# Testing for explosive bubbles: a review

• Anton Skrobotov
From the journal Dependence Modeling

## 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.

MSC 2010: 62F03; 62M10; 62P20; 91B84

## 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. [93] (hereafter PWY), consider a rational bubble by using the 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,

(1) P t = 1 1 + R E t ( P t + 1 + D t + 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 [14], the following solution is obtained:

(2) p t = p t f + b t ,

where

(3) p t f = κ γ 1 π + ( 1 π ) i = 0 π i E t d t + 1 + i ,

b t = lim t π i E t p t + i ,

(4) E t ( b t + 1 ) = 1 π b t = ( 1 + exp ( d p ¯ ) ) b t ,

with p t = log ( P t ) , d t = log ( D t ) , γ = log ( 1 + R ) , and π = 1 / ( 1 + exp ( d p ¯ ) ) , where ( d p ¯ ) is the average log dividend-price ratio, κ = log ( π ) ( 1 π ) log ( 1 / π 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 ( d p ¯ ) > 0 , the rational bubble b t is a sub-martingale and explosive in expectation. Under equation (4), we have:

(5) b t = 1 π b t 1 + ε b , t = ( 1 + g ) b t 1 + ε b , t ,

where E t 1 ( ε b , t ) = 0 , g = 1 π 1 = exp ( d p ¯ ) > 0 is the growth rate of the natural logarithm of the bubble, and ε b , t is a martingale difference sequence.

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

(6) y t = μ + ρ y t 1 + ε t ,

with δ > 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

(7) d t p t = κ γ 1 π i = 0 π i E t ( Δ d t + 1 + i ) .

If both p t and d t are 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, Δ p t is also explosive, because if p t is the explosive process, then p t and d t cannot be cointegrated. On the basis of these results, Diba and Grossman [34] proposed to test for stationarity in Δ p t or to test for cointegration between p t and d t for detecting the bubble. Phillips and Yu [95, Section 2.2] discussed that explosiveness in price is sufficient evidence for bubbles under various assumptions.

However, Diba and Grossman [34] showed that the impossibility of a downward rational bubble implies that the bubble will never begin again after collapsing. Evans [40] considered periodically collapsing bubbles and showed that the tests in Diba and Grossman [34] 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 behaviour 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).[1] Moreover, both time series, p t and d t , may be explosive and then they may be explosively cointegrated.[2] 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 Astill et al. [6,7], Bohl et al. [11], Breitung and Kruse [13], Chen and Tu [28], Fulop and Yu [44], Guo et al. [46], Harvey et al. [52,55,58], Kurozumi [66], Lin and Tu [71], Liu and Peng [72], Monschang and Wilfling [77], Pavlidis et al. [82], Phillips and Shi [89,90], Phillips et al. [93,97,98], Shi [104], Tao et al. [107], Wang and Yu [110], Whitehouse [112] Prices of cryptocurrencies Astill et al. [8], Bouri et al. [12], Cheah and Fry [26], Cheung et al. [30], Corbet et al. [31,32], Hafner [48], Harvey et al. [56,57] Real estate market (housing prices) Anundsen et al. [4], Banerjee et al. [9], Caspi [15], Chen et al. [27,29], Das et al. [33], Engsted et al. [35], Escobari and Jafarinejad [37], Harvey et al. [54], Horie and Yamamoto [60], Kivedal [63], Kurozumi [67], Pavlidis et al. [81], Pedersen and Schütte [84], Phillips and Yu [95], Shi [100], Shi and Phillips [102], Yiu et al. [114] Commodities prices Caspi et al. [16], Etienne et al. [38,39], Evripidou et al. [41], Fantazzini [42], Figuerola-Ferretti et al. [43], Gutierrez [47], Harvey et al. [53], Pavlidis et al. [83], Shi and Arora [101], Su et al. [106], Zhang and Yao [115] Exchange rates Bettendorf and Chen [10] Art market Kräaussl et al. [65] Long annual ratio of the US Debt/GDP series Kaufmann and Kruse [62] Credit risk in the European sovereign sector Phillips and Shi [91] Extreme Yugoslavian hyperinflation Nielsen [79]

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, Section 7 discusses the 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 to test for a bubble directly in the non-fundamental component. For example, Shi [100] investigated the bubbles in the housing market and proposed to calculate the fundamental component of price to rent ratios based on estimates of the five-variable vector autoregression 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 [102] 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 a fundamental component of the dividend-to-price ratio series (see also Shi and Phillips [103] for details). Pavlidis et al. [82,83] 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 Augmented Dickey-Fuller (SADF) test

Phillips et al. [93] proposed recursive tests which can detect evidence of explosive behaviour in time series { y t } , t = 1 , , T .[3] 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.[4]

Consider the following ADF-type regression as follows:

(8) y t = μ + ρ y t 1 + j = 1 k ϕ j Δ y t j + ε t .

