary behaviour relative to broad price movements in the economy. We thus label periods of positive deviations from such stationarity for sustained periods as episodes of price explosivity. This can be motivated conceptually that during periods where house prices rise at a significantly higher rate than general prices in the economy, we can feasibly expect it to be experiencing inflationary pressures resembling explosive behaviour.
One concern of this approach in anchoring the fundamental level of house prices relative to the aggregate price of goods in the economy, is the heterogeneity in the relative value of houses over time. As discussed in Knoll et al. (2014), the quality of houses and value of property increases over time. In our study, we make the plausible implicit assumptions that such changes occur over many decades and could be safely assumed to be less important in valuing homes in the short term viz-a-viz general aggregate prices.
Other studies have similarly used price trends to determine periods of high house prices. Studies by Lowe et al. (2002) and Goodhart and Hofmann (2008) define house price booms as periods where their real price indexes exceed some threshold relative to an Hodrick-Prescott (HP) filtered trend. Bordo and Jeanne (2002)'s definition, in contrast, attempts to calculate a long-run fair value, by measuring deviations of the 3-year moving average growth-rate from the series standard deviation. Other studies also focus on sustained peak-trough or trough-peak changes. 4 In a similarly long-dated study, Jordà, Schularick, and Taylor (2015) use a combination of the above, looking at the divergence of log real house prices from its trend rising above one standard deviation from the calculated HP filtered trend.
Our contribution to the above literature is to apply other novel techniques in defining possible periods of historical house price explosivity. A discussion of the literature on existing asset price bubble techniques follows.

Methodology literature
Accurately documenting the inflationary build up of asset prices has long interested economists and policy makers alike. A large literature has emerged that have tried to identify and explain the occurrence of asset price bubbles, leading to often divergent views on suitable policy responses following its detection (c.f. Gürkaynak (2008) for an in-depth discussion of the performance of various bubble detection techniques). Often the difficulty in testing for the presence of bubble-like behavior in asset price series lie in correctly identifying and date-stamping multiple periods of explosivity. Traditional unit root and co-integration tests aimed at identifying such periods (as e.g. proposed by Diba and Grossman (1988)), fail to identify the existence of bubbles that periodically collapse. Evans (1991), e.g., points out that ordinary stationarity tests remain exposed to the possibility of identifying pseudo stationary behaviour when a series in fact displays periodically collapsing bubbles.
Various techniques have been proposed that allow the detection of multiple periods of collapsing speculative bubble in asset prices. Al-Anaswah and Wilfling (2011) and Lammerding, Stephan, Trede, and Wilfling (2013), e.g., use Markov-switching models to differentiate between regimes of price stability and price explosivity (the latter authors also use a robust Bayesian estimation procedure). Another class of techniques use a sequential unit root testing procedure developed by  and Phillips, Wu, and Yu (2011), which built on the indirect stationarity tests suggested by Diba and Grossman (1984) and Hamilton and Whiteman (1985). As noted by Bettendorf and Chen (2013), the key advantage of sequential identification procedures, particularly relevant to our analysis, is that it detects periods of explosivity despite potential misspecifications of the market fundamental process. In this study, we will make use of the generalized version of the sequential ADF tests, developed by Phillips, Shi, and Yu (2013) (PSY hereafter), which is robust to the identification of multiple collapsing bubble episodes. It has since gained ground in its broad empirical applications (c.f. inter alia Bettendorf and Chen (2013); Etienne, Irwin, and Garcia (2014); Caspi, Katzke, and Gupta (2015)) and allows consistent date-stamping for the origination and termination of multiple asset price bubbles.
