 Thank you very much actually first of all let me thank organizers I think I'm here the second time and second time is very interesting conference. So it is definitely worthwhile for for being here Thank you a lot for that Okay, so this is I'm indeed Mikhail Anoufri if I'm from Sydney UTS This is the work with my quarter Valentin Panchenko who is from UNSW also Sydney based University We did this work because we were bored because we were waiting waited for some data set Which they promised us to give and so you could have a look at that, but we didn't have this data set because Australian Prudential Authority wanted to Check everything so we had to sign something. I don't know in any case. So we didn't know what to do. So essentially we tried to build a network of the Dependences between banks of financial entities without Looking in their own data. So the question was how to do this The presentation is called connecting the dots. So which dots are we connecting? We try to find some relationship between three area. So I Don't know how you are familiar with it. I'm not much familiar with First for example, but I will try to explain more or less what it is. So the first area is the econometric modeling financial econometrics modeling on Correlations, so okay, you can compute the correlation matrix of financial entities and what people often do When they look on the data, they try to reduce the dimensionality of the data because of course the dimension can be very large The number of time periods can be very large. That's the dimension essentially so what they try to do they try to look on so-called principal components or also Related factor analysis, so you often look at this several just few number of factors which are kind of directions where most of the variants of the data Emerges so it's useful to only look at those directions This is the first literature. There is big literature and then there is a second literature which we discovered It's called Gaussian graphical models. This is statistical literature. What these people are doing. They are building the network of Partial correlations not normal correlations as the first branch, but of the partial correlations In order to find parts of the networks which can be in a sense decomposed or can be Not affecting or not affected by each other Okay, so we are actually inspired mostly by this approach But what we want to do we try to attempt to relate it to the network theory in economics and in particular to those centrality measures About which for example Matt spoke yesterday So in terms of my presentation, I will start this explanation what is partial correlation Essentially, I will explain how it is related to the networks We'll talk a little bit what I think can be in relation with the network based measures and as an application as an example I will show The exercise of this applied to Australian banks economic sectors and international markets So what you will see in the end you will see a network of these entities Where every note will be some of the entity and where an inch age which imply that there is a some Non-zero partial correlation between these entities The motivation for the network approach for the financial data Kind of clear now. There are many possible citations the citation from the macroeconomist and policymaker He says in the current crisis We have seen that financial firms that become too interconnected to fail post serious problems for financial Stabilities and for regulators Due to the interconvectivity of today's financial markets the failure of a major counterparty has the potential to several disrupt Many other financial institutions. Okay, so this actually calls to the network Illustration and you would like to see this network as a way of Thinking of propagation of probably some shocks because the interconnectedness would be exactly captured by the network Number of contributions and these fields are large this just 2014 papers and those which I already read and so these are theoretical network models here And what usually theoretical network models do they of course they Either start with a network or they try to build a network and then on this network They they propose some some ways of starting how these shocks are propagating They apply sometimes to the some data sets But of course in order to apply to some data sets you have you have to have these data sets available and The data is always And so another approach is this empirical network approach. It's just called reconstructed networks So when you look on some data, which do not have a network, but you reconstruct it from From that date. And so what we are doing we are doing this differently from other people Okay, so let me tell you what the partial correlations Well, what what are the correlations between random variable? I think we all more or less know the measure tendency of this random variable to move in the same direction from their mean if this is positive or Different direction if it is negative So the partial correlations measures this linear dependence In a similar way, but conditional on the linear dependencies between all the other entities okay, so Suppose we have as an hour application about 20 financial entities We will observe the returns of these entities. We will be interested in the correlation between these returns over time and these correlations When you see them on the realizations they of course depend on the realizations of all other entities so in order to To think about network. It seems that it's more natural to condition on all the rest And so that the correlation between the 20s would just measure the linear dependence between them conditional all the rest Of course the total effect will be measured somehow by the total usual correlation, but the partial correlation We will measure this conditional effect Okay, so I will use this notation