Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.
35,750 characters · 7 sections · 25 citation commands
On the asymptotic behavior of bubble date estimators
\baselineskip= 6mm
The estimation of the dates of regime changes has a long history in the econometric literature (see, e.g., Casini2019 for a recent review). In this study, we address the issue of estimating the break dates in explosive bubble model. Such a bubble model is often generated (see PWY2011) as a unit root process followed by explosive process, which in turn followed by a unit root regime (with possible stationary recovering regime after the collapse of the bubble). For estimation of bubble dates, phillips2015a,phillips2015b (PSY hereinafter) proposed the recursive algorithm based on the right-tailed ADF test. Generally speaking, the origination of the bubble is taken at the date for which the test statistic begins to exceed the critical value, and the date of collapse is taken at the date for which the test statistic subsequently falls below the critical value (see Skrobotov2021 for review).
However, as demonstrated in harvey2017improving, the bubble dates estimates are more accurate if they are obtained by minimizing the sum of the squared residuals over all possible dates. For the model with only two regimes, chong2001structural investigated consistency and the limiting distribution of the break date estimator for the AR(1) model with a break in the coefficient. The break date estimator is based on minimizing the sum of the squared residuals (SSR) over all possible break dates. The results depend on the direction of the coefficient change. chong2001structural considered three cases, both regimes are stationary, the first regime is a unit root regime, and the second regime is stationary and vice versa. pang2018structural extended the result of chong2001structural, allowing the AR(1) coefficient to be dependent on the sample size. They also considered the case with change in persistence from a unit root to moderately explosive and vice versa. pang2021estimating (PDC hereinafter) considered the model with three regimes that reflect the explosive financial bubble model: the first regime is a unit root regime, followed by an explosive regime and then by a stationary collapsing regime. PDC investigated the sample splitting strategy: The first break date is estimated for the two regime AR(1) model. Results indicate that the break date estimator is consistent for the date of collapse (i.e., the second break). Then, one could consider the sample before the date of collapse and investigate the asymptotic behavior of the date of origination of the bubble -- the results are the same as that in pang2018structural. From a different perspective, the estimation procedure used by PDC is the same as that in chong1995 and bai1997multiple, in which multiple breaks are estimated one at a time in the regime-wise stationary model with level shifts.
In this study, we extend the results of PDC by allowing the forth unit root regime. The advantage of this extension is that we can investigate not only the emerging and collapsing dates of the bubble but also the recovering date to the normal market. As a result, we can totally investigate the abnormal market behavior. In our procedure, we first prove the consistency of the collapse date estimator by minimizing the sum of the squared residuals using the two-regime model, allowing nonstationary volatility. Second, as the estimated break date is consistent with the collapsing date, one could split the sample at the estimated break date and consider the estimator of the market recovery date (i.e., the date of change from stationary collapsing regime to unit root regime) using the subsample after the break date detected in the first step.\footnote{The asymptotics for the date of the beginning of the explosive regime were considered in PDC.} Third, we allow a weak dependence in the innovation errors. Our approach closely resembles that of harvey2017improving by minimizing the full SSR based on the four regimes model. However, harvey2017improving estimated all the break dates simultaneously, while we use the sample-splitting approach. As demonstrated in PDC, this performs better in three-regime case and computationally less involved (we perform the three SSR minimization with one break each with $O(T)$ computations, while harvey2017improving requires minimizing the three break model over all possible combinations of these breaks). Interestingly, the limiting behaviour of the market recovery date estimator depends on the relationship between the extent of the explosive regime and the stationary collapsing (recovering) regime.
It should be noted that our approach can be easily extended to the multiple bubbles context. harvey2020date proposed to initially identify the bubble regimes based on PSY approach and then obtain more precise bubble date estimates during each bubble episode. Subsequently, one can use our approach in the second step.
The remainder of this paper is organized as follows. Section 2 formulates the model and assumptions. In Section 3, we define the main procedure and provide the limiting behaviors of the break date estimators. The model with serially correlated shocks is considered in Section 4. The finite sample performance of the estimated break dates is demonstrated in Section 5, and the empirical example is given in Section 6. Section 7 concludes the paper. All proofs are collected in the appendix.
