International Review of Financial Analysis 110 (2026) 104848 Available online 4 December 2025 1057-5219/© 2025 The Author. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Log-periodicity: Fact or ction?☆ Klaus Grobys Finance Research Group, School of Accounting and Finance, University of Vaasa, Wolffintie 34, 65200 Vaasa, Finland A R T I C L E I N F O JEL classification: C22 C15 C58 C63 G12 Keywords: False positives Financial instability Log-periodic power law model Bubble detection S&P 500 A B S T R A C T A common empirical practice in LPPLS applications is to calibrate the model under parameter bounds and then declare an “LPPLS signature” when ADF/PP tests on calibration residuals reject a unit root at conventional tabulated critical values. We show that this procedure exhibits substantial size distortion. Using synthetic series that preserve the roughness and volatility of nancial data while excluding log-periodic structure, we compute bootstrap critical values by re-estimating the full two-stage procedure on each synthetic sample. Applied to S&P 500 monthly and daily data, conventional thresholds yield inated rejection rates. In contrast, the bootstrap restores empirical size to nominal levels and overturns many purported signatures. These ndings highlight the need for estimation-aligned inference in LPPLS diagnostics and call for a re-examination of published LPPLS evidence that may reect size-induced false positives. 1. Introduction In empirical nance, there is a dangerous temptation to mistake retrospective t for predictive insight. The Log-Periodic Power Law Singularity (LPPLS) model, introduced by Johansen et al. (2000), claims to reveal hidden regularities—oscillations accelerating toward a critical singularity—that supposedly signal market regime changes. Yet a model that captures the past with precision offers no guarantee of forecasting power. History can be overt as easily as it can be misunderstood. The LPPLS model’s apparent explanatory success demands a more funda- mental test: can it distinguish true structure from noise? In scientic modeling, credibility is earned not through the accommodation of known outcomes but through structured attempts at falsication (Edmans, 2024). This paper asks: does the LPPLS model reveal intrinsic market instabilities, or does it merely impose deterministic patterns upon stochastic uctuations? To confront this question, we simulate data that keep the familiar features of nancial returns—irregularity, heavy tails, volatile bursts—while removing any log-periodic component. This lets us probe directly the LPPLS model’s tendency to see patterns in noise. We examine two S&P 500 benchmarks—monthly (1871–2022) and daily (1980–1986)—to test robustness across horizons (e.g., Grobys, 2023; Sornette, 2017). In doing so, we follow common practice in the applied LPPLS literature by enforcing standard parameter bounds (see Sornette, 2017). In line with standard practice, we assess the stationarity of LPPLS residuals using Augmented Dickey–Fuller (ADF) tests (e.g., Grobys, 2023, 2025; Lin et al., 2014). We rst compare the residual statistics to conventional tabulated ADF critical values and record the share of samples labeled “LPPLS-consistent,” treating such rejections as false positives under the simulation-based null. We then implement an i.i.d. bootstrap, re-estimating the full two-stage procedure on each synthetic sample to obtain bootstrap critical values with correct empirical size. To benchmark these ndings within a familiar approach, we also apply a full-sample right-tailed ADF test on log prices (with an intercept and BIC lag selection) and compute both Monte Carlo and bootstrap critical values, thereby contrasting LPPLS-residual evidence with a standard test for mild explosiveness. Finally, to assess external validity, we replicate the estimation–testing procedure on gold futures (using the same data and settings as Grobys, 2025) as an additional robustness check. This design makes clear how tabulated critical values can inate bubble de- tections and, in turn, provides a more reliable framework for LPPLS- based diagnostics of speculative bubbles. Brée et al. (2013) and Brée and Joseph (2013) document an extensive literature applying various LPPLS models to identify nancial market bubbles. A phenomenon well-established in the study of nancial modeling, particularly in Generalized Autoregressive Conditional ☆ This paper was presented at the Erich-Schneider Seminar in April 2025 at Christian-Albrechts-Universität zu Kiel. The author gratefully acknowledges valuable comments received from Thomas Lux and Daniel Fehrle. Furthermore, the author thanks two anonymous referees for their constructive and helpful comments. E-mail address: klaus.grobys@uwasa.. Contents lists available at ScienceDirect International Review of Financial Analysis journal homepage: www.elsevier.com/locate/irfa https://doi.org/10.1016/j.irfa.2025.104848 Received 20 August 2025; Received in revised form 26 November 2025; Accepted 3 December 2025 International Review of Financial Analysis 110 (2026) 104848 2 Heteroskedasticity (GARCH) frameworks, manifests here: when the model fails, it is retrotted rather than discarded (Mandelbrot, 2008).1 The tendency of nancial theorists to revise parameters post hoc rather than reassess foundational assumptions has profound implications for the empirical validity of LPPLS model applications. Brée and Joseph (2013) tested the standard LPPLS model, originally introduced in Johansen et al. (2000), across 11 stock market crashes in the Hang Seng Index (1970–2008), nding that only seven conformed to the parameter constraints proposed ex post by Johansen and Sornette (2001a). The apparent inconsistency in these calibrations underscores the model’s selective alignment with observed bubbles rather than systematic vali- dation. Whereas Brée and Joseph (2013) center their analysis on true positives, Lin et al. (2014) undertake the only extensive examination of false positives, probing whether the LPPLS model falsely signals bubbles in non-speculative data. Their work extends the LPPLS framework by introducing the Volatility-Conned LPPLS (VC-LPPLS) model, incorpo- rating mean-reverting residuals and benchmarking its robustness against synthetic GARCH-driven datasets. Their ndings indicate an exceptionally low false positive rate (0.2 %) when tested on these arti- cially generated series. Yet, this inference rests upon the premise that GARCH processes adequately approximate nancial market dynam- ics—a premise contested by empirical studies revealing that real-world asset price movements frequently diverge from the volatility clustering intrinsic to GARCH frameworks. Thus, while Lin et al. (2014) assert a methodological improvement, their conclusions remain circumscribed by their reliance on simulated GARCH-driven benchmarks. This limita- tion highlights the need for a broader validation framework that sys- tematically evaluates the LPPLS model’s reliability in detecting speculative bubbles in nancial markets. Notably, the introduction of volatility constraints, as proposed by Lin et al. (2014), presents several methodological concerns. The standard LPPLS model is parsimonious, permitting robust calibration without additional constraints. Given that nancial markets inherently exhibit heteroskedasticity, the log-periodic component of LPPLS model already accommodates volatility clustering. Imposing an explicit volatility function risks overtting, introducing parameter uncertainty, and di- minishes predictive reliability. Empirical studies further suggest that log-periodic oscillations alone sufce for bubble detection, with vola- tility restrictions offering only marginal improvements, if any (Gustavsson et al., 2016; Shu& Song, 2024). Moreover, the predominant body of research employs the standard LPPLS model, reinforcing its centrality in empirical investigations. Prioritizing this formulation en- sures direct assessment of the model’s efcacy without confounding inuences from additional constraints. This choice preserves analytic tractability while avoiding excessive parameterization that could obscure fundamental log-periodic structures. Thus, the LPPLS model, in its original form, remains a fundamental and pragmatic tool for nancial market analysis, providing a reliable framework for identifying specu- lative regimes while sidestepping unnecessary complexities that may cloud empirical ndings. Moreover, using a GARCH model to simulate returns species con- ditional variance dynamics for returns: when α+ β < 1, returns are covariance-stationary and the conditional variance is mean-reverting. However, for markets such as the S&P 500, empirical work often points to highly persistent volatility (near-IGARCH), so a single low- order parametric GARCH—even with Student-t innovations—may underrepresent salient features of the data-generating process for the for the purposes of LPPLS residual ADF testing. Consequently, false-positive rates may be understated when LPPLS residual tests are judged against conventional tabulated critical values. To avoid embedding a particular volatility law in the null, we employ an i.i.d. bootstrap of returns—as opposed to using GARCH models—compounding to prices to preserve the empirical marginal distribution while excluding any imposed log- periodic structure. We then re-estimate the LPPLS models and re-test on each resample, constructing residual-ADF critical values that are aligned with the exact two-stage procedure (same bounds, starting values, and BIC lag selection). Our study reframes LPPLS diagnostics around size-controlled infer- ence. We follow Sornette’s (2017) practice of enforcing parameter bounds in LPPLS model calibration and then examine the common next step in the literature: applying ADF tests to the resulting residuals and consulting tabulated critical values (e.g., Lin et al., 2014). The premise is simple but consequential: ADF tables for raw data series do not generally apply to residual data from a nonlinear model with estimated and constrained regressors. Uncritically applying tabulated critical values may induce size distortion and encourage seeing structure in noise. Our contribution is to align inference with estimation. We construct the residual-ADF null distribution under the exact two-stage procedur- e—using the same bounds, starting values, and BIC-based lag selec- tion—via an i.i.d. bootstrap that re-estimates the LPPLS model and re- tests on every resample. This estimation-aligned bootstrap yields crit- ical values that restore empirical size to nominal levels and, in turn, reduce spurious detections of log-periodicity. We further contextualize the residual evidence against a standard ADF benchmark on log prices (BADF) and extend the design to gold futures to gauge external validity. The guiding message for empirical practice is straightforward: inference should be calibrated to the way the statistic is produced; when it is, apparent regularities may recede toward chance. Applied to the original S&P 500 data, the LPPLS t suggests mod- erate acceleration toward a critical point in the monthly series and a more pronounced acceleration in the daily series. However, using tabulated ADF critical values, the residual ADF indicates the residuals are stationary at conventional cutoffs. Yet the synthetic-data validation tells a different story. At the nominal 1 % level, the empirical rejection rate under the bootstrap null is 84.8 % (monthly) and 82.8 % (daily)—a severe oversizing. Using bootstrap-based critical values, the LPPLS re- sidual ADF statistics for the original datasets (λADF = 32277 monthly; λADF = 26194 daily) are not signicant (bootstrap p = 05960 and p = 08130, respectively). As a levels-based benchmark, we compute a full- sample BADF with an intercept and BIC-selected lags. Evaluated against the tabulated 1 % critical value, the empirical size under the bootstrap- simulated null is 13 %—still oversized, but well below the residual-ADF case. Using bootstrap inference, the BADF indicates mild explosiveness for the monthly S&P 500 sample (bootstrap p = 00370) and no such evidence for the daily sample (bootstrap p = 05930). A robustness check on gold futures points the same way. At the nominal 1 % level, the