1 Introduction

Between 2008 and 2013, home prices in Israel increased by 50 % in real terms, reaching 60 % in some regions. This increase is the highest among OECD member countries over the same period. Figure 1 provides data on two measures commonly used to gauge home price deviations from fundamentals, the price to rent and price to income ratios, at the national level for the period from January 1999 to July 2013. Both measures significantly deviate from their sample means (horizontal line, in red) at their current levels, and thus suggest a possible distortion in home prices. Despite their intuitive appeal, inferences about housing market conditions based on these measures might be misleading, since the measures do not explicitly account for possible changes in other fundamental factors besides rent and income Himmelberg et al. (2005).

Fig. 1
figure 1

Measures for deviations of home prices from fundamentals (January 1999–July 2013). Notes: Both measures are compared with their sample means (horizontal line, in red). The price to rent ratio is an index normalized to January \(2000 =1\). Income is measured as the annualized average wage per employee. Source: CBS, Dovman et al. (2012) and Bank of Israel calculations. a Price to rent ratio. b Price to income ratio. (Color figure online)

Israel is one of the few advanced economies to be mostly unaffected by the recent global financial crisis of 2007–2008. Additionally, there was no buildup of home prices in Israel prior to this crisis. Nonetheless, the recent increase in home prices occurred with an unprecedented and persistent drop in the short-term monetary policy rate during 2008. Theoretically, low interest rates should contribute to higher home prices (Poterba 1984). Yet, some relate prolonged periods of too-low interest rates with the emergence of a housing bubble [e.g., Taylor (2007)]. A failure to detect a housing bubble in real time may lead to damaging implications in the aftermath of its burst, such as overbuilding (Glaeser et al. 2008) or financial distress. It also has severe consequences for the real economy, including massive mortgage defaults (e.g., the subprime crisis in the USA).

This paper addresses the question of whether recent home price appreciation in Israel reflects the existence of a national or regional housing bubble or whether it is just the result of changes in fundamental factors. To answer this question, I integrate a housing market version of the dynamic Gordon growth model (Campbell et al. 2009), as well as advanced econometric bubble detection and monitoring strategies (Phillips et al. 2011, 2015b; Homm and Breitung 2012). The dynamic Gordon growth model decomposes changes in the price to rent ratio into changes in the expected paths of rent price growth rates, risk-free rates and risk premiums. A fourth “model consistent” factor that might affect the price to rent ratio is the rational bubble component. The model implies that if a bubble is present, then it must be expected to grow explosively in the sense that it has an autoregressive root greater than unity. Consequently, a price to rent ratio that embodies such an explosive bubble component also inherits its explosiveness. Phillips et al. (2011, 2015b), Phillips and Yu (2011) develop powerful test procedures that exploit this feature of explosiveness to identify bubbles.Footnote 1 Furthermore, they and Homm and Breitung (2012) propose methods to carry out real-time monitoring for bubbles.

I contribute to the empirical literature in three ways. First, I suggest a straightforward framework for incorporating leverage and mortgage rate elements into the Phillips et al. (2011, 2015b) bubble detection frameworks. Second, to the best of my knowledge, this paper is the first one to apply the Phillips et al. (2011, 2015b) frameworks to regional data. Conducting regional analysis is important, as it can potentially spot bubbles that exist in one, or several, of the regions and cannot be detected on the national level due to the averaging nature of aggregate national data. This is possible because Israel has readily available quality-adjusted data on home prices and rent at the regional level. Third, this study provides results from a thorough econometric analysis of housing bubbles in a country that is a prime candidate for this type of analysis, because of the recent developments in its housing market.

I use monthly national-level data on the quality-adjusted, price to rent ratio from January 1999 to July 2013. Additionally, I control for macroeconomic fundamental factors by using monthly data on the average wage, as well as the short- and long-term interest rates on Israeli government bonds and the average mortgage rate set by Israeli banks. I complement the national-level analysis by using regional-level price to rent data for nine regions between the first quarter of 1998 and the second quarter of 2013 to test for the possibility of a regional housing bubble. Using regional-level data accounts for the possibility that housing markets in different regions are not fully integrated.Footnote 2

I find that, essentially, recent developments in home prices are inconsistent with a housing bubble scenario. In particular, I cannot reject the null of a no-bubble scenario at the national and regional levels. The majority of the results hold under a variety of tests, alternate specifications, and leverage consideration. One exception is the Gush Dan region for which the results are inconclusive and depend on model specifications. I conclude that, overall, recent price movements in Israel are in line with the development of fundamental factors: mainly, lower interest rates and higher rent prices.

This study relates to the broad empirical literature on housing bubbles. In particular, it relates to a strand of this literature that uses econometric identification schemes based on time series. For example, Arshanapalli and Nelson (2008) apply cointegration tests to examine whether US housing prices and several fundamental factors share a common stochastic trend for first quarter of 2000 through the third quarter of 2007. They find evidence for a bubble. Similarly Taipalus (2006) applies unit root tests to the rent to price ratio for Finland, the USA, the UK, Spain, and Germany and concludes that, under the assumption that rent growth rates and expected returns are stationary, a bubble existed in nearly all markets.Footnote 3 In this paper I implement an empirical strategy that was recently used by Phillips and Yu (2011) for the US housing market, Yiu et al. (2013) for the Hong Kong local property market, Engsted et al. (2014) for housing markets in OECD countries, and by Pavlidis et al. (2013) to study data from the Dallas FED International House Price Database.Footnote 4 \({^{,}}\) Footnote 5

This study also relates to studies by Dovman et al. (2012) and Nagar and Segal (2010) that empirically assess recent developments in the Israeli housing market. Dovman et al. (2012) use multiple econometric bubble detection methods and report little evidence for a housing bubble as of August 2010.Footnote 6 Nagar and Segal (2010) estimate an econometric model of the Israeli housing market using cointegration methods and assert that in 2010, home prices deviated by 8–20 % from their long-run levels.

The remainder of this paper is organized as follows. In Sect. 2, I present a simple asset pricing model in the context of the housing market. Section 3 gives a technical description of the econometric bubble detection method I use. Section 4 briefly describes the data I use. In Sect. 5, I present and discuss the results of the tests. Section 6 presents a sensitivity analysis of my results, and Sect. 7 concludes.

2 Theoretical background

Himmelberg et al. (2005) provide the following definition for a housing bubble: “We think of a housing bubble as being driven by home buyers who are willing to pay inflated prices for houses today because they expect unrealistically high housing appreciation in the future.” The “unrealistically high” part refers to house price growth rates that are not related to housing market fundamentals, mainly expected rent payments and discount rates. Though this definition is quite intuitive, it is rather general and needs some more refinement.

