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Testing for an Explosive Bubble using High-Frequency Volatility
Keywords: Stochastic volatility model; Unit root test; Double asymptotics; Explosiveness; Asset price bubbles.
In recent years, initiated by the influential paper by {phillips2011explosive}, there has been a renewed interest in the empirical identification of explosive behavior in financial asset prices as a means to detect economic bubbles. In {phillips2011explosive}, the test statistic is the supremum of recursively implemented Dickey-Fuller (DF) $t$-statistics, and a right-tailed test, referred to as the PWY test, is conducted. This approach effectively addresses the power deficiency commonly encountered in simple right-tailed unit root tests, a problem extensively discussed in prior studies such as {Diba1987,Diba1988}, {Flood1980}, {Flood1986} and {evans1991pitfalls}. When applied to price series after removal of fundamental components, the presence of explosiveness indicates the existence of a bubble phenomenon.
Subsequent studies have explored various theoretical and practical aspects of the bubble testing problem. For example, {Phillips2011} study the behavior of bubbles during the subprime crisis. {homm2012testing} propose various supremum-type tests for bubbles based on alternative tests for changes in persistence in time-series models. They also propose to use the supremum of backwardly implemented recursive Chow-type tests to test for bubbles. Phillips2013 consider the problem of testing for multiple bubbles and propose a double supremum version of DF $t$-statistics over all possible subsamples as their test statistic. {Phillips2013a} study the limit theory of bubble date detectors in the context of multiple bubbles. See also harvey2017improving, chongtesting, {Breitung2013}, pavlidis2012new and shi2015identifying, among others. These studies collectively contribute to advancing our understanding of bubble detection techniques and provide valuable insights into the dynamics and implications of explosive behavior in financial markets.
Existing research on testing asset price bubbles has predominantly been based on classical discrete-time autoregressive (AR) models, related to the unit-root-testing literature. However, in the finance literature, it is commonly assumed that asset prices follow a continuous-time model, which can offer analytical simplicity for pricing derivatives. Moreover, from an empirical modeling standpoint, continuous-time models provide a natural framework for incorporating data sampled at various frequencies, which is particularly relevant for the problem addressed in this paper.
In this study, we consider the problem of testing for explosive behavior within the framework of a continuous-time stochastic volatility model with a linear drift and nonparametric stochastic volatility. Unlike previous approaches, we do not assume that the stochastic volatility process satisfies the Markovian property. When the volatility is constant, our model corresponds to the well-known Ornstein-Uhlenbeck (OU) process. Testing for unit roots in the context of the OU process has been explored in prior works such as {Phillips1987}, {perron1991continuous}, {Yu2014}, {Zhou2015} and {chen2015optimal}. To capture the explosive behavior in the data, we allow the persistence parameter to vary from the unit root regime to the explosive regime. While the existing literature on bubble testing accounts for time-varying volatility, the proposed methods primarily focus on unconditional heteroskedasticity, as discussed in harvey2015tests and HLZ20. However, relaxing the assumption of constant volatility and considering conditional heteroskedasticity and hence stochastic volatility is crucial, since volatility clustering is a stylized fact observed in asset prices.
Our testing procedure consists of two main steps. First, we utilize high-frequency data to estimate the integrated volatility over time intervals corresponding to a lower sampling frequency. This is achieved by employing the realized variance (RV) estimator, a widely used method in the literature (see, e.g., {ABDL01} and {NS02}). Second, we “devolatize” the log-price increments at the lower frequency using the estimated realized volatility. Subsequently, we apply the strategy introduced by {phillips2011explosive} (the PWY test strategy) to the devolatized series in order to test for explosive behavior in the data. We refer to this test as the RVPWY test. Under this framework, we show that the RVPWY statistic has the same asymptotic null distribution as the PWY statistic in the discrete-time model with a constant volatility. As a result, the critical values provided in {phillips2011explosive} be readily utilized for inference in our RVPWY test.
