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Path-Explosive Behaviour in Economic Time Series: A Realization-Centred Exploratory Framework Working Paper --- First Draft
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The detection of explosive dynamics in economic time series has been dominated for more than a decade by the recursive right-tailed unit root testing framework of PhillipsWuYu2011 and PhillipsShiYu2015, commonly known as the GSADF approach. These procedures test whether the autoregressive coefficient of a series exceeds unity over rolling subsamples, using bootstrap critical values derived from asymptotic Brownian motion theory. Their success in dating speculative asset price episodes has made them a reference tool in applied macroeconomics and financial stability analysis.
Yet the DGP-based approach carries assumptions that are quietly demanding. The underlying model posits a piecewise autoregressive process with a coefficient that switches between regimes according to a mechanism independent of the history of the series itself. The test asks whether that coefficient has exceeded unity---a question about an unobservable parameter inferred from an assumed parametric structure. In short samples the bootstrap critical values can be unreliable, and the procedure struggles to discriminate between explosive dynamics and I(2) accumulation in finite samples.
More fundamentally, the DGP perspective may be the wrong frame for a large class of economic phenomena. Consider a regional tourism destination in the early stages of development. Growth in tourist arrivals is not plausibly generated by a fixed-parameter autoregressive process. It is shaped by a sequence of discrete institutional decisions---infrastructure investment, land use zoning, air route licensing, marketing campaigns---each of which permanently alters the carrying capacity and growth potential of the destination. The generating mechanism at time $t$ depends on the entire accumulated history of decisions up to $t$, not just on the value of the series at $t-1$. The Markov property that underlies any standard autoregressive DGP fails structurally in such settings.
This observation motivates a different approach, which we develop in this paper. Rather than asking what process generated the series, we ask what the observable path looks like. Specifically, we ask whether the realised trajectory within a dateable episode exhibits the geometric signature of self-reinforcing multiplicative growth: positive and persistent normalised curvature, stable log growth rate, and sufficient absolute growth. We call this path-explosive behaviour---explosive in the sense that growth is self-reinforcing and multiplicative, but qualified by “path” to signal that this is a property of the observable realisation rather than of any assumed DGP.
The framework has four components: an endogenous window detection algorithm, a four-layer diagnostic battery of twelve path statistics, two absolute gate thresholds, and a weighted composite intensity score. For pairs of series, a co-explosive extension assesses temporal co-occurrence of path-explosive episodes via a Jaccard index and non-parametric intensity concordance. The paper proceeds by developing the theoretical motivation (Section (ref)), providing formal definitions (Section (ref)), specifying the full framework (Section (ref)), presenting the simulation study (Section (ref)), and reporting the empirical application (Section (ref)). Section (ref) concludes with an honest discussion of limitations and directions for future research.
Standard approaches to explosive dynamics proceed by positing a data-generating process and using the observable series to make inferences about its parameters. In the PSY/GSADF tradition, the DGP is a piecewise autoregressive process whose coefficient switches between a unit root and an explosive regime PhillipsShiYu2015. The observed path is evidence about the current value of this parameter.
This framework is internally consistent when the DGP is well-specified and its parameters are genuinely stable within regimes. For speculative asset price dynamics---where prices can deviate from fundamental values through self-fulfilling expectations---the assumption that an autoregressive coefficient has switched from below to above unity is at least a serviceable approximation. But there is a large class of economic phenomena for which it fails structurally: settings where growth trajectories are constituted by discrete, irreversible planning decisions rather than governed by a time-invariant stochastic mechanism.
The theoretical framework that best characterises these dynamics is path dependence. Formalised in economics by David1985 and Arthur1989, path dependence arises when small historical accidents or early decisions produce permanent lock-in to particular trajectories through self-reinforcing mechanisms. The key property is non-ergodicity: the long-run trajectory depends on initial conditions and the sequence of early events, not just on current state variables or transition probabilities. In Arthur1989's polya-urn models of technology adoption, increasing returns create positive feedback that amplifies early leads and makes reversal increasingly costly.
Pierson2000 extended path dependence to political and institutional settings, arguing that policy trajectories exhibit what he called “increasing returns to politics.” Once a development path is established, the constellation of actors who have organised their expectations and investments around that path creates a powerful constituency for its continuation. The costs of reversal increase over time not because physical infrastructure cannot be demolished, but because the entire social and economic system that has adapted to the existing trajectory cannot cheaply reorganise. This produces a ratchet effect: development decisions are practically irreversible once adaptation has occurred.