In this study, the focus is on bubbles of the rational type, commonly referred to as “rational bubbles”. This terminology refers to asset price bubbles that arise in models where all investors have rational expectations.Footnote 7 Though many historical episodes of booms and crashes in asset prices are labeled in retrospect as bubbles (as in the cases of the dot.com bubble in the late 1990s or the more recent the US housing bubble), the existence of bubbles within rational expectations models is still a matter of debate (Brunnermeier 2008). For instance, bubbles can be ruled out under rather weak assumptions within the framework of competitive general equilibrium models with infinitely lived representative agents (Santos and Woodford 1997). In contrast, overlapping generations models permit the existence of such bubbles. (One example is Galí 2014). In general, the theoretical feasibility of rational bubbles largely depends on underlying assumptions about the economy, such as the availability of information, trading constraints, liquidity considerations, etc.Footnote 8

Over the years, applied economists have tried resolving the conflict about the existence of bubbles by formulating econometric procedures designed to test the existence of such rational asset price bubbles. One strand of the literature utilizes predictions from rational asset pricing models to test the consistency of the data with the no-bubble hypothesis. This is generally done by comparing the stochastic properties found in the data with the dynamics implied by the no-bubble condition.

2.1 The model

To gain more insight on the rationale behind these econometric tests for bubbles, I follow Campbell et al. (2009) and present a theoretical home pricing model for the housing market. I first denote the definition of the realized real gross return for holding a home for one period by

$$\begin{aligned} V_{t+1}=\frac{P_{t+1}+R_{t+1}}{P_{t}}, \end{aligned}$$
(1)

where \(V_t\) denotes the real gross return on a home held from time t to time \( t+1 ,\, P_t \) is the real price of a home and at the end of period t , and \(R_{t+1}\) is the real payment received for renting the house from time t to \( t+1 \).

Using the Campbell and Shiller (1988) method we can express the log-linear approximation of Eq. (1) as:Footnote 9

$$\begin{aligned} v_{t+1}\approx \kappa +\rho p_{t+1}+\left( 1-\rho \right) r_{t+1}-p_{t} \end{aligned}$$
(2)

where \( p_t\equiv \log ({P_t}), \, r_t\equiv \log ({R_t}),\,v_t\equiv \log (V_t),\,\rho =1 \big / \left[ 1+e^{\left( \overline{r-p} \right) }\right] ,\,\overline{r-p}\) is the sample mean of the log rent to price ratio, and

$$\begin{aligned} \kappa =-\log (\rho )-(1-\rho )\log \left( \frac{1}{\rho }-1\right) . \end{aligned}$$

Solving Eq. (2) for the log price to rent ratio by forward iterations results in the following dynamic log-linear approximation of the present value formula:

$$\begin{aligned} p_t-r_t=\frac{\kappa }{1-\rho }+\sum _{j=0}^{\infty }\rho ^j \left( \varDelta r_{t+1+j}-v_{t+1+j}\right) +\lim \limits _{j\rightarrow \infty }\rho ^j\left( p_{t+j}-r_{t+j}\right) . \end{aligned}$$
(3)

I further assume that the single period return on a home is composed of the real risk-free rate, \( i_t \), and a risk premium, \( \varphi _t \), such that \( v_t=i_t+\varphi _t \).Footnote 10 Thus, Eq. (3) can be rewritten as

$$\begin{aligned} p_t-r_t=\frac{\kappa }{1-\rho }+\sum _{j=0}^{\infty }\rho ^j \left( \varDelta r_{t+1+j}-i_{t+1+j}-\varphi _{t+1+j}\right) +\lim \limits _{j\rightarrow \infty }\rho ^j\left( p_{t+j}-r_{t+j}\right) . \end{aligned}$$
(4)

Equation (4) holds ex post (since it follows from an identity). Hence, it must hold ex ante in expectations, conditioned on the information set available at time t . Thus, we can take conditional expectations and relate the current price to rent ratio to expected rent growth rates, risk-free rates and risk premiums.

$$\begin{aligned} p_t-r_t=\frac{\kappa -\varphi }{1-\rho }+{\mathbb {E}}_t\sum _{j=0}^{\infty }\rho ^j \left( \varDelta r_{t+1+j}-i_{t+1+j}\right) +{\mathbb {E}}_t\lim \limits _{j\rightarrow \infty }\rho ^j\left( p_{t+j}-r_{t+j}\right) , \end{aligned}$$
(5)

where \( {\mathbb {E}}_t \) is the expectation operator, and where I have assumed a constant expected risk premium, i.e., that \( {\mathbb {E}}_t(\varphi _{t+1})=\varphi \).Footnote 11 \({^{,}}\) Footnote 12

According to Eq. (3), home prices today are high relative to rent if investors expect some combination of high rent growth rates and low interest rates, or, investors expect prices to rise at a faster rate than rent forever. The latter case is commonly referred to as a rational bubble.

Equation (5) can be decomposed into two components,

$$\begin{aligned} p_t-r_t=f_t+b_t. \end{aligned}$$
(6)

The first component in the right-hand side of Eq. (6), \( f_t \), is the fundamental component, given by

$$\begin{aligned} f_t=\frac{\kappa -\varphi }{1-\rho }+\sum _{j=0}^{\infty }\rho ^j {\mathbb {E}}_t\left( \varDelta r_{t+1+j}-i_{t+1+j}\right) , \end{aligned}$$
(7)

which is stated only in terms of the fundamental factors–the risk premium and the expected paths of rent growth rates and risk-free rates. This relation commonly referred to as the Gordon growth model (Campbell and Shiller 1988).

The second component in the right-hand side of Eq. (6), \( b_t \), is the rational bubble, given by

$$\begin{aligned} b_t={\mathbb {E}}_t\lim \limits _{j\rightarrow \infty }\rho ^j \left( p_{t+j}-r_{t+j}\right) . \end{aligned}$$
(8)

If the transversality condition, \( \lim _{j\rightarrow \infty }\rho ^j\left( p_{t+j}-r_{t+j}\right) =0 \), holds, the log price to rent ratio does not explode.Footnote 13 That is, no bubble exists and the observed ratio equals the fundamentally implied ratio. In contrast, the existence of a bubble component is a situation where the price to rent ratio exceeds what is implied by fundamentals. The latter case is consistent with investors who expect to be compensated for overpayment by the expected appreciation of the bubble component. That is, investors buy homes since they expect to sell them for a higher price in the future. In essence, this behavior describes the general notion of a bubble quite intuitively.

