Estimation of Bilateral Connections in a Network: Copula vs. Maximum Entropy

Transcription

1 Estimation of Bilateral Connections in a Network: Copula vs. Maximum Entropy Pallavi Baral and Jose Pedro Fique Department of Economics Indiana University at Bloomington 1st Annual CIRANO Workshop on Networks in Trade and Finance November 9, 2012

2 Motivation We begin with examples of some complex networks.

3 Motivation We begin with examples of some complex networks. Figure: Fedwire Payment Network

4 Figure: Brazilian Interbank Network Motivation

5 Motivation Each network within this complex network, looks like a star.

6 Motivation Each network within this complex network, looks like a star. Figure: Star Interbank Network

7 Motivation However, often we do not have access to data on the interbank linkages.

8 Motivation However, often we do not have access to data on the interbank linkages. The interbank network data available usually looks like the following:

9 Motivation However, often we do not have access to data on the interbank linkages. The interbank network data available usually looks like the following: Figure: Interbank Network Data

10 Motivation So, we have the aggregate level data but not the micro level data.

11 Motivation So, we have the aggregate level data but not the micro level data. In terms of a matrix then, we have the following representation

12 Motivation So, we have the aggregate level data but not the micro level data. In terms of a matrix then, we have the following representation Figure: Matrix Representation of Interbank Linkage Data where X ij s denote the a connection between bank i and bank j, a i = j X ij and l i = i X ij.

13 Motivation In terms of a more generic setup we have, for any network, the following representation where a ij s denote a connection between nodes i and j, a i. = j a ij, a.i = i a ij and S G = i a i. = j a.i.

14 Motivation In terms of a more generic setup we have, for any network, the following representation where a ij s denote a connection between nodes i and j, a i. = j a ij, a.i = i a ij and S G = i a i. = j a.i. Figure: Matrix Representation of Any Network Data

15 Motivation And we only have access to the marginals of this matrix, i.e., a i. = j a ij and a.i = i a ij.

16 Motivation And we only have access to the marginals of this matrix, i.e., a i. = j a ij and a.i = i a ij. Figure: Example of Marginals of Network Data

17 Main Problem

18 Main Problem The primary question is then - Is there a way to estimate or simulate the joint distribution (a ij s) from these marginals?

19 Main Problem The primary question is then - Is there a way to estimate or simulate the joint distribution (a ij s) from these marginals? And the answer is YES.

20 Main Problem The primary question is then - Is there a way to estimate or simulate the joint distribution (a ij s) from these marginals? And the answer is YES. Upper and Worms (2004) proposed a method based on Maximum Entropy (ME, henceforth).

21 Maximum Entropy The standard ME estimation divides the total # or weight of connections equally among all other nodes.

22 Maximum Entropy The standard ME estimation divides the total # or weight of connections equally among all other nodes. Then uses the RAS algorithm proposed by Schneider (1990) to re-balance the matrix.

23 Maximum Entropy The standard ME estimation divides the total # or weight of connections equally among all other nodes. Then uses the RAS algorithm proposed by Schneider (1990) to re-balance the matrix. Matrix re-balancing is required to ensure that the sum of the individual elements amount to the original overall value of the connections.

24 Maximum Entropy The standard ME estimation divides the total # or weight of connections equally among all other nodes. Then uses the RAS algorithm proposed by Schneider (1990) to re-balance the matrix. Matrix re-balancing is required to ensure that the sum of the individual elements amount to the original overall value of the connections. Mistrulli (2010) provided a description of the shortcomings of ME, which serves as our motivation.

25 Maximum Entropy The standard ME estimation divides the total # or weight of connections equally among all other nodes. Then uses the RAS algorithm proposed by Schneider (1990) to re-balance the matrix. Matrix re-balancing is required to ensure that the sum of the individual elements amount to the original overall value of the connections. Mistrulli (2010) provided a description of the shortcomings of ME, which serves as our motivation. In particular, we will show that when we have complex systems such as networks, there are cases when ME doesn t perform well.

26 Maximum Entropy The standard ME estimation divides the total # or weight of connections equally among all other nodes. Then uses the RAS algorithm proposed by Schneider (1990) to re-balance the matrix. Matrix re-balancing is required to ensure that the sum of the individual elements amount to the original overall value of the connections. Mistrulli (2010) provided a description of the shortcomings of ME, which serves as our motivation. In particular, we will show that when we have complex systems such as networks, there are cases when ME doesn t perform well. Further, we propose an alternate methodology based on copulas.

27 Copulas A copula is a cumulative distribution function with uniform marginals.

28 Copulas A copula is a cumulative distribution function with uniform marginals. What makes a copula attractive?

29 Copulas A copula is a cumulative distribution function with uniform marginals. What makes a copula attractive? - A copula can express complex dependence structures using the marginals as a reference case and the corresponding dependence parameter.

30 Copulas A copula is a cumulative distribution function with uniform marginals. What makes a copula attractive? - A copula can express complex dependence structures using the marginals as a reference case and the corresponding dependence parameter. Raschke et al. (2010) use copulas to generate random networks for given dependence parameters.

31 Copulas Main differences between Raschke et al. (2010) and our work are as follows.

32 Copulas Main differences between Raschke et al. (2010) and our work are as follows. 1 They generate a random network whereas we propose a method applicable to directed networks via a sequential search using an estimated dependence parameter rather than assuming a value.

33 Copulas Main differences between Raschke et al. (2010) and our work are as follows. 1 They generate a random network whereas we propose a method applicable to directed networks via a sequential search using an estimated dependence parameter rather than assuming a value. 2 They do not address the role of the structure of data in affecting the performance of their methods.