Let us consider the following bubble's emerging and collapsing model for $t=1,2,\ldots,T$:
where $y_0=o_p(T^{1/2})$, $c_0\geq 0$, $\eta_0 > 1/2$, $\phi_a\coloneqq 1+c_a/T^{a}$ with $c_a > 0$ and $0<a<1$, $\phi_b\coloneqq 1-c_b/T^b$ with $c_b > 0$ and $0 < b < 1$, $c_1\geq 0$, and $\eta_1 > 1/2$. In this model, the process evolves up to time $k_e$ as a unit root process with a possibly positive drift shrinking to zero, and then $y_t$ behaves mildly explosive and next starts collapsing at $t=k_c+1$. Thereafter, the adjustment (collapsing and recovering) period lasts up to $k_r$ and then returns to a unit root process. We denote the mildly explosive regime with $\phi_a\coloneqq 1+c_a/T^a$ and the mildly stationary regime with $\phi_b\coloneqq 1-c_b/T^b$ for ease of exposition. However, we can also consider these AR(1) parameters in more general forms, such as $\phi_a\coloneqq 1+c_a/h_T$ and $\phi_b\coloneqq 1-c_b/k_T$, where $h_T+T/h_T\to \infty$ and $k_T+T/k_T\to \infty$. The break points $k_e$, $k_c$, and $k_r$ are unknown, and the corresponding break fractions are defined as $\tau_e\coloneqq k_e/T$, $\tau_c\coloneqq k_c/T$, and $\tau_r\coloneqq k_r/T$, respectively. Note that PDC considered a model up to $k_r$. Thus, our model is an extension of theirs to include the normal market behavior. Note that the result on the estimator of $k_r$ is new to the existing literature.
For model (ref), we make the following assumption.
Assumption (ref) implies that break dates are distinct and not quite close each other, which is typically assumed in the existing literature. Assumption (ref) is general enough to allow for nonstationary unconditional volatility.
Following CavaliereTaylor2007a,CavaliereTaylor2007b, the functional central limit theorem (FCLT) holds for the partial sum process of $\{\varepsilon_t\}$ under Assumption (ref):
for $0\leq \tau \leq 1$, where $\Rightarrow$ signifies weak convergence of associated probability measures, $W(\cdot)$ is a standard Brownian motion, $\kappa(\tau)\coloneqq \int_0^{\tau}\omega(s)ds/\int_0^1\omega^2(s)ds$ is called the variance profile, and $\tilde{\omega}^2\coloneqq \int_0^1\omega^2(s)ds$. Note that, when $\sigma_t$ is constant, $W^{\kappa}(\tau)$ reduces to a constant volatility times a standard Brownian motion.
Assumption (ref) is used mainly for the estimation of the collapsing date $k_c$ and the market recovering date $k_r$ for $a < b$. For the estimation of the emerging date of a bubble $k_e$ and the last break point $k_r$ for $a > b$, we impose the following stronger assumption:
Under Assumption (ref), the FCLT in (ref) becomes simpler and is given by
This result is used for deriving the limiting distributions of the estimators of $k_e$ and $k_r$ for $a > b$ in Theorems (ref)(iii) and (ref)(ii), respectively.
Following PDC, we estimate the model with a one-time break. Note that model (ref) can be expressed as
For a given $1 \leq k \leq T-1$, we denote the sum of the squared residuals as
where $\sum_{t=\ell}^m$ is abbreviated just as $\sum_{\ell}^m$ and \[ \hat{\phi}_a(k/T)=\frac{\sum_1^k y_{t-1}y_t}{\sum_1^k y_{t-1}^2} \quad\mbox{and}\quad \hat{\phi}_b(k/T)=\frac{\sum_{k+1}^T y_{t-1}y_t}{\sum_{k+1}^T y_{t-1}^2}. \] Defining the break point estimator as $\hat{k}\coloneqq \arg\min_{k} SSR(k/T)$, we have the following theorem.
Theorem (ref) is the same as Theorem 3(a) in PDC and implies that, even if the last regime with a unit root process is additionally included in the sample period, we can consistently estimate the bubble collapsing date. This may not be a surprising result because the explosive behavior in the second regime is quite different from those in the other regimes. Therefore, even if we estimate only a one time break date, the estimator is consistent with the collapsing date of the bubble. We also note that we allow nonstationary volatility in $\{\varepsilon_t\}$, general enough to apply our method to practical analyses. It might be possible to estimate the break dates by minimizing the weighted sum of the squared residuals using the volatility estimated by HLZ2022volatility, but this is beyond the scope of this paper.
