Speculative Bubbles in Present-Value Models: A Bayesian Markov-Switching State Space Approach∗

We estimate the dynamics of a speculative bubble subject to a surviving and a collapsing regime together with the dynamics of dividends and returns in a tractable state space specification of the present-value model. To estimate this new highdimensional model, we develop an efficient Markov chain Monte Carlo sampler to simulate from the joint posterior distribution. We find that real-world stock price bubbles show significant Markov-switching structure. Further, the results indicate that dividend growth rates are highly predictable. Finally, we find that bubble variation explains a large share of the variation in the price-dividend ratio and unexpected return.


Introduction
The 2008 meltdown of the global financial market has attracted renewed attention on speculative bubbles among academics and policy-makers. Speculative bubbles might be the answer to the question posed by Shiller (1980): If not dividend growth or expected returns, what does move prices? When investors share the belief that a variable or a group of variables, not related to fundamentals, influences prices, it is rational to include this piece of information into prices (Diba and Grossman, 1988). In this context, an explosive behaviour of stock prices is still consistent with a rational behaviour of economic agents. Experimental evidence has also confirmed that bubbles are fueled by symmetrically informed traders (Asako et al., 2020). Recently, Zheng (2020) has found that investors' coordination on fundamental strategy impacts the occurrence and burst of the bubble. This paper aims at detecting speculative bubbles in stock-price data by jointly studying the return and dividend dynamics. Specifically, we want to capture the information in the present-value relations among price-dividend ratios, expected returns, expected dividend growth rates and an eventual rational bubble. Thus, we contribute to both the present-value literature in the spirit of Campbell and Shiller (1988), and the literature on the identification of speculative bubbles.
Variation through time in the price-dividend ratio on corporate stocks conveys essential information about expected returns or expected dividend growth rates (Campbell and Shiller, 1988). Recently, Binsbergen and Koijen (2010) have pioneered a latent variables approach to estimate the expected returns and expected dividend growth rates of the aggregate stock market. They find that returns and dividend growth rates are predictable with R 2 values ranging from 8.2% to 8.9% for returns and 13.9% to 31.6% for dividend growth rates. More recently, Choi et al. (2017) have shown that incorporating regime shifts in the mean of price-dividend ratios into the present value model of Binsbergen and Koijen (2010) increases in-sample predictability. Another extension of Binsbergen and Koijen (2010) is considered by Piatti and Trojani (2017), who use a latent variables approach to estimate a present-value model with time-varying risk.
Despite the relevance of the phenomenon of periodically collapsing bubbles in stock prices, present-value approaches above do not account for it. This paper proposes to incorporate a speculative bubble subject to a surviving and a collapsing regime into the state space framework by Binsbergen and Koijen (2010). Specifically, we contribute to the literature on the present-value model in the spirit of Campbell and Shiller (1988) by allowing prices to deviate from fundamentals because of a latent rational bubble component subject to a surviving and a collapsing regime. Our framework allows us to estimate expected returns and expected dividend growth rates, as well as to identify bubble's collapse dates. This paper also contributes to the literature on the identification of speculative bubbles. Within the field, empirical papers have mainly proposed two different approaches for the detection of bubbles: indirect and direct bubble tests. 1 The first group of studies is based on the so-called indirect bubble tests. Here, the authors apply sophisticated cointegration and unit-root tests to a dividend-price relationship (see, e.g., Diba and Grossman, 1988;Evans, 1991;Froot and Obstfeld, 1991;Hall et al., 1999;Sarno and Taylor, 2003;Bohl, 2003;Bohl and Siklos, 2004;Kanas, 2005;Jiang and Lee, 2007;McMillan, 2007;Cerqueti and Costantini, 2011;Phillips et al., 2011;Chen et al., 2016). Among the indirect tests to detect bubbles, some researchers have proposed a Bayesian approach; among them Li and Xue (2009) ;Miao et al. (2015); Check (2014); Shi and Song (2014); Fulop and Yu (2017).
The second group of studies, which are more relevant to this work, implements direct tests for speculative bubbles by explicitly formulating the existence of a bubble in the alternative hypothesis (see West, 1987;Wu, 1997;Balke and Wohar, 2009;Al-Anaswah and Wilfling, 2011;Lammerding et al., 2013). The basic idea in the seminal paper of West (1987) is to compare two alternative estimators for the set of parameters needed to compute the expected present discounted values of a stock's dividend stream, where expectations is conditional on current and past dividends. Specifically, West (1987) constructs one set of estimates by regressing the stock price on a suitable set of lagged dividends. Instead, the other set of estimates is obtained using a pair of equations with one being an arbitrage equation yielding the discount rate, and the other being the ARIMA equation of the dividend process. Then the Hansen and Sargent (1980) formulas may be applied to this pair of equations to obtain a second set of estimates of the expected present discounted value parameters. Under the null hypothesis of no-bubble, the two sets of estimates should be the same, apart from sampling error. West (1987) finds that the test usually rejects the null hypothesis of no bubbles for the US market. More recently, Wu (1997) suggests a state space representation of the deviations of stock prices from the present-value model in which the bubble is included as an unobservable component. In Wu (1997), dividends are assumed to follow an autoregressive process. The analysis attributes large portions of stock price movements to speculative bubbles in the S&P 500. Al-Anaswah and Wilfling (2011); Lammerding et al. (2013) extend the state space model in Wu (1997) by allowing the bubble to switch between two alternative regimes, namely an explosive and a stationary regime. Al-Anaswah and Wilfling (2011) adopt the methodology of Kim and Nelson (1999) to identify regime-switching of speculative bubbles in stock price monthly data. Differently, Lammerding et al. (2013) propose a Bayesian approach to estimate the Markov-switching state space model of speculative bubbles in oil price data. This paper extends the state space model in Al-Anaswah and Wilfling (2011); Lammerding et al. (2013) by adding a bubble to the state space framework of Binsbergen and Koijen (2010) which model expected returns and expected dividend growth rates as latent variables. Thus, our state space model includes three latent variables namely, expected returns, expected dividend growth rates, and a rational bubble as opposed to one (Wu, 1997;Al-Anaswah and Wilfling, 2011;Lammerding et al., 2013). Moreover, we adopt a Markov-switching approach to identify the bubble's collapsing and surviving regimes. In contrast to Al-Anaswah and Wilfling (2011); Lammerding et al. (2013), we allow regime-switching also in fundamentals.