34 Copulas Main differences between Raschke et al. (2010) and our work are as follows. 1 They generate a random network whereas we propose a method applicable to directed networks via a sequential search using an estimated dependence parameter rather than assuming a value. 2 They do not address the role of the structure of data in affecting the performance of their methods. 3 We provide various scenarios in terms of different data structures and analyze the relative performance of Copula based methods as compared to the standard ME approach (our benchmark).

35 Proposed Methodology

36 Proposed Methodology We propose two methods based on copulas - direct and indirect.

37 Proposed Methodology We propose two methods based on copulas - direct and indirect. We simulate networks via Monte Carlo simulation technique and compare their performance to that of ME.

38 Proposed Methodology We propose two methods based on copulas - direct and indirect. We simulate networks via Monte Carlo simulation technique and compare their performance to that of ME. We use an error measure that is directly based on the distance between the true and the simulated bilateral connections.

39 Proposed Methodology The direct method directly fits a copula to the data and generates the joint distributions over connections based on the estimated dependence parameter.

40 Proposed Methodology The direct method directly fits a copula to the data and generates the joint distributions over connections based on the estimated dependence parameter. The indirect method based on constraint optimization of the error between the matrix to be estimated and the initial guess.

41 Proposed Methodology The direct method directly fits a copula to the data and generates the joint distributions over connections based on the estimated dependence parameter. The indirect method based on constraint optimization of the error between the matrix to be estimated and the initial guess. Fernandez-Vazquez (2010) use a similar method but with Cross Entropy.

42 Proposed Methodology The direct method directly fits a copula to the data and generates the joint distributions over connections based on the estimated dependence parameter. The indirect method based on constraint optimization of the error between the matrix to be estimated and the initial guess. Fernandez-Vazquez (2010) use a similar method but with Cross Entropy. However, the cross entropy fitness measure has a major disadvantage

43 Proposed Methodology The direct method directly fits a copula to the data and generates the joint distributions over connections based on the estimated dependence parameter. The indirect method based on constraint optimization of the error between the matrix to be estimated and the initial guess. Fernandez-Vazquez (2010) use a similar method but with Cross Entropy. However, the cross entropy fitness measure has a major disadvantage - in cases where connections are underestimated, it produces a negative number biasing the the overall assessment of the fitness of the model

44 Cross Entropy The main differences between Fernandez-Vazquez (2010) and our indirect methodology are as follows.

45 Cross Entropy The main differences between Fernandez-Vazquez (2010) and our indirect methodology are as follows. 1 We construct the initial guess as a hybrid of two copulas.

46 Cross Entropy The main differences between Fernandez-Vazquez (2010) and our indirect methodology are as follows. 1 We construct the initial guess as a hybrid of two copulas. 2 Instead of the Kullback-Leibler measure of divergence, we use our error measure.

47 Cross Entropy The main differences between Fernandez-Vazquez (2010) and our indirect methodology are as follows. 1 We construct the initial guess as a hybrid of two copulas. 2 Instead of the Kullback-Leibler measure of divergence, we use our error measure. 3 We use additional constraint(s) that imposes restrictions on the structure of the network.

48 Core-Periphey Model In order to impose structural constraints on our simulation process we use the Core-Periphery model setup.

49 Core-Periphey Model In order to impose structural constraints on our simulation process we use the Core-Periphery model setup. Suppose we are able to partition the entire network into blocks - { Block 1, Block 2, Block 3, Block 4}

50 Core-Periphey Model In order to impose structural constraints on our simulation process we use the Core-Periphery model setup. Suppose we are able to partition the entire network into blocks - { Block 1, Block 2, Block 3, Block 4} Figure: Dividing A Matrix into Blocks

51 Core-Periphey Model Core-Periphey Model assumes that there are certain key nodes in the network (the core) that have a relatively larger impact in the way connections are formed as well as the way connections are sustained.

52 Core-Periphey Model Core-Periphey Model assumes that there are certain key nodes in the network (the core) that have a relatively larger impact in the way connections are formed as well as the way connections are sustained. The rest of the nodes are more peripheral and have less of an impact in the manner in which dependencies arise within the network.

53 Core-Periphey Model Core-Periphey Model assumes that there are certain key nodes in the network (the core) that have a relatively larger impact in the way connections are formed as well as the way connections are sustained. The rest of the nodes are more peripheral and have less of an impact in the manner in which dependencies arise within the network. For example, we can term Block 1 as the core-core (CC) connections and Block 4 as periphery-periphery (PP) connections.

54 Core-Periphey Model Core-Periphey Model assumes that there are certain key nodes in the network (the core) that have a relatively larger impact in the way connections are formed as well as the way connections are sustained. The rest of the nodes are more peripheral and have less of an impact in the manner in which dependencies arise within the network. For example, we can term Block 1 as the core-core (CC) connections and Block 4 as periphery-periphery (PP) connections. Block 2 will then be core-periphery (CP) connections and Block 3 will be periphery-core (PC) connections.

55 Core-Periphey Model The sum of the number/value of connections in each block may be treated as an indicator of the relative importance of the block.

56 Core-Periphey Model The sum of the number/value of connections in each block may be treated as an indicator of the relative importance of the block. We call this the share of a block.

57 Core-Periphey Model The sum of the number/value of connections in each block may be treated as an indicator of the relative importance of the block. We call this the share of a block. And the ratio of the number of nodes in each block to the total number of nodes denotes the size of the block.