The consistency of the collapsing date is intuitively explained as follows. As given in the proof of Lemma 2, during the emergence of the bubble in the second regime, the process can be expressed as \[ y_k = \phi_a^{k-k_e}y_{k_e}+\sum_{j=k_e+1}^{k}\phi_a^{(k-j)}u_j \sim_a \phi_a^{k-k_e}y_{k_e} \] for $k_e+1\leq k \leq k_c$, where $y_{k_e}/\sqrt{T}$ weakly converges to a (time-transformed) Brownian motion. Therefore, the process evolves monotonically at a geometric rate (noting that $\phi_a > 1$) and attains its peak at $k=k_c$. Conversely, the process collapses at a geometrically decaying rate (noting that $\phi_b < 1$) because it can be expressed as
for $k_c+1\leq k \leq k_r$, where the first term on the right hand side can be shown to dominate the second term, at least, when $k$ is relatively close to $k_c$. These geometric explosive and collapsing behaviors help us identify the collapsing date $k_r$ consistently.
After we obtained the consistent estimator $\hat{k}$ of $k_c$, we decompose the whole sample into two sub-samples for $t=1,\ldots,\hat{k}$ and $\hat{k}+1,\ldots,T$ and estimate $k_e$ and $k_r$ from the first and second sub-samples, respectively. Because $\hat{k}$ is a consistent estimator of $k_c$, we can treat $k_c$ as a known break point. Thus, we use $k_c$, instead of $\hat{k}$, in the following.
Estimation of $k_e$ is based on the minimization of the sum of the squared residuals using the first sub-sample, and the estimator is defined as \[ \hat{k}_e\coloneqq \arg\min_{\underline{\tau}_e\leq k/T\leq \bar{\tau}_e} SSR_1(k/T) \] where $0 <\underline{\tau}_e < \tau_e < \bar{\tau}_e <{\tau}_c$ and \[ SSR_1(k/T) \coloneqq \sum_{1}^k\left(y_t-\hat{\phi}_c(k/T)y_{t-1}\right)^2+\sum_{k+1}^{k_c}\left(y_t-\hat{\phi}_d(k/T)y_{t-1}\right)^2 \] \[ \mbox{with}\quad \hat{\phi}_c(k/T)=\frac{\sum_{1}^k y_{t-1}y_t}{\sum_{1}^k y_{t-1}^2}\quad\mbox{and}\quad \hat{\phi}_d(k/T)=\frac{\sum_{k+1}^{k_c} y_{t-1}y_t}{\sum_{k+1}^{k_c} y_{t-1}^2}. \] The corresponding break fraction estimator is defined by $\hat{\tau}_e\coloneqq \hat{k}_e/T$.
The consistency of the $\hat{\tau}_e$ has been already established in Theorem 1.3 of pang2018structural and stated in the next theorem.
On the other hand, for the estimation of $k_r$, we minimize the sum of the squared residuals using the second sub-sample, and the estimator is defined as \[ \hat{k}_r\coloneqq \arg\min_{\underline{\tau}_r\leq k/T\leq \bar{\tau}_r} SSR_2(k/T) \] where $\tau_c <\underline{\tau}_r <\tau_r < \bar{\tau}_r <1$ and \[ SSR_2(k/T) \coloneqq \sum_{k_c+1}^k\left(y_t-\hat{\phi}_e(k/T)y_{t-1}\right)^2+\sum_{k+1}^T\left(y_t-\hat{\phi}_f(k/T)y_{t-1}\right)^2 \] \[ \mbox{with}\quad \hat{\phi}_e(k/T)=\frac{\sum_{k_c+1}^k y_{t-1}y_t}{\sum_{k_c+1}^k y_{t-1}^2}\quad\mbox{and}\quad \hat{\phi}_f(k/T)=\frac{\sum_{k+1}^T y_{t-1}y_t}{\sum_{k+1}^T y_{t-1}^2}. \] The corresponding break fraction estimator is defined by $\hat{\tau}_r\coloneqq \hat{k}_r/T$.
To investigate the asymptotic property of $\hat{k}_r$, we must distinguish the two cases: $a < b$ and $a >b$. In the case of $a < b$, the explosive behavior is faster than the collapsing (recovering) speed, and we continue to make Assumption (ref). On the other hand, to derive the limiting distribution, we need to impose stronger restrictions on the shocks given by Assumption (ref) in the case of $a > b$, in which the process recovers relatively faster than the evolution of the bubble.
