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arxiv: 2511.16172 · v2 · submitted 2025-11-20 · 💰 econ.EM · stat.ME

Confidence Sets for the Emergence, Collapse, and Recovery Dates of a Bubble

Eiji Kurozumi , Anton Skrobotov This is my paper

Pith reviewed 2026-05-17 21:12 UTC · model grok-4.3

classification 💰 econ.EM stat.ME
keywords bubble datesconfidence setsstructural breaksbreak date testslikelihood ratio testsElliott-Muller teststime serieseconometrics
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The pith

Confidence sets for bubble emergence, collapse, and recovery dates are built by inverting tests for structural break locations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a method to construct separate confidence sets for the dates when a bubble emerges, collapses, and recovers by inverting statistical tests that locate structural breaks in time series. It examines both likelihood ratio-type tests and Elliott-Muller-type tests, derives their limiting distributions under the null of no break, and proves consistency under the alternative that a break is present. Monte Carlo simulations show that combining the two families of tests achieves empirical coverage rates close to the nominal level while keeping the average length of the sets reasonably small. This approach matters because it supplies economists with explicit measures of uncertainty around the timing of bubble phases rather than point estimates alone.

Core claim

The central claim is that confidence sets for the three bubble dates can be obtained by inverting tests for the location of a break, with the limiting distributions derived under the null hypothesis of no break and asymptotic consistency established under the bubble-specific alternatives; finite-sample results indicate that combining likelihood-ratio and Elliott-Muller tests controls coverage while preserving short interval lengths.

What carries the argument

Inversion of likelihood ratio-type tests and Elliott-Muller-type tests for the location of structural breaks in the underlying data-generating process.

If this is right

  • Separate confidence sets are obtained for emergence, collapse, and recovery dates rather than a single break.
  • Combining the two test families improves finite-sample coverage without excessive widening of the sets.
  • The asymptotic results hold under both the null of no break and the specific alternatives that correspond to bubble phases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same inversion technique could be applied to other regime-shift problems such as dating the start and end of recessions.
  • Real-time updating of the confidence sets becomes feasible as new observations arrive, allowing dynamic monitoring of bubble risk.
  • Direct comparison with alternative break-date procedures would clarify whether the coverage gains are specific to the bubble setting or more general.

Load-bearing premise

The data-generating process for bubbles must produce distinct structural breaks at the emergence, collapse, and recovery dates whose locations the selected tests can detect consistently.

What would settle it

A Monte Carlo experiment or real-data application in which the combined tests produce empirical coverage rates for the true dates that fall substantially below the nominal level would falsify the method's practical reliability.

Figures

Figures reproduced from arXiv: 2511.16172 by Anton Skrobotov, Eiji Kurozumi.

Figure 1
Figure 1. Figure 1: Logarithm of Japanese stock price Note: The red vertical lines show the point estimates of Te, Tc, and Tr, respectively, while the gray, blue, and yellow areas correspond to the confidence set for Te, Tc, and Tr, respectively. 53 [PITH_FULL_IMAGE:figures/full_fig_p053_1.png] view at source ↗
read the original abstract

We propose constructing confidence sets for the emergence, collapse, and recovery dates of a bubble separately by inverting tests for the location of the break date. We examine both likelihood ratio-type tests and the Elliott-Muller-type (2007) tests for detecting break locations. The limiting distributions of these tests are derived under the null hypothesis, and their asymptotic consistency under the alternative is established. Finite-sample properties are evaluated through Monte Carlo simulations. The results indicate that combining different types of tests effectively controls the empirical coverage rate while maintaining a reasonably small length of the confidence set.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes constructing separate confidence sets for the emergence, collapse, and recovery dates of a bubble by inverting likelihood-ratio-type tests and Elliott-Muller (2007) tests for the location of a structural break. Limiting distributions of the tests are derived under the null hypothesis that the hypothesized date equals the true break date, asymptotic consistency is established under the alternative, and finite-sample coverage and length are assessed via Monte Carlo experiments. The central result is that combining the two classes of tests produces confidence sets with controlled empirical coverage and reasonably short length.

Significance. If the asymptotic validity holds, the approach supplies a statistically grounded way to date the three phases of a bubble rather than relying on point estimates or ad-hoc rules. The explicit combination of LR and EM tests together with Monte Carlo evidence on coverage-length trade-offs is a constructive contribution to the bubble-detection literature.

major comments (2)
  1. [§3] §3 (Asymptotic Theory): The limiting distributions under the null and the consistency statements under the alternative are stated, but the derivations appear to rely on standard I(1) or stationary break-date asymptotics. It is not shown that the same limiting expressions remain valid when the DGP contains a mildly explosive segment (ρ > 1) between the emergence and collapse dates; the change in convergence rate induced by the explosive root may invalidate the inversion step that produces the claimed coverage.
  2. [§4] §4 (Monte Carlo): The reported coverage probabilities are obtained from a specific bubble DGP. No results are presented that vary the explosiveness coefficient or the length of the explosive window, so it is unclear whether the good coverage of the combined procedure is robust to the bubble parameters that define the alternative.
minor comments (2)
  1. [Abstract] The abstract states that consistency under bubble-specific alternatives is established but supplies no further detail on the precise form of the alternative or on the maintained assumptions about the innovation process.
  2. [Notation] Notation for the three dates (emergence, collapse, recovery) should be introduced once and used uniformly in all subsequent sections and tables.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments regarding the asymptotic validity under explosive regimes and the robustness of the Monte Carlo design. We respond to each major comment below and indicate the revisions we will undertake.

read point-by-point responses

  1. Referee: [§3] §3 (Asymptotic Theory): The limiting distributions under the null and the consistency statements under the alternative are stated, but the derivations appear to rely on standard I(1) or stationary break-date asymptotics. It is not shown that the same limiting expressions remain valid when the DGP contains a mildly explosive segment (ρ > 1) between the emergence and collapse dates; the change in convergence rate induced by the explosive root may invalidate the inversion step that produces the claimed coverage.

