# Testing for explosive bubbles: a review111The author is grateful to Eiji
Kurozumi, Rob Taylor, Shuping Shi, Yang Zu, Madina Karamysheva, Rustam
Ibragimov, Artem Prokhorov and participants of the Center for Econometrics and
Business Analysis (CEBA, St. Petersburg State University) seminar series for
helpful comments and suggestions. Research was supported by a grant from the
Russian Science Foundation (RSF, Project No. 20-78-10113).

Anton Skrobotov
Russian Presidential Academy of National Economy and Public Administration
Saint Petersburg State University, Center for Econometrics and Business
Analysis

(May 2022)

###### Abstract

This review discusses methods of testing for explosive bubbles in time series.
A large number of recently developed testing methods under various assumptions
about innovation of errors are covered. The review also considers the methods
for dating explosive (bubble) regimes. Special attention is devoted to time-
varying volatility in the errors. Moreover, the modelling of possible
relationships between time series with explosive regimes is discussed.

_Keywords_ : rational bubble; testing for explosive bubble; explosive
autoregression; time-varying volatility; right-tailed unit root testing.

_JEL Codes_ : C12, C22

##  1 Introduction

The identification of rational bubbles has been explored in a substantial body
of recent research. This is explained by the causal relationship between
speculative bubbles and crises in banking systems, as well as subsequent
macroeconomic recessions. The most popular approach is the rational bubble
model which explains explosive behaviour in prices of financial assets. In
other words, following Phillips et al. (2011) (hereafter PWY), consider a
rational bubble by using present value theory, in which the fundamental price
of the asset is the sum of present discounted values of expected future
dividend. By using the no arbitrage condition

$P_{t}=\frac{1}{1+R}E_{t}(P_{t+1}+D_{t+1}),$ (1)

where $P_{t}$ is the observed real price of an asset, $D_{t}$ is the observed
real dividend (received from the asset for ownership between $t-1$ and $t$),
and $R$ is the real interest rate used for discounting expected future cash
flows. Using a log-linear approximation, as in both PWY and Campbell and
Shiller (1989), the following solution is obtained:

$p_{t}=p_{t}^{f}+b_{t},$ (2)

where

$p_{t}^{f}=\frac{\kappa-\gamma}{1-\pi}+(1-\pi)\sum_{i=0}^{\infty}{\pi^{i}E_{t}d_{t+1+i}},$
(3) $b_{t}=\lim_{t\rightarrow\infty}{\pi^{i}E_{t}p_{t+i}},$
$E_{t}(b_{t+1})=\frac{1}{\pi}b_{t}=(1+\exp(\overline{d-p}))b_{t},$ (4)

with $p_{t}=\log(P_{t})$, $d_{t}=\log(D_{t})$, $\gamma=\log(1+R)$,
$\pi=1/(1+\exp(\overline{d-p}))$, where $(\overline{d-p})$ is the average log
dividend-price ratio, $\kappa=-\log(\pi)-(1-\pi)\log(1/\pi-1)$.

Therefore, the asset price in equation (2) is composed of a fundamental term
$p_{t}^{f}$ (explained by expected dividends) and a rational bubble term
$b_{t}$. For $\exp(\overline{d-p})>0$, the rational bubble $b_{t}$ is a sub-
martingale and explosive in expectation. Under equation (4), we have:

$b_{t}=\frac{1}{\pi}b_{t-1}+\varepsilon_{b,t}=(1+g)b_{t-1}+\varepsilon_{b,t},$
(5)

where $E_{t-1}(\varepsilon_{b,t})=0$,
$g=\frac{1}{\pi}-1=\exp(\overline{d-p})>0$ is the growth rate of the natural
logarithm of the bubble, and $\varepsilon_{b,t}$ is a martingale difference.

Explosive behaviour can be statistically approximated by explosive
autoregression of the form

$y_{t}=\mu+\rho y_{t-1}+\varepsilon_{t},$ (6)

with $\delta>1$.

If there is no bubble, i.e. $b_{t}=0$, equation (2) implies that $p_{t}$ is
fully determined by $p_{t}^{f}$ and therefore by $d_{t}$. Then, (3) implies

$d_{t}-p_{t}=-\frac{\kappa-\gamma}{1-\pi}-\sum_{i=0}^{\infty}{\pi^{i}E_{t}(\Delta
d_{t+1+i})}.$ (7)

If $p_{t}$ and $d_{t}$ are both integrated, then (7) implies that they are
cointegrated with cointegrating vector $(1,-1)$. However, with the presence of
a bubble, (5) implies explosive behaviour in $b_{t}$, so the explosive
behaviour will be in $p_{t}$ regardless of the behaviour of $d_{t}$. In this
case, $\Delta p_{t}$ is also explosive, because if $p_{t}$ is the explosive
process, then $p_{t}$ and $d_{t}$ can not be cointegrated. Based upon these
results, Diba and Grossman (1988) proposed to test for stationarity in $\Delta
p_{t}$ or to test for cointegration between $p_{t}$ and $d_{t}$ for detecting
the bubble. Phillips and Yu (2011, Section 2.2) discussed that explosiveness
in price is sufficient evidence for bubbles under various assumptions.

However, Diba and Grossman (1988) showed that impossibility of a downward
rational bubble implies that the bubble will never begin again after
collapsing. Evans (1991) considered periodically collapsing bubbles and showed
that the tests in Diba and Grossman (1988) have low power to detect this type
of a recurring bubble. The reason for this is that periodically collapsing
bubbles imply a non-negligible probability of the bubble collapsing (i.e.
explosive behavior occurs only temporarily in a small part of the entire
sample) and behave as an $I(1)$ process and even an $I(0)$ process (i.e. the
collapse is similar to a mean-reversion process).222Tao and Yu (2020)
investigated information criteria to select the best model among the unit root
model, the local-to-unit-root model, the mildly explosive model and the
regular explosive model. But their analysis was restricted to the models which
use the full sample unit root model or the full sample explosive model,
without any switching in some subsamples. Tao and Yu (2020) also considered
indirect inference estimation. Moreover, both time series, $p_{t}$ and
$d_{t}$, may be explosive and then they may be explosively
cointegrated.333Explosive cointegration, or coexplosiveness, means that there
exists a linear combination of explosive time series which does not contain
explosive behaviour. See Nielsen (2010) and Engsted and Nielsen (2012) for
details. Then, if $d_{t}$ is not explosive, finding of explosive behaviour in
$p_{t}$ can be sufficient evidence of the presence of a bubble as the
explosive behaviour originates only from $b_{t}$.

This survey concentrates on unit root testing techniques for detecting and
dating explosive bubbles. Table 1 presents both classical and contemporary
methods that were used to detect explosive bubbles in prior literature,
grouped based on the area of application.

Table 1: Applications of tests for explosive bubbles  stock markets |  Phillips et al. (2011, 2015a, 2015b), Shi (2013), Bohl et al. (2013), Astill et al. (2017), Astill et al. (2018), Breitung and Kruse (2013), Fulop and Yu (2017), Guo et al. (2019), Harvey et al. (2015, 2017, 2019), Pavlidis et al. (2017), Kurozumi (2020a), Liu and Peng (2019), Lin and Tu (2020), Monschang and Wilfling (2020), Whitehouse (2019), Wang and Yu (2017), Tao et al. (2019), Phillips and Shi (2018, 2019), Chen and Tu (2019)
---|---
prices of cryptocurrencies |  Cheah and Fry (2015), Cheung et al. (2015), Corbet et al. (2018), Corbet et al. (2019), Bouri et al. (2019), Astill et al. (2021), Harvey et al. (2020c, b)
real estate market (housing prices) |  Phillips and Yu (2011), Yiu et al. (2013), Anundsen et al. (2016), Engsted et al. (2016), Pavlidis et al. (2016), Kivedal (2013), Das et al. (2011), Escobari and Jafarinejad (2016), Shi (2017), Caspi (2016), Banerjee et al. (2020), Harvey et al. (2020a), Kurozumi (2020b), Pedersen and Schütte (2020), Horie and Yamamoto (2016), Chen et al. (2017, 2019), Shi and Phillips (2020)
commodities prices |  Gutierrez (2013), Etienne et al. (2014, 2015), Zhang and Yao (2016), Shi and Arora (2012), Fantazzini (2016), Su et al. (2017), Figuerola-Ferretti et al. (2015), Caspi et al. (2018), Harvey et al. (2016), Pavlidis et al. (2018), Evripidou et al. (2022)
exchange rates |  Bettendorf and Chen (2013)
art market |  Kräussl et al. (2016)
long annual ratio of the US Debt/GDP series |  Kaufmann and Kruse (2013)
credit risk in the European sovereign sector |  Phillips and Shi (2020)
extreme Yugoslavian hyperinflation |  Nielsen (2010)

The large body of empirical research justifies the relevance of the methods we
will discuss. The remainder of this survey is structured as follows. Section 2
reviews various recursive right-tailed tests for explosive bubbles. Section 3
considers tests for explosive bubbles under the assumption of time-varying
volatility in the innovations. Different methods for estimating the dates of
exuberance and collapse as well as monitoring methods are discussed in Section
4. Section 5 discusses asymptotic results for the autoregressive parameter of
explosive processes. Section 6 describes various models of the relationship
between multiple time series with potentially explosive regimes. Finally, the
Conclusion discusses possible further research directions.

##  2 Testing for explosive behaviour

The tests we discuss in this section are intended for testing the unit root
null hypothesis in time series against the alternative of explosive behaviour
in some subsample of the series. These tests are usually based on (augmented)
Dickey-Fuller-type regression. In general, explosive periods are what is
investigated in price time series. But in some applications, for more
accuracy, one may need to decompose the price time series into fundamental and
non-fundamental parts, and test for a bubble directly in the non-fundamental
component. For example, Shi (2017) investigated the bubbles in the housing
market and proposed to calculate the fundamental component of rent to price
ratios based on estimates of the 5-variable VAR model of the US national
market. The non-fundamental component then is the difference between log
price-to-rent ratio and the calculated fundamental component. Shi and Phillips
(2020) suggested to first estimate the predictive regression of the dividend
to price ratio (in first differences) on payoffs of the asset, construct
fitted values of the dependent value, and cumulate them. The resulting time
series is fundamental component of the dividend to price ratio series (see
also Shi and Phillips (2021) for details). Pavlidis et al. (2017, 2018)
proposed another approach. They used a corrected version of the real time
series based on forward (futures) asset prices and market expectations of
future prices to exclude the possibility of explosiveness in market
fundamentals which can take place. Anyway, whether we leave or exclude a
fundamental part of the time series, we need to test for the explosive
behaviour of the series of interest.

###  2.1 Supremum ADF test

Phillips et al. (2011) proposed recursive tests which can detect evidence of
explosive behaviour in time series $\\{y_{t}\\}$, $t=1,\dots,T$.444In contrast
to the methods described below, Tao et al. (2019) and Banerjee et al. (2020)
considered random coefficient autoregressive processes for which it is
possible to construct probabilistic forecasts of bubbles and crashes. One can
use Markov-Swithing regressions in which two regimes correspond to unit root
and explosive behaviour. See Hall et al. (1999), Shi (2013) for details. This
approach, however, allows only two regimes: unit root and explosive, and
adding a third regime, stationary collapsing, would make the procedure
computationally expensive. See Fulop and Yu (2017) also for a Bayesian
perspective. The reason of using the recursive tests is that the price
behaviour is dominated by the explosive (i.e., bubble) component because it is
believed that the fundamental part of the price is at most $I(1)$. Therefore,
we can directly test for the bubbles in prices/dividend to price ratio, not in
the non-fundamental component directly.555As discussed in Shi and Phillips
(2021), we could apply the tests for bubbles to level variables, not logs, if
we allow for a time-varying discount factor.

