1 Introduction
Integration of global markets have resulted in higher interdependence between the financial markets. One emergent method of capturing the interdependence has been network analysis [28, 36, 10, 33]. Critical events like stock market crashes are likely to impact stock markets across the globe [9, 1, 14, 22].
The credit crisis of 2008 has propelled a volume of research on stock market interdependence and the resulting contagion. Kenourgios et al. [16], using a multivariate regimeswitching copula model found evidence of contagion effect between BRIC countries and US and UK stock markets during five financial crises. Markwat et al. [24] found ‘domino effect’ across global stock markets where a crash in a local market has impact on multiple stock markets. Boubaker et al. [11] confirmed contagion effect between the stock markets of the US and some selected developed and emerging countries during the subprime crisis. Longstaff [20] investigated subprime asset pricing during the subprime crisis and established contagion effect across markets. Proposing a waveletbased approach to test for financial contagion, Galllegati [12] concluded that global stock markets were impacted by the US subprime crisis. These and many other works [13, 8, 7] have shown that critical periods in stock price movement like stock market crashes always impact multiple markets across the globe. This means that the nature of the interdependence of global stock markets undergo changes during such periods. Both research and practice in stock markets need to focus on methods of modelling this interdependence and devise methodology to detect change points in the time evolution of such interdependence. Mantegna [23] provided one of the earlier works in exploring the dynamics among various stocks by constructing a minimal spanning tree (MST) using correlation between the stocks. Onnela et al. [29] explored the dynamics of the MST constructed from the correlation matrix of stock returns to detect market crises. Mizuno et al. [26] also studied correlation networks among currencies to construct a currency MST tree followed by clustering of the currencies. In a recent work, Kazemilari and Djauhari [15] analyzed the topological properties of stocks using the RV coefficient corresponding to multivariate time series data. Namaki et al. [27]
used random matrix theory to analyze stock market network and provided some useful insights on the interdependence of individual stock prices in US and Tehran stock markets.
An interesting problem in this regard is that of changepoint detection. Since we are considering networks which evolve over time, structural changes occur in the networks as the dynamics among the players of the network change significantly. These changes might be attributed to external factors affecting some or all of the players involved. Changepoint detection has been widely studied in the context of statistical process control; see [35, 21] and references therein. McCulloh and Carley [25] used concepts from process control for detecting changepoints in social networks varying over time. Aue et al. [2]
considered the problem in the context of low dimensional multivariate time series and developed a nonparametric test statistic for detecting change point in the coviance structure. Barnett and Onnela
[5] used a testing based approach as well using correlation networks for detecting multiple change points. Moving to the high dimensional set up, Roy et al. [32] developed a twostep algorithmic approach for single change point detection in networks corresponding to sparse Markov random fields. More recently, Avanesov and Buzun [3]developed a change point detection method in high dimensional covariance structure of the underlying variables using statistical hypothesis testing. Peel and Clauset
[31]adopted a testing of hypothesis approach as well to infer whether a change point has occured in a certain window of time, using a parametric family of distributions for the data. They used the concept of Bayes factor, and determined the change point using a
value approach, similar to concepts in [6]. Gaussian graphical models (GGMs), where the underlying random variables defined over the nodes of the network are assumed to be normally distributed, provide an utilatarian statistical framework for modeling dynamic networks
[37]. Kolar and Xing [18]consider the estimation problem of such networks, followed by another work where they developed the method for estimating networks with jumps using a varying coefficient varying structure (VCVS) model
[19]. In a recent work, Keshavarz et al. [17] considered a sequential change point detection algorithm for time varying sparse GGMs. A nonparametric sequential alorithm for detecting sudden changes in a network has been developed by Park and Sohn [30].We consider the problem of sequential change point detection in a dynamic network of a group of international stock markets to detect changes in the network dynamics during periods close to stock market crashes. Our work focuses on capturing the dynamics of the financial market via construction of the underlying network structure of the Asia – Pacific stock markets, which present an interesting combination of emerging and developed markets, along with US and UK stock markets using stock return data from April 2000 to April 2017. We use the Gaussian kernel function which provides a distribution free construction of underlying network while simultaneously preserving the topology. This gets rid of the assumption about the underlying variables being multivariate normal which for financial returns might not hold true in general. This is followed by a statistical testing approach for change point detection in timeevolving networks as proposed in [5].
This work contributes to the literature in twofold ways. First, we bring methodological innovation in network analysis using Gaussian kernel function, which renders the model more suited for nonnormal variables like financial returns. With the adoption of Gaussian radial basis kernel in the change detection method proposed by [5], we extend their model and bring in more efficiency in detecting change points. Additionally, we apply this method to financial returns of a group of stock markets of strategic importance and detect the impact of local crashes across markets. This opens the possibility of using our method as a warning signal. Moreover, we also illustrate that the kernel based network performs superior compared to the correlation network in terms of detecting valid change points.
2 Data description
We consider daily data for 11 stock market indices from Asia Pacific region: Australia (AORD), Hong Kong (HSI), Japan (NIKKEI), Singapore (STI), China (SSE), India (NIFTY), Indonesia (JKSE), South Korea (KOSPI), Malaysia (KLSE), Philippines (PSEI) and Taiwan (TWII), along with the US (DJI) and the UK (FTSE) stock market indices, for the period of April 2000 to April 2017. The data was taken from Thomson Reuters database.
In Figure 1, we plot the stock market returns corresponding to the indices – DJI, SSE and NIFTY, for the time period of January 2007 to January 2009. This timeline includes the 200708 global financial crisis period. Network based analysis will help in capturing the interplay among the various indices and explain the changing dynamics of the network before or after stock market crashes.
We construct the network based on the daily returns computed as the first difference of the level data. We observe that the following stock market crashes took place during the period.