bootstrap size is 81 % for the residual ADF. Evaluating the original re- sidual statistic against bootstrap cutoffs (λADF =  46215) yields p = 00580: reject at 10 % but not at 5 %—a marginal signal consistent with Grobys (2025). Taken together, the evidence is plain: empirical ADF quantiles relevant for LPPLS residuals sit far below tabulated critical values, and relying on standard tables induces substantial size distor- tion. Calibrating inference to the way the statistic is produced changes 1 The LPPLS model, rst formulated by Johansen et al. (2000), has not remained static; rather, it has undergone successive modications, each seeking to rene its predictive capabilities and extend its applicability within nancial markets. A notable revision is the approach of Shu and Song (2024), which sharpens critical-time estimation by incorporating singularity corrections. Another substantive renement is the Volatility-Conned LPPL model, embedding a mean-reverting volatility process to fortify robustness against market uctuations (Lin et al., 2014). The LPPL framework has further been expanded to encompass negative bubbles (anti-bubbles)—capturing self- reinforcing downward spirals and subsequent rebounds (Yan et al., 2010). Beyond structural enhancements, researchers have sought to integrate funda- mental economic factors, aiming to bridge the gap between speculative dy- namics and macroeconomic underpinnings (Zhou & Sornette, 2006). Additional developments include genetic algorithm optimization for parameter calibration (Filimonov & Sornette, 2013) and second-order LPPL models, designed to capture more intricate price uctuations (Sornette & Zhou, 2002). K. Grobys International Review of Financial Analysis 110 (2026) 104848 3 the picture; many apparent regularities recede toward chance. This paper is organized as follows: The next section provides an overview of recent literature on the application of the LPPLS model in various nancial markets. Section three presents the datasets, while section four describes the methodology. Section ve presents the results, followed by a discussion in section six. Finally, the last section concludes. 2. Literature review The LPPLS model, initially introduced in the seminal work of Johansen et al. (2000), has since been subjected to extensive empirical scrutiny across diverse nancial domains. This approach has been particularly employed to characterize speculative phenomena and detect instability in cryptocurrency, stock, energy, and carbon credit markets. Sornette (2017) provides a comprehensive survey of studies employing the LPPLS model to examine nancial bubbles, whereas the present discussion focuses on a selective yet representative set of recent contributions to the literature. One recent line of inquiry concerns the application of the LPPLS framework to cryptocurrency markets, where it serves as an instrument to identify speculative phases and abrupt shifts in valuation. The work of Ahn et al. (2024), Grobys (2024), and Zhang et al., 2024 exemplies this effort. These studies reveal that super-exponential price escalations in Bitcoin and Ethereum exhibit distinct LPPLS signatures prior to sub- stantial corrections. However, methodological variations exist. While Ahn et al. (2024), Van Eyden et al. (2023), Johansen and Sornette (2001b), and Grobys (2024) adopt the conventional LPPLS framework to analyze historical data, Zhang et al., 2024 incorporates wavelet analysis to enhance granularity in bubble detection. Further distinctions arise in the temporal scope of inquiry: whereas Grobys (2024) spans more than a decade of Bitcoin price movements, Ahn et al. (2024) centers on discrete speculative intervals. Despite these variations, the studies collectively underscore the utility of the LPPLS model in diagnosing instability within digital asset markets. A second avenue of recent research concerns equity markets, where the LPPLS model has been deployed to detect speculative surges and anticipate crises in various national stock indices (e.g., Grobys, 2023; Gupta et al., 2023, 2025; Johansen & Sornette, 2001b). Investigations such as those of Cepni et al. (2025) and Zhao and Sornette (2021) share the common objective of rening bubble detection by incorporating LPPLS condence metrics. Some studies, such as Ji and Zhang (2024), advance the methodology by introducing a Sequential Quadratic Pro- gramming algorithm to optimize parameter estimation, while others, such as Zhao and Sornette (2021), depart from traditional imple- mentations by integrating event-study techniques to analyze post- bubble dynamics. Furthermore, the contributions of Shu et al. (2021) and Song et al. (2022) differentiate between endogenous and exogenous crashes, illustrating the adaptability of the LPPLS methodology to distinct modeling paradigms. Collectively, this body of work afrms the relevance of the LPPLS model in equity market research, while also demonstrating the necessity of methodological renement to enhance reliability. Parallel efforts have been directed toward energy and commodity markets, where the LPPLS model is employed to discern speculative uctuations in crude oil, gold, and agricultural commodities. Empirical studies, including those by Gupta et al. (2024), Chang (2024), Cifarelli and Paesani (2021), and Grobys (2025), afrm the applicability of LPPLS in this domain. Nonetheless, the specic methodological ap- proaches diverge: whereas Chang (2024) and Xu et al., 2025 supplement the LPPLS model with econometric techniques such as GARCH and Markov regime-switching models, Cifarelli and Paesani (2021) intro- duce a heterogeneous-agent framework to distinguish between funda- mentalist and speculative market participants. Additionally, Grobys (2025) applies a traditional power-law framework to gold futures, diverging from oil-focused analyses. Yang et al. (2024), meanwhile, integrates machine-learning classication to assess the interaction be- tween monetary policy uncertainty and energy price bubbles. These methodological variations highlight both the versatility of the LPPLS model framework and the necessity of auxiliary statistical tools in spe- cic nancial contexts. A nal stream of recent literature explores speculative episodes in carbon credit and environmental markets, with studies such as those by Ghosh et al. (2021) and Huang and Wang (2024) investigating the in- uence of regulatory interventions on bubble formation. While both studies conrm that the LPPLS model is a viable instrument for detecting speculative surges in carbon credit pricing, their approaches differ. Huang and Wang (2024) employ Supremum Augmented Dickey-Fuller Test (SADF) and Generalized Supremum Augmented Dickey-Fuller Test (GSADF) unit root tests to validate bubble episodes, whereas Ghosh et al. (2021) relies exclusively on LPPLS condence indicators. Additionally, their scopes diverge: Huang and Wang (2024) examine multiple emissions trading schemes across different jurisdictions, whereas Ghosh et al. (2021) focuses specically on carbon credit ETFs. Despite these methodological distinctions, both studies converge on the conclusion that speculation in carbon markets is signicantly inuenced by policy regimes and macroeconomic conditions, reinforcing the importance of regulatory oversight. Across all domains of nancial research, the LPPLS model remains a central tool for characterizing speculative bubbles, yet considerable variation exists in its empirical implementation. A primary point of differentiation concerns validation strategies: whereas studies in equity markets frequently benchmark LPPLS model forecasts against historical market downturns, as in Shu et al. (2021) and Song et al. (2022), others, such as Zhao and Sornette (2021), emphasize post-bubble event anal- ysis. Additionally, robustness assessments vary; LPPLS condence in- dicators are prevalent in stock market research but less frequently utilized in cryptocurrency and energy market studies. Furthermore, while econometric supplements such as GARCH and Markov-switching models may enhance LPPLS model applications in commodity mar- kets, alternative enhancements—such as machine-learning algorithms and wavelet decomposition—have emerged in cryptocurrency research. In sum, while the LPPLS model appears to consistently detect specula- tive behavior across diverse markets, its predictive efcacy is contingent upon methodological renements tailored to specic nancial contexts. 3. Data We investigate log-periodicity usingmonthly and daily nancial data in the main analysis. Consistent with Grobys (2023), the monthly dataset for the S&P 500 index was sourced from Robert Shiller’s publicly accessible data library (www.econ.yale.edu/~shiller/data.htm). To maintain comparability with prior research, the present study utilizes the same data sample as that of Grobys (2023), encompassing the period from January 1871 to November 2022. In addition to the monthly data, daily observations of the S&P 500 were obtained for the period spanning January 2, 1980, to December 31, 1986, from www.investing.com. This dataset aligns with the sample employed by Sornette (2017) and Grobys (2023) in the calibration of the LPPLS model. Notably, this period concludes 202 trading days prior to the widely recognized stock market crash of October 19, 1987. Grobys (2023) identies the October 1987 crash as an enduring in- tellectual enigma for at least three principal reasons. First, the economic signicance of a single-day decline exceeding 20 % in equity market capitalization was extraordinary. Second, the probabilistic occurrence of such an event was anomalous relative to conventional nancial models, an observation echoing Mandelbrot’s assertion that it constituted “a number outside the scale of nature” (Mandelbrot, 2008, p. 4). Third, the crash transpired in the absence of discernible premonitory signals, a phenomenon that remains unexplained within the framework of tradi- tional nancial theories. Similarly, Sornette (2017) underscores that extensive research has sought to elucidate the underlying causes of the K. Grobys International Review of Financial Analysis 110 (2026) 104848 4 October 1987 crash, particularly by examining trading behaviors and market structures. Despite these efforts, no denitive cause has been identied. Notably, the sharp market decline observed in October 1987 was preceded by an exceptional market surge over the rst nine months of the year, a pattern evident across multiple economies. In the United States, for instance, stock prices recorded a substantial increase of 31.4 % during this period. Some analysts contend that the downturn in October was a consequence of the preceding speculative bubble, driven by excessively inated asset prices. The October 1987 stock market crash is widely regarded as among the most signicant nancial collapses in modern history to date. If the dependency structures present in the monthly and daily datasets mani- fest as periodic oscillations—oscillations that, according to theoretical models, culminate in nite-time singularities—then employing i.i.d. bootstrapped data, as formulated by Efron (1992), would effectively eliminate these dependencies. Therefore, for each data set j = (1,2), we compute b = 1,…,B synthetic data sets as follows: (a) First, for each given data set, we compute the log-returns of the data: rj,t = ln ( Pj,t Pj,t1 ) , where Pj,t denotes the price of the S&P 500 at time unit t for a given data set j. (b) Second, we bootstrap each data vector j of log-returns using random sampling with replacement giving us B bootstrapped log- return vectors for each data set: rj,1 rj,2 … rj,B  =   rj,1,2 rj,1,3 rj,2,2 rj,2,3 … ⋮ rj,B,2 rj,B,3 ⋮ ⋮ ⋮ rj,1,T rj,2,T … rj,B,T   (T1),B (c) Third, for each data set, we compute B synthetic S&P 500 indices by means of compounding as: Pj,b,t = P0 exp t s=1 rj,b,s  , with P0 denoting the initial price of the corresponding original data set j: Pj,1 Pj,2 … Pj,B  =   Pj,1 Pj,1,2 Pj,1 Pj,2,2 … ⋮ Pj,1 Pj,B,2 ⋮ ⋮ ⋮ Pj,1,T Pj,2,T … Pj,B,T   T,B (d) Finally, the synthetic S&P 500 indices Pj,1 Pj,2 … Pj,B  are transformed in terms of their natural logarithms. This procedure is employed to generate B = 1000 synthetic log- indices. Tables A.1 and A.2 present the descriptive statistics for the original