The presence of such a component is consistent with the rational expectations hypothesis, hence the term, “rational” bubble. In fact, adding any process that satisfies the following explosive (sub-martingale) property

$$\begin{aligned} {\mathbb {E}}_t(b_{t+1})=\rho ^{-1}b_t=\left[ 1+e^{\left( \overline{r-p} \right) }\right] b_t \end{aligned}$$
(9)

to \( f_t \) solves Eq. (2).Footnote 14 Footnote 15

The condition given in Eq. (9) implies that in the presence of a bubble component, \( p_t-r_t \) will manifest explosive autoregressive behavior. This is because the explosiveness property of the bubble component sooner or later will dominate the stochastic properties of \( \varDelta r_t \) and \( i_t \), regardless of whether they are stationary or integrated of order one. Hence, under the assumption of a constant expected risk premium, testing for a rational housing bubble in this model is equivalent to testing whether \( p_t-r_t \) has a root greater than one, while verifying that neither \( \varDelta r_t \) nor \( i_t \) have explosive roots.

Nonetheless, we cannot rule out other possible combinations of stochastic properties that may exist. For example, if evidence for explosiveness is found in \( p_t-r_t \) and in either one of the fundamental factors, no conclusive inference on the existence of a bubble in \( p_t-r_t \) can be made. Alternatively, finding that one of the fundamental factors is explosive, while the same does not apply to \( p_t-r_t \), may be interpreted as evidence against the underlying model.

2.2 Implications for econometric tests for bubbles

Several rational bubble detection strategies were developed over the past three decades based on insights arising from variations of the model described above.Footnote 16 Diba and Grossman (1988a) were among the first to suggest testing for bubbles by using unit root and cointegration tests on stock prices and dividends.Footnote 17 They find that stock prices and dividends are integrated of the same order (one) and that they are cointegrated. Based on these results, Diba and Grossman conclude that the no-bubble hypothesis cannot be rejected for US stock prices. Evans (1991) criticized the work of Diba and Grossman and the use of unit root and cointegration tests due to their power loss in the presence of a periodically collapsing bubble, i.e., a bubble that repeatedly emerges and bursts (but remains at positive levels at all times). Intuitively, this power loss phenomenon comes from the fact that a time series containing a complete cycle of a bubble tends to appear more like a stationary series rather than a unit root, due to the apparent ’mean-reversion’ caused by the tendency of the bubble to burst after the preceding run-up stage. This in turn biases unit root tests toward rejection of the null.

More recently, Phillips et al. (2011, hereinafter PWY) show how to overcome the low power problem in the presence of a periodically collapsing bubble. PWY’s method is based on using recursive right-tail unit root tests where the null of a unit root is tested against the alternative of a mildly explosive process.Footnote 18 In this case, the null hypothesis is of no-bubble and a rejection of this null is interpreted as evidence for a bubble.Footnote 19 PWY’s method is also designed to consistently estimate the origination and termination dates of a bubble (if it exists). This date-stamping feature can also be used as a real-time monitoring device.Footnote 20 Homm and Breitung (2012) compare PWY’s method to other common bubble detection methods and find, using Monte Carlo simulations, that it indeed has increased power in the detection of periodically collapsing bubbles and that it performs relatively well as a real-time monitoring device.

Before continuing, a comment is warranted. Econometric tests for rational bubbles, including PWY’s method, usually formulate the null hypothesis as ’no-bubble’. Thus, rejection of the null might lead one to conclude that a bubble is present in the data. Unfortunately, all that these bubble tests can show us is whether the data we observe are inconsistent with the null, since rejection is only possible within a specified model.Footnote 21

3 Econometric methodology

Implementing PWY’s test for bubbles is quite straightforward. The procedure involves recursive estimates of the Dickey and Fuller (1979) \( \tau \)-statistic, where the basic empirical specification used is the following standard Augmented Dickey–Fuller (ADF) auxiliary regression:

$$\begin{aligned} y_t=\mu +\delta {y_{t-1}}+\sum _{i=1}^{k}\phi _i\varDelta {y_{t-i}}+\varepsilon _t,\quad \varepsilon _t\sim \hbox {iid}\left( 0,\sigma ^2\right) \end{aligned}$$
(10)

where \( y_t \) is the time tested for explosiveness, \( \mu \) is the intercept, \( \delta \) is the autoregressive coefficient, k is the maximum number of lags, \( \varDelta \) is the difference operator, \(\phi _i \) for \( i=1\ldots k \) are the coefficients of the lagged first difference and \( \varepsilon _t \) is an iid error term.

Traditionally, Eq. (10) is used to test the null of a unit root against the alternative of stationarity. Nonetheless, the same equation can be used to carry out a test for a mildly explosive root.Footnote 22 Formally we test for:

$$\begin{aligned}&H_0: \delta =1 \,\text {(no-bubble)}\\&H_1: \delta >1 \,\text {(bubble)} \end{aligned}$$

using the ADF statistic, defined as

$$\begin{aligned} \text {ADF}=\frac{\hat{\delta }}{\text {SE}(\hat{\delta })}, \end{aligned}$$
(11)

where \( \hat{\delta } \) is the OLS estimate of \( \delta \) and SE stands for ’standard error’.

Before proceeding to a more detailed description of PWY’s testing procedure, some notation is required. First, assume a sample interval of [0, 1] .Footnote 23 Next, denote by \( \delta _{r_2}^{r_1} \) and by ADF \(_{r_2}^{r_1}\) the autoregressive coefficient from Eq. (10) and its corresponding ADF statistic, respectively, when both are estimated over the (normalized) sample \( [r_1,r_2] \), where \(r_1\) and \(r_2\) are fractions of the sample such that \( 0<r_1<r_2<1 \). Finally, denote by \( r_w \) the (fractional) window size of the regression, defined by \( r_w=r_2-r_1 \) and \( r_0 \) as the fixed initial window, set by the user.

The supremum ADF (SADF) test proposed by PWY is based on recursive calculations of the ADF statistics with an expanding window. The estimation procedure proceeds as follows (see Fig. 2): First, we set the first observation of the sample as the starting point of the estimation window, i.e., \( r_1=0 \). In the next step, we set the end point of the initial estimation window, \( r_2 \), according to a choice of a minimal window size, \( r_0 \), such that the initial window size is defined as \( r_w=r_2-r_1=r_2 \). Finally, we recursively estimate \( \delta ^0_{r_2} \) using Eq. (10) and calculate its corresponding ADF \(^0_{r_2} \) statistic, incrementing the window size, \( r_2\in [r_0,1] \), one observation at a time. In the final step, estimation is based on the whole sample (i.e., \( r_2=1 \) and the ADF statistic is ADF \(^0_1 \)). The SADF statistic, as defined by PWY, is the supremum value of the ADF \(^0_{r_2} \) sequence for \( r_2 \in [r_0, 1] \):

$$\begin{aligned} \begin{aligned} \text {SADF}(r_0) =&\sup _{r_2\in [r_0,1]}\lbrace \text {ADF}^0_{r_2}\rbrace . \end{aligned} \end{aligned}$$
(12)

The distribution of the SADF statistic under the null hypothesis has a nonstandard form. Asymptotic and finite sample critical values are obtained by Monte Carlo simulation methods. Accordingly, if the SADF statistic is larger than the corresponding critical value, we reject the null hypothesis of a unit root in \(y_t\) in favor of a mildly explosive process.