Consider the following ADF-type regression as

$y_{t}=\mu+\rho y_{t-1}+\sum_{j=1}^{k}{\phi_{j}\Delta
y_{t-j}}+\varepsilon_{t}.$ (8)

We want to test the null hypothesis of a unit root, $H_{0}:\rho=1$, against
the right-tailed alternative, $H_{1}:\rho>1$, at least in some subsample. PWY
proposed a recursive evolving test which consists of expanding the sample and
taking the supremum over all test statistics for each subsample. In other
words, we run all regressions for $t=k+1,\dots,\lfloor\tau T\rfloor$ for all
$\tau\in[\tau_{0},1]$ ($\lfloor\ \rfloor$ denotes the integer part of value)
with some preliminary chosen $\tau_{0}$,666Phillips et al. (2011) use
$\tau_{0}=0.1$ in empirical application while Phillips et al. (2015a)
recommend to use a more flexible choice $\tau_{0}=0.01+1.8/\sqrt{T}$. $T$ is
the sample size of the series. Consider the following ADF-type test statistic:

$ADF_{\tau}=\left(\frac{\sum_{j=1}^{\tau}{e_{j-1}^{2}}}{\hat{\sigma}_{\tau}^{2}}\right)^{1/2}\left(\hat{\rho}_{\tau}-1\right),$
(9)

where $\hat{\rho}_{\tau}$ being the OLS estimator of $\rho$ based on
regression (8) over the observations $t=k+1,\dots,\lfloor\tau T\rfloor$ (first
$\lfloor\tau T\rfloor$ observations), $\hat{\sigma}_{\tau}^{2}$ is the
corresponding variance estimator of $\sigma^{2}_{\varepsilon}$, and $e_{t}$
are OLS residuals from (8)777Note that $ADF_{1}$ is the standard full-sample
ADF test statistic.. Evidently, under the null hypothesis,

$ADF_{\tau}\Rightarrow\frac{\int_{0}^{\tau}{\tilde{W(r)}dW(r)}}{\left(\int_{0}^{\tau}{\tilde{W(r)}^{2}}\right)^{1/2}},$
(10)

where $W(r)\equiv W$ is standard Brownian motion, and
$\tilde{W}=W-\frac{1}{\tau}\int_{0}^{1}{W}$. The supremum-type test statistic
is

$SADF(\tau_{0}):=\sup_{\tau\in[\tau_{0},1]}{ADF_{\tau}}\Rightarrow\sup_{\tau\in[\tau_{0},1]}{\frac{\int_{0}^{\tau}{\tilde{W}dW}}{\left(\int_{0}^{\tau}{\tilde{W}^{2}}\right)^{1/2}}}.$
(11)

This test statistic can be used for testing for a unit root against explosive
behaviour in some subsample888Kaufmann and Kruse (2013) proposed the test
based on the bias corrected estimator of $\hat{\rho}_{\tau}$..

Lui (2019) allowed for a long memory dynamic in $\varepsilon_{t}$. If the
series exhibit long memory, then the standard SADF test diverges to infinity
at rate $n^{d}$, where $d\in(0,0.5)$ is the memory parameter, thus, the null
hypothesis of no explosive bubble is often falsely rejected. Lui (2019)
suggested to replace the estimator of $\sigma^{2}_{\varepsilon}$ in (9) by a
HAR (fixed-$b$) estimator. Critical values for the SADF test depend on the
estimate of the memory parameter $d$, $\hat{d}$.

###  2.2 More general data generating processes

Phillips et al. (2014) analysed and compared the limiting theory of the PWY
test under different hypotheses and model specifications. The question of
whether a constant and/or linear trend should be added to regression (8) was
investigated. Different specifications under the null are also allowed. That
is, PWY assumed

$y_{t}=y_{t-1}+\varepsilon_{t},$

while Diba and Grossman (1988) assumed

$y_{t}=\tilde{\mu}+y_{t-1}+\varepsilon_{t},$

so that $y_{t}$ has a deterministic trend if $\tilde{\mu}\neq 0$. Phillips et
al. (2014) considered the general specification which allows local-to-zero
constant as:

$y_{t}=\tilde{\mu}T^{-\eta}+y_{t-1}+\varepsilon_{t},\ \eta\geq 0.$ (12)

Here $y_{t}$ has a deterministic drift of the form $\tilde{\mu}t/T^{\eta}$,
whose magnitude depends on sample size and localizing parameter $\eta$. If
$\eta$ tends to zero or infinity, we obtain the two limiting cases considered
above.

Rewriting the model (12) as

$y_{t}=\frac{\tilde{\mu}t}{T^{\eta}}+\sum_{j=1}^{t}{\varepsilon_{j}}+y_{0},$
(13)

it can be seen that the drift is small in relation to the stochastic trend
when $\eta>1/2$ and equal to or stronger than the stochastic trend when
$\eta\leq 1/2$. Only in the last case $\eta$ can be consistently estimated
(see Phillips et al. (2014, Appendix A )999Consistent estimator of $\eta_{T}$
is $\hat{\eta}_{T}=-\log|\hat{\mu}_{T}|/\log T$ and
$\tilde{\eta}_{T}=-\log|\tilde{\mu}_{T}|/\log T$, where
$\hat{\mu}_{T}=\sum_{t=1}^{T}{ty_{t}}/\sum_{t=1}^{T}{t^{2}}$ and
$\tilde{\mu}_{T}=\sum_{t=1}^{T}{\tilde{t}y_{t}}/\sum_{t=1}^{T}{\tilde{t}^{2}}$,
$\tilde{t}=t-T^{-1}\sum_{s=1}^{T}{s}$ are consistent estimators of
$\mu_{T}=\tilde{\mu}T^{-\eta}$. However, there is a second order bias of
$-\log|\mu|/\log T$ for $\hat{\eta}_{T}$ which may be sufficiently large in
finite samples.), because the drift term is dominated by stochastic trend. In
other cases, the estimators of $\eta$ usually converge to 1/2 corresponding to
the rate of stochastic trend (see Phillips et al. (2014, Appendix A) for the
proof). Unfortunately, there is no finite sample comparison of the performance
of the estimator $\hat{\eta}_{T}$.

Phillips et al. (2014) also noted that under the alternative hypothesis,
adding a constant and/or a linear trend is not realistic for actual time
series (see, however, Wang and Yu (2017), who considered adding a linear
trend).

The limiting distributions of the ADF test under the null hypothesis were
obtained for $\eta>0.5$, $\eta<0.5$, and $\eta=0.5$ (these differ from (17)
because of local drift). Finite sample simulations demonstrated that for
$\eta>0.5$ the differences between the asymptotic and finite sample
distributions are negligible regardless of different $\eta$. This is not the
case for $\eta=0.5$, due to the dominating linear trend in the series, and the
differences vanishes with $\eta$ approaching to zero.

In summary, Phillips et al. (2014) recommended to always include constant term
in constructing recursive ADF tests, but to compare the actual test statistic
with different critical values (for $\eta>0.5$ and $\eta<0.5$) for
robustness.101010Sollis (2016) analyzed the behaviour of PWY procedure (SADF
test) when there may be a break in the drift parameter. Sollis (2016)
indicated a spurious rejection of unit root null under the break. Also, he
discussed the possible solutions based on pre-estimating the break date.

Phillips and Yu (2009) studied the following more general data generating
processes which specified the new initial value after the bubble episode, so
that the new unit root period begins not from the final value of the explosive
period, but from a different value:

$\displaystyle y_{t}$ $\displaystyle=$ $\displaystyle
y_{t-1}\mathbb{I}(t<T_{e})+\rho_{T}y_{t-1}\mathbb{I}(T_{e}\leq t\leq T_{c})$
$\displaystyle+\left(\sum_{k=T_{c}+1}^{t}{\varepsilon_{k}}+y^{*}_{T_{c}}\right)\mathbb{I}(t>T_{c})+\varepsilon_{t}\mathbb{I}(t\leq
T_{c}),$ $\displaystyle\rho_{T}$ $\displaystyle=$ $\displaystyle
1+\frac{c}{T^{\alpha}},\ c>0,\ \alpha\in(0,1),$

where $T_{e}=\lfloor\tau_{e}T\rfloor$ is the origination date of the bubble,
$T_{c}=\lfloor\tau_{c}T\rfloor$ is the date of its collapse, and thus the
period $[\tau_{e},\tau_{c}]$ is the bubble episode, the periods
$[1,T_{e})\cup(T_{c},T]$ are the normal market periods, $\mathbb{I}(\cdot)$
denotes an indicator function. At the moment of reinitialising, $T_{c}$, the
process is ‘‘jumping’’ to another level $y^{*}_{T_{c}}$, which can be written
as $y^{*}_{T_{c}}=y_{T_{e}}+y^{*}$ with $y^{*}=O_{p}(1)$. It is assumed that
$y_{0}=O_{p}(1)$.

Phillips and Shi (2018) (see also Harvey et al. (2017) and equations (53)-(54)
further in the text) considered a more reasonable mechanism that allows for
transitory collapse dynamics. So, an instantaneous collapse as in (2.2) may be
unrealistic, and some transient dynamics may be introduced after the peak —
the so-called collapse regime. The corresponding DGP can be written as

$y_{t}=\begin{cases}\tilde{\mu}T^{-\eta}+y_{t-1}+\varepsilon_{t},&t\in[1,T_{e})\cup(T_{r},T]\\\
(1+\delta_{1T})y_{t-1}+\varepsilon_{t},&t\in[T_{e},T_{c}]\\\
(1-\delta_{2T})\gamma_{T}y_{t-1}+\varepsilon_{t},&t\in(T_{c},T_{r}]\\\
\end{cases},$ (15)

where $T_{r}=\lfloor\tau_{r}T\rfloor$ denotes the end of the explosive regime
or the date of market recovery, so that the period $(T_{c},T_{r}]$ is the
collapse period and the periods $[1,T_{e})\cup(T_{r},T]$ are the normal market
periods. Also, $\delta_{1T}=c_{1}T^{-\alpha}$, $\delta_{2T}=c_{2}T^{-\beta}$,
$c_{1},c_{2}>0$ and $\alpha,\beta\in[0,1)$. The formulation of AR coefficients
follow moderate deviations from unity as in Phillips and Magdalinos (2007a):
the coefficient $\varphi_{T}$ deviates towards explosive behaviour, and the
coefficient $\gamma_{T}$ deviates towards stationary behaviour. Fortunately,
PWY procedure can consistently detect the bubble for this more general DGP.

###  2.3 Generalized SADF test

Phillips et al. (2015a, b) (hereafter PSY for both papers) considered the
following test statistic to account for multiple explosive regimes in time
series (focusing on the $\eta>0.5$ case for the drift term as more relevant in
empirical applications). Their Generalized Supremum ADF (GSADF) test is

$GSADF(\tau_{0})=\sup_{\tau_{2}\in[\tau_{0},1],\tau_{1}\in[0,\tau_{2}-\tau_{0}]}ADF_{\tau_{1}}^{\tau_{2}},$
(16)

where $ADF_{\tau_{1}}^{\tau_{2}}$ is the ADF-test statistic from (9) for
sample $t=\lfloor\tau_{1}T\rfloor+1,\dots,\lfloor\tau_{2}T\rfloor$. In this
form, $\tau_{\omega}=\tau_{2}-\tau_{1}$ is a window size. That is, for every
fixed $\tau_{2}$ the ADF test statistic is calculated over all possible
$\tau_{1}$ from 0 to $\tau_{2}-\tau_{0}$. The GSADF test is constructed as the
supremum over all possible subsample ADF test statistics with the sample size
not larger than $\tau_{0}$. It has the following asymptotic distribution:

$GSADF(\tau_{0})\Rightarrow\sup_{\tau_{2}\in[\tau_{0},1],\tau_{1}\in[0,\tau_{2}-\tau_{0}]}{\frac{\int_{\tau_{1}}^{\tau_{2}}{\tilde{W}dW}}{\left(\int_{\tau_{1}}^{\tau_{2}}{\tilde{W}^{2}}\right)^{1/2}}},$
(17)

where $W$ is standard Brownian motion, and
$\tilde{W}=W-\frac{1}{\tau_{2}-\tau_{1}}\int_{\tau_{1}}^{\tau_{2}}{W}$. PSY
allows minimum subsample window as $\tau_{0}$ (see also footnote 5) and
tabulated critical values for different $\tau_{0}$ and sample sizes. The SADF
test previously proposed by Phillips et al. (2011) is a special case of GSADF,
obtained by setting $\tau_{1}=0$ and $\tau_{2}=r_{\omega}\in[\tau_{0},1]$.
Phillips and Shi (2018) showed that although the GSADF procedure is designed
to detect bubble behaviour, it can also detect crisis periods (see also
Phillips and Shi (2019), Phillips and Shi (2020)) which are often observed in
empirical applications. GSADF outperforms SADF in terms of power because GSADF
is constructed in such a way that it obtains the most explosive subperiod in
maximization. The GSADF plays an important role in the subsequent discussion.

