SIGN-BASED UNIT ROOT TESTS FOR EXPLOSIVE FINANCIAL BUBBLES IN THE PRESENCE OF DETERMINISTICALLY TIME-VARYING VOLATILITY

This article considers the problem of testing for an explosive bubble in financial data in the presence of time-varying volatility. We propose a sign-based variant of the Phillips, Shi, and Yu (2015, International Economic Review 56, 1043–1077) test. Unlike the original test, the sign-based test does not require bootstrap-type methods to control size in the presence of time-varying volatility. Under a locally explosive alternative, the sign-based test delivers higher power than the original test for many time-varying volatility and bubble specifications. However, since the original test can still outperform the sign-based one for some specifications, we also propose a union of rejections procedure that combines the original and sign-based tests, employing a wild bootstrap to control size. This is shown to capture most of the power available from the better performing of the two tests. We also show how a sign-based statistic can be used to date the bubble start and end points. An empirical illustration using Bitcoin price data is provided.


INTRODUCTION
Empirical identification of explosive behavior in financial asset price series is closely related to the study of rational bubbles, with a rational bubble deemed to have occurred if explosive characteristics are manifest in the time path of prices, but not for the dividends. Consequently, methods for testing for explosive time series behavior have been a focus of much recent research. Phillips et al. (2015) [PSY] model potential bubble behavior using a time-varying autoregressive specification, which allows an explosive autoregressive regime over a subset of the data, and suggest testing for such a property using a double supremum of forward and backward recursive right-tailed Dickey-Fuller (DF) unit root tests, a generalization of the original and widely used Phillips, Wu, and Yu (2011) [PWY] test that employed a single supremum of forward-only recursive DF tests.
These articles assume constant unconditional volatility in the underlying error process, yet, in practice, time-varying volatility is a well-known stylized fact observed in empirical financial data (see, for example, Rapach, Strauss, and Wohar, 2008). Harvey, Leybourne, Sollis, and Taylor (2016) [HLST] demonstrate that the asymptotic null distribution of the PWY test depends on the nature of the volatility, so if the test is compared to critical values derived under a homoskedastic error assumption, its size is not controlled under time-varying volatility. This lack of size control typically leads to serious over-sizing, and consequently frequent spurious identification of a bubble. HLST propose a wild bootstrap method to provide critical values for the PWY test, which delivers correct asymptotic size in the presence of time-varying volatility (while retaining the same local asymptotic power as the original PWY test, were it infeasibly size-corrected to account for the time-varying volatility). An entirely similar bootstrap approach can be applied to the PSY test. 1 In this article, we suggest a new approach to obtaining heteroskedasticityrobust inference in the presence of a bubble. Instead of calculating the PSY statistic (denoted P SY ) directly from the observed data series, y t say, we calculate it from the series of cumulated signs of the first differences of the data, i.e., a cumulation of sign( y t ) = y t / | y t | (for nonzero y t ), which is clearly invariant to the variance of y t (assuming a zero mean for y t ). As a direct consequence of this, the sign-based PSY statistic (denoted s P SY ) is then exact invariant to the pattern of time-varying volatility and therefore, unlike P SY , requires no wild bootstrap procedure to control size. Sign-based approaches to testing for unit roots against stationary autoregressive models have been considered by, inter alios, Campbell and Dufour (1995) and So and Shin (2001), although these are not based on the cumulations of sign( y t ). Unit root testing using cumulated standardized differences is also considered by Beare (2018) (in a context of full sample testing against a stationary alternative), but our method is quite distinct in that we standardize by | y t |, rather than using a nonparametric estimator of the spot standard deviation, resulting in the sign of y t .
We derive the asymptotic distribution of the sign-based test under the unit root null and alternative of a local to unit root explosive regime. Here, we derive a stochastic expansion of the partial sum process (PSP) of signed first differences, allowing for time-varying volatility and a time-varying autoregressive coefficient, thereby extending the results of Boldin (2013) for a homoskedastic constant coefficient model. We then use this result to establish the asymptotic properties of our test statistics.
Using a number of different specifications for the bubble process and pattern of time-varying volatility, we show that the local asymptotic power of s P SY compares very well with that of P SY and it is, rather more often than not, the more powerful of the two test procedures, sometimes by a significant margin. Although s P SY has a good deal of merit as a stand alone test, because for some bubble process and volatility pattern settings the power of P SY is higher than that of s P SY , we then proceed to consider a union of rejections approach (cf. Harvey, Leybourne, and Taylor, 2009), whereby the null is rejected in favour of explosive behavior if either P SY or s P SY rejects. We find that the union of rejections testing strategy performs very well across the full range of volatility and bubble specifications that we consider, capturing much of the power available from either test. In common with P SY , a feasible variant of the union test does require a (joint) wild bootstrap to ensure asymptotic size control. In the article, we refer mainly to the test of PSY and its sign-based counterpart, but we simultaneously consider variants appropriate for the original test of PWY.
We then move on to consider how a modified variant of our sign-based statistic can be used to date the start and end of a bubble. We propose a new dating strategy based on maximizing a dating statistic. Under a mildly explosive assumption for the bubble magnitude, we show that our proposed dating strategy is consistent for estimating the start and end of the bubble. This, of course, is a relevant property to establish from the viewpoint of an applied researcher who is interested in characterizing the timeline of a historical bubble episode in relation to, say, economic or financial events that are known to have occurred.
The rest of the article is organized as follows. Section 2 outlines the heteroskedastic bubble model, describes the PSY testing approach, and introduces our sign-based version of the PSY test. Here, we also establish the limit distributions of these tests under local bubble alternatives. Asymptotic size (where relevant) and local powers are compared in Section 3. The union of rejections procedure and the associated wild bootstrap method are outlined in Section 4. Finite sample properties of the tests are explored in Section 5. Our sign-based dating methodology is described and its consistency properties shown in Section 6. A generalization of our sign-based test to account for possible asymmetry of the innovation distribution is given in Section 7. Section 8 briefly discusses extensions to the basic model. An empirical illustration of our new testing and dating procedures, using Bitcoin price data, is provided in Section 9, with Section 10 concluding the article. Proofs of our asymptotic results are provided in an appendix. We use the following notation: 1(.) denotes the indicator function; · the integer part; ⇒ weak convergence; p ⇒ weak convergence in probability, and p → convergence in probability. D = D[0, 1] denotes the space of right continuous with left limit (càdlàg) processes on [0, 1]. Finally, 'x := y' ('x =: y') indicates that x (y) is defined by y (x). In this article, we study two types of models for the explosive behavior in data, and we use the following terminology: (i) locally explosive refers to the alternative where the autoregressive root is 1 + cT −1 , with c a positive constant and T the sample size; (ii) mildly explosive refers to the alternative specified in Phillips and Magdalinos (2007), where the root is 1 + cT −α with α ∈ (0, 1). Formally, our sign function is defined as sign(x) = −21(x 0) + 1.
For the majority of our analysis, under H 1 we will consider locally explosive alternatives (and collapses) of the form δ i.T = c i T −1 , c i > 0, i = 1, 2; the scaling by T −1 providing the appropriate Pitman drift for asymptotic power comparisons of the tests. In Section 6 below, when we consider dating the start and end points of the bubble, a slightly stronger, mildly explosive, bubble magnitude will be assumed for δ 1,T . For the innovation process ε t , we make the following assumptions: The volatility term σ t satisfies σ t = σ (t/T ), where σ (·) ∈ D is nonstochastic and strictly positive.
Under Assumption A2, the innovation variance is nonstochastic, bounded, and displays a countable number of jumps. It also allows for variance processes displaying (possibly) multiple one-time volatility shifts (which need not be located at the same point in the sample as the putative regimes associated with bubble behavior), polynomially (possibly piecewise) trending volatility and smooth transition variance breaks, among other things. The conventional homoskedasticity assumption, that σ t = σ for all t, is also permitted, since here σ (s) = σ for all s. Assumption A3 ensures F(z) is continuously differentiable in a small neighbourhood around zero, and that the density f (z) exists, is strictly positive, and is bounded from above. Assumption A4 implies that E(sign(z t )) = 0, which is necessary for the invariance principle of the partial sum of the signs to hold. Assumption A4 also implies the median of z t is zero, in addition to the zero mean assumption from A1; the imposed distributional assumption on z t is only slightly weaker than assuming the distribution of z t is symmetric about zero. Note that monthly financial returns, which are often used in a bubble testing context, are usually found to be symmetric about zero; see, for example, Tsay (2010 , Table  1.2), Christoffersen (2012, Sect. 2). In Section 7 below, we will consider relaxing this symmetry assumption.
Under Assumptions A1-A2, the following invariance principle holds for the PSP of ε t : Under Assumptions A1-A2 and A4, we have the following invariance principle for the PSP of sign(ε t ) := −21(ε t ≤ 0) + 1: Here, W (r ) and W s (r ) are standard Brownian motion processes, with the correlation coefficient being the constant −2E{1(z t ≤ 0)z t }. Notice that W σ (r ) is a stochastic integral dependent on the volatility function σ (s). Also note that sign(ε t ) is exact invariant to σ t .

