Estimating Global Bank Network Connectedness Mert Demirer (MIT) Francis X. Diebold (Penn) Laura Liu (Penn) Kamil Yılmaz (Koç) September 22, 2016 1 / 27
Financial and Macroeconomic Connectedness Market Risk, Portfolio Concentration Risk (return connectedness) Credit Risk (default connectedness) Counterparty Risk, Gridlock Risk (bilateral and multilateral contractual connectedness) Systemic Risk (total directional connectedness, total system-wide connectedness) Business Cycle Risk (local or global real output connectedness) 2 / 27
Covariance So pairwise... So linear... So Gaussian... 3 / 27
A Very General Environment x t = B(L) ε t ε t (0, Σ) C(x, B(L), Σ) 4 / 27
A Natural Financial/Economic Connectedness Question: What fraction of the H-step-ahead prediction-error variance of variable i is due to shocks in variable j, j i? Non-own elements of the variance decomposition: d H ij, j i C(x, H, B(L), Σ) 5 / 27
Variance Decompositions for Connectedness N-Variable Connectedness Table x 1 x 2... x N From Others to i x 1 d11 H d12 H d1n H Σ N j=1 d 1j H, j 1 x 2 d21 H d22 H d2n H Σ N j=1 d 2j H, j 2........ x N dn1 H dn2 H dnn H Σ N j=1 d Nj H, j N To Others Σ N i=1 d i1 H Σ N i=1 d i2 H Σ N i=1 d in H Σ N i,j=1 d ij H From j i 1 i 2 i N i j Upper-left block is variance decomposition matrix, D Connectedness involves the non-diagonal elements of D 6 / 27
Connectedness Measures From others to i: C H i = To others from j: C H j = N j=1 j i N i=1 i j d H ij d H ij ( i s total imports ) ( j s total exports ) Pairwise Directional: Ci j H = d ij H ( i s imports from j ) Net: Cij H = Cj i H C i j H ( ij bilateral trade balance ) - Total Directional: Net: Ci H = C i H C i H ( i s multilateral trade balance ) - Total System-Wide: C H = 1 N dij H ( total world exports ) N i,j=1 i j 7 / 27
Background Recent paper: Diebold, F.X. and Yilmaz, K. (2014), On the Network Topology of Variance Decompositions: Measuring the Connectedness of Financial Firms, Journal of Econometrics, 182, 119-134. Recent book: Diebold, F.X. and Yilmaz, K. (2015), Financial and Macroeconomic Connectedness: A Network Approach to Measurement and Monitoring, Oxford University Press. With K. Yilmaz. 8 / 27
Network Representation: Graph and Matrix 0 1 1 1 1 0 1 0 0 1 1 0 A = 1 0 0 1 0 1 1 1 1 0 0 0 1 1 0 0 0 1 0 0 1 0 1 0 Symmetric adjacency matrix A A ij = 1 if nodes i, j linked A ij = 0 otherwise 9 / 27
Network Connectedness: The Degree Distribution Degree of node i, d i : N d i = j=1 Discrete degree distribution on 0,..., N 1 Mean degree, E(d), is the key connectedness measure A ij 10 / 27
Network Representation II (Weighted, Directed) 0.5.7 0 0 0 0 0 0 0.3 0 A = 0 0 0.7 0.3.3.5 0 0 0 0.5 0 0 0 0.3 0 0 0 0 0 0 to i, from j 11 / 27
Network Connectedness II: The Degree Distribution(s) A ij [0, 1] depending on connection strength Two degrees: d from i = d to j = N j=1 N i=1 from-degree and to-degree distributions on [0, N 1] Mean degree remains the key connectedness measure A ij A ij 12 / 27
Variance Decompositions as Weighted, Directed Networks Variance Decomposition / Connectedness Table x 1 x 2... x N From Others x 1 d11 H d12 H d1n H j 1 d 1j H x 2 d21 H d22 H d2n H j 2 d 2j H........ x N dn1 H dn2 H dnn H j N d Nj H To Others i 1 d H i1 i 2 d H i2 i N d H in i j d H ij Total directional from, Ci H = N j=1 dij H: from-degrees j i Total directional to, C j H = N i=1 dij H: to-degrees i j Total system-wide, C H = 1 N N i,j=1 i j dij H : mean degree 13 / 27
Relationship to MES MES j mkt = E(r j C(r mkt )) Sensitivity of firm j s return to extreme market event C Market-based stress test of firm j s fragility Total directional connectedness from (from-degrees) From others to j 14 / 27
Relationship to CoVaR VaR p : p = P (r < VaR p ) ( ) CoVaR p,j i : p = P r j < CoVaR p,j i C (r i ) ( ) CoVaR p,mkt i : p = P r mkt < CoVaR p,mkt i C (r i ) Measures tail-event linkages Leading choice of C (r i ) is a VaR breach Total directional connectedness to (to-degrees) From i to others 15 / 27