The following theorem reveals that the convergence order of $\hat{\tau}_r$ depends on whether $a < b$ or $a > b$.
From Theorem (ref), we can observe that the break date from the collapsing (recovering) regime to the normal market can be consistently estimated if $a<b$; that is, the explosive speed of the process is faster than the collapsing speed. In this case, as given in (ref), the process decays at geometric rate for $k_c+1\leq k \leq k_r$ (because the first term of (ref) dominates the second one) and reaches at
Note that $\phi_a^{(k_c-k_e)}\phi_b^{(k_r-k_e)}\to \infty$ when the explosive speed is faster ($a<b$) because
because $a < b$. That is, the explosive effect induced by $\phi_a$ remains in the process during the whole collapsing (recovering) regime, after which the process evolves as \[ y_k = y_{k_r}+c_1\frac{k-k_r}{T^{\eta_1}}+\sum_{j=k_r+1}^{k}\varepsilon_j \sim_a y_{k_r} \] because we can observe from (ref) that $T^{-\alpha}\phi_a^{(k_c-k_e)}\phi_b^{(k_r-k_c)}\to \infty$ for any $\alpha > 0$. This monotonically geometrical decay during the collapsing regime followed by the asymptotically flat behavior makes $\hat{k}_r$ consistent.
On the other hand, when the collapsing speed is faster ($a>b$), the effect of the explosive component in $y_k$ in the recovering regime diminishes to zero as $k$ approaches $k_r$ because $\phi_a^{(k_c-k_e)}\phi_b^{(k_r-k_c)}\to 0$ when $a > b$, as can be observed in (ref). That is, the initial value effect by $y_{k_c}$ in the recovering regime gradually disappears and the process behaves as if it started from the small initial value. Therefore, we obtain the same result as Theorem 1 of pang2018structural.
The asymptotically different convergence orders in Theorem (ref) reflect the finite sample performance of $\hat{k}_r$, as presented in Section 5.
In this section, we extend model (ref) with $\{\varepsilon_t\}$ being serially correlated. We consider the case where $\{\varepsilon_t\}$ is a linear process given by
For the innovations $\{v_t\}$, we make the following assumption.
Linear process (ref) can be expressed as, by the BN decomposition,
and $\psi=\sum_{j=0}^{\infty}\psi_j$. It is also well known that the summability condition in (ref) implies $\sum_{\ell=0}^{\infty}|\tilde{\psi}_{\ell}|<\infty$. In this case, we have the following corollary.
By Corollary (ref), we can estimate the three break dates by using the misspecified AR(1) model with the same convergence orders of the estimators as before, although we may expect that the finite sample accuracy of the estimators would be improved by correctly specifying the correlation structure by considering the ADF type regression with an additional lag structure.
In this section, we examine the performance of the estimates of the bubble regimes dates in finite samples. Our purpose is to demonstrate how the finite sample properties of the estimated break dates reflect the asymptotic results obtained in the previous section.