A key challenge when using PSY's approach to identify asset price bubbles, is specifying the true definition of a fundamental level from which prices deviate. Typically, the return to holding the asset, in the form of dividend yields for equities (c.f. PSY, (2013)) and the convenience yield for commodities (c.f. Pindyck (1993); Lammerding et al. (2013); Gilbert (2010); Shi and Arora (2012)), is first defined in a pricing equation. Then a bubble component is specified, which, at times, displays explosive behaviour. Although several papers critique this identification of bubble components (e.g. Cochrane (2009);Pástor and Veronesi (2006); Cooper (2010) offer critical discussions on this), explosive or mildly explosive behaviour in asset price series indicate possible market exuberance during the inflationary phase of a bubble, a feature that can be uncovered from recursive testing procedures on time-series data (Phillips et al., 2013;Phillips and Magdalinos, 2007). Caspi, Katzke, and Gupta (2015) also use the GSADF approach to identify periods where oil prices deviate from the general price level in the US, as well as levels of oil inventory supplies, respectively. Their use of these measures as proxies for the fundamental price of oil follow from a similar lack of data on historical oil price derivatives used to calculate the convenience yield. Instead, they study periods where the nominal price of oil displays periods of significant build-up relative to the general price level and stock of US oil supply, which both act as credible alternatives to the standard convenience yield.
The second approach that we will use in this study to identify periods of explosivity tests the null of a unit root process against the alternative of fractionally integrated orders which exceed one. Several studies have in the past used a long memory process to test for explosivity in asset price series using a test statistic developed by Robinson (1994) (e.g. Cuñado, Gil-Alana, andGracia (2007); Gil-Alana (2003; Balcilar, Ozdemir, and Cakan (2015). A key consideration in defining explosive periods are controlling for structural breaks, which, as highlighted by Perron (1989), may lead to the non-rejection of the unit-root hypothesis. Gil-Alana (2003) assumed known structural break dates in their analysis, while Gil-Alana (2008) employed a residuals sum squared approach where a single structural break date was allowed at an unknown time. Our approach follows that of Balcilar et al. (2015) in allowing multiple structural breaks at unknown dates. We then use Robinson (1994)'s LM test statistic to determine the fractional order of integration of the US house price series after controlling for endogenously determined level and trend shifts. We then recursively identify periods where the lower bound of the fractional order exceeds unity, and subsequently return to levels below unity, to allow us to identify explosive periods equivalent to those determined using PSY, (2013)'s GSADF approach. Both approaches are robust to multiple periods of periodically collapsing bubbles, less sensitive to the specific definition of the underlying fundamental process and able to provide recursive date-stamping of explosive periods in the underlying data.

Data Description
Our metric of interest in this study is the real house price (RHP) over the annual period of 1830-2013, with the start and end date being purely driven by data availability on house prices at the time of writing. The nominal house price index used is the Winans International U.S. Real Estate Index, which tracks the price of new homes back to 1830. This index is then deflated by the Consumer Price Index (CPI) to derive the real house price (RHP) index. 5 The RHP is then transformed into its natural logarithmic form.
Our motivation for the use of the Winans Real Estate Index (WIREI)follows from its robust design and tracking of a wide geographical sample of US house values. The emphasis of the index design is to allow researchers to study US real estate as an asset class. This is particularly useful to our analysis, as we seek to identify periods in the RHP series where US house prices as an asset class, broadly, experienced explosivity. The WIREI aggregates house prices across all the major geographic areas in the US, while aggregating price reports from the Census Bureau, Bureau of Labor and Statistics, as well as work done by Long (1869Long ( -1936, Gottlieb (1837 -1868) and Riggleman (1830 -1836). 6 The benefit of this is that future research can use our results for RHP explosivity to compare it to corresponding periods of explosivity in other asset classes (such as stocks and bonds). The dataset we use has a correlation of roughly 80% for overlapping periods with the shorter dated Shiller House price index, although the latter has a sharper rise and decline in the early 2000s than the WIREI.