###  2.4 Extensions

There are some approaches and modifications related to SADF and GSADF tests.
Homm and Breitung (2012) proposed to consider the supremum of the recursive
Chow test through the following regression:

$\Delta\tilde{y}_{t}=\phi_{HB}\mathbb{I}(t>\lfloor\tau
T\rfloor)\tilde{y}_{t-1}+e_{t},$ (18)

where $y$ is preliminary de-meaned as $\tilde{y}_{t}=y_{t}-\bar{y}$ where
$\bar{y}=T^{-1}\sum_{t=1}^{T}y_{t}$.111111The lagged values of
$\Delta\tilde{y}_{t}$ can be added to (18). The Chow-type test statistic,
$C_{\tau}$, is defined as $t$-ratio for $\phi_{HB}$. Then the test of Homm and
Breitung (2012) is defined as

$HB=\sup_{\tau\in[0,1-\tau_{0}]}C_{\tau}.$ (19)

This statistic is actually the supremum of a sequence of backward recursive
statistics. Harvey et al. (2015) developed local-to-unit root asymptotic
distribution of the HB test as well as the SADF test and found that the HB
test outperformed SADF if the explosive regime belongs to the end of the
sample and does not terminate. Harvey et al. (2015) suggested to use a so-
called union of rejection testing strategy to utilize the advantages of both
tests. This strategy is based on rejection at least by one of the tests, and
can be written as

$\text{Reject $H_{0}$ if $\\{SADF>\psi_{\xi}q^{SADF}_{\xi}\text{ or
}HB>\psi_{\xi}q^{HB}_{\xi}\\}$},$

with $q^{SADF}_{\xi}$ and $q^{HB}_{\xi}$ being critical values at level $\xi$,
and $\psi_{\xi}$ being the scaling constant intended to ensure correct
(asymptotic) size of the composite procedure.

Korkos et al. (2019) extended the covariate ADF (CADF) unit root testing
approach of Hansen (1995) for testing for explosive bubbles to improve the
power of the test. In this approach, the model is generated as

$\displaystyle y_{t}$ $\displaystyle=$ $\displaystyle\mu+u_{t},$ (20)
$\displaystyle\Delta u_{t}$ $\displaystyle=$ $\displaystyle\delta
u_{t-1}+\varepsilon_{t},$ (21) $\displaystyle\Phi(L)\varepsilon_{t}$
$\displaystyle=$ $\displaystyle b(L)^{\prime}\Delta x_{t}+\nu_{t},$ (22)

where $\Delta x_{t}$ is an $m$-vector of stationary covariates, and $\Phi(L)$
and $\beta(L)$ are some lag operators. It is assumed that $\Psi(L)\Delta
x_{t+k_{1}+1}=z_{t}$ where $\Psi(L)$ is some autoregressive lag polynomial of
order $l$.

The main idea is to add leads and lags of stationary covariates to regression
(8) as

$\Delta y_{t}=\mu+\delta y_{t-1}+\sum_{j=1}^{k}{\phi_{j}\Delta
y_{t-j}}+\sum_{j=-k_{1}}^{k_{2}}{\beta_{j}\Delta x_{t-j}}+\varepsilon_{t}$
(23)

and calculate the $CADF_{r}$ statistic which is simply $t$-ratio for testing
$\delta=0$ over the observations $t=k+1,\dots,\lfloor\tau T\rfloor$. The final
$SCADF$ test statistic of Korkos et al. (2019) is defined as

$SCADF(\tau_{0}):=\sup_{\tau\in[\tau_{0},1]}{CADF_{\tau}}.$ (24)

The limiting distribution of the $CADF_{r}$ has the following form:

$SCADF(\tau_{0})\Rightarrow\sup_{\tau\in[\tau_{0},1]}{\frac{\int_{0}^{\tau}{\tilde{Q}(r)dP(r)}}{\left(\int_{0}^{\tau}{\tilde{Q}(s)^{2}}\right)^{1/2}}},$
(25)

where $\tilde{Q(r)}=Q(r)-\frac{1}{\tau}\int_{0}^{\tau}{Q(r)}$,
$Q(r)=b(1)\Psi(1)W_{z}(r)+W_{\nu}(r)$ and $P(s)=W_{\nu}(r)/\sigma_{\nu}$. It
can be shown that the limiting distribution is based on a convex mixture of
the standard normal and the Dickey-Fuller distribution with the nuisance
parameter $\varrho^{2}$ (the value of $\varrho^{2}$ determines the weights and
measures the relative contribution of the covariate $\Delta x_{t}$ to the
error term $\varepsilon_{t}$). The estimator of $\varrho^{2}$ is given as
$\hat{\varrho}^{2}=\hat{\sigma}^{2}_{\varepsilon\nu}/(\hat{\sigma}^{2}_{\varepsilon}\hat{\sigma}^{2}_{\nu})$,
where $\sigma^{2}_{\varepsilon\nu}$, $\sigma^{2}_{\varepsilon}$, and
$\sigma^{2}_{\nu}$ are, respectively, the covariance between $\varepsilon$ and
$\nu$, the variance of $\varepsilon$, and the variance of $\nu$. All of them
are estimated via the HAC approach.

Korkos et al. (2019) proposed a bootstrap algorithm similar to Chang et al.
(2017) to obtain critical values for the $SCADF$ test and to avoid estimating
$\varrho^{2}$ in each subsample.

Whitehouse (2019) considered a GLS-based version of the PWY (SADF) test.
Earlier, Harvey and Leybourne (2014) investigated the OLS and GLS-based right-
tailed unit root tests and found that in contrast to left-tailed tests, the
GLS-based test has higher power when the magnitude of the initial condition of
the series is large.121212The initial condition of the series is the deviation
of the first observation from the deterministic component of the process. The
GLS-based test follows from the auxiliary regression

$\Delta\tilde{u}_{t}=\delta\tilde{u}_{t-1}+\varepsilon_{t},$ (26)

where $\tilde{u}_{\tau,t}=y_{t}-z_{t}^{\prime}\tilde{\theta}$,
$\tilde{\theta}$ is the OLS estimator from the (quasi) GLS regression
$y_{\bar{c}}=(y_{1},y_{2}-\bar{\rho}y_{1},\dots,y_{\tau}-\bar{\rho}y_{\tau
T-1})^{\prime}$ on
$z_{\bar{c}}=(z_{1},z_{2}-\bar{\rho}z_{1},\dots,z_{\tau}-\bar{\rho}z_{\tau
T-1})^{\prime}$, where $\bar{\rho}=1+\bar{c}/T$ and $z_{t}=1$ or
$z_{t}=(1,t)^{\prime}$ is the deterministic component.131313$\bar{c}=1.6$ in
the constant case and $\bar{c}=2.4$ in the trend case were recommended
following Harvey and Leybourne (2014). Let $ADF\text{-}GLS_{\tau}$ be a simple
$t$-ratio from the regression (26) for $t=1,\dots,\lfloor\tau T\rfloor$. Then
the supremum GLS-based test proposed by Whitehouse (2019) is of standard form:

$SADF\text{-}GLS(\tau_{0}):=\sup_{\tau\in[\tau_{0},1]}{ADF\text{-}GLS_{\tau}}.$
(27)

This test has higher local asymptotic power than the conventional SADF test
when the explosive period is large relative to the full sample size (i.e., the
proportion of the sample for which the data follows an explosive process is
large). Moreover, the GLS-based test becomes better if the magnitude of
initial condition increases. The initial condition does not affect the ranking
of the two test types. Whitehouse (2019) also proposed a union of rejection
testing strategy based on two tests, $SADF\text{-}GLS(\tau_{0})$ and
$SADF(\tau_{0})$ for both cases, with or without trend.

###  2.5 Testing for end-of-sample bubble

Astill et al. (2017) considered the situation when the explosive bubble is
both ongoing at the end of the sample, and of finite length. They adopted end-
of-sample instability tests of Andrews (2003) and Andrews and Kim (2006) for
testing the null of no end-of-sample bubble against the bubble alternative.
The model considered has the following form:

$\displaystyle y_{t}$ $\displaystyle=$ $\displaystyle\mu+u_{t},\
t=1,\dots,T+m,$ (28) $\displaystyle u_{t}$ $\displaystyle=$
$\displaystyle\begin{cases}u_{t-1}+\varepsilon_{t},\ t=1,\dots,T\\\ \delta
u_{t-1}+\varepsilon_{t},\ t=T+1,\dots,T+m,\end{cases}$ (29)

where $\varepsilon_{t}$ is a mean zero, stationary and ergodic process. The
series follows a unit root process before the moment $T$ and possibly
explosive process during the following $m$ observations with $m$ being
substantially smaller than $T$ and of finite length. The null hypothesis
corresponds to no bubble, $\rho=1$, and the alternative hypothesis corresponds
to a bubble during the end-of-sample, $\rho>1$. Astill et al. (2017) noted
that under the null, $\Delta y_{t}=\varepsilon_{t}$ during the full sample,
while under the alternative, $\Delta y_{t}=\varepsilon_{t}$ up to time $T$ and
$\Delta y_{t}=\Delta
u_{t}=\delta(1+\delta)^{t-T-1}u_{T}+\sum_{j=0}^{t-T-1}{(1+\delta)^{j}\Delta\varepsilon_{t-j}}$,
where $\delta=\rho-1$, and the first term, which is $O_{p}(T^{1}/2)$,
dominates the second term, $O_{p}(1)$. By the first order Taylor series
expansion of $(1+\delta)^{t-T-1}$ around $\delta=0$,
$(1+\delta)^{t-T-1}\approx 1+(t-T-1)\delta$, we have the following
approximation:

$\Delta y_{t}=\delta(1-\delta)u_{T}+\delta^{2}u_{T}(t-T)+e_{t},$ (30)

where $e_{t}$ contains the higher order terms in the Taylor series expansion
and $O_{p}(1)$ term. Then, the instability test is simply the $t$-test for
upward trend in regression of $\Delta y_{t}$ on a linear trend. Omitting the
constant is correct for rolling sub-sample statistics because they are
calculated before the moment $T$. The test statistic may be simply the
numerator of the $t$-test for upward trend in regression of $\Delta y_{t}$ on
a linear trend:

$S_{m}=\sum_{t=j+1}^{j+m}{(t-j)\Delta y_{t}}\text{ or
}\sum_{t=j+1}^{j+m}{\sum_{s=t}^{j+m}\Delta y_{s}}.$ (31)

This is Andrews $S$ type statistic. The Andrews-Kim $R$ type statistic is
defined as follows:

$R_{m}=\sum_{t=j+1}^{j+m}{\left(\sum_{s=t}^{j+m}\Delta y_{s}\right)^{2}}.$
(32)

The asymptotic size of $S_{m}$ and $R_{m}$ will not be affected by finite
number of bubbles of finite length in the period before moment $T$. The
critical values are obtained using sub-sampling techniques applied to the
first $T$ observations.141414Astill et al. (2017) also recommended $m=10$
based on numerical simulations.

To account for possible unconditional variance on innovations $\varepsilon$,
Astill et al. (2017) proposed a studentised White-type version of (33):

$S_{m}^{\star w}=\frac{S_{m}}{\sqrt{\sum_{t=j+1}^{j+m}((t-j)\Delta
y_{t})^{2}}}.$ (33)

Astill et al. (2017) demonstrated that their methods dominate PSY for the case
of short-lived end-of-sample bubble.

##  3 Testing for explosive bubbles under time-varying volatility

PWY and other papers discussed above assumed that the unconditional variance
of the innovation process is stationary under both the null unit root and
explosive alternative hypothesis. However, a general decline in the
unconditional volatility of the shocks driving macroeconomic series has been a
commonly observed phenomenon. Some classical unit root tests are severely
oversized because their limiting distributions depend on a particular
function, the so-called variance profile, of the underlying volatility process
(see Cavaliere (2004), Cavaliere and Taylor (2007, 2008a, 2009) and references
therein). It should be noted that supremum based ADF-type test (PWY, PSY) are
still robust to conditional heteroskedasticity as demonstrated by PSY.

Harvey et al. (2016) addressed this issue in an explosive bubble context.
Consider the following DGP for $\\{y_{t}\\}$ in time-varying parameter form

$y_{t}=(1+\delta_{t})y_{t-1}+\varepsilon_{t}\qquad\mbox{or}\qquad\Delta
y_{t}=\delta_{t}y_{t-1}+\varepsilon_{t}$ (34)

with obvious definition of $\delta_{t}$: $\delta_{t}$ may be $>0$ for an
explosive regime, $<0$ for a stationary collapsing regime and $=0$ for a unit
root regime. Harvey et al. (2016) and subsequent papers considered local-to-
unit root behaviour of $\delta_{t}$: $\delta_{t}=c_{1}/T$ for explosive period
and $\delta_{t}=c_{2}/T$ for stationary collapse period with $c_{1}>0$ and
$c_{2}\leq 0$. The process (34) can be seen as different reparametrization of
(15) except to behaviour of $\delta_{t}$. The non-stationary volatility is
generated as $\varepsilon_{t}=\sigma_{t}z_{t}$, where $\\{z_{t}\\}$ is a
martingale difference sequence with respect to natural filtration, and the
volatility $\sigma_{t}$ is defined as $\sigma_{\lfloor sT\rfloor}=\omega(s)$
for $s\in[0,1]$, where $\omega(\cdot)\in{\cal D}$ is a non-stochastic and
strictly positive function satisfying
$0<\underline{\omega}<\omega(s)<\bar{\omega}<\infty$. An assumption about the
volatility function allows a general class of volatility processes, such as
breaks in volatility, trending volatility, and regime switching volatility.
Harvey et al. (2016) demonstrated that, similar to classical unit root tests,
asymptotic inference of the PWY (SADF) test will be affected by the presence
of time-varying volatility: for the most natural cases of non-stationary
volatility behaviour such as single and double breaks in volatility, and
trending volatility, the SADF test is badly oversized so that it often
spuriously rejects the null hypothesis of no bubble against an explosive
alternative in some sub-period. To take into account this issue, Harvey et al.
(2016) use the following wild bootstrap. We note that although Harvey et al.
(2016) proposed their algorithm only for the SADF test, their methodology can
be easily implemented for the GSADF test.