In tourism and urban economics, these dynamics operate through agglomeration economies Krugman1991. A destination that crosses a critical mass threshold becomes more attractive than alternatives regardless of its underlying natural endowments, because of thick labour markets, specialised suppliers, accumulated brand capital, and knowledge spillovers. Each period's growth becomes proportional to the current level of accumulated development rather than a fixed additive increment. This self-reinforcing multiplicative structure is precisely what the path-explosive framework is designed to detect.
The implications for methodology are direct. When the DGP is itself path-dependent---when its parameters are functions of the accumulated history of decisions and their consequences---asking “has the autoregressive coefficient exceeded unity?” has no stable answer, because the generating mechanism is evolving along the very path it is generating. What does have a stable and observable answer is whether the realised trajectory exhibits the geometric signature of multiplicative self-reinforcing growth. The normalised curvature statistic and the log growth rate stability statistic that anchor our framework answer this question directly from the observable path, without requiring knowledge of the underlying DGP.
A further pragmatic consideration reinforces the case for a realization-centred approach in our target settings. Planning-intensive growth processes are typically recorded annually with short samples---$T \leq 100$ is common for subnational tourism, urban, and infrastructure series. The GSADF procedure has severely limited power in these settings, as Phillips, Shi and Yu (2015) themselves acknowledge in their finite-sample evaluations. Our framework was designed and validated precisely for this data environment.
We work with two definitions at different levels of abstraction, linked explicitly by the gate mechanism of Section (ref).
The qualifier “path” signals that explosiveness is assessed as a property of the observable realisation. Definition (ref) does not require the series to have been generated by an autoregressive process with root exceeding unity; it requires only that the observable trajectory behaves as if self-reinforcing multiplicative growth were operating over the episode. This distinction matters most in planning-intensive settings where the generating mechanism cannot be characterised by a fixed autoregressive coefficient.
Path-explosive behaviour is not synonymous with speculative bubble behaviour. A bubble in the asset pricing tradition BlanchardWatson1982 is defined by its eventual collapse toward fundamental value---transience is built into the definition. Path-explosive behaviour may be permanent: a structural transformation that places a system on a higher growth trajectory is path-explosive during the transition but need not collapse. The Balearic Islands tourism expansion of 1964--1971 is a case in point.
The operational implementation of Definition (ref) distinguishes three qualitatively different types of sustained growth, which are frequently conflated in applied work.
Type II dynamics characterise planning-led structural transformations where self-reinforcing mechanisms operate consistently over a sustained interval. Type III dynamics characterise speculative bubbles and rapid but irregular accelerations: the convex curvature is present, but the log growth rate is not constant, typically because the rate of acceleration varies as sentiment, capacity constraints, or policy responses evolve. Standard asset price bubbles are generally Type III. Type I dynamics are the primary confound that the gate mechanism excludes.
This definition does not require cointegration or a common stochastic trend. It requires that the dated intervals of self-reinforcing multiplicative growth are temporally aligned and that explosive intensity co-moves across those intervals. This is a more conservative and more directly interpretable concept than linear cointegration in explosive regimes EngstedNielsen2012.
All series are index-normalised prior to any computation: $\tilde{y}_t = y_t/y_1$. This preserves all shape, ratio, and growth rate properties while ensuring numerical stability at large absolute levels. Without normalisation, the theoretical convergence $\overline{NC} \to (\rho-1)^2$ degenerates numerically when series reach large values after many explosive periods.
Candidate episode windows are detected from the second-difference sequence of the normalised series without imposing any external window size. A window opens at $t_0$ when four consecutive periods of positive curvature acceleration are observed: $\Delta^2\tilde{y}_{t_0-3},\ldots,\Delta^2\tilde{y}_{t_0}>0$. From $t_0$ the window expands forward until two consecutive negative second differences close it, or until the maximum width $w_{\max}=15$ is reached. A retained window $[t_0,t_1]$ must satisfy: width $\geq w_{\min}=5$; absolute growth $|\tilde{y}_{t_1}-\tilde{y}_{t_0}|/\tilde{y}_{t_0}\geq 0.10$; and a minimum gap of five periods between consecutive windows. At most two non-overlapping windows per series are retained.
Within each detected window, twelve path statistics are computed across four layers addressing distinct geometric dimensions of the trajectory.