Our proposed testing and date-stamping procedure involves applying the PWY test to a devolatized time series. This approach has significant implications for the outcomes of the testing and date-stamping procedure. To illustrate this idea, we plot the daily BitCoin log-prices from January 1, 2018 to August 31, 2021 in the left panel of Figure (ref), and the devolatized log-price series (using the daily RV calculated from 5-minute returns) in the right panel, taken from our later empirical analysis. The differences between the two plots are striking. In the devolatized sample, sudden large drops in the BitCoin price are less apparent. Moreover, the magnitude of price decreases is considerably reduced, while the magnitude of price increases is largely preserved, making them more pronounced in a relative sense. This effect is primarily due to the higher volatility during periods of price drops in the original data. Since the devolatization strategy removes heteroskadasticity from the data, our intuition suggests that the devolatized sample provides a better foundation for more accurately identifying and date-stamping explosive behavior. This advantage is supported by our Monte Carlo simulations. Moreover, empirical analyses highlight some key differences between the proposed method and the existing method.
Empirically, existing tests for explosive bubbles (e.g. phillips2011explosive and Phillips2013) are often applied to monthly asset prices.\footnote{Exceptions include Laurent_Shi_2022 and ATZ23. Also see the recent progress in this direction by COR22.} Widely available daily or intra-day asset price data are not used. Based on an empirically justifiable approximation strategy that we propose, we show that the methodology developed in this paper can be used to exploit the information of monthly volatility in asset price data from daily asset prices, which in turn help us in detecting explosiveness at the monthly frequency. In one of our empirical examples, we apply this idea to detect bubbles in the real S&P500 index.
In a similar vein, when the objective is to detect explosiveness in daily asset prices, having access to intra-day price data allows for accurate estimation of daily integrated volatility. By leveraging these highly precise daily volatility measures, we can enhance the reliability of our inference for explosiveness at the daily frequency. Consequently, our methodology offers a valuable tool for detecting and studying explosive behavior in daily asset prices. In our first empirical investigation, we apply our method to detect explosive behavior in cryptocurrencies at the daily level. Cryptocurrencies, being highly volatile assets, present an interesting and relevant context for studying explosiveness. By analyzing daily price data, we can gain insights into the presence and characteristics of potential explosive episodes in this market.
It is important to note that despite utilizing high-frequency intraday data in this paper, the primary interest lies in testing explosiveness at the low-frequency level, i.e.\ at the monthly or daily level. While it is theoretically possible to test for explosiveness in high-frequency data, the data-generating process at higher frequencies tends to be more complex compared to low-frequency data. Factors such as diurnal effects and market micro-structure noise need to be taken into account in high-frequency analyses, which can introduce complications that may invalidate existing methods. In practice, policymakers and investors often seek to understand whether an economic time series exhibits a bubble at the low-frequency level, even if they have access to high-frequency data. This preference can be attributed to various reasons, including the high costs associated with implementing new policies or rebalancing portfolios. As a result, our motivation is to test for explosiveness using low-frequency data, while assuming that higher-frequency data is available.
In terms of testing for explosiveness, the present paper is related to several strands of literature. First, it is related to CJ20, JK21 and JPM22. In particular, JK23 propose to use quadratic variation only to identify explosive behavior in asset prices. Second, it is related to ATZ23 where a different devolatizing technique and a different test were used. In particular, ATZ23 use implied volatility obtained from options to devolatize the price series in the first stage and a CUSUM test was applied to the devolatized prices. Also, ATZ23 assume that volatility follows a Markov process. Finally, our study is related to PY09twostage, where the realized volatility is used to estimate parameters in the diffusion function in the first stage of a two-stage approach. Related work, focusing on parameter estimation, is by Corradi2006 and Todorov2009.
The structure of the paper is as follows. Section 2 considers a continuous-time model, defines the realized variance estimator, and discusses how to devolatize data. Section 3 considers the bubble testing problem, defines the RVPWY statistic, and derives the asymptotic null distribution. Section 4 performs Monte Carlo simulations to study the finite sample size and power properties of the proposed statistic, in comparison with currently available tests. Section 5 considers the real-time date-stamping problem. Section 6 applies the proposed methodology to two empirical datasets. Section 7 concludes the paper and discusses possible future research. Technical lemmas and mathematical proofs are collected in the Appendix.
