Testing for explosive bubbles: a review
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Anton Skrobotov
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.
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,
where
where
with
Therefore, the asset price in equation (2) is composed of a fundamental term
where
Explosive behaviour can be statistically approximated by explosive autoregression of the form
with
If there is no bubble, i.e.,
If both
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
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.
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
Consider the following ADF-type regression as follows:
We want to test the null hypothesis of a unit root,
where
where
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
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
while Diba and Grossman [34] assumed
so that
Here,
Rewriting the model (12) as follows:
it can be seen that the drift is small in relation to the stochastic trend, when
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
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
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:
where
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:
where
2.3 Generalised SADF test
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
where
where
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:
where
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:
with
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:
where
The main idea is to add leads and lags of stationary covariates to regression (8) as
and calculate the
The limiting distribution of the
where
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
where
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,
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:
where
where
This is Andrews
The asymptotic size of
To account for possible unconditional variance on innovations
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
with obvious definition of
Assumption 1
The non-stationary volatility is generated as
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)
Generate the vectors of bootstrap innovations as
Construct the bootstrap sample data via recursion
Using the bootstrap sample,
Bootstrap
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
Harvey et al. [55] proposed a weighted least squares-based modification of the PWY test. Transform the model as follows:
This regression is infeasible because we do not observe variance function
where
Here,
where
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
or, equivalently,
where
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:
where
Harvey et al. [57] proposed another method which controls the size under time-varying volatility. This method is based on cumulated signs
where
over subsample from
where
for
The special case of sGSADF test is the sSADF with the restriction
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
Hafner [48] modified wild bootstrap algorithm of Harvey et al. [53] to allow a skeweness of the distribution of the series. He replaced
Kurozumi et al. [69] proposed to use a transformation of the series according to the volatility behaviour in function
The so-called (asymptotic) average innovation variance is defined as follows:
Note that
for
More precisely, because the variance profile is a strictly monotonically increasing function, we have the unique inverse given by
because
By taking into account (43), Kurozumi et al. [69] proposed the following test statistics based on the time-transformed ADF statistics:
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
where
This guarantee that the length of the bubble is economically significant, and the bubble episode is at least
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
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
For practical implementation, Phillips and Yu [94] suggest to use the sequence of critical values
For the estimator
Besides
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,
while for the latter, the BIC is defined as follows:
where
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
where
PSY extended their procedure for the case of multiple bubbles. They suggest to obtain the dates
PSY proved the consistency of the bubble dates estimators by assuming that
Lui [73] extended the PSY procedure to allow a long memory behaviour of the errors. For this, he proposed a HAR (fixed-
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)
Estimate regression (8) using the full sample period, imposing
For the sample
initialised at
Using the bootstrap sample
Bootstrap critical values of the PSY procedure are obtained from the distribution of
Phillips and Shi [91] in their empirical section used
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)):
This DGP implies a unit root process until the time
where
Let
where
Harvey et al. [57] 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. [57] suggested to maximize the sign-based statistic over all possible dates of a bubble as follows:
where
with
Monschang and Wilfling [77] 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 [77] considered the standard sign-based procedure (56) without correction of the standard errors as in (57).
The approach of Harvey et al. [52] 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. [80] 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 minimising the sum of squared residuals for the two-regime model, and the estimated break date
Harvey et al. [54] combined the approach of PSY and Harvey et al. [52] and proposed the two-step approach. 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. [54] 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. [61] 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. [61] 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 the study by Phillips and Magdalinos [86,87]. The authors considered a mildly explosive process of the form:
which initialises at
The sequence of autoregressive parameter
where
The results (59) allow us to construct a confidence interval for
where
Guo et al. [46] generalised the Phillips and Magdalinos [86] results by allowing an deterministic drift term
Chan et al. [23] (see also Liu and Peng [72]) 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 [75] where the authors developed a testing procedure based on instrumental variable estimation and pre-testing for stationary or explosive direction. Phillips [85] considered estimation and inference of localising parameters
All the aforementioned methods should be implemented for the sample interval with a stable autoregressive parameter. For example, 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. [46] and Xiang et al. [113] 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. [46] 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. [93] that the explosive autoregressive root not greater than 1.05 means moderate explosiveness. Xiang et al. [113] considered the Bitcoin market capitalisation 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. [80] and Kurozumi and Skrobotov [68], 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 [95] propose a test which detects whether the bubble migrates form one market to another. Consider two series,
Let a date-stamping strategy detect the origination date of the bubble in
Similarly, define
Let
Let the autoregressive parameter of
i.e., after the origination the bubble, the behaviour of
Phillips and Yu [95] consider a simplified linear version of (60), where
that is, the data cover the collapse period of
Greenaway-McGrevy and Phillips [45] propose an another method to find the evidence of contagion from one time series to others. Their method implies to first compute recursive coefficients
where core denotes core series from which the bubble originates by assumption. The delay parameter
Regression (62) is functional, so the functional coefficient
Chen et al. [29] 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 follows:
where
where the factor
Chen et al. [29] 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
There are other approaches operating the term co-exposiveness. Magdalinos and Phillips [76] developed the asymptotic theory for cointegrating systems with possibly mildly explosive variables.[20] Chen et al. [27] extended the theory of Magdalinos and Phillips [76] to a continuous time framework (see also Tao et al. [107]). Chen and Tu [28] developed the asymptotic theory for spurious regression with a moderately explosive processes (see also Lin and Tu [71]). Nielsen [79] 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:
where
Evripidou et al. [41] considered the so-called co-bubble behaviour via the following regression model:
where
Evripidou et al. [41] suggested to use a Kwiatkowski-Phillips-Schmidt-Shin-type [70] (KPSS) statistic of the following form:
where
7 Conclusion
This article 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 the study by Cavaliere et al. [18]). For now, only one paper, by Lui [73] accounted for a long memory dynamic in the innovation error of the explosive bubble model. Second, penalised methods such as least absolute shrinkage and selection operator (LASSO) can be used for identification of bubble regimes. Indeed, the methods based on minimisation 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 [99] inter alia), and extensions for the multiple bubble model could be investigated.
Acknowledgements
The author is grateful to Eiji Kurozumi, Rob Taylor, Shuping Shi, Yang Zu, Madina Karamysheva, Rustam Ibragimov, Artem Prokhorov, three anonymous referees and participants of the Center for Econometrics and Business Analytics (CEBA, St. Petersburg State University) seminar series and St. Petersburg Economic seminar (HSE, EU, PDMI) for helpful comments and suggestions.
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Funding information: The article was written on the basis of the RANEPA state assignment research programme.
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Conflict of interest: The authors state no conflict of interest.
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Artikel in diesem Heft
- Research Articles
- Joint lifetime modeling with matrix distributions
- Consistency of mixture models with a prior on the number of components
- Mutual volatility transmission between assets and trading places
- Functions operating on several multivariate distribution functions
- An optimal transport-based characterization of convex order
- Test of bivariate independence based on angular probability integral transform with emphasis on circular-circular and circular-linear data
- A link between Kendall’s τ, the length measure and the surface of bivariate copulas, and a consequence to copulas with self-similar support
- Review Article
- Testing for explosive bubbles: a review
- Interview
- When copulas and smoothing met: An interview with Irène Gijbels
- Special Issue on 10 years of Dependence Modeling
- On copulas with a trapezoid support
- Characterization of pre-idempotent Copulas
- Abel-Gontcharoff polynomials, parking trajectories and ruin probabilities
- A nonparametric test for comparing survival functions based on restricted distance correlation
- Constructing models for spherical and elliptical densities