11Sep01, Economic effects arising from the September 11 attacks, USA.
The September 11 attacks caused global stock markets to drop sharply. The attacks themselves caused approximately $40 billion in insurance losses, making it one of the largest insured events ever.

09Oct02, Stock market downturn of 2002, USA.
A downturn in stock prices during 2002 was observed in stock exchanges across the United States, Canada, Asia, and Europe. After recovering from lows reached following the September 11 attacks, indices slid steadily starting in March 2002, with dramatic declines in July and September leading to lows last reached in 1997 and 1998.

27Feb07, Chinese Stock Bubble of 2007, China.
The SSE Composite Index of the Shanghai Stock Exchange tumbled 9% from unexpected selloffs, the largest drop in 10 years, triggering major drops in worldwide stock markets.

11Oct07, United States bear market of 2007  09, USA.
Till June 2009, the Dow Jones Industrial Average, Nasdaq Composite and S&P 500 all experienced declines of greater than 20% from their peaks in late 2007.

16Sep08, Financial crisis of 2007  08
On September 16, 2008, failures of large financial institutions in the United States, primarily due to exposure of securities of packaged subprime loans and credit default swaps issued to insure these loans and their issuers, rapidly devolved into a global crisis resulting in a number of bank failures in Europe and sharp reductions in the value of equities (stock) and commodities worldwide. The failure of banks in Iceland resulted in a devaluation of the Icelandic króna and threatened the government with bankruptcy. Iceland was able to secure an emergency loan from the IMF in November. Later on, U.S. President George W. Bush signed the Emergency Economic Stabilization Act into law, creating a Troubled Asset Relief Program (TARP) to purchase failing bank assets. This had disastrous effects on the world economy along with world trade.

27Nov09, 2009 Dubai debt standstill, United Arab Emirates.
Dubai requested a debt deferment following its massive renovation and development projects, as well as the Great Recession. The announcement caused global stock markets to drop.

27Apr10, European sovereign debt crisis, Europe.
Standard & Poor’s downgraded Greece’s sovereign credit rating to junk four days after the activation of a 45billion Euros EUIMF bailout, triggering the decline of stock markets worldwide and of the Euro’s value, and furthering a European sovereign debt crisis.

06May10, 2010 Flash Crash, USA.
The Dow Jones Industrial Average suffered its worst intraday point loss, dropping nearly 1,000 points before partially recovering.

01Aug11, August 2011 stock markets fall, USA.
Stock markets around the world plummeted during late July and early August, and remained volatile for the rest of the year.

12Jun15, 2015 – 16 Chinese stock market crash, China.
China stock market crash started in June and continued into July and August. In January 2016, Chinese stock market experienced a steep selloff which sets off a global rout.