######  Algorithm 1 (Bootstrap Tests)

>   * Step 1:
>
> Generate the vectors of bootstrap innovations as $e_{t}^{*}=w_{t}\Delta
> y_{t}$ for $t=2,\dots,T$ initialized at $e_{1}^{*}=0$, where
> $\\{w_{t}\\}_{2}^{T}$ be IID sequence of N(0,1) random variates.
>
>   * Step 2:
>
> Construct the bootstrap sample data via recursion $\Delta
> y_{t}^{*}=e_{t}^{*}$ for $t=2,\dots,T$ initialized at $y^{*}_{0}=0$.
>
>   * Step 3:
>
> Using the bootstrap sample, $\\{y_{t}^{*}\\}$, compute the bootstrap $GSADF$
> statistic denoted as $GSADF^{*}$ exactly as was done for the original data
> for fixed lag length $k=0$.
>
>   * Step 4:
>
> Bootstrap $p$-values are then defined as:
> $P_{GSADF,T}^{\ast}:=G_{GSADF,T}^{\ast}(GSADF^{\ast})$, where
> $G_{GSADF,T}^{\ast}(\cdot)$ denote the conditional (on the original sample
> data) cumulative distribution functions (cdfs) of $GSADF^{\ast}$. In
> practice, the cdfs required here will be unknown, but can be approximated in
> the usual way via numerical simulation.
>
>

This algorithm allows a very general form of innovation variance. Although
Harvey et al. (2016) assumed that this variance is non-stochastic, bounded and
displays a countable number of jumps, their approach still holds for the
assumptions made in Cavaliere and Taylor (2009) (they allow stochastic
limiting variance including, e.g., nonstationary autoregressive stochastic
volatility, models with random volatility jumps, near-integrated GARCH
processes, and explosive, nonstationary volatility). Note that in this
algorithm, in Step 3, we set $k=0$ because the wild bootstrap scheme
annihilates any weak dependence presented in $\Delta y_{t}$ in Step 1.
However, Pedersen and Schütte (2020) proposed a sieve based implementation of
Step 2 and 3, similar to Chang and Park (2003), which improve the size
properties.

Harvey et al. (2019) proposed a weighted least squares based modification of
PWY test. Transform the model as

$\frac{\Delta y_{t}}{\sigma_{t}}=\rho_{t}\frac{y_{t-1}}{\sigma_{t}}+z_{t},\
t=2,\dots,T.$ (35)

This regression is infeasible because we do not observe variance function
$\sigma_{t}$. If $\sigma_{t}$ would be known, we could construct the supremum
based test as PWY:

$SBZ(\tau_{0})=\sup_{\tau\in[\tau_{0},1]}BZ_{\tau},$ (36)

where $BZ_{r}$ is calculated from regression (35) over subsample
$\\{y_{1},\dots,y_{\lfloor\tau T\rfloor}\\}$

$BZ_{\tau}=\frac{\sum_{t=1}^{\lfloor\tau
T\rfloor}{\Delta\tilde{y_{t}}\tilde{y}_{t-1}/\sigma_{t}^{2}}}{\left(\sum_{t=1}^{\lfloor\tau
T\rfloor}{\tilde{y}_{t-1}^{2}/\sigma_{t}^{2}}\right)^{1/2}}.$ (37)

Here $\tilde{y_{t}}=y_{t}-y_{1}$ to guarantee the invariance to non-zero mean.
The limit distribution under the null and local alternative depends on
limiting volatility process $\omega(r)$. To make the test feasible, Harvey et
al. (2019) use non-parametric kernel smoothing estimator of $\sigma_{t}$:

$\hat{\sigma}_{t}^{2}=\frac{\sum_{i=2}^{T}{K_{h}\left(\frac{i-t}{T}\right)(\Delta
y_{i})^{2}}}{\sum_{i=2}^{T}{K_{h}\left(\frac{i-t}{T}\right)}},$ (38)

where $K_{h}(s)=K(s/h)/h$ and $K(\cdot)$ is a kernel function with a bandwidth
parameter $h$.

Because the limiting distribution of the SBZ test still depends on the
volatility function, Harvey et al. (2019) utilise the wild bootstrap
implementation of Harvey et al. (2016) to guarantee control of size. Moreover,
Harvey et al. (2019) suggest to use a bootstrap-based union of rejections
testing strategy because neither of the tests, SBZ and SADF, dominate each
other across all volatility specifications (and SBZ displays non-monotonic
power in some cases). This strategy has the form

$\text{Reject $H_{0}$ if $\\{SADF>\psi_{\xi}q_{\xi}^{SADF}\text{ or
}SBZ>\psi_{\xi}q_{\xi}^{SBZ}$\\}}.$

or, equivalently,

$\text{Reject $H_{0}$ if
$U=\max\left(SADF,\frac{q_{\xi}^{SADF}}{q_{\xi}^{SBZ}}SBZ\right)>q_{\xi}^{U}$}.$

where $\psi_{\xi}$ is a (asymptotic) scaling constant to ensure correct
(asymptotic) size of the composite procedure,and $q_{\xi}^{U}=\psi_{\xi}\times
q_{\xi}^{SADF}$ is an (asymptotic) critical value for $U$ test.

To control size for this procedure, because the limiting distribution of the
SADF and SBZ tests and therefore asymptotic critical values depend on the
volatility function, the following modification of decision rule is used:

$\text{Reject $H_{0}$ if
$U=\max\left(SADF,\frac{q_{\xi}^{\ast,SADF}}{q_{\xi}^{\ast,SBZ}}SBZ\right)>q_{\xi}^{\ast,U}$},$

where $q_{\ast,\xi}^{SBZ}$ and $q_{\ast,\xi}^{SADF}$ are the bootstrap based
critical values for $SADF$ and $SBZ$ test statistics, and $q_{\ast,\xi}^{U}$
is the bootstrap based critical value for the $U$ test statistic.

Harvey et al. (2020c) proposed another method which controls size under time-
varying volatility. This method is based on cumulated signs
$C_{t}=\sum_{i=2}^{t}{sign(\Delta y_{t})}$, $t=2,\dots,T$. The supremum sign-
based test is defined as

$sGSADF(\tau_{0})=\sup_{\tau_{2}\in[\tau_{0},1],\tau_{1}\in[0,\tau_{2}-\tau_{0}]}sADF_{\tau_{1}}^{\tau_{2}}.$
(39)

where $sADF_{\tau_{1}}^{\tau_{2}}$ is a $t$-ratio in the regression

$\Delta C_{t}=\hat{\delta}(\tau_{1},\tau_{2})C_{t-1}+e_{t}$ (40)

over subsample from $\lfloor\tau_{1}T\rfloor$ to $\lfloor\tau_{2}T\rfloor$.
I.e.,

$sADF_{\tau_{1}}^{\tau_{2}}=\frac{\hat{\delta}(\tau_{1},\tau_{2})}{\sqrt{\hat{s}^{2}(\tau_{1},\tau_{2})/\sum_{t=\lfloor\tau_{1}T\rfloor+1}^{\lfloor\tau_{2}T\rfloor}C_{t-1}^{2}}}$
(41)

where
$\hat{s}^{2}(\tau_{1},\tau_{2})=(\lfloor\tau_{2}T\rfloor-\lfloor\tau_{1}T\rfloor-1)^{-1}\sum_{t=\lfloor\tau_{1}T\rfloor+1}^{\lfloor\tau_{2}T\rfloor}{e_{t}^{2}}$.
Because $sign(\Delta y_{t})=sign(z_{t})$ under the null hypothesis, the test
is exact invariant to the volatility function $\sigma_{t}$. Moreover, the
sGSADF test is exact invariant to the constant in DGP. If we allow a weak
dependence of errors, we should just replace $sign(\Delta y_{t})$ by
$sign(\Delta y_{t}-\sum_{j=1}^{k}{\hat{\phi}_{j}(t)\Delta y_{t-j}})$, where
$\hat{\phi}_{j}(t)$ are obtained from the following recursive regression

$\Delta
y_{i}=\hat{\alpha}(t)+\hat{\delta}(t)y_{i-1}+\sum_{j=1}^{k}{\hat{\phi}_{j}(t)\Delta
y_{i-j}}+e_{i}$

for $i=k+5,\dots,t$.

The special case of $sGSADF$ test is the $sSADF$ with the restriction
$r_{1}=0$. Simulations of Harvey et al. (2020c) demonstrated that standard
$GSADF$ and $SADF$ tests are more powerful (in terms of size-adjusted power)
than $sGSADF$ and $sSADF$ tests respectively for large deviations from the
null. Also, the $sGSADF$ is more powerful than the $sSADF$ in contrast to
relationship between standard $GSADF$ and $SADF$.

Harvey et al. (2020c) also proposed a union of rejection testing strategy with
wild bootstrap implementation with $GSADF$ and $sGSADF$ tests exactly in the
same way as Harvey et al. (2019). To address the issue of asymmetric errors,
Harvey et al. (2020c) replace $sign(\Delta y_{t})$ by its recursive demeaned
version as $sign(\Delta y_{t})-(t-1)^{-1}\sum_{t=2}^{t}{sign(\Delta y_{t})}$.
The bootstrap algorithm is modified accordingly.

Hafner (2020) modified wild bootstrap algorithm of Harvey et al. (2016) to
allow a skeweness of the distribution of the series. He replaced $w_{t}$ in
Step 1 by $w_{t}=u_{t}/\sqrt{2}+(v_{t}^{2}-1)/2$ where $u_{t}\sim N(0,1)$ and
$v_{t}\sim N(0,1)$, so that $E(w_{t})=0$, $E(w_{t}^{2})=1$ and
$E(w_{t}^{3})=1$. Hafner (2020) also use sieve-based recolouring in Step 2.

Kurozumi et al. (2020) proposed to use a transformation of the series
according to the volatility behaviour in function $\sigma_{t}$, similar to
Cavaliere and Taylor (2008b) in the classical unit root testing context. Under
non-stationary volatility assumption, the partial sum process of
$\\{\varepsilon_{t}\\}$ is asymptotically characterized by the variance
profile, termed by Cavaliere and Taylor (2008b), which is defined as

$\eta(s):=\left(\int_{0}^{1}\omega(r)^{2}dr\right)^{-1}\int_{0}^{s}\omega(r)^{2}dr.$

The so called (asymptotic) average innovation variance is defined as

$\bar{\omega}^{2}:=\int_{0}^{1}\omega(r)^{2}dr.$

Note that $\eta(s)=s$ under homoskedasticity. Then, we have the following weak
convergence due to Theorem 1 of Cavaliere and Taylor (2007):

$\frac{1}{\sqrt{T}}y_{\lfloor rT\rfloor}=\frac{1}{\sqrt{T}}\sum_{t=1}^{\lfloor
rT\rfloor}\varepsilon_{t}\Rightarrow\bar{\omega}W(\eta(r))=:\bar{\omega}W^{\eta}(r)\qquad(0\leq
r\leq 1),$ (42)

for $y_{\lfloor\cdot\rfloor}$ defined in (34), where $\Rightarrow$ denote weak
convergence in $D[0,1]$ and $W(\cdot)$ is a standard Brownian motion, while
$W^{\eta}(\cdot)$ is called a variance transformed Brownian motion (Brownian
motion under a modification of the time domain). In the case of a constant
variance with $\sigma_{t}=\sigma$, we have $\eta(r)=r$ and thus $W^{\eta}(r)$
reduces to a standard Brownian motion. The time transformation is based on the
variance profile $\eta(s)$. Roughly, we should take the sampling interval
longer in the low volatility regime, whereas we take it shorter for large
values of $\sigma_{t}$.