Markov-switching models (Hamilton, 1989) have been used extensively in the bubble literature. Driffill and Sola (1998) allow fundamentals to switch between alternative regimes in a stock price model which includes an intrinsic bubble. Hall et al. (1999) propose a univariate Markov-switching Augmented-Dickey-Fuller test to detect bubble episodes, later extended by Shi (2013) to allow for heteroskedasticity. Further, Brooks and Katsaris (2005) show that a three-regime model that allows for dormant, explosive and collapsing speculative behaviour can explain the dynamics of the S&P 500. Shi and Song (2014) propose an infinite hidden Markov model, which allows for an infinite number of regimes to detect, date stamp and estimate speculative bubbles. In a recent contribution, Fulop and Yu (2017) have developed a new model for real time bubbles detection where the dynamic structure of the asset price, after the fundamental value is removed, is subject to two different regimes.
A common drawback of the bubble literature is that rejection of the present-value model that are interpreted as evidence of the presence of bubbles can still be explained by alternative structures of the fundamentals. In this paper we mitigate this issue in two ways. First, we restrict our analysis to rational bubbles which impose fairly strong restrictions on the dynamics of the bubble component. Hence these restrictions can help us identify the non-fundamental component in the data. Second, we use a less restrictive fundamentals model. Indeed, our econometric procedure allows us to analyse a more complex model with time-varying discount rates and regime-switching in fundamentals and the bubble. In doing so, we allow the fundamentals part to fit the data better, leaving less room for a bubble. This is an improvement with respect to the model in Al-Anaswah and Wilfling (2011), which allows regime-switching only in the bubble process and assumes constant return rates. Moreover, in their specification the dividend process follows a pure random walk.
In line with previous research on the identification of speculative bubbles, we employ artificial as well as real-world datasets. The artificial bubble processes are defined in the sense of Evans (1991), whereas the real-world datasets are drawn from Datastream. We consider 20 years of monthly data (November 1997-October 2017 for the price index, the dividend yield and the market value. We use data for five countries: United States, United Kingdom, Malaysia, Japan and Brazil. We choose this set of countries because of their economic relevance and the severe bubble episodes experienced in the past (Kindleberger and Aliber, 2003). The advantage of the artificial datasets with respect to real-world data is that the bubbles' collapse dates are known, hence they allow us to assess the accuracy of our bubble-detection method. For the real-world datasets, we rely on what economic historians have classified as bubble periods (Kindleberger and Aliber, 2003).
In order to estimate this new high-dimensional model, we adopt the Bayesian approach and use Markov chain Monte Carlo (MCMC) methods to simulate from the joint posterior distribution. Indeed, when the number of model parameters is large, standard maximum likelihood estimation tends to be numerically unstable and may result in estimates that are locally, but not globally, maximal. In contrast, MCMC methods are numerically more robust and can handle a large number of parameters and latent variables. In addition, one novel feature of our implementation is that it builds upon the band and sparse matrix algorithms for state space models developed in Chan and Jeliazkov (2009), McCausland et al. (2011) and Chan (2013, which are shown to be more efficient than the conventional Kalman filter-based algorithms. We find that our new bubble-detection method is able to correctly identify 92.27% of all the bubble collapsing dates in the artificial datasets. Moreover, it never signals a bubble when there is none in the price process. These results represent an improvement with respect to the methodology discussed in Al-Anaswah and Wilfling (2011) which correctly identifies around 50% of all the bubble collapsing dates. Also, we find that our framework is able to identify most of the bubble periods classified as such by Kindleberger and Aliber (2003). Consistent with Al-Anaswah and Wilfling (2011) and Lammerding et al. (2013), we document the existence of statistically significant Markov-switching in the data-generating process of real-world stock price bubbles.
Our framework is also able to predict dividend growth rates as well as returns with R 2 values ranging from 74.07% to 78.89% for dividend growth rates and 4.04% and 20.71% for returns in the artificial datasets. In the real-world datasets, the R 2 values for dividend growth rates are quite high, the highest value is recorded for the US where it is equal to 70.49% while the lowest value is registered for Brazil where it is equal to 49.10%. However, the R 2 values for returns are less than 1% with the exception of Brazil where it is above 3%.
We show that present-value models should not ignore the bubble component of stock prices. Indeed, we find that in the surviving bubble regime most of the variation in the price-dividend ratio is related to the bubble variation. Specifically, bubble variation accounts for more than 50% of the price-dividend variation in all the countries under study with the exception of Brazil where it accounts for about 36%. Further, bubble variation explains also a large share of unexpected return variation in the surviving bubble regime. 2 The paper is structured as follows: next section reviews the present-value model by Campbell and Shiller (1988). In Section 3, we present the econometric model, and Section 4 discusses the posterior sampler. In Section 5, we present the data sources and some descriptive statistics. Section 6 discusses the results, and Section 7 concludes.