58 Core-Periphey Model The sum of the number/value of connections in each block may be treated as an indicator of the relative importance of the block. We call this the share of a block. And the ratio of the number of nodes in each block to the total number of nodes denotes the size of the block. We assume that the size of CC is smaller than that of PP.

59 Data Asymmetry We define asymmetry of data in terms of variable block share and block size.

60 Data Asymmetry We define asymmetry of data in terms of variable block share and block size. Equal Block Share = Symmetry in Shares.

61 Data Asymmetry We define asymmetry of data in terms of variable block share and block size. Equal Block Share = Symmetry in Shares. Equal Block Sizes = Symmetry in Sizes.

62 Data Asymmetry We define asymmetry of data in terms of variable block share and block size. Equal Block Share = Symmetry in Shares. Equal Block Sizes = Symmetry in Sizes. This notion can be applied appropriately to represent underlying community structures of networks as well.

63 Main Findings of the Paper

64 Main Findings of the Paper We apply our methods to both sparse and dense networks.

65 Main Findings of the Paper We apply our methods to both sparse and dense networks. For sparse networks, we find that ME and our methods do equally well.

66 Main Findings of the Paper We apply our methods to both sparse and dense networks. For sparse networks, we find that ME and our methods do equally well. For dense networks, we find that the degree of asymmetry in the data affects the performance of ME and our methods.

67 Main Findings of the Paper We apply our methods to both sparse and dense networks. For sparse networks, we find that ME and our methods do equally well. For dense networks, we find that the degree of asymmetry in the data affects the performance of ME and our methods. In particular, higher the asymmetry better our methods perform upto a threshold size and share of CC.

68 Direct Method - Algorithm

69 Direct Method - Algorithm First, we present the steps involved in fitting a copula to the data.

70 Direct Method - Algorithm First, we present the steps involved in fitting a copula to the data. Since we have two vectors as our marginals, we use a bivariate copula.

71 Direct Method - Algorithm First, we present the steps involved in fitting a copula to the data. Since we have two vectors as our marginals, we use a bivariate copula. Specifically, we use the functional form given by a Gumbel Copula.

72 Direct Method - Algorithm The Gumbel Copula belongs to the extreme values family.

73 Direct Method - Algorithm The Gumbel Copula belongs to the extreme values family. For a bivariate case it may be expressed as

74 Direct Method - Algorithm The Gumbel Copula belongs to the extreme values family. For a bivariate case it may be expressed as C G(a.i ;a i. ) = exp( [( ln(a.i )) θ + ( ln(a i. )) θ ]) 1 θ The parameter θ controls the strength of dependence.

75 Direct Method - Algorithm The Gumbel Copula belongs to the extreme values family. For a bivariate case it may be expressed as C G(a.i ;a i. ) = exp( [( ln(a.i )) θ + ( ln(a i. )) θ ]) 1 θ The parameter θ controls the strength of dependence. The upper tail dependence for this copula is given by θ.

76 Direct Method - Algorithm The Gumbel Copula belongs to the extreme values family. For a bivariate case it may be expressed as C G(a.i ;a i. ) = exp( [( ln(a.i )) θ + ( ln(a i. )) θ ]) 1 θ The parameter θ controls the strength of dependence. The upper tail dependence for this copula is given by θ. And the lower dependency is 0.

77 Standard Copula Algorithm Plot the densities of the marginal sums of rows and columns (i.e, the available data).

78 Standard Copula Algorithm Plot the densities of the marginal sums of rows and columns (i.e, the available data). Based on these plots of the marginals, infer the nature of distribution of the marginals (e.g., perhaps, it is a mixture or it is not).

79 Standard Copula Algorithm Plot the densities of the marginal sums of rows and columns (i.e, the available data). Based on these plots of the marginals, infer the nature of distribution of the marginals (e.g., perhaps, it is a mixture or it is not). Transform these marginals into uniform distribution (using kernel density estimation), which is required for it to be able to serve as an input into the copula function.

80 Standard Copula Algorithm Plot the densities of the marginal sums of rows and columns (i.e, the available data). Based on these plots of the marginals, infer the nature of distribution of the marginals (e.g., perhaps, it is a mixture or it is not). Transform these marginals into uniform distribution (using kernel density estimation), which is required for it to be able to serve as an input into the copula function. Thereafter, fit a copula to the transformed data using maximum likelihood estimation that estimates the dependency parameter.

81 Standard Copula Algorithm The ML estimation undertaken in this step makes a parametric assumption that the copula is a function of some dependence parameter, θ.

82 Standard Copula Algorithm The ML estimation undertaken in this step makes a parametric assumption that the copula is a function of some dependence parameter, θ. Our aim is estimate the θ in order to deduce the nature of dependence using the ML estimation method.

83 Standard Copula Algorithm The ML estimation undertaken in this step makes a parametric assumption that the copula is a function of some dependence parameter, θ. Our aim is estimate the θ in order to deduce the nature of dependence using the ML estimation method. The copula distribution function is given by C(a i., a.j ; θ).

84 Standard Copula Algorithm The ML estimation undertaken in this step makes a parametric assumption that the copula is a function of some dependence parameter, θ. Our aim is estimate the θ in order to deduce the nature of dependence using the ML estimation method. The copula distribution function is given by C(a i., a.j ; θ). We denote copula density as c θ (a i. ; a.j ).