We want to test the null hypothesis of a unit root, H 0 : ρ = 1 , against the right-tailed alternative, H 1 : ρ > 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 , , τ T for all τ [ τ 0 , 1 ] ( denotes the integer part of value) with some preliminary chosen τ 0 ,[5] T is the sample size of the series. Consider the following ADF-type test statistic:

(9) ADF τ = j = 1 τ e j 1 2 σ ˆ τ 2 1 / 2 ( ρ ˆ τ 1 ) ,

where ρ ˆ τ is the ordinary least squares (OLS) estimator of ρ based on regression (8) over the observations t = k + 1 , , τ T (first τ T observations), σ ˆ τ 2 is the corresponding variance estimator of σ ε 2 , and e t are OLS residuals from (8).[6] Evidently, under the null hypothesis,

(10) ADF τ 0 τ W ( r ) ˜ d W ( r ) 0 τ W ( r ) ˜ 2 1 / 2 ,

where W ( r ) W is standard Brownian motion, and W ˜ = W 1 τ 0 1 W . The supremum-type test statistic is

(11) SADF ( τ 0 ) sup τ [ τ 0 , 1 ] ADF τ sup τ [ τ 0 , 1 ] 0 τ W ˜ d W 0 τ W ˜ 2 1 / 2 .

This test statistic can be used for testing for a unit root against explosive behaviour in some subsample[7].

Lui [73] allowed for a long memory dynamic in ε t . If the series exhibits long memory, then the standard SADF test diverges to infinity at rate n d , where d ( 0 , 0.5 ) is the memory parameter; thus, the null hypothesis of no explosive bubble is often falsely rejected. Lui [73] suggested to replace the estimator of σ ε 2 in (9) by a heteroskedasticity and autocorrelation robust (HAR) (fixed- b ) estimator. Critical values for the SADF test depend on the estimate of the memory parameter d , d ˆ .

### 2.2 More general data generating processes

Phillips et al. [92] 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 + ε t ,

while Diba and Grossman [34] assumed

y t = μ ˜ + y t 1 + ε t ,

so that y t has a deterministic trend if μ ˜ 0 . Phillips et al. [92] considered the general specification which allows local-to-zero constant as follows:

(12) y t = μ ˜ T η + y t 1 + ε t , η 0 .

Here, y t has a deterministic drift of the form μ ˜ t / T η , whose magnitude depends on sample size and localising parameter η . If η tends to zero or infinity, we obtain the two limiting cases considered earlier.

Rewriting the model (12) as follows:

(13) y t = μ ˜ t T η + j = 1 t ε j + y 0 ,

it can be seen that the drift is small in relation to the stochastic trend, when η > 1 / 2 and equal to or stronger than the stochastic trend when η 1 / 2 . Only in the last case, η can be consistently estimated (see Phillips et al. [92, Appendix A ][8]), because the drift term is dominated by the stochastic trend. In other cases, the estimators of η usually converge to 1/2 corresponding to the rate of stochastic trend (see Phillips et al. [92, Appendix A] for the proof). Unfortunately, there is no finite sample comparison of the performance of the estimator η ˆ T .

Phillips et al. [92] 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 [110], who considered adding a linear trend).

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

In summary, Phillips et al. [92] recommended to always include constant term in constructing recursive ADF tests, but to compare the actual test statistic with different critical values (for η > 0.5 and η < 0.5 ) for robustness.[9]

Phillips and Yu [94] 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:

(14) y t = y t 1 I ( t < T e ) + ρ T y t 1 I ( T e t T c ) + k = T c + 1 t ε k + y T c I ( t > T c ) + ε t I ( t T c ) , ρ T = 1 + c T α , c > 0 , α ( 0 , 1 ) ,

where T e = τ e T is the origination date of the bubble, T c = τ c T is the date of its collapse, and thus the period [ τ e , τ c ] is the bubble episode, the periods [ 1 , T e ) ( T c , T ] are the normal market periods, and I ( ) 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 [89] (see also Harvey et al. [52] and equations (53) and (54) further in the text) considered a more reasonable mechanism that allows for transitory collapse dynamics. So, an instantaneous collapse as shown in (14) may be unrealistic, and some transient dynamics may be introduced after the peak – the so-called collapse regime. The corresponding data generating process (DGP) can be written as follows:

(15) y t = μ ˜ T η + y t 1 + ε t , t [ 1 , T e ) ( T r , T ] ( 1 + δ 1 T ) y t 1 + ε t , t [ T e , T c ] ( 1 δ 2 T ) γ T y t 1 + ε t , t ( T c , T r ] ,

where T r = τ r T 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 ) ( T r , T ] are the normal market periods. Also, δ 1 T = c 1 T α , δ 2 T = c 2 T β , c 1 , c 2 > 0 and α , β [ 0 , 1 ) . The formulation of autoregression (AR) coefficients follow moderate deviations from unity as shown in Phillips and Magdalinos [86]: the coefficient φ T deviates towards explosive behaviour, and the coefficient γ T deviates towards stationary behaviour. Fortunately, PWY procedure can consistently detect the bubble for this more general DGP.

Phillips et al. [97,98] (hereafter PSY for both papers) considered the following test statistic to account for multiple explosive regimes in time series (focusing on the η > 0.5 case for the drift term as more relevant in empirical applications). Their generalised supremum ADF (GSADF) test is

(16) GSADF ( τ 0 ) = sup τ 2 [ τ 0 , 1 ] , τ 1 [ 0 , τ 2 τ 0 ] ADF τ 1 τ 2 ,

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

(17) GSADF ( τ 0 ) sup τ 2 [ τ 0 , 1 ] , τ 1 [ 0 , τ 2 τ 0 ] τ 1 τ 2 W ˜ d W τ 1 τ 2 W ˜ 2 1 / 2 ,

### 2.4 Extensions

There are some approaches and modifications related to SADF and GSADF tests. Homm and Breitung [59] proposed to consider the supremum of the recursive Chow test through the following regression:

(18) Δ y ˜ t = ϕ HB I ( t > τ T ) y ˜ t 1 + e t ,

where y is preliminary de-meaned as y ˜ t = y t y ¯ , where y ¯ = T 1 t = 1 T y t .[10] The Chow-type test statistic, C τ , is defined as t -ratio for ϕ HB . Then the test of Homm and Breitung [59] is defined as follows:

(19) HB = sup τ [ 0 , 1 τ 0 ] C τ .

This statistic is actually the supremum of a sequence of backward recursive statistics. Harvey et al. [58] 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. [58] 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 follows:

Reject H 0 if { SADF > ψ ξ q ξ SADF or HB > ψ ξ q ξ HB } ,

with q ξ SADF and q ξ HB being critical values at level ξ , and ψ ξ being the scaling constant intended to ensure correct (asymptotic) size of the composite procedure.

Korkos et al. [64] extended the covariate ADF (CADF) unit root testing approach of Hansen [50] for testing for explosive bubbles to improve the power of the test. In this approach, the model is generated as follows:

(20) y t = μ + u t ,

(21) Δ u t = δ u t 1 + ε t ,

(22) Φ ( L ) ε t = b ( L ) Δ x t + ν t ,

where Δ x t is an m -vector of stationary covariates, and Φ ( L ) and β ( L ) are some lag operators. It is assumed that Ψ ( L ) Δ x t + k 1 + 1 = z t , where Ψ ( 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

(23) Δ y t = μ + δ y t 1 + j = 1 k ϕ j Δ y t j + j = k 1 k 2 β j Δ x t j + ε t ,

and calculate the CADF r statistic which is simply t -ratio for testing δ = 0 over the observations t = k + 1 , , τ T . The final SCADF test statistic of Korkos et al. [64] is defined as follows:

(24) SCADF ( τ 0 ) sup τ [ τ 0 , 1 ] CADF τ .

The limiting distribution of the CADF r has the following form:

(25) SCADF ( τ 0 ) sup τ [ τ 0 , 1 ] 0 τ Q ˜ ( r ) d P ( r ) 0 τ Q ˜ ( s ) 2 1 / 2 ,

where Q ( r ) ˜ = Q ( r ) 1 τ 0 τ Q ( r ) , Q ( r ) = b ( 1 ) Ψ ( 1 ) W z ( r ) + W ν ( r ) and P ( s ) = W ν ( r ) / σ ν . 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 ϱ 2 (the value of ϱ 2 determines the weights and measures the relative contribution of the covariate Δ x t to the error term ε t ). The estimator of ϱ 2 is given as ϱ ˆ 2 = σ ˆ ε ν 2 / ( σ ˆ ε 2 σ ˆ ν 2 ) , where σ ε ν 2 , σ ε 2 , and σ ν 2 are, respectively, the covariance between ε and ν , the variance of ε , and the variance of ν . All of them are estimated via the heteroskedasticity and autocorrelation consistent (HAC) approach. Korkos et al. [64] proposed a bootstrap algorithm similar to Chang et al. [25] to obtain critical values for the S C A D F test and to avoid estimating ϱ 2 in each subsample.