row ij Just short notation. That's the definition of partial correlation Then the network which you will see will have nodes nodes will correspond to the returns Xi so these are the nodes and the edges will correspond to those nodes which have non-zero partial correlation between them Okay to proceed a little bit Further, I think it's useful to Establish some relationship with linear regressions, which we actually are all familiar so we have remembered so the x is a Random variable which has a number of random variable components there the financial entities So take one of them and regress it Dimin and regress it on the all the others So the equation which you see here I can okay, so here This is this exercise you for any i you regress this variable on all the others and then you look also You have some Residuals, so that ij is the effect of xj in this regression on the side Okay, so it turns out that when you write this for any i you have the system of equation In order to have the orthogonality condition so in order to have that epsilon the residuals really are independent on xj You should have these Equalities so you can show it that for random process Multivariate random variable this components x1 and so on xn These regressions will have orthogonal residuals Only if betas are related to those raw which were partial correlations through this equality Okay, so what is here epsilon is the residual sigma is the variance covariance of these residuals and These are variances of epsilon thing which stays on the diagonal of that big sigma So what for example you see here that if this residuals and the regression would have exactly the same variances Then the partial correlation would be exactly the betas and betas would be there for symmetric So when you run regression of xi on all others and you look on a coefficient In front of xj and vice-versa you would have exactly the same bit But generally we will have different betas so the matrix of betas will be asymmetric not symmetric by the matrix of raw the asymmetric so this this Normalization of betas takes care of it Okay, so this is just a statistical fact Now what I am going to do I'm going to rescale each of these variables and I'm going to rescale them in a some way in a strange way so What this rescaling says that you take the random variable you subtract them in well so so far so good Then you divide on the standard it looks like standard deviation But this is not a standard deviation of xi Because usually you rescale it and you divide on unconditional standard deviation of xi This is the conditional standard deviation of xi that is conditional on all others the variance of epsilon i is the conditional variance of xi and so this is a little bit of Strange, I mean, I don't know. Well, this is the object which just looks like important here And I think they can the division of the Division by the conditional variance Exactly is here natural because you want to Condition on all other components so you look on the conditional variances In the same way after you Normalize epsilon i you get ei and so that equation above is just this equation Okay, so I'm now going to look at this equation I can also write them in it in matrix form and so here p is the matrix of partial correlations the Matrix which has zero on the diagonals and these raw coefficients Which are usually small in our applications. They are almost always positive between zero and one but mostly closer to zero of this partial correlation Okay, so this is the statistical equality of Of the data Now I have to be careful here because Whatever I'm going to tell here you should all remember that I remember that correlation doesn't imply the causation So the problem is that Looking in the correlations. We don't know what causes what of course But observationally We know that if one of the variables is Like taken out of the Nature of state Then others will be affected and the relationship of of this effect is captured by this equation Where p is the matrix of partial correlation so imagine that Initially all the variables were in mean and then e that you may think as a shock hit the system Well, actually maybe you just observed that there was something with the variables and you think that it was a shock Which caused this Okay, so of course then p times c would be the first effect of this shock. So what is p times c? p times c is just the Well, you know this the sum of this partial correlations over the vectors of the shock So if the shock for example comes in particular by one component to one to this to this node one Then and and one has an edge with three. So it means that they have correlation between them then three You'll get this shock in one period of time so In one step and p will measure this And then p square e will measure the second step. So it may also go back and may go here or may go here The third p to the third e is the third Effect third order effect and so on and so we may say that p power k times c This is case order effect of this shock um, and when you sum them up In principle should get the total effect, which is probably the observable Okay, so the sum of all this shock Should be equal to this well I think mathematically it may not because this may diverge But suppose it converges And then you get that as the result and you of course can also get this directly from that equation You see that x depends on this in on e in this way Okay, so this matrix would give you the total effect of the shock But with the partial correlation matrix we are able to decompose it on some way On the direct less direct and so on Okay Now the total effect of the shock probably should be related if you ask Admetricians we tested it should be related to variance covariance matrix That is not the partial correlation matrix, but just the correlation matrix. Okay. Well, actually it is And this is how Another