datasets, whereas Tables A.3 and A.4 summarize the corre- sponding statistics for the sample means obtained from i.i.d. boot- strapped samples,  rj,1 rj,2 … rj,B . A comparative analysis between Tables A.1 and A.3 reveals that the mean of the bootstrapped sample means (0.3710) closely approximates the sample mean of the original data (0.3696), a pattern similarly observed when comparing Table A.2 with Table A.4. This outcome is anticipated, as the boot- strapping methodology inherently preserves statistical properties, ensuring consistency in sample means. Figs. 1–2 display alternative historical paths of the S&P 500 generated from bootstrapped log- compounded returns for the monthly and daily samples, respectively. Because the monthly panel spans a long horizon, compounding mag- nies small return differences into large dispersion in price levels. The upper tail of the bootstrap (e.g., the 97.5 % quantile) therefore explodes and dominates the vertical scale, obscuring the bulk of the distribution. To preserve readability, Fig. 1 is plotted in log prices. In both gures, the realized series and the bootstrap median lie very close at the terminal date T—as expected under our resampling scheme: starting from the same initial value and drawing returns from the empirical distribution, the median compounded path tracks the data’s unconditional drift and thus aligns with the observed endpoint up to sampling variation. Fig. 1. Alternative historical trajectories for the S&P 500 based on log-compounded returns for January 1871–November 2022. This gure illustrates 1000 synthetic trajectories of the S&P 500 index generated by compounding monthly log-returns resampled with replacement from the his- torical data (January 1871–November 2022, source: Shiller database). Prices are plotted in natural logarithms to preserve readability. The red solid line denotes the realized historical path, while the other lines depict the empirical dispersion of the bootstrap simulations. The proximity of the realized and median synthetic paths at the terminal point reects that the bootstrap preserves unconditional drift but removes log-periodic dependencies. (For interpretation of the references to colour in this gure legend, the reader is referred to the web version of this article.) K. Grobys International Review of Financial Analysis 110 (2026) 104848 5 4. Methodology 4.1. Main analysis 4.1.1. Implementing the LPPLS model using log-prices of the S&P 500 A plain power law model for nancial log-prices is given by the following: ln[Pt ] = A+ B(tc  t)β, (1) where lnPj,t  denotes the logarithm of the value of the S&P 500 index at time t, tc is the critical time, A is the expected value of the logarithmic S&P 500 when approaching tc, B denes the exposure to faster-than- exponential growth, and β is the power law exponent controlling faster-than-exponential price growth (Sornette, 2017). The critical time tc indicates the end of the accelerating oscillations, which results in a nite-time singularity manifested in a regime change (Zhang et al., 2016). According to Sornette (2017), the simple power law model of Eq. (1) needs to be extended by accounting for periodic oscillations: ln[Pt ] = A+ B(tc  t)β[1+ Ccos(ωln(tc  t) + ϕ ) ], (2) where C denotes the exposure of the log-periodic oscillations around the power law singular growth, ω denotes the angular log-frequency of os- cillations during the formation of the bubble, ϕ is the phase parameter, and all other notations are as previously dened. The LPPLS model of Eq. (2) is rst implemented for the original data sets on the S&P 500 using the following set of constraints (Sornette, 2017): ∞ < A < ∞, 10 < B < 10, 01 ≤ β ≤ 09, C < 1, 5 ≤ ω ≤ 15, T ≤ tc ≤ 2T,  π < ϕ < π Note that 10 < B < 10 allows growth/decay exibility without excessive divergence, 01 ≤ β ≤ 09 ensures valid power-law behavior (prevents innite variance), C < 1 ensures oscillations remain moder- ate, 5 ≤ ω ≤ 15 prevents extreme oscillations while allowing exibility, T ≤ tc ≤ 2T ensures the critical time is in the future but within a reasonable range, and π < ϕ < π constrains the phase shift to avoid numerical instability. Furthermore, the LPPLS model is calibrated using the following initial starting values: A = ln[PT ] + 100, B =  010, β = 050, C = 001, ω = 600, tc = T+10, ϕ = 000 Note that, B = 010 is consistent with the expected super- exponential growth, β = 050 is chosen within a valid theoretical range corresponding to 01 ≤ β ≤ 09, C = 001 initiates small oscilla- tory contribution to start, ω = 600 corresponds to a common log- periodic oscillation frequency, tc = T + 10 ensures the predicted event arrives beyond the dataset, and ϕ = 000 is initially set to zero, but optimized during the calibration. After calibrating the LPPLS model using the original data sets, the model is calibrated for the synthetic datasets of S&P 500 log-prices. To ensure comparability, each LPPLS model calibration uses the same sample constraints and initial parameter values. 4.1.2. Testing the LPPLS signatures for statistical significance Lin et al. (2014) test the LPPLS hypothesis within a Generalized Autoregressive Conditional Heteroskedasticity (GARCH) framework. They rst generate synthetic nancial time series from GARCH(1,1) processes and calibrate LPPLS on rolling windows. Crucially, they then condition on LPPLS ts whose parameters fall within the canonical bounds (e.g., on β,ω,C, tc). For those parameter-admissible ts, they test residuals, lnPt  l̂nPt = ut , (3) for stationarity using Augmented Dickey–Fuller (ADF) regressions, Fig. 2. Alternative historical trajectories for the S&P 500 based on log-compounded returns for January 2, 1980–December 31, 1986. This gure presents 1000 synthetic trajectories of the S&P 500 index obtained from daily i.i.d. bootstrap resampling of returns from January 2, 1980, to December 31, 1986 (source: Investing.com). The red line indicates the observed price series, and the other lines show the range of bootstrapped paths. The median bootstrap path tracks the realized series closely, demonstrating that the procedure preserves rst-moment properties while eliminating periodic structure. (For interpretation of the references to colour in this gure legend, the reader is referred to the web version of this article.) K. Grobys International Review of Financial Analysis 110 (2026) 104848 6 Δut = δut1 + γ1 Δut1 + ⋯ + γp Δutp + εt , (4) where εt is assumed i.i.d., and rejection of a unit root at conventional tabulated levels (e.g., 5 %, 1 %, 0.1 %) is interpreted as evidence of a statistically signicant LPPLS “signature.” Under this conditioned pro- cedure, Lin et al. (2014) report a very low false-positive rate in GARCH- generated series (about 0.2 % of samples satisfy the bounds and yield stationary residuals). In their study, unit-root tests on residuals reject stationarity in non-bubble periods, whereas parameter-admissible, LPPLS-identied episodes exhibit stationary residuals—suggesting, at least preliminarily, that the LPPLS model can distinguish bubble regimes from heteroskedastic but otherwise standard dynamics. Note that in the study of Lin et al. (2014), the GARCH model is used to simulate returns, which are then compounded to prices before cali- brating the LPPLS model under parameter bounds; residuals from ad- missible LPPLS model ts are subsequently tested with ADF tests using tabulated critical values. This design treats GARCH as a generator of conditional variance dynamics for returns—under α+ β < 1, returns are covariance-stationary and conditional variance is mean-reverting—- while the LPPLS framework species a deterministic structure in (log) price levels with a power-law acceleration and log-periodic oscillations. However, for asset markets like the S&P 500, empirical work often points to highly persistent volatility (near-IGARCH), so a single low- order parametric GARCH model (or even GARCH models with t- distributed innovations) may underrepresent salient features of the data- generating process for the purposes of LPPLS residual ADF testing. To avoid embedding a particular volatility law in the null, our study uses an i.i.d. bootstrap of returns, compounding to prices to preserve the empirical marginal distribution while excluding any imposed log- periodic structure. We then re-estimate LPPLS and re-test on each resample, constructing residual-ADF critical values that are aligned with the exact two-stage procedure (same bounds, starting values, and BIC lag selection). As outlined in Section 3, the rst dataset comprises bootstrapped monthly observations spanning January 1871 to November 2022, while the second dataset consists of bootstrapped daily observations covering the period from January 2, 1980, to December 31, 1986. As noted earlier, our i.i.d. resampling removes serial dependence by construction, enabling a more rigorous evaluation of the LPPLS signature (log-peri- odic structure) versus noise. Consequently, the ADF test statistics are estimated for the residuals of LPPLS models calibrated to synthetic data and compared with those derived from the original data. 4.2. Additional analysis 4.2.1. Benchmarking LPPLS-residual ADF false-positive rates against full- sample BADF tests To benchmark the false-positive rates from the LPPLS-residual ADF test we perform additional analysis by implementing a full-sample ADF test for mild explosiveness on log prices. Specically, we assess whether a plain ADF test on levels—implemented with a comparable sample length T and the same BIC lag-selection rule—yields higher, lower, or similar false-positive rates relative to the ADF test applied to LPPLS residuals. For both tests, we report two sets of critical values: (i) Monte- Carlo “tabulated” values computed under a generic unit-root null (a model-agnostic baseline), and (ii) bootstrap-based values obtained by i. i.d. resampling from the same synthetic data-generating process used earlier to calibrate the LPPLS results, re-estimating the full estima- tion–test sequence on each resample. Using an identical synthetic generator ensures a methodologically aligned comparison across testing procedures. This design directly contextualizes the estimated false pos- itive rates for the LPPLS residual tests against a widely used ADF-type benchmark under matched sample and lagging conditions. It is important to note that standard ADF response-surface tables (e. g., MacKinnon) report left-tailed critical values for tests of stationarity under various deterministic specications, but they do not provide right- tailed cutoffs for the explosive alternative H1 : ρ > 0. In addition, our implementation uses nite-sample BIC lag selection, which alters the null distribution relative to xed-lag tables. For these reasons, and to supply a standard, model-agnostic benchmark, we compute right-tailed ADF critical values by Monte Carlo under a generic unit-root null cali- brated to our sample length T, deterministic specication, and lag- selection rule. This follows common practice in the explosive-root literature, where inference for right-tailed ADF-type tests is obtained by simulation rather than pre-tabulated values. We then apply the full- sample right-tailed ADF to log prices and evaluate signicance using these Monte-Carlo critical values. In parallel, we also report bootstrap- based critical values obtained by i.i.d. resampling (re-estimating the procedure on each resample) for both the LPPLS-residual ADF and the ADF benchmark, to examine whether size is restored when inference matches the nite-sample procedure. Furthermore, our empirical design chosen for our main analysis es- timates a single LPPLS model specication on a xed window (monthly or daily) and then makes one test decision based on the residual ADF test—there is no scanning over sub-samples. The most appropriate benchmark is the full-sample right-tailed ADF on log prices (equiva- lently, the backward ADF (BADF) evaluated at the terminal date T in the Phillips–Shi–Yu framework): it applies the same decision logic (one test on one sample), uses the same finite-sample ingredients (BIC lag selection, given T), and targets the same alternative (mild explosiveness, H1 : ρ > 0). By contrast, SADF and especially GSADF are search pro- cedures: they maximize ADF test statistics