estimating global bank network connectedness · pdf filereading and web materials two papers:...
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Estimating Global Bank Network Connectedness
Mert Demirer (MIT)Francis X. Diebold (Penn)
Kamil Yılmaz (Koc)
September 18, 2014
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Financial and Macroeconomic Connectedness
I Market Risk, Portfolio Concentration Risk(return connectedness)
I Credit Risk(default connectedness)
I Counterparty Risk, Gridlock Risk(bilateral and multilateral contractual connectedness)
I Systemic Risk(system-wide connectedness)
I Business Cycle Risk(local or global real output connectedness)
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A Natural Financial/Economic Connectedness Question:
What fraction of the H-step-ahead prediction-error variance ofvariable i is due to shocks in variable j , j 6= i?
Non-own elements of the variance decomposition: dHij , j 6= i
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Reading and Web Materials
Two Papers:
Diebold, F.X. and Yilmaz, K. (2014), “On the Network Topology ofVariance Decompositions: Measuring the Connectedness of FinancialFirms,” Journal of Econometrics, 182, 119-134.
Demirer, M., Diebold, F.X. and Yilmaz, K. (2014), “Estimating GlobalBank Network Connectedness,” Manuscript, in progress.
Book in Press:
Diebold, F.X. and Yilmaz, K. (2015), Financial and Macroeconomic
Connectedness: A Network Approach to Measurement and Monitoring,
Oxford University Press, in press. With K. Yilmaz.
www.FinancialConnectedness.org
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Network Representation: Graph and Matrix
A =
0 1 1 1 1 01 0 0 1 1 01 0 0 1 0 11 1 1 0 0 01 1 0 0 0 10 0 1 0 1 0
Symmetric adjacency matrix AAij = 1 if nodes i , j linkedAij = 0 otherwise
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Network Connectedness: The Degree Distribution
Degree of node i, di :
di =N∑j=1
Aij
Discrete degree distribution on 0, ..., N − 1
Mean degree, E (d), is the key connectedness measure
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Network Representation II (Weighted, Directed)
A =
0 .5 .7 0 0 00 0 0 0 .3 00 0 0 .7 0 .3.3 .5 0 0 0 0.5 0 0 0 0 .30 0 0 0 0 0
“to i , from j”
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Network Connectedness II: The Degree Distribution(s)
Aij ∈ [0, 1] depending on connection strength
Two degrees:
d fromi =
N∑j=1
Aij
d toj =
N∑i=1
Aij
“from-degree” and “to-degree” distributions on [0,N − 1]
Mean degree remains the key connectedness measure
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Variance Decompositions as Weighted, Directed Networks
Variance Decomposition / Connectedness Table
x1 x2 ... xN From Others
x1 dH11 dH
12 · · · dH1N
∑j 6=1 d
H1j
x2 dH21 dH
22 · · · dH2N
∑j 6=2 d
H2j
......
.... . .
......
xN dHN1 dH
N2 · · · dHNN
∑j 6=N dH
Nj
ToOthers
∑i 6=1 d
Hi1
∑i 6=2 d
Hi2 · · ·
∑i 6=N dH
iN
∑i 6=j d
Hij
Total directional connect. “from,” CHi←• =
∑Nj=1
j 6=idHij : “from-degrees”
Total directional connect. “to,” CH•←j =
∑Ni=1i 6=j
dHij : “to-degrees”
Systemwide connect., CH = 1N
∑Ni,j=1
i 6=jdHij : mean degree
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Relationship to MES
MES j |mkt = E (rj |C(rmkt))
I Sensitivity of firm j ’s return to extreme market event C
I Market-based “stress test” of firm j ’s fragility
“Total directional connectedness from” (from-degrees)
“From others to j”
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Relationship to CoVaR
VaRp : p = P (r < −VaRp)
CoVaRp,j |i : p = P(rj < −CoVaRp,j |i | C (ri )
)CoVaRp,mkt|i : p = P
(rmkt < −CoVaRp,mkt|i | C (ri )
)
I Measures tail-event linkages
I Leading choice of C (ri ) is a VaR breach
“Total directional connectedness to” (to-degrees)
“From i to others”
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Estimating Global Bank Network Connectedness
I Daily range-based equity return volatilities
I Top 150 banks globally, by assets, 9/12/2003 - 2/7/2014
I 96 banks publicly traded throughout the sampleI 80 from 23 developed economiesI 14 from 6 emerging economies
I Market-based approach:
I Balance sheet data are hard to get and rarely timelyI Balance sheet connections are just one part of the storyI Hard to know more than the market
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Many Interesting Issues / Choices
I Approximating model: VAR? Structural DSGE?
I Identification of variance decompositions:Cholesky? Generalized? SVAR? DSGE?
I Time-varying connectedness: Rolling estimation? SmoothTVP’s? Regime switching?
I Estimation: Classical? Bayesian? Hybrid?
I Selection: Information criteria? Stepwise? Lasso?I Shrinkage: BVAR? Ridge? Lasso?
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Selection and Shrinkage via Penalized Estimationof High-Dimensional Approximating Models
β = argminβ
T∑t=1
(yt −
∑i
βixit
)2
s.t.K∑i=1
|βi |q ≤ c
β = argminβ
T∑t=1
(yt −
∑i
βixit
)2
+ λK∑i=1
|βi |q
Concave penalty functions non-differentiable at the origin produceselection. Smooth convex penalties produce shrinkage. q → 0produces selection, q = 2 produces ridge, q = 1 produces lasso.
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Lasso
βLasso = argminβ
T∑t=1
(yt −
∑i
βixit
)2
+ λ
K∑i=1
|βi |
βALasso = argminβ
T∑t=1
(yt −
∑i
βixit
)2
+ λ
K∑i=1
wi |βi |
βEnet = argminβ
T∑t=1
(yt −
∑i
βixit
)2
+ λ
K∑i=1
(α|βi |+ (1− α)β2
i
)βAEnet = argminβ
T∑t=1
(yt −
∑i
βixit
)2
+ λ
K∑i=1
(αwi |βi |+ (1− α)β2
i
)where wi = 1/βν
i , βi is OLS or ridge, and ν > 0.
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Still More Choices (Within Lasso)
I Adaptive elastic net
I α = 0.5 (equal weight to L1 and L2)
I OLS regression to obtain the weights wi
I ν = 1
I 10-fold cross validation to determine λ
I Separate cross validation for each VAR equation.
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A Final Choice: Graphical Display via “Spring Graphs”
I Node size: Asset size
I Node color: Total directional connectedness “to others”
I Node location: Average pairwise directional connectedness(Equilibrium of repelling and attracting forces, where (1) nodes repeleach other, but (2) edges attract the nodes they connect accordingto average pairwise directional connectedness “to” and “from.”)
I Edge thickness: Average pairwise directional connectedness
I Edge arrow sizes: Pairwise directional connectedness “to” and“from”
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Individual Bank Network, 2003-2014
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Individual Bank / Sovereign Bond Network, 2003-2014
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Dynamic System-Wide Connectedness
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Conclusions: Connectedness Framework and Results
I Flexible approximating models, static or dynamic estimation
I Firmly grounded in network theory
I Directional, from highly granular to highly aggregated
(Pairwise “to” or “from”; total directional “to or “from”;systemwide)
I Results
I MES and CoVaR perspectives are effectively special cases
I Substantive results for global bank equity volatilityconnectedness
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