85 Standard Copula Algorithm The ML estimation undertaken in this step makes a parametric assumption that the copula is a function of some dependence parameter, θ. Our aim is estimate the θ in order to deduce the nature of dependence using the ML estimation method. The copula distribution function is given by C(a i., a.j ; θ). We denote copula density as c θ (a i. ; a.j ). ln L(θ a i., a.j ) = N c θ ( ˆF 1 (a ik ), ˆF 2 (a kj )) (1) k=1

86 Standard Copula Algorithm The ML estimation undertaken in this step makes a parametric assumption that the copula is a function of some dependence parameter, θ. Our aim is estimate the θ in order to deduce the nature of dependence using the ML estimation method. The copula distribution function is given by C(a i., a.j ; θ). We denote copula density as c θ (a i. ; a.j ). ln L(θ a i., a.j ) = N c θ ( ˆF 1 (a ik ), ˆF 2 (a kj )) (1) k=1 We then use this estimate of the dependency parameter,ˆθ, to generate a matrix of cummulative probabilities.

87 Direct Method Algorithm Begin by rescaling the matrix of probabilities derived from the copula estimation into a stochastic matrix.

88 Direct Method Algorithm Begin by rescaling the matrix of probabilities derived from the copula estimation into a stochastic matrix. Then, apply the RAS algorithm to the matrix to rebalance it.

89 Direct Method Algorithm Begin by rescaling the matrix of probabilities derived from the copula estimation into a stochastic matrix. Then, apply the RAS algorithm to the matrix to rebalance it. RAS algorithm is an iterative proportional fitting procedure for estimating the bilateral connections (or a ij s) of the connection matrix such that the marginal sums remain unchanged.

90 Direct Method Algorithm Begin by rescaling the matrix of probabilities derived from the copula estimation into a stochastic matrix. Then, apply the RAS algorithm to the matrix to rebalance it. RAS algorithm is an iterative proportional fitting procedure for estimating the bilateral connections (or a ij s) of the connection matrix such that the marginal sums remain unchanged. Then calculate the error measure.

91 Direct Method Algorithm Begin by rescaling the matrix of probabilities derived from the copula estimation into a stochastic matrix. Then, apply the RAS algorithm to the matrix to rebalance it. RAS algorithm is an iterative proportional fitting procedure for estimating the bilateral connections (or a ij s) of the connection matrix such that the marginal sums remain unchanged. Then calculate the error measure. i j ɛ = aˆ ij a ij i j a ij where â ij is the estimated a ij. (2)

92 Direct Method Algorithm Begin by rescaling the matrix of probabilities derived from the copula estimation into a stochastic matrix. Then, apply the RAS algorithm to the matrix to rebalance it. RAS algorithm is an iterative proportional fitting procedure for estimating the bilateral connections (or a ij s) of the connection matrix such that the marginal sums remain unchanged. Then calculate the error measure. i j ɛ = aˆ ij a ij i j a ij where â ij is the estimated a ij. The error measure returns the sum of the differences between the estimated and the true values as a percentage of total connections. (2)

93 Direct Method Algorithm Then estimate matrix of connections using the maximum entropy method.

94 Direct Method Algorithm Then estimate matrix of connections using the maximum entropy method. Calculate the error measure.

95 Direct Method Algorithm Then estimate matrix of connections using the maximum entropy method. Calculate the error measure. Compare both error measures.

96 Direct Method Algorithm Then estimate matrix of connections using the maximum entropy method. Calculate the error measure. Compare both error measures. The fit with the lower error measure is a better fit.

97 Indirect Method Algorithm

98 Indirect Method Algorithm First we estimate the matrix of connections using the maximum entropy method.

99 Indirect Method Algorithm First we estimate the matrix of connections using the maximum entropy method. Now, suppose we believe that the data is a combination of two different copulas in the ratio of x and 1 x.

100 Indirect Method Algorithm First we estimate the matrix of connections using the maximum entropy method. Now, suppose we believe that the data is a combination of two different copulas in the ratio of x and 1 x. Then, use the direct copula estimation method to fit each copula to the data separately.

101 Indirect Method Algorithm First we estimate the matrix of connections using the maximum entropy method. Now, suppose we believe that the data is a combination of two different copulas in the ratio of x and 1 x. Then, use the direct copula estimation method to fit each copula to the data separately. Using these estimates we construct Q 0 as follows.

102 Indirect Method Algorithm First we estimate the matrix of connections using the maximum entropy method. Now, suppose we believe that the data is a combination of two different copulas in the ratio of x and 1 x. Then, use the direct copula estimation method to fit each copula to the data separately. Using these estimates we construct Q 0 as follows. 1 First insert values extracted from the appropriate copula estimates in x% of the number of cells in Q 0.

103 Indirect Method Algorithm First we estimate the matrix of connections using the maximum entropy method. Now, suppose we believe that the data is a combination of two different copulas in the ratio of x and 1 x. Then, use the direct copula estimation method to fit each copula to the data separately. Using these estimates we construct Q 0 as follows. 1 First insert values extracted from the appropriate copula estimates in x% of the number of cells in Q 0. 2 Then for the (1 x)% we use the other copula estimates.

104 Indirect Method Algorithm For example, if we were looking at a 5x5 Q 0 matrix with, say, 60% of values from the first copula.

105 Indirect Method Algorithm For example, if we were looking at a 5x5 Q 0 matrix with, say, 60% of values from the first copula. Then we take the first 15 (0.6 25) estimates, i.e., the first 3 rows and 5 columns from the matrix of the first copula.

106 Indirect Method Algorithm For example, if we were looking at a 5x5 Q 0 matrix with, say, 60% of values from the first copula. Then we take the first 15 (0.6 25) estimates, i.e., the first 3 rows and 5 columns from the matrix of the first copula. Use them to be the first 3 rows and 5 columns of Q 0.

