The interdependency structure in the Mexican stock exchange: A network approach
TThe interdependency structure in the Mexicanstock exchange: A network approach
Erick Trevino Aguilar [email protected] ∗ April 15, 2020
Abstract
Our goal in this paper is to study and characterize the interdependency struc-ture of the Mexican Stock Exchange (mainly stocks from BMV) in the period2000-2019 and provide visualizations which in a one shot provide a big-picturepanorama. To this end, we estimate correlation / concentration matrices from dif-ferent models and then compute metrics from network theory including eigencen-tralities and network modularity. In this paper we investigate the interdependency structure of daily returns in the Mex-ican stock exchange market. To this end, we build a database of free and publiclyavailable time series of main stocks for the period 2000-2019 and conduct our studyin two stages that are then put together to give a unified treatment to our main topic ofinterest here which is the interdependency structure of daily log-returns in the Mexicanstock exchange.In the first stage we focus on the estimation of partial correlations of log returnsof daily prices. The reason to focus on partial correlations is the following. Given acollection of Gaussian series A , . . . , A n a zero partial correlation between A and A implies that A and A are conditionally independent meaning that A and A couldstill be (unconditionally) correlated but only through a third factor adapted to theother series A , . . . , A n . There are of course di ff erent methods to estimate a covari-ance / correlation / concentration matrix and we have selected a estimation based on aspecific class of Markovian Random Fields (MRF) which in the statistical literature arewell known under the name Gaussian Graphical model (GGm). The adjective “graph-ical” emphasizes the fact that attached to the probabilistic model there is a graph inwhich edges expresses conditional dependencies, from which a very convenient visualrepresentation is obtained. There are three reasons to work with this model. First of all,the benefit of the already mentioned visual representation provided by the model. The ∗ Unidad Cuernavaca del Instituto de Matem´aticas, UNAM a r X i v : . [ q -f i n . T R ] A p r econd is that we have decided to study the period 2000-2019 in a yearly basis. Thereis a trade-o ff to this treatment. On the one hand, short periods of time reduce problemswith heavy tails. On the other, the number of stocks in each year is a significant pro-portion of the available observations. Hence, a lasso -regularized estimation is usefulin this context which is inbuilt in the estimation of a GGm. Third, we want an estima-tion that filters out a “noisy” correlation selecting only clear relationships between twoseries, again this is provided by the lasso -regularized estimation. Loosely speaking,we follow a partial correlations selection approach which conceptually is comparableto a covariance selection approach [8]. Once partial correlations matrices have beenestimated we provide a list of stylized facts from them. Then, taking the graphs con-structed from the matrix of partial correlations as its adjacency matrix, we computedegree and eigen centralities and rank di ff erent stocks accordingly.In the second stage we estimate correlation matrices of time series (estimated througha Multivariate Dynamic Conditional Correlation GARCH specification). Then, apply atechnique from network-theory based on those correlation matrices: The maximizationof a modularity objective function. This procedure will provide with a partition on thestocks list for a community structure.After this introduction the paper is organized in the following form. Section 2 givessome background on the approach of random networks in finance and economics. Italso provides details on the data used to feed the models. In Section 3 we report on theestimated partial correlation matrices from GGm’s. In Section 4 we report centralitiesof networks based on the partial correlation matrices from the previous Section 3. InSection 5 we report on correlation matrices computed from a multivariate GARCHspecification and then maximize a modularity objective function of networks based onthese matrices. This will define groups (communities) of stocks. Section 6 concludesthe paper with a financial discussion based on the main findings of estimations. The classical Markowitz theory of portfolio selection illustrates the correlation matrixrelevance for financial decisions. However, it has been longly known the nontrivial-ity of correlation estimation from empirical data. Moreover, in contexts where sparsecorrelation (specially for partial correlation) matrices are expected, it is desirable tohave a systematic method to discard “non-clear correlations” and account for a parsi-monious model as motivated by [8]. As we mentioned in the introduction, in this paperwe choose to apply a GGm for a parsimonious estimation of concentration / partial-correlation matrices. For the estimation of correlation matrices we apply a multivariateGARCH model.Beyond the estimation problem, it is useful to have tools that starting from matricesare able to generate metrics providing snapshots of the market from which quick buttrustable diagnosis are possible. Situations in which this is desirable include, from thepoint of view of an investor, the decision of re-balancing a portfolio, and from the point2f view of a regulator, interventions in the market in order to lessen the contagion of ashock in a specific sector.We find those tools in the theory of random networks. Specially in the form oflocal metrics (centralities computed from concentration matrices) to classify the inter-connectedness of stocks and a global metric (the modularity computed from correlationmatrices) to detect communities of stocks.The described approach is not new in finance and economics, however for the Mex-ican stock exchange there are few works in this direction. In the next section we presentrelated literature. Note however that we do not pretend to give an exhaustive list on thisactive topic which deserves a survey by its own, but to give a brief panorama on activityfor this line of work. Gaussian graphical models, Random graphs, and Network theory approaches in a fi-nancial context is an active research area attracting more and more attention with anincreasing number of papers. The following is a non exhaustive list just exhibiting thedi ff erent approaches and applications.Papers with a GGm approach in finance include [14], [1], [15]. Theoretical back-ground on graphical models can be found on [50], [22], [3]. Some studies determinepower laws for degree connectedness defined by assets correlation matrices; see [28],[47], [20], [5], [33], [6]. Papers studying community detection in a financial contextinclude [28], [25], [2], [18], [36]. See [12] for a survey on methods for communitydetection. Minimal Spanning Trees applied to financial market ranking include [17],[23], [51], [6], [26]. Spillover e ff ects and shocks contagion, [42], [29], [21], [4], [9].Portfolio selection, [35], [38]. Detection of stock prices manipulation [44]. Portfo-lio diversification [5]. Also relevant for statistical analysis, random matrix theory forcorrelation matrices has been studied in [4], [25], [41], [37]. According to [45] there indeed existed an impact from the 2007-2008 subprime crisisin the Mexican economy mainly due to two shocks, first, a decline in Mexico’s exportsand second, a constrained access to international financial markets. Thus, an eventevidencing an integration of the Mexican stock exchange and the US market. A phe-nomenon documented by some authors; see e.g., [39, 40, 49]. Figure 1 illustrates pricelevels for some selected stocks in the Mexican stock market for the years 2006, 2007and 2008. It can be argued that on 2006 bullish stocks dominate the market while on2008 they turn bearish. Not surprising and reported by some authors [34, 19]. Laterwe will go beyond a visual examination and confirm by a multivariate GARCH modelthrough a shift from positive to negative intercepts on log returns of each time seriesof the period; see Section 5.2 below. However, quite interesting, we will show thatthe partial-correlations interdependency structure of the Mexican financial market does3ot show a dramatic change as response to that shock for price levels, see Figure 2 andSection 3.3. . . . Subprime crisis period
Figure 1: Filtered list of stock time series for the period 2006-2008.