Fig. 2
figure 2

Illustration of the SADF test procedure. Notes: Set \( r_1=0 \) and \(r_2\in [r_0,1]\). Next, use \( [0,r_2] \) as the initial window and vary \( r_2 \). At each step, \( r_w=r_2 \) is the window width

3.1 Date-stamping bubble periods and monitoring

As mentioned in the previous section, the PWY procedure can also be used to consistently estimate the origination and termination dates of a bubble. Thus, if the null hypothesis of no-bubble is rejected, we can, under general regularity conditions, consistently estimate the bubble period (Phillips and Yu 2009). Moreover, Homm and Breitung (2012) and Phillips et al. (2015b) show that these date-stamping procedures can be used not only as an ex post dating strategy but also for real-time monitoring of bubbles.

The date-stamping procedure is based on comparing each element of the ADF\(^0_{r_2}\) sequence to its corresponding right-tailed critical value which is based on a sample size of \({Tr_2} \) observations.Footnote 24 The estimated origination point of a bubble, denoted by \({r_e}\), is the first chronological observation in which ADF\(^0_{r_2}\) crosses its corresponding critical value from below. The estimated termination point, denoted by \( {r_f} \), is the first chronological observation which comes after \( {r_e} \) in which the ADF\(^0_{r_2}\) crosses its critical value from above. Formally, the estimates of the bubble period are given by

$$\begin{aligned} \hat{r}_e&=\inf _{r_2\in [r_0,1]}\left\{ r_2:\text {ADF}^0_{r_2}>cv_{r_2}^{\beta _{T}} \right\} \end{aligned}$$
(13)
$$\begin{aligned} \hat{r}_f&=\inf _{r_2\in [\hat{r}_e,1]}\left\{ r_2:\text {ADF}^0_{r_2}<cv_{r_2}^{\beta _{T}} \right\} \end{aligned}$$
(14)

where \(cv_{r_2}^{\beta _{T}}\) is the \( 100(1-\beta _{T})\% \) critical value of the standard ADF statistic based on \( [Tr_2] \) observations.Footnote 25 \(^{,}\) Footnote 26

Another procedure used for monitoring purposes is the CUSUM test suggested by Homm and Breitung (2012). This test is designed to detect a regime shift between a unit root process and an explosive root process in real time. Let \( t_0=\lfloor Tr_0 \rfloor \) be the training sample and let \([t_0+1,t_2]\) be the monitoring interval, where \( t_2=\lfloor Tr_2 \rfloor \) is the latest observation of the monitoring interval. The CUSUM statistic is defined as

$$\begin{aligned} \text {CUSUM}^{t_2}_{t_0}=\frac{1}{\hat{\sigma }_{t_0}^2}\sum _{j=t_0+1}^{t_2}\varDelta y_j=\frac{1}{\hat{\sigma }_{t_0}^2}\left( y_{t_2}-y_{t_0}\right) , \end{aligned}$$
(15)

where \(\hat{\sigma }_{t_0}^2\) is a consistent estimate of the variance of \( \varDelta y_t \) over the sample \( [1,t_0] \). Accordingly, detection of a shift toward a bubble regime is made when the CUSUM statistic crosses its critical values sequence (at some predefined significance level) from below.

3.2 Indirect inference and confidence intervals

Statistical inference on the least squares (LS) estimate of \( \delta \) suffers from two drawbacks. First, under the mildly explosive alternative we cannot use the standard confidence intervals. Instead, as shown by (Phillips and Magdalinos 2007), a correct \(100(1-\alpha )\% \) confidence interval for the LS estimator of \( \hat{\delta }_n \) is given by

$$\begin{aligned} \left[ \hat{\delta }_n-\frac{(\hat{\delta }_n)^2-1}{(\hat{\delta }_n)^n}C_\alpha ,\, \hat{\delta }_n+\frac{(\hat{\delta }_n)^2-1}{(\hat{\delta }_n)^n}C_\alpha \right] , \end{aligned}$$
(16)

where \( C_\alpha \) is the two-tailed percentile critical value of the standard Cauchy distribution.Footnote 27

The second drawback comes from the fact that the LS estimate of \( \delta \) is known to be biased downward in finite samples. Hence, using the confidence intervals shown above with the LS point estimate of \( \delta \) might be misleading.Footnote 28 Phillips et al. (2011) show how to correct the bias by applying the indirect inference method. Accordingly, H paths of an AR(1) process for \( y_t \) are simulated for different values of \(\delta \in \varPhi \), where \( \delta \) is the autoregressive coefficient and \( \varPhi \) is the parameter space. Let \( \hat{\delta }^\mathrm{LS}_h(\delta ) \) denote the LS estimator of \( \delta \), given a path h , where \( h=1,\ldots ,H \) and let \( \bar{\delta }^\mathrm{LS}(\delta )\) be the mean of \( \hat{\delta }^\mathrm{LS}_h(\delta ) \) over h , i.e.,

$$\begin{aligned} \bar{\delta }^\mathrm{LS}_H(\delta )=\frac{1}{H}\sum _{h=1}^{H} \hat{\delta }^\mathrm{LS}_h(\delta ). \end{aligned}$$
(17)

The indirect estimator, \( \tilde{\delta }_H \), is defined as

$$\begin{aligned} \tilde{\delta }_H=\underset{\delta \in \varPhi }{\text {argmin}}\parallel \hat{\delta }^\mathrm{LS}-\bar{\delta }^\mathrm{LS}_H(\delta ) \parallel , \end{aligned}$$
(18)

where \( \parallel \cdot \parallel \) is some finite dimensional distance metric and \( \hat{\delta }^\mathrm{LS} \) is the LS estimate of \( \delta \) from the actual data.