over expanding or moving windows and reject if any window is explosive. That supremum struc- ture introduces a built-in multiple-testing component and a different null distribution; using SADF/GSADF as the benchmark would conate differences in window-search multiplicity with differences in model performance, and would not reect how LPPLS is applied here. Finally, among alternatives, ADF-type tests are the most widely used and transparent baseline for (non-)stationarity and mild explosiveness in nancial time series, whereas Markov-switching models require stron- ger parametric assumptions (regime number, transition structure), are sensitive to starting values and identication, and yield results that are less directly comparable to a single-window LPPLS decision rule. For these reasons, we use the full-sample ADF (BADF) as the benchmark and report both Monte-Carlo “tabulated” and bootstrap-based critical values to place our LPPLS ndings in a standard, methodologically aligned context. We design the Monte Carlo experiment for the null data-generating process as follows: For each replication b = 1,…,B we generate a unit- root series of the length T = 2000 similar as in the empirical application2: y(b)0 = 0, y(b)t = y(b)t1+ u(b)t , u(b)t ∼ N (0, 1), t = 1,…,T The innovation variance is without loss because the ADF t-statistic is scale-invariant under H0. On each simulated path y(b)1:T we estimate the standard ADF regression Δyt = α+ γt+ ρyt1 + ∑p i=1 ϕi Δyti + εt , with the deterministic specication xed ex ante to a constant term that is α ∕= 0 and γ = 0. It is noteworthy that the ADF test regression used 2 Using the exact sample lengths T = 1770and T = 1823 leads to the same critical values, up to negligible Monte Carlo error. K. Grobys International Review of Financial Analysis 110 (2026) 104848 7 here deviates from the main analysis and this is for the following reason: The null model used to obtain critical values determines their distribu- tion. Our Monte Carlo “tabulated” critical values are generated under a generic Gaussian unit-root process with no drift and the same BIC lag rule. By contrast, the bootstrap critical values are generated from i.i.d. resampled S&P 500 returns, which are chained back to price levels, thereby inheriting the marginal features of the data (e.g., non-zero mean, heavy tails) but not serial dependence. Under a right-tailed ADF test without an intercept (AR specication), even a modest positive drift and fat-tailed innovations shift mass into the right tail of the statistic, producing much larger critical values—this reects the chosen null, not an error. Moreover, with long samples and BIC typically selecting few lags, standard errors are small, so slight positive deviations of ρ from zero can yield large t-statistics, further thickening the right tail under the bootstrap null. These considerations motivate us to employ the bench- mark ADF test on log prices with an intercept to account for drift in equity indices, providing a specication that is methodologically aligned with the data-generating features. The lag length p is selected by BIC over p = 0,…,K, where we use a Schwert-style cap K = ⌊12(T100)14 ⌋. For each replication we record the ADF t-statistic on yt1 (the coefcient ρ). Using the B simulated t-statistics, we estimate the left- and right-tailed critical thresholds via the empirical quantiles of the simulated distribution. • Left-tail (stationarity alternative H1 : ρ < 0): the 10th, 5th, and 1st percentiles (Q010,Q005,Q001). • Right-tail (mild explosiveness H1 : ρ > 0): the 90th, 95th, and 99th percentiles (Q090,Q095,Q099). The Monte-Carlo “tabulated” critical values are calibrated to our empirical design—they match the sample length, impose the presence of a constant term, and implement the identical BIC lag-selection rule—yet they remain model-agnostic, as they are generated under a generic unit- root null rather than via the LPPLS model estimation procedure. Note again that all inference is conducted on a single, prespecied full sample per dataset. No rolling-window search is performed; consequently, no multiple-testing adjustment is required. Our benchmark BADF refers to the full-sample ADF with lag length selected by BIC, against which we report empirical size and rejection behavior. 4.2.2. Gold futures: cross-market robustness and replication To assess whether our conclusions extend beyond U.S. equities, we replicate the entire estimation–testing procedure on gold futures. Gold is an instructive comparison for three reasons. First, it is a large and sys- temically relevant market that attracts sustained attention from in- vestors and policymakers. Second, it has been the focus of renewed interest in recent years, with research documenting structural shifts in demand and pricing dynamics. Third, exchange-traded futures provide liquid, high-frequency price series with transparent contract specica- tions, which are well suited to the bootstrap resampling and re- estimation design used here. Using the same daily gold futures data as in the study of Grobys (2025)—December 2, 2015 to November 6, 2024—and applying the same LPPLS model estimation settings as in our S&P 500 analysis (Section 4.1.1), we ensure results that are directly comparable across asset classes and, at the same time, conduct a sci- entic replication of the study of Grobys (2025). Scientic replication is particularly important in empirical nance (e.g., Hou et al., 2020), where evidence is primarily observational and reliability is best assessed against specications that are similar but not identical. The objective here is a robustness check of our main ndings; a broader examination of emerging markets or cryptocurrencies is left for future research. 4.2.3. Dependence-aware resampling via the stationary (geometric) block bootstrap To assess robustness to serial dependence and volatility clustering in returns, we complement the i.i.d. bootstrap with the stationary (geo- metric) block bootstrap applied to log-returns. The procedure generates resamples of length T by concatenating contiguous blocks whose lengths are geometrically distributed with expected value m. Operationally, at each step t > 1 a new block is initiated with probability p = 1m at a randomly drawn start index (sampling start indices with replacement); otherwise the resample advances to the next observation in the current block, wrapping around at T. This design preserves generic short-range dependence without imposing a specic parametric structure. We set the expected block length to m = ⌈T13⌉, a conventional choice in the block-bootstrap literature, and keep the total length of synthetic samples equal to the original sample size T. For each resam- pled return path, we compound to a synthetic price series, re-estimate LPPLS under the same parameter bounds and initialization as in the baseline specication, compute residuals, and apply the residual ADF test with lag order selected by BIC. For illustration, we implement this dependence-aware resampling for the monthly S&P 500 series using B = 1, 000 bootstrap replications. 4.2.4. Start–value robustness: randomizing starting values Nonlinear least squares estimation of the LPPLS model can exhibit sensitivity to initial conditions because the objective function is non- convex and the nonlinear parameters (β,ω,ϕ, tc) may admit multiple local minima. To ensure that the reported calibration is not an artefact of a particular initialization, we conduct a start–value robustness check on the original sample. The aim is twofold: (i) to verify that the solution we report is representative within the admissible parameter region dened in the present study, and (ii) to document basic convergence diagnostics under a transparent and replicable initialization scheme. Methodologically, we carry out a multi–start calibration in which only the nonlinear parameters are randomized, while the linear co- efcients are set by conditional least squares. Specically, for each run k = 1,…,K we draw β(0) ∼ Unif  β  , β  , ω(0) ∼ Unif [ ω  ,ω ] , ϕ(0) ∼ Unif[ π, π], t(0)c ∼ Unif(tmax + 1, tmax +T], using the same bounds ( β  , β) and (ω  ,ω) as in the main analysis (Section 4.1.1), where tmax is the last in–sample time point and T the sample length (so that τ = tc  t > 0 for all observations). Conditional on the draw  β(0) ,ω(0) ,ϕ(0) , t(0)c  , we construct the LPPLS regressors and obtain the linear coefcients—denoted A(0),B(0),C(0)—by ordinary least squares. The resulting vector θ(0) = A(0) ,B(0) , β(0) ,C(0) ,ω(0) , t(0)c ,ϕ(0)  is then passed to the constrained nonlinear least squares estimation procedure under the same parameter bounds dened in section 4.1.1. This “nonlinear–randomized, linear–OLS” initialization concentrates randomization where it is empirically consequential (the nonlinear block) while exploiting separability of the model to set the linear block efciently. We repeat this procedure for K independently generated starts. For each run we examine the optimizer’s exit ag, objective value (sum of squared residuals), and the estimated parameter vector. Convergence is dened ex ante by a positive solver exit ag together with nite, in–bounds estimates. 4.2.5. Profiling the objective in the critical time tc LPPLS model estimation is known to exhibit weak curvature in the K. Grobys International Review of Financial Analysis 110 (2026) 104848 8 critical time tc, which can translate into wide dispersion of tc across admissible initializations without materially changing t quality. To examine this directly—and to document identication in tc without altering our study’s inferential focus—we construct a prole of the least- squares criterion in tc. The idea is standard: for each xed value of tc, we re-estimate all remaining parameters to (locally) minimize the sum of squared residuals (SSE), and we then view the minimized SSE as a function of tc. A broad, shallow trough in this prole indicates weak identication of tc. Let tmax denote the last in-sample time point and T the sample length. We evaluate the prole on a uniform grid of 200 points over the ad- missible interval tc ∈ (tmax + 1, tmax +T], which ensures τ = tc  t > 0 for all observations. For each grid value tc = gi, we x tc by setting its lower and upper bounds equal to gi and then constrained-optimize the remaining parameters under the same bounds and numerical tolerances dened in Section 4.1.1. The starting vector for the rst grid point is the unconstrained optimum θ from the original monthly S&P 500 data sample; for subsequent points we retain the same starting vector. We record, for each grid point, the solver exit ag, the estimated parameter vector, and the minimized SSE. A grid evaluation is classied as successful if the solver returns EF > 0 (success ag), the estimates are nite and within bounds, and SSE is nite. Otherwise it is treated as a failure and omitted from the prole. This treatment isolates the geometry of the criterion from occasional local non-convergence when tc is exogenously xed. To further aid interpretation of the pro- le, we plot a 0.5 % tolerance line as a horizontal reference at 1005× minuSSE(u), where minuSSE(u) is the minimum SSE attained over the tc grid. Formally, if SSEmin = minuSSE(u), the threshold is SSEthr = 1005SSEmin. Grid points with SSE(tc) ≤ SSEthr deliver ts that are within 0.5 % of the best attainable on the grid and can be regarded as practically indistinguishable in t quality. 