We constructed a database of close prices from publicly available information at Ya-hoo.Finance website. The complete list of analyzed stocks can be found in AppendixA while the database and all estimations are themselve available upon request. Thefrequency was daily in a span of time comprising 01-01-2000 to 31-12-2019. Alwaysconsidered time series of log returns: R t ( i ) = log (cid:16) S t + ( i ) S t ( i ) (cid:17) where S ( i ) is the price levelof stock labeled i .We breakup the data in windows of one year (from january to december) and applied afiltering process in two steps. In the first step, for each year, stocks in the market withthe most complete information were selected. The criterion was that only stocks withmore than 90% of all the available dates were selected. Then, in a second step, stocksprices not having a minimum of variance in moving windows spanning 30 dates werediscarded. This filtering process already presents the interesting fact of a positive evo-lution of the Mexican market for equities in the sense of an increase of activity. Indeed,4s we go forward along the years, more and more time series of stocks prices satisfythe filtering process evidencing a grow up in terms of more activity in the market withmore variability of prices and more quotes. Visual evidence can be found in Figure2 and Section A. An important aspect of this work will be to consider how industrialsectors are interconnected. Here we consider a list of sectors obtained from a BMV’sclassification. These are listed in Table 1.Table 1: Industrial sectors Energy Industry The IPC Index Materials Basic consummingHealth Telecommunications Financial services Non basic consumming Information Technologies
Let Σ be the covariance matrix of a random vector ( R , . . . , R n ) with multivariate Gaus-sian distribution. A zero component Σ i , j = R i and R j . On the other hand, the inverse matrix J : = Σ − , the so-called concentrationmatrix, has the property that a zero component J i , j = J . Systematic presentations forgraphical models can be found in [50], [22], [3]. In this section we start with the basic definition of a Markovian Random Field (MRF)which is the fundamental probabilistic concept from which a GGm is defined. Letus introduce a graph G = ( V , E ) with a set of nodes V = { , . . . , n } and edges E .Recall that a complete subgraph of G is called a clique . We denote by C the classof maximal cliques of the graph G . Let be given a random vector (cid:126) X = ( X , . . . , X n )with multivariate accumulative distribution function p . Then, the vector (cid:126) X has a Gibbs istribution compatible with the graph G if it has a representation p ( x , . . . , x n ) = Z (cid:89) C ∈C ψ C ( x C ) , where { ψ c } C ∈C are suitable functions and x C denotes a vector in which only the indexesof C appear. Gibbs distribution are characterized through di ff erent Markov properties.Similarly X A for A ⊂ V , A = ( A i , . . . , A i k ) denotes the vector ( X i , . . . , X i k ). Let usintroduce them:1. (cid:126) X is a MRF with respect to G if it has the Markov property: For any pair i , j ∈ V with i (cid:44) j and non adjacent in the graph G , the random variables X i and X j areconditionally independent on all the other variables. We denote this conditionalindependency by: X u (cid:121) X v | X V / { u , v } . (cid:126) X is locally a MRF with respect to G if: For each v ∈ V , the random variable X v is conditionally independent of all other variables which are not neighboors(they are not adjacent). We denote this by X v (cid:121) X V / neighborhood ( v ) | X neighborhood ( v ) . (cid:126) X is globally a MRF if: For two disjoint subsets A , B ⊂ V , the vectors (cid:126) X A , (cid:126) X B areconditionally independent on a separating set S ⊂ V . We denote this by: X A (cid:121) X B | X S . The next is a fundamental equivalence result; see [16, Chapter 7].
Theorem 1 (Hammersley-Cli ff ord) . Assume that the distribution p of (cid:126)
X is defined ina finite state space and is positive valued. Then p is a Gibbs distribution if and only (cid:126)
Xsatisfies any of the Markov properties.
For a list of Gibbs distributions see e.g., [50, Section 3]. In this paper we will workwith the following specific Gibbs distribution (hence, specific MRF and specific GGm) p θ ( x ) = exp θ · x + m (cid:88) i = m (cid:88) i = Θ i , j x i x j − A ( θ ) . (1)where A ( · ) is a normalizing constant; see [50, Example 3.3] for more details. The MRFmodel in (1) specifies also the GGm we will work with. Indeed, (1) does not apriorispecify any graph, but from the set of parameters Θ i , j ∈ R we derive a partial correlationmatrix which indeed can be seen as the adjacency matrix of a weighted graph.6 .2 The GGm Now we explain the specification of the GGm we are going to estimate. Let Σ bethe covariance matrix of the log returns time series R (1) , . . . , R ( n ). Denote by J theconcentration matrix, J : = Σ − . Indeed, the components of the matrix J are given interms of the coe ffi cients Θ i , j in equation (1). Denote by ρ i , j the partial correlation of R ( i ) and R ( j ). Consider the linear regressions defining partial correlations: R ( i ) − µ ( i ) = (cid:88) j (cid:44) i β i , j ( R ( j ) − µ j ) + (cid:15) ( i ) (2)where µ i is the unconditional mean of R ( i ) and (cid:15) ( i ) is a residual. Then β i j = ρ i , j (cid:112) var ( (cid:15) ( i )) var ( (cid:15) ( j )) . (3)It is also true that ρ i , j = − J i , j (cid:112) J ii J j j . The adjacency matrix P = ( P i , j ) is defined by P i , i = P i , j = ρ i , j . (4) Remark 1.
Let us emphasize now that the estimation of the GGm (1) will ultimatelyresult in the matrix P and this matrix is our main input for this section. In this section we report the results on estimating a GGm with underlying MRF (foreach year in the period 2000-2019) with specification (1). From this estimation ex-ercise, proposed as able of capturing conditional (in)dependencies for logreturns timeseries, we get a list of partial correlation matrices P for the years in the period 2000-2019. Such matrices are available as supplementary material. A graphical represen-tation of partial correlations is displayed in the panel of Figure 2 and a complete listof most strong partial correlations in the interval [0 . ,
1] and [0 . , .
3] can be found inAppendix B. The complete list is available as supplementary material. From them, wehave the following stylized facts: • First of all we see in Figure 2 a stable continuous evolution of partial-correlationsinterdependence structure. In particular, at this step of a visual evidence, if thereindeed exist impact of global crisis episodes (e.g., dot.com bubble, Subprimecrisis, etc.) it doesn’t seem to a ff ect at large for the partial-correlations interde-pendence structure. • Many links above 0.2 and frequently include the main index from the Mexicanstock Exchange BMV denominated IPC (quoted as MXX in Yahoo.Finance); seethe tables in Appendix B. 7
As we move forward in time, the market grows (with more nodes of stocks con-sistently quoted by year). However, it does not seem to be evidence that inter-connectedness in the market changes drastically from one year to the other, evenfor the subprime crisis period. • Stock connections must be due to exogenous factors to the market, since thegraph is based on partial correlations. However, for links that involve as a nodethe IPC, it typically happens that the other node is a stock involved in the con-struction of the index. • A large number of links between stocks in di ff erent sectors. An empirical factreported for other markets; see e.g., [28]. To our best knowledge, not previouslydocumented for the Mexican market. Nonetheless, satisfactorily, intrasectorialpartial correlations are also present. • We see persistent links between pair of stocks that along the twenty years periodappear frequently but not systematically; see Apendix C. • Negative (partial) correlations appear only seldom. • For the year 2000 we see a partial correlation of 0.98 between ICA and ELEK-TRA which apriori looks as an odd finding. But this is actually supported bydata; see Figure 3. • The strongest links above 0.3 are those typically having as one of its nodes theIPC. Also for the rank [0 . , .