4 Data

The data on home and rent prices are taken from the Israeli Central Bureau of Statistics (CBS). I use seasonally unadjusted monthly observations on the price to rent ratio at the national level, composed of the Prices of Dwellings Index (hereinafter: the home prices index) and on the Owner Occupied Dwellings Services Price Index (hereinafter: the rent index).Footnote 29 I use data on the 1-year real risk-free rate, measured by the difference between the Bank of Israel (BOI) nominal interest rate (for 1 year) and 1-year expected inflation, both obtained from the Bank of Israel. The latter is measured by the yield spread between inflation-adjusted and nominal Israeli government bonds with 1 year maturity.Footnote 30 \(^{,}\) Footnote 31. The sample I use covers the period from January 1999 to July 2013 and includes 175 observations. The choice of this specific sample is due to the availability of data. For further details on the data, see “Appendix 1”.

Table 1 presents summary statistics for the log price to rent index (denoted as \( p_t-r_t \)), log real rent index (\( r_t \), deflated by the CPI) and the short-term real risk-free rate (\( i_t \)). Several notable observations arise from the table. First, the price to rent ratio reached a peak in February 2013. The latest observation available (July 2013) is around 2 % lower than the peak. Second, the real risk-free rate reached a record low of \(-\)1.9 % in the midst of the recent global financial crisis (June 2009), due to the drop in the Bank of Israel interest rate (notwithstanding the drop in expected inflation.) Third, real rent is at its peak at the end of the sample, reflecting a 20 % increase since July 2008. And finally, all series possess a high degree of persistence, even at the 12th lag (see the last three columns of Table 1). This high degree of autocorrelation is evident in the apparent nonstationary nature of these series.

Table 1 Summary statistics (1999:M1–2013:M7)

Figure 3a–c depicts the developments in real home price, real rent and the price to rent ratio over the sample period (all presented in natural logarithms). The motivation behind this study is based on the steep rise seen in the home prices index circa 2008 (see Fig. 3a).Footnote 32 Home prices increased during 2008–2013 by approximately 50 % in real terms, averaging an annual growth rate of nearly 10 %. However, prior to the recent run-up, there was a continuous period of real price depreciation. Though not presented in this figure, this real depreciation lasted for more than a decade. Despite following an upward trend since 2008, the rent index does not show the same rapid expansion pattern as prices. The short-term risk-free rate is depicted in Fig. 3d. The series appear to follow a downward sloping trend as of the beginning of the 2000s. This trend is most likely the result of the disinflation process undergone by the Israeli economy following the stabilization program of 1985.Footnote 33 Arguably, the most important feature in the context of the recent developments in home prices is the big decline of the risk-free rate seen right after the outbreak of the global financial crisis.

Fig. 3
figure 3

National-level time series plots (1999:M1–2013:M7). Notes: Log home prices (\( p_t \)) and log rent (\( r_t \)) indices are deflated by the CPI. Log price to rent ratio (\( p_t-r_t \)) is an index, normalized to January \(2000 =0\). The real risk-free rate is the difference between the Bank of Israel interest rate and expected inflation (see “Appendix 1” for further details). a Log real price \( p_t \). b Log real rent \( r_t \). c Log price to rent ratio \( p_t-r_t \). d Log gross real risk-free rate \( i_t \)

For the regional-level analysis I use quarterly data on the mean home prices and rent (the latter is transformed into annual terms) sorted by nine regions and given in current shekel prices (I use seasonally unadjusted data.) The regional data’s sample covers the period from 1998:Q1 to 2013:Q2 and includes 62 quarterly observations. (See “Appendix 1” for further details.) In this study I focus on the 3.5–4 room apartments segment. I do so since quality-adjusted data (such as a regional hedonic price indexes) are not available. I argue that using this specific segment roughly controls for quality. Moreover, the 3.5–4 room apartments segment represents the median apartment (out of the stock) in Israel, and also constitutes the vast majority of transactions.

Fig. 4
figure 4

Log price to rent at the regional level (1998:Q1–2013:Q2). Notes: The data on prices and rent used for constructing the price to rent ratio for each region are for 3.5–4 room apartments within each specific region

The nine regional log price to rent ratios are plotted in Fig. 4. As we can see from the graphs, the price to rent ratio in all nine regions is currently high compared to historical levels, where the upward trend for most regions started somewhere during the mid to late 2000s. However, the dynamics of the regional price to rent ratios in the last 5 years are quite heterogeneous. Some of the regions–namely, Center, Gush-Dan, Haifa and North–exhibit the same pattern seen at the national level, i.e., a rapid rise in the ratio, while the other regions display a rather stable growth path during the same period. The presence of heterogeneous dynamic patterns highlights the importance of regional-level analysis since it potentially enables us to detect regional housing bubbles that otherwise would have been missed within a national-level analysis because of the averaging nature of the aggregate ratio.Footnote 34

5 Results

5.1 National level

Before I provide a description of the main findings at the national level, I briefly discuss the specifications used for deriving these results. The SADF statistic and critical values are calculated for the log price to rent ratio, log real rent and log gross real risk-free rate by recursive estimations of Eq. (10) for each individual variable. The conduct of all these tests and critical values simulations are performed using the ‘rtadf’ EViews Add-in (Caspi 2013) and MATLAB. The optimal lag length is chosen by the Schwartz information criterion (SIC) when estimating Eq. (10) for the whole sample (with the maximum number of lags set to 12). Accordingly, the lag length in the recursive procedure is set to 3 for the log ratio and for log real rent, and to 2 for the log gross rate.Footnote 35 \({^{,}}\) Footnote 36 The SADF statistic is recursively estimated with an initial widow size of 36 observations, i.e., 3 years, which constitutes 20 % of the sample. This choice of initial window size relies on Phillips et al. (2011) and Phillips et al. (2015b) who use a window size of approximately 3 years for monthly data. Though my choice of minimal window size is arbitrary and not data driven, my results nonetheless are shown to be robust to different choices of window sizes and lags. (See Sect. 6.1.) In deriving the critical values for the SADF statistic, I set the data generating process (DGP) for the null to a random walk without a drift as in Phillips et al. (2011).Footnote 37

Table 2 Results of the ADF and SADF tests

Table 2 presents the standard ADF statistic and the SADF statistic for all variables, as well as their corresponding right-tail critical values for the sample of 1999:M1–2013:M7. The table also shows the date where the ADF\( _0^{r_2} \) sequence has reached its maximum (i.e., the date which corresponds to the SADF statistic.) The ADF statistic is estimated over the whole sample and is mostly used for comparison reasons and not for inference on bubbles. As we can see, according to the SADF test statistic, the null of no-bubble in the log price to rent ratio cannot be rejected at conventional significance levels—the SADF statistic is well below the 90 % critical value of 0.592 needed to reject the null. Furthermore, the same null cannot be rejected for both of the fundamental factors—log rent and the risk-free rate. The SADF statistics valued at \(-\)1.495 for \( r_t \) and 0.983 for \( i_t \) are also well below their corresponding 90 % critical values.Footnote 38 \({^{,}}\) Footnote 39