The PSY Test
The PSY statistic is used to test H 0 against H 1 , the alternative being that y t behaves as an explosive AR(1) process for at least some subperiod of the sample.
In this context, and in the absence of knowledge concerning the timing of any potential explosive behavior, PSY propose a test based on the double-supremum of recursive right-tailed DF tests. Specifically, the statistic is given by where DF(λ 1 ,λ 2 ) denotes the standard DF test, that is the t-ratio forφ(λ 1 ,λ 2 ) in the fitted ordinary least squares (OLS) regression calculated over the subsample period t = λ 1 T ,..., λ 2 T . That is The P SY statistic is therefore the supremum of a double sequence of statistics with minimum sample length π T . The singlesupremum statistic of PWY arises as a special case of the P SY statistic: PW Y := sup λ 2 ∈[π,1] DF(0,λ 2 ).
We now state the large sample behavior of P SY under a locally explosive H 1 for DGP 4. Its behavior under DGPs 1-3, and under H 0 , arise as special cases.
THEOREM 1. For model (1), under H 1 with δ i.T = c i T −1 , c i > 0, i = 1, 2 and Assumptions A1-A2, Corresponding limiting distributions under DGP 1, DGP 2, or DGP 3 are obtained by imposing the relevant restrictions on τ 2 and τ 3 . The limit distribution of P SY under the null hypothesis H 0 is given by M M 0,0 (or, equivalently, on setting τ 1 = 1 such that U (r ) = W σ (r )). The limit of the PW Y test is sup λ 2 ∈[π,1] L c 1 ,c 2 (0,λ 2 ) =: M c 1 ,c 2 , with distribution M 0,0 under H 0 . The limits of both P SY and PW Y are dependent on the (limit) form of heteroskedasticity σ (s) under the null and alternative hypotheses.

The Sign-Based PSY Test
Let C t be the cumulated sum of signs C t := t i=2 sign( y t ), t = 2,... , T . The sign-based analogue of (3) is then given by where s DF(λ 1 ,λ 2 ) denotes the t-ratio forρ(λ 1 ,λ 2 ) in the fitted (without intercept) OLS regression C t =ρ(λ 1 ,λ 2 )C t −1 + e t calculated over the period t = λ 1 T ,... , λ 2 T , i.e., The sign-based analogue of the PW Y test arises as a special case of the s P SY test: s PW Y := sup λ 2 ∈[π,1] s DF(0,λ 2 ). Under the null hypothesis, since sign( y t ) = sign(z t ), these tests are exact invariant to the pattern of heteroskedasticity σ t .
For DGP 4, the next Theorem gives the large sample behavior of s P SY under a locally explosive H 1 . and where V 1 (r ) and V 2 (r ) are as defined in Theorem 1.
Once more, the corresponding limiting distributions under DGP 1, DGP 2, or DGP 3 obtain by imposing the relevant restrictions on τ 2 and τ 3 , with the limit distributions of s P SY under the null hypothesis H 0 being given by M M s 0,0 (or on setting τ 1 = 1 so that U s (r ) = W s (r )). The limit of the PW Y sign test, s PW Y , is given by sup λ 2 ∈[π,1] L s c 1 ,c 2 (0,λ 2 ) =: M s c 1 ,c 2 , with distribution M s 0,0 under H 0 . Note that M M s 0,0 and M s 0,0 are invariant to σ (s), while under the alternative hypothesis, the limits of s P SY and s PW Y depend on the pattern of heteroskedasticity σ (s), and also the density of z t via the appearance of f (0).
For π = 0.1, limit null critical values for s P SY and s PW Y , for the standard significance levels, are given in Table 1 under "T = ∞". These are computed using direct simulation of the limiting functionals of Theorem 2, using 2,000 Monte Carlo replications, and approximating the Brownian motion process involved using N I I D(0, 1) random variates, with the integrals approximated by normalized sums of 1,000 steps. Also shown in Table 1 are finite sample critical values for s P SY and s PW Y based on generating ε t as N I I D(0, 1) (with u 1 = ε 1 ) for T = 100, 200, and 400. It is clear that convergence of the finite sample critical values to their asymptotic counterparts is fairly slow (particularly for s P SY ), but this is not uncommon for extremum statistics based on subsamples.
Remark 1. By construction, both the original P SY and PW Y statistics, and the sign-based variants s P SY and s PW Y , are numerically invariant to the nuisance parameter μ in the DGP (1). For P SY and PW Y , this follows due to the inclusion of an intercept term in the Dickey-Fuller regressions (3), while for s P SY and s PW Y the statistics only make use of C t , which, being based on the (sign of) y t , does not depend on μ. Hence the finite sample and limit distributions of these statistics, and consequently their finite sample and asymptotic sizes and local powers, do not depend on μ. One could also envisage tests of the form of P SY and PW Y but based on Dickey-Fuller regressions that exclude an intercept term. Such tests would have finite sample distributions that depend on the nuisance parameter μ under both the null and alternative, while it can easily be shown that their asymptotic null and local alternative distributions would be invariant to μ provided μ = o(T 1/2 ).