\paragraph{Layer 1 --- Level geometry.} Three statistics capture convexity in levels: the normalised quadratic acceleration $\tilde{\alpha}_2 = \hat{\alpha}_2/\bar{y}$ from a within-window quadratic regression; convexity persistence $CP$, the fraction of periods with positive second differences; and mean growth rate $MG$.
\paragraph{Layer 2 --- Growth rate dynamics.} Three statistics characterise the period-on-period growth rate $g_t = \Delta\tilde{y}_t/\tilde{y}_{t-1}$: the normalised growth rate trend $\tilde{\beta}_1 = \hat{\beta}_1/|\bar{g}|$; growth rate sign persistence $GP$; and the ratio to the pre-episode baseline $GR$, capped at $\pm 10$.
\paragraph{Layer 3 --- Normalised curvature.} Define $nc_t = \Delta^2\tilde{y}_t/\tilde{y}_{t-2}$, Winsorised at within-window 1st/99th percentiles. The three statistics are: mean $\overline{NC}$; positivity rate $NCP$; and normalised trend $NCT = \hat{\gamma}_1/|\overline{NC}|$. The central theoretical motivation is:
Property (ref) is what makes $\overline{NC}$ the pivot of the framework. The normalised second difference is bounded and positive under explosive dynamics; it shrinks to zero under I(2) accumulation regardless of how large the series becomes.
\paragraph{Layer 4 --- Log-space behaviour.} Three statistics exploit the linearity of geometric growth in log-space: log trajectory linearity $LL = 1 - \mathrm{sd}(\hat{\nu}_t)/(|\hat{\delta}_1| (t_1-t_0))$ where $\hat{\nu}_t$ are residuals from a within-window log-linear regression; log growth rate stability $LGS$; and log growth rate trend $LGT = \hat{\phi}_1/|\bar{\ell}|$. Under geometric growth $LGS\approx 1$ and $LGT\approx 0$; under I(2), $LGS\approx 0$.
\paragraph{Stage A --- Gate.} Before any intensity scoring, all three gate conditions must hold:
The threshold $(\rho_{\min}-1)^2 = 0.001024$ is theoretically anchored: it is the minimum $\overline{NC}$ consistent with an economically meaningful explosive root ($\rho_{\min} = 1.032$). The threshold $\tau_{LGS}=0.70$ is the decisive empirical discriminator: our simulation study establishes that $LGS$ averages $0.997$ for strong explosive ($\rho=1.10$), $0.921$ for mild explosive ($\rho=1.04$), $0.009$ for unit root, and $0.338$ for I(2). A threshold at 0.70 creates clean separation without distributional assumptions. For settings where growth is self-reinforcing but geometrically irregular (Type III), the empirical gate at $\tau_{LGS}=0.35$ allows classification with explicit acknowledgement that strict geometric regularity is not met.
\paragraph{Stage B --- Intensity score.} Gate-passing episodes are scored against regime-level 75th-percentile thresholds $\boldsymbol{\tau}$ calibrated from 500 replications of a mild explosive process at the target sample length:
where $d_j$ is the fraction of Layer $j$ statistics exceeding their calibration thresholds. Layer 3 receives weight 3, reflecting its unique theoretical grounding. Classification boundaries map $\tilde{S}$ to $\{\text{None},\,\text{Mild},\,\text{Moderate},\,\text{Strong}\}$ at $\tilde{S}\in\{0.36,\,0.57,\,0.75\}$.
The endogenous windowing and gate procedure is applied independently to each series in a pair, yielding gate-passing episode sets $\mathcal{W}_1^*$ and $\mathcal{W}_2^*$. The Jaccard co-occurrence index is: \[ J = \frac{|\mathcal{C}|}{|\mathcal{W}_1^*|+|\mathcal{W}_2^*|-|\mathcal{C}|} \] where $\mathcal{C}$ collects co-occurring gate-passing episode pairs. For pairs in $\mathcal{C}$, intensity scores are compared via Spearman rank correlation $\rho_S$, Kendall's $\tau$, and sign concordance $SC$. Classification follows Definition (ref).