3] links with IPC dominate but with a little declinein comparison with the interval [0 . , • For the rank [0 . , . • FEMSA indeed has a persistent relationship with IPC with partial correlationsabove 0.3; see the table in Appendix C. In this same table we do not see animportant stock as AMX. This is an interesting confirmation for GGm model’strength, since it captures a realistic fact; see e.g., the news stories expansion andel economista, etc. 8
CC ALFAMX ARABAC BIMCEMCMR CYDELE FEM GFIGFNICAICH KIM KUO MXXPINSOR
ACC ALFAMX ARAAZT BAC BIMCEMCMR CYDELE FEM GFIGFNGME ICAICH KIM MXXPE& SOR
ACC ALFAMX ARAAZT BAC BIMCEMCIDELE FEM GFIGFNGME ICAICH KIM MXXPE& SOR
ACC ALFAMX ARAAZT BAC BIMCEMCID CYD DINELE FEMGCC GFIGFNGISGME HERICAICH KIM MXXPAPPE& SORWAL
ACC ALFALS AMX ARAAZT BAC BIMCEMCID CMOCMR CYDELE FEMGCC GFIGFNGMDGME HERICAICH KIM KUO MXXPAPPE&RCE SORWAL
ALFALS AMX ARAAZT BAC BIMCEM CERCIDCMR CULCYDELE FEMGCC GFIGFNGISGMDGME HER HOMICAICH KIM KUO MXXPAPPE&RCE SOR URBWAL
ALFALS AMX ARAAXTAZT BAC BIMCEM CMOCMR CYDELE FEMGCC GFIGFNGISGMDGME HER HOMICAICH KIM MXXPAP PASPE& PINRCE SOR URBWAL
ALFALS AMX ARAAXTAZT BAC BIMCAB CEMCID CMOCMR CULCYDELE FEM GAPGCCGFA GFIGFNGISGMDGME HER HOMICAICH KIM MXXOMAPAP PASPE& PINRCE SOR URBWAL llllll ll llllll llll llllllllll llll llllll ll ll llllllllll llllll llllll ll llllll llllllll llll llllll llll llll llll llllllll ll
ACALFALS AMX ARAASUAUT AXTAZT BACBIMCAB CEMCIE CMOCMR CULCYDELE FEM FIN FRAGAPGCAGCCGFA GFIGFNGIG GISGMDGME GRU HOMICAICH IDEKIMLAMMAX MEDMEG MXXOMAPAP PASPE& PINPOC SARSIM SORTMMURBVIT WAL llllll ll llllll llll llll llllll llll llll llllll ll llllllllll llllllll llllll llllll llll llllll llll ll llllllll llllll llll llllllll ll
ACALFALS AMX ARAASUAUT AXTAZT BACBIM BOLCABCEM CERCIE CMOCMR CULCYDELE FEM FINGAPGCAGCCGFA GFIGFNGIGGME GPRGRUHER HOMICAICH IDEKIM LABLAMMAX MEDMEG MIN MXXOMAPAPPE& PINPOCRCE SARSIM SORTMMURBVIT WAL llllll ll llllll llll llllll llllll llllll ll llllllllll llllll llll llll llllll llll ll llllll llll llllll llll llllll llll llllllll
ACALFALS AMX ARAASUAUT AXTAZT BACBAFBIM BOLCEMCID CMOCYDELE FEM FINGAPGCAGCCGFA GFIGFNGIG GMDGME GRUHER HOMICAICH IDEKIM KUO LABLAMLIV MEDMEG MXXOMAPAP PASPE& PINPOCRCE SARSIM SORTMMURBWAL
AC ACTALE ALFALS ARAASUAUTAXTAZT BACBIM BOLCEM CHDELE FEM FINGAPGCA GENGFA GFIGFNGMDGME GRUHERHOMICAKIM LABMXXOMAPAPPE& PINPOC SARSIM SORSPO URBWAL ll llllllllll ll llllll llll llll llll llllll llll ll llllllllll llll llllllllll llllllllll llll llll llllll ll llllll llll llll llllll llll llllll ll
AC ACTAERALEALFALS AMX ARAASUAUT AXTAZT BACBIM BOLCEM CHDCMOCMR CYDELE FEM FINFRAGAPGCAGCC GENGFA GFIGFNGISGMDGME GRUHER HOMICAICH IDEKIM LABLAM MEDMEG MFR MON MXXOMAPAP PASPE& PINPOC RASARSIM SORSPO TMMURBVIT WAL ll llllllllllll ll llllllllll llllll llll llll ll llllllll ll llllll llllll llll llllllllll ll llll llllll llll llllll llllll ll llllllll llll llll llllll llll ll ll llll llll ll
AC ACTAERALEALFALPALS AMX ARAASUAUT AXTAZT BACBAFBIM BOLCEM CHDCIE CMO CRECULCYDELE FEM FINFRAGAP GBMGCAGCC GENGFA GFIGFNGISGMDGME GRU GSAHER HOMICAICH IDEKIM LABLAMLIV MEDMEG MFR MON MXXOMAORBPAP PASPE& PINPOC RASARSIM SORSPO TEA TMM VALVAS VESVIT WAL ll llllllllllll ll llllll llll llll llll llllll llllllll ll llllll llllll llll llllllllll llllll llllll ll llll llllll llllll llllllllllll llll llllll llllll ll llllll ll ll
AC ACTAERALEALFALPALS AMX ARAASUAUT AXTAZT BACBIM BOLCEM CHDCMOCMR CRECULCYDELE FEM FINFRAGAP GBMGCAGCC GENGFA GFIGFNGISGMDGME GRUGSAHCI HERICAICH IDE IENKIM LABLALLIV MEDMEGMFR MONMXXOMAORBPAPPE& PINPOC RASARSIM SORSPO TEA TMM VALVESVIT VOL WAL ll llllllllllllll ll llllll llll llll llll llll llllllll ll llllll llllll llll llllllllll llllll llll llll ll llll llllll llllll llllllllll llll llll llll llllll llllll ll ll
AC ACTAERAGUALEALFALPALS AMX ARAASUAUT AXTAZT BACBIM BOLCEM CHDCMO CRECULCYDELE FEM FINFRAGAP GBMGCAGCC GENGFA GFIGFNGISGMDGME GRUGSAHCI HERHOT ICAICH IDE IENKIM LABLALLAM MEDMEGMFR MXXOMAORBPAPPE& PINPOC RARCE SARSIM SORSPO TEA TMMVESVIT VOL WAL ll llllllllllllll ll llllll llll llll llllll llllll llllllllll ll llllll llllll llllll llllllllll llllll llllll llll llll ll llllllll llllll llllll llllllll llll llllllllll ll llllll ll ll
AC ACTAERAGUALEALFALPALS AMX ARAASUAUT AXTAZT BACBIM BOLCADCEM CHDCMOCMR CRECULCYDELEKELEM FEM FINFRAGAP GBMGCAGCC GENGEOGFA GFIGFNGICGISGME GRUGSAHCI HERHOMHOT ICAIDE IENJAV KIM LABLACLALLAM MEDMEGMFR MONMXXNEM OMAORBPAPPE& PINPOC QRASORSPOTEA TMM UNIVESVIT VOL WAL llllllllllllll ll llllll llll llll llllll llllll llllllll ll llll llllll llllll llllllllll llllll llllllll ll llllll llllllllll ll llllll llllll llllllll ll llllll ll llllll ll llllllllll llll
ACAERAGUALEALFALPALS AMX ARAASUAUT AXTAZT BACBIM BOLCADCEM CHDCIECMO CRECYDELEKELEM FEM FINGAP GBMGCAGCC GENGEOGFA GFIGFNGICGISGME GRUGSAHCI HERHOMHOTICH IDE IENJAVKIM LABLACLALLAMLIV MAX MEDMEGMFR MONMXXNEM OMAORBPAPPE& PIN QRASIM SIT SORSPOTEA TMM UNIURBVESVINVIT VOLWAL llllllllllll ll llllll llll ll llll llllllll llllll llllllllll llll llllll llll llllllllllllll llllllll llllllll ll llll llllllll ll llllll llllll llllllll ll llllll ll llllll llll llllllllll llll
ACAERAGUALFALPALS AMX ARAASUAUT AXTAZT BAC BBABIM BOLBSMCADCEM CHDCIECMO CRECUECYDELEKELEM FEMGAP GBMGCAGCC GENGFA GFIGFNGICGIGGISGMDGME GMXGRUGSAHCI HERHOMHOTICH IDE IENKIM LABLACLALLIV MAX MEDMEGMFR MONMXXNEM OMAORBPAPPE& PIN QRASIM SIT SORSPOTEA TMMTRA UNIURBVESVINVIT VOLWAL ll llllllllllll ll llllll llll ll llll llllllll llll llllllllllll ll llll llllll llll llllllllll llllllll llll ll ll llll llllllll ll llllll llll llllllll llll llllll ll llll llll llll llllll llll llll
AC ACTAERAGUALFALPALS AMX ARAASUAUT AXTAZT BAC BBABIM BOLBSMCADCEM CHDCMO CRECUECULCYDELEKELEM FEM FRAGAP GBMGCAGCC GENGFA GFIGFNGICGISGME GMXGRUGSAHCI HERHOT ICH IDE IENKIM LABLACLALLAM MAX MEDMEGMFR MXXNEM OMAORBPAPPE& PINPOC QRASIM SIT SORTEA TMMTRA UNIURB VALVESVIN VISVIT VOLWAL
Figure 2: Graphs associated to partial correlation matrices by year in the period 2000-2019. For a given edge the green color (resp. red color ) represents a positive (resp.negative) relationship. Edge’s width represents strength of correlation. A list of par-tial correlations in di ff erent ranks can be found in Appendix B. Vertexes are groupedaccording to its industrial sector.In the tables from Appendix C we see the most “persistent” relationships betweenstocks for which partial correlations in absolute value were in a given interval for nineor more years. Quite notoriously they are rare and always involves the index IPC. Weare particularly interested in obtaining metrics (centralities) from network theory to seea possible e ff ect of the afore mentioned global episodes. Centrality is a measure conceptually designed in such a way that a vertex with high cen-trality is arguably highly influential. The first concept of centrality we use is the degreecentrality which for a vertex in a weighted network is just the sum of all connectingedge’s weights. For our graphs of partial correlations, the degree centrality gives in-formation of the pattern of a shock’s transmission. The idea is that for an influential(i.e., with high centrality) stock in the financial network, a bad day is accompanied9igure 3: In purple the time series for ELEKTRA and in blue the time series for ICA.Prices in logarithmic scale for the year 2000. Source: Yahoo.Finance.with many other stocks in the same situation. Note that there is no causation claimedhere. The second measure of centrality that we estimate is the eigencentrality . This isa global measure in that scores for each node are assigned by a contrast of the qualityof its links. For example a node with just one link to another influential node couldhave a highest eigencentrality than a node with two or more links. The computation ofeigencentralities transfers to a spectral analysis of the adjacency matrix and in crucialsteps is substantiated by Perron-Frobenius theory (see e.g., [43, Chapter 17]).