Whitehouse [112] considered a generalised least squares (GLS)-based version of the PWY (SADF) test. Earlier, Harvey and Leybourne [51] 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.[11] The GLS-based test follows from the auxiliary regression

(26) Δ u ˜ t = δ u ˜ t 1 + ε t ,

where u ˜ τ , t = y t z t θ ˜ , θ ˜ is the OLS estimator from the (quasi) GLS regression y c ¯ = ( y 1 , y 2 ρ ¯ y 1 , , y τ ρ ¯ y τ T 1 ) on z c ¯ = ( z 1 , z 2 ρ ¯ z 1 , , z τ ρ ¯ z τ T 1 ) , where ρ ¯ = 1 + c ¯ / T and z t = 1 or z t = ( 1 , t ) is the deterministic component.[12] Let ADF-GLS τ be a simple t -ratio from the regression (26) for t = 1 , , τ T . Then the supremum GLS-based test proposed by Whitehouse [112] is of the standard form:

(27) SADF-GLS ( τ 0 ) sup τ [ τ 0 , 1 ] ADF-GLS τ .

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 [112] also proposed a union of rejection testing strategy based on two tests, SADF-GLS ( τ 0 ) and SADF ( τ 0 ) for both cases, with or without trend.

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

Astill et al. [7] 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 [2] and Andrews and Kim [3] for testing the null of no end-of-sample bubble against the bubble alternative. The model considered has the following form:

(28) y t = μ + u t , t = 1 , , T + m ,

(29) u t = u t 1 + ε t , t = 1 , , T δ u t 1 + ε t , t = T + 1 , , T + m ,

where ε 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, ρ = 1 , and the alternative hypothesis corresponds to a bubble during the end-of-sample, ρ > 1 . Astill et al. [7] noted that under the null, Δ y t = ε t during the full sample, while under the alternative, Δ y t = ε t up to time T and Δ y t = Δ u t = δ ( 1 + δ ) t T 1 u T + j = 0 t T 1 ( 1 + δ ) j Δ ε t j , where δ = ρ 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 + δ ) t T 1 around δ = 0 , ( 1 + δ ) t T 1 1 + ( t T 1 ) δ , we have the following approximation:

(30) Δ y t = δ ( 1 δ ) u T + δ 2 u T ( t T ) + e t ,

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 Δ 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 Δ y t on a linear trend:

(31) S m = t = j + 1 j + m ( t j ) Δ y t or t = j + 1 j + m s = t j + m Δ y s .

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

(32) R m = t = j + 1 j + m s = t j + m Δ y s 2 .

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.[13]

To account for possible unconditional variance on innovations ε , Astill et al. [7] proposed a studentised White-type version of (33):

(33) S m w = S m t = j + 1 j + m ( ( t j ) Δ y t ) 2 .

Astill et al. [7] 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 earlier 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 [17], Cavaliere and Taylor [19,20,22], and the references therein). It should be noted that supremum-based ADF-type tests (PWY, PSY) are still robust to conditional heteroskedasticity as demonstrated by PSY.

Harvey et al. [53] addressed this issue in an explosive bubble context. Consider the following DGP for { y t } in time-varying parameter form

(34) y t = ( 1 + δ t ) y t 1 + ε t or Δ y t = δ t y t 1 + ε t ,

with obvious definition of δ t : δ t may be > 0 for an explosive regime, < 0 for a stationary collapsing regimes and = 0 for a unit root regime. Harvey et al. [53] and subsequent papers considered local-to-unit root behaviour of δ t : δ t = c 1 / T for explosive period and δ t = c 2 / T for stationary collapse period with c 1 > 0 and c 2 0 . The process (34) can be seen as different reparametrisation of (15) except to behaviour of δ t . In the following, we state the assumption on regression errors.

## Assumption 1

The non-stationary volatility is generated as ε t = σ t z t , where { z t } is a martingale difference sequence with respect to natural filtration, and the volatility σ t is defined as σ s T = ω ( s ) for s [ 0 , 1 ] , where ω ( ) D is a non-stochastic and strictly positive function satisfying 0 < ω ̲ < ω ( s ) < ω ¯ < .

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. [53] 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. By taking into account this issue, Harvey et al. [53] use the following wild bootstrap. We note that although Harvey et al. [53] proposed their algorithm only for the SADF test, their methodology can be easily implemented for the GSADF test.

## Algorithm 1

(Bootstrap tests)

1. Generate the vectors of bootstrap innovations as e t = w t Δ y t for t = 2 , , T initialised at e 1 = 0 , where { w t } 2 T be IID sequence of N ( 0 , 1 ) random variates.