From figure 1 below we see that the CPI-deflated price of houses in the US remained roughly stable from the beginning of our sample until the mid-part of the 20th century.  Knoll et al. (2014) comprehensively discuss the difficulties in defining an appropriate index for housing prices. Jordà et al. (2015) also alludes to the difficulty of distinguishing between the value of the structure and the underlying land in such indexes. We refer interested readers to these studies to gain insight into such challenges. The WIREI data set aggregates the prices of new homes in the US dating back to 1830. 8 We believe, in this regard, that new house prices provide a fair estimate of the value of existing (unsold) house prices, which would otherwise be exceedingly difficult to estimate without adding a long list of other potential calculation pitfalls. 9 In light of the above concerns, the WIREI index adjusts the price data for average house size over time (price per square foot).
The first step in using the GSADF date-stamping procedure is to apply the summary right-tailed GSADF tests to the series. Table 1 shows that for both series, at the 5% level (with the smallest window size of 15), we find that our GSADF test statistics exceed the 10% and 5% right-tailed critical values respectively, rejecting the hypothesis in favour of a root exceeding unity at some point. This provides evidence that RHP experienced periods of explosivity for the full sample. Using this approach to locate the bubbles, we compare the SADF statistic sequence with the 95% SADF critical value sequence, obtained using Monte Carlo simulations.Details of this approach are contained in Phillips et al. (2013). The existence of possible structural breaks in the series would merely serve to strengthen the argument for roots exceeding unity, and so we do not control for such events here. As can be seen from figure 2 in the appendix, RHP shows sustained growth in the post-war era, reaching its peak in 2005. Over the sample period, there were three episodes identified by the GSADF approach as explosive. Our fractional integration approach also provides evidence for the presence of several periods of explosivity for the RHP. These results and their economic relevance will be discussed in the next section.

Methodological Discussion
The first technique that we use to label episodes of price explosivity builds on the work pioneered by  and Phillips, Wu, and Yu (2011), and in particular the generalized form of the sup ADF (GSADF) proposed by Phillips, Shi, and Yu (2013). This method uses a flexible moving sample test procedure to consistently and efficiently detect and date-stamp periods where a price series displays a root exceeding unity. Bubbles are so identified in a consistent manner with false identifications seldom given even in modest sample sizes. 10 The test procedure suggested by PSY recursively implements an ADF-type regression test using a rolling window procedure. Suppose the rolling interval begins with a fraction r 1 and ends with a fraction r 2 , with the size of the window given as r w = r 2 − r 1 . Then, let: where µ, δ and φ are parameters estimated using OLS. We then test null of H 0 : δ = 1 against the right sided alternative H 1 : δ > 1. The number of observations used in equation 1 is then T w = [r w T ], where [.] is the integer part. The ADF statistic corresponding to equation 1 is thus denoted by ADF r 2 r 1 . Building on this approach, PSY formulated a backward sup ADF test where the end point of the subsample remains fixed at a fraction r 2 of the entire sample, with the window size expanding from an initial fraction r 0 to r 2 . This backward sup ADF (SADF) procedure can thus be defined as: PSY then suggested repeatedly implementing the SADF procedure of equation 2 for each r 2 ∈ [r 0 , 1], leading to a generalized form (GSADF) written as: The supremum form of the recursively estimated ADF is motivated by the observation that asset price bubbles generally collapse periodically. 11 In this scenario, the sup ADF test delivers efficient bubble detection capabilities where one or two bubbles emerge, with the generalized form performing well even in the presence of multiple bubble episodes. The initial minimum fraction in the SADF approach of equation 2, r w = r 0 , is selected arbitrarily, keeping in mind the issue of estimation efficiency. Thereafter, we expand the sample window forward until r w = r 1 = 1, the full sample, and we have a recursive estimate of ADF defined as ADF r k , ∀k ∈ (r 0 , r 1 ). From the sequence of ADF statistics (SADF) so produced, we can then identify the supremum value that can be used to test the null hypotheses of unit root against its right-tailed (mildly explosive) alternative by comparing it to its corresponding critical values. If the right tailed alternative to the unit root null is thus accepted, we can infer mild explosivity of the series, indicated by δ r 1 ,r 2 .