Let us explain with more detail about eigencentrality and at the same time also clarifyabout shocks transmission. Let V = { , . . . , n } denote our set of stocks and recall thematrix P defined in (4). The eigencentrality is a function f : V → R satisfying f ( v ) = r (cid:88) w ∈ N ( v ) P v , w f ( w ) , v ∈ V , (5)where r is a non negative constant and N ( v ) denotes the neighbors of v . Note that f ( v ) = r (cid:88) w ∈ V P v , w f ( w ) , since by definition w ∈ N ( v ) if and only if P v , w (cid:44)
0. Now this can be written in matricialnotation as f ( V ) = r P f ( V ) T , where f ( V ) = ( f (1) , . . . , f ( n )). Hence, f ( V ) is an eigenvector of P attached to r as itseigenvalue.To continue we follow the discussion in [4], recall the coe ffi cients β i , j in equation(3). The matrix of coe ffi cients B = ( β i , j ) with β ii = B = diag ( J ) − P diag ( J ) . We can write the linear regression in a compactmatricial notation as R − µ = B ( R − µ ) + (cid:15) = diag ( J ) − P diag ( J ) ( R − µ ) + (cid:15). (6)10et ˜ R ( i ) : = R ( i ) − µ ( i ) and ˜ R = ( R (1) − µ (1) , . . . , R ( n ) − µ ( n )). Then, diag ( J ) ˜ R = P diag ( J ) ( R − µ ) + diag ( J ) (cid:15). Hence the vector X : = diag ( J ) ˜ R satisfies X = P X + η where η : = diag ( J ) (cid:15) .Now assume that between times t and t there is a shock ∆ = (0 , . . . , , δ, , . . . , ff ecting X ( i ). Then, X at time t is given by P ( X + ∆ ) + η and the change is then P ∆ . Note that P ∆ does not need to be a scalar of ∆ , meaning that the shock a ff ect-ing originally to X ( i ) is also a ff ecting to other components indicating that the shockpropagates.The spectral decomposition of P helps on assessing the reach of propagation andrationalizes the definition of eigencentrality. Let W , . . . , W n denote the set of eigenvec-tors of P and Λ = { λ , . . . , λ n } the corresponding set of eigenvalues, which we assumeis decreasingly ordered by its modulus. Here is a main assumption: there is a uniqueeigenvalue attaining the spectral radius. This means | λ | > | λ | ≥ | λ | . . . ≥ | λ n | . If thematrix P has only nonnegative components, Perron-Frobenious theory guarantees weare in this situation and even more properties; see e.g., [43, Chapter 17]. Represent ∆ by ∆ = (cid:80) i α i W i . Then, for k ∈ NP k ∆ = λ k α W + n (cid:88) i = (cid:32) λ i λ (cid:33) k W i . Hence lim k →∞ λ k P k ∆ = α W . Then, as time runs the leading term indicating the e ff ect of the initial shock ∆ takes theform λ k α W . In Figure 4 we see estimated centralities for our networks. The blue line is the largestmodulus per year of eigenvalues. The green line represents the maximum degree cen-trality for each year. Very unsurprising this maximum is always attained by the IPC.The red (resp. red and dashed) line represents the average of each node’s degree cen-trality (resp. average of each node’s absolute value degree centrality).11
000 2005 2010 2015 . . . . . . . year Figure 4: The blue line is the largest eigenvalue and dashed blue line is its absolutevalue. The green line is the maximum degree centrality by year, always attained by theIPC. The red lines are averages of degree centralities, resp. absolute value of degreecentralities.These are the facts we observe from Figure 4: • First of all, spectral radius are approximately bounded by two, which coincideswith the range documented for other markets; see e.g., [28]. • The red line and the red dashed lined are almost indistinguishable. This happensas a consequence to the fact that almost all partial correlations are non negative.We also observe the extraordinary stability on the metric represented by this line. • The patterns of the green line and blue line are similar. As we mentioned, thegreen line is attained by the IPC. So it could be expected that also the blue lineis related to this index. Although we do not go into this claim, assuming it iscorrect, in order to capture e ff ects beyond the IPC it might be necessary in thiscase to complement with the second eigenvalue together with its eigenvector forcentrality and the analysis for a shock contagion. Indeed, Figure 5 shows that inmany cases the dominant eigenvalue has multiplicity two or more, and in othercases that the second eigenvalue turns out to be close to the first. Certainly, theidea of considering beyond the dominant eigenvector for eigencentrality is notnew; see e.g., [32]. Analysis for the Mexican case will be addressed elsewhere. • There is indeed variability for centralities, but changes from one year to the other,are indeed in units. Thus, changes are subtle. For example, for the subprimecrisis period, we see in the blue line small upwards jumps from 2005 ( 0.95) to2006 (1.36) and then a decline, from 2007 (1.08) to 2008 (0.88). Analogously formax degree centrality in the green line: we see small upwards jumps from 2005(2.79) to 2006 (3.31) and then a decline, from to 2007 (3.54) to 2008 (2.47).12
Continuing with the previous point. We see an abrupt upwards movement forthe green line which is reasonable to associate with the dot.com bubble’s burst:From the year 2001 (1.49) to 2002 (3.12). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 5: Eigenvalue’s modulus per year.In Figure 6 we see a panel of barplots for degree centralities separated into di ff erentranges for all stocks in their respective period and in Figure 7 histograms per year.This is what we would like to remark. First of all, as we already mentioned for thered lines in Figure 4, links with negative values are few in quantity and magnitudeas more precisely illustrated in Figure 6a. This is also evidenced in Figure 7 wherehistograms for all years in the period 2000-2019 are illustrated. In Figure 6b we seea quite homogenous distribution in the range [0 . , . . , . . ,
1] we see inFigure 6d a more heterogenous situation with some dominating stocks.13 year2
NEGATIVE DEGREE CENTRALITIES (a) Only negative values are plotted. year
DEGREE CENTRALITIES[0.01,0.1] (b) Values in the range [0 . , .