Throughout the paper, we use $\xrightarrow{d}$ to denote convergence in distribution and use $\Rightarrow$ to denote weak convergence of a stochastic process. $\sim$ denotes “is distributed as”. $\lfloor x \rfloor$ denotes the integer part of a non-negative real number $x$.
Consider the following continuous-time process with stochastic volatility for the log asset price process $\{y_t \}_{t \geqslant 0}$
where $y_0=0$, $W$ is a Brownian motion process, $\sigma_t$ is a strictly positive stochastic volatility process, independent of the $W$ process, and $\kappa$ is the persistence parameter. Note that $\sigma_t$ is nonparametrically specified and need not satisfy the Markov property.\footnote{Well-known volatility processes that do not satisfy the Markov property are the fractional Brownian motion and the fractional Ornstein-Uhlenbeck process; see gatheral2018 and wang2023.} In ((ref)), $\kappa = 0$ corresponds to the unit root model, and $\kappa > 0$ corresponds to an explosive model. This model is an extension of the continuous-time Ornstein-Uhlenbeck (OU) model considered in {perron1991continuous} and {Zhou2015} to allow for stochastic volatility.
First, assume the asset price process $y_t$ is observed on a low-frequency time grid $t_i=iH$, where $i= 0, 1, 2, \ldots, n$, and $H$ is the sampling interval (e.g., one day or one month). Discretizing model ((ref)) at the observational frequency, we have
$i = 1, \ldots, n$, where $\varepsilon_i$ is defined as $\varepsilon_i:= \int_{t_{i-1}}^{t_i} \sigma_u \mathrm{d} W_u /\sqrt{\int_{t_{i-1}}^{t_i} \sigma_u^2 \mathrm{d} u}$. Notice that this is an exact discretization: no approximation is used. Conditional on the $\sigma$ process, $ \int_{t_{i-1}}^{t_i} \sigma_u \mathrm{d} W_u$ follows a $N(0, \int_{t_{i-1}}^{t_i} \sigma_u^2 \mathrm{d} u)$ distribution, and hence $ \varepsilon_i$'s are identically and independently distributed (i.i.d.) as $N(0,1)$. By the independence between $\sigma$ and $W$ process, we therefore have $\varepsilon_i \overset{\mathrm{i.i.d.}}{\sim} N(0,1)$, independent of $\sigma$. With these properties, the discretized model ((ref)) can be read as a linear regression model of the sequence $\{y_{t_i} - y_{t_{i-1}}\}_{i=1}^{n}$ against $\{\int_{t_{i-1}}^{t_i} y_u \mathrm{d} u\}_{i=1}^{n}$ with coefficient $\kappa$, and with innovations given by the product of an i.i.d.\ standard normal sequence $\{\varepsilon_i \}_{i=1}^n$ and the daily conditional standard deviations $\{ (\int_{t_{i-1}}^{t_i} \sigma_u^2 \mathrm{d} u )^{1/2} \}_{i=1}^{n}$. In this paper, we follow the convention to refer to the quantity $\int_{t_{i-1}}^{t_i} \sigma_u^2\mathrm{d} u$ as the integrated variance and it square root as the integrated volatility.
Denote the unknown integrated volatility over the period from $t_{i-1} $ to $t_i$ as $\omega_i := \sqrt{\int_{t_{i-1}}^{t_i} \sigma_u^2 \mathrm{d} u}$. When $\kappa = 0$, the model ((ref)) becomes
We can then construct an infeasible pseudo-sample $x_i^{\ast}$ by setting $x_0^{\ast} = 0$ and
Clearly, if $\kappa = 0$ then
the cumulative sum of i.i.d.\ standard normal variables, and hence a random walk, by construction:
with $x_0^{\ast} = 0$. Therefore, by testing the existence of a unit root in the pseudo-series $\{x_i^{\ast}\}_{i=0}^{n}$ (e.g.\ using the classical Dickey-Fuller test), one can potentially test the unit root hypothesis in the original model ($\ref{OU}$). The least squares estimation of the AR coefficient is asymptotically efficient for the random walk model.