More precisely, because the variance profile is a strictly monotonically
increasing function, we have the unique inverse given by $g(s):=\eta^{-1}(s)$.
Then, consider the time-transformed series
$\tilde{y}_{t}=y_{t^{\prime}}-y_{t^{\prime}=0}$ with a non-decreasing sequence
$t^{\prime}=\lfloor g(t/T)T\rfloor$. We note that
$\tilde{y}_{0}=y_{0}-y_{0}=0$ and $\tilde{y}_{T}=y_{T}-y_{0}$. As shown by (9)
in Cavaliere and Taylor (2007), we have, under the null hypothesis,

$T^{-1/2}\tilde{y}_{\lfloor rT\rfloor}\approx T^{-1/2}y_{\lfloor g(\lfloor
rT\rfloor/T)T\rfloor}\approx T^{-1/2}y_{\lfloor
g(r)T\rfloor}\Rightarrow\bar{\omega}W^{\eta}(g(r))=\bar{\omega}W(r)$ (43)

because $W^{\eta}(g(r))=W(\eta(g(r)))=W(r)$, and thus the time-transformed
series behaves as if it were a unit root process with a constant variance.

By taking into account (43), Kurozumi et al. (2020) proposed the following
test statistics based on the time-transformed ADF statistics:

$STADF=\sup_{\tau_{2}\in[\tau_{0},1]}TADF_{0}^{\tau_{2}}\qquad\mbox{and}\qquad
GSTADF(\tau_{0})=\sup_{\tau_{2}\in[\tau_{0},1],\tau_{1}\in[0,\tau_{2}-\tau_{0}]}TADF_{\tau_{1}}^{\tau_{2}},$
$\mbox{where}\quad
TADF_{\tau_{1}}^{\tau_{2}}=\frac{\tilde{y}_{\lfloor\tau_{2}T\rfloor}^{2}-\tilde{y}_{\lfloor\tau_{1}T\rfloor}^{2}-\bar{\omega}^{2}(\lfloor\tau_{2}T\rfloor-\lfloor\tau_{1}T\rfloor)}{2\bar{\omega}\sqrt{\sum_{t=\lfloor\tau_{1}T\rfloor+1}^{\lfloor\tau_{2}T\rfloor}\tilde{y}_{t-1}^{2}}}.$
(44)

The limiting distributions of these test statistics are the same as in the
case of homoskedasticity, therefore we do not need any bootstrap procedures to
control the size. The transformation of the time series is based on volatility
profile which can be estimated from data. For this, Kurozumi et al. (2020)
utilise the approach of Harvey et al. (2020b) and estimate non-parametrically
time-varying autoregressive coefficient, collect residuals, and use them for
estimating the variance profile.

Kurozumi et al. (2020) performed Monte-Carlo simulations with comparison of
$STADF$, bootstrap-based $SADF$ test of Harvey et al. (2016), sign-based test
of Harvey et al. (2020c), and SBZ test of Harvey et al. (2019), the latest two
with union of rejection and bootstrap. Kurozumi et al. (2020) demonstrated
that none of the tests dominates the others, but size of $STADF$ test is well
controlled compared to the others.

##  4 Detecting the dates of origination and collapse of the bubbles

###  4.1 Real-time monitoring and date estimation

The $\sup{ADF}$ test considered in the previous section is unable to determine
the date of origination ($T_{e}$) and collapse ($T_{c}$) of the bubble. In
this subsection, we consider the methods for estimating the dates which can be
used in real time. PWY proposed estimators for these two dates as

$\hat{\tau}_{e}=\inf_{s\geq\tau_{0}}{\\{s:ADF_{s}>cv_{\beta_{T}}^{adf}(s)\\}},\
\hat{\tau}_{c}=\inf_{s\geq\tau_{e}}{\\{s:ADF_{s}<cv_{\beta_{T}}^{adf}(s)\\}},$
(45)

where $cv_{\beta_{n}}^{adf}(s)$ is a right-tailed critical values of $ADF_{s}$
(defined in (9)) corresponding to the significance level $\beta_{T}$. That is,
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. The size $\beta_{T}$ should satisfy $\beta_{T}\rightarrow 0$
as $T\rightarrow\infty$. This guarantee that
$cv_{\beta_{n}}^{adf}(s)\rightarrow\infty$ under the null hypothesis of no
bubble, so the method will not falsely detect the bubble if it actually do not
present in the data. Phillips and Yu (2009) suggest to estimate the date of
collapse at least after some period after the origination date as

$\hat{\tau}_{c}=\inf_{s\geq\tau_{e}+\frac{\log(T)}{T}}{\\{s:ADF_{s}<cv_{\beta_{n}}^{adf}(s)\\}}.$
(46)

This guarantee that the length of the bubble is economically significant, and
the bubble episode is at least $O(\log(T))$.

PWY in their empirical study also used rolling regression for bubble dates
estimation, that is, each regression was based on fixed length of a lower
order than $T$. This method leads to the same estimate of the oridination date
of the bubble, but leads to the earlier date of collapse.

Phillips and Yu (2009) considered the asymptotic theory for the dates of
origination and collapse of the bubble in terms of moderate deviation from a
unit root process as in Phillips and Magdalinos (2007a, b). They established
that under the null hypothesis of no explosive bubble, and under
$cv^{adf}_{\beta_{T}}\rightarrow\infty$, the probability of detecting the date
of origination of a bubble is equal to zero as $T\rightarrow\infty$, i.e.
$Pr(\hat{\tau}_{e}\in[\tau_{0},1])\rightarrow 0$. Under the alternative
hypothesis (mildly explosive process in model (5)) the estimator
$\hat{\tau}_{e}$ is consistent for $\tau_{e}$ by assuming that
$1/cv^{adf}_{\beta_{T}}+cv^{adf}_{\beta_{T}}/T^{1-\alpha/2}\rightarrow 0$. The
idea of consistency of the date of the bubble origination is that observations
from the explosive sub-period are included in the estimation of the
autoregressive coefficient, these observations dominate ones from a unit root
process before the explosive sub-period. The difference in the signal between
the two periods then provides information and explains why the test procedure
consistently estimates the dating.

For practical implementation, Phillips and Yu (2009) suggest to use the
sequence of critical values $cv^{adf}_{\beta_{T}}=\frac{2}{3}\log\log^{2}T$
(see Phillips and Yu (2009, Theorem 3.2) for details). This guarantee that the
critical value converges to infinity (the Type I error is asymptotically
negligible) with lower rate than $T^{1-\alpha/2}$.

For the estimator $\hat{\tau}_{c}$, under the null hypothesis of no explosive
bubble similarly $Pr(\hat{\tau}_{f}\in[r_{0},1])\rightarrow 0$. Under the
alternative hypothesis and assuming
$1/cv^{adf}_{\beta_{T}}+cv^{adf}_{\beta_{T}}/T^{1-\alpha/2}\rightarrow 0$, the
estimator $\hat{\tau}_{c}$ is consistent for $\tau_{c}$. In this case,
$\hat{\delta}_{T}(\tau)\rightarrow_{p}1$ for $\tau>\tau_{c}$, but the limiting
distribution of this estimator has second order downward bias (i.e. bias
toward stationary process). This bias is explained by the data under
estimation which includes an explosive sub-regime as well as the period after
a collapse resulting in the mean-reverting behaviour. The $ADF_{\tau}$ test
diverges to $-\infty$ in this case.

Besides $ADF_{\tau}$ test, Phillips and Yu (2009) also consider the
coefficient based test, $ADF_{\tau}^{\delta}=\tau(\hat{\delta}_{\tau}-1)$. The
asymptotic properties of this test are similar to the $ADF_{\tau}$ except for
latter converges to $-\infty$ for $\tau>\tau_{c}$ with higher rate (under
$\alpha>1/3$). The difference is due to the sensitivity of the standard error
to the collapse of a moderate explosive process. Therefore in some cases the
$ADF_{\tau}$ can better estimate the date of collapse than the
$ADF_{\tau}^{\delta}$. This is confirmed by simulations in finite samples. At
the other hand, the estimates of the origination of a bubble are similar for
both tests.

To improve the finite sample properties of the estimators, Phillips and Yu
(2011) proposed to select an initial condition (selected to the first
observation in the previous setup) for initialisation of the recursive
procedure based on the Schwartz-Bayesian information criterion (BIC). If the
non-explosive regime moves to explosive regime, the most powerful procedure is
based on the recursive statistics which are calculated by using only the
observations from the explosive regime (because the observations from unit
root process are not taken into account). In other words, let the estimate of
origination date of a bubble, $\hat{T}_{e}=\lfloor\hat{\tau}_{e}T\rfloor$, is
identified by recursive procedure (45). Let $n_{\min}$ is the number of
observations in the sample
$\\{y_{\hat{T}_{e}-n_{\min}+1},\dots,y_{\hat{T}_{e}}\\}$. This sample can be
constructed as some fraction (say, 10%) from the sample before $\hat{T}_{e}$.
Further, values of BIC for two competing models are compared: the unit root
model and autoregressive model. For the former, the BIC is defined as

$BIC_{UR}=\log\left(\frac{\sum_{t=\hat{T}_{e}-n_{\min}-n_{k}}^{\hat{T}_{e}}{(\Delta
y_{t}-\bar{y})^{2}}}{n_{k}+n_{\min}}\right)+\frac{\log(n_{k}+n_{\min})}{n_{k}+n_{\min}},$

while for the latter, the BIC is defined as

$BIC_{AR}=\log\left(\frac{\sum_{t=\hat{T}_{e}-n_{\min}-n_{k}}^{\hat{T}_{e}}{(y_{t}-\hat{\mu}-\hat{\delta}y_{t-1})^{2}}}{n_{k}+n_{\min}}\right)+\frac{2\log(n_{k}+n_{\min})}{n_{k}+n_{\min}},$

where
$\bar{y}=(n_{k}+n_{\min})^{-1}\sum_{\hat{T}_{e}-n_{\min}-n_{k}}^{\hat{T}_{e}}{y_{t}}$,
$\hat{\mu}$ and $\hat{\delta}$ are OLS estimators in the regression
$y_{t}=\mu+\delta y_{t-1}+\varepsilon_{t}$, both BIC values are calculated for
the sample $\\{y_{\hat{T}_{e}-n_{\min}-n_{k}+1},\dots,y_{\hat{T}_{e}}\\}$. If
the BIC value for the unit root model is higher and point estimate of $\delta$
is greater than one, we redefine the initial condition $\hat{T}_{e}-n_{\min}$
as $\hat{T}_{e}-n_{\min}-1$, that is add one observation to the beginning of
the sample and again compare the two BICs for larger samples. We repeat this
procedure until the BIC for unit root model is lower than the BIC for
autoregressive model. After this, we denote final initial condition as
$\hat{T}_{0}$ and apply PWY procedure for the sample from $\hat{T}_{0}$ as in
(46). The resulting estimate of the origination date is defined as
$\hat{T}_{e}(\hat{T}_{0})$. If the sample after repeated comparing the BICs
becomes $\\{y_{1},\dots,y_{\hat{T}_{e}}\\}$, and the BIC for unit root model
is larger than BIC for autoregressive model, we set the initial condition at
$t=1$, so that the procedure becomes exactly the same as PWY, and the
resulting estimate of the origination date $\hat{T}_{e}(\hat{T}_{0})$
coincides with $\hat{T}_{e}$. In general case, it is expected that
$\hat{T}_{e}(\hat{T}_{0})\leq\hat{T}_{e}$.