Economic Model
In this section we briefly review the log-linearized present-value model in the spirit of Campbell and Shiller (1988) in which both expected returns and expected dividend growth rates are treated as latent variables as suggested by Binsbergen and Koijen (2010).
Let denote with pd t and ∆d t+1 respectively the log price-dividend ratio and the log dividend growth rate The log gross return, denoted as r t+1 , is defined as follows Equation (1) is nonlinear since it involves the log of the sum of price and dividend. However, using the first order Taylor expansion it can be well approximated by where κ and ρ are parameters of linearizations, κ = log(1 + exp(pd)) − ρpd and ρ = exp(pd) 1 + exp(pd) ,pd = E[pd] (Campbell and Shiller, 1988).
(2) and imposing the transversality condition, we obtain the unique no-bubble solution where Ψ t denotes the economic agents' information set at time t. Similar to Binsbergen and Koijen (2010), we assume that both expected returns (µ t ≡ E t [r t+1 | Ψ t ]) and expected dividend growth rates (g t ≡ E t [∆d t+1 | Ψ t ]) follow an AR(1) process The dividend growth rate and the return rate are respectively equal to their expected value plus an orthogonal shock: Assuming that lim j→∞ ρ j pd t+j = 0 and taking expectations conditional upon time t we obtain the fundamental price-dividend ratio: which uses provided that Finally, the fundamental price-dividend ratio can be written It is important to stress that if the transversality solution does not hold, the no-bubble solution pd f in (11) represents only a particular solution to the difference equation (2), and the general solution has the form where b t is a rational speculative bubble, that is a deviation of the stock price from fundamentals generated by extraneous factors or rumors and driven by self-fulfilling expectations. The bubble component of the price-dividend ratio satisfies the homogeneous difference equation In line with the literature (i.e., Wu, 1997;Al-Anaswah and Wilfling, 2011;Lammerding et al., 2013), we assume that the bubble component follows a linear AR(1) process When estimating the price-dividend equation (12), we are confronted with the problem that expected returns, expected dividend growth rates, and the bubble component are unobservables. Hence, we have to express our model in state space form.

Econometric Model
Bubbles are empirically plausible only if they are likely to collapse after reaching high levels. For instance, Al-Anaswah and Wilfling (2011) and Lammerding et al. (2013) allow the bubble in the present-value model in Wu (1997) to switch between two alternative regimes: an explosive and a stationary regime. Using both stock and oil price data, they document statistically significant Markov-switching between these two regimes. Their findings motivate us to extend the present-value model of Binsbergen and Koijen (2010) to incorporate a speculative bubble that switches between two regimes. The two regimes aim to represent the two distinct phases in the bubble process, namely, one in which the bubble survives and one in which it collapses.
Differently from Al-Anaswah and Wilfling (2011), we allow all the model parameters in the bubble and fundamentals equations to switch between two distinct regimes S t ∈ {1, 2}.
The regime indicator S t , which is independent of all the other shocks in our model, is governed by a first-order Markov process with constant transition probabilities, where p 11 = P (S t = 1|S t−1 = 1) and p 22 = P (S t = 2|S t−1 = 2) are between 0 and 1.
By the end of time t or at the beginning of time t + 1, economic agents observe S t but not future states. Thus, economic agents' information set at the end of time t is specified as where I t consists of the observed data up to time t.
The model transition equations can be written as: The dividend growth rate is then equal to and the price-dividend equation is where, for i, j ∈ {1, 2} and j = i, we have defined: Notice that we have added to the equation for the price-dividend an orthogonal error e t+1 . Indeed when we substitute the transition variables at t + 1, we are confronted with the fact that next period regime is unknown. Hence, we use their expectation conditioned on the information available at time t + 1 which generate an error, measured by e t+1 .
In line with Al-Anaswah and Wilfling (2011) and Lammerding et al. (2013), we express the price-dividend equation in first difference to circumvent nonstationarity problems: Concerning the return process, approximation (2) together with equation (12) imply that the return shock has the following form: Since the series of market returns is fully described by the dividend growth rates and the price-dividend ratio, we omit it from our state space specification. Let α t denote the 7 × 1 vector of unobservable variables, and y t be the 2 × 1 vector of observable variables: We can now express the model in matrix form: where α 1 ∼ N (0, ΓV 1 Γ ), G, M , and Z are time invariant matrices of the appropriate dimensions, and ξ t and η t are (4 × 1) and (2 × 1) vector of disturbances, respectively The model matrices are defined as follows: Given that the bubble process is exogenous, we have assumed σ gb = σ µb = σ db = 0.