piece of statistical analysis for us revealed the following about this partial correlation. There is A magic relationship between partial correlation raw And the inverse of variance covariance matrix. So if omega is a variance covariance matrix of x Call k, they call it often concentration matrix. So this is the inverse of it and Form this Formula from the elements of this k Turns out that this ratio will be exactly partial correlation through aj And in matrix form you can write exactly the same by this equation Okay, so what happens is that you look at the Non-diagonal elements of aj and you Normalize them with the diagonal elements of k. So while I don't know The effect what what really measures k aj I actually know what measures k i and k aj. It turns out that they are exactly the ever the inverse of variance of Epsilon so those Residuals in the in the linear regression which was here a couple of slides before so you actually can write the diagonal decay There as this one and so Eventually from that equation you The simple algebra you get that So remember this was the matrix which was in front of the shock and that would measure the total effect So let's call this matrix of the total effect It turns out that this is related to the variance covariance matrix, but this is not exactly the variance covariance matrix. That's the matrix which is normalized by variances of of epsilon That is by conditional Variance of x's conditional So now we know that if partial correlation matrix would have an interpretation of one step Propagation of the shock then the total effect will be measured by this Okay, so How does it How is it connected with the centrality measures? I I I don't know Fully I I agree actually that there are many centrality measures and it's Very useful to know And to have a direct interpretation for each of that Okay, so in this case what we can say of course we can compute degree of centrality of the network of partial correlations that will be the Probably the sum of weights of all adjacent edges because we have the weighted matrix the partial correlations are weights So the degree would be the sum of these weights Okay, in principle if you have a note with the high degree it means that Either it has a high weight for some entity or it has many entities with Maybe small weights or both and so it basically means that it should be Important in getting the shocks and also transmitting the shocks intuitive Now eigenvector centrality takes into account the centrality of neighbors And it is defined like the eigenvector corresponding to the largest eigenvalue of matrix and in this case of matrix p of partial correlation matrix So let's Actually see a little bit on what it is if we remember this is the case effect of the shock You Define you so u1 and so on u n these are eigenvectors Right is zero as the sum of them Make some obvious computations use the definition of eigenvalues So what you find here you find that if lambda one this is the eigenvalue corresponding This is the largest eigenvalue Then all these ratios apart from the ratio for i equal to one would would go to zero When k goes to infinity and so you would have only lambda one here Power k and then power k again So this this would be only this lambda one times this sum Where you have the u1 on the place of ui And so u1 Would be the vector which would accumulate the shock and the k saturation when k goes to infinity Okay, what does it mean so if the Some entity has a high centrality it may mean Two things at least Well on the one hand it means that if the shock e has a High coordinate for the For the Entity with high centrality then it should have actually high effect for everybody else in the future But it also may mean that this entity Will get also high shock eventually So again because we don't know about the causation here it's hard to interpret but In principle this eigenvector would accumulate the shocks on the k iteration Now you may then think about this generalization you may take the first p Eigen vectors the first p in the sense corresponding to the largest p eigenvalues How many p's you should take well it depends probably but you would like to take those if they are close to each other This largest you would like p which are quite close to each other And then in principle the space spun by these vectors would asymptotically accumulate the shocks Now this thing that's actually Think which is what is done in principal components analysis Again, so I said that they are applying principal components usually to variance covariance matrix But what we can show that the eigen Vectors of p are exactly the same as eigenvectors of these matrix And eigenvalues are not the same but they are exactly in the same order So that means actually that if you apply the centrality In this p centrality space For the matrix p it's the same as to apply principal component analysis To to these matrix But not to the covariance covariance Okay, so it seems that there is this now. I don't know what exactly I maybe I I should ask Matt, but I I don't know what is exactly the interpretation of the centrality space. Of course the eigenvector centrality you would like to get it positive So that's one of the reason why you take the eigenvector corresponding to the largest eigenvalue for the positive matrices, but These would of course have negative components But still if you are able to imagine somehow the the space this is the space where the shocks would go So It might be useful Now also there is another centrality measure, which is bonacic centrality. You can write the definition So here you accumulate the degree centrality and then you have a dumping factor for the neighbors and you look So this is alpha this is dumping factor and and so on and so eventually you get Something like that and this some computations we found that if you take alpha equal to 1 which