2. Construct the bootstrap sample data via recursion Δ y t = e t for t = 2 , , T initialised at y 0 = 0 .

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 .

4. Bootstrap p -values are then defined as: P GSADF , T G GSADF , T ( GSADF ) , where G GSADF , T ( ) denotes the conditional (on the original sample data) cumulative distribution functions (cdfs) of GSADF . 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. [53] 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 [22] (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 Δ y t in Step 1. However, Pedersen and Schütte [84] proposed a sieve-based implementation of Step 2 and 3, similar to Chang and Park [24], which improve the size properties.

Harvey et al. [55] proposed a weighted least squares-based modification of the PWY test. Transform the model as follows:

(35) Δ y t σ t = ρ t y t 1 σ t + z t , t = 2 , , T .

This regression is infeasible because we do not observe variance function σ t . If σ t would be known, we could construct the supremum-based test as PWY:

(36) SBZ ( τ 0 ) = sup τ [ τ 0 , 1 ] BZ τ ,

where BZ r is calculated from regression (35) over subsample { y 1 , , y τ T }

(37) BZ τ = t = 1 τ T Δ y t ˜ y ˜ t 1 / σ t 2 t = 1 τ T y ˜ t 1 2 / σ t 2 1 / 2 .

Here, 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 ω ( r ) . To make the test feasible, Harvey et al. [55] use non-parametric kernel smoothing estimator of σ t :

(38) σ ˆ t 2 = i = 2 T K h i t T ( Δ y i ) 2 i = 2 T K h i t T ,

where K h ( s ) = K ( s / h ) / h and K ( ) 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. [55] utilise the wild bootstrap implementation of Harvey et al. [53] to guarantee control of size. Moreover, Harvey et al. [55] 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

Reject H 0 if { SADF > ψ ξ q ξ SADF or SBZ > ψ ξ q ξ SBZ } .

or, equivalently,

Reject H 0 if U = max SADF , q ξ SADF q ξ SBZ SBZ > q ξ U .

where ψ ξ is a (asymptotic) scaling constant to ensure correct (asymptotic) size of the composite procedure, and q ξ U = ψ ξ × q ξ 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:

Reject H 0 if U = max SADF , q ξ , SADF q ξ , SBZ SBZ > q ξ , U ,

where q , ξ SBZ and q , ξ SADF are the bootstrap based critical values for SADF and SBZ test statistics, and q , ξ U is the bootstrap-based critical value for the U test statistic.

Harvey et al. [57] proposed another method which controls the size under time-varying volatility. This method is based on cumulated signs C t = i = 2 t sign ( Δ y t ) , t = 2 , , T . The supremum sign-based test is defined as follows:

(39) s GSADF ( τ 0 ) = sup τ 2 [ τ 0 , 1 ] , τ 1 [ 0 , τ 2 τ 0 ] s ADF τ 1 τ 2 ,

where s ADF τ 1 τ 2 is a t -ratio in the regression

(40) Δ C t = δ ˆ ( τ 1 , τ 2 ) C t 1 + e t ,

over subsample from τ 1 T to τ 2 T . That is,

(41) s ADF τ 1 τ 2 = δ ˆ ( τ 1 , τ 2 ) s ˆ 2 ( τ 1 , τ 2 ) / t = τ 1 T + 1 τ 2 T C t 1 2 ,

where s ˆ 2 ( τ 1 , τ 2 ) = ( τ 2 T τ 1 T 1 ) 1 t = τ 1 T + 1 τ 2 T e t 2 . Because sign ( Δ y t ) = sign ( z t ) under the null hypothesis, the test is exact invariant to the volatility function σ t . Moreover, the sGSADF test is the exact invariant to the constant in DGP. If we allow a weak dependence of errors, we should just replace sign ( Δ y t ) by sign ( Δ y t j = 1 k ϕ ˆ j ( t ) Δ y t j ) , where ϕ ˆ j ( t ) are obtained from the following recursive regression:

Δ y i = α ˆ ( t ) + δ ˆ ( t ) y i 1 + j = 1 k ϕ ˆ j ( t ) Δ y i j + e i

for i = k + 5 , , t .

Harvey et al. [57] 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. [55]. To address the issue of asymmetric errors, Harvey et al. [57] replace sign ( Δ y t ) by its recursive demeaned version as sign ( Δ y t ) ( t 1 ) 1 t = 2 t sign ( Δ y t ) . The bootstrap algorithm is modified accordingly.