Bayesian Estimation
In this section we describe a Bayesian approach for estimating our Markov-switching state space model. Since the number of model parameters is quite large, standard maximum likelihood estimation tends to be numerically unstable and may result in estimates that are locally, but not globally, maximal. For this reason we apply MCMC methods which are numerically more robust. A key novel feature of our approach is that it builds upon the band and sparse matrix algorithms for state space models developed in Chan andJeliazkov (2009), McCausland et al. (2011) and Chan (2013), which are shown to be more efficient than conventional Kalman filter-based algorithms.
We assume the following prior distributions: i.
We define Regime 1 as the bubble surviving regime, while Regime 2 represents the bubble collapsing regime. The main model parameter is ρ, which governs the growth rate of the bubble process. When ρ increases, the bubble's growth rate decreases. In particular, when ρ ≤ 1, the bubble is explosive; when ρ > 1, the bubble follows a stationary AR(1) process. We assume a Normal prior for ρ with mean equal to 0.75 in Regime 1 and 1.25 in Regime 2. For other parameters we assume the same priors across the two regimes. The values of the hyperparameters are informed by the the estimation results of previous studies (Binsbergen and Koijen, 2010;Choi et al., 2017;Piatti and Trojani, 2017). Finally, we adopt a conjugate prior Beta(15, 1) for the transition probabilities p 11 and p 22 . Table  1 summarizes the priors and starting values for the MCMC algorithm.
We implement the following 7-block Metropolis-within-Gibbs sampler to simulate from the joint posterior distribution: 1. Sample from f (α|Y, Θ, Ω, σ 2 b , σ 2 e , S, p 11 , p 22 ). It can be shown that the full conditional distribution of α is Gaussian. As a first step, we rewrite the transition and the measurement equations in matrix form: where To simulate from N (α, P −1 ), we first obtain the Cholesky factor C of P such that C C = P . Then, given u ∼ N (0, I), we solve Cx = u for x by back substitution and take α =α + x. It can be shown that α ∼ N (α, P −1 ); see, e.g., Chan and Jeliazkov (2009).
2. Sample from f (Ω|y, α, Θ, σ 2 b , σ 2 e , S, p 11 , p 22 ). This step is standard as Ω follows an Inverse-Wishart distribution: where IW stands for the Inverse-Wishart distribution, e t = ( g of the variances follows an Inverse-Gamma distribution: where IG stands for the Inverse-Gamma distribution. 4. Sample from f (Θ 1 |y, α, Θ 2 , Ω, σ 2 b , σ 2 e , S, p 11 , p 22 ). This step is also standard as Θ 1 follows a Normal distribution. To see that, we first write the measurement equation as We can then express the constant term Using standard linear regression results, one can show that the conditional posterior The matrixX is a 2T ×4 matrixX = (X S 1 ,S 0 , ..., X S T ,S T −1 ) ,C = (C S 1 ,S 0 , ..., C S T ,S T −1 ) , and Y , α are the stacked vectors of y t and α t respectively, H ZG and Σ are defined as: 5. Sample from f (Θ 2 |Y, α, Θ 1 , Ω, σ 2 b , σ 2 e , S, p 11 , p 22 ). Since this conditional distribution is nonstandard, we sample Θ 2 using an adaptive Random Walk Metropolis-Hastings algorithm (Roberts and Rosenthal, 2009). In particular, we update each element of Θ 2 at a time. Given the current draw Θ (s) 2 , we update the j-th variable by adding a normal random variable centered at zero to obtain the candidate draw Θ * 2 . 6 The candidate is then accepted with probability We impose stationarity conditions for expected dividend growth rates and expected returns, i.e., −1 < γ 1 < 1, −1 < δ 1 < 1.
6. Sample from f (S|Y, α, Θ, Ω, σ 2 b , σ 2 e , p 11 , p 22 ). This step can be done using the algorithm proposed by Chib (1996); see also Kim and Nelson (1999). Specifically, we use the following decomposition of the joint conditional density: 7 where Y 1:t denotes all the data up to time t, and α 1:t is similarly defined.
To compute each of these conditional distributions, we first run the Hamilton filter (Hamilton, 1989) to get the filtered distributions f (S t |Y 1:t , α 1:t ), t = 1, 2, ..., T . The last iteration of the filter provides f (S T |Y, α). More specifically, these filtered distributions are defined by where f (y t |S t , α t−1 , y t−1 ) is a multivariate normal distribution defined by the model.
Then, the conditional distribution f (S t |S t+1 , Y 1:t , α 1:t ) can be computed by using: where f (S t+1 |S t ) is the transition probability and f (S t |Y 1:t , α 1:t ) is calculated using the Hamilton filter as described above. Note that the probability P r(S t = 6 For the first batch of 50 iterations, we update each variable j by adding a N (0, 0.1 2 ) distributed random variable. Then, after the l-th batch of 50 iterations, we update the logarithm of the standard deviation of the proposed normal increment log(s j ), by adding or subtracting an adaptation amount δ(l) = min(0.01, l −1/2 ). Specifically, if the fraction of acceptances of variable j was greater than 0.44 on the l-th batch, we increase log(s j ) by δ(l); otherwise we decrease it by the same amount. Note that Roberts et al. (1997) and Roberts et al. (2001) show that in various one-dimensional settings the optimal acceptance rate is around 0.44. 7 For notational convenience, in what follows we suppress the dependence on the model parameters.
2|S t+1 , Y 1:t , α 1:t ) can be obtained after the normalization: Finally, to obtain a draw from f (S t |S t+1 , Y 1:t , α 1:t ), we generate a random number from a uniform distribution between 0 and 1. If the generated number is less than P r(S t = 2|S t+1 , Y 1:t , α 1:t ), we set S t = 2; otherwise we set it equal to 1.
7. Sample from f (p 11 , p 22 |Y, α, Θ, Ω, σ 2 b , σ 2 e , S). Conditional on S, the transition probabilities p 11 and p 22 are independent of the data y, the state variables α, and other model parameters.
Since we choose conjugate priors for both p 11 and p 22 , we only need to calculate the number of switches between the regimes in order to derive the posterior distributions of p 11 and p 22 . The posterior distributions are two independent beta distributions: p 11 ∼ Beta(u 11 + n 11 , u 12 + n 12 ), p 22 ∼ Beta(u 22 + n 22 , u 21 + n 21 ), where n ij refers to the transitions from state i to j; i, j ∈ {1, 2}.