The alternative procedure of detecting the dates of a bubble was proposed by
Phillips et al. (2015a, b) (hereafter PSY). Contrast to (45) and (46), the
estimates of the dates $T_{e}=\lfloor\tau_{e}T\rfloor$ and
$T_{c}=\lfloor\tau_{c}T\rfloor$ are based on

$\hat{\tau}_{e}=\inf_{\tau_{2}\in[\tau_{0},1]}{\\{\tau_{2}:BSADF_{\tau_{2}}(r_{0})>scv_{\beta_{T}}(\tau_{2})\\}},$
(47)
$\hat{\tau}_{c}=\inf_{\tau_{2}\in[\hat{\tau}_{e}+\delta\frac{\log(T)}{T},1]}{\\{\tau_{2}:BSADF_{\tau_{2}}(\tau_{0})<scv_{\beta_{T}}(\tau_{2})\\}},$
(48)

where $BSADF_{\tau_{2}}(\tau_{0})$ is the backward sup ADF test defined as

$BSADF_{\tau_{2}}(\tau_{0})=\sup_{\tau_{1}\in[0,\tau_{2}-\tau_{0}]}ADF_{\tau_{1}}^{\tau_{2}}.$

PSY extended their procedure for the case of multiple bubbles. They suggest to
obtain the dates $T_{e}=\lfloor\tau_{e}T\rfloor$ and
$T_{c}=\lfloor\tau_{c}T\rfloor$ for the second bubble just repeating the
procedure above, but starting with collapse of the first bubble. More
precisely, for the case of two bubbles the dates of originations and collapses
are based on

$\hat{\tau}_{1e}=\inf_{\tau_{2}\in[\tau_{0},1]}{\\{\tau_{2}:BSADF_{\tau_{2}}(\tau_{0})>scv_{\beta_{T}}(\tau_{2})\\}},$
(49)
$\hat{\tau}_{1c}=\inf_{\tau_{2}\in[\hat{\tau}_{1e}+\delta\frac{\log(T)}{T},1]}{\\{\tau_{2}:BSADF_{\tau_{2}}(\tau_{0})<scv_{\beta_{T}}(\tau_{2})\\}},$
(50)
$\hat{\tau}_{2e}=\inf_{\tau_{2}\in[\hat{\tau}_{1f},1]}{\\{\tau_{2}:BSADF_{\tau_{2}}(\tau_{0})>scv_{\beta_{T}}(\tau_{2})\\}},$
(51)
$\hat{\tau}_{2c}=\inf_{\tau_{2}\in[\hat{\tau}_{2e}+\delta\frac{\log(T)}{T},1]}{\\{\tau_{2}:BSADF_{\tau_{2}}(\tau_{0})<scv_{\beta_{T}}(\tau_{2})\\}}.$
(52)

PSY proved the consistency of the bubble dates estimators by assuming that
$1/scv_{\beta_{T}}(r_{2})+scv_{\beta_{T}}(r_{2})/T^{1/2}\rightarrow 0$. The
PWY procedure consistently estimates the dates of the first bubble, but not
the second bubble if the duration of the second bubble is shorter than that of
the first bubble ($\tau_{1c}-\tau_{1e}>\tau_{2c}-\tau_{2e}$). If the duration
of the second bubble is longer, then the PWY procedure detects the second
bubble with some delay depending on the duration of the first bubble. PSY
proposed a possible modification, called sequential PWY. This modification
imply detect the first bubble by PWY procedure, and then start the PWY
procedure again from the observation of the estimated date of the collapse
(imply it as a first observation) to the end, and so on. One drawback of the
sequential PWY procedure is when the minimum window length $\tau_{0}$ is
larger than the distance between the termination dates of the two bubbles.
Overall, the PSY detection procedure outperforms the sequential PWY procedure.

Lui (2019) extended the PSY procedure to allow a long memory behaviour of the
errors. For this, he proposed a HAR (fixed-$b$) approach to correctly dating
the explosive bubble.

For the case of data displaying time-varying volatility behaviour, Phillips
and Shi (2020) proposed a modification of the wild bootstrap algorithm of
Harvey et al. (2016). Because the dating procedure is sequential, Phillips and
Shi (2020) also address the multiplicity issue. They considered the following
algorithm.

######  Algorithm 2 (Composite Bootstrap Procedure)

>   * Step 1:
>
> Estimate regression (8) using the full sample period, imposing $\delta=0$,
> and obtain residuals $e_{t}$.
>
>   * Step 2:
>
> For the sample $\lfloor\tau_{0}T\rfloor+T_{b}-1$ ($T_{b}$ is the number of
> observations in the window over which size is to be controlled) generate a
> bootstrap sample via recursion
>
> $\Delta y_{t}^{*}=\sum_{j=1}^{k}{\hat{\phi}_{j}\Delta
> y_{t-j}^{*}}+e_{t}^{*}$
>
> initialized at $y_{i}^{*}=y_{i}$ with $i=1,\dots,j+1$, $\hat{\phi}_{j}$ are
> OLS estimates obtained in the fitted regression from Step 1. The bootstrap
> innovations are generated as $e_{t}^{*}=w_{t}e_{t}$, where $\\{w_{t}\\}$ be
> IID sequence of N(0,1) random variates.
>
>   * Step 3:
>
> Using the bootstrap sample $\\{y_{t}^{*}\\}$, compute the PSY test statistic
> sequence
>
> $\\{BSADF_{t}^{*}(\tau_{0})\\}_{t=\lfloor\tau_{0}T\rfloor}^{\lfloor\tau_{0}T\rfloor+T_{b}-1}$
> and the maximum value of this test statistic sequence
> $M_{t}^{*}=\max_{t\in[\lfloor\tau_{0}T\rfloor,\lfloor\tau_{0}T\rfloor+T_{b}-1]}BSADF_{t}^{*}(\tau_{0})$.
>
>   * Step 4:
>
> Bootstrap critical values of the PSY procedure are obtained from the
> distribution of $M_{t}^{*}$.
>
>

Phillips and Shi (2020) in their empirical section used $T_{b}=24$ in monthly
data which means that the empirical size is controlled over a two-year period.
Phillips and Shi (2020) also developed the R package psymonitor to implement
the proposed bootstrap procedure. Phillips and Shi (2020) noted that the date-
stamping procedure can also identify periods of crises.

Some papers are devoted to real-time detection of explosive bubbles in
financial time series. Astill et al. (2018) extended the test proposed by
Astill et al. (2017) to a real-time monitoring scheme. The problem is that the
Astill et al. (2018) test or the PSY test do not control size during
sequential testing. The statistic of Astill et al. (2018) detects a bubble if
any statistic in the monitoring period exceeds the largest value of the
statistic calculated over a training period of data. Kurozumi (2020b)
considered PWY and PSY monitoring test statistics151515Actually, Kurozumi
(2020b) considered a sequence of ADF statistics and a sequence of BSADF
statistics., provided a new set of critical values for monitoring period and
developed local asymptotic theory for the detecting statistics and CUSUM
monitoring statistics of Homm and Breitung (2012) (see also Breitung and Kruse
(2013)). Kurozumi (2020b) demonstrated that although the ADF-type monitoring
scheme outperforms the CUSUM-type monitoring scheme under moderate deviation
from a unit root asymptotic, this is not the case under local-to-unit root
asymptotic. This is confirmed in finite sample simulations. It turns out that
the CUSUM test outperforms the ADF-type tests when the bubble emerges early in
the monitoring period and the bubble episode is short. Astill et al. (2021)
extended the CUSUM test of Homm and Breitung (2012) to allow for time-varying
volatility. They suggested to replace standard variance estimator by its non-
parametric counterpart. Kurozumi (2020a) investigated the asymptotic
properties of the stopping time, which is the detecting date of a bubble. He
obtained that the CUSUM-type test detects a bubble sooner than the ADF-type
tests when the bubble emerges early in the monitoring period. Kurozumi (2020a)
also proposed a union of rejection type procedure to monitor the bubble. This
union of rejection includes both BSADF and CUSUM statistics.

###  4.2 Bubble date estimation from historical data

In this subsection, we consider the methods with more accurate break date
estimation. However, these methods can not be useful for real-time monitoring.
Harvey et al. (2017) proposed an approach based on minimization of sum of
squared residuals. They considered a more general DGP as Phillips and Shi
(2018) (see (15)):

$\displaystyle y_{t}$ $\displaystyle=$ $\displaystyle\mu+u_{t},$ (53)
$\displaystyle u_{t}$ $\displaystyle=$
$\displaystyle\begin{cases}u_{t-1}+\varepsilon_{t},&t=2,\dots,\lfloor\tau_{1,0}T\rfloor,\\\
(1+\delta_{1})u_{t-1}+\varepsilon_{t},&t=\lfloor\tau_{1,0}T\rfloor+1,\dots,\lfloor\tau_{2,0}T\rfloor,\\\
(1-\delta_{2})u_{t-1}+\varepsilon_{t},&t=\lfloor\tau_{2,0}T\rfloor+1,\dots,\lfloor\tau_{3,0}T\rfloor,\\\
u_{t-1}+\varepsilon_{t},&t=\lfloor\tau_{3,0}T\rfloor+1,\dots,T,\\\
\end{cases}$ (54)

This DGP implies a unit root process until the time
$\lfloor\tau_{1,0}\rfloor\equiv\lfloor\tau_{e}\rfloor$, then followed by an
explosive process until
$\lfloor\tau_{2,0}\rfloor\equiv\lfloor\tau_{c}\rfloor$. After
$\lfloor\tau_{2,0}\rfloor\equiv\lfloor\tau_{c}\rfloor$, there may be a
stationary collapsing regime (which is interpreted as the return to normal
market behavior) until the time
$\lfloor\tau_{3,0}\rfloor\equiv\lfloor\tau_{r}\rfloor$. After the collapsing
regime, the series follows a unit root process until the end of the sample.
There are some special cases. If
$\lfloor\tau_{2,0}\rfloor=\lfloor\tau_{3,0}\rfloor$, the explosive regime
instantly changes to a unit root regime. If, moreover, $\tau_{2,0}=1$, the
explosive regime is not terminated and continues to the end of the sample
period. If $\tau_{3,0}=1$, the collapsing regime is not terminated and
continues to the end of the sample period as well. Actually, Harvey et al.
(2017) proposed to choose the bubble model depending on the restrictions:
$\tau_{2,0}=1$ (Model 1), $\tau_{2,0}=\tau_{3,0}$ (Model 2), or $\tau_{3,0}=1$
(Model 2). Model 4 is unrestricted. For Model 4, we can estimate the
corresponding regression of the form

$\Delta
y_{t}={\mu}_{1}D_{t}(\tau_{1},\tau_{2})+\hat{\mu}_{2}D_{t}(\tau_{2},\tau_{3})+{\delta}_{1}D_{t}(\tau_{1},\tau_{2})y_{t-1}+{\delta}_{2}D_{t}(\tau_{2},\tau_{3})y_{t-1}+e_{t},$
(55)

where $D_{t}(a,b)=\mathbb{I}(\lfloor aT\rfloor<t\leq\lfloor bT\rfloor)$. The
Model 3 corresponds to the restriction $\tau_{3}=1$, the Model 2 corresponds
to the restrictions $\mu_{2}=\beta_{2}=0$, and the Model 2 corresponds to the
restrictions $\mu_{2}=\beta_{2}=0$ and $\tau_{2}=1$.

Let $SSR_{j}(\cdot)=\sum_{t=2}^{T}{\hat{e}_{t}}$ be the sum of squared
residuals for the Model $j$ with the constraints
$y_{\lfloor\tau_{2}T\rfloor}>y_{\lfloor\tau_{2}T\rfloor}$ and
$y_{\lfloor\tau_{2}T\rfloor}>y_{\lfloor\tau_{3}T\rfloor}$.161616These
constraints guarantee that the explosive regime is upward and the collapsing
regime is downward. Then the dates
$T_{e}=\lfloor\tau_{e}T\rfloor\equiv\lfloor\tau_{1}T\rfloor$,
$T_{c}=\lfloor\tau_{c}T\rfloor\equiv\lfloor\tau_{2}T\rfloor$, and
$T_{r}=\lfloor\tau_{r}T\rfloor\equiv\lfloor\tau_{3}T\rfloor$ can be estimated
based on minimisation of the corresponding $SSR$ over all possible locations.
Under the correct model and fixed magnitudes of $\delta_{1}$ and $\delta_{2}$,
the dates of a bubble can be consistently estimated. If the type of the model
is unknown, Harvey et al. (2017) suggest to choose it by comparing the BIC
criteria of the form

$BIC=T\log[T^{-1}SSR(\cdot)]+k\log(T),$

where $k=2+1$ for Model 1, $k=2+2$ for Model 2, $k=4+2$ for Model 3, and
$k=4+3$ for Model 4. The minimum BIC approach always selects the appropriate
model asymptotically and often outperforms the PSY procedure for dating the
bubble in finite samples. It should be noted that the BIC approach must be
performed only after the detecting a bubble (e.g. by GSADF test).

