— intervals of growth and contraction in output or monetary markets — lie on the coronary heart of macroeconomic evaluation. When international locations share a typical foreign money, as within the eurozone, synchronized cycles are essential for a one-size-fits-all financial coverage to work. This concept was first put forth by Robert Mundell (1961), the daddy of Optimum Foreign money Space idea. If, for instance, Germany is in disaster and Spain is booming, as was the case after the flip of the millennium, the European Central Financial institution (ECB) can not set the correct rate of interest for each international locations. A decrease rate of interest would result in overheating the financial system in Spain, and a better rate of interest would exacerbate the disaster in Germany.
Conventional measures of cycle synchronization typically depend on easy correlations. However what occurs if two economies are following very comparable enterprise or monetary patterns, however one is simply “forward” or “behind” the opposite by a number of quarters?
Enter Dynamic Time Warping (DTW), a method initially developed for speech recognition however more and more in style in knowledge science for evaluating time collection with comparable shapes but completely different timings. In our paper, “Warpings in Time: Enterprise and Monetary Cycle Synchronization within the Euro Space” (Bugdalle & Pfeifer, 2025), we assemble composite indices of euro-area enterprise and monetary cycles after which use DTW to measure how carefully these cycles align throughout international locations. Our Optimum Foreign money Space (OCA) monitor that makes it potential to trace cycle divergence in actual time — and to identify part lags with out penalizing them as harshly as conventional metrics would.
Capturing part shifts and amplitude variations
Most current research of cycle synchronization do three issues that may be problematic:
- Static therapy of cycles: For instance, trend-extraction strategies (just like the HP filter) take away the cyclical parts from the information. Even in additional complicated frameworks — resembling state-space fashions that do permit for cyclicality — the cycle frequency itself typically stays fastened.
- Use of the imply: Commonplace dispersion indicators like variance or customary deviation at all times misread the common because the “optimum”. In different phrases, distances will not be measured between pairs of cycles, however relative to a imply or reference cycle. This obscures multimodality. As an illustration, if our cycles truly fall into two (or extra) well-separated clusters, the centroid will lie between them — in a area the place no actual knowledge exist — and all of the cycle-to-mean distances will look average, though cycles from completely different clusters are literally extraordinarily distant.
- Section shifts: Most distance measures are Euclidean. For instance, two cycles could also be barely shifted in time but nonetheless completely synchronized. This level could also be notably essential for financial coverage. Many OCA indicators find yourself overstating divergence, particularly in intervals when economies are “practically” in sync however offset by a number of months or quarters.
Dynamic Time Warping (DTW) for cycle synchronization
DTW is a non-parametric algorithm that finds the optimum alignment (or “warping”) between two time collection by permitting one collection to stretch or compress in time to match the opposite. In our case, DTW is utilized to every sort of the smoothed cycle indices, that means one measure of similarity is estimated for every sort of cycle. Inside every cycle class, DTW computes the alignment path πij for every pair of nations i and j that minimizes the cumulative distance between two cycles:
[D(mathbf{x}_i, mathbf{x}_j) = min{pi_{ij}} sum_{(t, s) in pi_{ij}} left| mathbf{x}_{i,t} – mathbf{x}_{j,s} right|^2,]
the place xi and xj is the smoothed cycle values at time t and s for international locations i and j, respectively. The ensuing distance D(xi, xj) captures the diploma of similarity, with smaller values indicating nearer alignment of the 2 cycles. To make sure that the DTW comparability displays the timing of cyclical actions, the alignment is carried out over an area window (Sakoe-Chiba Band) outlined by the common cycle length. Lastly, to combination all pairwise DTW distances into one euro‐space indicator, we compute a GDP‐weighted imply of D(xi, xj). This weighted common is the divergence index proven under (Determine 3).
Key advantages of DTW with Sakoe-Chiba Bands in an economic-cycle context:
- Section-invariance. Small lags or leads don’t robotically set off massive divergence scores. A one-quarter shift gained’t severely penalize the space if the underlying patterns stay virtually an identical.
- Form sensitivity. DTW preserves details about amplitude, pattern reversals, and the relative “form” of booms and busts. Two international locations that each expertise a pointy credit score growth — even when one is forward by 1 / 4 — will nonetheless be deemed extremely comparable.
- Time-varying flexibility. By making use of DTW over a rolling window (e.g., an area band of ±5 quarters for enterprise cycles, ±6 quarters for monetary cycles), the tactic adapts to altering cycle durations with out imposing a hard and fast frequency.