Data and Descriptive Statistics
We apply our two-regime Markov-switching state space model to artificial as well as realworld datasets. The artificial bubble processes are defined in the sense of Evans (1991), whereas the real-world datasets are drawn from Datastream. For computational reasons we focus our analysis on the last 20 years of monthly data (November 1997 -October 2017) for the price index (PI), the dividend yield (DY) and the market value (MV). 8 We use data for five countries: United States (US), United Kingdom (UK), Malaysia (MY), Japan (JP) and Brazil (BR). We choose this set of countries because of their economic relevance and the severe bubble episodes experienced in the past (Kindleberger and Aliber, 2003). Moreover, this allow us to compare our results with those of previous studies. The model can also be used to study bubbles in investment styles such as industry, size and value, and in individual stocks. The model can be applied to stocks not paying dividends, however it needs to use alternative measures of fundamentals such as earnings. Evans (1991) considers a class of rational bubbles that are positive and periodically collapsing defined as follows: where δ and λ are real positive parameters such that 0 < δ < λ(1 + r). The variable u t is assumed to be independent and identically distributed (iid) lognormally with unit mean. Specifically, we assume u t = exp(x t − τ 2 /2) with x t iid N (0, τ 2 ). The process θ t is an exogenous iid Bernoulli process which assumes the value 1 with probability π and the value 0 with probability 1 − π, 0 < π ≤ 1. The parameters δ, λ, and π govern the frequency with which bubbles erupt, the scale of the bubble and the average length of time before collapse.

Simulated Datasets
As long as the bubble process is below λ, the bubble grows at the mean rate 1 + r. When B t > λ the bubble grows at the faster mean rate (1 + r)π −1 and it can collapse with a probability 1 − π in each period. Whenever the bubble collapses, it falls to δ and the process starts again. For the sake of comparison, we use the parameter specifications of the Evans-bubble process adopted by Al-Anaswah and Wilfling (2011) (see Panel A of  Table 2).
We simulate expected returns and expected dividend growth rates according to equations (4) and (5) respectively. We use equations (6) and (11) to generate the dividend growth rate and the fundamental price-dividend ratio series. We generate the bubble stock pricedividend ratio by adding the logarithm of the  to the fundamental price-dividend ratio: Figure 1 shows a realization for the log-price-dividend ratio and the log-Evans-bubble. Finally, the time series of returns is built from the series of the dividend growth rate and the price-dividend ratio using approximation (2). In Panel B of Table 2 we present the parameters used for generating the time-series of our state space model. We generate five artificial datasets with either 100 or 200 observations, for space constraint we report only the results for the biggest datasets of 200 observations.

Real-World Datasets
In line with the literature, we use monthly data on the price index (PI), the dividend yield (DY) and the market value (MV) for the Datastream country indices for: United States (US), United Kingdom (UK), Malaysia (MY), Japan (JP), and Brazil (BR). 9 We consider data from November 1997 to October 2017, a total of 20 years of monthly data. All the data are sourced from Datastream (see Appendix A for further details on the data used).  Table 3 reports some descriptive statistics for the log-price-dividend ratio of the country indices. The US, the UK and Japan have an average log-price-dividend ratio of about 6.50, while the average for Malaysia is 5.08 and 4.26 for Brazil, with the latter being more volatile. The correlations among the developed countries (US, UK and Japan) are quite high, and they range from 0.75 between the US and the UK, to 0.52 between Japan and the UK. The price-dividend ratio of Malaysia is positively correlated with the other countries with the exception for Japan where the correlation is negative. The price-dividend ratio of Brazil is negatively correlated to Japan and it shows almost no correlation with the US. The series for Japan hits a maximum in year 2000, then it decreases and it maintains levels comparable to those of the US and the UK. The price-dividend ratio for Malaysia and Brazil is lower compared to those of the other economies, and they are highly correlated.
value and availability of data.
We can observe that all the countries have experienced a sharp drop in the price-dividend ratio in 2008 in correspondence with the Global Financial Crises.