107 Indirect Method Algorithm For example, if we were looking at a 5x5 Q 0 matrix with, say, 60% of values from the first copula. Then we take the first 15 (0.6 25) estimates, i.e., the first 3 rows and 5 columns from the matrix of the first copula. Use them to be the first 3 rows and 5 columns of Q 0. The remaining part of Q 0 then comes from the other copula in a similar way.

108 Indirect Method Algorithm So Q 0 may be represented as

109 Indirect Method Algorithm So Q 0 may be represented as Figure: Hybrid of Copulas As Initial Guess

110 Indirect Method Algorithm Once Q 0 has been constructed, conduct a constrained minimization of

111 Indirect Method Algorithm Once Q 0 has been constructed, conduct a constrained minimization of G Q0 (3)

112 Indirect Method Algorithm Once Q 0 has been constructed, conduct a constrained minimization of G Q0 (3) subject to R.P = B (4) 1 G is the matrix to be estimated with typical element {a ij }

113 Indirect Method Algorithm Once Q 0 has been constructed, conduct a constrained minimization of G Q0 (3) subject to R.P = B (4) 1 G is the matrix to be estimated with typical element {a ij } 2 R is the vector of restrictions imposed on G

114 Indirect Method Algorithm Once Q 0 has been constructed, conduct a constrained minimization of G Q0 (3) subject to R.P = B (4) 1 G is the matrix to be estimated with typical element {a ij } 2 R is the vector of restrictions imposed on G 3 P is the matrix of probabilities.

115 Indirect Method Algorithm Once Q 0 has been constructed, conduct a constrained minimization of G Q0 (3) subject to R.P = B (4) 1 G is the matrix to be estimated with typical element {a ij } 2 R is the vector of restrictions imposed on G 3 P is the matrix of probabilities. 4 B is the target vector.

116 Indirect Method Algorithm Thereafter, extract the Q that is the minimizer.

117 Indirect Method Algorithm Thereafter, extract the Q that is the minimizer. Calculate the error measure of the Q and compare it to the one derived from maximum entropy.

118 Indirect Method Algorithm Thereafter, extract the Q that is the minimizer. Calculate the error measure of the Q and compare it to the one derived from maximum entropy. The better performer will have a lower error measure.

119 Monte Carlo Simulations - Data Generating Processes

120 Monte Carlo Simulations - Data Generating Processes We concentrate on intra- and inter-block asymmetries.

121 Monte Carlo Simulations - Data Generating Processes We concentrate on intra- and inter-block asymmetries. We introduce parameters to vary these called inter and intra.

122 Monte Carlo Simulations - Data Generating Processes We concentrate on intra- and inter-block asymmetries. We introduce parameters to vary these called inter and intra. We use two different DGPs - DGP 1 and DGP 2

123 Monte Carlo Simulations - DGP 1 It imposes that entries in the PP block are all 0 s.

124 Monte Carlo Simulations - DGP 1 It imposes that entries in the PP block are all 0 s. The values in the CC block are high (e.g., between 5000 and 10000)

125 Monte Carlo Simulations - DGP 1 It imposes that entries in the PP block are all 0 s. The values in the CC block are high (e.g., between 5000 and 10000) The values in the CP and PC blocks are lower than CC but higher than PP.

126 Monte Carlo Simulations - DGP 1 It imposes that entries in the PP block are all 0 s. The values in the CC block are high (e.g., between 5000 and 10000) The values in the CP and PC blocks are lower than CC but higher than PP. These are varied (except the PP and CC block) using the inter and intra parameters.

127 Monte Carlo Simulations - DGP 1 Figure: Graphical Representation of DGP 1

128 Monte Carlo Simulations - DGP 2 It imposes that entries in the PP block are NOT 0 s.

129 Monte Carlo Simulations - DGP 2 It imposes that entries in the PP block are NOT 0 s. The values in the CC block are higher (e.g., between and )

130 Monte Carlo Simulations - DGP 2 It imposes that entries in the PP block are NOT 0 s. The values in the CC block are higher (e.g., between and ) These are varied (except the CC block) using the inter and intra parameters.

131 Monte Carlo Simulations - DGP 2 Figure: Graphical Representation of DGP 2

132 Monte Carlo Simulations - Findings Under DGP 1

133 Monte Carlo Simulations - Findings Under DGP 1 We conducted 1000 simulations for fixed networks sizes but variable core sizes.

134 Monte Carlo Simulations - Findings Under DGP 1 We conducted 1000 simulations for fixed networks sizes but variable core sizes. We conducted 1000 simulations for variable networks sizes but fixed core sizes.

135 Monte Carlo Simulations - Findings Under DGP 1 Under the direct method with variable intra-block asymmetry the following represents difference between the error measure of ME and Copula based direct method.

136 Monte Carlo Simulations - Findings Under DGP 1 Under the direct method with variable intra-block asymmetry the following represents difference between the error measure of ME and Copula based direct method. Figure: Performance of Gumbel vs. ME under DGP 1 with Variable Inra Block Asymmetry

137 Monte Carlo Simulations - Findings Under DGP 1 The graphs clearly indicate that the difference is negative, implying ME performs better.

138 Monte Carlo Simulations - Findings Under DGP 1 The graphs clearly indicate that the difference is negative, implying ME performs better. However, as the network size increases, there seems to an upward trend.

139 Monte Carlo Simulations - Findings Under DGP 1 The graphs clearly indicate that the difference is negative, implying ME performs better. However, as the network size increases, there seems to an upward trend. Potential Reasons:

140 Monte Carlo Simulations - Findings Under DGP 1 The graphs clearly indicate that the difference is negative, implying ME performs better. However, as the network size increases, there seems to an upward trend. Potential Reasons: 1 Share of PP is 0.