In practice, the integrated volatility $\omega_i$ is not observable and such a strategy is therefore not feasible. However, by replacing the integrated volatility with its well-known realized volatility estimator when data at a finer grid is available (see {ABDL01} and {NS02}), a feasible way of constructing the pseudo-series can be proposed as follows.
Assume now that for each time interval $[t_{i-1}, t_i]$, we also observe $y$ over a finer time grid $t_{i-1,j} = t_{i-1}+j h$, with $j=0,1,\ldots,M$ and $h = H/M$, such that $t_{i-1,0} = t_{i-1}$ coincides with the left end-point and $t_{i-1,M} = t_{i}=t_{i,0}$ coincides with the right end-point of the interval. Thus, we have a total of $M + 1$ high-frequency observations with sampling interval $h$ (e.g., 5 minutes) in the time interval $[t_{i-1}, t_i]$. Define
the realized variance estimator over the period $[t_{i-1},t_i]$. It is known from {ABDL01} and {NS02} that in the absence of discontinuities (jumps) in the volatility path, and as $M\rightarrow\infty$, this is a consistent estimator of the integrated variance over the interval $[t_{i-1},t_i]$. Using the realized volatilities to scale the corresponding increments of the original series, we can then construct a feasible discrete-time pseudo-sample $x_i$, $i = 0, 1, \ldots, n$ as
for $i = 1, \ldots, n$ and $x_0 = 0$.
With the realized volatility instead of the integrated volatility in its construction, the feasible pseudo-series $\{x_i\}_{i=0}^n$ no longer follows an exact random walk model due to estimation errors. We now show that the feasible pseudo-series has the same asymptotic behaviour the infeasible series, in the sense that the partial sum of the feasible series converges weakly to the same Brownian motion limit as the infeasible series. For this result we will need the following assumptions:
We now give our first result, the weak limit of the partial sum of the devolatized series.
To consider the explosiveness testing problem, we extend the continuous-time OU process for the log-price in ((ref)) to allow for time-varying $\kappa$, i.e.,
where we consider a model with a one-time change from the unit root regime to the explosive regime at $t_{i^{\ast}}$:
for some integer $i^{\ast}<n$. That is, we assume that the change of regime only happens at a low-frequency time point. In principle, the change can also happen at a time point between two low-frequency time points. However, since the high frequency observations are only used in the realized volatility estimator, which is asymptotically unaffected by the change of value of $\kappa_t$ from 0 to a fixed $\kappa^{\ast}$ due to the in-fill asymptotic scheme, we make the assumption that the change of regime only happens at the observational frequency where it matters.\footnote{ An exception is when the change of parameter induces explosiveness in the integrated drift in the high observational frequency, i.e., the drift burst case; in that case, the realized volatility estimator will be affected. See COR22.} The explosiveness testing problem considered is as follows:
That is, we wish to test if a change of regime happened within the sample period. The null hypothesis is that no such change happened.
Assume we have already estimated the realized volatility within all the low-frequency intervals and have constructed a feasible pseudo-sample $\{ x_1, \ldots, x_n \}$. Define $\mathrm{RVDF} _{\tau}$ to be the “with constant" version of the Dickey-Fuller statistic calculated using the subsample $\{ x_{1}, \ldots, x_{\lfloor \tau n\rfloor} \}$, where $0\leqslant\tau\leqslant 1$. Then analogous to PWY, we can define the RVPWY statistic as
where $[\tau_0, 1]$ is a fixed time span and $\tau_0$ is a small fraction to ensure a reasonable sample size in the smallest subsample $\{ x_1, \ldots, x_{\tau_0} \}$.