ASYMPTOTIC SIZE AND POWER OF THE TESTS
We now consider the asymptotic size and power of the P SY and PW Y tests, and asymptotic powers of s P SY and s PW Y tests. The sizes and powers are computed via direct simulation of the limiting functionals in Theorems 1 and 2, again using 2,000 Monte Carlo replications.

Size
Sizes for P SY and PW Y are examined in the case of volatility shifts of the form We simulate the asymptotic sizes of upper-tail nominal 0.05-level tests, and use the limit null critical value which would be obtained under homoskedasticity, i.e., from the distributions M M 0,0 and M 0,0 evaluated assuming σ (s) = 1, which is akin to ignoring any possibility of heteroskedasticity. We consider the range of values σ 1 ∈ {1, 1/6, 1/3, 3, 6}. The results are given in Table 2. We do not show size results for s P SY and s PW Y as they are always correctly sized asymptotically. Panel (a) of Table 2 sets τ σ 1 ∈ {0.4, 0.8} and τ σ 2 = 1. This represents a single volatility shift at time fraction τ σ 1 , which might be thought of as being akin to DGP 1 with the bubble episode being replaced by a heteroskedastic one. It is evident that P SY and PW Y are both badly oversized when σ 1 > 1, this oversize  being particularly serious for σ 1 = 6. Comparing P SY and PW Y , we see that the length of the heteroskedastic episode, as measured by τ σ 1 , actually has little effect on the degree of oversize present in P SY , while for PW Y we see a modest decrease in size with increasing τ σ 1 . In Panel (b), we set τ σ 1 ∈ {0.1, 0.5} and τ σ 2 = 0.7. Here, there is a change in volatility between time fractions τ σ 1 and τ σ 2 , which is now akin to DGP 2 with the bubble episode being replaced by a heteroskedastic one. Here, we see that P SY is badly oversized for both σ 1 < 1 and σ 1 > 1, and for all τ σ 1 . While PW Y is similarly oversized for σ 1 > 1, for σ 1 < 1 (modest) oversize is only evident when τ σ 1 = 0.1. This represents something of a departure in behavior between the two tests, indicating that the size of P SY is more sensitive to the presence of heteroskedasticity.

Power
We now examine the asymptotic power of the tests under a locally explosive H 1 , for both a benchmark case of homoskedasticity, and also in the presence of heteroskedasticity. We do this in the context of DGP 1, DGP 2, and a representative DGP involving a collapse regime (specifically, DGP 4), noting that we find the specification of the collapse regime to have relatively little bearing on the powers of the tests. We simulate the asymptotic powers of upper-tail nominal 0.05-level tests. For P SY and PW Y , we infeasibly size-correct when a particular pattern of heteroskedasticity is present by taking critical values from the σ (s)-dependent M M 0,0 and M 0,0 limit distributions. For s P SY and s PW Y , the critical values are the limit ones from Table 1. To evaluate the powers of these tests, we (implicitly) assume that z t ∼ N I I D(0, 1) and correspondingly set f (0) = 1/ √ 2π = 0.399. The model parameter settings we consider are as follows: (unit root, then bubble, then unit root to sample end)