Constructing composite enterprise and monetary cycles
For instance DTW’s energy, we first assemble two composite cycle indices for every euro-area nation:
- Enterprise Cycle Index: Quarterly actual GDP progress, personal consumption progress, gross fastened capital formation progress, and unemployment progress.
- Monetary Cycle Index: Quarterly actual credit score progress (financial institution lending), house-price progress, stock-price progress, and authorities bond-price progress.
Utilizing a non-parametric strategy launched by Schüler et al. (2020), we extract every nation’s underlying cycle — an index that alternates between 0 and 1 to replicate expansionary versus contractionary phases, however with time-varying amplitude and length. This avoids inflexible detrending and retains turning factors intact.
Figures 1 and a couple of
Notice: Indices of enterprise cycle and monetary cycles as deviations from their historic median progress — 0.5 corresponds to the long-term median progress fee of every index. The composite monetary cycle combines quarter-on-quarter progress in credit score, home costs, fairness costs and bond costs — exhibiting each the uncooked (unfiltered) collection and the band-pass-filtered collection utilizing country-specific frequency bands. The filtered enterprise cycle combines quarter-on-quarter progress in GDP, consumption, funding, and unemployment
From pairwise DTW distances to an combination divergence monitor
As soon as every nation’s enterprise and monetary cycles are estimated, we compute pairwise DTW distances between each pair of nations (e.g., Germany vs. Spain, France vs. Italy, and so on.). To type a single euro-area “divergence index,” they take a GDP-weighted common of all pairwise DTW distances. The next index worth means larger divergence throughout nationwide cycles; a decrease worth means tighter synchronization.
Determine 3
Notice: The determine reveals quarterly measures of cycle divergence within the euro space from 1985Q1–2023Q4. The dashed purple line plots the imply dynamic-time-warping (DTW) distance throughout all pairwise comparisons of nationwide financial-cycle indices; the dashed blue line reveals the equal for business-cycle indices. The stable black line is the GDP-weighted common of those two collection, our composite Divergence Monitor. Shaded gray bands mark OECD recession intervals for the eurozone. Larger values point out larger divergence throughout member-state cycle
Whenever you plot this collection (Determine 3), a number of patterns emerge:
- Nineties convergence: Enterprise-cycle divergence fell sharply as convergence standards below Maastricht took maintain.
- Pre-2008 monetary divergence: Monetary cycles truly diverged nicely earlier than the worldwide monetary disaster — that peak in divergence is nearly invisible to correlation or amplitude-based metrics.
- Submit-2010 re-alignment: ECB unconventional financial coverage (OMT, QE) coincided with convergence in each enterprise and monetary cycles.
- Late-2021 uptick: For the reason that COVID-19 shock, divergence has begun creeping again up, as some international locations seemingly recovered quicker than others.
Takeaways for knowledge scientists and economists
- Versatile sample matching: When evaluating financial (or any) time collection that will share the identical “form” however are out of part, DTW is commonly a greater similarity measure than Euclidean distance or easy correlation.
- Dealing with non-stationary frequencies: Enterprise and monetary cycles don’t are available in neat, fixed-length packages. DTW’s capacity to adapt to various cycle durations preserves real-world turning factors.
Inquisitive about exploring the code or following the dwell “Divergence Monitor” for the euro space? Take a look at https://github.com/Moritz-Pfeifer/Divergence_Monitor for knowledge, Python notebooks, and https://moritz-pfeifer.github.io/eurozone-divergence-monitor/ for an interactive visualization that allows you to see how synchronization has advanced because the Eighties.
References:
Bugdalle, T., Pfeifer, M. (2025). Warpings in time: Enterprise and monetary cycle synchronization within the euro space. SSRN preprint. Link to working paper
Sakoe, H., Chiba, S. (1978). Dynamic programming algorithm optimization for spoken phrase recognition. IEEE Transactions on Acoustics, Speech, and Sign Processing, 26(1), 43–49. Link to paper
Schüler, Y. S., P. P. Hiebert, and T. A. Peltonen (2020). Monetary cycles: Characterisation and real-time measurement. Journal of Worldwide Cash and Finance 100. Link to paper
Mundell, R. (1961). A idea of optimum foreign money areas. American Financial Assessment, 51(4), 657–665. Link to paper