1] are plotted. year
DEGREE CENTRALITIES[0.1,0.5] (c) Values in the range [0 . , .
5] are plotted. year
DEGREE CENTRALITIES[0.5,1] (d) Values in the range [0 . ,
1] are plotted.
Figure 6: A comparison of degree centralities by year at di ff erent ranges.14 . . . . . . . . . . . . . . . . . . . −0.5 1.5 . . −0.5 1.0 . . . . −0.5 1.5 . . . . −0.5 1.5 . . −0.5 1.5 . . . . . . . . . . Figure 7: Histograms of degree centrality per year.
Let y : = { y t } Nt = denote a one dimensional time series with N observations. A GARCHspecification for its volatility usually starts with a flux of information determined bya filtration {F t } Nt = in which F t is a σ -algebra representing information at time t and y follows the dynamic y t = E [ y t | F t − ] + (cid:15) t ( θ ) . Here θ is a parameter vector whose specification specializes the model, µ ( θ ) is theconditional mean of the time series at time t , usually modeled through an ARMA timeseries. For example an ARMA(1,1) (as we will consider here) is specified by µ t = µ + ε t + φµ t − + ψε t − , (7)where φ , ψ are parameters to be estimated and ε is white noise, i.e. an uncorrelatedcentered time series. The residual (cid:15) ( θ ) captures the conditional volatility of y : var ( y t | F t ) = E [( y t − µ t ( θ )) | F t ] = E [( (cid:15) t ( θ )) | F t ] = var ( (cid:15) t ( θ )) . Its specification is the essence of a GARCH model. We will consider the standardGARCH(1,1) model: (cid:15) t = σ t z t (8) σ t = α + α (cid:15) t − + β σ t − , (9)15here { z t } Nt = is white noise.Now consider a set of univariate time series y (1),. . . , y ( n ). A class of models in the multivariate GARCH literature known as Dynamic Conditional Correlation (DCC) wasintroduced by [11] and [48]. The DCC class builds up on univariate GARCH modelsand then specifies the dynamic of time varying conditional covariance matrix of thetime series y (1),. . . , y ( n ). It has a general dynamics H t = D t R t D t . Here D t is a diagonal matrix of time varying standard deviations from univariate GARCHmodels and R t is a time varying correlation matrix. For estimation, the matrix R t is de-composed as R t = ( Q ∗ t ) − Q t ( Q ∗ t ) − where Q is specified in [10, Equation (2)]. In Table 2 we report the coe ffi cient µ in the specification (7) for each stock in the year2006, analogously for Table 3 in the year 2008. The estimation of these coe ffi cientsprovides further support to the claim reported in Section 2.2 after the visual evidenceof Figure 1. Table 2: The coe ffi cient µ for the year 2006.Stock mu value Stock mu value1 ALFAA 0.0006 GISSAA 0.00112 ALSEA 0.0026 GMD 0.00363 AMXA 0.0017 GMEXICOB 0.00214 ARA 0.0027 HERDEZ 0.00165 AXTELCPO 0.0013 HOMEX 0.00276 AZTECACPO 0.0006 ICA 0.00267 BACHOCOB 0.0012 ICHB 0.00368 BIMBOA 0.0018 KIMBERA 0.00119 CEMEXCPO 0.0013 MXX 0.002110 CMOCTEZ 0.0016 PAPPEL 0.001811 CMRB 0.0013 PASAB -0.001512 CYDSASAA 0.0016 PE&OLES 0.002713 ELEKTRA 0.0017 PINFRA 0.006514 FEMSAUBD 0.0024 RCENTROA 0.003915 GCC 0.0022 SORIANAB 0.002116 GFINBURO 0.0007 URBI 0.001817 GFNORTEO 0.0033 WALMEX 0.002316able 3: The coe ffi cient µ for 2008 year.Stock mu value Stock mu value Stock mu value1 AC -0.0013 ELEKTRA 0.0006 IDEALB-1 -0.00082 ALFAA -0.0019 FEMSAUBD 0.0019 KIMBERA 0.00023 ALSEA -0.0013 FINDEP -0.0036 LAMOSA -0.00164 AMXA -0.0023 FRAGUAB 0.0004 MAXCOMA -0.00285 ARA -0.0018 GAPB -0.0017 MEDICAB -0.00016 ASURB -0.0014 GCARSOA1 -0.0003 MEGACPO -0.00287 AUTLANB 0.0037 GCC -0.0032 MXX -0.00108 AXTELCPO -0.0053 GFAMSAA -0.0023 OMAB -0.00249 AZTECACPO 0.0000 GFINBURO 0.0008 PAPPEL -0.004710 BACHOCOB -0.0020 GFNORTEO -0.0003 PASAB -0.003011 BIMBOA 0.0002 GIGANTE -0.0026 PE&OLES -0.001012 CABLECPO 0.0000 GISSAA -0.0009 PINFRA -0.001613 CEMEXCPO -0.0030 GMD -0.0053 POCHTECB -0.004814 CIEB -0.0023 GMEXICOB -0.0032 SAREB -0.003315 CMOCTEZ -0.0005 GRUMAB -0.0001 SIMECB 0.000716 CMRB -0.0005 HOMEX 0.0007 SORIANAB 0.001017 CULTIBAB -0.0001 ICA -0.0006 TMMA -0.004018 CYDSASAA -0.0029 ICHB 0.0010 URBI -0.0018 Assume we are given an undirected and unweighted graph G with vertexes V = { , . . . , n } and edges E . Community structure in the graph means that there exists a partition of V in groups of vertexes in such a way that within groups vertexes are highly connectedand more edges exists among them, while at the same time, edges between groups areless observed; see [12] for a survey of methods in community detection. The aforedescription presents a general idea and to make it operative, it is necessary to give amore quantitative formulation. A popular approach is through the famous concept ofmodularity as introduced by [31] and further developed in [30]. Following the nota-tion of [30] we introduce the following objects. Let A be the adjacency matrix of G and let m = (cid:80) i k i where k i denotes the degree of vertex i so k i = (cid:80) j A i , j . Furtherdenote by s ∈ { , . . . , n } n a vector having the same dimension of A , and representingan allocation of vertexes to communities. Thus, s i represents the community assignedto vertex i . Now the idea is to compare the graph G with a graph G (cid:48) having no com-munity structure. A group V k = { i ∈ V | s i = k } possess an accumulated weight of (cid:80) i , j ∈ V k A i , j . Now for G (cid:48) , assuming it is a random instance of an Erd˝os-R´enyi graph,the set V k should have an accumulated weight of (cid:80) i , j ∈ V k k i k j m . Hence, the di ff erence (cid:80) i , j ∈ V k A i , j − k i k j m quantifies how distant is the immersion of community V k in the graph G