then bubble, then collapse, then unit root to sample end).
We set c 1 ∈ {2, 4, 6, 8} and σ 1 ∈ {1, 1/6, 1/3, 3, 6}; σ 1 = 1 representing the benchmark homoskedastic case. In each DGP, the heteroskedastic episode is made coincident with the bubble (or bubble and collapse) regime(s), which seems a reasonable restriction to impose and it limits the number of cases to consider. The results are given in Table 3. As there are still a large number of table entries, as a simple gauge of the broad relative power performance of P SY compared with s P SY , and PW Y with s PW Y , entries where the power of one test exceeds that of the other by at least 0.04 are underlined. The results for DGP 1 appear in Table 3(a). Considering τ 1 = 0.4, it is fairly evident that, outside of the homoskedastic case, P SY is generally less powerful than s P SY , and PW Y is less powerful than s PW Y . It is also evident that PW Y can perform very poorly compared to s PW Y when σ 1 < 1. When τ 1 = 0.8, P SY is dominated by s P SY for σ 1 < 1 but now shows some gains when σ 1 > 1. PW Y is generally now more powerful than s PW Y unless σ 1 < 1, where s PW Y can offer substantial gains. In Table 3(b), we give the results for DGP 2. For τ 1 = 0.1, P SY is less powerful than s P SY and it is noticeable that P SY can have much lower power for σ 1 < 1 -something which was not observed under DGP 1. The same is also true when we compare PW Y with s PW Y . For τ 1 = 0.5, P SY remains inferior to s P SY when σ 1 < 1 while PW Y is now better performing than s PW Y unless σ 1 < 1, where the ranking is reversed. Under DGP 4 in Table 3(c), the results are throughout very similar to those found for DGP 2 in Table 3(b), suggesting that the addition of the collapse period, in itself, has very little effect on the power of these suprema-based tests. As such, similar comments apply here as made under DGP 2. Under heteroskedasticity, then, our results clearly demon-  strate that s P SY and s PW Y can, in terms of asymptotic power, be considered as very worthy competitors to their standard counterparts. Particularly, but by no means exclusively, they have better power properties for downward volatility shifts; a case which proves to be a distinct weakness for PW Y throughout. Tables 3(a)-3(c) also report (infeasibly size-adjusted) local asymptotic power results for variants of P SY and PW Y that exclude an intercept in the  Dickey-Fuller regressions (cf. Remark 1), which we denote by P SY 0 and PW Y 0 . These results are obtained under the assumption that μ = o(T 1/2 ), in which case the limit distributions for P SY 0 and PW Y 0 are invariant to μ and take the same form as those for P SY and PW Y , respectively, as given in Section 2.1, but with U (r ) replaced by U (r ). Comparing P SY 0 and PW Y 0 with P SY and PW Y , it is clear that, as would be expected, exclusion of the intercept term results in su-  perior local asymptotic power when μ = o(T 1/2 ) is satisfied (apart from a few minor exceptions for PW Y ). What is noteworthy is that in cases where s P SY and s PW Y display power gains over their P SY and PW Y counterparts, s P SY and s PW Y can often also achieve power gains over P SY 0 and PW Y 0 . In what follows, we retain our main emphasis on the original P SY and PW Y tests rather than the P SY 0 and PW Y 0 variants, due to the potential for the latter to have finite sample behavior influenced by the unknown nuisance parameter μ, an issue we revisit in the finite sample results of Section 5. One interesting observation from the local asymptotic power results is that under homoskedasticity, P SY and PW Y are in general more powerful than s P SY and s PW Y , respectively, for large c 1 , while the sign-based tests are in general more powerful for smaller values of c 1 . The latter finding may appear surprising since the sign-based tests are motivated by heteroskedasticity considerations, but there is no theoretical reason why these procedures cannot perform better than the original P SY and PW Y tests, since there are no optimality claims associated with P SY and PW Y in terms of power under homoskedasticity. What is also interesting is that (on the basis of these limit simulations at least), s P SY is always found to be more powerful than s PW Y , under homoskedasticity and heteroskedasticity, while there is no such unambiguity present between P SY and PW Y .
For s P SY and s PW Y , as a robustness check, we also evaluated powers of these tests under an (implicit) assumption that z t ∼ t (5) (a fat tailed distribution) setting Since this value of f (0) is little different to 0.399, the powers change very little, but are always slightly smaller than for z t ∼ N I I D(0, 1) because the offset terms in (6)  Of the four statistics, arguably then, s P SY seems to emerge as the one with the best overall performance, followed by P SY . However, since there is no unique ranking between these two tests, we can consider a simple method which attempts to harness the better power of each for a given DGP setting, via a union of rejections strategy, which we detail in the next section.

A UNION OF REJECTIONS STRATEGY
Our approach is fundamentally based around that of Harvey et al. (2009), who consider the problem of testing for a unit root in the presence of uncertainty surrounding whether or not a linear trend is present in the deterministic component by combining tests which do and do not allow for trends, rejecting the unit root null if either test rejects. In the current context, we consider a combination of s P SY and P SY , although the same method is directly applicable to a combination of s PW Y and PW Y . Specifically, denoting the asymptotic ξ level null critical value of s P SY by cv s ξ (from the σ (s)-invariant M M s 0,0 distribution) and that of P SY by cv ξ (from the σ (s)-dependent M M 0,0 distribution) a union of rejections strategy can be written as the decision rule An application of the continuous mapping theorem (CMT) along with the results in Theorems 1 and 2 yields the asymptotic distribution of u P SY as Note that this union of rejections strategy as it stands is doubly infeasible as the u P SY statistic itself uses the σ (s)-dependent cv ξ , and also the critical value cv u ξ is σ (s)-dependent via ψ ξ . The scaling constant ψ ξ can be determined from the limit distribution of u P SY with c 1 = c 2 = 0, but there is actually no need to calculate it explicitly since, for a given value of cv s ξ /cv ξ , all we actually require is the critical value cv u ξ which is obtained directly from the null limit distribution of u P SY .
Infeasibly size-corrected limit powers for u P SY and its s PW Y /PW Y -based counterpart, denoted u PW Y , are also shown in Tables 3(a)-3(c). The immediate feature we observe for the union procedures is that, throughout, their power levels are always really quite close to the higher of the two constituent tests. This is never something that can be guaranteed in general with such union-based procedures, due to the implicit scaling constant ψ ξ essentially having the effect of always inflating the critical values applied to each constituent test. Here, however, the impact of this scaling appears to be really rather modest, thereby rendering the union a rather effective tool in this particular instance.
Thus far only s P SY and s PW Y represent properly feasible test procedures as they are asymptotically size controlled without requiring knowledge of σ t . For P SY and PW Y, and u P SY and u PW Y , asymptotic size control can be obtained by employing a wild bootstrap scheme to construct the relevant critical values. This is shown to be valid in the context of the PW Y test in HLST, and we now outline how this applies to P SY and u P SY .
The wild bootstrap algorithm is: 1. Generate a wild bootstrap sample {y b t } T t =1 by setting where the w t are N I I D(0, 1) variates. 2. Use the wild bootstrap sample to compute the pair of statistics P SY and s P SY .
3. Repeat step 1 and step 2 M times, denoting the resulting pairs of statistics by Note that under H 0 , since sign( The next Theorem details the joint asymptotic distribution of P SY b m and s P SY b m under a locally explosive H 1 . The marginal convergence result regarding P SY b m follows directly from HLST. Noting that sign( y b t ) = sign(z t w t ) + o p (1), the proof of the marginal convergence result for s P SY b m follows the same strategy as HLST and the proof of Theorem 2 of this article. The joint convergence occurs because both statistics are calculated from the same bootstrap sample (this result is needed below for the asymptotic validity of the union of rejections strategy). The proof of Theorem 3 is therefore straightforward and omitted for the sake of brevity. The Theorem demonstrates that the wild bootstrap procedure is first order valid in approximating the asymptotic joint null distribution of the P SY and s P SY statistics under a locally explosive H 1 (which includes H 0 as a special case).
The ξ level bootstrap critical values are obtained from the empirical distribution functions of P SY b m and s P SY b m calculated from M bootstrap replications. Denoting these critical values as cv b ξ and cv b,s ξ , a rejection of H 0 for P SY is obtained if P SY > cv b ξ and a rejection of H 0 for s P SY is obtained if s P SY > cv b,s ξ . As T, N → ∞, it follows that cv b ξ and cv b,s ξ converge in probability to cv ξ and cv s ξ , so these individual bootstrap procedures are correctly sized in the limit. Consequently, P SY inherits exactly the same asymptotic local power properties under H 1 as its infeasibly size-corrected counterpart of Section 3.2 (this is trivially true of s P SY as cv s ξ does not depend on σ t ). The wild bootstrap counterpart of the union statistic u P SY is given by The ξ level bootstrap critical value for the union is obtained from the empirical distribution function of u P SY b m , and denoting this critical value as cv b,u ξ we reject Here u P SY b is a feasible variant of u P SY that replaces cv s ξ /cv ξ with cv b,s ξ /cv b ξ . Note that this approach does not require knowledge of the scaling constant ψ ξ , as the size control is obtained implicitly using the bootstrap critical values. As T, N → ∞, u P SY b is correctly sized in the limit under H 0 because cv b,u ξ converges in probability to cv u ξ , and it has the same limiting local power function as u P SY under H 1 . An entirely analogous wild bootstrap approach can be implemented for PW Y (as in HLST), s PW Y and u PW Y .