The Monte-Carlo simulations reported in this section are based on the series generated by (ref) with $y_0=0$ and $\{\varepsilon_t\}\sim IIDN(0,1)$. Data were generated from this DGP for samples of $T=(400,800)$ with $50,000$ replications.\footnote{All simulations were programmed in R with rnorm random number generator.} We set the drift terms in the first and fourth regimes to $c_0T^{-\eta_0}=1/800$ and $c_1T^{-\eta_0}=1/800$, respectively, following PDC. In this experiment, we focus on two cases. In the first case, the explosive coefficient $\phi_a$ changes with a fixed $\phi_b$; $\phi_b$ is set to $0.96$ while $\phi_a$ takes values among $(1.01, 1.05, 1.09)$. In the second case, the collapsing coefficient $\phi_b$ varies with a fixed $\phi_a$; $\phi_a$ is set to 1.05 while $\phi_b$ varies among $(0.98, 0.96, 0.94)$. We set the localising parameters $c_a=c_b=1$; then, the values of $a$ and $b$ are uniquely determined based on the definitions given by $\phi_a=1+1/T^a$ and $\phi_b=1-1/T^b$. For the dates of bubble regimes, we use $(\tau_e,\tau_c,\tau_r)$ to be equal to (0.4,0.6,0.7).\footnote{To compare the estimator of $k_c$ with that of PDC, we also set $(\tau_e,\tau_c,\tau_r)$ to be equal to (0.4,0.6,1), that is, without the last unit root regime (the normal market behavior after the collapse and recovering). The results are entirely similar and omitted for brevity.}
Further, in the minimization of $SSR(k/T)$, $SSR_1(k/T)$, and $SSR_2(k/T)$, we excluded the first and last 5% observations from the permissible break date $k$. For example, when estimating $k_r$ based on $SSR_2(k/T)$, the permissible break date $k$ ranges from $\hat{k}_c+0.05T+1$ to $0.95T$. If the break date estimate $\hat{k}_c$ exceeds $0.95T$, then we cannot estimate $k_r$; we do not include such a case in any bins of the histogram and thus the sum of the heights of the bins does not necessarily equal one for $\hat{k}_r$ in some cases. Similarly, we cannot estimate $\hat{k}_e$ when $\hat{k}_c < 0.05T$. In the following, we investigate only $\hat{k}_c$ and $\hat{k}_r$ because, once $\hat{k}_c$ is obtained, the estimate of $k_e$ in the four regime model is the same as that in the three regime model, which has been already investigated by PDC.
Figure 1 presents the histograms of $\hat{k}_c$ when $\phi_b=0.96$ and $\phi_a=(1.01, 1.05, 1.09)$. As expected, when $\phi_a$ becomes larger and/or the sample size increases, the $\hat{k}_c$ becomes more accurate. When $a>b$ ($|\phi_a-1| < |\phi_b-1|$ in our setting) as in Figures 1(a) and 1(b), the frequency of selecting the true break date is not very high (it is around 30% for $T=400$ and 65% for $T=800$), but it increases to almost 100% as in Figures 1(c)--1(f).
In the second case (Figure 2) where $\phi_a=1.05$ and $\phi_b$ varies among $(0.98, 0.96, 0.94)$, the estimate $\hat{k}_c$ is quite accurate for all the three cases with the almost 100% frequency of selecting the true break date.
Figure 3 demonstrates the histograms of $\hat{k}_r$ for the fixed value of $\phi_b=0.96$ and the different values of $\phi_a$. For the small value of $\phi_a=1.01$, as shown in Figures 3(a) and 3(b), which corresponds to the case of $a >b$, we can observe that the accuracy of $\hat{k}_r$ deteriorates, while for $a<b$ ($\phi_a=1.05$ and $\phi_a=1.09$) the estimate becomes more accurate. In particular, for the large sample size of $T=800$, the frequency of selecting the true break date is very close to 100%.
In the case where $\phi_a$ is fixed at $1.05$ and $\phi_b$ varies, we can observe the good performance of the estimate $\hat{k}_r$ even for the case of $a>b$ ($\phi_a=1.05$, $\phi_b=0.94$) as shown in Figures 4(e) and 4(f). Reflecting the result in Theorem (ref), the estimate $\hat{k}_r$ is most accurate when $\phi_a=1.05$ and $\phi_b=0.98$; the frequency of selecting the true break date is approximately $75$% and 100% for $T=400$ and $800$, respectively.
We next consider the case where the explosive and collapsing regimes are shorter such as $(\tau_e,\tau_c,\tau_r)=(0.5,0.55,0.6)$, which is relatively close to NASDAQ data in empirical application. Figures are collected in the appendix and, as expected, the finite sample properties deteriorate in most case, although they improve as $T$ gets larger and/or the explosive speed gets relatively faster (Figures D.1--D.4).
We also experiment with shorter minimum separation periods; we exclude the first and last 1% samples from the permissible break date $k$. The results provided in the appendix demonstrate that there is virtually no significant difference between two trimming parameters by comparing Figures 1--4 with D.5--D.8.
Finally, we investigate the finite sample behavior of the break date estimates when the volatility process $\sigma_t$ follows the single shift model, $\omega(s) = \sigma_{0} + (\sigma_{1} - \sigma_{0})1(s > \tau_{\sigma})$, where $\tau_{\sigma}=0.5$ and $\sigma_1/\sigma_0\in\{1/3,3\}$. Note that $\sigma_1/\sigma_0=1$ corresponds to the case of constant unconditional volatility.