Results
We estimate the present-value model in equations (16) to (18) using the MCMC method described in Section 4. We run a total of 1,000,000 iterations, and discard the first 100,000 as burn-in. To improve the efficiency of the sampler, we perform thinning every 100 draws. 10 Using the sample of the posterior draws, we report the sample means as point estimates. 95% credible intervals are constructed using the 2.50% and 97.50% sample quantiles.
The parameter ρ is the main variable of interest, indeed 1/ρ is the autoregressive parameter of the bubble process. A value of ρ less than one implies that the bubble is explosive, while a value greater than one means that the bubble process is stable.
p 11 = P (S t = 1|S t−1 = 1) and p 22 = P (S t = 2|S t−1 = 2) are the transition probabilities of the two-regime Markov process. If p 11 > p 22 , then the bubble surviving regime is more persistent than the bubble collapsing regime. Table 4 reports the posterior estimates of the parameters for our state space model for the artificial datasets. The main variables are presented in bold characters. Table 5 shows the bubble 95% credible intervals for the parameters ρ 1 and ρ 2 , we can observe that the two parameters are statistically significant and significantly different from each other. In particular, ρ 1 is significantly less than one, while ρ 2 is significantly greater than one, meaning that we correctly find significant regime-switch in our artificial datasets. Moreover, in Regime 1 the bubble is explosive while in Regime 2 it collapses. The transition probability p 11 is always greater than p 22 , suggesting that the surviving bubble regime is more persistent than the collapsing bubble regime. Table 6 reports the posterior estimates of the real-world datasets, while Table 7 shows the 95% credible intervals for the autoregressive bubble parameter ρ. Again, we observe ρ to be significantly different in the two regimes for all the real-world datasets except Malaysia. Moreover, we find that ρ 1 is not significantly greater than one and ρ 2 is significantly greater than one at the 5%-level for the US, the UK, Japan and Brazil. Instead, in Malaysia ρ 1 does not appear to be significantly smaller than 1 at 5%-level, meaning that we do not find statistical significance of an explosive behaviour of the bubble process in this country. Consistent with Al-Anaswah and Wilfling (2011), and Lammerding et al. (2013), we find that the transition probability p 11 is always greater than p 22 , meaning We present the estimation results of the present-value model in equations (16) to (18) for the five artificial datasets. The model is estimated according to the procedure described in Section 4. Note that 1/ρ is the autoregressive parameter of the bubble process. When ρ is less than one the bubble is explosive, when it is greater than one the bubble is stable. p 11 = P (St = 1|S t−1 = 1) and p 22 = P (St = 2|S t−1 = 2) are the transition probabilities of the two-regime Markovswitching model. We present the 95% credible intervals for the bubble parameter ρ in Regime 1 (ρ 1 ) and Regime 2 (ρ 2 ). The 95% credible intervals consists of the 2.50% and 97.50% quantiles of the posterior distribution of ρ 1 and ρ 2 .
that the surviving bubble regime is more persistent than the collapsing bubble regime in all the real-world datasets. In particular, the probability of remaining in Regime 1 (p 11 ) is higher than 99% in the US, the UK and Japan, while in Malaysia it is 98.40% and it equals 97.23% in Brazil. The estimates of the transition probability p 22 , instead, vary between 80.29% (Brazil) and 92.89% (Japan). These results also imply that the expected duration of the surviving regime 1/(1 − p 11 ) is higher than the expected duration of the collapsing regime 1/(1 − p 22 ). Concerning the remaining model parameters, we find that they are not significantly different across the two regimes. The unconditional expected log dividend growth rate (γ 0 ) and the unconditional expected log return (δ 0 ) are not significantly different from zero. Consistently with Fama and French (1988), Campbell and Cochrane (1999), Ferson et al. (2003), Pástor and Stambaugh (2009), Binsbergen and Koijen (2010) and others, we find expected returns to be highly persistent with a monthly persistence coefficient (δ 1 ) of above 0.75 for all the datasets. The estimated persistence of expected dividend growth rates (γ 1 ) instead is generally lower, it ranges between 0.49 and 0.56. Furthermore, shocks to expected dividend growth rates and expected returns are generally positively correlated.
We next examine the identification of speculative bubbles, the prediction of returns and dividend growth rates, and the variance decomposition of the price-dividend ratio and unexpected returns in Sections 6.1, 6.2, and 6.3, respectively. We present the estimation results of the present-value model in equations (16) to (18) for the five real-world datasets. The model is estimated according to the procedure described in Section 4. Note that 1/ρ is the autoregressive parameter of the bubble process. When ρ is less than one the bubble is explosive, when it is greater than one the bubble is stable. p 11 = P (St = 1|S t−1 = 1) and p 22 = P (St = 2|S t−1 = 2) are the transition probabilities of the two-regime Markovswitching model. We present the 95% credible intervals for the bubble parameter ρ in Regime 1 (ρ 1 ) and Regime 2 (ρ 2 ). The 95% credible intervals consists of the 2.50% and 97.50% quantiles of the posterior distribution of ρ 1 and ρ 2 .

Bubble Identification
In this section, we analyze the smoothed surviving-probabilities P (S t = 1|Ψ T ) that the bubble process has been in Regime 1 at time t, (t = 1, ..., T ), in order to distinguish between dates on which either the surviving bubble Regime 1 or the collapsing bubble Regime 2 has been in force. The business-cycle literature generally suggests using the following decision rule: Regime 1 has been in force if P (S t = 1|Ψ T ) >0.50, while Regime 2 has been in force if P (S t = 1|Ψ T ) ≤ 0.50 (see for instance Goodwin, 1993). However, we expect the bubble process to be in the surviving Regime 1 most of the time, and in the collapsing Regime 2 only for few short periods. This intuition is confirmed by our results which show that the expected duration of the surviving Regime 1 is higher than the expected duration of the collapsing Regime 2, moreover it is confirmed by the findings of Al-Anaswah and Wilfling (2011), and Lammerding et al. (2013). Hence, the threshold value of 0.50 may fail to correctly identify regime switch dates. We follow Al-Anaswah and Wilfling (2011) who suggest to take into consideration the first two moments of   Figure 3 refers to the artificial datasets 1 to 4. For datasets 1 and 2 our procedure is able to identify eleven out of twelve bubbles. In dataset 3 (third panel Figure 3) we identify twelve out of fourteen bubbles. The bottom panel of Figure 3 refers to dataset 4, in this case all the bubbles are detected correctly, and in dataset 5 (Figure 4) we can identify twelve out of thirteen bubbles.
Summing up, as can be seen from the figures above, our methodology correctly identifies 57 out of 62 (92.27%) of all the bubble collapsing dates in the five artificial datasets. We also observe that our procedure may fail to recognize bubbles of smaller size, in particular we may fail to identify those bubbles which emerge after a bubble of a bigger size. Further, our procedure never signals a bubble which has no counterpart in the price process.
We now turn our attention to the results for the real-world datasets. Figure 5 graphs the smoothed surviving-probabilities and the log-price-dividend ratio for the US. We identify only one collapsing period from October 2008, that is after the collapse of the investment bank Lehman Brothers, until May 2009. The smoothed surviving-probabilities slightly decrease in two episodes at the beginning of the series and again after the 2008 collapse, however the decrease is not sharp enough to be signaled as a bubble collapse. Consistent with Balke and Wohar (2009), we cannot say that the decline in stock prices in the 2000s has been caused by a bubble bursting. Similar comments apply to the United Kingdom ( Figure 6) for which the series of the price-dividend ratio is strongly correlated with that of the US.  September 1998, and from December 1999 until January 2000. These episodes reflect the Brazilian stock-market and currency crisis which culminated in a sharp devaluation of the Brazilian currency (the real) to the US dollar in 1999. The effect was caused by the 1997 Asian financial crisis which led Brazil to increase interest rates and to institute spending cuts and tax increases in an attempt to maintain the value of its currency. The devaluation also precipitated fears that the ongoing economic crisis in Asia would spread to South America, as many South American countries were heavily dependent on industrial exports from Brazil. We register also a drop in the smoothed surviving-probabilities in May 2001 in correspondence of the Brazil energy crises, and in November 2008, one month after the US, the United Kingdom and Japan collapse for the 2008 global financial crises. However, we do not detect a collapsing regime associated with the 2014 Brazilian economic crises.