    Authors: We acknowledge that the derivations in Section 3 rely on standard break-date asymptotics and do not explicitly re-derive the limiting distributions under a DGP that includes a mildly explosive segment with ρ > 1. The inversion step for coverage is justified under the null that the hypothesized date equals the true break date, but the referee correctly notes that the explosive root may alter convergence rates. We will add a dedicated subsection in the revised version that establishes the relevant limiting expressions under the full bubble DGP, confirming that the inversion continues to deliver the claimed asymptotic coverage. revision: yes

  2. Referee: [§4] §4 (Monte Carlo): The reported coverage probabilities are obtained from a specific bubble DGP. No results are presented that vary the explosiveness coefficient or the length of the explosive window, so it is unclear whether the good coverage of the combined procedure is robust to the bubble parameters that define the alternative.

    Authors: We agree that the Monte Carlo design in Section 4 uses a fixed parameterization and does not explore sensitivity to the explosiveness coefficient or explosive-window length. To demonstrate robustness, we will expand the simulation study in the revision to include grids over ρ > 1 and varying lengths of the explosive phase, reporting coverage and length results for the combined LR-EM procedure under these alternatives. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation of bubble-date confidence sets

full rationale

The paper constructs confidence sets by inverting LR-type and Elliott-Muller (2007) tests for break locations, derives the limiting distributions under the null hypothesis of a fixed break date, establishes asymptotic consistency under the bubble alternatives, and evaluates finite-sample coverage via Monte Carlo simulations on external DGPs. No equation or step reduces the reported confidence sets or coverage results to a fitted quantity defined inside the paper or to a self-citation chain; the inversion procedure and limiting results are presented as standard applications of existing break-date asymptotics, with the Monte Carlo serving as external validation rather than a tautological check.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on standard domain assumptions from the structural-break literature applied to a bubble model; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption Bubble phases correspond to distinct structural breaks in the observed time series whose locations are identifiable by the chosen tests.
    This premise underpins both the test inversion and the derivation of limiting distributions under the null.

pith-pipeline@v0.9.0 · 5385 in / 1173 out tokens · 46498 ms · 2026-05-17T21:12:01.522102+00:00 · methodology

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Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

  1. [1]

    σ2 2b ,(81) where the convergence in probability holds by Lemma 2(a) of Kurozumi and Hayakawa (2009) by noting that the term in the parentheses corresponds to the mildly integrated process, and the convergence in probability holds uniformly overϵ≤λ ∗ 2 ≤λ ∗ 1 −ϵby Lemma A.10 of Hansen (2000). In addition, asy 2 T1 =O p(T β) andy 2 T2 =O p(T β) whereas T1X...

    work page 2009

  2. [2]

    σ2 2b +T 2 CU σ2 Z λ∗ 2 λ∗r (W(s ∗)−W(λ ∗ r))2ds∗,(84) 44 while T2X t=T1+1 yt−1∆yt = TrX t=T1+1 yt−1∆yt + T2X t=Tr+1 yt−1∆yt ∼a −TCU σ2 2 (λ∗ r −λ ∗

    work page

  3. [3]

    On the contrary, forT 1 < T 2 ≤T r, only the first terms of (84) and (85) matter and we can observe thatt T1,T2 → −∞

    +T CU σ2 2 h (W(λ ∗ 2)−W(λ ∗ r))2 −(λ ∗ 2 −λ ∗ r) i ,(85) which impliest T1,T2 =O p(1) forT 2 > T r. On the contrary, forT 1 < T 2 ≤T r, only the first terms of (84) and (85) matter and we can observe thatt T1,T2 → −∞. Therefore, we conclude that EM a,12,EM b,12 → −∞. The behavior of LR a,21, EM a,21, and EM b,21 is obtained similarly to the proof of Theo...

    work page

  4. [4]

    σ2 2b +T 2 CU Z λ∗ 1 λ∗r (W(s ∗)−W(λ ∗ r))2 ds∗ (86) and T1X t=T2+1 yt−1∆yt = TrX t=T2+1 yt−1∆yt + T1X t=Tr+1 yt−1∆yt ∼a −TCU σ2 2 (λ∗ r −λ ∗

    work page

  5. [5]

    On the contrary,t T2,T1 =O p(1) forT 2 < T r and the same order is observed forT r ≤T 2

    +T CU σ2 2 h (W(λ ∗ 1)−W(λ ∗ r))2 −(λ ∗ 1 −λ ∗ r) i .(87) Then, we observe that LR a,21 → ∞. On the contrary,t T2,T1 =O p(1) forT 2 < T r and the same order is observed forT r ≤T 2. As the sign oft T2,T1 is indeterministic while the scalars go to zero from the definitions of EM r a,21 and EMr b,21, we conclude that they diverge to∞or −∞.■ 45 Table 1: Coef...

    work page 2012