The generalized form of this approach defined in equation 3, uses a variable window width approach which allows both the starting and ending points to change within a predefined range, [r 0 , 1]. This allows the identification of multiple periods of explosivity and allows us to consistently date-stamp the starting and ending points. The starting points are identified as the periods, T re , at which the backward sup ADF sequence crosses the corresponding critical value from below. The corresponding ending point to an explosive period is similarly defined as the period, T r f , where the backward sup ADF sequence crosses the critical value point from above.
We can formally define identified periods of explosivity using the GSADF approach as:r Where cv β T r 2 is the 100(1 − β t )% critical value of the sup ADF statistic based on [T r 2 ] observations. We also set β t to a constant value, 5%, as opposed to letting β T → 0 as T → 0. The BSADF(r 0 ) for r 2 ∈ [r 0 , 1] is the backward sup ADF statistic that relates to the GSADF statistic by noting that: 12 The second approach that we use also tests the right tailed alternative to a unit root null hypothesis, but unlike standard right-tailed tests, focuses on the fractional order of integration. The approach that we follow is similar to Balcilar et al. (2015), who built on the procedure developed by Robinson (1994) in determining the fractional order of integration. They also allow for the identification of multiple endogenously determined structural breaks in the form of level and trend shifts at endogenously determined dates. The identification approach is based on the procedure suggested by Gil-Alana (2008) and built on the principles suggested in Bai and Perron (1998). Balcilar et al. (2015) also construct statistical tests for the different orders of fractional integration for each regime, using Robinson (1994)'s LM test to determine the most likely order of integration. To explain this procedure, consider the following multiple regression form: where y t is the house price index series, β a k × 1 vector of unknown parameters and z t a k × 1 vector of observable variables, which includes a constant, polynomials in time trends (t) and structural break dummies, depending on the deterministic structure imposed. As noted in Balcilar et al. (2015), the presence of such deterministic regressors does not affect the limiting null and local distribution of the Robinson test statistic.
We consider the general case where z t includes a constant, a linear time trend and m = 2k level, as well as trend shift dummies, DLT tl t,i = (DL tl t,i , DT tl t,i ) at the dates i = T tl b,1 , ..., T tl b,k . We then set DL tl t,i = 1 if t > T tl b,i and zero otherwise, and also DT tl t,i = t−T tl b,i and zero otherwise. Here we will also follow the notation of Balcilar et al. (2015) by defining T k as the set of disjoint break dates, T k = {T tl b,1 , ..., T tl b,k }. We also define β z t as follows: with the regressor errors, x t , assumed driven by the following process as: with L the lag operator, u t covariance stationary, integrated of order zero, I(0), and having a spectral density function that is positive and finite at zero frequency. Allowing for a fractional order of integration in equation 8, implies that d can assume any value on the real line. The model structure above is based on the least squares principle first proposed by Bai and Perron (1998). The estimation is carried out as follows: first, a grid of values for the fractional integration parameter, d, is chosen as, e.g., d 0 = [0.00, 0.01, ..., 1.20]. The least squares estimates of µ, δ, φ i and θ i in equation 8 are then obtained for each k-partition of {T 1 , ..., T k }, denoted as {T k }, by minimizing the sum of squared residuals in the d 0 difference models. This implies, minimizing the following residuals sum of squares (RSS): over all value of T 1 , ..., T k , yielding the parameter estimatesμ,δ,φ i andθ i , ∀i ∈ [1, ...k], and also the break dates, {T k }. We also employ Schwarz' (1978) Bayesian information criterion (BIC) to select the number of breaks, k, prior to running the procedure. 