It is interesting to see that the asymptotic null distribution of the RVPWY statistic under a general continuous-time process with nonparametrically specified stochastic volatility is the same as the PWY statistic under a discrete time AR model with homoskedastic errors. Since the critical values of the PWY test have been tabulated in {phillips2011explosive}, the result of Theorem (ref) implies that those critical values can be used for the RVPWY test.
In this section, we perform Monte Carlo simulations to study the size and power properties of the RVPWY test under the Heston model in finite samples, and to compare these to the classical PWY test. We also include the wild bootstrap PWY proposed in harvey2015tests in our comparison. Although harvey2015tests only demonstrate the validity of the wild bootstrapped PWY test under unconditional heteroskedasticity, recent work by BCGR21 leads us to conjecture that the wild bootstrap can also deliver asymptotically valid inference for the PWY test under stochastic volatility.
We considered the following Heston model with time-varying $\kappa_t$:
where $W_t^1$ and $W_t^2$ are two independent Brownian motions. The volatility model parameter values used are $a =0.05$, $b = 0.25$, $c = 0.30$ with a unit time interval corresponding to 1 day. Taking 252 trading days for a year, we simulate 1 year of daily observations, so we set $n = 252$. We take daily observations as the low frequency sampling interval: $H=1$. The high frequency sampling interval is taken as $h = 1 / 78$, to simulate 78 five-minute observations within each trading day (assumed to contain 6.5 trading hours). Under the null hypothesis, it is assumed that $\kappa_t = 0$ for all $t$. The number of replications is always set to 5000.
Using the finite sample critical values for $n=200, r_0=0.137$ in phillips2011explosive, we calculate the empirical rejection frequency of the RVPWY and the classical PWY tests. The empirical size is given in Table (ref) under various combinations of the volatility parameter specifications (i.e., $a,b,c$). We observe that the original PWY test is moderately to severely oversized (depending on the volatility parameters), while the RVPWY test shows good size control under stochastic volatility. The positive size distortions of the PWY test indicate that stochastic volatility could lead to wrongly identified or spurious explosiveness. Consistent with our conjecture, the wild bootstrap indeed delivers a valid size correction for the PWY test under stochastic volatility, although some very mild overrejection may occur.
Since the PWY test is oversized under stochastic volatility, its power is not directly comparable to the wild bootstrap PWY test and the RVPWY test. We therefore compare the size-corrected power of the PWY test with the power (without size correction) of the wild bootstrap PWY test and the RVPWY test. In general, we anticipate that the wild bootstrap PWY test will have similar power performance as the size-corrected PWY test. For simplicity, we report the power of the tests at 5% level in all the following tables.
Table (ref) studies the power of the tests for different values of $\kappa^{\ast}$. As expected, we see that the power of the wild bootstrap PWY test is indeed similar to that of the size-corrected PWY test. The power of all tests increases as the magnitude of $\kappa^{\ast}$ increases. The RVPWY test seems to have a clear power advantage over the PWY tests for all $\kappa^{\ast}$ values.
Table (ref) studies the effect of the location of the bubble regime on the test power. Since the bubble in our simulation design runs towards the end of the sample, an earlier starting time of the bubble means that the explosive regime lasts longer. Consistent with our intuition, the power of all tests is higher when the starting time of the bubble regime is earlier. The relative performance of the three tests is the same as discussed before.
Table (ref) studies the effect of the volatility mean-reversion parameter $a$. It seems that a larger $a$ is associated with higher testing power for all tests. Table (ref) studies the effect of the mean volatility parameter $b$. It seems that a larger $b$ is associated with higher testing power for all tests. Finally, Table (ref) studies the effect of the volatility-of-volatility parameter $c$. It seems that a higher volatility-of-volatility parameter is associated with higher testing power for all tests. In all the scenarios studied above, the relative power performance of the three test is the same as seen before: the wild bootstrap PWY test has similar power performance as the size-corrected PWY test, and the RVPWY test has uniformly higher power than both the PWY tests.