141 Monte Carlo Simulations - Findings Under DGP 1 The graphs clearly indicate that the difference is negative, implying ME performs better. However, as the network size increases, there seems to an upward trend. Potential Reasons: 1 Share of PP is 0. 2 Size and Share of CC are high but not enough to induce to make the data significantly asymmetric.

142 Monte Carlo Simulations - Findings Under DGP 1 The graphs clearly indicate that the difference is negative, implying ME performs better. However, as the network size increases, there seems to an upward trend. Potential Reasons: 1 Share of PP is 0. 2 Size and Share of CC are high but not enough to induce to make the data significantly asymmetric. So, we check with DGP 2, which addresses both these points.

143 Monte Carlo Simulations - Findings Under DGP 2 The following graph clearly indicates an upward trend as CC and PP values are increased under DGP 2, implying that the direct method performs much better.

144 Monte Carlo Simulations - Findings Under DGP 2 The following graph clearly indicates an upward trend as CC and PP values are increased under DGP 2, implying that the direct method performs much better. Figure: Performance of Gumbel vs. ME under DGP 2 with Variable Intra Block Asymmetry

145 Monte Carlo Simulations - Findings Under DGP 2 However, as we reduce the PP values towards 0, one starts noticing a downward trend!

146 Monte Carlo Simulations - Findings Under DGP 2 However, as we reduce the PP values towards 0, one starts noticing a downward trend! Figure: Performance of Gumbel vs. ME under DGP 2 with Variable Intra Block Asymmetry with Lower PP

147 Monte Carlo Simulations - Findings On the other hand, under DGP 1 and DGP 2 as we change inter-block asymmetry we observe a distinct negative trend.

148 Monte Carlo Simulations - Findings On the other hand, under DGP 1 and DGP 2 as we change inter-block asymmetry we observe a distinct negative trend. Figure: Performance of Gumbel vs. ME under DGP 1 and DGP 2 with Variable Inter Block Asymmetry

149 Monte Carlo Simulations - Findings Potential Reasons:

150 Monte Carlo Simulations - Findings Potential Reasons: 1 Beyond a point (in particular, under DGP 2), as we increase the parameter inter, we end up balancing the shares of all blocks.

151 Monte Carlo Simulations - Findings Potential Reasons: 1 Beyond a point (in particular, under DGP 2), as we increase the parameter inter, we end up balancing the shares of all blocks. 2 As such, we end up making the data more symmetric.

152 Monte Carlo Simulations - Findings Potential Reasons: 1 Beyond a point (in particular, under DGP 2), as we increase the parameter inter, we end up balancing the shares of all blocks. 2 As such, we end up making the data more symmetric. 3 As the data becomes more symmetric, ME starts to perform better.

153 Monte Carlo Simulations - Findings Potential Reasons: 1 Beyond a point (in particular, under DGP 2), as we increase the parameter inter, we end up balancing the shares of all blocks. 2 As such, we end up making the data more symmetric. 3 As the data becomes more symmetric, ME starts to perform better. 4 Until the data is asymmetric, the copula-based method performs better.

154 Monte Carlo Simulations - Findings Potential Reasons: 1 Beyond a point (in particular, under DGP 2), as we increase the parameter inter, we end up balancing the shares of all blocks. 2 As such, we end up making the data more symmetric. 3 As the data becomes more symmetric, ME starts to perform better. 4 Until the data is asymmetric, the copula-based method performs better. 5 Observations are similar even when we vary core sizes.

155 Monte Carlo Simulations - Findings For Smaller Networks We observe the following for varibale shares of CC and performance of copula-based vs. ME methods under DGP 1.

156 Monte Carlo Simulations - Findings For Smaller Networks We observe the following for varibale shares of CC and performance of copula-based vs. ME methods under DGP 1. Figure: Performance of Gumbel vs. ME under DGP 1 with Variable Inter Block Asymmetry For Small Networks

157 Monte Carlo Simulations - Findings For Smaller Networks We observe the following for varibale shares of CC and performance of copula-based vs. ME methods under DGP 2.

158 Monte Carlo Simulations - Findings For Smaller Networks We observe the following for varibale shares of CC and performance of copula-based vs. ME methods under DGP 2. Figure: Performance of Gumbel, Hybrid vs. ME under DGP 2 with Variable Inter Block Asymmetry For Small Networks

159 Monte Carlo Simulations - Findings For Smaller Networks We observe that with DGP 1, we continue to get mixed results.

160 Monte Carlo Simulations - Findings For Smaller Networks We observe that with DGP 1, we continue to get mixed results. Whereas with DGP 2, we have that both the direct as well as indirect methods do better than ME even with smaller network sizes.

161 Monte Carlo Simulations - Findings For Smaller Networks We observe that with DGP 1, we continue to get mixed results. Whereas with DGP 2, we have that both the direct as well as indirect methods do better than ME even with smaller network sizes. In particular, as we increase the value of PP block keeping the values of CC constant, the copula based methods perform much better.

162 Monte Carlo Simulations - Findings For Smaller Networks We observe that with DGP 1, we continue to get mixed results. Whereas with DGP 2, we have that both the direct as well as indirect methods do better than ME even with smaller network sizes. In particular, as we increase the value of PP block keeping the values of CC constant, the copula based methods perform much better. Potential Reasons:

163 Monte Carlo Simulations - Findings For Smaller Networks We observe that with DGP 1, we continue to get mixed results. Whereas with DGP 2, we have that both the direct as well as indirect methods do better than ME even with smaller network sizes. In particular, as we increase the value of PP block keeping the values of CC constant, the copula based methods perform much better. Potential Reasons: 1 When we increase the values of PP, we increase the asymmetry in the data.