5. Results 5.1. Main results 5.1.1. Results from original data sets on the S&P 500 First, we t the original monthly S&P 500 log-price series (January 1871–November 2022) to the LPPLS specication in Eq. (2), using the constraints and starting values in Section 4.1.1. The estimates in Table 1 indicate a power-law exponent β = 04884, consistent with moderate super-exponential acceleration as the critical time is approached. The critical time is estimated at tc = 18596166, i.e., about 36.62 months beyond the sample endpoint T = 1823. Next, we t the daily S&P 500 log-price series (January 2, 1980–December 31, 1986) to the same LPPLS model with identical constraints and initial values. As reported in Table 1, the estimated exponent β = 02195 implies a pronounced super-exponential acceleration, with tc = 18794155, approximately 109.42 days beyond the daily sample endpoint T = 1770. In both fre- quencies, β ∈ (0, 1) and tc > T, delivering qualitatively similar LPPLS signatures; we therefore turn to residual diagnostics in terms of unit-root testing to assess their statistical credibility. Whereas Figs. 3 and 4 show the evolutions of the log-prices for the original data as well as the LPPLS model, Figs. 5 and 6 plot the estimated model residuals over the sample. Visual inspection of the estimated model residuals suggests stationarity. Using the residuals as dened in Eq. (3), we carry out ADF tests for the original data sets in line with Eq. (4). Consistent with Grobys (2023&2025), we choose the lag-order in line with the Schwarz-Criterion. From Table 2 we observe that the re- siduals from both data sets are stationary even on a 1 % level. Since Lin et al. (2014) assess LPPLS consistency by applying ADF tests to residuals at conventional signicance levels (e.g., 5 %, 1 %, 0.1 %), our 1 % rejections—under tabulated critical values—would likewise be classi- ed as statistically signicant LPPLS signatures. 5.1.2. Results from synthetic monthly S&P 500 data We begin by calibrating the LPPLS model in Eqs. (1–2) to the syn- thetic monthly datasets. Table 3 reports descriptive statistics for the estimated parameters. As expected under hard bounds, several estimates lie near the admissible limits. In particular, the power-law exponent β spans most of the constrained interval: recall that smaller β (closer to 0) implies more pronounced super-exponential acceleration as t→tc, whereas larger β (closer to 1) approaches exponential growth and in- dicates weaker acceleration. The median estimated critical time is tc = 20539337, about 19.24 years beyond the sample endpoint, and Table 1 Calibration of the LPPLS model on original S&P 500 datasets. Data set A B β C ω tc ϕ Monthly 10.3330 0.2454 0.4884 0.0534 7.2116 1859.6166 3.1416 Daily 6.6236 0.3575 0.2195 0.0292 5.6505 1879.4155 3.1416 This table reports estimated parameters of the LPPLS model applied to the original monthly (1871–2022) and daily (1980–1986) log-price series. Parameters are dened in Eq. (2); estimation follows bounded nonlinear least squares with standard parameter constraints. The critical time (tc ) denotes the inferred termination point of super-exponential growth, measured relative to the sample endpoint. Fig. 3. Calibration of the LPPLS model on monthly S&P 500 data (January 1871–November 2022). The gure plots the observed log S&P 500 prices (blue solid line) alongside the tted Log-Periodic Power Law Singularity (LPPLS) curve (grey line). The cali- bration applies the constrained parameter set described in Section 4.1.1. The estimated trajectory exhibits moderate super-exponential acceleration toward the critical time, consistent with the model’s expected behavior under bounded parameters. (For interpretation of the references to colour in this gure legend, the reader is referred to the web version of this article.) K. Grobys International Review of Financial Analysis 110 (2026) 104848 9 substantially later than thetc obtained from the original data. Moreover, the earliest critical time estimate is essentially at the lower bound—the month when the sample ends (tc = T = 1823)—while the latest sits at the upper bound (tc = 2T = 3646). Thus, the synthetic-data estimates effectively span the imposed window T < tc < 2T, underscoring the role of the constraints and the limited identication of tc in this setting. Table 4 reports the correlation matrix of the LPPLS parameters estimated from the synthetic monthly data. Nineteen of the twenty-one pairwise correlations are statistically signicant at the 5 % level. For example, the correlation between A and tc is 0.54 with a t-statistic of 20.13, indicating signicance at any conventional level; a larger esti- mated critical time naturally coincides with a higher tted terminal log- level, yielding a positive association between the level parameter and tc. By contrast, the phase ϕ and the angular log-frequency ωare negatively correlated (0.45, t =  1581), consistent with the view that phase shifts and frequency adjustments can partially substitute in tting oscillatory structure. While many remaining correlations are modest in magnitude, relatively strong negative associations—such as (ϕ, ω), (A, B), and (A, β)—point to parameter trade-offs that merit further investigation, particularly in relation to identication strength and the role of hard bounds. How often do “signicant” LPPLS signatures arise by chance? If the 1 % level used in practice (e.g., Lin et al., 2014) were appropriate for residual ADF testing, then under synthetic monthly log-price paths the empirical rejection rate should be about 1 %—i.e., only ~1 % of left-tail ADF statistics should fall below the tabulated 1% critical value (≈2.57 for the model specication used in the ADF test regression). Table 5 summarizes the distribution of ADF statistics from 1000 LPPLS cali- brations on synthetic monthly data. The median is λADF =  34158; since the tabulated 1% cutoff is2.57, at least half of the samples would be labeled “LPPLS-consistent” under conventional thresholds. In fact, 848/1000 statistics satisfy λADF <  257, implying a false-positive rate of 84.8 % at the 1 % level. Consistently, the bootstrap left tail critical values at the 10 %/5 %/1 % signicance levels are 4.3753/4.6774/ 5.2444, all much more negative than the tabulated values. At the 5 % level, 969/1000 statistics lie below the tabulated cutoff (≈ 1.94), implying a 96.9 % false-positive rate under table thresholds. Applying Fig. 4. Calibration of the LPPLS model on daily S&P 500 data (January 2, 1980–December 31, 1986). This gure compares daily log-prices of the S&P 500 with the tted LPPLS curve under the standard parameter bounds. The estimation captures stronger acceleration toward the inferred critical point relative to the monthly sample, suggesting that short-horizon data amplify the LPPLS curvature. Fig. 5. Residuals of the LPPLS model for monthly S&P 500 data (1871–2022). The gure shows the residuals from the LPPLS t in Fig. 3. Visual inspection suggests approximate stationarity, a property formally assessed via Augmented Dickey–Fuller tests reported in Table 2. These residuals form the basis for evaluating the statistical signicance of LPPLS “signatures.” Fig. 6. Residuals of the LPPLS model for daily S&P 500 data (1980–1986). This gure presents the residuals from the LPPLS calibration in Fig. 4. The residual series appears mean-reverting, consistent with the stationarity results in Table 2, though later bootstrap analyses demonstrate that conventional ADF thresholds overstate signicance. Table 2 Augmented Dickey–Fuller (ADF) tests of LPPLS residuals from original datasets. Monthly data Daily data ADF test statistic Lags ADF test statistic Lags 3.2277*** 2 2.6194*** 1 *** Statistically signicant on a 1 % level. ADF statistics test the null of a unit root in residuals from the LPPLS ts in Table 1. Critical values correspond to MacKinnon one-sided thresholds (10 %, 5 %, 1 %). The critical values for statistical signicance at the 10 %, 5 %, and 1 % levels are 1.62, 1.94, and  2.57, respectively. Lag order is selected by Schwarz criterion. Signicant rejections under tabulated values imply apparent LPPLS “signatures,” though later bootstrap analysis reveals size distortion. K. Grobys International Review of Financial Analysis 110 (2026) 104848 10 the size-correct bootstrap critical values to the original monthly residual statistic, λADF =  32277, we do not reject even at 10 % (bootstrap p = 05960). These results document a large gap between empirical ADF quantiles relevant for LPPLS residuals and tabulated thresholds, with table values inducing substantial size distortion. 5.1.3. Results from synthetic daily S&P 500 data Table 6 reports descriptive statistics for the LPPLS parameters esti- mated from the synthetic daily data. As with the monthly simulations, the power-law exponent spans the admissible range, 01 ≤ β ≤ 09. The median critical time is tc = 19584728, which is 79.0573 trading days later than thetc obtained from the original daily sample. Consistent with the imposed bounds, the parameters β, C, ω, ϕ, and tc collectively cover the full constrained ranges. In particular, the earliest tccoincides with the sample endpoint (tc = T = 1770), while the latest sits at the upper boundary (tc = 2T = 3540), indicating that the synthetic-data estimates effectively span the identication window T < tc < 2T. A key distinction between the two samples is informational: for the daily pre-crash window, the subsequent collapse on 19 October 1987 is an ex-post fact, whereas for the monthly long-horizon series no com- parable terminal event is observed within the sample. The question, therefore, is whether the LPPLS—when paired with residual ADF tests—distinguishes fact from ction in resampled daily data. Table 7 reports ADF statistics from 1000 LPPLS calibrations on synthetic daily series. Mirroring the monthly results, the median is λADF =  33332, well below the tabulated 1 % left-tail cutoff (≈ 2.57), and 828/1000 statistics satisfy λADF <  257, implying an 82.8 % false-positive rate at the nominal 1 % level under table values. The bootstrap left-tail critical values for the daily sample are 4.3808/4.6898/5.1854 at the 10 Table 3 Calibration of the LPPLS model on synthetic monthly data. Statistic A B β C ω tc ϕ Minimum 3.4434 10.0000 0.1000 0.9990 5.0000 1823.0000 3.1416 1 % Qnt. 5.2267 10.0000 0.1055 0.2990 5.0000 1823.0000 3.1416 2.5 % Qnt. 5.6948 10.0000 0.1332 0.2219 5.0000 1823.0000 3.1416 5 % Qnt. 6.4211 10.0000 0.1527 0.1609 5.0000 1823.0000 3.1416 10 % Qnt. 7.2450 5.9002 0.1945 0.1049 5.0000 1823.0020 3.1416 Median 10.8462 0.0180 0.8070 0.0516 5.8752 2053.9337 1.2299 90 % Qnt. 27.3032 0.0063 0.9000 0.1534 6.6062 3223.0592 3.1416 95 % Qnt. 37.1718 0.0053 0.9000 0.2060 7.0268 3646.0000 3.1416 97.5 % Qnt. 41.4985 0.0044 0.9000 0.2547 7.5551 3646.0000 3.1416 99 % Qnt. 48.3577 0.0032 0.9000 0.3527 8.3848 3646.0000 3.1416 Maximum 53.8685 0.0016 0.9000 0.7852 15.0000 3646.0000 3.1416 Mean 13.9631 1.1505 0.6955 0.0420 5.8897 2276.5283 0.2223 Standard Deviation 9.2136 2.9747 0.2528 0.1255 0.7787 536.2921 2.9485 Excess Kurtosis 4.3123 4.5739 0.1012 12.6518 22.6485 1.1311 1.8925 Skewness 2.1609 2.5225 1.0894 0.9446 2.8434 1.4725 0.1252 T 1000 1000 1000 1000 1000 1000 1000 Table 3 summarizes empirical distributions of LPPLS parameters estimated from 1000 bootstrapped monthly datasets. We report extreme values (minimum/ maximum), selected quantiles (1 %, 2.5 %, 5 %, 10 %, 50 %, 90 %, 95 %, 97.5 %, 99 %), the mean and sample standard deviation, skewness, excess kurtosis (kurtosis relative to the normal distribution), and the sample size T. Quantiles are computed from the empirical distribution. Table 4 Correlation matrix of estimated LPPLS parameters derived from monthly synthetic data. Panel A. Estimated correlation matrix of LPPLS parameters. A B β C ω tc ϕ A 1.00 0.87*** 0.69*** 0.06** 0.31*** 0.54*** 0.25*** B 1.00 0.79*** 0.06* 0.23*** 0.29*** 0.24*** β 1.00 0.06** 0.12*** 0.17*** 0.21*** C 1.00 0.09*** 0.04 0.08*** ω 1.00 0.38*** 0.45*** tc 1.00 0.19*** ϕ 1.00 Panel B. Estimated t-statistics. A – 55.40 30.29 2.03 10.48 20.13 8.25 B – 40.71 1.86 7.32 9.63 7.96 β – 1.96 3.93 5.54 6.86 C – 2.74 1.40 2.63 ω – 13.06 15.81 tc – 6.07 ϕ – Panel A of this table presents the correlation matrix of estimated LPPLS parameters derived from monthly synthetic data. For each correlation rij in Panel B, we report the corresponding t-statistic for testing H0 : ρij = 0 against H1 : ρij ∕= 0, based on the usual small-sample approximation under joint normality: tij = rij ̅̅̅̅̅̅̅̅̅̅̅̅̅̅ Nij  2 1 r2ij √ ,with degrees of freedom df = Nij  2, where Nij is the number of paired observations for series iand j. Two-sided p-values can be obtained from the Student-t distribution with Nij  2 degrees of freedom. K. Grobys International Review of Financial Analysis 110 (2026) 104848 11 %/5 %/1 % levels, respectively—very close to those from the synthetic monthly data (4.3753/4.6774/5.2444), underscoring that the empirical quantiles relevant for LPPLS residuals are much more negative than the tabulated thresholds. At the 5 % level, 959/1000 statistics lie below the tabulated cutoff (≈ 1.94), yielding a 95.9 % table-based false-positive rate. Finally, applying the size-correct bootstrap critical values to the original daily residual statistic (λADF =  26194), we do not reject even at 10 % (bootstrap p = 08130). Taken together, these daily results reinforce the monthly evidence: there is a large gap be- tween the empirical ADF quantiles relevant for LPPLS residuals and tabulated values, and reliance on table thresholds induces substantial size distortion. 