Key takeaways
  • Observed path over hidden model
  • Four layers read level, speed, bend, and logs
  • Two gate checks screen weak episodes
  • Pair tools compare shared explosive episodes
  • Tests cover prices, debt, and tourism

When prices or debt start climbing faster and faster, the real question is whether the series is just noisy or genuinely self-reinforcing. This paper argues that the answer should come from the observed path itself, not only from a hidden data-generating process. It introduces a descriptive framework for “path-explosive behaviour,” using four diagnostic layers: level geometry, growth rate dynamics, normalised curvature, and log-space behaviour. Those layers are designed to separate true multiplicative growth from I(2) dynamics, and they do so without distributional assumptions or asymptotic critical values. The method also adds two absolute gate thresholds before assigning a composite intensity score. For pairs of series, it checks co-explosive behaviour with a Jaccard co-occurrence index and non-parametric intensity concordance measures. The paper tests the framework in simulation across four DGP regimes and applies it to real house prices, commodity prices, public debt, and Spanish tourism destinations. Its broader claim is simple: in settings where discrete institutional decisions shape growth, a realization-centred description can be more informative than a traditional unit-root test.

A house price can look calm for years. Then the climb turns self-feeding. That is the puzzle here. Most readers expect a test to start with a hidden model. This framework starts with the path you can already see. It asks what the observed series does. It does not begin with what a hidden process may have done. That shift matters when prices, debt, or tourism spend start to bend upward. It also matters when a choice cannot be undone. The surprise is simple. You can sort explosive episodes from I(2) dynamics. I(2) dynamics need two rounds of differencing to calm down. Reading the path does the rest.

Reading the curve before the model

The framework uses four diagnostic layers. They are level geometry, growth rate dynamics, normalised curvature, and log-space behaviour. Each layer checks a different sign of self-reinforcing growth. Two absolute gate thresholds come next. They screen episodes before the framework assigns a composite intensity score. For pairs of series, the method tracks shared explosive episodes. It uses a Jaccard co-occurrence index. It also uses intensity concordance measures that make no shape assumptions. In simulation, the framework showed discriminating power and stayed conservative across four data-generating process regimes. It then applies the framework to real house prices, commodity prices, public debt, and Spanish tourism destinations.

4diagnostic layers

plus 2 gate thresholds

recursive unit root testing
  • Level geometry watches how the series bends upward.
  • Growth rate dynamics check whether gains keep speeding up.
  • Normalised curvature measures how sharp the bend becomes.
  • Log-space behaviour checks whether multiplying growth looks straighter in logs.

“the approach operates directly on observable path properties of the realised series.”

From the abstract

How the framework reads a series

The method reads a series like a trail of footprints. Level geometry asks whether the line bends up more and more. Growth rate dynamics asks whether each step grows faster than the last. Normalised curvature measures how sharp that bend is after scale control. Log-space behaviour checks the series after a log transform. A log transform squeezes big numbers closer together. The framework then looks for episodes that pass the two gate thresholds. Only then does it score intensity. That keeps the process descriptive first and selective second.

Why the path matters

A debt series can keep rising after one bad turn. House prices can do the same. Path dependence means earlier choices steer later outcomes. Planning irreversibility means some choices cannot be undone easily. In those settings, the observed path may tell more than a hidden model. That is why the framework stays close to what you can see. Its pair tools also compare how two series flare together. The Jaccard co-occurrence index counts shared episode overlap. The concordance measures check whether intensity moves in step. They make no shape assumptions. That makes the pair check easy to read.

What this changes

House prices, commodity prices, public debt, and Spanish tourism destinations all leave a path. The framework reads that path first. Instead of asking only whether a hidden process crossed a line, it asks how the path itself behaves. That helps when the history of a series carries the real clue. It also helps when two series rise together in linked episodes. The episode-level view can separate a single flare from a shared climb. That matters for those four settings. It also separates path-explosive behaviour from simple bubble talk.

The next hard check

Spanish tourism destinations may be the cleanest next test. That is where planning decisions can lock in growth paths. If the same episode rules still work there, the surprise becomes more than a neat idea. It becomes a way to read future risk from the shape of the past. That is the promise of a path-first view. It does not wait for a hidden model to speak first. It watches the path and asks what kind of growth it already shows.