Prediction of Returns and Dividend Growth Rates
We now investigate the in-sample predictability of our state space model. We use the estimated series of expected dividend growth rates (ĝ t ) and expected returns (μ t ) as if they were observables and we regress them on realized dividend growth rates and realized returns.
(33) Table 8 shows the results for the artificial datasets, for dividend growth rates the R 2 ranges from 74.07% to 78.89% while for returns it is between 4.04% and 20.71%. We report regression results for respectively dividend growth rates and returns on their estimated expected values represented byĝt andμt. Standard errors are reported in parenthesis. Note: * * p ≤ 0.05, * p ≤ 0.1.  Expected dividend growth rates explains a large fraction of actual dividend growth, while the fraction of explained return variability is lower. Also, given that the goodness-of-fit measures of returns vary substantially in the five datasets, we argue that the return predictability features of this model are less robust with respect to the dividend predictability features.
Figures 10, and 11 plot the realized and estimated expected dividend growth rates on the left, and the realized and estimated expected returns on the right for the five artificial datasets. The realized and expected dividend growth series are strongly correlated, however the latter series is more stable. Concerning returns, their expectation is less volatile than the series of realized returns. Further, the correlation between realized and expected returns is lower than that for dividend growth rates. We report regression results for respectively dividend growth rates and returns on their estimated expected values represented byĝt andμt. Standard errors are reported in parenthesis. Note: * * p ≤ 0.05, * p ≤ 0.1. Table 9 reports the results for the real-world datasets. The R 2 values for dividend growth rates are quite high in all the datasets, the highest value is recorded for the US where it is equal to 70.49% while the lowest value is registered for Brazil where it is equal to 49.10%. 11 Instead, the coefficient β r in the regressions for returns is significant only for Brazil and the goodness-of-fit measure is equal to 3.12%. In the other countries the R 2 values for returns are less than 1% meaning that the estimated series of expected returns (μ t ) do not help in predicting stock returns.
Visual inspection of Figures 12-16 confirms what we have observed in the regression results. 11 Since the dividend growth predictability we find may be driven by the seasonality in dividend payments, we apply a stable seasonal filter to estimate the seasonal component in the series of dividend growth rates and expected dividend growth rates. Then we estimate the regression equation using the deseasonalized series. The dividend growth predictability is confirmed and the results are available upon request.

Variance Decomposition
In this section we derive the variance decomposition of both the price-dividend ratio and unexpected returns. The variance decomposition of the price-dividend ratio is given by The first term, B 2 2 V ar(g t ), represents the variation in the price-dividend ratio due to expected dividend growth rate variation. The second term, B 2 1 V ar(µ t ), measures the variation in the price-dividend ratio due to expected return variation. The third term, V ar(b t ), accounts for the variation in the price-dividend ratio due to bubble variation. The remaining terms represent the covariation among these three components. The unexpected returns can be written as As before, we decompose the unexpected return variation into the influence of dividend growth variation, discount rate variation, bubble variation and the covariance among these terms. Table 10 summarizes the results for the variance decomposition of the price dividend ratio and unexpected returns for the real-world datasets. We use sample covariances We report the decomposition of the price-dividend variation and the unexpected returns variation into discount rate variation, dividend growth variation, bubble variation and the covariances among these three terms.
and we standardize all terms so that the sum is equal to 100%. We find that in the surviving bubble regime, most of the variation in price-dividend ratio is related to the bubble variation. Specifically, bubble variation accounts for more than 50% of the pricedividend variation in all the countries under study with the exception of Brazil where it accounts for about 36%. Instead, consistent with Binsbergen and Koijen (2010), in the collapsing bubble regime discount rate variation accounts for most of the variation in the price-dividend ratio. Again, consistent with Binsbergen and Koijen (2010) we document that dividend growth plays a major role in explaining unexpected returns variation in the surviving bubble regime, while discount rate variation accounts most of the variation in unexpected returns in the collapsing bubble regime with the exception for Brazil. Also, bubble variation explains a large share of unexpected return variation in the surviving bubble regime. These results are consistent with Balke and Wohar (2009) which find that the bubble component is substantially important in explaining fluctuations in the log price-dividend ratio when there are no permanent components in market fundamentals.