164 Monte Carlo Simulations - Findings For Smaller Networks We observe that with DGP 1, we continue to get mixed results. Whereas with DGP 2, we have that both the direct as well as indirect methods do better than ME even with smaller network sizes. In particular, as we increase the value of PP block keeping the values of CC constant, the copula based methods perform much better. Potential Reasons: 1 When we increase the values of PP, we increase the asymmetry in the data. 2 As the intra-block asymmetry rises, we are essentially increasing the asymmetry in the data which helps (1).

165 Application to Dense Network - BIS

166 Application to Dense Network - BIS Bank of International Settlements data provides a bilateral description of interbank exposures at a country level.

167 Application to Dense Network - BIS Bank of International Settlements data provides a bilateral description of interbank exposures at a country level. Figure: Structure of BIS Data

168 Application to Dense Network - BIS Since the BIS data has a given degree of asymmetry, we analyze its various sub-matrices.

169 Application to Dense Network - BIS Since the BIS data has a given degree of asymmetry, we analyze its various sub-matrices. We look at the following matrices extracted from the BIS matrix which is a matrix of connections:

170 Application to Dense Network - BIS Since the BIS data has a given degree of asymmetry, we analyze its various sub-matrices. We look at the following matrices extracted from the BIS matrix which is a matrix of connections: 6 6, 7 7, 8 8, 10 10, 11 11, 13 13, 15 15, and the full dataset,

171 Application to Dense Network - BIS For instance, a matrix of the BIS data looks like the following.

172 Application to Dense Network - BIS For instance, a matrix of the BIS data looks like the following. Figure: Structure of BIS Matrix

173 Application to Dense Network - BIS We obseve that for 6 6 with 5 nodes in the core, the share of CC is around 96%,

174 Application to Dense Network - BIS We obseve that for 6 6 with 5 nodes in the core, the share of CC is around 96%, CP = 0%,

175 Application to Dense Network - BIS We obseve that for 6 6 with 5 nodes in the core, the share of CC is around 96%, CP = 0%, PC = %,

176 Application to Dense Network - BIS We obseve that for 6 6 with 5 nodes in the core, the share of CC is around 96%, CP = 0%, PC = %, PP = %,

177 Application to Dense Network - BIS We obseve that for 6 6 with 5 nodes in the core, the share of CC is around 96%, CP = 0%, PC = %, PP = %, i.e.,the data is highly asymmetric.

178 Application to Dense Network - BIS We obseve that for 6 6 with 5 nodes in the core, the share of CC is around 96%, CP = 0%, PC = %, PP = %, i.e.,the data is highly asymmetric. At 21 21, the share of the blocks are very similar,

179 Application to Dense Network - BIS We obseve that for 6 6 with 5 nodes in the core, the share of CC is around 96%, CP = 0%, PC = %, PP = %, i.e.,the data is highly asymmetric. At 21 21, the share of the blocks are very similar, i.e., CC = 34.5%,

180 Application to Dense Network - BIS We obseve that for 6 6 with 5 nodes in the core, the share of CC is around 96%, CP = 0%, PC = %, PP = %, i.e.,the data is highly asymmetric. At 21 21, the share of the blocks are very similar, i.e., CC = 34.5%,CP = %,

181 Application to Dense Network - BIS We obseve that for 6 6 with 5 nodes in the core, the share of CC is around 96%, CP = 0%, PC = %, PP = %, i.e.,the data is highly asymmetric. At 21 21, the share of the blocks are very similar, i.e., CC = 34.5%,CP = %,PC = %

182 Application to Dense Network - BIS We obseve that for 6 6 with 5 nodes in the core, the share of CC is around 96%, CP = 0%, PC = %, PP = %, i.e.,the data is highly asymmetric. At 21 21, the share of the blocks are very similar, i.e., CC = 34.5%,CP = %,PC = % and PP = %,

183 Application to Dense Network - BIS We obseve that for 6 6 with 5 nodes in the core, the share of CC is around 96%, CP = 0%, PC = %, PP = %, i.e.,the data is highly asymmetric. At 21 21, the share of the blocks are very similar, i.e., CC = 34.5%,CP = %,PC = % and PP = %, implying the data is highly symmetric.

184 Application to Dense Network - BIS We obseve that for 6 6 with 5 nodes in the core, the share of CC is around 96%, CP = 0%, PC = %, PP = %, i.e.,the data is highly asymmetric. At 21 21, the share of the blocks are very similar, i.e., CC = 34.5%,CP = %,PC = % and PP = %, implying the data is highly symmetric. As the data becomes more asymmetric, we observe that the copula based methods do better.