5.2. Additional results 5.2.1. Results from benchmarking with the BADF test The preceding sections established that ADF tests applied to LPPLS residuals are subject to size distortion when evaluated against conven- tional (tabulated) critical values, and that bootstrap critical values aligned with the two-stage procedure restore size. To place those nd- ings in context, we now report a benchmark based on log prices: a full- sample right-tailed ADF (BADF) implemented on the same monthly and daily S&P 500 samples (with comparable sample lengths). This bench- mark does not search over windows and therefore mirrors our single- window LPPLS decision rule, while targeting a different alternative (mild explosiveness in levels rather than residual stationarity). For completeness and comparability, we provide both sets of thresholds for BADF—Monte-Carlo “tabulated” critical values matched to T and specication, and bootstrap critical values obtained by re-estimating the BADF on i.i.d. resamples. The goal is not episode dating, but to assess whether a standard, widely used ADF-type test on levels ags mild explosiveness in the same samples and under the same nite-sample choices. Using T = 2, 000 and B = 10,000 in the Monte-Carlo design, the right-tailed ADF critical values—i.e., the 90th, 95th, and 99th percen- tiles of the null distribution—are  04150,  00755, and 05909, respectively. Applying the full-sample ADF (BADF) to the log S&P 500 price series (model specication including constant terms, BIC lag se- lection) yields test statistics of 1.1775 (monthly) and  0.5560 (daily). That is, using the Monte-Carlo “tabulated” right-tail cutoffs, we reject Table 5 Distribution of ADF statistics for residuals from synthetic monthly data. Statistic ADF statistic Lag Length Minimum 6.0201 0 1 % Qnt. 5.2444 0 2.5 % Qnt. 4.9290 0 5 % Qnt. 4.6774 0 10 % Qnt. 4.3753 0 Median 3.4158 0 90 % Qnt. 2.4143 0 95 % Qnt. 2.1398 0 97.5 % Qnt. 1.8772 1 99 % Qnt. 1.4904 4 Maximum 0.1412 13 Mean 3.4052 0.1090 Standard Deviation 0.7880 0.7885 Excess Kurtosis 0.3515 129.1491 Skewness 0.0360 10.5582 T 1000 1000 This table presents the descriptive statistics for estimated ADF-test statistics for residuals derived from synthetic monthly data. The critical values for 10 %, 5 %, and 1 % statistical signicance levels for the standard ADF test are 1.62, 1.94, and  2.57. The lag-order is chosen in line with the Schwarz-Criterion. The large fraction of statistics below tabulated critical values indicates ina- ted false-positive rates when conventional thresholds are used. Table 6 Calibration of the LPPLS model on synthetic daily data. Statistic A B β C ω tc ϕ Minimum 2.3015 6.7295 0.1000 0.9990 5.0000 1770.0000 3.1416 1 % Qnt. 4.4415 5.0184 0.1000 0.9990 5.0000 1770.0000 3.1416 2.5 % Qnt. 4.6750 4.0701 0.1000 0.6280 5.0000 1770.0000 3.1416 5 % Qnt. 4.9269 2.8071 0.1000 0.2940 5.0000 1770.0000 3.1416 10 % Qnt. 5.1661 1.5280 0.1000 0.1518 5.0000 1770.0026 3.1416 Median 5.9403 0.0057 0.6647 0.0541 6.0280 1958.4728 1.7520 90 % Qnt. 8.7485 0.0006 0.9000 0.3484 7.5760 2761.7118 3.1416 95 % Qnt. 11.0187 0.0004 0.9000 0.6301 8.3212 3495.5588 3.1416 97.5 % Qnt. 13.9107 0.0002 0.9000 0.9990 9.4924 3540.0000 3.1416 99 % Qnt. 15.9048 0.0583 0.9000 0.9990 11.7616 3540.0000 3.1416 Maximum 20.1571 1.2081 0.9000 0.9990 15.0000 3540.0000 3.1416 Mean 6.5924 0.4068 0.5915 0.0736 6.2174 2147.6659 0.3259 Standard Deviation 2.1519 1.0428 0.3085 0.3054 1.2607 470.5718 2.9139 Excess Kurtosis 10.1349 11.0132 1.3007 4.2051 12.1262 2.1382 1.8510 Skewness 2.9543 3.2527 0.4848 0.0701 2.7723 1.6771 0.1898 T 1000 1000 1000 1000 1000 1000 1000 Table 6 summarizes empirical distributions of LPPLS parameters estimated from 1000 bootstrapped daily datasets. We report extreme values (minimum/maximum), selected quantiles (1 %, 2.5 %, 5 %, 10 %, 50 %, 90 %, 95 %, 97.5 %, 99 %), the mean and sample standard deviation, skewness, excess kurtosis (kurtosis relative to the normal distribution), and the sample size T. Quantiles are computed from the empirical distribution. Table 7 Descriptive statistics for estimated ADF-test statistics for residuals derived from synthetic daily data. Statistic ADF Statistic Lag Length Minimum 6.0721 0 1 % Qnt. 5.1854 0 2.5 % Qnt. 4.9596 0 5 % Qnt. 4.6898 0 10 % Qnt. 4.3808 0 Median 3.3332 0 90 % Qnt. 2.2983 0 95 % Qnt. 2.0368 0 97.5 % Qnt. 1.6306 1 99 % Qnt. 1.2951 1 Maximum 0.5915 4 Mean 3.3261 0.0360 Standard Deviation 0.8224 0.2622 Excess Kurtosis 0.1678 128.7633 Skewness 0.0783 10.2625 T 1000 1000 This table presents the descriptive statistics for estimated ADF-test statistics for residuals derived from synthetic daily data. The critical values for 10%, 5%, and 1 % statistical signicance levels for the standard ADF test are 1.62, 1.94, and  2.57. The lag-order is chosen in line with the Schwarz-Criterion. The lag- order is chosen in line with the Schwarz-Criterion. K. Grobys International Review of Financial Analysis 110 (2026) 104848 12 H0 : ρ = 0 for monthly data by a substantial margin, including at the 1 % level, whereas for daily data, we fail to reject H0 : ρ = 0 even on a 10 % level (see Table 8). These ndings indicate that the BADF test detects mild explosiveness only for the monthly sample. Table 9 reports descriptive statistics for the BADF test statistics from B = 1,000 i.i.d. bootstrap replications based on the monthly S&P 500 log-price series used in the main analysis. That is, in each replication, we resample returns i.i.d., reconstruct the price path, take log prices, and re- estimate the BADF statistic under the bootstrap null using the same specication (intercept included; BIC lag selection). The bootstrap dis- tribution is centered to the left of the tabulated 10 % right-tail cutoff: its median Q050 = 04787 is below λADF,090 =  04150. The bootstrap right-tailed critical values are 0.7566, 1.0305, and 1.7350 at the 10 %, 5 %, and 1 % levels, respectively. At the nominal 1 % level (critical value 0.5909), the empirical rejection rate under the bootstrap null is 13 %, i. e., the test is oversized. When evaluated against the bootstrap critical values (Table 9), the BADF statistic for the original monthly sample, λADF = 11775, exceeds the 5 % cutoff (1.0305) but not the 1 % cutoff (1.7350); thus, we reject H0 : ρ = 0 at the 5 % level but not at 1 % (the bootstrap p-value is 0.037). Table 10 reports descriptive statistics for the BADF test statistics from B = 1,000 i.i.d. bootstrap replications based on the daily S&P 500 data. In each replication, we resample returns i.i.d., reconstruct the price path, take log prices, and re-estimate the BADF statistic under the bootstrap null using the same specication (intercept included; BIC lag selection). As in the monthly case, the bootstrap distribution is shifted left relative to the tabulated cutoff: its median, Q050 =  08711, lies below the Monte-Carlo ‘tabulated’ 10 % right-tail critical value, λADF,090 =  04150. The bootstrap right-tailed critical values are 0.4935, 0.8216, and 1.5796 at the 10 %, 5 %, and 1 % levels, respec- tively. Consistent with the monthly results, at the nominal 1 % level (tabulated critical value 0.5909), the empirical rejection rate under the bootstrap null is 8.7 %, i.e., the test is oversized. By contrast, when evaluated against the bootstrap critical values (Table 13), the BADF statistic from the daily sample, λADF =  05560, does not exceed the 5 % cutoff (0.8216), so we fail to reject H0 : ρ = 0 at the 5 % level (p-value 0.3920). Knowing that stock market crash as of October 1987 arrived only a few months in the ex-post sample of our daily data sample, a reader might wonder how come that the BADF does not assert a bubble for- mation in the period preceding the crash. Two possible aspects of the testing framework could explain why the full-sample BADF statistic on the daily pre-crash window does not exceed the simulated critical values: (i) Tested alternative versus LPPLS dynamics. The BADF is a linear unit-root test targeting mild explosiveness in the sense of ρ > 0 in Δyt = α+ ρyt1 + p i=1ϕiΔyti + εt. By construction, it does not target the log-periodic (oscillatory) structure of the LPPLS spec- ication. A time series can exhibit LPPLS-type curvature and accelerating oscillations without generating a sufciently large positive ADF coefcient to trigger a right-tailed rejection. Hence, failure to reject with BADF does not contradict evidence consis- tent with LPPLS; it reects that the two procedures test different alternatives. (ii) Size–power trade-off with an intercept. For the benchmark on log prices we include an intercept to absorb deterministic drift, which may improve nite-sample size but generally reduces power to detect short explosive episodes unless they dominate the sample. At daily frequency, the explosive run-up preceding October 1987 Table 8 Right-tailed Augmented Dickey-Fuller tests on the original data sets using a constant in the test regression. Monthly data Daily data ADF test statistic Lags ADF test statistic Lags 1.1775*** 2 0.5560 0 *** Statistically signicant on a 1 % level. This table presents the results of Augmented Dickey-Fuller (ADF) tests con- ducted on the log-prices of the original datasets. The model used for test regression accounts for a constant term. The critical values for the right-tailed test derived from Monte Carlo for statistical signicance at the 10 %, 5 %, and 1 % levels are 0.4150, 0.0755, and 0.5909, respectively. The lag order is selected according to the Schwarz criterion. The monthly series exhibits mild explosiveness under Monte Carlo derived inference, whereas the daily series does not. Table 9 Descriptive statistics of BADF (ADF with intercept) test statistics for synthetic monthly log-price series of the S&P 500. Statistic ADF Statistic Lag Length Minimum 4.1310 0 1 % Qnt. 2.8071 0 2.5 % Qnt. 2.4146 0 5 % Qnt. 2.1145 0 10 % Qnt. 1.7631 0 Median 0.4787 0 90 % Qnt. 0.7566 0 95 % Qnt. 1.0305 0 97.5 % Qnt. 1.2808 1 99 % Qnt. 1.7350 4 Maximum 2.7851 13 Mean 0.5066 0.1130 Standard Deviation 0.9795 0.8005 Excess Kurtosis 0.0532 121.7901 Skewness 0.1027 10.2100 T 1000 1000 The table reports summary statistics of the full-sample right-tailed ADF statistic estimated with a constant term on synthetic monthly log-price paths for the S&P 500. For reference, Monte-Carlo “tabulated” critical values matched to the sample length and specication are 0.4150, 0.0755, and 0.5909 at the 10 %, 5 %, and 1 % signicance levels, respectively. In each estimation, the lag order p is selected by the Schwarz (BIC) criterion. At the 5 % signicance level, the monthly series exhibits mild explosiveness under size-corrected (bootstrap) inference. Table 10 Descriptive statistics of BADF (ADF with intercept) test statistics for synthetic daily log-price series of the S&P 500. Statistic ADF Statistic Lag Length Minimum 3.6289 0 1 % Qnt. 3.0044 0 2.5 % Qnt. 2.6905 0 5 % Qnt. 2.5222 0 10 % Qnt. 2.2173 0 Median 0.8711 0 90 % Qnt. 0.4935 0 95 % Qnt. 0.8216 0 97.5 % Qnt. 1.1271 0 99 % Qnt. 1.5796 1 Maximum 2.3144 4 Mean 0.8491 0.0320 Standard Deviation 1.0339 0.2550 Excess Kurtosis 0.3894 144.1863 Skewness 0.0816 10.9887 T 1000 1000 The table reports summary statistics of the full-sample right-tailed ADF statistic estimated with a constant term on synthetic daily log-price paths for the S&P 500. For reference, Monte-Carlo “tabulated” critical values matched to the sample length and specication are 0.4150, 0.0755, and 0.5909 at the 10 %, 5 %, and 1 % signicance levels, respectively. In each estimation, the lag order p is selected by the Schwarz (BIC) criterion. The daily series does not exhibit mild explosiveness under size-corrected inference, reinforcing evidence derived from Monte Carlo “tabulated” critical values. K. Grobys International Review of Financial Analysis 110 (2026) 104848 13 may be relatively short compared with the length of the pre-crash window, and BIC lag selection further attenuates persistence by absorbing short-run dynamics. These features lower the BADF statistic relative to its critical values, even in periods widely regarded as bubbly. 