Conclusions
We have shown that in the surviving bubble regime, most of the variation in the pricedividend ratio is related to the bubble variation. Specifically, bubble variation accounts for more than 50% of the price-dividend variation in all the countries under study with the exception of Brazil where it accounts for about 36%. Further, bubble variation explains also a large share of unexpected return variation in the surviving bubble regime.
These results suggest that present-value models should not ignore the bubble component of stock prices. This paper proposes to incorporate a speculative bubble subject to a surviving and a collapsing regime into the present-value model by Binsbergen and Koijen (2010), who adopts a latent variables approach to estimate expected returns and expected dividend growth rates. Further, we suggest an econometrically robust Bayesian MCMC methodology to estimate our model.
This study applies our two-regime Markov-switching state space model to artificial as well as real-world datasets. The artificial bubble processes are defined in the sense of Evans (1991), we show that our bubble-detection methodology is able to identify 92.27% of all the bubble collapsing dates in the artificial datasets, moreover it never signals a bubble which has no counterpart in the price process. These results represent an improvement with respect to the approach discussed in Al-Anaswah and Wilfling (2011) which correctly identifies around 50% of all the bubble episodes. The real-world datasets consist of the monthly time series data for the price index, the dividend yield and the market value for the United States, United Kingdom, Malaysia, Japan and Brazil. We find that our framework is able to identify most of the bubble periods classified as such by Kindleberger and Aliber (2003).
In line with Al-Anaswah and Wilfling (2011) and Lammerding et al. (2013), we document the existence of statistically significant Markov-switching in the data-generating process of real-world stock price bubbles. Furthermore, our methodology is also able to predict dividend growth rates as well as returns with R 2 values ranging from 74.07% to 78.89% for dividend growth rates and 4.04% and 20.71% for returns in the artificial datasets. In the real-world datasets, we find that dividend growth rates are predictable with R 2 values ranging from 70.49% for the US to 49.10% for Brazil. However, the R 2 values for returns are less than 1% with the exception of Brazil where it is above 3%.
A common drawback of the bubble literature is that rejection of the present-value model that are interpreted as evidence of the presence of bubbles can still be explained by alternative structures of the fundamentals. In this paper we mitigate this issue in two ways. First, we restrict our analysis to rational bubbles which impose fairly strong restrictions on the dynamics of the bubble component. Hence these restrictions can help us to identify the non-fundamental component in the data. Second, we use a less restrictive fundamentals model, indeed our econometric procedure allows us to analyse a more complex model with time-varying discount rates and regime-switching in fundamentals and the bubble. Doing so, we allow the fundamentals part to fit better the data, leaving less room for a bubble.
In sum, our setup allows to model jointly expected dividend growth rates, expected returns and the bubble component of stock prices. As such it may improve conventional methods for the detection of real-time stock-price bubbles allowing an early detection of future bubbles. Moreover, this methodology allows for hypothesis testing of some features of expected dividend growth rates and expected returns such as their persistence.

A Datastream Data
Datastream country indices include a representative list of stocks for each country. The sample covers a minimum of 75 -80% of total market capitalisation. Suitability for inclusion is determined by market value and availability of data. The aggregate price index (PI), dividend yield (DY), and market value (MV) for each country index are calculated as follows: where P i,t is the unadjusted price of asset i in month t, N t is the number of shares in issue on month t, f adjustment factor for capital actions, D i,t is dividend per share of asset i in month t, and M is the number of constituents in index. We use the above variables to compute the log price-dividend ratio (pd t ), dividend growth rate (∆d t ) and returns (r t ).

B Efficiency of the Metropolis-within-Gibbs Sampler
Since the Metropolis-within-Gibbs sampler used to simulate from the joint posterior distribution may produce parameter draws that are highly autocorrelated, in Table 12 we report the average autocorrelation of the ith parameter draw with the (i + 1)th draw, (i + 10)th draw, and (i + 100)th draw. To further assess the efficiency of the sampler, in the last two columns we report the average inefficiency factors of the posterior draws which are defined as follow where Corr(θ i , θ i+l ) is the sample autocorrelation of parameter θ at lag length l, L is the maximum lag and it is chosen to be large enough so that the autocorrelation tapers off. The inefficiency factor measures the number of extra draws needed to obtain results equivalent to the ideal case of independent draws. Consider for example an inefficiency factor of 50, this means that around 5000 posterior draws are needed to have the same information of 100 independent draws. The inefficiency factor of independent draws is one.
From Table 12 we can observe that some of the parameters show high autocorrelations, with large inefficiency factors as well. The maximum inefficiency factor is reported for the parameter δ 1 in regime 2 and it is equal to 1160.8113. We present the average autocorrelation of the posterior draws of the model in equations (16) to (18) for the artificial datasets. The model is estimated according to the procedure described in Section 4. The last two columns report the inefficiency factors.
To improve the efficiency of the sampler we perform thinning of the posterior draws and keep only every 100th draws. Table 13 reports the average autocorrelation of the posterior draws and the inefficiency factors after performing thinning of the draws. We can see that thinning significantly reduce the inefficiency factors. We present the average autocorrelation of the posterior draws with thinning every 100 draws of the model in equations (16) to (18) for the artificial datasets. The model is estimated according to the procedure described in Section 4. The last two columns report the inefficiency factors.