Does the growth of military hard power back up the growth of monetary soft power via data-driven probabilistic optimal relations?

The study examines whether military hard power (proxied by military expenditure as a share of GDP) causally supports monetary soft power (proxied by the ratio of exchange-rate growth to broad-money growth). Building on distinctions between coercive hard power and non-coercive soft power, and noting ongoing currency and influence competition among nations, the paper focuses on the interplay between military spending and monetary dynamics. The central research question is whether the growth rate of monetary soft power is positively proportional to the growth rate of real military hard power. The authors outline a data-driven causal analysis using probabilistic and geometric tools to benchmark and assess directional relationships over time across countries.

Prior work links economic growth and military expenditure with mixed findings: military spending may be boosted by economic growth, may foster growth, hinder it, or have negligible effects. Currency competition and monetary policy’s influence on the real exchange rate are also well documented, as are the implications of exchange rates for development. Determinants of military expenditure include GDP and resource availability, and much exchange-rate research examines trade-exchange rate linkages. The paper cites: Nye on hard/soft power; Robertson and Sin on measuring hard power via real military spending; Beckley on economic development and military effectiveness; Dunne and Smith; Santamaría et al.; Su et al.; Manamperi on varied growth effects of defense spending; Liss on currency war; Pham on monetary policy and exchange rate; Gala on real exchange rate and development; Looney and Fredericksen; Odehnal et al. on determinants; Kang and Dagli on trade and exchange rates.

Variables: Military hard power (MHP) is operationalized as the growth rate of military expenditure over GDP: MHP_ct = mg,ct+1 / mg,ct − 1, where mg,ct is military expenditure as a share of GDP for country c at time t. Monetary soft power (MSP) is defined as the ratio of the annual growth of the exchange rate (expressed as USD per domestic currency, i.e., the reciprocal of LCU per USD) to the annual growth in broad money: MSP_ct = Δζ_ct^−1 / Δm_ct, where Δm_ct = m_ct − m_ct−1. Data: Annual country-level time series from 1961–2019 for three indicators (broad money, official exchange rate, military expenditure as % of GDP) from the World Bank. The period is divided into 58 adjacent year-to-year intervals. For each period t, the sample S_t comprises countries with non-missing data for all indicators in that period; sample size varies by t. Preprocessing and categorization: For each period t, compute X_t = {MHP_ct} and Y_t = {MSP_ct} for all c in S_t. Apply k-means clustering to X_t (k=5) and Y_t (k=4) to obtain centroids (KMX_t for X, KMY_t for Y) and cluster memberships (KCX_t, KCY_t). De-mean the centroid vectors to obtain dKMX_t and dKMY_t. Probabilistic structure and causal metrics: Construct empirical joint structure via cluster co-occurrences to estimate conditional probabilities p(y|x) and p(x|y), forming matrices J_{Y|X} and J_{X|Y}. Define directed average values AV_{Y|X} and AV_{X|Y} using the conditional probabilities and de-meaned centroids. Introduce the causal product and the causal correlation coefficient (CCC) to quantify direction-specific linear-preserving causality from X to Y (CCC_{X→Y}) and from Y to X (CCC_{Y→X}); CCC values lie in [−1, 1]. Optimal empirical relations: Enumerate total surjective relations mapping categories of X to categories of Y (and vice versa); in this application there are 693,601 such relations. Identify optimal empirical relations by maximizing the difference in directional CCCs (i.e., selecting surjective mappings that maximize CCC_{X→Y} − CCC_{Y→X} for the X→Y analysis, and analogously for Y→X). Benchmark relations and distance measures: Define absolute causal relations (ACR) as totally ordered, surjective relations that serve as probability-independent linear benchmarks. Compute Hausdorff distances between the sets of optimal empirical relations and the set of absolute causal relations, using both average and minimal distance variants. Analyze the time series (across the 58 periods) of these distances and of the maximal directional CCCs to assess the stability, strength, and asymmetry of causality between MHP and MSP. Implementation details and examples (definitions, matrices, and a worked example) are provided in the text and the Supplementary Appendix; algorithms are detailed in Supplementary Appendix A.

- Directional strength and stability: The maximal optimal causal correlation coefficients (MaxCCC) for both directions vary little over time, with differences mainly around 1, indicating noticeable and analyzable causal features built on a stable foundation. - Distinct separation in distances: Hausdorff distances between optimal empirical and absolute causal relations show a clear separation between the MHP→MSP and MSP→MHP directions under both average and minimal distance metrics, indicating clear causality between the variables with few abnormalities in the data. - Magnitude of distances: Under the Hausdorff average distance, fluctuations are centralized around approximately 5.5 and 4.5 for the two directions; under the minimal distance, around 5 and 3. These levels suggest substantial and persistent causality. - Asymmetry: There is a consistent asymmetry (gap) between MSP→MHP and MHP→MSP distances, indicating one-way, universal causality: one factor leads the other, but not vice versa. - Directionality: The MHP→MSP relation is strictly stronger than MSP→MHP (also reflected in comparative Hausdorff distances and summarized by Fig. 5). Overall, military hard power causally backs up monetary soft power, but monetary soft power contributes very little to military hard power.

The analysis addresses the core question by demonstrating a persistent, directional causal linkage from military hard power to monetary soft power across nearly six decades and many countries. The stronger alignment of empirical optimal relations with absolute linear benchmarks in the MHP→MSP direction implies that increases in military expenditure relative to GDP tend to support or stabilize a country’s currency dynamics relative to broad money growth. Policy-wise, this suggests that appropriate levels of military spending may help underpin monetary credibility and currency valuation, even if the reverse (currency-related factors driving military spending growth) is weak. Methodologically, the approach’s reliance on probabilistic structures and distance-based linearity preservation allows analysis without strict parametric assumptions, accommodates varying sample sizes across time, and provides intuitive geometric interpretations of causality strength and stability. These features enhance interpretability and robustness when dealing with multi-country, time-varying data.

The study develops a mathematical-probabilistic framework—combining k-means categorization, conditional-probability-based causal correlation coefficients, optimization over surjective relations, and Hausdorff distance benchmarking—to assess directional causality between military hard power and monetary soft power. Empirically, results show a clear, one-way causal relation: military hard power supports monetary soft power; the reverse effect is minimal. This finding aligns with the notion that suitable military spending over GDP can help stabilize and promote currency value. Future research directions include: relaxing the fixed numbers of k-means categories via data-driven or alternative clustering methods; augmenting indicators of hard power (e.g., force structure, nuclear warheads, personnel, capital assets) and soft power (e.g., interest rates, GDP, other macro-financial variables); incorporating non-numeric or fuzzy data; and combining this linearity-preserving framework with regression or predictive models to quantify effect sizes and enable forecasting.

- Dependence on optimal surjective relations: Inferences are based on selected total surjective mappings; considering broader or fuzzy relations may yield less concise causality or different nuances. - Indicator scope: Only three indicators are used (military expenditure/GDP, exchange rate, broad money). Additional military or monetary variables could refine measurement of hard and soft power. - Clustering choices: The numbers of clusters (k=5 for MHP, k=4 for MSP) are chosen a priori, which may influence results; alternative selection criteria or methods could be explored. - Model form: The approach emphasizes linearity preservation and does not estimate explicit functional forms (e.g., regressions), limiting direct effect-size interpretation and predictive use without complementary methods. - Data limitations: Country coverage varies by period due to missing data; while the framework accommodates dynamic sample sizes, this variability may affect generalizability across time and regions.