from G (cid:48) . The modularity function is now defined as the sum over all communities: Q ( s ) : = (cid:88) k (cid:88) i , j ∈ V k (cid:32) A i , j − k i k j m (cid:33) = (cid:88) i , j ∈ V (cid:32) A i , j − k i k j m (cid:33) δ ( s i , s j ) , δ ( s i , s j ) = s i = s j in which case δ ( s i , s j ) = Q ( · ) is defined for unweighted undirected graphs.In particular, for graphs obtained from a correlation matrix, which indeed is weighted,the modularity function Q ( · ) requires to be adjusted. Moreover, the null model (thegraph G (cid:48) ) is critical for the well-functioning of modularity; see e.g., the discussion in[12]. Hence, to couple with this problem, we choose to work with the formulation of[25] where correlation matrix is filtered and modularity is adjusted for the right “nullmodel” G (cid:48) . The analysis is again based on a spectral analysis as we now explain. Let C be a correlation matrix and consider the set of eigenvalues λ , . . . , λ n which we assumeare displayed in increasing order. Let v , . . . , v n be the corresponding eigenvectors.Moreover, let T be the number of observations and the critical values λ − : = (cid:32) − (cid:114) nT (cid:33) , λ + : = (cid:32) + (cid:114) nT (cid:33) . The values λ − , λ + are parameters for Marcenko-Pastur distribution in random matrixtheory which is given by ρ ( λ ) = Tn √ ( λ + − λ )( λ − λ + )2 πλ . Define the matrices C r : = (cid:88) λ i ≤ λ + λ i v tr i · v i (10) C g : = (cid:88) λ + <λ i <λ n λ i v tr i · v i (11) C m : = λ n v tr n · v n . (12)We have a decomposition of the correlation matrix C given by C = C m + C g + C r . (13)From the ordering of the eigenvalues, the matrix C r represents some random noise, C m a global signal which in our financial context is attached to the market as a whole and C g represents information in a mesoscopic scale just between C r and C m . However, theset of eigenvalues λ i satisfying λ + < λ i < λ n could be empty (as we will find) and inthis case there makes no sense to consider C g . Next, we explain how the modularityfunction Q ( · ) is adjusted. Accordingly, focusing in the matrix C g , and taking intoaccount the decomposition (13), the null model is C r + C m and the modularity functionstakes the form Q ( s ) : = C norm (cid:88) i , j [ C i , j − C ri , j − C mi , j ] δ ( s i , s j ) = C norm (cid:88) i , j C gi , j δ ( s i , s j ) (14)for C norm = (cid:80) i , j C i , j a normalizing constant. However, as we mentioned before, forsome empirical correlation matrices, the matrix C g will be null. Hence, it also makessense to consider a decomposition C = C s + C r with C s : = (cid:80) λ + <λ i λ i v tr i · v i Q ( s ) : = C norm (cid:88) i , j [ C i , j − C ri , j ] δ ( s i , s j ) = C norm (cid:88) i , j C si , j δ ( s i , s j ) . (15)Hence, in this section we maximize the modularity functions Q and Q in order todefine communities and report on them. It is known that the maximization of modu-larity functions is a NP-hard problem; see [7]. Hence, the optimization is approachedthrough several heuristic algorithms. We implement the popular Louvian algorithm,adjusted as described by [25] accordingly to the modularity functions Q and Q . Q In Figure 8 we see the resulting communities obtained with the Louvian algorithmapplied to the modularity function Q defined in (15). In all of the years of the periodthere are two communities. The first community is a “giant component” and the othercommunity consist of a small number of isolated vertexes. Hence, at this scale ourprocedure does not detect a complex community structure. This is what we expected.Note however the stylized fact: • The turmoil at the Subprime crisis period is captured by a visually evident in-crease on interconnectedness particularly for the years 2007 to 2009.19
000 2001 2002 20032004 2005 2006 2007 lll l l ll ll ll ll ll l l lllll lll ll l lll lllll lll ll ll ll ll ll l lll l l ll ll l ll ll ll l lllll lll lll lll ll llll l llll lll ll l ll l l ll l lllll l lll ll l l ll lll ll l llll ll lllll ll lll ll ll ll l l lll ll ll lll ll lll l l llllll l lll ll l ll ll ll l llll l lllll ll llll l ll lll ll lll ll l l llll ll ll lll ll l l ll ll l l llllll l lll ll ll ll lll llll l ll lll ll lllll lll lll l ll lll lll llllll ll lll lll l lll l l l lllllll l lll ll ll lll ll lllll l llll lll lllll lll lll ll ll l llll lll lll llll ll lllll l lll l l lllllll lll ll ll lll ll llll l ll lll lll lllll lll llll l ll lllll lll lll llll l lll l lll llll ll ll l lll l Figure 8: Communities obtained from modularity function Q . The color of the nodesrepresent community, which is equivalently represented by vertex’ shape. For visualityonly edges with weights in absolute value in the interval [ . , ∞ ) are shown. Onlyweights above 0 . Q For the definition of the modularity function Q the matrix C g is necessary and shouldnot be a null matrix. For our data, this is the case for only a few years: 2000, 2010,2016, 2018 and 2019. For these years, a representation of communities can be seenfrom Figure 9. This is what we observe: • First of all, in each year, there are only two communities as can be seen from thecolor of the vertexes, or equivalently from their shape. Interestingly, there is noclear larger community. • Second, and also interesting, for our data, industrial sector is non determinant for20he community assignment. More clearly, each industrial sector have vertexes ineach community. This fact should be compared with the finding of Section 3.3based on partial correlations where there