phillips2011explosive propose a strategy to date-stamp the start and the end of a bubble in real time. Analogous to phillips2011explosive, a date-stamping strategy to locate the origination and conclusion dates of the explosive period can be based on our RVDF statistic, by comparing the time series of test statistics $\text{RVDF}_{r}$, $r\in[r_0,1]$, to the right-tailed critical values of the asymptotic distribution of the standard Dickey-Fuller $t$-statistic. In particular, letting $r_e$ denote the origination date and $r_f$ the conclusion date of the explosive period, these dates can be estimated as follows:
where $\ensuremath{\operatorname{cv}}_{\beta_n}$ is the right-side critical value of DF corresponding to a significance level of $\beta_n$. As noted in phillips2011explosive, to achieve consistent estimation of the date stamps $\{\hat{r}_e,\hat{r}_f\}$, the significance level $\beta_n$ needs to approach zero asymptotically, and correspondingly $\ensuremath{\operatorname{cv}}_{\beta_n}$ must diverge to infinity in order to eliminate type I errors as $n\rightarrow\infty$. In practical implementations, it is conventional to set the significance level in the 1--5% range.
To study the statistical properties of our date-stamping procedure in the possible existence of explosiveness, we consider a continuous-time OU model similar to that considered in phillips2011explosive, where the time-varying mean-reversion parameter satisfies
with $i_1 := \lfloor \tau_1 n \rfloor$ and $i_2 := \lfloor \tau_2 n \rfloor$, and where $0 < \tau_1 < \tau_2 < 1$ are the origination time and the conclusion time of the bubble regime in the normalized time scale; and $c > 0$ and $0 < \alpha < 1$ are constants. With this specification of the time-varying parameter, the continuous-time process starts with a unit root regime, then changes to be mildly explosive at time $t_{i_1}$, and reverts back to an unit root regime at time $t_{i_2}$. As in phillips2011explosive, we assume that the process is reinitialized at time $t_{i_2}$ to a level close to the start of the explosive regime, i.e., $y_{t_{i_2}} = y_{t_{i_1}} + y^{\ast}$, with $y^{\ast}=O_p(1)$. Therefore, this is an instant crash model where the bubble collapses completely. In this model, we show that our date stamping procedure is consistent for the origination and collapse dates over the normalized time scale, as in phillips2011explosive.
When intra-day asset price data is available, it offers the opportunity to derive highly accurate daily volatility measures. In this empirical application, we demonstrate the use of daily realized volatility within our methodology to detect explosive behavior in cryptocurrencies. Specifically, we focus on two cryptocurrencies, Bitcoin (BTC) and Ethereum (ETH), during the period from January 1, 2018, to August 31, 2021. We employ the RVPWY test and the RVDF detector to test and date-stamp potentially explosive behavior in the cryptocurrencies. To facilitate comparison, we also include the results obtained from the PWY test and its associated DF detector. Our analysis involves computing daily realized volatility by utilizing 5-minute log-returns over each 24-hour trading day. Subsequently, we apply the tests for explosive behavior to the log-price series of the cryptocurrencies.\footnote{Note that unlike stock price data, cryptocurrencies are traded 24 hours a day and 7 days a week. Moreover, since the data that we have are daily log-returns, we calculate cumulative sum of the log-returns to recover the log-price series. Since the PWY test uses a “with constant” version of the DF test, it therefore makes no difference whether the original log-price or the cumulative sum is used.} We use $r_0=0.1$ in all our calculations.
In Figure (ref), we plot the log-price series of the two cyptocurrencies. To be transparent about the devolatization using realized volatility in our proposed methodology, we also plot the pseudo log-price series for comparison purpose for the two currencies. It can be seen that for the BTC series, the decrease in price at the beginning of the original sample becomes milder in magnitude, while the increase in the latter sample observed in the original sample is largely preserved. The plots of the ETH series and its reconstructed sample show a similar pattern. The difference is due to the higher volatility during the decreasing phase of the price. The PWY and PSY statistics Phillips2013, Phillips2013a are known to flag fast downturns as explosive regimes phillipsshi2020,wangyu2023. Our intuition is that some of the fast downturns may not be flagged as explosiveness after adjusting for volatility.