5.2.2. Results from daily data on gold futures Consistent with Grobys (2025), we obtain daily gold futures data from 2 December 2015 to 6 November 2024 (2147 observations). Tables A.5–A.6 report descriptive statistics for daily returns and for the bootstrapped sample means, and Fig. 7 plots the original price series alongside alternative historical trajectories generated by resampling. As with the S&P 500 simulations, the bootstrap median is very close to the realized series at the terminal date T, as expected under our resampling design. We next t the original daily log-price series of gold futures to the LPPLS specication in Eq. (2), using the constraints and starting values described in Section 4.1. The estimates in Table 11 yield a power- law exponent β = 07081, consistent with less pronounced super- exponential acceleration (relative to the S&P 500 ts) as the critical time is approached. The estimated critical time is tc = 21779168, i.e., about 30.92 trading days beyond the sample endpoint T = 2147. Fig. 8 overlays the tted LPPLS curve on the log-price data, while Fig. 9 plots the corresponding residuals over the sample. Visual in- spection suggests stationarity, which we formally evaluate using the ADF regression in Eq. (4) applied to residuals dened in Eq. (3). The lag order is chosen by the Schwarz (BIC) criterion. As reported in Table 12, the ADF statistic for the original residuals is  46215, which—under tabulated left-tailed critical values—would be deemed statistically sig- nicant at the 1 % level, consistent with the convention in Lin et al. (2014). Turning to the resampling evidence, Table 13 reports descriptive statistics for LPPLS parameters estimated from synthetic daily gold se- ries. As in the S&P 500 simulations, β spans the admissible range 01 ≤ β ≤ 09. The median critical time is tc = 23942255, which is 247.23 trading days later than the tc from the original t. Consistent with the imposed bounds, β, C, ω, ϕ, and tc collectively cover the full constrained ranges. In particular, the earliest tc coincides with the sample endpoint (tc = T = 2147), while the latest occurs at the upper boundary (tc = 2T = 4294), indicating that the synthetic estimates span the identication window T < tc < 2T. Table 14 reports ADF statistics from 1000 LPPLS calibrations on synthetic daily gold series. Mirroring the S&P 500 results, the median is λADF =  33047, well below the tabulated 1 % left-tail cutoff (≈ 2.57), and 810/1000 statistics satisfy λADF <  257, implying an 81.0 % false-positive rate at the nominal 1 % level under table values. The bootstrap left-tail critical values for the daily gold sample are¡4.3515 / ¡4.6634 / ¡5.0516 at the 10 % / 5 % / 1 % levels, respectively—very close to those obtained for the S&P 500 synthetic monthly and daily data—reinforcing that the empirical quantiles relevant for LPPLS re- siduals are far more negative than tabulated thresholds. At the 5 % level, 947/1000 statistics lie below the tabulated cutoff (≈ 1.94), yielding a 94.7 % table-based false-positive rate. Finally, when we evaluate the original daily residual statistic against the bootstrap critical values, λADF = 46215 produces a bootstrap p-value of 0.0580. Thus, we reject at the 10 % level but not at 5 %. Interpreted cautiously, this is marginal evidence—at the 10 % level—of an LPPLS-consistent episode in gold, consistent with the narrative in Grobys (2025). Taken together, these results corroborate the evidence from the S&P 500: there is a pronounced gap between empirical ADF quantiles for LPPLS residuals and the tabulated values, and reliance on table thresholds induces substantial size distortion. 5.2.3. Dependence-aware resampling: results and comparison with i.i.d. bootstrap (monthly S&P 500) Appendix Tables A.7–A.8 report, respectively, the empirical distri- butions of the LPPLS parameter estimates and the residual ADF statistics obtained under the stationary (geometric) block bootstrap applied to monthly S&P 500 returns. These results are intended to assess the sensitivity of our inference to serial dependence and volatility clus- tering. For comparability, the resampling design mirrors the baseline specication except for the block structure (expected block length m = ⌈T13⌉); all estimation bounds, initialization, and lag-order selection (BIC) are held xed. Relative to the i.i.d. bootstrap results in Tables 3 and 5, the block- bootstrap distributions of the LPPLS parameters exhibit very similar location and overall shape. Any observed differences are of second order and consistent with the introduction of weak dependence—e.g., modest changes in dispersion for some parameters—without altering the qual- itative features of the empirical distributions. This indicates that the calibration step is not materially affected by the choice between i.i.d. and dependence-aware resampling at the monthly frequency. The residual ADF statistics (Appendix Table A.8) likewise remain broadly comparable to those in Table 5. While dependence-aware resampling can admit slight shifts in quantiles and a marginally different distribution of BIC-selected lag orders—consistent with pre- serving serial correlation—the substantive conclusions are unchanged. In particular, conventional tabulated ADF critical values continue to over-reject in this two-stage setting, whereas the estimation-aligned bootstrap yields empirical rejection rates in line with nominal size. Hence, the central nding—that size distortion arises when applying tabulated ADF cutoffs to residuals from the constrained nonlinear rst stage—is robust to the adoption of a block-bootstrap design on monthly data. Overall, the dependence-aware analysis corroborates the baseline i.i. d. evidence. Allowing for generic short-range dependence through sta- tionary blocks does not modify the interpretation of our results, but it strengthens their credibility by demonstrating that the conclusions do not hinge on the i.i.d. resampling assumption. 5.2.4. Multi-start robustness on the original sample (monthly S&P 500) This subsection documents the start-value robustness check—for illustration—on the original monthly S&P 500 sample, using K = 500 independently generated initializations for the nonlinear block (β,ω,ϕ, tc), with the linear coefcients (A,B,C) set by conditional OLS before constrained NLS re-estimation. Table A.9 reports distributional Fig. 7. Alternative historical trajectories for gold futures (December 2, 2015–November 6, 2024). This gure displays 1000 synthetic trajectories of gold futures prices, generated by compounding daily log-returns resampled with replacement from the orig- inal dataset (source: Grobys, 2025; Investing.com). The realized price path (red line) and the median bootstrap trajectory (grey line) align at the sample endpoint, conrming that the bootstrap reproduces unconditional drift while excluding deterministic oscillations. (For interpretation of the references to colour in this gure legend, the reader is referred to the web version of this article.) K. Grobys International Review of Financial Analysis 110 (2026) 104848 14 summaries for the optimized parameters in the order [A,B, β,C,ω, tc,ϕ] together with the objective value (SSE). Two features emerge clearly. First, the optimization landscape is at in economically relevant regions. The best objective value is virtually identical to the median across converged runs—SSE(best)/SSE(median) = 0.999. Consistent with this, the minimum SSE coincides with its 10 % quantile (both 17.742), whereas the median SSE is 17.762. These di- agnostics indicate a benign, at basin around the selected solution rather than a sharp, start-specic optimum; consequently, the reported t is not an artefact of a particular initialization. Second, the dispersion of the critical time is wide but well within the admissible window, as expected for LPPLS. The optimized tc spans 1852.3 to 3646.0 (i.e., from just beyond the sample end to one sample length ahead, tmax < tc ≤ tmax + T), with a median of 2967.1. This con- rms that the admissible search region is effectively explored and that tc is weakly pinned down within it, a property already noted in the liter- ature and consistent with the at objective. Regarding convergence, the SSE is nite for every run, implying that non-convergence is captured by the solver’s exit ag (EF) alone under our rule. In our implementation, 263 of 500 starts returned EF > 0 (converged), while 237 of 500 were classied as failures by the solver’s criterion (EF ≤ 0). Thus, Converged/K = 263/500 (~52.6 %): roughly half of the randomized initializations reach an acceptable so- lution (EF > 0 with nite, in-bounds estimates). The remaining ~47 % either enter infeasible regions, violate admissibility checks, or terminate without a success ag—behavior that is common in nonconvex cali- brations with broad start bounds. Importantly, the atness jdocumented here (SSE(best) ≈ SSE(median)) shows that the multiplicity of starts is not generating materially different solutions of comparable quality; rather, many initializations land in essentially the same trough of the objective. Across the other parameters, the summaries are in line with the imposed bounds and standard calibrations: β concentrates in the upper half of [01,09] (median 0.7197), ω covers [5, 15] with a median of 11.306, and ϕ spans [ π, π] as designed. 5.2.5. Objective profile in tc: evidence of weak curvature To evaulate the prole in tc, we use the starting vector for the rst grid point is the unconstrained optimum θ from the original sample (see Section 5.1.1): Table 11 Calibration of the LPPLS model on original data set for Gold futures. Dataset A B β C ω tc ϕ Daily 7.7315 0.0026 0.7081 0.2718 5.0000 2177.9168 0.9473 This table reports estimated parameters of the LPPLS model applied to daily gold futures (2015—2024) log-price series. Parameters are dened in Eq. (2); estimation follows bounded nonlinear least squares with standard parameter constraints. The critical time (tc ) denotes the inferred termination point of super-exponential growth, measured relative to the sample endpoint. Fig. 8. Calibration of the LPPLS model on daily gold futures (2015–2024). This gure plots observed daily gold futures log-prices together with the tted LPPLS curve. The estimated model yields a critical time approximately 30 days beyond the sample end, indicating moderate super-exponential growth consistent with transient speculative dynamics. (For interpretation of the ref- erences to colour in this gure legend, the reader is referred to the web version of this article.) Fig. 9. Residuals of the LPPLS model for daily gold futures (2015–2024). This gure shows residuals from the LPPLS calibration in Fig. 8. The pattern appears weakly mean-reverting; formal ADF results in Table 12 conrm sta- tistical stationarity at the 1 % level under conventional thresholds. (For inter- pretation of the references to colour in this gure legend, the reader is referred to the web version of this article.) Table 12 Augmented Dickey-Fuller tests of the residuals from original data set on Gold futures. ADF test statistic Lags 4.6215*** 0 *** Statistically signicant on a 1 % level. ADF statistics test the null of a unit root in re- siduals from the LPPLS ts in Table 11. Critical values correspond to MacKinnon one-sided thresholds (10 %, 5 %, 1 %). The critical values for statistical signicance at the 10 %, 5 %, and 1 % levels are 1.62, 1.94, and  2.57, respec- tively. Lag order is selected by Schwarz criterion. Signicant rejections under tabulated values imply apparent LPPLS “signatures.” K. Grobys International Review of Financial Analysis 110 (2026) 104848 15 θ(0) =  A(0) ,B(0) , β(0) ,C(0) 1 ,ω(0) , t(0)c ,ϕ(0)  = (103330,  02454,04884,  00534,72116,18596166,  31416); Fig. A.1 displays the prole of the least-squares criterion as a func- tion of the critical time tc, obtained by xing tc on a grid and re- estimating all remaining parameters at each grid point. The plotted curve is tc↦SSE(tc), i.e., the minimized criterion conditional on tc. From Fig. A.1 we observe that the prole exhibits a broad, shallow trough: a substantial fraction of the tc grid points lie below the 0.5 % tolerance line, indicating that many distinct tc values yield virtually identical t quality. This directly evidences weak curvature in tc and explains the dispersion of tc observed in the multi-start robustness check (see Section 5.2.4). The vertical line at the unconstrained estimate tc (≈ 1859.6) falls in a region with slightly higher SSE than the lowest portion of the trough; however, the difference is well within the 0.5 % tolerance, so it is economically negligible. Occasional spikes reect isolated non- convergence when tc is xed and do not alter the conclusion that the criterion is at over a wide set of tc values. Taken together with the multi-start robustness check in Section 5.2.4 (which showed SSE(best)SSE(median) ≈ 1 across converged runs), the prole conrms that dispersion in tc reects the atness of the criterion rather than instability of the estimation. Importantly, none of these features affect our main conclusion on estimation-aligned residual-ADF size control: the bootstrap calibration remains robust regardless of the precise tc chosen within the at region. 