Harvey et al. (2020c) considered their sign-based test (39) for detecting the
dates of the explosive bubble. The recursive implementation of the sign-based
test a la PSY can give the consistent estimator of the origination date (under
mildly explosive bubble magnitude), but the date of the collapse can not be
consistently estimated. Instead, Harvey et al. (2020c) suggested to maximize
the sign-based statistic over all possible dates of a bubble as

$\\{\hat{\tau}_{e},\hat{\tau}_{c}\\}=\arg\max_{\tau_{2}\in[\tau_{0},1],\tau_{1}\in[0,\tau_{2}-\tau_{0}]}sADF_{\tau_{1}}^{\tau_{2}},$
(56)

where $sADF_{\tau_{1}}^{\tau_{2}}$ is defined in (41), $\hat{\tau}_{e}$
corresponds to $\tau_{1}$ and $\hat{\tau}_{c}$ corresponds to $\tau_{2}$ in
maximization. It turns out that $\hat{\tau}_{c}$ is the consistent estimator
for $\tau_{c}$, but $\hat{\tau}_{e}$ is not. To obtain the consistent
estimator of the origination and collapse dates of the bubble, Harvey et al.
(2020c) used the modified version of (40) replacing
$\hat{s}^{2}(\tau_{1},\tau_{2})$ by
$\tilde{s}^{2}(\tau_{1},\tau_{2}))^{\varepsilon}$, where

$\tilde{s}^{2}(\tau_{1},\tau_{2})=\frac{\lfloor\tau_{2}T\rfloor\hat{s}^{2}(0,\tau_{2})-\lfloor\tau_{1}T\rfloor\hat{s}^{2}(0,\tau_{1})}{\lfloor\tau_{2}T\rfloor-\lfloor\tau_{1}T\rfloor-1}$
(57)

with $\varepsilon=0.01$. Consistency results hold under moderate deviation
from the unit root in (53) and obviously require
$\tau_{2}-\tau_{1}\geq\tau_{0}$.

Monschang and Wilfling (2020) investigated a robustness of the PSY date
stamping procedure and sign-based date stamping procedure for a leverage
effect (via the TGARCH model). They demonstrated via Monte-Carlo simulations
that the PSY procedure gives more accurate estimates of the dates of a bubble
than the sign-based procedure, while PSY frequently detects non-existing
bubbles. Note that Monschang and Wilfling (2020) considered the standard sign-
based procedure (56) without correction of the standard errors as in (57).

The approach of Harvey et al. (2017) is difficult to implement for the case of
multiple bubbles because the number of potential models to be considered grows
exponentially in the number of bubble regimes. Pang et al. (2020) proposed to
consider a sample-splitting method for estimating the dates of structural
breaks: for one bubble model, initially the date of collapse is (consistently)
estimated by minimizing the sum of squared residuals for the two-regime model,
and the estimated break date $\hat{T}_{2}=\hat{T}_{c}$ is consistent for the
date of collapse; then, treating $\hat{T}_{2}=\hat{T}_{c}$ as a known break
point, the date of origination of the bubble is estimated for the sample
before the estimated date of collapse. Although the origination break date
estimator is not consistent, the corresponding break fraction is. However,
Pang et al. (2020) do not allow unit root behaviour after the collapse regime.
Kurozumi and Skrobotov (2021) extended the model of Pang et al. (2020) by
allowing a unit root regime after recovering stationary regime. It turned out
that as in Pang et al. (2020), the estimated break date in two-regime model is
still consistent for the date of collapse. Considering the samples before and
after the estimated date of collapse, the date of origination and the date of
recovery can be estimated. The limiting results for the date of origination
are similar to ones obtained in Pang et al. (2020). The asymptotic of the date
of recovery depends on the relative extent of the explosive bubble and
stationary recovery regime. More precisely, if $\delta_{1}=1+c_{1}/T^{a}$ and
$\delta_{2}=1-c_{b}/T^{b}$ in (54), then the estimated date of recovery
$\hat{T}_{3}=\hat{T}_{r}$ will be consistent only if $a<b$. Otherwise,
$(1-\delta_{2})T(\hat{\tau}_{3}-\tau_{3,0})$ has a non-degenerate limiting
distribution. The simulation results of Kurozumi and Skrobotov (2021) confirm
the asymptotic results obtained. Also, Kurozumi and Skrobotov (2021)
investigated the performance of the sample-splitting approach under non-
stationary volatility in the shocks. The demonstrated that if volatility
increases, then $T_{c}$ and $T_{r}$ can be in a volatile regime. Thus, it
would be difficult to distinguish between the shift in the parameter and large
shocks.

Harvey et al. (2020a) combined the approach of PSY and Harvey et al. (2017)
and proposed the two-step procedure. At the first step, the PSY procedure is
applied to obtain preliminary dates of the bubble. At the second step, the
full sample is split based on the preliminary dates of the bubbles (so that
the subsample begins from the mid-point between the end of the previous
explosive regime and the start of the current explosive regime, and subsample
ends at the mid-point between the end of the current explosive regime and
starts of the next explosive regime), and BIC based approach is applied for
each of the parts of the split sample. Harvey et al. (2020a) demonstrated via
Monte-Carlo simulation that their two-step method is more accurate in
comparison to the PSY procedure; the latter often estimates the origination
dates of the bubbles later than they actually occur.

Jiang et al. (2020) obtained the exact distribution of the maximum likelihood
estimator of the structural break point in model with two regimes under
continuous record asymptotic. In particular, they considered a model changing
from a unit root regime to explosive regime. The distributions turn out
asymmetric and trimodal regardless of the break date. Jiang et al. (2020) also
developed in-fill asymptotic distributions of break dates and demonstrated the
performance of the in-fill asymptotic distribution over the long-span
counterpart. Despite the difficulties of the obtained distribution, the
authors suggested the method for constructing a confidence interval for the
break date based on highest density region.

##  5 Asymptotics for autoregressive parameter

What should we do after detecting an explosive bubble and identifying the
dates of its exuberance and collapse? We might want to estimate the growth
rate of the bubble and then construct its confidence interval. The asymptotic
theory needed for the explosive behaviour inference was developed in Phillips
and Magdalinos (2007a, b). Authors considered a mildly explosive process of
the form:

$y_{t}=\rho_{T}y_{t-1}+\varepsilon_{t},\ \rho_{T}=1+\frac{c}{k_{T}},\ c>0,$
(58)

which initialises at $y_{0}=o_{p}(\sqrt{k_{T}})$, independent on
$\\{\varepsilon_{t},t\geq 1\\}$, where $\\{k_{T}\\}_{T\geq 1}$ is the sequence
tending to infinity so that $k_{T}=o(T)$ with $T\rightarrow\infty$. The model
(58) does not include the constant to exclude the presence of
deterministically explosive component in $y_{t}$. The error process may be
weakly dependent as in Phillips and Magdalinos (2007b), conditionally
heteroskedastic as in Arvanitis and Magdalinos (2018), or anti-persistent as
in Lui et al. (2020).

The sequence of autoregressive parameter $\rho_{T}=1+\frac{c}{k_{T}}>1$ has a
local character, that is, $\rho_{T}\rightarrow 1$ with $T\rightarrow\infty$.
Such sequence tends to infinity with a slower rate than $T$, and represent
moderate deviations from unity (the autoregressive roots belong to larger
neighborhoods of one than conventional local to unity roots). One can consider
a special case for $k_{T}=T^{\alpha}$, $\alpha\in(0,1)$. When
$\alpha\rightarrow 1$, we obtain a local to unit root behaviour while for
$\alpha\rightarrow 0$, we obtain stationary or explosive process. Under some
regularity conditions, Phillips and Magdalinos (2007a) obtained the following
results:

$\frac{k_{T}(\rho_{T})^{T}}{2c}(\hat{\rho}_{T}-\rho_{T})\Rightarrow C,\text{
and
}\frac{(\rho_{T})^{T}}{(\rho_{T})^{2}-1}(\hat{\rho}_{T}-\rho_{T})\Rightarrow
C,$ (59)

where $C$ is a standard Cauchy distribution. Note that under standard (fixed)
asymptotics, the limiting Cauchy distribution is hold only under Gaussian
errors, see Anderson (1959), White (1958).171717The standard $t$ statistic for
$\delta_{T}$ has an asymptotically standard normal distribution, see Guo et
al. (2019). Phillips et al. (2010) considered $\delta_{m,n}=1+\frac{cm}{T}>1$
and obtained the same result if $T\rightarrow\infty$ followed by
$m\rightarrow\infty$.

The results (59) allow us to construct a confidence interval for $\rho_{T}$ as

$\left(\hat{\rho}_{T}\pm\frac{(\hat{\rho}_{T})^{2}-1}{(\hat{\rho}_{T})^{n}}C_{\alpha}\right),$

where $C_{\alpha}$ is the two-side $\alpha$-percentile of Cauchy
distribution181818The corresponding values are equal to $C_{0.10}=6.315$,
$C_{0.05}=12.7$ and $C_{0.01}=63.65674$..

Guo et al. (2019) generalized the Phillips and Magdalinos (2007a) results by
allowing an deterministic drift term $\mu_{T}$ in (58) with
$\mu_{T}\sqrt{k_{T}}\rightarrow\nu\in[0,\infty)$ (small drift) or
$\mu_{T}\sqrt{k_{T}}\rightarrow\infty$ (large drift). Under i.i.d. errors
$\varepsilon_{t}$, the $t$-statistic of $\delta_{T}$ under the null of
moderate explosiveness is asymptotically standard normal regardless of the
magnitude of the drift.191919See also Wang and Yu (2015) with the theory of
fixed $\rho_{T}=\rho>1$. Under weakly dependent errors, the HAC based
$t$-statistic also has standard normal limit. However, HAR-based $t$-statistic
asymptotically has Student’s $t$ distribution regardless of the magnitude of
the drift. Finally, Guo et al. (2019) proposed the constructiion of confidence
interval for $\rho_{T}$ based on $t$ statistic inversion: if the $t$ test
fails to reject the null value $\rho_{T}$, it should be included in the
confidence interval. Xiang et al. (2021) extended the approach of Guo et al.
(2019) by allowing a structural change in drift.

Chan et al. (2012) (see also Liu and Peng (2019)) proposed to construct a
confidence interval for an autoregressive parameter with correct coverage
based on empirical likelihood methods. Their approach is robust to all types
of autoregressive processes: stationary, unit root, near unit root, moderate
deviations from a unit root (towards stationary or explosive sides), or
explosive. Similar robustness is achieved in Magdalinos and Petrova (2021)
where the authors developed a testing procedure based on instrumental variable
estimation and pre-testing for stationary or explosive direction. Phillips
(2021) considered estimation and inference of localizing parameters $c$ and
$\alpha$ in the representation $\rho_{T}=1+c/T^{\alpha}$. Considering possibly
explosive sample intervals in time series (just 4 years before the end of the
sample), Phillips (2021) argued that the method may be seen as an alternative
of recursive bubble identification methods.

All methods above should be implemented for the sample interval with a stable
autoregressive parameter. E.g., in the bubble model with unit root, explosive,
and stationary part, we wish to do inference for the growth rate of the
explosive period. For this, we first should identify the dates of exuberation
and collapse of a bubble. These dates may be identified with error, and, in
general, the estimated dates are not equal to the true dates. Guo et al.
(2019) and Xiang et al. (2021) considered the sample just 100 periods before
the highest point of time series while PWY used the sample with the highest
value of the Dickey-Fuller test statistic. Guo et al. (2019) found the
confidence intervals for the autoregressive parameter for seven of ten stock
indices in range from 1.001 (lower bound) to 1.079. The degree of
explosiveness is quite mild according the rule of Phillips et al. (2011) that
the explosive autoregressive root not greater than 1.05 means moderate
explosiveness. Xiang et al. (2021) considered the Bitcoin market
capitalization in the fourth quarter of 2017 and found the confidence interval
for autoregressive parameter to be [1.003, 1.061] which is wider if we do not
pay attention to the structural change.

It seems natural to exclude some boundary observations after the estimated
date of exuberation and before the estimated date of collapse for estimation
and inference of the explosive period. Also, in light of the results of Pang
et al. (2020) and Kurozumi and Skrobotov (2021), the date of collapse is often
estimated more accurately (for a wider set of parameters), so that there is no
need to remove a lot of observations before the estimated date of collapse.

##  6 Investigating the relationship between different bubbles

A question whether bubbles in different markets move together or with some
lags has also been addressed in the literature. Phillips and Yu (2011) propose
a test which detects whether the bubble migrates form one market to another.
Consider two series, $x_{t}$ and $y_{t}$, and we want to test the migration
from $x_{t}$ to $y_{t}$. Let $\theta_{X}(\tau)$ be an autoregressive
coefficient characterizing $\\{x_{t}\\}_{t=1}^{\lfloor\tau T\rfloor}$. It can
be recursively estimated from OLS regression as $\hat{\theta}_{X}(\tau)$ (i.e.
$\theta_{X}(\tau)$ and $\hat{\theta}_{X}(\tau)$ are actually time series
depending on $\tau$).

Let a date-stamping strategy detect the origination date of the bubble in
$x_{t}$ as $T_{eX}=\lfloor\tau_{eX}T\rfloor$, and the autoregressive estimate
$\hat{\theta}_{X}(\tau)$ has a maximum at $T_{pX}=\lfloor\tau_{pX}T\rfloor$.
It is assumed that $T_{pX}$ is not necessarily equal to $T_{eX}$, the date of
collapse, in general.