13 We then calculate the test statistic of Robinson (1994) for each value of d 0 in the grid, a procedure that can be summarized as follows (following again the notation of Balcilar et al. (2015)). In order to test the null hypothesis: Robinson (1994) developed the following score statistic: with I(λ j ) the periodogram ofû t . Parameter estimates forη are derived from the Whittle Maximium Likelihood (WML) method: with g(λ j ; η) the known function of the parametric spectral density of u t . The model in equation 6 is completed by specifying a parametric form for u t . In our analysis, we choose a general specification for u t nested within an Autoregressive Moving Average (ARMA) model. This implies that by definition that x t is characterized by a fractionally integrated ARMA (ARFIMA) model, which is a commonly used parametric specification for measuring long memory. The ARMA(p, q) model is denoted as: while the ARFIMA(p, d, q) model for x t can be written as: where ε t is a white noise process with variance, σ 2 , and φ(L) = 1− p j=1 φ j L j and Ψ(L) = 1 − q j=1 Ψ j L j are polynomials in the lag operator L, with degrees of freedom p and q respectively. Furthermore, we assume that φ(Z) and Ψ(Z) share no common roots and φ(Z) = 0 and Ψ(Z) = 0, ∀Z ≤ 1. The spectral density functions of these models, respectively, are given by: with η a l × 1 vector of unknown parameters estimated by maximum likelihood, assuming that the orders p, q are known a priori. 14 Note also that the fractional parameter, d, is fixed under the null, thus equation 15 above is relevant to our empirical estimations. Our approach can thus be summarized as follows. We select a value d 0 in the grid d 1 0 + i∆ d , with ∆ d the grid increment and i = 1, ..., s. Then an initial disjoint break date, T 1 , is selected and the residuals, , are thus obtained. This is then used to calculate ther statistic of equation 11, with break dates then updated using the Bai and Perron (1998) algorithm. These steps are then repeated until T t=1û t 2 is minimized, and done for all the grid increments. At each step in the process, we minimize the RSS(T k ) for a given d 0 , with the parametersβ and nuisance parametersη estimated sequentially.
An approximate one-sided test of H 0 : d = d 0 is then rejected in favor of H a : d > d 0 (d < d 0 ) at the 100α% level whenr > z α (r < −z α ), with α the probability that the standard normal distribution exceeds z α . In the empirical implementation, we allow structural breaks in the full sample estimation. We use this procedure in the same fashion as the rolling window ADF regression of Phillips, Shi, and Yu (2013). In the rolling implementation, the sample interval begins with a fraction r 1 and ends with a fraction r 2 , with the size of the window given as r w = r 2 − r 1 . We do not allow structural breaks in the rolling estimation since a small window size of r w is unlikely to include structural break impacts.

Empirical Results
Below follows a discussion of the explosive periods identified using the approaches outlined above. The purpose of this section is not to fully discuss the causes and reactions to these periods of explosivity (this would be an interesting follow up to this paper), but instead to concisely summarize the environment surrounding these episodes and highlight the extent of real house price declines following such periods. Our second technique identifies more periods of explosivity in the post-war period (4), whereas the first technique identifies only three such periods in total. Figure 2 displays the results of the GSADF procedure over the sample period, with starting periods of explosivity labeled when the blue line (BSADF sequence) exceeds the red line (95% critical values), and ends where it dips below the red line. These periods of explosivity are summarized in table 2. We see that for the RHP series, there are three periods of explosivity with relatively short durations. 15 The first episode of explosivity was preceded by the five year depression following the panic of 1873, and saw the US Congress require a form of quantitative easing in the late 1870s. 16 This was followed by a spike in asset prices broadly, with real housing prices rising by 149% between 1878 and 1880. After this period of explosivity, real prices declined by roughly a third within three years.
The second period was between 1956 and 1957, where real house prices rose by over 43% between 1955 -1957. This was driven by a decade of prosperity where the US economy grew significantly and employment were at all-time lows. This period was followed by a house price correction of 12% within two years, before again experiencing a sustained price increase (with weak explosivity identified by the end of the 1960s again). 17 .