All simulations use $T=80$, $\sigma=0.10$, burn-in of 50 periods, and $n=500$ replications. Individual series regimes: strong explosive ($\rho=1.10$), mild explosive ($\rho=1.04$), unit root ($\rho=1.00$), and I(2) ($\Delta^2 y_t=\varepsilon_t$). Co-explosive scenarios use pure AR processes with bivariate correlated innovations generated via Cholesky decomposition: strong co-explosive (both $\rho=1.10$, $r=0.80$), mild co-explosive ($\rho_1=1.10$, $\rho_2=1.04$, $r=0.80$), independent explosive (both $\rho=1.10$, detection restricted to non-overlapping halves), and spurious I(2) (two independent I(2) series). Intensity thresholds are calibrated from 500 replications of a mild explosive process at $T=80$.
Table (ref) shows that the LGS gate is the decisive discriminator. Unit root series are entirely blocked (0.0% pass) while both explosive regimes pass at near-100% rates. The I(2) pass rate of 29.8% correctly reflects the genuine ambiguity between I(2) and explosive behaviour in finite samples: these are episodes where I(2) dynamics locally resemble geometric growth before the denominator in $nc_t$ grows large enough to suppress $\overline{NC}$. The $\overline{NC}$ statistic matches its theoretical prediction of $(\rho-1)^2$ after index normalisation: the mean for strong explosive is $0.01000$, matching $(0.10)^2$ to four decimal places, confirming that Property (ref) holds empirically at the relevant sample sizes. Unit root series receive a composite score of zero in every single replication. I(2) reaches Mild or above in only 4.3% of replications despite a 29.8% gate pass rate, because gate-passing I(2) episodes score low on the corroborating intensity layers.
Three findings emerge from Table (ref). First, the framework achieves 70% power for detecting strong co-explosive behaviour with zero false positives in both null scenarios---the conservative design means non-classified cases receive no label rather than a wrong one. Second, mild co-explosive detection at 3.4% is a genuine power limitation: when $\rho_1=1.10$ and $\rho_2=1.04$, index-normalised trajectories diverge at rate $(1.10/1.04)^T$ exceeding a factor of 100 over $T=80$ periods, so the window detector locates the most explosive segments at different calendar positions in the two series regardless of innovation correlation. Third, false positive rates are exactly zero in both null scenarios---the temporal co-occurrence requirement (Jaccard) and the intensity concordance requirement jointly prevent spurious classification.
We apply the framework to four datasets spanning the range of phenomena for which path-explosive behaviour is a theoretically meaningful concept. Dataset A covers real residential property prices (index 2015=100) for Spain, Ireland, Germany, and the USA over 1975--2023, compiled from OECD and BIS sources. Dataset B covers nominal commodity prices for crude oil (USD/barrel), gold (USD/troy oz), and copper (USD/metric ton) over 1970--2023, from the World Bank Pink Sheet. Dataset C covers general government gross debt as a percentage of GDP for Spain, Italy, Greece, and Ireland over 1980--2023, from IMF World Economic Outlook data. Dataset D---the paper's primary empirical focus---reports annual tourist arrivals for four Spanish coastal destinations: Málaga, Alicante, Baleares, and Barcelona over 1960--2023.
For Datasets A and C we apply the empirical gate ($\tau_{LGS}=0.35$), reflecting the irregular but self-reinforcing character of house price and debt dynamics. For Dataset B we report results under both the full sample and a pre-collapse subsample truncated at the peak year (Oil: 1970--2008; Gold: 1970--2012; Copper: 1970--2011) to illustrate the effect of the post-peak reversal on detection. For Dataset D we apply the strict gate ($\tau_{LGS}=0.70$), consistent with the evidence in Table (ref) that the 1960s tourism episodes exhibit Type II geometric regularity. Intensity thresholds are calibrated from 500 replications of a mild explosive process at $T=64$ to match the tourism series length.
Table (ref) reports the complete diagnostic statistics for every detected window across all four datasets. This transparency is central to the framework's philosophy: the reader can assess the evidence directly rather than accepting a binary classification.
\paragraph{Type II path-explosive behaviour: Spanish tourism, 1964--1972.} The clearest path-explosive episodes are the Balearic Islands (1964--1971, $LGS=0.717$, Moderate, $\tilde{S}=0.702$) and Málaga (1965--1972, $LGS=0.799$, gate pass, $\tilde{S}=0.095$, None). Both pass the strict gate with $LGS$ values well above the 0.70 threshold and close to the simulation means for genuine explosive processes. The $\overline{NC}$ values (0.055 and 0.060) imply explosive roots in the range $\hat{\rho}\approx 1.24$--$1.25$, consistent with the dramatic annual growth rates recorded during the foundational decade of Spanish mass tourism---the period of rapid airport expansion, package holiday industrialisation, and concentrated foreign direct investment in coastal hotel infrastructure.