For $\sigma_1/\sigma_0=1/3$, with which the volatility in the first half of the sample is higher than in the later sample, $\hat{k}_c$ and $\hat{k}_r$ perform similarly to the case of the constant unconditional volatility; the frequency of correctly selecting the true collapsing date $k_c$ is relatively high, whereas the performance of $\hat{k}_r$ is not satisfactory when $\phi_a=1.01$ and $\phi_b=0.96$ (Figures D.9--D.12).
In the case where the low volatility regime shifts to the high volatility one at the middle of the sample ($\sigma_1/\sigma_0=3$), it becomes more difficult to select the true collapsing and recovering dates than in the case of $\sigma_1/\sigma_0=1/3$, in particular, when $\phi_a=1.01$ and $\phi_b=0.96$. The possible reason is that both $k_c$ and $k_r$ are in the volatile regime when $\sigma_1/\sigma_0=3$ and thus it would be difficult to distinguish between the shifts in the parameters and the large shocks. However, the performance of the estimates improves for the larger values of $\phi_a$ and/or as $T$ gets larger (Figures D.13--D.16).
In this section, we demonstrate the application of the bubble dates estimation method for the four regime model investigated in the previous sections to two time series data sets. It is worth noticing that even though we may detect emergence and collapse of a bubble by some tests, we are not sure whether the final observation available for estimation is included in the collapsing regime or the unit root regime. Therefore, before estimating the break dates, we need to determine whether the three or four regime model is appropriate. Following harvey2017improving, we implement the BIC with the penalty term given by the number of estimated parameters plus the number of breaks to determine the model. In the following, we define by $BIC_2$ the BIC obtained from the model without the collapsing regime; by $BIC_3$ the BIC obtained from the model without the recovering regime; by $BIC_4$ the BIC obtained from the full four-regime model.
The first example is the close prices of monthly data from January 1985 to August 2013 of NASDAQ Composite Index.\footnote{The data was downloaded from https://finance.yahoo.com} First, we select the model using the BICs; $BIC_{2}$, $BIC_{3}$, and $BIC_4$ equals 3409.296, 3404.268, and 3387.727, respectively. Thus, the four regime model has the minimum BIC and is selected. As explained in the previous section, we fit the AR(1) model with two regimes and the estimated break date corresponds to the collapsing date, which is February 2000, as depicted in Figure 5. Next, by splitting the whole sample at the estimated collapsing date, the date of origination of the bubble is detected at August 1998 from the first subsample, while the date of recovery is estimated at September 2001 from the seconod subsample. As is observed in Figure 5, our method can detect the explosive and collapsing behavior very well.
The second example is an application of our method to the logarithm of the U.S.\ house price index from January 1991 to December 2012, provided by the Federal Housing Finance Agency, adjusted by the consumer price index.\footnote{The house price index and the CPI are available from https://www.fhfa.gov/DataTools/Downloads/pages/house-price-index.aspx and https://fred.stlouisfed.org/} Again, because the BIC selects the four regime model ($BIC_{2}$, $BIC_{3}$, and $BIC_4$ equal $-2809.591$, $-2895.905$, and $-2918.223$, respectively), we first estimate the collapsing date, which is estimated at November 2006. As is seen in Figure 6, the explosive behavior becomes mild around early in 2006 but the collapsing date is estimated just before the series staring crashing. By splitting the whole sample at the estimated collapsing date, the emergence date of the explosive behavior is estimated at September 1997, while the recovering date is at May 2011. These estimated dates are consistent with the visual inspection and the proposed method works very well.
In this paper, we considered the four regime bubble model and investigated the break date estimators using the sample splitting approach; the break dates are estimated one at a time. We showed that the break date estimated initially is consistent with the collapsing date of the bubble. We used the second subsample after the first estimated break date to estimate the break date returning to the normal market. The results revealed that this estimator is consistent with the recovering date if the explosive speed of the process during the bubble period is faster than the collapsing (recovering) speed, whereas in the case where the explosive speed is relatively slower, only the corresponding break fraction estimator is consistent. We need further investigation when the two speeds are the same.
Although we considered the case where the bubble occurred only once, we can extend our approach to the multiple bubble model in conjunction with the approach used by harvey2020date. In this case, we first identify the bubble regimes based on the PSY approach and then implement our method to each estimated regime.