Connecting the dots: Econometric methods for uncovering networks with an application to the Australian financial institutions

Transcription

1 Connecting the dots: Econometric methods for uncovering networks with an application to the Australian financial institutions Mikhail Anufriev a b a Economics, University of Technology, Sydney b Business School ACE 2015

2 Which dots are we connecting? Correlations and principal components from financial econometrics Gaussian graphical models from statistics Centrality measures from network theory Overview: Modeling framework Correlations from DCC Partial correlations and relations to linear regressions Reconstructed network based on partial correlations Network-based measures and their relations Empirical Application: Australian banks, economic sectors, and international markets

3 Motivation for Network Approach Financial Crisis led to increased interest in the relationship between systemic risk and network effects In the current crisis, we have seen that financial firms that become too interconnected to fail pose serious problems for financial stability and for regulators. Due to the complexity and interconnectivity of todays financial markets, the failure of a major counterparty has the potential to severely disrupt many other financial institutions, their customers, and other markets. Charles Plosser, 03/06/09

4 Recent Contributions: just to name a few... Theoretical network models Acemoglu, Ozdaglar and Tahbaz-Salehi (2014, AER). "Systemic Risk and Stability in Financial Networks". Elliott, Golub and Jackson (2014, AER). Financial networks and contagion. Glasserman and Young (2014, JBF). How likely is contagion in financial networks? Empirical networks (Reconstructed networks) Diebold and Yilmaz (2014, JoE) "On the network topology of variance decompositions: Measuring the connectedness of financial firms". Dungey, Luciani and Veredas (2014, JBF). "Googling CIFIs". Barigozzi and Brownlees (2013, WP). "Nets: Network estimation for time series."

5 Partial correlations Let X = (X 1,..., X n ) be a random variable; V = {X 1,..., X n }. Partial correlation measures linear dependence between any two components of X after controlling for linear dependence with all other components ρ ij ρ Xi,X j = Corr ) (X V\{Xi,X j } i V\{X i, X j }, X j V\{X i, X j } Network of Partial Correlations: X i : node i (bank / economic sector / international market) edges connect nodes with non-zero partial correlations

6 Relation to linear regression X can be represented with a system of linear regressions (for all i) X i µ i = j i β ij (X j µ j ) + ε i, ε i N(0, Σ) Orth. cond. E(ε i X j ) = 0 is satisfied iff β ij = ρ ij Var(εi ) Var(ε j ). rescale x i = (X i µ i )/ Var(εi ) and e i = ε i / Var(εi ) x i = j i ρ ij x j + e i.

7 Possible Interpretation x i = j i ρ ij x j + e i in matrix form, where P has 0 on-diagonal and ρ ij off-diagonal x = Px + e kth-order effect of shock e is defined as P k e total effect of e is e + Pe + P 2 e + = (I P) 1 e Note: this works when lim k P k = 0

8 Relation to Variance-Covariance matrix Define Ω = Cov(X) Set K = {k ij } Ω 1, concentration matrix By inverse of partitioned variance Whittaker (2009) shows Matrix P = I D 1/2 Note: k ij ρ ij =. kii k jj K KD 1/2 K D K = diag{k 11,..., k nn } = diag and so the matrix of the total effect is is adjacency matrix of network (I P) 1 = D 1/2 Σ Ω D 1/2 Σ { } 1 Var(ε 1 ),..., 1 = D 1 Σ Var(ε n ),

9 Relation to Variance-Covariance matrix Define Ω = Cov(X) Set K = {k ij } Ω 1, concentration matrix By inverse of partitioned variance Whittaker (2009) shows Matrix P = I D 1/2 Note: k ij ρ ij =. kii k jj K KD 1/2 K D K = diag{k 11,..., k nn } = diag and so the matrix of the total effect is is adjacency matrix of network (I P) 1 = D 1/2 Σ Ω D 1/2 Σ { } 1 Var(ε 1 ),..., 1 = D 1 Σ Var(ε n ),

10 Centrality measures (network theory) degree centrality - the sum of weights over all adjacent edges eigenvector centrality - takes into account centrality of neighbors the eigenvector u 1 corresponding to the largest eigenvalue λ 1 of P Let e0 be shock, its k-order effect is P k e 0 = P k b i u i = b i P k u i = ( ) k b i λ k i u i = λ k λi 1 b i u i λ i i i i 1 Thus u 1 gives the asymptotic direction for P k e 0 when k Intuitively: vector u 1 shows in which node any initial shock will asymptotically accumulate

11 Centrality measures Bonacich centrality - accumulation of degree centrality, degree centrality of neighbors (dampened by α [0, 1]), degree centrality of neighbors of neighbors (dampened by α 2 ), and so on c B (α) = P 1 + αp α 2 P = (I αp) 1 P 1, when α = 1 Bonacich centrality is equal to c B (1) = (I P) 1 P 1 = (I P) sum of indirect effects (i.e., first-orders, second-order, etc.) of unit shock e = 1 compare with total effect of e e + Pe + P 2 e + = (I P) 1 e

12 Conditional Correlations (Bollerslev, 1990, Engle, 2002) Start with each entity and filter its returns (Y t ) to remove conditional mean/variance we use AR(1)-GARCH (1,1) model Y t = c + by t 1 + u t, u t N(0, σ 2 t ) σ 2 t = γ + δσ 2 t 1 + φu 2 t 1 Form matrix X t by taking standardized innovations u t /σ t for all entities; and compute correlation R from X t Reconstruct P by inverting R.

13 Data Source/classification: Datastream 6 publicly traded banks: Big Four (ANZ, CBA, NAB, Westpac) + Macquarie and Suncorp 2 regional: Bank of Qnd and Bendigo and Adelaide bank sectors of the real economy: basic materials, industrials, etc. Asian market. Time-span 6/11/2000 to 22/08/2014 3, 600 daily observations in total Network is built for 22/08/2014 pre 2008 post 2008

14 Consumer Oil Gas Gds Telecom Asia Market Consumer Svs Basic Materials Utilities Health Care Technology Industrials Macquarie Insurance Suncorp Bendigo& Westpac Ad. Bank Real Estate Bank of Qlnd. ANZ NAB CBA Network for 22-Aug-2014, cut-off level κ = 0.075

15 Stability of the networks Oil&Gas Consumer Gds Telecom Oil&Gas Consumer Gds Telecom Utilities Asia Market Basic Utilities Consumer Svs Health Care Asia Market Basic Materials Consumer Svs Health Care Materials Technology Industrials Technology Industrials Insurance Macquarie Insurance Macquarie Bendigo& Suncorp Westpac Bendigo Ad. Bank Suncorp Westpac Ad. Bank Real Estate Real Estate Bank of Qlnd. ANZ Bank of Qlnd. ANZ NAB NAB CBA CBA (a) pre 2008 (b) post 2008

16 R 2 Degree Eigenvec. Bonacich full full full full ANZ Westpac Industrials NAB CBA Basic Materials Consumer Svs Oil & Gas Insurance Macquarie Asia Market Real Estate Suncorp Bank of Qlnd Bend&Ad.Bank Utilities

17 Summary linked Graphical Gaussian models with theoretic network literature established link between partial correlations and shock propagation discussed links with network-based measures reconstructed the network for Australia financial institutions and linked them with other domestic sectors and world economy