FINITE SAMPLE SIZE AND POWER OF THE TESTS
We now turn to an examination of the finite sample properties of the various wild bootstrap procedures. Our simulations are based on the model (1) with T = 100. Here, we set μ = 0 and u 1 = ε 1 , where ε t = σ t z t with the z t generated as N I I D(0, 1) random variates. Table 4 shows 0.05-level finite sample sizes; Tables 5(a)-5(b) report powers for the constellations of parameter settings used in our previous asymptotic size and power simulations given in Tables 2 and 3(a)-3(b). For brevity, we omit results pertaining to DGP 4. Here, the limit volatility functions σ (s) are discretised to σ t (t/T ) in an obvious way. Once again 2,000 Monte Carlo replications are used and we employ M = 499 bootstrap replications.
As regards finite sample size accuracy, there is a definite tendency for (bootstrap) P SY to be undersized, with size often dropping below 0.03 (and occasionally below 0.01, including in the homoskedastic case). In comparison, the size of (bootstrap) s P SY (whose size is invariant to σ 1 ) is reasonably accurate at 0.059. Interestingly, the undersize of P SY does not translate into substantial undersize of the (bootstrap) union u P SY ; its size is never below 0.04, so s P SY is clearly having an offsetting effect within the combination. PW Y also has some tendency to undersize (unless σ 1 = 6 when it can be modestly oversized), but to a lesser degree than P SY . The size of s PW Y is very accurate at 0.054 and u PW Y offers better size control than PW Y .
Considering finite sample power, Table 5(a) gives results for DGP 1. When τ 1 = 0.4, P SY is generally less powerful than s P SY . In fact, the latter's superiority in this regard is more readily apparent here than in the asymptotic context (Table 3(a)); this is probably a manifestation of the undersizing of P SY noted above. There is no clear winner when comparing PW Y and s PW Y , unlike in the asymptotic case where s PW Y was generally the better performing test. However, TABLE 4. Finite sample empirical size of nominal 0.05-level tests: T = 100, σ (s) = 1(0 ≤ s ≤ τ 1 ) + σ 1 1(τ 1 < s ≤ τ 2 ) + 1(τ 2 < s ≤ 1) it is still the case that PW Y can have very low power compared to s PW Y when σ 1 < 1. When τ 1 = 0.8, P SY is dominated by s P SY unless σ 1 > 1 and PW Y outperforms s PW Y unless σ 1 < 1 which is similar to the asymptotic case although the magnitudes involved can differ between the finite sample and asymptotic cases. The finite sample results for DGP 1 would reasonably suggest that s P SY is a better performing test than P SY overall; while there is no clear ranking between PW Y and s PW Y , PW Y can have very low power for σ 1 < 1. Results for DGP 2 are given in Table 5(b). When τ 1 = 0.1, we see that P SY and PW Y are, respectively, less powerful than s P SY and s PW Y , and the differences are often particularly marked. For τ 1 = 0.5, P SY is again generally inferior to s P SY , while PW Y performs better than s PW Y unless σ 1 < 1, in which case s PW Y can offer substantial gains. These findings are again largely in accordance with the asymptotic results (Table 3(b)). Throughout Table 5, we also see that the union procedures have power levels close to whichever constituent test is displaying the higher power under given circumstances. Overall then, it would appear reasonable to conclude that our asymptotic power simulation results provide a decent indicator of how the various bootstrap procedures will perform in practice, even when only data series of modest length are available. Lastly, we note that while we have implemented the s P SY and s PW Y tests using bootstrap critical values here, we could of course simply  Table 6, we report finite sample size and power results for wild bootstrap versions of the P SY 0 and PW Y 0 tests that exclude an intercept from TABLE 5(b). Finite sample powers of nominal 0.05-level tests: T = 100, DGP 2, τ 2 = 0.7, σ (s) = 1(0 ≤ s ≤ τ 1 ) + σ 1 1(τ 1 < s ≤ τ 2 ) + 1(τ 2 < s ≤ 1)   be modestly oversized. As μ increases, we observe an increase in the upward size distortion for P SY 0 when σ 1 ≤ 1 and a decrease in size when σ 1 > 1, to the point that undersize can be displayed for large σ 1 and μ. On the other hand, PW Y 0 remains approximately correctly sized when σ 1 ≤ 1 but can be very undersized for σ 1 > 1 and large μ. Turning to the test powers, even greater sensitivity to μ can be observed, and we observe that an increase in μ corresponds to a decrease in finite sample power for P SY 0 and PW Y 0 . Even for μ = 10, the power of P SY 0 and PW Y 0 can be reduced by up to 0.175 and 0.137, respectively, compared to the μ = 0 case. These reductions in power become more marked as μ increases, with P SY 0 and PW Y 0 powers for μ = 50 reduced by up to 0.470 and 0.414, respectively, compared to when μ = 0. Overall, we observe that the finite sample properties of the P SY 0 and PW Y 0 tests are highly sensitive to the unknown DGP parameter μ, reinforcing our recommendation for use of the P SY /PW Y and s P SY /s PW Y procedures which offer robustness (exact invariance) to μ.