18Aug15, 2015 – 16 stock market selloff, United States.
The Dow Jones fell 588 points during a twoday period, 1300 points from August 18 – 21. On Monday, August 24, world stock markets were down substantially, wiping out all gains made in 2015, with interlinked drops in commodities such as oil, which hit a sixyear price low, copper, and most of Asian currencies, but the Japanese yen, losing value against the United States dollar. With this plunge, an estimated ten trillion dollars had been wiped off the books on global markets since June 3.
3 Network construction
We have a fixed set of nodes for constructing the financial network corresponding to different stocks. This results in dimensional observations at fixed time points . Let
denote the vector of observations for stock
, and the resulting data matrix be denoted by Based on the observed data matrix , we aim to construct a similarity graph or network , where is the set of nodes corresponding to and is the collection of edges. The resulting network may be an unweighted or a weighted one, depending on whether the ‘similarity’ associated with any pair of edges is binary, or some positive quantity , respectively. We say that two nodes are connected with edgestrength if In this context, the weighted adjacency matrix of a network is defined as the symmetric matrix .3.1 Similarity Networks
We now discuss different types of similarity networks that can be constructed in this context.
neighborhood networks: An neighborhood network is constructed by connecting the nodes for which the distance between two nodes exceeds a certain threshold The distance measure can be any welldefined metric, for example, the distance or the distance. The resulting network is an unweighted one.
Correlation/Partial Correlation networks: Correlation networks are constructed using the absolute value of the pairwise correlation between any two nodes as the corresponding edgeweights, that is, being the sample correlation coefficient between and . Correlation networks have been extensively studied in various fields including analyzing protein networks (see [34] and references therein), and financial networks [5, 26, 33]. Partial correlation networks are conceptually similar, with the main difference being the choice of the weight function as the partial correlation between pairs of nodes instead of correlations. Partial correlation between any two nodes adjust for the effect of the remainder of the nodes and hence might be more appropriate. Banerjee et al. [4] used partial correlation graphs to model proteomic data in the context of graph clustering.
Kernel based networks: Kernelbased networks result in a fully connected graph where the underlying similarity measure captures the local neighborhood relationships of the network. The kernel function acts as a similarity measure between any pair of nodes, such that similar behaving nodes have higher edge weights. For example, for a Gaussian radialbasis kernel, the strength of an edge between two nodes and is given by
(3.1) 
The parameter above controls the width of the nodeneighborhoods in the network. Other choices of the kernel function include Laplacian kernel, polynomial kernel, Spline kernel, Bessel kernel, etc.
We shall be using the Gaussian kernel function to construct our financial network. Note that the network can be constructed using any subvector of as well. One major reason behind the choice is that the network corresponding to the nodes is constructed under a nonparametric framework without any distributional assumptions on the columns of In comparison, partial correlation networks effectively capture the conditional independence structure of the underlying variables in the nodes, but only under the assumption that the variables are multivariate Normal. Hence kernelbased approaches provide a distributionfree construction of the underlying network while simultaneously preserving the neighborhood topology.
4 Changepoint detection method
We adopt the method proposed in [5] for multiple change point detection. While their method was developed for correlation networks, we shall be primarily working with the network constructed using the Gaussian radial basis kernel as described in the previous section. For two contiguous time windows and where , we say that a changepoint exists at the boundary if the networks constructed using the respective time windows are significantly different.
Let us define the adjacency matrix of the network constructed using the subvector of observations in the time range as . The presence of a changepoint at the time point in a given time interval , should result in a different network topology in the respective time intervals and This change in the topology may be captured using some suitable distance metric like the Euclidean norm, defined as
(4.1) 
If and respectively represent the ‘true’ adjacency matrices corresponding to the network in the given time intervals, then our problem boils down to testing the hypothesis
for some We propose to use the distance metric defined above as the test statistic. However, the exact sampling distribution of the same is not known. Hence we shall adopt a resampling approach to generate values of
under the assumption that the null hypothesis
is true. For this, we first generate bootstrap samples of the rows of the data matrix corresponding to the observations in the time interval . Let us denote the bootstrap resamples of the corresponding data matrix as where is the total number of bootstrap resamples. For each such data matrix, we construct the bootstrap resampled network using the Gaussian radial basis kernel function for the time intervals and , where . The quantity is chosen in such a way that we do not search for a potential change point near the endpoints or . Thus we finally get resampled values of the test statistic as(4.2) 
where and respectively denote the weighted adjacency matrices in the time intervals and based on the resampled data matrix A score corresponding to the potential changepoint is given by
(4.3) 
where and .
Note that, if autocorrelations are present among the observations, then resampling using an independent bootstrap as above might lead to a bias in approximating the null distribution. In that case, a sieve bootstrap may be used as outlined in [5].
A potential changepoint is then identified as the time point having the maximum score, so that we define We calculate a similar score for the observed data , given by
(4.4) 
The resulting pvalue of the test is thus given by
(4.5) 
where If the resulting value is lesser than some chosen level of significance , then we reject the null hypothesis and conclude that a change point exists at the time point
For detecting multiple change points over a longer interval, we can search for potential change points by splitting the interval into smaller intervals and then repeating the above method for each of the different segments. Apart from this, a chosen subinterval might also have multiple change points. In that case, as discussed in [5], after we have detected a certain change point at the time point , we can search for potential change points within the subintervals and individually, and repeat it further over smaller subintervals.
5 Results
We applied our change point detection method on different windows of length 200 days with choice of to be 50, so as to detect a change point for We checked for the presence of autocorrelations among the daily stock returns using a DurbinWatson testing procedure but didn’t get sufficient evidence in favor of the same. This lead us to use independent bootstrap for resampling the rows of the data matrix.
Our change point analyses reveal that the dates where the network dynamics undergoes change, are all close to some stock market crash. Table 1 list the dates and the related stock market events. In Figure 2
, we show the plots of the zscores for the observed data (blue line) against the zscores corresponding to the bootstrapped samples (black lines), for networks analysed over a 200day time period of 21April2008 to 30June2009. Our method scanned for change points in a set of 101 contiguous dates, excluding
days on either side of the aforementioned dates. The true zscores reach the peak on 06October2008, and has a value of zero, thus confirming a changepoint in the network around that time period. The date corresponds to the global finanical crisis of 200708.If we carefully observe the change points, we find that the cases where no change point were detected, that is, the 9/11 and 2010 Flash crash – were exogenous crashes where stock markets took a temporary dip due to unforeseen circumstances. In all other cases, however, there were stock market bubbles building over a time and the shocks were endogenous in nature. The only exception is the Dubai 2009 debt standstill. That may be expected as most of the major stock markets like FTSE shrugged off the effect of the Dubai crisis within a few days. A closer inspection reveals that the time gap between the change point and the reported crash dates is different for different events. That may be explained from the fact that the impact of different crashes on the network dynamics of multiple stock markets is not same for all events. In some cases, the stock markets of countries other than the place of origin of crash react with a lag while in others the spread of contagion is instantaneous. What is evident from our studies that most stock market crashes significantly impact multiple stock markets, albeit in varying degrees. Our method has detected the break in network dynamics in all the cases where there was a known contagion effect.
Name  Date  Country  Change Point  


