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

1. Problem Statement

While significant literature exists on detecting the presence of rational bubbles in financial asset prices (e.g., Phillips et al., 2011; Phillips et al., 2015), there is a notable gap in the methodology for constructing confidence sets (intervals) for the specific dates when a bubble emerges ($T_e$), collapses ($T_c$), and recovers ($T_r$).

Existing methods primarily rely on:

1. Point Estimators: Consistent estimators for break fractions (e.g., minimizing sum of squared residuals).
2. Asymptotic Distributions of Estimators: Constructing intervals based on the limiting distribution of the break-point estimator.

The Core Issue: The authors demonstrate via preliminary simulations that constructing confidence sets based on the limiting distribution of the break-point estimator performs poorly for bubble dates. These intervals tend to be excessively wide (often covering the entire permissible range) and fail to provide meaningful inference, particularly when the explosive speed is not sufficiently fast.

2. Methodology

The authors propose a novel approach to constructing confidence sets by inverting hypothesis tests for the location of the break date, rather than relying on the asymptotic distribution of the estimator. This follows the framework of Elliott and Müller (2007) and Eo and Morley (2015).

A. The Model

The study utilizes a four-regime model for a time series $y_t$:

1. Regime 1 ($1 \le t \le T_e$): Random walk ($\phi=1$).
2. Regime 2 ($T_e < t \le T_c$): Explosive autoregression ($\phi_a = 1 + a/T^\alpha, a>0$).
3. Regime 3 ($T_c < t \le T_r$): Mean-reverting collapse ($\phi_b = 1 - b/T^\beta, b>0$).
4. Regime 4 ($T_r < t \le T$): Return to random walk.

The parameters $\alpha$ and $\beta$ govern the speed of explosion and collapse, respectively. The authors assume $\alpha, \beta \in (0, 1)$, covering "moderate deviation" from unity.

B. Testing Framework

For each break date (e.g., $T_e$), the authors test the null hypothesis $H_0: T = T_1$ against the alternative $H_1: T = T_2 \neq T_1$. By inverting these tests over a permissible range of dates, they construct a confidence set with nominal coverage $1-\delta$.

Two primary classes of test statistics are developed:

1. Likelihood Ratio (LR) Type Tests:

   • Derived under the assumption of Gaussian errors (though asymptotic theory does not require normality).
   • The test statistic compares the likelihood of the data under the null break date versus a specific alternative break date.
   • The authors propose specific scalings for the statistics depending on whether the alternative date is before or after the null date.
   • Finite-Sample Correction: To address over-sizing in small samples, the authors propose a modification (e.g., $LR^e_{b,12}$) that shifts the statistic to correct for finite-sample bias without altering the asymptotic null distribution.

2. Elliott-Müller (EM) Type Tests:

   • Inspired by optimal tests for break dates in stationary regressions.
   • These tests maximize weighted average power by integrating the likelihood ratio over priors for the break size and break date.
   • The resulting statistics are weighted averages of $t$-statistics (or squared $t$-statistics) for the break coefficient.
   • The authors utilize one-sided statistics where the direction of the break is known (e.g., unit root to explosive).

C. Specific Test Combinations

Recognizing that individual tests may be consistent only in one direction (e.g., detecting a break later than the null but not earlier), the authors propose combining tests to form robust confidence sets:

• $LE^e$ (Emergence): Combines a corrected LR test ($LR^e_{b,12}$) and an EM test ($EM^e_{a,21}$).
• $LE^c$ (Collapse): Combines $LR^c_{a,21}$ and $EM^c_{a,12}$.
• $LE^r$ (Recovery): Combines $EM^r_{b,12}$ and $LR^r_{b,21}$.

3. Key Contributions

1. First Confidence Sets for Bubble Dates: This is the first paper to systematically develop and derive the asymptotic theory for confidence sets specifically for the emergence, collapse, and recovery dates of bubbles.
2. Inversion of Tests: The paper shifts the paradigm from using estimator distributions to inverting break-location tests, proving this approach yields tighter and more reliable intervals.
3. Asymptotic Theory: The authors derive the limiting distributions of the proposed LR and EM-type statistics under both the null and alternative hypotheses. They show that the distributions depend on Brownian motions and ADF-type functionals, often requiring response-surface regressions to tabulate critical values.
4. Finite-Sample Corrections: The paper introduces specific corrections to the LR statistics to mitigate over-rejection (over-sizing) in finite samples, a common issue in bubble detection.
5. Handling Asymmetry: The methodology explicitly addresses the asymmetry in detection power (e.g., a test might be consistent for detecting a later break but inconsistent for an earlier one) by combining complementary tests.

4. Results

A. Monte Carlo Simulations

The authors evaluate performance using 2,000 replications with varying bubble sizes (small, medium, large) defined by the parameter $a$.

• Coverage Rates:
  • Standard intervals based on estimator distributions fail (coverage $\approx 100\%$ but interval length is the whole sample).
  • The proposed $LE$ tests (combinations) achieve coverage rates close to the nominal 90% level, particularly for medium and large bubbles.
  • For "small" bubbles (slow explosiveness), coverage can be slightly liberal, but the methods still outperform existing approaches.
• Interval Length:
  • The confidence sets are significantly shorter than those derived from estimator distributions.
  • As the explosive speed ($a$) increases, the length of the confidence set decreases, indicating higher precision.
  • The combined tests ($LE^e, LE^c, LE^r$) generally offer the best trade-off between coverage accuracy and set length.

B. Empirical Application

The methods were applied to the Nikkei 225 daily stock prices (2012–2013), a period associated with the Bank of Japan's monetary policy shifts.

• Point Estimates: The estimated emergence was Nov 14, 2012; collapse May 22, 2013; recovery June 13, 2013.
• Confidence Sets:
  • The 90% confidence set for the emergence date included January 23, 2013 (the day after the BOJ announced a 2% inflation target), suggesting a strong link between policy and the bubble's start.
  • The set for the collapse date was very narrow (10 days), indicating high precision in identifying the peak.
  • The set for the recovery date was wide and extended significantly past the point estimate, suggesting the market may have taken longer to normalize than the point estimate suggested.
• Comparison: The new method reduced the confidence set size by 44 percentage points compared to the traditional estimator-based interval.

5. Significance

This paper provides a rigorous statistical framework for policymakers and researchers to quantify the uncertainty surrounding the timing of financial bubbles.

• Policy Relevance: By providing confidence sets rather than just point estimates, the method allows analysts to assess whether specific policy interventions (like the BOJ's inflation target) fall within the statistically plausible window of a bubble's emergence.
• Robustness: The proposed tests are robust to the specific speed of explosiveness and offer a practical solution to the "wide interval" problem that has plagued previous attempts at dating bubbles.
• Future Directions: The authors note that while they treat the three break dates separately, joint confidence sets remain a challenge due to the complexity of the joint null hypothesis, leaving this for future research.

In summary, Kurozumi and Skrobotov successfully bridge the gap between bubble detection and precise dating, offering a statistically sound tool for identifying the lifecycle of financial bubbles with quantifiable uncertainty.