DATING BUBBLE START AND END POINTS USING SIGN-BASED STATISTICS
We now consider how to consistently estimate the bubble start and end points, τ 1 and τ 2 when δ 1,T > 0. For simplicity, we examine this issue within the context of DGP 2. Consistent estimation is not possible using the current Pitman drift bubble magnitude δ 1.T = c 1 T −1 , c 1 > 0, so in what follows we replace this with a stronger, mildly explosive bubble magnitude of the form δ 1,T = c 1 T −α where α ∈ (0, 1). 3 PWY and PSY propose dating strategies for the start and end points of the bubble based on repeated implementation of their recursive tests over expanding samples, using critical values that diverge to infinity but at a rate slower than the derived divergence rate of the statistics over the mildly explosive regime. We considered implementing the PWY/PSY dating approach using our sign-based statistics, and it is not difficult to show that the sign-based statistics diverge at the rate T 1/2 in the mildly explosive regime, allowing τ 1 to be estimated consistently. 4 However, the sign-based statistics are unable to consistently estimate τ 2 with the PWY/PSY dating strategy. 5 In view of this, we pursue a dating strategy based on maximizing subsample statistics. By way of motivation, in view of the s P SY statistic of (4), fairly intuitive estimators of (τ 1 ,τ 2 ) are provided by the maximisers 3 The consistency results in this section also hold when α = 0, which represents a fixed magnitude (rather than mildly explosive) bubble. 4 The PWY and PSY dating statistics diverge at a rate dependent on the mildly explosive parameter α. By construction, our sign-based statistics are independent of the actual magnitude of the mildly explosive bubble, yielding a fixed rate of divergence T 1/2 for all α ∈ (0, 1). 5 Based on our later results in Lemma 1, it is not difficult to see that the order of divergence of our sign-based statistics is unchanged before and after τ 2 , hence this change point cannot be identified by directly applying the PWY/PSY dating strategy.
In essence, replacingŝ 2 (λ 1 ,λ 2 ) withs 2 (λ 1 ,λ 2 ), which is a weighted average of s 2 (0,λ 2 ) andŝ 2 (0,λ 1 ), creates a kink in the limiting function, making (τ 1 ,τ 2 ) a unique maximiser of it, while raisings 2 (λ 1 ,λ 2 ) to the power ε relates to finite sample considerations. Since Theorem 1 holds for all 0 < ε ≤ 1, it might be tempting simply to set ε = 1 for any practical application. However, unreported simulation evidence suggests the finite sample properties of this choice can be very poor, with the estimatorτ 1 (τ 2 ) being badly biased upwards (downwards). Setting ε to a much smaller value, such as ε = 0.01, was found to yield much less biased estimators. The results of Theorem 4 can be shown to continue to hold in the presence of any form of bubble collapse (as in DGP 3 or DGP 4, or indeed an instantaneous collapse). They also hold under DGP 1, i.e., for τ 2 = 1. Note that the consistency result in Theorem 4 requires τ 2 − τ 1 ≥ π, so that the length of the bubble regime is at least as long as the minimum window width for which s DF * (λ 1 ,λ 2 ) is computed. To allow for bubbles of short length, for example a bubble that emerges late in the sample period (under DGP 1), a relatively small value of π may be appropriate for accurate dating. Note that the setting for π here does not need to coincide with the setting for π in the testing procedure.
In principle, we could also use s DF * (λ 1 ,λ 2 ) in place of s DF(λ 1 ,λ 2 ) in our testing setup. This would unify the testing and dating aspects of our procedure rather conveniently. Unfortunately, the finite sample size and power properties of s DF * (λ 1 ,λ 2 ) we found to be somewhat inferior in comparison to those of s DF(λ 1 ,λ 2 ) and we therefore cannot recommend such a strategy.

ASYMMETRIC ERRORS
Assumption A4 implies that the mean and median of z t are the same (zero). It is possible that this assumption could fail to hold, for example Campbell, Lo, and MacKinlay (1997 , Table 1.1) find (very) mild asymmetries in daily financial returns which is likely to imply violation of Assumption A4. Suppose, then, that E(z t ) = 0 but F(0) = 1/2. In this case, under the null hypothesis, E{sign(z t )} = 0 and the invariance principle (2) clearly breaks down. It is then obvious that some form of de-meaning of sign(z t ) is required to make progress. This could be carried out in a number of different ways, but a convenient method is to employ recursive de-meaning of sign( y t ) before cumulating to form C t . Specifically, we replace sign( y t ) in the construction of s DF(λ 1 ,λ 2 ) with which is invariant to E[sign(z t )]. The advantage of recursive de-meaning is that (7) only involves data up to time t, which is, of course, relevant for any kind of real-time bubble monitoring exercise (full-sample de-meaning, for example, would not have this property). 6 Under a locally explosive H 1 and Assumptions A1-A3, we can show (along the lines of the proof of Theorem 2) that where σ 2 sz = V ar[sign(z t )] and U s (r ) is a copy of the distribution of that given in (6) (we duplicate the notation only to avoid repeating each expression). Then, denoting the new statistic bys P SY , we find that 6 Note that if a bubble is present from the beginning of the sample period, it is theoretically possible (although unlikely in practice) that all the values of sign( yt ) are equal to one, in which case the recursively de-meaned series would be zero for all time periods. While this causes problems for calculation of the test statistic, it is clear that such an occurrence should be taken as evidence of a bubble.

s P SY ⇒ sup
Asymptotic and finite sample critical values for this test, and its s PW Y counterpart, denoteds PW Y , are given in Table 6. The corresponding bootstrap statistics are based on recursive de-meaning of sign( y b t ). The consistency results of Theorem 4 can also be shown to hold in the case of recursive de-meaning.
With traditional left-tail unit root testing, it is well known that any form of de-trending of the data to account for deterministic terms in the observed series reduces the power of the test relative to the case where no de-trending is required. We would have little reason to suggest the same will not happen in the current context of right-tail testing. Since recursive de-meaning of sign( y t ) is de facto equivalent to de-trending of C t , we should therefore expect to finds P SY and s PW Y to have lower power than s P SY and s PW Y , respectively. We examine the extent to which this occurs both asymptotically and in finite samples. In addition to the results for s P SY and s PW Y , Table 3 also reports asymptotic local powers fors P SY ands PW Y , along with results for the corresponding union of rejections proceduresū P SY (union ofs P SY and P SY ) andū PW Y (union ofs PW Y and PW Y ). In Tables 3(a) and 3(c), we observe a loss in power through usings P SY ors PW Y compared to s P SY or s PW Y , as anticipated, with these being most apparent for smaller values of c 1 . Although some of the power gains thats P SY ands PW Y offered over P SY and PW Y are removed, there are still many cases where the sign-based approach outperforms the standard tests, sometimes by a substantial margin. In Table 3(b), we observe the unexpected feature that, for some DGP settings,s P SY ands PW Y can have higher local asymptotic power than the nonrecursively demeaned variants s P SY and s PW Y , respectively. Given that the latter tests were already seen to outperform P SY and PW Y in a majority of cases for DGP 2, it follows thats P SY ands PW Y offer valuable power gains relative to the original P SY and PW Y tests also. Throughout Table 3, the union of rejections proceduresū P SY andū PW Y behave in a similar way to the union procedures of Section 4, with power levels displayed that are close to the better of the two constituent tests that comprise each union.
Tables 4 and 5 report finite sample size and power results for the recursively de-meaned variants of the tests (and the corresponding unions), using bootstrap critical values throughout as in Section 5. Table 4 shows the sizes ofs P SY and s PW Y to be close to the nominal level in finite samples, and the correspondingū P SY andū PW Y union sizes are similar to those for u P SY and u PW Y . In Table 5, the finite sample power results fors P SY ands PW Y follow broadly similar patterns to the asymptotic results, although for many settings the finite sample powers can be considerably lower than their asymptotic counterparts. This is particularly noticeable in Table 5(b) where the unexpected result thats P SY ands PW Y had higher local asymptotic power than s P SY and s PW Y is now reversed in finite samples. While our finite sample results are limited to T = 100, unreported simulations using larger finite sample sizes confirm that the finite sample powers of the tests converge to the local asymptotic results in Table 3. Table  5 also shows that, once again, each union of rejections procedure has power close to the better of its constituent tests.

