Collective dynamics of stock market efficiency

The study addresses whether and how the informational efficiency of stock markets varies over time, challenging the common assumption of constant efficiency in empirical finance. While the efficient market hypothesis (EMH) asserts that prices fully reflect available information, real markets exhibit autocorrelations, fat-tailed return distributions, crashes, and profitable strategies, indicating time-varying inefficiencies. The authors aim to quantify time-dependent efficiency for major global stock indices and uncover collective behaviors, stability of efficiency rankings, and market interdependencies. By focusing on sliding-window permutation entropy of log-returns, they seek to reveal both long-term hierarchical organization and short-term dynamics, and to construct a network view capturing global inter-market efficiency patterns and systemic implications.

Foundational EMH literature posits price changes as random and unpredictable under ideal efficiency (Fama; Cootner; Malkiel). Empirical anomalies include non-Gaussian return distributions, predictability around crashes, and successful strategies (e.g., Sornette; Stanley et al.). Prior studies quantified efficiency or inefficiency across equities, sovereign bonds, commodities, and cryptocurrencies, using tools such as permutation entropy, forbidden patterns, complexity-entropy planes, and multifractals. However, most works treat efficiency as time-invariant over study periods, leaving the temporal evolution and collective, networked aspects of efficiency insufficiently explored. This gap motivates a dynamic, sliding-window approach and inter-market clustering/network analysis.

Data: Daily adjusted closing prices for 43 major world stock market indices from January 1, 2000 to October 31, 2020 (5204 data points each) were obtained via Yahoo! Finance (yfinance), Wall Street Journal market data, and investing.com (investpy). Log-returns R(t) were computed as R(t) = log P(t) − log P(t−1).

Time-varying efficiency measure: Informational efficiency H(t) was defined as the normalized permutation entropy of log-returns within 500-trading-day sliding windows advanced by one day. Permutation entropy (embedding dimension d = 4, chosen given window length) quantifies the randomness of ordinal patterns; H ≈ 1 indicates high efficiency (random-like), lower H indicates more regularity/inefficiency.

Long-term similarity and clustering: For each pair of markets i,j, the correlation distance between their entropy time series Hi(t), Hj(t) was computed as d = sqrt(2(1 − ρ)), where ρ is the Pearson correlation. Ward’s minimum variance hierarchical clustering was applied to this distance matrix. The dendrogram was cut by maximizing the silhouette score to select the number of clusters.

Short-term dynamics and stability: The entropy series H(t) for each market was sampled with a 1-year sliding window. Within each window, markets were ranked by average H and clustered using the same correlation distance and Ward linkage; the silhouette score determined cluster counts per window. Stability of rankings across windows was assessed with Kendall’s rank correlation (Kendall-τ). Stability of clusterings across windows was evaluated with the adjusted Rand index (ARI).

Network construction: A weighted, undirected network was built where nodes are markets. An edge between two markets exists if the pair appeared in the same short-term cluster at least once; edge weight equals the number of windows in which the pair co-clustered. Centrality was evaluated using PageRank. Edge weight inequality was summarized by the Gini coefficient.

Community detection: Stochastic block models (SBM) were fitted to the weighted network using graph-tool, considering usual SBM, degree-corrected SBM (DCSBM), nested SBM, and nested DCSBM. Model selection used minimal description length. The best model was the nested SBM without degree correction, yielding the most likely modular structure (two modules) and node membership probabilities via MCMC sampling.

• The average efficiency across markets, H(t) aggregated, is smoother than individual series and dips around the 2007–2008 global financial crisis, indicating decreased efficiency during that period.
• Long-term hierarchical structure: Correlation distances among full H(t) series reveal hierarchical organization. Maximizing silhouette score yields 16 clusters: 15 small clusters (largest has 5 markets) and one singleton, indicating diverse long-term profiles but no large cohesive long-term groups.
• Short-term stability: Efficiency rank stability (Kendall-τ across 1-year windows) shows small diagonal blocks of width about 1–2 months, indicating ranks remain stable only briefly. Clustering stability (ARI across windows) shows blocks of about 4 months, indicating groups with similar short-term efficiency profiles persist only for a few months.
• Network of short-term co-movements: The co-clustering network among the 43 markets is a complete graph (all pairs co-cluster at least once), evidencing strong global entanglement of efficiency dynamics. Edge weights are unequal (Gini coefficient = 0.18), implying some market pairs co-move more frequently.
• Influential markets: PageRank centrality identifies Amsterdam AEX (Netherlands) and KOSPI Composite (South Korea) as most influential for efficiency dynamics; the MOEX Russia Index and RTS Index (Russia) are among the least influential.
• Modular structure: Nested SBM uncovers two modules. Module 1 (24 indices) comprises 6 USA indices, 14 Asia-Pacific indices, and markets from Brazil, Italy, Israel, and Mexico. Module 2 (19 indices) includes 12 European indices and markets from Argentina, Canada, Indonesia, Russia, Saudi Arabia, South Africa, and Thailand. Geography appears to play a partial role, with notable exceptions.
• Systemic implication: The dense, weighted network indicates potential for global spreading of low-efficiency states (risk) as well as global emergence of high-efficiency states.

The findings demonstrate that informational efficiency is not static; instead, markets exhibit time-varying efficiency with collective short-term co-movements. The hierarchical clustering of full-period entropy series captures broad, long-term similarities but misses rapidly changing short-term structures. Stability analyses show that both efficiency rankings and clusters are transient, persisting only for months at best. The network approach aggregates these transient co-movements into a global picture, revealing complete connectivity with heterogeneous interaction strengths and a clear two-module structure. This supports the view that efficiency is a collective, globally entangled phenomenon. Such entanglement implies that inefficiency (predictability) episodes can propagate systemically, posing risk, while collective dynamics can also sustain globally high efficiency. Identification of influential markets and modular partitions provides insight into where and how such states may originate and propagate, with practical implications for monitoring systemic risk and understanding regional or structural interdependencies.

This work introduces a time-varying measure of stock market informational efficiency based on permutation entropy of log-returns and reveals that: (i) global markets can be hierarchically grouped by long-term efficiency profiles; (ii) short-term efficiency ranks and clusters are unstable, lasting only months; and (iii) aggregating short-term co-movements yields a fully connected yet modular financial network with identifiable influential markets and two principal modules. These contributions underscore that efficiency is a collective, dynamic property with systemic implications. Future research could integrate dynamic efficiency into risk and trading models to quantify transaction risks; incorporate causality-oriented measures to infer directionality of influence; extend analyses to longer historical horizons and high-frequency data; adopt multi-scale frameworks; and move from indices to large sets of individual stocks to refine network granularity.

• The co-clustering network encodes association frequency but not causal direction; influence directionality remains unresolved. - The study period, while long (2000–2020), may omit earlier structural regimes; results could vary with longer histories. - Daily data may miss intra-day dynamics; high-frequency and multi-scale analyses could yield different structures. - Using market indices aggregates across constituents; networks built from individual stocks might reveal different or richer patterns. - The complete connectivity suggests strong globalization, but edge weights, noise, and windowing choices (500-day and 1-year windows, d=4) may affect results and stability estimates.