11Sep01  United States 



09Oct02  United States  13Nov02  

27Feb07  China  27Dec06  

11Oct07  United States  09Oct07  

16Sep08  United States  06Oct08  

27Nov09 




27Apr10  Europe  09Feb2010  

06May10  United States 



01Aug11  USA  27Jul11  

12Jun15  China  16Jun15  

18Aug15  United States  19Aug15 
For comparison, we applied the changepoint detection procedure for a correlation network as well. The method based on correlation network failed to detect almost all the major stock market crises, except for the financial crisis of 2007  2008 (change point detected on 12August2008). Additionally, it detected two change points on 28July2006 and 10August2016. With no reported stock market crash during this time it is very unlikely that the global stock market correlation underwent any significant change during that period.
6 Discussion
Literature has shown that crashes at local stock markets often impact multiple markets and spread across them. A natural corollary of this phenomenon is that the network dynamics of stock markets across the globe are likely to undergo critical shift. In this paper we create a network of AsiaPacific stock markets along with US and UK stock markets using daily index data. We use the Gaussian kernel function which provides a distribution free construction of underlying network while simultaneously preserving the topology. To understand the impact of critical regimes we detect change points in the network adopting the method presented in [5] with a modification allowing for accommodation of Gaussian kernel. On inspection, the dates corresponding to the change points detected are found close to reported crash dates of any of the stock markets in the network, especially those crashes which were endogenous in nature. Our work contributes both to the methodology and application of network change point detection. On the methodological side we contribute by modifying the method presented in [5]. This allowed us to build the network with gaussian kernel which has its obvious advantages as explained above. On the application side the ability of our method to detect change points in case of all the known endogenous stock market crashes makes it useful in distinguishing endogenous and exogenous crashes. This may also serve as a warning bell for change in market dynamics for investors exposed to multiple stock markets across globe.
Acknowledgements
S.B. is partially supported by DST INSPIRE Faculty Award, Govt. of India, Grant No. 04/2015/002165.
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