Higher Order Dynamics
We have assumed thus far that ε t is serially uncorrelated. More generally, we may consider it to have an autoregressive representation of the form with ρ i such that ε t is stationary under homoskedasticity. In this case, in the spirit of the recursive de-meaning described above, we fit the recursive OLS regressions We then construct s DF(λ 1 ,λ 2 ) using sign( y t − p i=1ρ i (t) y t −i ). The null limit distribution of s P SY can be shown to remain the same as given in Section 2.2. There is no need to alter the bootstrap data generation scheme nor the form of s P SY b m because the wild bootstrap removes any weak dependence present in y t . In practice, p is unknown but could be determined using standard information criteria, for example BIC.

More General Deterministic Terms
If the constant deterministic term μ in (1) is replaced by a process undergoing a finite number, n say, of deterministic level shifts, the limit distributions of the sign-based statistics are unchanged. This occurs because only n of the sign( y t ) are affected, so the effect is asymptotically negligible. Moreover, there is no restriction on the magnitudes of the level shifts due to the sign transformation. The limit distributions of P SY and PW Y are also unchanged, but only provided the level shift magnitudes are o(T 1/2 ). The practical consequence of this is that, in finite samples, the sizes and powers of P SY and PW Y will be rather more sensitive to large level shifts than those of the sign-based tests. Phillips, Shi, and Yu (2014) consider the possibility of a local-to-zero drift term. In the context of our model, this translates to replacing μ with μ + βT −d t, where d is a positive constant. It can be shown that the null limit distributions for the sign-based statistics continue to hold provided d > 1/2, coinciding with the restriction that PSY and PWY require for their tests to be asymptotically invariant to the local drift. To see this, define It is straightforward to calculate that using a similar argument. A simple application of Markov's inequality then shows that T = o p (1) when d > 1/2.

AN EMPIRICAL ILLUSTRATION
By way of a practical illustration of the use of our sign-based tests and dating methods, we apply them to Bitcoin price data (measured in pounds sterling) to study the possibility of explosive behavior being present in Bitcoin prices from late 2017. Bitcoin is a digital asset designed to work as a medium of exchange that uses cryptography (a so-called "cryptocurrency") and is considered a speculative asset among economists. The data range we choose is for the period 1/9/2017 to 28/1/2018. Bitcoin is traded 24/7 globally so price observations are available on all days, giving 149 observations. The data, which is plotted in Figure 1, is the daily closing price and was obtained from the website https://finance.yahoo.com/quote/BTC-GBP. In what follows, testing and dating are based on setting π = 0.1. Table 7 shows the value of the statistics P SY , s P SY ands P SY , with p = 1 in (8) (selected by BIC assuming a maximum value of p = 4) and one lagged difference included in the OLS regressions underlying P SY (a small number of observations are lost through accounting for serial correlation; for consistency, we compute all tests over the same effective sample size). The entries in round brackets are bootstrap p-values for the tests based on M = 9, 999 bootstrap replications. The P SY test clearly fails to reject the null hypothesis (measured at any conventional significance level), while both s P SY ands P SY show strong rejections. The strength of rejection obtained from s P SY is slightly higher than for s P SY , which might be expected in view of the simulation results of Section 7 above. Table 7 also reports results for the feasible union of rejections procedures u P SY b andū P SY b . The p-values associated with these procedures imply rejection of the null in both cases, albeit at a slightly higher significance level than was found for the s P SY ands P SY tests, as would be expected. It can also be seen that the values of the u P SY b andū P SY b statistics coincide with the s P SY and s P SY statistics, consistent with the rejections coming from the sign-based tests rather than the original test.
The additional entry for P SY in square brackets is the bootstrap p-value obtained when we do not account for any heteroskedasticity, which we carry out by constructing the increments of y b t using w t instead of y t w t . This approach is then essentially the same as using standard finite sample critical values obtained using N I I D(0, 1) errors. Interestingly, this leads to a complete overturn of the previous nonrejection by P SY . That this occurs, however, we take as a potential indication of substantial levels of heteroskedasticity being present in the data, and therefore an indication of the need to correct for it before we are in a position to make size-controlled inference. These contrasting findings are perhaps particularly pertinent given that changing volatility is widely considered to be a trademark characteristic of Bitcoin price data. The plot of absolute price changes shown in Figure 1 would seem to support such a view.
Having provided significant evidence for the presence of explosivity in the data on the basis of s P SY ands P SY , we can proceed to date it (we do not attempt to date using the PSY approach given that P SY failed to reject). Using the dating statistic s DF * (with ε = 0.01), the start date for the explosive regime is identified as 13/11/2017. The Bitcoin price suffered a small crash from 8/11/2017, and was undergoing a continuous 5-day decrease until 12/11/2017, after which it started a rapid increase period until mid December 2018. As such, the s DF * statistic seems to be reasonably accurate in identifying 13/11/2017 as the start of the period of rapid increase. The s DF * statistic finds that 7/12/2017 is the end date of the explosive regime. On 7/12/2017, the Bitcoin price reached a local maximum after continuously increasing for about 3 weeks from £4,379, closing at £12,882. Then it suffered a short one-day crash, before it was pushed to its historical high on 22/12/2017. Our s DF * therefore seems to be rather accurate in identifying this crash, placing the end time on the crash day (so no dating delay is inherent). We also apply the same dating strategy to the recursively de-meaned data, denoting this statistic ass DF * . We find that thes DF * statistic places the start date rather earlier than s DF * , at 4/10/2017. This essentially treats the oneweek crash starting from 8/11/2017 as a random shock and also picks up the relatively gradual increase in the Bitcoin price from 10/2017 as an explosive regime, which also seems reasonable to us. Notice that the end date identified bys DF * is identical to that from s DF * . As such, s DF * suggests an explosive regime that is concentrated around the period of most intensive upward movement in the prices, whiles DF * is suggestive of a more gradually emerging explosive regime. Both these scenarios seem plausible and we would not wish to take a stance in favour of one or other without conducting deeper analysis that lies outside of the remit of this article.