Table (ref) gives the values of the test statistics. The PWY test does not identify explosive behavior in both prices, while the RVPWY test finds explosive behavior in the ETH series at the 10% level. Although the RVPWY test does not find explosive behavior in the BTC series, the test statistic is very close to the 10% critical value.
We also plot the evolution of the RVDF detector and the DF detector in Figure (ref), together with the critical values of the detector, which is -0.08, the 5% critical value of the DF distribution. This helps us better understanding the difference in the values of the RVDF and RVPWY statistics and the corresponding DF and PWY statistics. For the BTC series, the price decrease in the beginning the sample becomes less obvious in the reconstructed sample, such that the price increase in the latter sample becomes more pronounced. This explains why the RVDF detector crosses the $-0.08$ horizontal line after 2021, but the DF detector always stays below the line. However, the maximum of the RVDF detector (1.069 as in Table (ref))) is still lower than the 10% critical value of the RVPWY test, such that we cannot conclude that explosive behavior exists in this series. For the ETH series, the DF detector identifies the downturn of the price as explosive behaviour, but fails to find explosive behavior during the time when the ETH price increases quickly from 2020. With the RVDF detector, we find results more in line with our intuition of the data: the downturn part shows less evidence of explosiveness, but the explosive behavior after 2021 can be detected now. The maximum of the RVDF detector (1.282 as in Table (ref)) is large enough to guarantee a rejection of the RVPWY test at 10% level now.
For the specific dates of the explosive regime, since the PWY test does not find any explosive behavior in both series and the RVPWY test only finds explosive behavior in the ETH series, we only report the starting date March 5, 2021 of the explosive behavior found by the RVDF detector. There are also a number of explosive regimes identified before March 5, 2021. However, the longest explosive regime found by the RVDF detector during the downturn of ETH price in 2018 was from October 12, 2018 to November 20, 2018. Since these durations are shorter than the smallest window we choose to calculate all the statistics, we do not report them.
The purpose of this empirical application is to illustrate how daily level stock index data can be exploited to estimate a monthly volatility measure, which can then be used in our methodology to help testing and date-stamping explosive behavior at the monthly frequency.
The dot-com bubble was a historic economic bubble and period of excessive speculation that occurred in the 1990s, a period of extreme growth in the usage and adaptation of the Internet by businesses and consumers. phillips2011explosive studied the monthly real Nasdaq index, and Phillips2013 studied the price-dividend ratio of S&P500 during the dot-com bubble, and both find explosive behavior. In this empirical application, we study the potential explosive behavior during this period in the nominal and real S&P500 indices using the proposed methodology, by exploiting the volatility information in the daily S&P500 price data, which is widely available but not used in the previous analysis. To study the explosive behavior in the nominal S&P500 index, our methodology can be applied directly. For the real S&P500 index, our realized volatility estimator cannot be applied because the Consumer Price Index (CPI) is not available at the daily level. We will first discuss an empirically meaningful approximation strategy for the monthly volatility of the real S&P500 index. Given the estimated volatility, we apply the testing and date-stamping methodology developed in this paper to the data, and compare the outcomes to those obtained by the PWY method. The sample period we use is January 1990 to June 2005, the same as in phillips2011explosive. The daily level S&P500 index data has been downloaded from Yahoo Finance. The monthly US CPI data has been downloaded from the US Federal Reserve website. Since we are using daily S&P500 returns to estimate monthly volatility, it is necessary to account for the mean estimate, therefore the realized variance estimator we use in this example is based on demeaned daily S&P500 returns.
We consider testing the explosiveness in both the nominal and the real S&P500 index at the monthly frequency. Using the daily nominal S&P500 index data, it is straightforward to estimate the monthly volatility in the nominal S&P500 series. However, for the monthly volatility of the real S&P500 index, the daily real S&P500 index is not available due to the unavailability of the daily CPI data. Here we introduce an approximation strategy, where the monthly volatility of the real S&P500 series can be approximated by the monthly volatility of the nominal S&P500.