Hafner [48] modified wild bootstrap algorithm of Harvey et al. [53] to allow a skeweness of the distribution of the series. He replaced w t in Step 1 by w t = u t / 2 + ( v t 2 1 ) / 2 , where u t N ( 0 , 1 ) and v t N ( 0 , 1 ) , so that E ( w t ) = 0 , E ( w t 2 ) = 1 and E ( w t 3 ) = 1 . Hafner [48] also use sieve-based recolouring in Step 2.

Kurozumi et al. [69] proposed to use a transformation of the series according to the volatility behaviour in function σ t , similar to Cavaliere and Taylor [21] in the classical unit root testing context. Under non-stationary volatility assumption, the partial sum process of { ε t } is asymptotically characterised by the variance profile, termed by Cavaliere and Taylor [21], which is defined as follows:

η ( s ) 0 1 ω ( r ) 2 d r 1 0 s ω ( r ) 2 d r .

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

ω ¯ 2 0 1 ω ( r ) 2 d r .

Note that η ( s ) = s under homoskedasticity. Then, we have the following weak convergence due to Theorem 1 of Cavaliere and Taylor [19]:

(42) 1 T y r T = 1 T t = 1 r T ε t ω ¯ W ( η ( r ) ) ω ¯ W η ( r ) ( 0 r 1 ) ,

for y defined in (34), where denotes weak convergence in D [ 0 , 1 ] and W ( ) is a standard Brownian motion, while W η ( ) is called a variance transformed Brownian motion (Brownian motion under a modification of the time domain). In the case of a constant variance with σ t = σ , we have η ( r ) = r , and thus, W η ( r ) reduces to a standard Brownian motion. The time transformation is based on the variance profile η ( s ) . Roughly, we should take the sampling interval longer in the low volatility regime, whereas we take it shorter for large values of σ t .

More precisely, because the variance profile is a strictly monotonically increasing function, we have the unique inverse given by g ( s ) η 1 ( s ) . Then, consider the time-transformed series y ˜ t = y t y t = 0 with a non-decreasing sequence t = g ( t / T ) T . We note that y ˜ 0 = y 0 y 0 = 0 and y ˜ T = y T y 0 . As shown by equation (9) in the study by Cavaliere and Taylor [19], we have, under the null hypothesis,

(43) T 1 / 2 y ˜ r T T 1 / 2 y g ( r T / T ) T T 1 / 2 y g ( r ) T ω ¯ W η ( g ( r ) ) = ω ¯ W ( r ) ,

because W η ( g ( r ) ) = W ( η ( 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. [69] proposed the following test statistics based on the time-transformed ADF statistics:

(44) STADF = sup τ 2 [ τ 0 , 1 ] TADF 0 τ 2 and GSTADF ( τ 0 ) = sup τ 2 [ τ 0 , 1 ] , τ 1 [ 0 , τ 2 τ 0 ] TADF τ 1 τ 2 , where TADF τ 1 τ 2 = y ˜ τ 2 T 2 y ˜ τ 1 T 2 ω ¯ 2 ( τ 2 T τ 1 T ) 2 ω ¯ t = τ 1 T + 1 τ 2 T y ˜ t 1 2 .

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. [69] utilise the approach of [56] and estimate non-parametrically time-varying autoregressive coefficient, collect residuals, and use them for estimating the variance profile.

Kurozumi et al. [69] performed Monte-Carlo simulations with comparison of STADF, bootstrap-based SADF test of Harvey et al. [53], sign-based test of Harvey et al. [57], and SBZ test of Harvey et al. [55], and the latest two with union of rejection and bootstrap. Kurozumi et al. [69] demonstrated that none of the tests dominates the others, but the 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 follows:

(45) τ ˆ e = inf s τ 0 { s : ADF s > c v β T adf ( s ) } , τ ˆ c = inf s τ e { s : ADF s < c v β T adf ( s ) } ,

where c v β n adf ( s ) is a right-tailed critical values of ADF s (defined in (9)) corresponding to the significance level β 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 β T should satisfy β T 0 as T . This guarantee that c v β n adf ( s ) 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 [94] suggest to estimate the date of collapse at least after some period after the origination date as follows:

(46) τ ˆ c = inf s τ e + log ( T ) T { s : ADF s < c v β n adf ( s ) } .

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 the 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 [94] 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 the study by Phillips and Magdalinos [86,87]. They established that under the null hypothesis of no explosive bubble, and under c v β T adf , the probability of detecting the date of origination of a bubble is equal to zero as T , i.e., Pr ( τ ˆ e [ τ 0 , 1 ] ) 0 . Under the alternative hypothesis (mildly explosive process in model (5)), the estimator τ ˆ e is consistent for τ e by assuming that 1 / c v β T adf + c v β T adf / T 1 α / 2 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, and 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 [94] suggest to use the sequence of critical values c v β T adf = 2 3 log log 2 T (see Phillips and Yu [94, 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 α / 2 .