185 Application to Dense Network - BIS We obseve that for 6 6 with 5 nodes in the core, the share of CC is around 96%, CP = 0%, PC = %, PP = %, i.e.,the data is highly asymmetric. At 21 21, the share of the blocks are very similar, i.e., CC = 34.5%,CP = %,PC = % and PP = %, implying the data is highly symmetric. As the data becomes more asymmetric, we observe that the copula based methods do better. In particular, the threshold is 11 11

186 Application to Dense Network - BIS We obseve that for 6 6 with 5 nodes in the core, the share of CC is around 96%, CP = 0%, PC = %, PP = %, i.e.,the data is highly asymmetric. At 21 21, the share of the blocks are very similar, i.e., CC = 34.5%,CP = %,PC = % and PP = %, implying the data is highly symmetric. As the data becomes more asymmetric, we observe that the copula based methods do better. In particular, the threshold is with block shares for CC = %,

187 Application to Dense Network - BIS We obseve that for 6 6 with 5 nodes in the core, the share of CC is around 96%, CP = 0%, PC = %, PP = %, i.e.,the data is highly asymmetric. At 21 21, the share of the blocks are very similar, i.e., CC = 34.5%,CP = %,PC = % and PP = %, implying the data is highly symmetric. As the data becomes more asymmetric, we observe that the copula based methods do better. In particular, the threshold is with block shares for CC = %, CP = %,

188 Application to Dense Network - BIS We obseve that for 6 6 with 5 nodes in the core, the share of CC is around 96%, CP = 0%, PC = %, PP = %, i.e.,the data is highly asymmetric. At 21 21, the share of the blocks are very similar, i.e., CC = 34.5%,CP = %,PC = % and PP = %, implying the data is highly symmetric. As the data becomes more asymmetric, we observe that the copula based methods do better. In particular, the threshold is with block shares for CC = %, CP = %, PC = %

189 Application to Dense Network - BIS We obseve that for 6 6 with 5 nodes in the core, the share of CC is around 96%, CP = 0%, PC = %, PP = %, i.e.,the data is highly asymmetric. At 21 21, the share of the blocks are very similar, i.e., CC = 34.5%,CP = %,PC = % and PP = %, implying the data is highly symmetric. As the data becomes more asymmetric, we observe that the copula based methods do better. In particular, the threshold is with block shares for CC = %, CP = %, PC = % and PP = %

190 Application to Dense Network - BIS We obseve that for 6 6 with 5 nodes in the core, the share of CC is around 96%, CP = 0%, PC = %, PP = %, i.e.,the data is highly asymmetric. At 21 21, the share of the blocks are very similar, i.e., CC = 34.5%,CP = %,PC = % and PP = %, implying the data is highly symmetric. As the data becomes more asymmetric, we observe that the copula based methods do better. In particular, the threshold is with block shares for CC = %, CP = %, PC = % and PP = % below which the Copula based methods do better and above which the ME approach does better.

191 Application to Dense Network - BIS This is represented in the following figure.

192 Application to Dense Network - BIS This is represented in the following figure. Figure: Performance of Copula Based Methods w.r.t BIS data, Variable Core Share

193 Application to Dense Network - BIS This is represented in the following figure.

194 Application to Dense Network - BIS This is represented in the following figure. Figure: Performance of Copula Based Methods w.r.t BIS data, Variable Core Size

195 Application to Sparse Network - emid

196 Application to Sparse Network - emid e-mid is the Italian based European reference electronic market for liquidity trading.

197 Application to Sparse Network - emid e-mid is the Italian based European reference electronic market for liquidity trading. This platform provides anonymized data for euro-denominated unsecured interbank transactions.

198 Application to Sparse Network - emid e-mid is the Italian based European reference electronic market for liquidity trading. This platform provides anonymized data for euro-denominated unsecured interbank transactions. We use a sample of emid data comprised of 43 trading days recorded on the last two months of 2011.

199 Application to Sparse Network - emid e-mid is the Italian based European reference electronic market for liquidity trading. This platform provides anonymized data for euro-denominated unsecured interbank transactions. We use a sample of emid data comprised of 43 trading days recorded on the last two months of We compare the performances of copula based methods, the ME approach as well as a random network generation method where we used a binomial probability distribution of connecting two nodes randomly.

200 Application to Sparse Network - emid e-mid is the Italian based European reference electronic market for liquidity trading. This platform provides anonymized data for euro-denominated unsecured interbank transactions. We use a sample of emid data comprised of 43 trading days recorded on the last two months of We compare the performances of copula based methods, the ME approach as well as a random network generation method where we used a binomial probability distribution of connecting two nodes randomly. Our preliminary results are reported here.

201 Application to Sparse Network - emid Data Description

202 Application to Sparse Network - emid Data Description The following figure shows the number of non-zero entries (our nodes).

203 Application to Sparse Network - emid Data Description The following figure shows the number of non-zero entries (our nodes). Figure: emid Data Description

204 Application to Sparse Network - emid Data Description Here is what we find:

205 Application to Sparse Network - emid Data Description Here is what we find: Figure: Performance of ME, Copula Based Direct Method and Random Network Generation Method

206 Application to Sparse Network - emid Data Description We observe that both ME and Copula based direct method outperform the random network generation method.

207 Application to Sparse Network - emid Data Description We observe that both ME and Copula based direct method outperform the random network generation method. It may be deduced that the random network generation approach may not be sufficient to solve the problem.

208 Application to Sparse Network - emid Data Description We observe that both ME and Copula based direct method outperform the random network generation method. It may be deduced that the random network generation approach may not be sufficient to solve the problem. ME and Copula-based direct method perform almost exactly the same!

209 Application to Sparse Network - emid Data Description We observe that both ME and Copula based direct method outperform the random network generation method. It may be deduced that the random network generation approach may not be sufficient to solve the problem. ME and Copula-based direct method perform almost exactly the same! Does that mean that impact of asymmetry vanishes as the matrices become very sparse?

210 Application to Sparse Network - emid Data Description We observe that both ME and Copula based direct method outperform the random network generation method. It may be deduced that the random network generation approach may not be sufficient to solve the problem. ME and Copula-based direct method perform almost exactly the same! Does that mean that impact of asymmetry vanishes as the matrices become very sparse? Intuitively, YES.