Denote the nominal S&P500 index as $S_t$, and the corresponding CPI by $P_t$. Then the monthly real S&P500 index $X_t$ is defined as $X_t=S_t/P_t$, so the monthly real S&P500 log-return is $\Delta \log X_t = \Delta \log S_t - \Delta \log P_t$. Its variance is \[ \mathrm{Var}(\Delta X_t)= \mathrm{Var}(\Delta \log S_t) + \mathrm{Var}(\Delta \log P_t) - 2\mathrm{Cov}(\Delta \log S_t,\Delta \log P_t). \] Looking at the empirical data at the monthly frequency, we find that the CPI, which represents the price level for consumer goods and services, has much smaller variability than the the stock price index. This can be seen in the variance-covariance matrix of monthly nominal S&P500 returns and monthly CPI inflation, given in Table (ref). From the table, we can see that at the monthly level, the CPI variance and the CPI-S&P500 covariance are about a factor 500 smaller than the S&P500 variance. This empirical observation motivates the use of the approximation $\mathrm{Var}(\Delta \log X_t )\approx \mathrm{Var}(\Delta \log S_t)$.
Using this approximation, the variance of the real index can be estimated by the realized variance using the daily nominal stock index data. Indeed, although CPI does have a nontrivial effect on the level of real stock index, its effect on the variance is negligible. Since the daily stock index data are widely available from 1970s, these data can be used to improve our understanding of the volatility structure also in the real index.
Applying the realized variance estimator, we estimate the monthly volatility of S&P500 from January 1990 to June 2005. The estimated realized volatility path, together with the plot of the nominal and the real S&P500 indices, is given in Figure (ref). From the volatility estimate, the volatility seems in general higher during the bubble build-up period and obviously higher when the price drops.
We then apply the testing and date-stamping strategies developed in this paper to both the nominal and the real S&P500 series, and compare the result with those obtained by the PWY method. The results are summarized in Table (ref). First, it seems that the RVPWY test gives slightly higher statistic values than the PWY test for both series, such that the RVPWY test rejects the null hypothesis at the 1% level in both the nominal and the real S&P500 series; while the PWY test also rejects the null hypothesis, but only at the 5% level in both series.
The results for date-stamping are also given in Figure (ref). In the nominal series, the RVDF detector finds the explosive behavior from February 1995, 5 monthly earlier than the DF detector by PWY. In the real S&P500 series, the RVDF detector finds the explosive behavior from March 1995, which is 8 months earlier than PWY. For the conlusion time of the explosive regime, the nominal S&P500 index reaches its maximum in September 2000, and the corresponding real index peaked in April of the same year. In hindsight, it is reasonable to take the month when the series reached its maximum (i.e. when it started to crash) as the end of the bubble regime and hence, in the nominal series, the RVDF detector realizes the end of the bubble one month earlier than the DF detector; while in the real series, the DF detector finds the end of bubble one month earlier than the RVDF detector. Overall, it appears that the RVDF detector is more sensitive in detecting the start of a bubble; while the performance of the two detectors is similar in identifying the end of a bubble.
In a new continuous-time framework, by incorporating the volatility information in data sampled at a higher frequency, we develop a powerful yet easy-to-implement test for the explosive behavior in data sampled at a lower frequency. The presence of stochastic volatility makes OLS estimation of the persistence parameter inefficient, and existing asymptotic null distributions of test statistics (derived under homoskedasticity) invalid. To achieve valid inference for the explosive behavior, we propose to devolatize the price series sampled at the lower frequency using realized volatility from data sampled at the higher frequency. We conduct simulation studies to compare the performance of our method relative to some existing methods. Simulation results suggest that our test has well-controlled size and using high-frequency volatility measures to devolatize data also brings efficiency improvement, in the sense that our test has markedly improved power performance than the existing methods. Empirical applications using the cryptocurrencies and the S&P500 prices allow us to find some interesting empirical results.