The last episode of explosivity is identified between 2004 and 2006. This follows a period where real house prices rose by roughly 26% from 2000-2006. The explosive episode identified was preceded by the Fed funds rate being lowered significantly 18 , and characterized by sharply increasing house prices, large scale deregulation of institutions able to provide mortgage products, and a proliferation of investment vehicles designed by leveraged institutions to magnify the property market returns. This culminated in a period of credit-driven mortgage price increases. Prices corrected within two years by more than 15%, and by 21.5% within five years.
The next technique used in this study to label periods of explosiveness is the procedure proposed by Balcilar et al. (2015). The estimation was carried out as follows: for each chosen value of d we use the statistic forr, given in equation 11, to test whether the fractional parameter, d, exceeds 1. This would be indicative of an explosive period, making it comparable to the sequential unit root tests above. We first test for various fractional orders d in the full ample. In our estimation for, the full sample we use two deterministic structures for z t , with z 1,t corresponding to a constant and trend, and z 2,t corresponding to the general case in equation 7. The estimation procedure detailed in section 4, identified two endogenously determined linear trend and level breaks (denoted DT t,l t,i and DL t,l t,i , respectively), which occurred at 1877 and 1954. These breaks correspond to periods of explosivity defined using the GSADF approach. The procedure is then used in a rolling estimation fashion with fixed window size of r w = 15. Rolling estimation does not allow structural break dummies, since a small window size does not suffer from structural break impacts.
The fit of the structural break model for the full sample can be viewed in figure 3 in the appendix. As can be seen, the model tracks the broad trend 17 Weak explosivity here implies a single month of explosivity identified in 1964 and 1968) 18 The Fed funds rate was lowered from 6.5% to 1.75% in 2001, following fears of a deflationary trap following the DotCom crash.
of the data rather well. Table 3 provides the estimated full sample fit of the structural break model, using deterministic structure z 2,t .
From table 3, we see that nearly all of the parameters for the second model structure, z 2,t , are significant. The significant structural break dummy estimates confirm the existence of significant breaks in both trend and levels of the RHP series at 1877 and 1954.
In order to validate the use of the GSADF procedures earlier (as structural breaks could lead to the shifting up of orders of integration), we also include z 1,t 's estimates in table 4. From it we see firstly that when not controlling for the structural breaks, the lower bound of significance for the fractional order of integration estimate exceeds unity at the 1% level. When controlling for the structural breaks using z 2,t , we see non-rejection covers the range 0.94 to 1.00 at the 5% level, and 0.92 to 1.01 at the 1% level. 19 This indicates that there is strong evidence that RHP experienced periods of explosivity, when comparing the null of a unit root to the more flexible test of a fractional order of integration, even when not controlling for structural breaks. This validates the use of PSY, (2013)'s approach, as it indicates that such breaks do not significantly account for explosivity in the full sample for RHP.
In order to date-stamp periods of explosivity using this approach, we employ a rolling window procedure to calculate ther statistic. We use a fixed length window size of 15 sequentially from the beginning to the end of the sample, adding a single observation and dropping the last at each step. We then calculate at each step a range ofr statistics, enabling us to estimate a lower and upper bound limit for d (using a one sided test with 5% significance level). The benefit of using this approach to identify periods of explosivity is that, firstly, it allows for a changing structure of the underlying data, and secondly it is robust to possible structural breaks. This implies we use the rolling window identification technique on the z 1,t deterministic form, as opposed to the form accounting for the breaks explicitly. Although there are differing views on the appropriate size of such fixed window techniques 20 , our chosen window size reflects our desire to optimize the representativeness of the model, particularly as we identified two breaks in the series. Figure 4 shows our rolling window estimations.
From figure 4, we identify periods of explosivity as starting when the lower bound (blue line) cross 1, and ends when it dips below 1. Table 5 summarizes the periods of explosivity so identified. As before, we ignore episodes shorter than 1 period in duration, while also excluding periods of potential negative explosivity, as our focus is on price build-ups.