The contrast between Baleares (Moderate) and Málaga (gate pass but None) is directly readable from Table (ref) and is the most instructive result of the application. Both destinations have virtually identical gate statistics across all three gate conditions. Their divergence lies entirely in the intensity layers---specifically in $NCT$, which is $+0.389$ for Baleares (explosive curvature intensifying within the episode) and $-0.432$ for Málaga (explosive curvature decelerating), and in $\tilde{\beta}_1$, which is $+0.073$ for Baleares and $-0.071$ for Málaga (growth rate trending up vs down within the window). The framework correctly identifies Baleares as the destination where the self-reinforcing mechanism was operating most powerfully. This is consistent with the historical record: the Palma de Mallorca airport underwent major capacity expansion before the Málaga region reached comparable infrastructure levels, giving Baleares a structural first-mover advantage that sustained its internal momentum longer.
Alicante and Barcelona do not pass the gate ($LGS\approx 0.46$ in both cases). Their detected windows are convex and growing---the window detector correctly identifies this---but the log growth rate was insufficiently stable for Type II classification. These are plausibly Type III episodes: growth driven by a combination of organic development and intermittent policy intervention rather than by a consistently self-reinforcing mechanism.
\paragraph{Borderline path-co-explosive: Málaga and Baleares.} The co-explosive analysis finds $J=1.000$ for the Málaga--Baleares pair: their single gate-passing episodes overlap perfectly in calendar time, spanning 1964--1972. Intensity concordance statistics cannot be computed because there is only one co-occurring episode pair, falling below the minimum of two pairs required for reliable concordance assessment. We classify this as borderline path-co-explosive: the temporal co-occurrence condition is satisfied but the concordance condition is undetermined.
The economic interpretation is substantive. The two destinations that drove the foundational decade of Spanish mass tourism entered their path-explosive episodes simultaneously---both responding to the same external driver, the opening of the European package holiday market to Mediterranean destinations in the early 1960s---while their specific growth dynamics were shaped by their respective infrastructure endowments and planning contexts. The simultaneous takeoff reflects the common driver; the different intensity profiles reflect path-specific factors. This is exactly the pattern that the path dependence framework predicts: a common trigger activates self-reinforcing growth in multiple locations, but the intensity and duration of each episode is determined by the locally accumulated endowments.
\paragraph{The commodity sample split.} The pre-collapse truncation for commodities produces no change in results: Oil detects the same window (2000--2005) with $LGS=0.000$ in both the full sample and the truncated sample; Gold detects the same window (1975--1987) in both cases. The detected windows close well before the respective peak years (2008 for oil, 2012 for gold), so the post-peak collapse cannot be contaminating the within-window statistics. The $LGS=0.000$ result is intrinsic to the episode dynamics. Commodity price surges are driven by sequential demand shocks interacting with supply constraints---growth that is large and convex but highly irregular in its log growth rate. The framework correctly refuses to classify these as Type II path-explosive. The comparison table makes the reason transparent: the curvature ($\overline{NC}$) is there, but the geometric regularity ($LGS$) is not.
\paragraph{The distinction from standard bubble detection.} The house price and debt results highlight the most important conceptual distinction between path-explosive detection and standard bubble detection. The Spanish house price cycle of 2000--2008 and the Irish Celtic Tiger boom of 1995--2007---paradigmatic cases in the PSY/GSADF literature EngstedEtAl2016---are not detected as path-explosive. Ireland finds windows in 1978--1985 and 1989--1994, not the 1995--2007 bubble. Spain finds no gate-passing windows at all.
This is not a failure. The 1995--2007 house price surge in both countries grew monotonically but with a declining log growth rate throughout: prices grew fast in the late 1990s and then grew fast but at a declining rate through 2007. This is Type III dynamics. The PSY/GSADF procedure detects it as explosive because the autoregressive coefficient crosses unity. Our framework does not detect it as path-explosive because the geometric regularity required for Type II classification---and even for empirical-gate classification---is absent. The two frameworks answer different questions. PSY/GSADF asks whether the autoregressive coefficient has exceeded unity. Our framework asks whether the realised trajectory exhibits sustained multiplicative self-reinforcing growth. For the 2000s house price cycle, the answers differ, and the difference is informative: those episodes were rapid accumulations that lacked the internal self-reinforcing momentum of a genuine structural transformation.