Similarly, define $\hat{\theta}_{Y}(\tau)$, $T_{eY}=\lfloor\tau_{eY}T\rfloor$
and $T_{pY}=\lfloor\tau_{pY}T\rfloor$ for $\\{y_{t}\\}_{t=1}^{\lfloor\tau
T\rfloor}$. It is assumed that $\tau_{pY}>\tau_{pX}$, and this inequality
should be confirmed by date-stamping algorithm.

Let $m=T_{pY}-T_{pX}$ be a number of observations in the interval
$(T_{pY},T_{pX}]$. Phillips and Yu (2011) formulate the null hypothesis that
the coefficient $\theta_{Y}(\tau)$ for $y_{t}$ moves from a unit root
behaviour before $T_{eY}$ (i.e. $\theta_{Y}(\tau)=1$) to a moderately
explosive behaviour after $T_{eY}$ (i.e.
$\theta_{Y}(\tau)=1+c_{Y}/T^{\alpha}$). The alternative hypothesis is that the
value $\theta_{Y}(\tau)$ depends on the corresponding recursive coefficient
$\theta_{X}(\tau)$ of $x_{t}$. Thus, the value $\theta_{Y}(\tau)$ is defined
by the value $\theta_{X}(\tau)$, i.e., when the series $x_{t}$ is collapsing,
and $\theta_{X}(\tau)$ decreases, the bubble migrates from $x_{t}$ to $y_{t}$
and reveals the increase of $\theta_{Y}(\tau)$, which is larger than unity.

Let the autoreressive parameter of $x_{t}$ behave like
$\theta_{X}(\tau)=1+c_{X}\left(\frac{\lfloor\tau T\rfloor-T_{pX}}{m}\right)/T$
for $\lfloor\tau T\rfloor>T_{pX}$, where the function $c_{X}(\cdot)<0$ to
guarantee the mean reversion during the collapse. The dependence of
$\theta_{Y}(\tau)$ on $\theta_{X}(\tau)$ can be expressed as:

$\theta_{Y}(\tau)=1+\frac{c_{Y}+dc_{X}\left(\frac{\lfloor\tau T\rfloor-
T_{pX}}{m}\right)}{T^{\alpha}},\ \lfloor\tau T\rfloor\geq\tau_{eY},$ (60)

i.e., after the origination the bubble, the behaviour of $x_{t}$ affects
$y_{t}$ through the localising parameter $c_{X}(\cdot)$. As the bubble
collapses, $x_{t}$ returns to a near martingale behaviour, and the localising
parameter $c_{X}(\cdot)$ affects the autoregressive parameter $\theta_{Y}$.

Phillips and Yu (2011) consider a simplified linear version of (60), where
$c_{X}$ is a negative constant:

$\theta_{Y}(\tau)-1=\beta_{0T}+\beta_{1T}(\theta_{X}(\tau)-1)\frac{\lfloor\tau
T\rfloor-T_{pX}}{m}+\epsilon_{t},\ \lfloor\tau
T\rfloor=\lfloor\tau_{pX}T\rfloor+1,\dots,\lfloor\tau_{pY}T\rfloor,$ (61)

that is, the data covers the collapse period of $x_{t}$ and the growth of the
bubble of $y_{t}$. The null hypothesis of no bubble migration is equivalent to
$d=0$ in equation (60) or $\beta_{1T}=0$ in equation (61). Under the
alternative, $d<0$ ($\beta_{1T}<0$) because $dc_{X}(\cdot)>0$, which
stimulates explosive behaviour of $y_{t}$. By using recursive coefficients
$\hat{\theta}_{X}(\tau)$ and $\hat{\theta}_{Y}(\tau)$, the regression
estimates of $\beta_{0T}$ and $\beta_{1T}$ can be obtained from regression
(61). Phillips and Yu (2011) propose an asymptotically conservative and
consistent test based on standardized test statistics
$Z_{\beta}=\hat{\beta}_{1T}/L(m)$, where
$1/L(m)+L(m)/n^{\varpi}\rightarrow\infty$ for all $\varpi>0$ and for some
slowly varying function $L(m)$ like $a\log(m)$ with $a>0$, where
$\lfloor\tau_{pY}T\rfloor-\lfloor\tau_{pX}T\rfloor=O(n)$. Standard normal
critical values can be used, and the test rejects the null for large values of
the test statistic. However, Phillips and Yu (2011) caution against
interpreting the test results as causal.

Greenaway-McGrevy and Phillips (2016) propose an another method to find the
evidence of contagion from one time series to others. Their method implies to
first compute recursive coefficients $\delta$ from regression (8) over the
sample period. Let $i$ be the number of the series of interest. Authors
recommended to use a fixed window width subsample in which the sequence
$\\{\hat{\delta}_{i,s}\\}_{s=S}^{T}$ is recursively estimated by least square
regression using a moving window of data of length $S$. In other words,
$\hat{\delta}_{i,s}$ is estimated for the range from $t=s-S+1$ to $s$ for
$s=S,S+1,\dots,T$. Then the contagion regression is

$\hat{\delta}_{i,s}=\theta_{1i}+\theta_{2i}\left(\frac{s}{T-S+1}\right)\hat{\delta}_{core,s-d}+\varepsilon_{s},\
s=S,\dots,T,$ (62)

where $core$ denotes core series from which the bubble originates by
assumption. The delay parameter $d$ captures the lag in market contagion from
the core series on other series. Greenaway-McGrevy and Phillips (2016) allowed
$d$ to be an integer from 0 to 12 and estimated by nonlinear least squares
regression to achieve the largest $R^{2}$.

Regression (62) is functional, so the functional coefficient $\theta_{2i}$ is
time-varying. Greenaway-McGrevy and Phillips (2016) suggest to estimate this
coefficient by a local level kernel regression (see their Technical Appendix
for details). This allows to characterize the continuous change of the effect
of the core series on the other series before the bubble, during the bubble
and after the bubble. The larger estimate of $\theta_{2i}$, the stronger the
effect. For the bubble example, the behavior of time-varying coefficient
$\theta_{2i}$ may have the form of $\cap$, indicating increasing $\theta_{2i}$
during the exuberance of the bubble (contagion effect increases) and
decreasing $\theta_{2i}$ during the collapse of the bubble (contagion effect
decreases).

Chen et al. (2019) developed a new theory for testing and date-stamping the
common bubble factor in large-dimensional time series. More precisely, the
dynamics for time series is generated as

$X_{it}=f_{0,t}\lambda_{0,i}+e_{it,}$ (63)

where $f_{0,t}$ follows a unit root process202020See also Horie and Yamamoto
(2016) who allowed also the possible explosive behaviour in idiosyncratic
component.. This factor captures only fundamentals. Under the alternative
hypothesis,

$X_{it}=f_{0,t}\lambda_{0,i}+f_{1,t}\lambda_{1,i}\mathbb{I}(t>T_{e})+e_{it,}$
(64)

where the factor $f_{1,t}$ follows a mildly explosive autoregressive process,
and $T_{e}=\lfloor\tau_{e}T\rfloor$ is the date of market exuberance, and the
market exuberance lasts until the end of the sample period.

Chen et al. (2019) suggested to estimate the leading common factor by
principal components. After that, the PSY procedure to identification the
bubble is applied to the estimated first factor. The identified bubble in this
first factor is equivalent to a common bubble in large-dimensional time series
$X_{it}$. The limiting distribution of the PSY test statistic is the same as
original one despite of a factorisation, therefore this approach allows to
consistently estimate the origination date of the common bubble.

There are other approaches operating the term co-exposiveness. Magdalinos and
Phillips (2009) developed the asymptotic theory for cointegrating systems with
possibly mildly explosive variables.212121 Phillips and Magdalinos (2013) and
Nielsen (2008) also described the inconsistency results in vector
autoregression with common explosive roots.  Chen et al. (2017) extended the
theory of Magdalinos and Phillips (2009) to a continuous time framework (see
also Tao et al. (2019)). Chen and Tu (2019) developed the asymptotic theory
for spurious regression with a moderately explosive processes (see also Lin
and Tu (2020)). Nielsen (2010) proposed a cointegrated vector autoregressive
model with single explosive root (co-explosive model), so that the
cointegrating and co-explosive vectors eliminate random walk and explosively
growing behaviour respectively via the following error correction model:

$\Delta_{1}\Delta_{\rho}X_{t}=\Pi_{1}\Delta_{\rho}X_{t-1}+\Pi_{\rho}\Delta_{1}X_{t-1}+\sum_{j=1}^{k-2}{\Phi_{j}\Delta_{1}\Delta_{\rho}X_{t-j}}+\mu+\varepsilon_{t},$
(65)

where $\Delta_{\rho}X_{t}=X_{t}-\rho X_{t-1}$ and
$\Delta_{1}X_{t}=X_{t}-X_{t-1}$, and $\Pi_{1}$, $\Pi_{\rho}$ and $\Phi_{j}$
are $2\times 2$ parameter matrices. Engsted and Nielsen (2012) demonstrated
how this model can be helpful for investigating the rational bubbles in asset
prices.

Evripidou et al. (2022) considered the so-called co-bubble behaviour via the
following regression model:

$y_{t}=\mu_{y}+\beta_{x}x_{t-i}+\beta_{z}z_{t}+\varepsilon_{y,t},$ (66)

where $\varepsilon_{y,t}$ is a mean zero stationary I(0) process and $x_{t}$
is generated as a unit root process with bubble and collapsing regimes (see
equations (53)-(54)). If $\beta_{z}=0$ and $\beta_{x}>0$, then $x_{t}$ and
$y_{t}$ co-bubble, that is, their linear combination is I(0). If $i<0$, $i=0$
and $i>0$, then the bubble in $x_{t}$ precedes one in $y_{t}$ (bubble in
$x_{t}$ migrates to $y_{t}$), the bubbles occur at the same time, and the
$y_{t}$ bubble precedes one in $x_{t}$ (bubble in $y_{t}$ migrates to
$x_{t}$), respectively. The variable $z_{t}$ is unobserved, the bubble process
is generated in a similar way to $x_{t}$. So, if $\beta_{z}>0$ and
$\beta_{x}=0$, then the bubble process in $x_{t}$ is not affected by one in
$y_{t}$. Therefore, Evripidou et al. (2022) proposed testing the null
$H_{0}:\beta_{x}>0,\beta_{z}=0$ (co-bubble behaviour between $y_{t}$ and
$x_{t}$) against $H_{1}:\beta_{x}=0,\beta_{z}>0$.

Evripidou et al. (2022) suggested to use a KPSS-type statistic of the
following form:

$S=\hat{\sigma}_{y}^{-2}t^{-2}\sum_{t=i\mathbb{I}(i>0)+1}^{T+i\mathbb{I}(i<0)}{\left(\sum_{s=i\mathbb{I}(i>0)+1}^{t}{\hat{e}_{y,s}}\right)^{2}},$
(67)

where $\hat{e}_{y,t}$ is the OLS residual from a regression of $y_{t}$ on a
constant and $x_{t}$, and $\hat{\sigma}_{y}^{2}=\\\
T^{-1}\sum_{t=i\mathbb{I}(i>0)+1}^{T+i\mathbb{I}(i<0)}{\hat{e}_{y,t}^{2}}$.
This test is actually the test for I(0) for $\varepsilon_{y,t}$. To allow a
time varying volatility in $\varepsilon_{y,t}$, Evripidou et al. (2022)
adopted a wild bootstrap implementation. The limiting distribution of the test
statistic is free from precise properties of the regressor series $x_{t}$. The
choice of the delay parameter $i$ should be made carefully, and different
values should be experimented with in empirical applications.

##  7 Conclusion

This paper has reviewed the development of testing for explosive bubbles.
Recently, many methods have been proposed to account for various features of
data such as weak dependence, non-stationary conditional and unconditional
volatility, and long memory. Other methods have addressed different
specifications of the model generating the bubble. Overall, for empirical
applications, the researcher needs to first test for an explosive bubble via
tests with very general assumptions, and second (if a bubble is detected)
estimate the dates of the bubble depending on model specification under
similar very general assumptions.

However, some important issues remain to be addressed and could be
investigated in further research. First, long memory dynamic can be
accommodated in data generating processes coupled with heteroskedasticity (as
in Cavaliere et al. (2017)). For now, only one paper Lui (2019) accounted for
a long memory dynamic in the innovation error of the explosive bubble model.
Second, penalised methods such as LASSO can be used for identification of
bubble regimes. Indeed, the methods based on minimization of the sum of
squared residual are computationally expensive, especially in multiple bubble
episodes. LASSO methods have already been developed for the multiple breaks
model (see Qian and Su (2016) inter alia), and extensions for the multiple
bubble model could be investigated.

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