Estimating Connectedness Thus far we ve worked under correct specification, in population: C(x, H, B(L), Σ) Now we want: ( ) Ĉ x, H, B(L), Σ, M(L; ˆθ), and similarly for other variants of connectedness 16 / 27
Many Interesting Issues / Choices x objects: Returns? Return volatilities? Real activities? x universe: How many and which ones? (Major banks) x frequency: Daily? Monthly? Quarterly? Specification: Approximating model M: VAR? DSGE? Estimation: Classical? Bayesian? Hybrid? Selection: Information criteria? Stepwise? Lasso? Shrinkage: BVAR? Ridge? Lasso? Identification (of variance decompositions): Assumptions: Cholesky? Generalized? SVAR? DSGE? Horizon H: Match VaR horizon? Holding period? Understanding: Network visualization 17 / 27
Selection and Shrinkage via Penalized Estimation of High-Dimensional Approximating Models ˆβ = argmin β T t=1 ( ( T ˆβ = argmin β y t t=1 i y t ) 2 β i x it s.t. i β i x it ) 2 + λ K β i q c i=1 K β i q i=1 Concave penalty functions non-differentiable at the origin produce selection. Smooth convex penalties produce shrinkage. q 0 produces selection, q = 2 produces ridge, q = 1 produces lasso. 18 / 27
Lasso ( T ˆβ Lasso = argmin β y t t=1 i β i x it ) 2 + λ K β i ( T ˆβ ALasso = argmin β y t ) 2 K β i x it + λ w i β i t=1 i i=1 ( T ˆβ Enet = argmin β y t ) 2 K ( β i x it + λ α βi + (1 α)βi 2 t=1 i i=1 ( T ˆβ AEnet = argmin β y t ) 2 K ( β i x it + λ w i α βi + (1 α)βi 2 t=1 i i=1 i=1 where w i = 1/ ˆβ i ν, ˆβ i is OLS or ridge, and ν > 0. ) ) 19 / 27
Still More Choices (Within Lasso) Adaptive elastic net α = 0.5 (equal weight to L 1 and L 2 ) OLS regression to obtain the weights w i ν = 1 10-fold cross validation to determine λ Separate cross validation for each VAR equation. 20 / 27
A Final Choice: Graphical Display via Spring Graphs Node size: Asset size Node color: Total directional connectedness to others Node location: Average pairwise directional connectedness (Equilibrium of repelling and attracting forces, where (1) nodes repel each other, but (2) edges attract the nodes they connect according to average pairwise directional connectedness to and from. ) Edge thickness: Average pairwise directional connectedness Edge arrow sizes: Pairwise directional connectedness to and from 21 / 27
Estimating Global Bank Network Connectedness Market-based approach: Balance sheet data are hard to get and rarely timely Balance sheet connections are just one part of the story Hard to know more than the market Daily range-based equity return volatilities Top 150 banks globally, by assets, 9/12/2003-2/7/2014 96 banks publicly traded throughout the sample 80 from 23 developed economies 14 from 6 emerging economies 22 / 27
Individual Bank Network, 2003-2014 23 / 27
Individual Bank / Sovereign Bond Network, 2003-2014 24 / 27
Estimating Time-Varying Connectedness Earlier: C(x, ( B, Σ) ) Ĉ x, M(ˆθ) ( Now: ) Ĉ t x, M(ˆθ t ) Yet another interesting issue/choice: Time-varying parameters: Explicit TVP model? Regime switching? Rolling? 25 / 27
Dynamic System-Wide Connectedness 150-Day Rolling Estimation Window 26 / 27
Conclusions: Connectedness Framework and Results Use network theory to summarize and visualize large VAR s, static or dynamic Directional, from highly granular to highly aggregated (Pairwise to or from ; total directional to or from ; total system-wide) For one asset class (stocks), network clustering is by country, not bank size For two asset classes (stocks and government bonds), clustering is first by asset type, and then by country Dynamically, there are interesting low-frequency and high-frequency connectedness fluctuations Most total connectedness changes are due to changes in pairwise connectedness for banks in different countries 27 / 27