The SADF statistics of the log price to rent ratio differ from the ADF statistic estimated using the whole sample. That is, the latest value of the ADF\(^0_{r_2} \) sequence (July 2013) is not the maximal value of the sequence. Accordingly, the ADF\(^0_{r_2} \) statistic, valued at \(-\)0.237, corresponds to the sample that ends at 2002:M6. The point estimate of the autoregressive coefficient for the sample 1999:M1–2013:M2 is 0.987 and the (bias adjusted) indirect inference estimator is 1.009 where its 95 % confidence interval, calculated according to Eq. (16), lies between 1.057 and 0.962.Footnote 40 This result is in accordance with the results of the SADF test, namely the non-rejection of the null of unit root (no-bubble.)Footnote 41

Figure 5 plots the sequence of ADF\( ^0_{r_2} \) statistics (solid, blue) together with its corresponding sequence of critical values (dotted, red). As we can see, despite not being currently at its peak, the ADF \( ^0_{r_2} \) is relatively high compared to its historical level. In addition, we see that the test statistics sequence for the price to rent ratio has recently gotten ’closer’ to the 95 % critical value threshold. Hence, although we are unable to reject the null of no-bubble, we do see an upward rising trend toward this threshold ever since late 2009. This highlights the importance of the real-time monitoring aspects of the PWY strategy. Crossing the rejection threshold at some point in the future may serve as an early warning of price distortions.Footnote 42

5.2 Interest rates

Recall that in the previous section I proxy the risk-free rate using the difference between the Bank of Israel interest rate and expected inflation. Though this seems like the reasonable “real world” counterpart to the theoretical \( v_t \), there may be other relevant interest rates investors face, each capturing some of the special features of the housing market. Other rates include the longer-term real rate and/or the mortgage rate. The former rate is justified by the fact that buying a home is a long-term decision that must incorporate more forward looking behavior, while the latter is the explicit interest rate most home buyers face due to their ability to use leverage.

Fig. 5
figure 5

Results of the SADF date-stamping procedure for The log price to rent ratio (1999:M1–2013:M7). Notes: The figure presents the results of the SADF (\( r_0 = 36 \)) procedure for the natural logarithms of log price to rent ratio index (dashed, green) for the sample period of 1999:M1–2013:M7. The recursive ADF sequence (solid, blue) was estimated with a 3-lag specification. The sequence of critical values (dotted, red) is derived using a Monte Carlo simulation with 2000 replications where the underlying data are generated by a random walk with normal iid errors. (Color figure online)

To verify the robustness of the main results I apply the same explosiveness test procedure used earlier on the (zero coupon) real interest rate on 10-year government bonds, which represents the long-term risk-free alternative-yield to purchasing a home, and on the average fixed real rate on new mortgages. The data on the zero coupon rate is obtained from the Bank of Israel.Footnote 43 The data on the average mortgage rate are obtained from the Bank of Israel Banking Supervision Department. The average mortgage interest rate is a weighted average of interest rates on new fixed-rate mortgages, where the weights are proportional to the new mortgages’ face value.

Table 3 reports the results of the ADF and SADF tests for the noted above different rates. The SADF statistic cannot reject the null of no bubble for either of these rates at conventional significance levels. These findings reinforce the lack of explosiveness found by using the short-term real risk-free rate described in Sect. 5.1. Nonetheless, these findings are less important for now since the SADF test for the log price to rent ratio does not point to the existence of a bubble.

Table 3 Results of the ADF and SADF tests for different interest rates

5.3 Leverage

Another important issue which we have ignored thus far is the fact that most home purchases use some amount of leverage (mostly mortgage loans from banks). The question arises as to whether incorporating leverage rates in the present value model affects the previous analysis. To answer this question, I follow Dovman et al. (2012) and present a modified version of the present value model that incorporates leverage.

Consider the following definition of the gross one-period return, \( \tilde{V}_t \), on holding a home that is partly financed by taking a mortgage:

$$\begin{aligned} \tilde{V}_t=\frac{P_{t+1}-I^m_t\lambda _t P_t+R_{t+1}}{\left( 1-\lambda _t\right) P_t}, \end{aligned}$$
(19)

where \( I^m_t \) is the gross mortgage rate and \( \lambda _t \) is the leverage rate. Eq. (19) states that the ex post one-period gross return \( \tilde{V}_t \) is the ratio between the income from period \( t+1 \)—future selling price plus rent minus the interest rate paid on the mortgage that covered a fraction \( \lambda _t \) of the home value at time t , and the equity paid in time t . Rearranging Eq. (19) yields

$$\begin{aligned} {V}_t=\frac{P_{t+1}+R_{t+1}}{P_t}, \end{aligned}$$
(20)

where now

$$\begin{aligned} {V}_t=\left( 1-\lambda _t\right) \tilde{V}_t+\lambda _t I^m_t \end{aligned}$$
(21)

is the gross return adjusted to leverage. In other words, \( {V}_t \) is the gross return left for the investor after paying down the mortgage (principle plus interest). Equation (20) is nearly identical to Eq. (1) apart for the definition of the gross return. The solution for (20) is thus similar to the one for the model without leverage, only now \( V_t \) is defined according to Eq. (21). Formally, the leverage-adjusted log price to rent ratio is given as

$$\begin{aligned} p_t-r_t=\frac{\kappa }{1-\rho }+\sum _{i=0}^{\infty }\rho ^i E_t\left( \varDelta r_{t+1+i}-{v}_{t+1+i}\right) +\lim \limits _{i\rightarrow \infty }\rho ^iE_t\left( p_{t+i}-r_{t+i}\right) , \end{aligned}$$
(22)

where now, \( {v}_t= \log \left[ \left( 1-\lambda _t\right) \tilde{V}_t+\lambda _t I^m_t\right] \).

In order to verify that \( {v}_t \) is not explosive we can use the simple fact that the leverage-adjusted gross return in the model is a convex linear combination of the risk-free rate (and the risk premium) \( \tilde{V}_t \) and the mortgage interest rate \( I^m_t \), where the weights are determined by the leverage rate \( \lambda _t \) such that \( 0\le \lambda _t\le 1 \). Thus, in order to conclude that \( {v}_t \) is non-explosive it is sufficient to verify that the risk-free rate and the mortgage rate are non-explosive.Footnote 44 Recalling the results presented in Table 3 that ruled out explosiveness in the average mortgage interest rate and the fact that none of the proxies for the risk-free rate are found explosive, we can conclude that \({v}_t\) is not explosive.

