— intervals of enlargement and contraction in output or monetary markets — lie on the coronary heart of macroeconomic evaluation. When nations share a standard foreign money, as within the eurozone, synchronized cycles are needed 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 principle. 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’t set the proper rate of interest for each nations. A decrease rate of interest would result in overheating the economic system in Spain, and a better rate of interest would exacerbate the disaster in Germany.
Conventional measures of cycle synchronization usually depend on easy correlations. However what occurs if two economies are following very related enterprise or monetary patterns, however one is simply “forward” or “behind” the opposite by a couple of quarters?
Enter Dynamic Time Warping (DTW), a way initially developed for speech recognition however more and more in style in knowledge science for evaluating time collection with related shapes but totally 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 nations. 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 info. Even in additional complicated frameworks — similar to state-space fashions that do enable for cyclicality — the cycle frequency itself usually stays fastened.
- Use of the imply: Commonplace dispersion indicators like variance or commonplace deviation at all times misread the typical because the “optimum”. In different phrases, distances are usually not 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 reasonable, though cycles from totally 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 significantly essential for financial coverage. Many OCA indicators find yourself overstating divergence, particularly in intervals when economies are “practically” in sync however offset by a couple 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 kind of the smoothed cycle indices, that means one measure of similarity is estimated for every kind 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 nations 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 a neighborhood window (Sakoe-Chiba Band) outlined by the typical cycle length. Lastly, to mixture 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 received’t severely penalize the space if the underlying patterns stay nearly an identical.
- Form sensitivity. DTW preserves details about amplitude, pattern reversals, and the relative “form” of booms and busts. Two nations that each expertise a pointy credit score increase — even when one is forward by 1 / 4 — will nonetheless be deemed extremely related.
- Time-varying flexibility. By making use of DTW over a rolling window (e.g., a neighborhood band of ±5 quarters for enterprise cycles, ±6 quarters for monetary cycles), the tactic adapts to altering cycle durations with out imposing a set frequency.
Constructing composite enterprise and monetary cycles
As an 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 price 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 mixture 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 forth.). To type a single euro-area “divergence index,” they take a GDP-weighted common of all pairwise DTW distances. A better index worth means higher divergence throughout nationwide cycles; a decrease worth means tighter synchronization.
Determine 3
Notice: The determine exhibits 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 exhibits 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. Increased values point out higher divergence throughout member-state cycle
While you plot this collection (Determine 3), a number of patterns emerge:
- Nineties convergence: Enterprise-cycle divergence fell sharply as convergence standards beneath 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.
- Put up-2010 re-alignment: ECB unconventional financial coverage (OMT, QE) coincided with convergence in each enterprise and monetary cycles.
- Late-2021 uptick: Because the COVID-19 shock, divergence has begun creeping again up, as some nations possible 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 usually a greater similarity measure than Euclidean distance or simple correlation.
- Dealing with non-stationary frequencies: Enterprise and monetary cycles don’t are available in neat, fixed-length packages. DTW’s means to adapt to various cycle durations preserves real-world turning factors.
Thinking about exploring the code or following the stay “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 permits you to see how synchronization has advanced for the reason that Nineteen 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 principle of optimum foreign money areas. American Financial Evaluation, 51(4), 657–665. Link to paper