From table 5, we see that the fractional integration rolling window approach offers greater insight into periods of explosiveness during the 1800s, particularly as there is a much shorter burn-in period. We see, e.g., several periods of explosiveness in real house prices during the 1849 -1855 California gold rush 21 , which saw an increase in real house prices of over 70% during this period. 22 The US gold rush continued until 1864, during which time another period of explosivity can be identified towards the latter part. The next period of RHP explosivity is labeled between 1866 -1873, right after the US civil war which ended in 1865. Prices during this phase peaked in 1867, 81% higher than during the war in 1864. The next episode of explosivity identified is from 1926 -1929, during which time real house prices rose by over 48%. This coincided with unprecedented asset price inflation across nearly all US asset classes. Real house prices peaked in 1928, and were down 85% by 1932, following the start of the Great Depression in 1929.
This approach then identifies two short-lived periods of explosivity during the mid 1980s and late 1990s. The first episode transpired in the build up to what is today known as the Savings and Loan crisis, which started in 1986, and saw credit-driven real house prices rise by roughly 30% from 1984-1987. This was caused in part by large scale deregulation of lending standards and a reduction in capital reserve requirements in the US, which both served to drive large scale credit creation, particularly in financing mortgages. The 1990s saw RHP first decline substantially (after peaking in 1989, it fell by roughly 21% by 1993), while picking up in the late 1990s and reaching its 1989 peak again in 2001. Real prices then surged in the early 2000s, peaking in 2004 at 27% higher than in 2000. The RHP correction came after the 2008 global financial crisis, with a sharp turnaround in RHP between 2009 -2011, identified as a period of significant decline in RHP using our statistics. These results echo the findings by Shiller (2015), who show real estate asset price increased during the early and mid 1980s and mid-1990s, with a subsequent contraction in the late 2000s.

Conclusion
This paper set out to identify periods of US house price explosivity from 1830 -2013. In order to identify house price fundamentals, we make use of the general price level (measured as the US CPI index). The implicit assumption thus made is that house prices tend to reflect general movements in prices across the economy. Large deviations from past levels could therefore be considered as explosive in the short term as it could feasibly lead to higher allocation towards houses as assets experiencing high capital growth. This, in turn, feeds into more demand and even higher prices, potentially driving an episode of unsustainable asset price increases, particularly as a result of factors inherent to property purchases (such as typically high transaction costs and low ability to short-sell) that make it uniquely prone to bubbletype episodes. Although other measures have been suggested for use as fundamentals, we are constrained by data availability for our long dated sample. 23 The first technique used to identify periods of explosivity, is the recursive GSADF test suggested by Phillips et al. (2013). This test allows the effective date-stamping of periodically collapsing bubble-like periods, allowing us to label several historical periods of significant real house price build-ups. For the RHP measure, we define three short periods of explosivity, during the late 1800s, mid 1950s and the mid 2000s.
The second measure used to test right tailed alternatives to unit root testing, focuses on the fractional order of integration, d. The procedure uses Robinson (1994)'sr statistic to define confidence bands for likely values of d. We also allow for the identification of multiple periods of endogenously determined structural breaks in the form of level and trend shifts at unknown dates. We then use a rolling window approach to date-stamp periods of likely explosivity in the series, identified as periods where the lower bound of the 95% confidence interval of d exceeds unity. The periods so identified suggest several periods of explosivity during the 1800s, particularly surrounding the US gold rush, as well as immediately following the Civil War. Significant and unsustainable build-ups in real house prices are then also observed in the 1920s shortly before the Great Depression, the 1980s during the period preceding the S&L crisis, as well as during the late 1990s and early 2000s, with a correspondingly significant negative price adjustment following the global financial crisis. Our results suggest that the more flexible, long memory approach of using fractional integration to test the alternative hypothesis, provides a richer set of dates of where prices likely deviated from mean reversion toward aggregate prices in the US.