5.4 A comparison with other detection methods

To gain more perspective about the plausibility of my main findings, I compare the results with those of other bubble detection strategies. First, I apply the CUSUM procedure developed by Homm and Breitung (2012) to the log price to rent ratio.Footnote 45 To be consistent with the SADF procedure, I set the training sample to 36 observations. The 95 % critical values sequence of the test are obtained by Monte Carlo simulation with 10,000 replications, where the underlying data are generated by a random walk with normal iid errors.

The results of the CUSUM test are presented in Fig. 6. As we can see, the CUSUM test statistic (solid, blue) does not cross the 95 % critical value threshold anywhere during the sample, thus not indicating any shift toward an explosive regime. This, of course, is in line with the results of the SADF test presented in the previous section.

Fig. 6
figure 6

Results of the CUSUM monitoring procedure for the log price to rent ratio (2003:M12–2013:M7). Notes: The figure presents the results of the CUSUM test for the natural logarithms of log price to rent ratio index (dashed, green) for the sample period of 1999:M1–2013:M7. The CUSUM test statistics sequence (solid, blue) is calculated according to Eq. (15) where the training sample is set to 36 observations. The sequence of 95 % critical values (dotted, red) is derived using a Monte Carlo simulation with 10,000 replications, where the underlying data are generated by a random walk with normal iid errors. (Color figure online)

Next, I compare the results to Dovman et al. (2012) who also test for a housing bubble in Israel during 2008–2010.Footnote 46 Dovman et al. (2012) use several bubble identification schemes, and find no strong evidence of a bubble (as of August 2010). These schemes include: a direct estimate of the fundamental price using ex post or forecast rent payments and ex post and future interest rates (depending on the availability of the data); an estimate of the bubble as an unobserved component using a Kalman filter, as in Wu (1995); and an estimate of the fundamental price and rent to price ratio using linear regression and filtering techniques.

Before I turn to a comparison with the study by Dovman et al. (2012), a remark about the methodological differences is worth making. While Dovman et al. (2012) estimate different measures of the fundamental price and use it to derive the size of the bubble component, the PWY method I use indirectly identifies a bubble based on the dynamic properties of the data. Hence, my results can only give an answer as to whether the null of no-bubble can be rejected or not, without specifying a measure of magnitude of the bubble. In contrast, the methods used by Dovman et al. (2012) result in quantitative measures of the bubble component itself, without providing a clear statistical threshold to test whether such a bubble exists.

Figure 7 presents two of the housing bubble indicators developed by Dovman et al. (2012) (and updated to July 2013), based on the direct and the unobserved component methods, for the period of January 1999 to July 2013. Both these indicators are essentially estimates of the bubble components measured in terms of the percent deviations of the observed price from the fundamental price (in this case a value of zero indicates that no bubble exists). As we can see, and in contrast with my findings, both indicators point to the existence of a bubble over most of the sample in question. Measured price distortion reaches up to a 40 % overvaluation in the beginning of the sample, according to the direct method (blue, solid), and up to a 20 % overvaluation in July 2013 according to the unobserved component method (red, dotted). Moreover, there seem to be periods where the bubble component is found to be negative, thus violating the argument made by Diba and Grossman (1988b) about the impossibility of negative bubbles. This highlights the difficulty in measuring bubbles as residuals since these can also be the result of a misspecification of the model, and there is no easy way of distinguishing between the two (Gürkaynak 2008).

Fig. 7
figure 7

Alternative housing bubble indicators for Israel based on Dovman et al. (2012). Notes: Monthly data for the period January 1999–July 2013. Both indicators are in terms of percent deviations of observed prices from the fundamental price. These two bubble indicators are calculated in Dovman et al. (2012). In the direct method (solid, blue), the fundamental price is calculated as the sum of ex post rent payments discounted by forward rates. (The authors used forecast values for out-of-sample rent data.) In the unobserved component method (dashed, red), the bubble component is extracted as an unobserved variable using the Kalman filter. Source:Dovman et al. (2012), updated by the Bank of Israel. (Color figure online)

5.5 Regional level

Regional-level analysis is quite common in the empirical literature on housing bubbles (Himmelberg et al. 2005; Case and Shiller 2003; Smith and Smith 2006; Clark and Coggin 2011, to name a few.). This type of analysis takes into account the possibility of housing markets in different regions not being fully integrated. Accordingly, inference based on national-level data might be biased toward not rejecting the null of no-bubble in cases where only a small portion of the national housing market manifests explosive behavior. Hence, though we may be able to rule out the possibility of there being a housing bubble at the national level, we need to make sure we are not missing a housing bubble in one or more of Israel’s regions.

I now turn to an analysis of the log price to rent ratio at the regional level. I estimate the ADF and SADF statistics for nine regional log price to rent ratios. The minimal window size is set to 20 observations (5 years, \(\approx \)30 % of the sample).Footnote 47 The optimal lag length in all of the recursions is set according to the SIC when applying it to the whole sample (with the maximum number of lags set to 10.) Accordingly, the SIC selects an optimal lag length of zero for all regions but one. The exception is the Krayot region where the selected optimal number of lags is one. The SADF test is not applied to the risk-free rate since the same interest rate is relevant to all regions, and the results of the SADF test for the interest rate have already been presented above. Explosiveness test results for mean rent by region are not presented to preserve space, but none of them points toward a rejection of the null hypothesis.

Table 4 reports the results of right-tailed ADF and SADF (\( r_0=20 \)) tests for the regional log price to rent ratios. According to the SADF test statistics, reported in the column (2), the no-bubble hypothesis cannot be rejected for the pooled mean price to rent ratio of all nine regions (under the label “ALL” in the table) at conventional significance levels. The SADF statistic, valued at \(-\)1.002 (second column), is well below its 90 % right-tail critical value of 0.794. This finding is consistent with the results for the national level. Moreover, we cannot reject the null of no-bubble for all regions at conventional significance levels. The majority of the results of the SADF test for most regions are shown to be robust to different choices of lag length and window size. (See “Appendix 1”.) The only exception is the Gush Dan region where results are sensitive to the choice of lag length. For example, the null of no-bubble in the Gush Dan region is rejected at the 95 % significance level when the test equation includes 3 lags and at the 90 % level when it includes 4 lags.