6. Discussion 6.1. Synthesis of results and comparison with earlier research The results offer three overarching messages. First, point estimates of the critical time tc differ from values reported in prior studies (e.g., Grobys, 2023; Sornette, 2017). Using the same daily S&P 500 sample as in the present study—while noting that the LPPLS specications may differ slightly—Grobys (2023) and Sornette (2017) estimate the critical time (nite-time singularity) to be 294 and 158 trading days ahead, respectively. The actual crash occurred 202 trading days after December 31, 1986 (October 19, 1987). By contrast, the present study places the crash time at approximately 109 trading days after the sample endpoint. This is not anomalous; it is consistent with Brée et al.’s (2013) argument that tc is a “sloppy” parameter—highly sensitive to seemingly minor changes in specication, sample endpoints, and constraints. In our setting, tc often lies well beyond the sample endpoint T, which suggests that within the admissible window T < tc < 2T the objective is relatively at in the tc direction. That geometry helps explain why different sam- ples and bound sets can deliver materially different tc without implying instability in the rest of the t. In other words, disagreement in tc across studies is expected once one acknowledges its weak curvature and the role of hard bounds. Second, the bootstrap critical values for the residual ADF tests are strikingly similar across the monthly, daily, and gold applications at the 10 %, 5 %, and 1 % levels. This clustering has a methodological origin. First, we hold the empirical design xed: the same constrained LPPLS calibration, the same residual ADF specication, and BIC for lag selec- tion. Second, the large sample lengths in all cases mean BIC typically selects low lag orders, yielding comparable nite-sample distributions for the test statistic. Third, the i.i.d. resampling of returns preserves the empirical marginal (including tail thickness); once compounded to levels and residualized, the resulting null features are broadly similar. In short, the near alignment of bootstrap quantiles reects design Table 13 Calibration of the LPPLS model on synthetic daily Gold futures data. Statistic A B β C ω tc ϕ Minimum 5.0551 6.6377 0.1000 0.9990 5.0000 2147.0000 3.1416 1 % Qnt. 6.4257 4.1983 0.1000 0.9990 5.0000 2147.0000 3.1416 2.5 % Qnt. 6.6721 3.1396 0.1000 0.9990 5.0000 2147.0000 3.1416 5 % Qnt. 6.9696 2.0450 0.1000 0.6215 5.0000 2147.0000 3.1416 10 % Qnt. 7.2823 0.9996 0.1000 0.2944 5.0000 2147.0268 3.1416 Median 8.1531 0.0025 0.7179 0.0069 5.9557 2394.2255 1.6752 90 % Qnt. 10.0117 0.0004 0.9000 0.4087 7.8196 3205.8131 3.1416 95 % Qnt. 12.1669 0.0001 0.9000 0.8294 9.2329 3833.1623 3.1416 97.5 % Qnt. 14.2763 0.0046 0.9000 0.9990 10.7265 4294.0000 3.1416 99 % Qnt. 16.5522 0.0627 0.9000 0.9990 13.1727 4294.0000 3.1416 Maximum 22.0894 0.6664 0.9000 0.9990 15.0000 4294.0000 3.1416 Mean 8.5730 0.2745 0.6179 0.0249 6.3030 2570.8822 0.3887 Standard Deviation 1.7841 0.8060 0.3042 0.3707 1.4991 516.2677 2.8635 Excess Kurtosis 13.6120 17.2837 1.1853 2.1652 9.2866 3.0398 1.7961 Skewness 3.1505 3.9168 0.5910 0.0480 2.6599 1.8186 0.2443 T 1000 1000 1000 1000 1000 1000 1000 Table 13 summarizes empirical distributions of LPPLS parameters estimated from 1000 bootstrapped daily gold-futures datasets. We report extreme values (minimum/ maximum), selected quantiles (1 %, 2.5 %, 5 %, 10 %, 50 %, 90 %, 95 %, 97.5 %, 99 %), the mean and sample standard deviation, skewness, excess kurtosis (kurtosis relative to the normal distribution), and the sample size T. Quantiles are computed from the empirical distribution. Table 14 Descriptive statistics for estimated ADF-test statistics for residuals derived from synthetic daily data on Gold futures. Statistic ADF Statistic Lag Length Minimum 6.2405 0 1 % Qnt. 5.0516 0 2.5 % Qnt. 4.8230 0 5 % Qnt. 4.6634 0 10 % Qnt. 4.3515 0 Median 3.3047 0 90 % Qnt. 2.2518 0 95 % Qnt. 1.9218 0 97.5 % Qnt. 1.4935 1 99 % Qnt. 1.0906 2 Maximum 0.2698 4 Mean 3.2864 0.0490 Standard Deviation 0.8407 0.3077 Excess Kurtosis 0.3577 66.2071 Skewness 0.1429 7.6062 T 1000 1000 This table presents the descriptive statistics for estimated ADF-test statistics for residuals derived from synthetic daily data on Gold futures. The critical values for 10 %, 5 %, and 1 % statistical signicance levels for the standard ADF test are 1.62, 1.94, and  2.57. The lag-order is chosen in line with the Schwarz- Criterion. The lag-order is chosen in line with the Schwarz-Criterion. The bootstrap-based inference restores empirical size, conrming that tabulated thresholds overstate signicance in LPPLS diagnostics. K. Grobys International Review of Financial Analysis 110 (2026) 104848 16 invariance rather than market-specic peculiarities. Third, when inference is aligned with the estimation procedure (i.e., using bootstrap critical values from the full two-stage pipeline), the LPPLS-residual tests reject far less often than when judged against conventional tables. In our main samples, only gold exhibits marginal signicance (10 % level). The daily S&P 500 pre-1987 window does not reject under bootstrap thresholds despite the ex-post crash that follows shortly after T. This pattern is consistent with limited power of a single full-sample unit-root test to detect short, late-sample explosive episodes, especially with an intercept absorbing drift and BIC absorbing short-run dynamics. As a result, failure to reject in that daily window does not contradict the historical “bubble” narrative; it reects the test’s power properties given the design we match to the LPPLS pipeline. A potential concern is weak identication in the calibrated LPPLS models. Two clarications are in order. (i) In our re-estimation, the critical time tc typically lies well beyond T, and the ts are stable under the chosen bounds; this pattern does not indicate pervasive weak iden- tication of the full parameter vector. (ii) We do observe boundary hits for the phase parameter, ϕ ≈  π. This is a standard outcome when the oscillatory amplitude ∣C∣ is small: the log-periodic term carries little weight, the objective function is nearly at in ϕ, and a constrained optimizer tends to park ϕ at the boundary. Importantly, our inference does not rely on ϕ. It rests on residual ADF tests, which summarize whether the residuals are stationary and mean-reverting in the original ts. Hence, boundary behavior in ϕreects an identication nuance of a weakly weighted component, not a failure of the residual-based diag- nostic used to assess statistical signicance. 6.2. Implications The implication is methodological. A common practice calibrates LPPLS under hard bounds and then declares a “signature”when residual ADF tests reject at tabulated cutoffs. Our results show that this practice is oversized in the constrained, two-stage setting: table values do not reect the nite-sample distribution of an ADF statistic computed on estimated residuals from a nonlinear, bounded rst stage. In effect, the calibration can superimpose oscillatory structure that is then “certied” by inappropriate critical values, inating false positives. When inference is aligned with estimation—by recomputing critical values via a bootstrap that mirrors the full procedure—many purported “signatures” recede. As a complementary benchmark, the full-sample BADF on log prices is less oversized under our design and therefore produces fewer false positives than the LPPLS-residual ADF. This does not elevate BADF to a universal standard; rather, it shows that bench- marking against a familiar ADF-type test helps contextualize LPPLS ndings and separate size issues from claims about detection power. For the literature, the message is direct: results obtained from LPPLS residual tests against tabulated ADF thresholds warrant re-examination. Studies that enforce bounds and then rely on conventional tables for residual stationarity may reect size distortion. Reassessing those nd- ings with estimation-aligned critical values (bootstrap or Monte Carlo, as appropriate) is a natural and necessary next step. 6.3. Limitations and avenues for future research Our approach prioritizes size-corrected inference for the LPPLS re- sidual tests, but it does so at a cost: like any unit-root test, the bootstrap- calibrated ADF inherits the size–power trade-off. In particular, a single full-sample test has limited power for short or late-sample explosive episodes. Exploring window-based procedures (e.g., SADF/GSADF) under an analogous estimation-aligned calibration would be informa- tive, though it introduces multiple-testing adjustments and departs from the single-window logic of the LPPLS application we analyze. An additional avenue is to rene identication via theory-guided bounds or a Bayesian specication with informative priors. While a Bayesian treatment (e.g., shrinkage onm,ω or priors on tc) could further formalize parameter uncertainty, such extensions are deferred to future work. Our cross-market check is limited to gold futures. While this repli- cation is informative, a broader assessment across emerging equity markets, cryptocurrencies, and other commodities would help map where LPPLS residual-ADF inference is most susceptible to size distor- tion and where levels-based benchmarks such as BADF are most infor- mative. We view this broader mapping as an important agenda for future research. 7. Conclusion This study revisits the empirical practice of diagnosing bubbles with the LPPLS model by aligning inference with the way statistics are pro- duced. The central message is straightforward. When the LPPLS model is calibrated under parameter bounds and its residuals are judged against conventional ADF test tables, the resulting tests are substantially over- sized for the constrained, two-stage setting. Using synthetic series that preserve the roughness and tail behavior of nancial returns while excluding log-periodic structure, we show that tabulated critical values can label as “signicant” a large share of episodes that are, by con- struction, devoid of log-periodicity. When critical values are re- computed by re-estimating the full procedure on each resample, the empirical size is restored and many apparent “signatures” recede. Applied to S&P 500 monthly and daily samples, tabulated critical values yield inated rejection rates for LPPLS residual-ADF tests; boot- strap critical values calibrated to the exact estimation design overturn these ndings. A benchmark based on the full-sample ADF on log prices (with an intercept and BIC lag selection) provides additional context: while that benchmark is not a universal standard, it is less oversized under our design and produces far fewer false positives than LPPLS residual-ADF tests. The cross-market application to gold futures points in the same direction: empirical quantiles relevant for residual tests lie far from tabulated critical values, and size-correct inference changes the inference of apparent regularities. The results also c