would be not bold But normal then bonacic centrality would be equal to this Now what is this this is the total effect of the shock of units So every every get one unit shock every entity and then you subtract this shock So this is something like the sum of the indirect effects So it seems that bonacic centrality with alpha is one Would be high for those who are accumulating these indirect effects, especially strong And I don't know if other alpha would have any interpretation I think not because in some sense matrix of partial correlations already Uh Dumpens the the the dependencies Okay, so this is what I had about this first part the theoretical part then we applied it we applied using Financial entities in australia. We took the publicly available data. We filter returns using some econometric technique This is called dynamical conditional correlation in order to get the correlation matrix being You know dependent on the previous in order to take into account some time effects Which are known in the financial kinematics and then when we get this r So we started with some returns they called y here every y Is a vector So I mean it is of course it has time but you also have many of these y's because You have many integers and then eventually you find this r Which is the correlation between them and but but it is updated. So it has r t Okay, so from here we get p As I said, we did not have any any data. So we use the data stream which we are available So this is a publicly traded banks six publicly traded banks in australia big four Two regional banks. There are a number of sectors of real economy and major international markets for australia. So we have this 14 years of data 3600 observations, but what we did so first of all we we we build the network and the day when we did it So that was 22nd of august of 2014 And then to see you how how it is robust We also build some networks in different time periods And each time when we build this we have for our dcc procedure We use this size 1215 previous periods in order to evaluate dynamic correlation Okay, so that's the data. This is the network which we got Again, so the these are the entities you see for example These four are the four large banks in australia. These are two Regional banks and then you see the sectors you have the asian market here europe market and norse american market There's a cutoff level here in terms of partial correlations equal to zero point zero 75. So others are not shown Okay, so in the the the width of the of the link is of course the size of this partial correlation So what we see is that for example the banks the four big banks are Related to each other Quite strongly They are not all related to other sectors, but they are related indirectly to other sectors for example through financial service sectors and Other sectors are Over here the small banks are related to each other related a little bit to the big four and to the financial sectors We have a very strong relation with the real estate and financial sectors which Okay, so just just finding which makes sense. I think interestingly here. There are these other markets. So you see that for example Australia all the sectors of Australia not really Correlated to the norse american or to european market, but they do have an impact through them right so through this channel This is probably because the trading in asia market Is overlapping in time is the trading in the australian market It overlaps partially with europe and europe overlaps partially with norse america So only in this case we may say that this is actually this directional causality because europe Is late and norse america is even later But but again, I remind there are no any causality here Okay, so that we checked a little bit the stability of it and you know that this is pretty stable You probably don't see it from here, but Well, the the four banks are always there, so that's that's good Well in terms of centrality measures we found that eigenvector centrality is So if you scale them the largest they'll be for financial services Financial services there nine z and west park. These are two banks in australia But for example, bonaccio centrality is is maybe different. It is actually high for north american and for asian market And the financial services as well Okay, so that's the example of this application But to conclude What we try to do we we try to link the graphical gaussian models those Use partial correlation networks obese network literature We we established some link between partial correlation and shock propagation in our interpretation of this network And I discussed some network based measures Okay, so what you found at the end you so you so the reconstructed networks for australian financial institutions Now what what we want to do next I mean one of Natural step would be to try to see if we have actual data or if we have more direct data To see whether whether the network from them would be similar to this perceived network We call it perceived because it's a network which comes from the market return So it actually reflects the perception of probably of people about the market And that's that's would be probably the next step. Okay. That's it. Thank you for Partial correlation that leaves to be common So there was something that was hitting every one of these entities That wouldn't show up that no What percentage do you have any way of measuring what percentage of variation is completely common in the So The partial correlation thing Quite of common factor, would you say or something like that? Like No, we don't so to answer this shortly we don't and I I think we have to look at that but Yeah, of course we look on the variances and covariances and also we look on conditional on what is on our System, so you probably can answer this by including something else in the system and see what changes Yeah