For the estimator τ ˆ c , under the null hypothesis of no explosive bubble similarly Pr ( τ ˆ f [ r 0 , 1 ] ) 0 . Under the alternative hypothesis and assuming 1 / c v β T adf + c v β T adf / T 1 α / 2 0 , the estimator τ ˆ c is consistent for τ c . In this case, δ ˆ T ( τ ) p 1 for τ > τ c , but the limiting distribution of this estimator has second-order downward bias (i.e., bias towards 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 τ test diverges to in this case.

Besides ADF τ test, Phillips and Yu [94] also consider the coefficient based test, ADF τ δ = τ ( δ ˆ τ 1 ) . The asymptotic properties of this test are similar to the ADF τ except for latter converges to for τ > τ c with higher rate (under α > 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 τ can better estimate the date of collapse than the ADF τ δ . 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 [95] 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, T ˆ e = τ ˆ e T , is identified by recursive procedure (45). Let n min is the number of observations in the sample { y T ˆ e n min + 1 , , y T ˆ e } . This sample can be constructed as some fraction (say, 10%) from the sample before T ˆ 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 follows:

BIC UR = log t = T ˆ e n min n k T ˆ e ( Δ y t y ¯ ) 2 n k + n min + log ( n k + n min ) n k + n min ,

while for the latter, the BIC is defined as follows:

BIC AR = log t = T ˆ e n min n k T ˆ e ( y t μ ˆ δ ˆ y t 1 ) 2 n k + n min + 2 log ( n k + n min ) n k + n min ,

where y ¯ = ( n k + n min ) 1 T ˆ e n min n k T ˆ e y t , μ ˆ , and δ ˆ are OLS estimators in the regression y t = μ + δ y t 1 + ε t , both BIC values are calculated for the sample { y T ˆ e n min n k + 1 , , y T ˆ e } . If the BIC value for the unit root model is higher and point estimate of δ is greater than one, we redefine the initial condition T ˆ e n min as T ˆ 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 the unit root model is lower than the BIC for the autoregressive model. After this, we denote final initial condition as T ˆ 0 and apply PWY procedure for the sample from T ˆ 0 as in (46). The resulting estimate of the origination date is defined as T ˆ e ( T ˆ 0 ) . If the sample after repeated comparing the BICs becomes { y 1 , , y T ˆ e } , and the BIC for unit root model is larger than BIC for the 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 T ˆ e ( T ˆ 0 ) coincides with T ˆ e . In general case, it is expected that T ˆ e ( T ˆ 0 ) T ˆ e .

The alternative procedure of detecting the dates of a bubble was proposed by Phillips et al. [97,98] (PSY). Contrast to (45) and (46), the estimates of the dates T e = τ e T and T c = τ c T are based on

(47) τ ˆ e = inf τ 2 [ τ 0 , 1 ] { τ 2 : BSADF τ 2 ( r 0 ) > scv β T ( τ 2 ) } ,

(48) τ ˆ c = inf τ 2 [ τ ˆ e + δ log ( T ) T , 1 ] { τ 2 : BSADF τ 2 ( τ 0 ) < scv β T ( τ 2 ) } ,

where BSADF τ 2 ( τ 0 ) is the backward sup ADF test defined as follows:

BSADF τ 2 ( τ 0 ) = sup τ 1 [ 0 , τ 2 τ 0 ] ADF τ 1 τ 2 .

PSY extended their procedure for the case of multiple bubbles. They suggest to obtain the dates T e = τ e T and T c = τ c T for the second bubble just repeating the aforementioned procedure, 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

(49) τ ˆ 1 e = inf τ 2 [ τ 0 , 1 ] { τ 2 : BSADF τ 2 ( τ 0 ) > scv β T ( τ 2 ) } ,

(50) τ ˆ 1 c = inf τ 2 [ τ ˆ 1 e + δ log ( T ) T , 1 ] { τ 2 : BSADF τ 2 ( τ 0 ) < scv β T ( τ 2 ) } ,

(51) τ ˆ 2 e = inf τ 2 [ τ ˆ 1 f , 1 ] { τ 2 : BSADF τ 2 ( τ 0 ) > scv β T ( τ 2 ) } ,

(52) τ ˆ 2 c = inf τ 2 [ τ ˆ 2 e + δ log ( T ) T , 1 ] { τ 2 : BSADF τ 2 ( τ 0 ) < scv β T ( τ 2 ) } .

