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This study investigates the presence of deterministic dependencies in international stock markets using recurrence plots and recurrence quantification analysis (RQA). The results are based on a large set of free float-adjusted market capitalization stock indices, covering a period of 15 years. The statistical tests suggest that the dynamics of stock prices in emerging markets is characterized by higher values of RQA measures when compared to their developed counterparts. The behavior of stock markets during critical financial events, such as the burst of the technology bubble, the Asian currency crisis, and the recent subprime mortgage crisis, is analyzed by performing RQA in sliding windows. It is shown that during these events stock markets exhibit a distinctive behavior that is characterized by temporary decreases in the fraction of recurrence points contained in diagonal and vertical structures.

The question of whether the seemingly random behavior exhibited by the price of financial assets and commodities is partially explained by chaotic nonlinear deterministic processes has received considerable attention by financial economists. In classical finance theory, fluctuations in asset prices are driven either by homoscedastic random walks or heteroscedastic martingale difference sequences. However, simple nonlinear deterministic processes can emulate price dynamics that are indiscernible from stochastic processes, providing an alternative model for the behavior of asset prices. Furthermore, nonlinear determinism can potentially explain large movements in financial data that linear stochastic models cannot account for Ref. [1]. While evidence of violations of the random walk and martingale hypotheses has been found in financial markets (see, e.g., Refs. [2], [3] and [4]) and despite the profusion of tests devised for detecting chaotic determinism in time series data (such as the Grassberger-Procaccia and BDS tests) there is little agreement whether the dynamics of financial data is consistent with stochastic or chaotic processes [5]. Despite that, strong evidence of non-chaotic nonlinear dependencies has been found in financial data (see, e.g., Ref. [6]). Recurrence plots [7] and recurrence quantification analysis [8], [9] and [10] are nonlinear time series analysis techniques that detect deterministic dependencies in time series. A recurrence plot (RP) is a visual representation of recurrences (similar system states attained at distinct times) that reveals complex deterministic patterns in dynamical systems. Recurrence quantification analysis (RQA) provides the instruments for quantification of these structures and detect critical transitions in the system. Although RPs and RQA originated in physics, they have been successfully employed in a large number of scientific disciplines [11]. These techniques are particularly appropriate for modeling financial and economics time series since they require no assumptions on stationarity, statistical distribution and minimum number of observations. In recent years, several articles employed RPs and RQA to study deterministic dependencies in financial data. These investigations contemplated various markets such as stocks [12], [13], [14], [15], [16], [17] and [18], exchange rates [19], [20] and [21] and electricity prices [22]. However, the research on stocks has focused on the largest market capitalization indices, including the Dow Jones [12] and [16], the S&P 500 [14] and [17], the NASDAQ and the DAX [15], and little empirical work has been done on the behavior of stocks in emerging markets and smaller developed markets. In fact, to the best of the authors’ knowledge, applications of RPs to emerging markets only considered the Warsaw stock index (WIG) [13] and the Indian stock index (NIFTY) [18]. This void in the extant literature is significant, given that smaller developed economies and many emerging economies progressively enjoy a greater role in the global economy, due to expanding capital and trade movements, and understanding deterministic dependencies in global stock markets is relevant not only to finance theorists but to portfolio managers who use international diversification to reduce risk.1 The absence of studies on emerging markets and smaller developed markets leaves several research questions unanswered. First, it is well-known that stocks in emerging markets have distinct features from stocks in their developed counterparts, such as higher average returns and unconditional volatility, and greater levels of predictability of stock returns. Furthermore, emerging markets are typically characterized by small numbers of listed companies, low market capitalization, trading volumes and liquidity, and high levels of political risk and regulatory restrictions. Accordingly, it is important to understand whether these differences are reflected in recurrence plots and the corresponding RQA measures. Second, while smaller developed markets and emerging markets underwent a remarkable development and a greater integration in global capital markets, a substantial share of the integration may have occurred at a regional level. Thus, similarities in recurrence plots of markets across the same economic region should be investigated. Third, critical financial events increasingly affect both developed and emerging economies. Therefore, it is essential to understand the impact of these events on RQA measures and compare how they affect developed and emerging markets. This paper attempts to address these questions by performing a comprehensive examination of the behavior of a large number of stock markets using recurrence plots and recurrence quantification analysis. The analysis is based on 15 years of daily prices of free float-adjusted market capitalization stock indices from 46 countries, representing about 70% of the world’s population and 90% of the world’s GDP. These indices are constructed and maintained by Morgan Stanley Capital International (MSCI) and are commonly adopted as the benchmark against which the performance of international equity portfolios are compared. Because the construction and maintenance of the MSCI index family follows a consistent methodology, idiosyncrasies associated to local stock exchange indices are avoided. The data employed in this study covers the period from January 1995 to December 2009. This period witnessed the 1997 Asian currency crisis, the 2000 burst of the dot-com bubble, and the 2008–09 subprime mortgage crisis. The dynamics of some selected indices during these financial events is analyzed by computing RQA measures in sliding windows. The remainder of this paper is organized as follows. The next section describes the database of equity indices employed in this study. Section 3 briefly reviews the recurrence plot methodology and shows several plots of stock indices across different economic regions. The patterns on these plots are also analyzed. The recurrence quantification analysis measures for the complete data set are reported and discussed in Section 4. Statistical tests comparing RQA measures in developed and emerging stock markets are also presented. In Section 5, the temporal evolution of RQA measures during critical financial events is addressed using a windowed version of RQA. Finally, Section 6 presents some concluding remarks.

In this study, a comprehensive investigation of the dynamics of 46 stock markets was performed using recurrence plots and recurrence quantification analysis. The analysis covered the period between January 1995 and December 2009. Distance plots of several stock markets were presented. The analyzed plots suggest that stock markets in countries with strong economic interdependence tend to display similar features in recurrence plots. For instance, while distance plots of Western markets exhibited a “butterfly” shaped structure, Southeast Asian market displayed an “arrow” shaped structure. On the other hand, the plots for Eastern European and Latin American markets are characterized by small distances in the lower left corner of the plot and larger distances towards the upper right corner. Several RQA measures and corresponding 95% confidence intervals were computed for the complete period. With respect to measure DET, which provides an indication of determinism in the price-generating system, the two largest markets in the world, Japan and the United States, exhibited the first and third lowest values, respectively. Other large European stock markets, such as France, Italy, the Netherlands, and the United Kingdom also showed relatively low values of DET. However, the confidence intervals of the RQA measures are large and prudence is needed when interpreting the relative order of the markets. In the emerging markets group, the stock market of Taiwan clearly stands out as having the lowest values of DET. The measure ENTR provided similar results to those of measure DET. In the group of developed markets, Japan and the United States exhibited the lowest values of ENTR, while in the group of emerging markets Taiwan presented the lowest ENTR. In fact, the value of ENTR for Taiwan is smaller that those of many developed markets. Furthermore, the results provided by measures based on vertical structures, LAM and TT, essentially replicated those of measures based on diagonal structures. Two-group mean comparison TT-tests and median comparison Wilcoxon–Mann–Whitney UU-tests indicated that the differences between developed and emerging markets in terms of RQA measures are statistically significant. These results substantiate the notion that the dynamics of stock markets with large trading volumes and liquidity, and fewer problems of information asymmetry and opaqueness, are normally less predictable. A time-dependent RQA was performed, focusing on the behavior of stock markets during stock market collapses, such as the burst of the technology bubble, the Asian currency crisis and the subprime mortgage crisis. This analysis showed that measures DET and LAM can vary substantially over long periods of time. In particular, during these critical events significant declines in the levels of DET and LAM are observed.