This paper has proposed a realization-centred framework for detecting and characterising path-explosive behaviour in economic time series. The framework addresses a gap in the existing literature: the absence of tools designed specifically for settings where growth trajectories are constituted by discrete irreversible planning decisions rather than governed by a fixed stochastic mechanism. In these settings, the DGP-based approach of the GSADF tradition is not merely an approximation but is structurally inappropriate. What is appropriate is asking whether the observable realised path exhibits the geometric signature of self-reinforcing multiplicative growth---which is precisely what our framework does.
The central empirical finding is the confirmed path-explosive classification of the Balearic Islands tourism episode of 1964--1971 (Moderate, $\tilde{S}=0.702$, strict gate) and the borderline path-co-explosive relationship with the contemporaneous Málaga episode. The diagnostic statistics in Table (ref) make the evidence transparent: both destinations exhibit $LGS$ values above 0.70 and $\overline{NC}$ values consistent with implied explosive roots around 1.24--1.25, while the intensity score correctly distinguishes Baleares (accelerating internal momentum) from Málaga (decelerating despite gate passage). These results are consistent with the planning history of Spanish mass tourism and illustrate the capacity of the framework to detect structural transformations that standard bubble detection procedures are not designed to identify.
The comparison across datasets yields a substantive finding that goes beyond methodology: Type II path-explosive behaviour is qualitatively rarer than the rapid convex growth that characterises speculative bubbles and fiscal crises. Of the fifteen episodes detected across four datasets, only two pass the strict gate. The geometric regularity required for Type II classification is demanding, and correctly so: it identifies growth episodes where the self-reinforcing mechanism operated consistently enough to produce a nearly constant log growth rate over a sustained interval. This is the exception, not the rule, in macroeconomic data.
\paragraph{Retrospective characterisation.} The framework is primarily a retrospective tool. The window detector requires four consecutive periods of positive acceleration before opening, meaning an episode must have begun before it can be characterised. Gate statistics require the window to be substantially complete for reliable assessment. This is appropriate for the primary application---dating historical path-explosive episodes---but limits real-time utility.
A natural extension is sequential monitoring: applying the window detector and provisional gate statistics as new annual observations arrive, generating early warning signals when curvature conditions are met but $LGS$ has not yet stabilised, and upgrading to confirmed classification when all gate conditions are satisfied. This creates a two-state monitoring system (early warning; confirmed) with explicit uncertainty in the early warning phase. For planning authorities operating on annual data, early warning two to three years before confirmed classification would have direct operational value. Once a window is confirmed, the implied growth rate $\hat{\rho}=\exp(\bar{\ell})$ supports conditional trajectory extrapolation under the assumption of regime continuation---useful for infrastructure capacity planning, though carrying no information about when the explosive regime will end.
\paragraph{Cross-sectional lead-lag indicators.} The co-explosive component of the framework creates conditions for identifying systematic temporal lead-lag relationships between series that regularly exhibit path-co-explosive episodes. If destination A consistently enters its path-explosive episodes one or two periods before destination B across multiple historical cycles, then A's current trajectory is a leading indicator for B's future explosive onset. For datasets with multiple historical episodes per series---commodity markets, metropolitan housing markets, competing tourism destinations---this cross-sectional approach would have genuine forecast content. Developing the inference framework for estimating and testing lead-lag structures in co-explosive systems is a natural direction for future work.
\paragraph{Calibration and threshold sensitivity.} The intensity score is calibrated against simulated explosive processes, and the ordinal classification boundaries are empirically motivated rather than theoretically derived. Sensitivity analysis around these thresholds, and investigation of empirical calibration from datasets with known episode classifications, would strengthen the robustness of the intensity scoring. The distinction between the strict and empirical LGS gates is theoretically motivated by the Type II/III taxonomy, but a more formal criterion for selecting between gates---perhaps based on a preliminary assessment of whether pure geometric growth is plausible in the institutional context---would improve replicability.
\paragraph{Multivariate extension.} The current co-explosive analysis is pairwise. A multivariate extension analogous to cointegration rank---characterising the number of common path-explosive components in a system of series---would be valuable for applications where the researcher is interested in the systemic rather than pairwise co-explosive structure, such as panels of destinations, cities, or commodity markets.