PSY proved the consistency of the bubble dates estimators by assuming that 1 / scv β T ( r 2 ) + scv β T ( r 2 ) / T 1 / 2 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 ( τ 1 c τ 1 e > τ 2 c τ 2 e ). 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 is two-step: first, detect the first bubble by PWY procedure; second, 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 τ 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 [73] 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.

If the data are characterised by time-varying volatility behaviour, Phillips and Shi [91] proposed a modification of the wild bootstrap algorithm of Harvey et al. [53]. Because the dating procedure is sequential, Phillips and Shi [91] also address the multiplicity issue. They considered the following algorithm.

### Algorithm 2

(Composite bootstrap procedure)

1. Estimate regression (8) using the full sample period, imposing δ = 0 , and obtain residuals e t .

2. For the sample τ 0 T + 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

Δ y t = j = 1 k ϕ ˆ j Δ y t j + e t

initialised at y i = y i with i = 1 , , j + 1 , ϕ ˆ 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.

3. Using the bootstrap sample { y t } , compute the PSY test statistic sequence { BSADF t ( τ 0 ) } t = τ 0 T τ 0 T + T b 1 and the maximum value of this test statistic sequence M t = max t [ τ 0 T , τ 0 T + T b 1 ] BSADF t ( τ 0 ) .

4. Bootstrap critical values of the PSY procedure are obtained from the distribution of M t .

Phillips and Shi [91] in their empirical section used T b = 24 in monthly data which means that the empirical size is controlled over a 2-year period. Phillips and Shi [91] also developed the R package psymonitor to implement the proposed bootstrap procedure. Phillips and Shi [91] 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. [6] extended the test proposed by Astill et al. [7] to a real-time monitoring scheme. The problem is that the Astill et al. [6] test or the PSY test do not control the size during sequential testing. The statistic of Astill et al. [6] 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 [67] considered PWY and PSY monitoring test statistics,[14] 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 [59] (see also Breitung and Kruse [13]). Kurozumi [67] demonstrated that although the ADF-type monitoring scheme outperforms the CUSUM-type monitoring scheme under moderate deviation from a unit root asymptotic, and 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. [8] extended the CUSUM test of Homm and Breitung [59] to allow for time-varying volatility. They suggested to replace standard variance estimator by its non-parametric counterpart. Kurozumi [66] 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 [66] 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. [52] proposed an approach based on minimisation of sum of squared residuals. They considered a more general DGP as Phillips and Shi [89] (see (15)):

(53) y t = μ + u t ,

(54) u t = u t 1 + ε t , t = 2 , , τ 1 , 0 T , ( 1 + δ 1 ) u t 1 + ε t , t = τ 1 , 0 T + 1 , , τ 2 , 0 T , ( 1 δ 2 ) u t 1 + ε t , t = τ 2 , 0 T + 1 , , τ 3 , 0 T , u t 1 + ε t , t = τ 3 , 0 T + 1 , , T .

This DGP implies a unit root process until the time τ 1 , 0 τ e , then followed by an explosive process until τ 2 , 0 τ c . After τ 2 , 0 τ c , there may be a stationary collapsing regime (which is interpreted as the return to normal market behaviour) until the time τ 3 , 0 τ r . After the collapsing regime, the series follows a unit root process until the end of the sample. There are some special cases. If τ 2 , 0 = τ 3 , 0 , the explosive regime instantly changes to a unit root regime. If, moreover, τ 2 , 0 = 1 , the explosive regime is not terminated and continues to the end of the sample period. If τ 3 , 0 = 1 , the collapsing regime is not terminated and continues to the end of the sample period as well. Actually, Harvey et al. [52] proposed to choose the bubble model depending on the restrictions: τ 2 , 0 = 1 (Model 1), τ 2 , 0 = τ 3 , 0 (Model 2), or τ 3 , 0 = 1 (Model 3). Model 4 is unrestricted. For Model 4, we can estimate the corresponding regression of the form

(55) Δ y t = μ 1 D t ( τ 1 , τ 2 ) + μ ˆ 2 D t ( τ 2 , τ 3 ) + δ 1 D t ( τ 1 , τ 2 ) y t 1 + δ 2 D t ( τ 2 , τ 3 ) y t 1 + e t ,

where D t ( a , b ) = I ( a T < t b T ) . The Model 3 corresponds to the restriction τ 3 = 1 , the Model 2 corresponds to the restrictions μ 2 = β 2 = 0 , and the Model 2 corresponds to the restrictions μ 2 = β 2 = 0 and τ 2 = 1 .

Let SSR j ( ) = t = 2 T e ˆ t be the sum of squared residuals for the Model j with the constraints y τ 2 T > y τ 2 T and y τ 2 T