CONCLUSIONS
In this article, we have proposed a sign-based variant of the PSY test for explosive autoregressive behavior in financial time series. In contrast to the original test, this test does not require bootstrap-type methods to control size in the presence of heteroskedastic innovations, thereby offering computational efficiency gains when applied in practice. Under a locally explosive bubble alternative, we also find that the sign-based test has appealing asymptotic power properties, with the potential to deliver substantially greater power than the original test for many volatility and bubble model specifications. However, because the original test may still outperform the sign-based test for some specifications, we also suggested a union of rejections procedure that combines the sign-based and original tests and employs a joint wild bootstrap to control size. This union is seen to succeed in capturing most of the power available from the better performing of the two tests for a given alternative. Our finite sample simulations indicate that our new procedures should work well in practice. We have also shown how a slight variant of the sign-based test can be used to consistently date the start and end points of a mildly explosive bubble, and how a recursively de-meaned variant of the test can allow for asymmetry in the innovations. We applied our sign-based testing and dating procedures to recent Bitcoin price data and uncovered robust evidence for the existence of an explosive regime in this data, subsequently identifying what we consider to be plausible start and end dates for this regime.
Finally, we consider some areas for future research. First, our assumptions consider only deterministically time-varying volatility, and it would be interesting to extend our work to ARCH-type and stochastic volatility dynamics, investigating conditions under which our tests remain asymptotically valid. Secondly, we recognise that the DGP we have chosen to analyse in this article is not the only specification capable of generating bubble-type behavior. Random coefficient autoregressive models (e.g., Blanchard and Watson, 1982;Granger and Swanson, 1997) and certain types of noncausal model (e.g., Gourieroux and Jasiak, 2018) represent plausible alternative specifications. An investigation of the performance of our sign-based tests in the context of such alternative models would be interesting. Finally, in principle the procedures developed in this article could be extended to the context of real-time detection and dating of possibly multiple bubbles. Additional issues arise when considering a real-time analysis, for example the need to control size for tests applied at multiple sequential points in time (see, e.g., Homm and Breitung, 2012), and a full development of such real-time monitoring procedures would also be of value.

A.1. Proof of Theorem 1
It follows from HLST that (where W σ (r ) corresponds to the variance-transformed Brownian motion processωW η (r ) in the notation of that article). Also, since y t = ε t + O p (T −1/2 ) for all t, we find that A little straightforward manipulation allows us to write DF(λ 1 ,λ 2 ) in the form Then, following the arguments of the proof of Theorem 1 in PSY, DF(λ 1 ,λ 2 ) can be interpreted as a continuous functional of the partial sum process T −1/2 y rT andσ 2 (λ 1 ,λ 2 ), allowing application of the continuous mapping theorem to give The stated limit for the P SY statistic is then obtained from a further continuous mapping argument following the proof of Theorem 1 in PSY, since the double sup operator can be written as a continuous functional over the space of functions {DF(λ 1 ,λ 2 ), (λ 1 ,λ 2 ) ∈ ([0, 1 − π], [λ 1 + π, 1])} with respect to the uniform norm. See also Shi, Hurn, and Phillips (2018a, 2018b) for similar arguments in deriving sup-type limit results.

(A.2)
= U s (r ) from which the main result follows easily. The result in (A.2) extends Theorem 1 of Boldin (2013) to allow for time-varying volatility and a time-varying coefficient mean level model. In what follows, we only demonstrate the result for the last regime where r > τ 3 ; the results in the other regimes can be obtained in the same way.
First note that under H 1 , We first examine the difference T −1/2 C rT − T −1/2 rT t =2 sign(ε t ). Using the definition sign(x) = −21(x 0) + 1, we have where δ t = δ 1,T 1( τ 1 T < t τ 2 T ) − δ 2,T 1( τ 2 T < t τ 3 T ). Next, we make decompositions of the above two sums of indicator functions around the corresponding distribution function F(.) of z t : where A-C are defined implicitly. Looking at terms A and B terms together and denoting Our aim is to evaluate the mean and variance of A + B. First notice that {H t } T t =2 is a martingale difference sequence with respect to the natural filtration: This implies E(A + B) = 0. Next, where we have used the inequality Var(X|F t −1 ) E(X 2 |F t −1 ) in the second step; in the last step, the cross product term of the quadratic expansion E 1 −δ t y t −1 σ t < z t 0 1 0 < z t −δ t y t −1 σ t |F t −1 = 0 as the two sets considered in the indicator functions are mutually exclusive. In the previous derivation for Var(H t |F t −1 ), we have also used the result that E(1(A) 2 ) = E(1(A)) for any set A. We have Since {H t } is a martingale difference sequence, by Burkholder's inequality (e.g., Hall and Heyde (1980) Thm. 2.10), for a generic constant C > 0, it is satisfied that where we have substituted in the expression derived in (A.3). Since δ t y t −1 p → 0 in the locally explosive regime, δ t = 0 identically in the unit root regimes, and δ t y t −1 p → 0 in the stationary regime, the set {−δ t y t −1 /σ t < z t 0} converges to a null set and we have E 1 −δ t y t −1 σ t < z t 0 1 −δ t y t −1 σ t 0 → 0.
Note that for the first term, τ 2 is in the mildly explosive regime so Applying the functional central limit theorem for the second term, we have T −1/2 (ε rT + ε rT −1 + ··· + ε τ 2 T +1 ) ⇒ Clearly, the first term dominates and we obtain

B.4. Proof of Lemma 2
Part (a) relates to a unit root regime, and the claimed weak convergence is known from the proof of Theorem 2.
For (b), by definition The first term satisfies T −1/2 τ 1 T i=2 sign(ε i ) ⇒ W s (τ 1 ), while for the second term, notice that