Table 4 Results of the SADF test for the regional price to rent ratios

The ADF test statistic, reported in column (1), cannot reject the null of no-bubble at conventional significance levels for all regions. (The region which is “closest” to rejection is Jerusalem, where the p-value for such rejection is 0.14.) Moreover, the SADF statistic exceeds the ADF statistic for all regions. That is, the latest value of the ADF\(^0_{r_2} \) statistic sequence for all regions is below the maximal value of the sequence. Nonetheless, as is evident in the fourth column of the table, six out of the nine regions reached a peak somewhere during the last 2–3 years. This is consistent with the recent rapid price appreciation, yet it does not point to the existence of a bubble in any of these regions.

6 Sensitivity analysis

6.1 Lag length and minimal window size

It is well known that unit root test results may be sensitive to the specification of the test equation. Moreover, the need to set a minimum window size for the recursive procedure of the SADF test adds more complexity, which might result in additional sensitivity of the results to various chosen minimal window sizes.

Recall that in deriving the results for the national level (Sect. 5.1) I chose to estimate the ADF auxiliary regression with a lag length of 3 and to set the minimal window size to 36 observations. I now check the sensitivity of these results to different choices of lag length and window size. Before proceeding we must set some upper limit of the number of lags I experiment with. To do so, I note that when conducting the ADF test for the whole sample the AIC and SIC suggest using an optimal lag length of 3 and 6, respectively. Thus, for this analysis I set the what is chosen by the AIC as the upper bound for the lag length. As for the minimal window size, I choose a minimum length of 12 observations (i.e., 1 year). The choice of 12 observations as a lower bound is quite arbitrary, yet using less than 12 observations, especially when there are up to six lags included in the equation, seems unreasonable.

Table 5 shows the robustness of the main results, presented in Sect. 5.1, to various selections of lag lengths and window sizes.Footnote 48 As we can see, neither of the SADF statistics for a particular choice of lag length and window size is significant at the 95 % level. Moreover, we can see that the choice of lag length does not matter at the window size used to derive the main results (\( r_0 = 36 \)). The only case were we are able to reject the null at the 90% significance level is for \( r_0 = 24 \) and \( k=6 \). The evidence shown in Table 5 leads me to conclude that the rejection of the null of no-bubble at the national level is robust to these alternative specifications.

Table 5 Sensitivity of the SADF test for the log price to rent ratio to specifications of lag length and window sizes

6.2 Sample period

My choice of the sample period was dictated by the availability of the data. More specifically, January 1999 is the first month when the rent index in its current format of renewed contracts took effect. However, the home prices index in its current format is available since January 1994.Footnote 49 To see whether using a longer sample has any effect on the main findings, I follow Dovman et al. (2012) and concatenate data from the existing rent contracts index for the period of January 1994 to December 1998 to the Owner Occupied Dwellings Services Price Index (which includes new and renewed contracts).Footnote 50 Figure 8 shows the extended price, rent and price to rent ratios (in logs) for the period starting from January 1994. One interesting feature shown by the graphs is that the current level of the log price to rent ratio is lower than its level in the mid-1990s.

Fig. 8
figure 8

Time series plots—extended sample (1994:M1–2013:M7). Notes: Log home prices (\( p_t \)) and log rent (\( r_t \)) indices are deflated by the CPI. Log price to rent ratio (\( p_t-r_t \)) is an index, normalized to January \(2000 = 0\). The real risk-free rate (\( v_t \)) is given by the difference between BOI interest rate and expected inflation (see “Appendix 1” for further details). a Log real price \( p_t \). b log real rent \( r_t \). c log price to rent ratio \( p_t-r_t \)

The results of the right-tail ADF and SADF test statistics and critical values for the sample starting from January 1994 are reported in Table 6 along with a comparison with the results from the previous section. None of the test statistics for the ratio, rent and interest rate are able to reject the null of no-bubble in the longer sample.

Table 6 Results of the test for explosive behavior using an extended sample

Next, I check whether the results within the original sample (1999:M1–2013:M7) are sensitive to the choice of the initial observation. To do so I implement the generalized SADF (GSADF) suggested by Phillips et al. (2015b, hereinafter: PSY). This strategy generalizes the SADF test by allowing a more flexible estimation window, wherein, unlike the SADF procedure, the starting point, \(r_1\), is not fixed at 0 but rather is allowed to vary within the range \( [0,r_2-r_0] \). Formally, the GSADF statistic is defined as

$$\begin{aligned} \begin{aligned} \text {GSADF}(r_0)&= \sup _{\begin{array}{c} r_2\in [r_0,1] \\ r_1\in [0,r_2-r_0] \end{array}} \lbrace \text {ADF}_{r_1}^{r_2}\rbrace . \end{aligned} \end{aligned}$$
(23)

Essentially, the GSADF procedure estimates all possible subsamples of some arbitrary minimum size and above, calculates the ADF\( _{r_2}^{r_1} \) statistic for each of these subsamples (note that now the ADF statistic also depends on the starting point of the window, \( r_1 \)), and finds the maximal value of the ADF\( _{r_2}^{r_1} \) sequence. This maximal value is defined as the GSADF statistic. Phillips et al. (2015b) show that the distribution of the GSADF statistic exists and has a nonstandard form. Thus, critical values are obtained by using Monte Carlo simulation methods. Accordingly, if the GSADF statistic is larger than the corresponding critical value, we reject the null hypothesis of a unit root in favor of a mildly explosive process.Footnote 51

The GSADF statistic and its corresponding critical values for the log price to rent ratio are presented in Table 7 along with a comparison to the previously calculated standard ADF statistic and the SADF statistic. To be consistent with the SADF procedure, I set the lag length to 3 lags and the minimum window size to 36 observations. As we can see, the GSADF statistic for the price to rent ratio is 0.392, well below the 90 % critical value of 1.281. Thus, the null of no-bubble hypothesis can also not be rejected by the GSADF statistic.

Table 7 Initial starting point sensitivity check—the GSADF test

7 Conclusion

This paper examines whether the run-up in Israeli home prices during 2008–2013 (10 % average real annual growth rate) reflects a housing bubble. I address this question by applying the dynamic Gordon growth model and econometric bubble detection and monitoring strategies proposed by Phillips et al. (2011), Phillips et al. (2015b) and Homm and Breitung (2012) to Israeli housing market data at the national level (1999:M1–2013:M7) and to nine regions (1998:Q1–2013:Q2). Overall, the null hypothesis of no-bubble cannot be rejected for the Israeli data. The results for the national-level data and for most of the regional-level data are robust to a variety of tests, model specifications and to consideration of leverage and mortgage rates. One exception is the Gush Dan region, where results are rather inconclusive due to their sensitivity to the choice of lag length in the test equation. I conclude that the recent run-up in home prices is likely to be the outcome of changes in fundamental factors (rent and interest rates). The integrated theoretical and empirical frameworks presented here for the Israeli case could be applied to other countries sharing similar developments in their housing markets.