also existed intersectorial links. • This is our explanation of the years in which there existed a non trivial matrix C g . First of all recall that this matrix represents structure between individualstocks and the market as a whole, while in crisis periods this last structure iswhat prevails since stocks tend to be highly correlated at those times. In theyear 2000 we find the peak (and burst) of the dot com bubble from which forthe years 2001 and 2002 bearish markets prevailed. What we see from Figure8 for the network constructed from the matrices C s is an increase in intercon-nectedness while in Figure 9 we see that in the period 2000:2002 there existeda “mesoscopic” structure for the year 2000 in which there is a “local minimum”for graphs interconnectedness. Analogously for the year 2010 in Figure 9 whichcoincides with a local minimum in Figure 8 for the “extended” subprime crisisperiod 2007-2010. • Now we compare the years 2016, 2018 and 2019 in Figures 8 and 9. Those areyears in which global events occurred, to mention some of them: The Brexit(starting from its referendum in june 2016), US elections for the period 2017-2020, China-US trade war starting from july 2018. However, none of these seemsto be comparable to the levels of the dot.com bubble and the subprime crisis. Inparticular for the Mexican stock market they didn’t have such an impact as tohide the e ff ects of a mesoscopic structure and inducing all stocks as movingaccording to a unique factor. ll l lll llll ll l lll ll lll l l ll ll lll ll l llllll ll ll l lll lll lllll ll lll lll ll ll lll ll ll l llll l ll lll l l llll l ll lllll lll ll llll ll l ll ll ll l lll ll l lll lll l llll llll lll lll lll llll ll llll l Figure 9: Communities from modularity function Q . The color of nodes identifiesmembership to the same community and equivalently for the vertex’ shape. For vis-ibility only edges with weights in absolute value in the intervals [ . , ∞ ) are shown.Only weights above 0 . Conclusion
In global crisis periods, price levels of stocks in the Mexican stock exchange indeedpresent obvious changes which are visually evident and confirmed by econometricmodels as we have shown here and is also documented by other authors. However,the interdependency structure is a more complex phenomenon and much less studied.Our findings show that as long as partial correlations are concerned, the interdepen-dency structure is quite stable and only centrality metrics from network theory havethe fine sensibility to quantify changes. Degree and eigen centralities indeed presenta discontinuous variation, an upwards jump at the peak of the crisis and then a down-wards jump when the shock of the crisis has been absorbed in the market. Anotherinteresting finding of studying interdependency structure from partial correlations isthat only a small number of negative partial correlations which are also in magnitudesmall are present. We argue this is an indicator of a positive synergy of an integratedmarket. Reinforcing this claim, we find that industrial sectors are strongly intercon-nected even at the level of partial correlations, which is a less established property. Ingeneral and in particular for the Mexican case.Interdependency from the point of view of (“absolute”) correlations confirms find-ings from partial correlations. It also provides evidence of an integrated market forthe Mexican case. Indeed, this is what we learned from the estimation of modularitieswhich determined community structure with no separation of industrial sector. Fromfiltered matrices with noise filtered out (the matrices C s ) a single giant componentemerged. Moreover, here the e ff ect of global episodes for interdependency structurewas quite clear even for visual appreciation. This is what we learned in Figure 8 andis perfect as evidence for the modeling strength. Indeed, correlations are more sensibleto trading activity than partial correlations, and capture relationships among stocks dueto such activity which is even more pronounced at crisis periods. We also studied com-munity structure from the matrices C g which are the correlation matrices after noiseand the global market mode have been filtered out. At this scale it happens that onlya few observed years present a mesoscopic structure. For the years 2000 and 2010 inwhich mesoscopic structure is present, we observe a “local minimum” for interconnec-ctedness in Figure 8. For the years 2016, 2018, 2019 we note also a turmoil of stressperiods (e.g., the Brexit, China-US trade a ff air, etc) which nevertheless are not to becompared in serverity with the episodes of the dot.com bubble and the subprime crisisso are not able to blur the presence of structure at the mesoscopic level. A Filtered list of stocks
In this appendix we ilustrate the collection of stocks selected for each year in the period2000-2019. 22 AC x ACCELSAB
ACTINVRB
AEROMEX
AGUA
ALEATIC x ALFAA
ALPEKA
ALSEA x AMXA x ARA
ASURB
AUTLANB
AXTELCPO
AZTECACPO x BACHOCOB
BAFARB
BBAJIOO x BIMBOA
BOLSAABSMXBCABLECPOCADUA x CEMEXCPOCERAMICBCHDRAUIBCIDMEGACIEBCMOCTEZ xxxxxxxx xxxxxxxxx xxxxxxxxx xxxxxxxxxxx xxxxxxxxxx xxxxxxxxxx xxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx
Figure 10: Filtered list of stocks 1 / x CMRB
CREAL
CUERVO
CULTIBAB x CYDSASAA
DINEA x ELEKTRA
ELEMENT x FEMSAUBD
FINDEP
FRAGUAB
GAPB
GBMINTBO
GCARSOA1
GCC
GENTERA
GEOB
GFAMSAA x GFINBURO x GFNORTEOGICSABGIGANTEGISSAAGMDGMEXICOBGMXTGPROFUTGRUMABGSANBORB−1 xxxxxxx xxxxx xxxxxxxxx xxxxxxxxx xxxxxxxxxxx xxxxxxxxxx xxxxxxxxxxxxx xxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxx xxxxxxxxxxxx xxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx
Figure 11: Filtered list of stocks 2 / HCITY
HERDEZ
HOMEX
HOTEL x ICA x ICHB
IDEALB−1
IENOVA
JAVER x KIMBERA x KUOA
LABB
LACOMERUBC
LALAB
LAMOSA
LIVEPOL1
MAXCOMA
MEDICAB
MEGACPO
MFRISCOA−1MINSABMONEXB x MXXNEMAKAOMABORBIAPAPPELPASABPE&OLES x PINFRA xxxxx xxxxx xxxxxxx xxxxxxxx xxxxxxxxx xxxxxxxxxx xxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxx xxxxxxxxxx xxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx
Figure 12: Filtered list of stocks 3 / POCHTECB Q RA RCENTROA
SAREB
SIMECB
SITESB−1 x SORIANAB
SPORTS
TEAKCPO
TMMA
TRAXIONA
UNIFINA
URBI
VALUEGFO
VASCONI
VESTA
VINTE
VISTAA
VITROAVOLARAWALMEX x x xx xxx xxxx xxxx xxxx xxxxxxxx xxxxxxxxx xxxxxxxx xxxxxxx xxxxxxxxxx xxxxxxxxxxxxx xxxxxxxxxxxxx xxxxxxxxxxxxx xxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxx
Figure 13: Filtered list of stocks 4 / B Partial correlations
In Table 4 we display partial correlations above the threshold 0.3 in absolute value.26able 4: Links in the rank [0 . ,
1] for the period 2000-2009. tick1 tick2 Par.Corr year tick1 tick2 Par.Corr yearELEKTRA ICA 0.98 2000 HOMEX URBI 0.32 2012FEMSAUBD MXX 0.4 2000 ICHB SIMECB 0.43 2012AZTECACPO SORIANAB 0.33 2001 AMXA MXX 0.87 2013AMXA MXX 0.37 2002 FEMSAUBD MXX 0.44 2013AZTECACPO MXX 0.38 2002 GMEXICOB MXX 0.48 2013CEMEXCPO MXX 0.38 2002 ICHB SIMECB 0.34 2013FEMSAUBD MXX 0.42 2002 MXX WALMEX 0.31 2013MXX SORIANAB 0.32 2002 CEMEXCPO MXX 0.33 2014AMXA MXX 0.57 2003 FEMSAUBD MXX 0.59 2014AZTECACPO MXX 0.32 2003 ASURB GAPB 0.35 2015CEMEXCPO MXX 0.36 2003 CEMEXCPO MXX 0.33 2015FEMSAUBD MXX 0.34 2003 FEMSAUBD MXX 0.42 2015MXX WALMEX 0.76 2003 GFNORTEO MXX 0.37 2015AMXA MXX 0.7 2004 GMEXICOB MXX 0.39 2015CEMEXCPO MXX 0.52 2004 ICHB SIMECB 0.36 2015MXX WALMEX 0.43 2004 CEMEXCPO MXX 0.46 2016CEMEXCPO MXX 0.44 2005 FEMSAUBD MXX 0.61 2016MXX WALMEX 0.57 2005 GFNORTEO MXX 0.36 2016CEMEXCPO MXX 0.5 2006 MXX WALMEX 0.34 2016FEMSAUBD MXX 0.42 2006 ASURB GAPB 0.33 2017GMEXICOB MXX 0.37 2006 CEMEXCPO MXX 0.78 2017MXX WALMEX 0.83 2006 FEMSAUBD MXX 0.57 2017CEMEXCPO MXX 0.5 2007 GEOB HOMEX 0.36 2017GAPB OMAB 0.38 2007 GFNORTEO MXX 0.46 2017GMEXICOB MXX 0.31 2007 GMEXICOB MXX 0.3 2017MXX WALMEX 0.58 2007 ICHB SIMECB 0.37 2017ALFAA ARA 0.3 2008 ASURB GAPB 0.33 2018GAPB OMAB 0.31 2008 CEMEXCPO MXX 0.74 2018ICHB SIMECB 0.41 2008 FEMSAUBD MXX 0.84 2018MXX WALMEX 0.52 2008 GFNORTEO MXX 0.65 2018CEMEXCPO MXX 0.37 2009 GIGANTE LIVEPOL1 0.3 2018GAPB OMAB 0.34 2009 MXX WALMEX 0.37 2018ICHB SIMECB 0.34 2009 ASURB GAPB 0.31 2019MXX WALMEX 0.49 2009 CEMEXCPO MXX 0.58 2019CEMEXCPO MXX 0.35 2010 FEMSAUBD GFNORTEO -0.33 2019ICHB SIMECB 0.34 2010 FEMSAUBD MXX 0.79 2019MXX WALMEX 0.36 2010 GAPB OMAB 0.33 2019FEMSAUBD MXX 0.35 2011 GFNORTEO MXX 0.83 2019MXX WALMEX 0.33 2011 GMEXICOB MXX 0.54 2019
Partial correlations in absolute value in the interval [0 . , .
3] are displayed in Table5. 27able 5: Links in the rank [0 . , .
3] for the period 2000-2009. tick1 tick2 Par.Corr year tick1 tick2 Par.Corr yearCEMEXCPO MXX 0.26 2000 GCARSOA1 GFINBURO 0.26 2010GFINBURO MXX 0.21 2000 GFNORTEO MXX 0.21 2010GFNORTEO MXX 0.24 2000 GMEXICOB MXX 0.29 2010MXX SORIANAB 0.25 2000 HOMEX ICA 0.21 2010ARA GFNORTEO 0.25 2001 ALFAA MXX 0.22 2011AZTECACPO ELEKTRA 0.21 2001 CEMEXCPO ICA 0.27 2011CEMEXCPO FEMSAUBD 0.29 2001 CEMEXCPO MXX 0.21 2011ALFAA MXX 0.2 2002 ELEKTRA MXX 0.22 2011AZTECACPO ELEKTRA 0.22 2002 GFNORTEO MXX 0.3 2011GFINBURO MXX 0.29 2002 GMEXICOB MXX 0.29 2011GFNORTEO MXX 0.23 2002 HOMEX URBI 0.28 2011ARA MXX 0.22 2003 CEMEXCPO MXX 0.27 2012GFINBURO MXX 0.21 2003 FEMSAUBD MXX 0.28 2012MXX SORIANAB 0.26 2003 GMEXICOB MXX 0.26 2012ALFAA MXX 0.29 2004 MXX WALMEX 0.22 2012AMXA CEMEXCPO -0.23 2004 ALFAA MXX 0.27 2013AZTECACPO MXX 0.2 2004 AMXA FEMSAUBD -0.24 2013GFINBURO MXX 0.24 2004 CEMEXCPO MXX 0.22 2013GMEXICOB MXX 0.26 2004 CULTIBAB TEAKCPO -0.22 2013GMEXICOB PE&OLES 0.22 2004 GFNORTEO MXX 0.26 2013MXX SORIANAB 0.23 2004 GMEXICOB PE&OLES 0.23 2013ALFAA MXX 0.23 2005 HOMEX SAREB 0.22 2013AMXA MXX 0.21 2005 ALFAA MXX 0.24 2014ARA URBI 0.21 2005 ALSEA CULTIBAB 0.25 2014FEMSAUBD MXX 0.22 2005 ASURB GAPB 0.21 2014GMEXICOB MXX 0.25 2005 GFNORTEO MXX 0.26 2014KIMBERA MXX 0.2 2005 GMEXICOB MXX 0.3 2014ALFAA MXX 0.23 2006 MFRISCOA-1 PE&OLES 0.27 2014AMXA MXX 0.25 2006 ALFAA MXX 0.21 2015GFINBURO MXX 0.27 2006 GFINBURO MXX 0.22 2015GFNORTEO MXX 0.23 2006 MXX WALMEX 0.2 2015GMEXICOB WALMEX -0.2 2006 AC BIMBOA 0.21 2016MXX PINFRA 0.21 2006 ALFAA GFINBURO 0.22 2016BIMBOA MXX 0.23 2007 ASURB GAPB 0.2 2016GFNORTEO MXX 0.23 2007 GENTERA PINFRA 0.23 2016HOMEX MXX 0.2 2007 GFNORTEO RA 0.21 2016ICA MXX 0.24 2007 MFRISCOA-1 PE&OLES 0.26 2016MXX URBI 0.23 2007 MXX ORBIA 0.2 2016ALFAA AXTELCPO 0.25 2008 ALFAA ALPEKA 0.24 2017AMXA MXX 0.23 2008 CEMEXCPO FEMSAUBD -0.27 2017CEMEXCPO MXX 0.24 2008 CEMEXCPO GFNORTEO -0.24 2017FEMSAUBD MXX 0.26 2008 GAPB OMAB 0.2 2017GCARSOA1 MXX 0.21 2008 GFNORTEO GMEXICOB -0.21 2017GMEXICOB MXX 0.28 2008 HOMEX URBI 0.27 2017MXX PE&OLES 0.25 2008 MXX WALMEX 0.24 2017BIMBOA MXX 0.21 2009 CEMEXCPO FEMSAUBD -0.26 2018FEMSAUBD MXX 0.21 2009 FEMSAUBD GFNORTEO -0.25 2018GCARSOA1 MXX 0.21 2009 GAPB OMAB 0.23 2018GMEXICOB MXX 0.26 2009 GMEXICOB MXX 0.26 2018HOMEX MXX 0.25 2009 ICHB SIMECB 0.25 2018ASURB GAPB 0.24 2010 CEMEXCPO FEMSAUBD -0.24 2019AXTELCPO GFAMSAA 0.21 2010 CEMEXCPO WALMEX -0.21 2019 Persistent links from partial correlations
Table 6: Persistent links in the rank [0 . ,
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