Issues in
Connectedness Measurement
for Large Networks
Francis X. Diebold
April 4, 2014
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Diebold-Yilmaz Connectedness via Variance Decomposition
Connectedness Table
x2
...
xN
x1
x1
x2
..
.
H
d11
H
d21
..
.
H
d12
H
d22
..
.
···
···
..
.
H
d1N
H
d2N
..
.
xN
H
dN1
H
dN2
···
H
dNN
H
i6=2 di2
···
To
Others
H
i6=1 di1
P
H =
Ci←•
P
PN
H =
C•←j
CH =
1
N
PN
j=1
j6=i
i6=j
i6=N
H
diN
P
i6=j
dijH
dijH , are the “from degrees”
PN
i,j=1
P
From Others
P
dH
Pj6=1 1j
H
j6=2 d2j
..
P . H
j6=N dNj
i=1
i6=j
dijH , are the “to degrees”
dijH , is the mean degree (to or from)
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Static Pairwise Connectedness
Country Banking Systems, 2004-14, Adaptive Elastic Net
3/9
Dynamic Total Connectedness
Country Banking Systems, 2004-14, Adaptive Elastic Net
100
90
80
70
60
50
04
05
06
07
08
09
10
11
12
13
4/9
Network Analysis with a VAR Approximating Model
yt = Ayt−1 + εt
εt ∼ iid(0, Σ)
I
Diebold-Yilmaz variance decomposition-based (V ).
Based on variance decomposition matrix D = f (A).
What fraction of the H-step-ahead prediction-error variance of
variable i is due to shocks in variable j?
I
Brownlees et al. coefficient-based (C ).
Based directly on A.
How is yi,t connected to yj,t−1 ?
5/9
V vs. C Issues
I
V is readily calculated for any approximating model, not just
AR(1). Not obvious how to do C .
I
V requires an identifying assumption, whereas C does not.
But C just skirts the issue by ignoring Σ.
I
V facilitates examination of multi-step connectedness via H,
whereas C is tied to 1-step. (We could also do IRF’s.)
I
V cleanly breaks down connectedness from granular to
aggregate, with pieces adding to 100%. Not clear for C .
I
Sparse A estimation methods do not force a sparse D, since
D = f (A) for nonlinear f (·). Sparse A estimation methods do
force a sparse A (by construction). Hence V is more
appealing than C .
6/9
V and C Issues
I
Recovering degrees of freedom. Selection (e.g., forward
stepwise), shrinkage (e.g., ridge), or both (e.g., lasso).
I
System estimation methods vs. equation-by-equation
I
Time-variation: Smooth evolution (e.g. rolling) vs. sharp
breaks (e.g. fused ridge, fused lasso). Bandwidth selection.
I
Many flavors of lasso
(e.g., lasso, adaptive lasso, elastic net, adaptive elastic net)
7/9
Lasso I: Penalized Estimation
β̂ = arg min
β
T
X
!2
yt −
t=1
X
βi xit
s.t.
i
K
X
(|βi |q )1/q ≤ c
i=1
Equivalently,

β̂ = arg min 
β
T
X
t=1
!2
yt −
X
i
βi xit

K
X
+λ
(|βi |q )1/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.
8/9
Lasso II: Flavors of Lasso
β̂Lass


!2
T
K
X
X
X
= arg min 
yt −
βi xit
+λ
|βi |
β
t=1

β̂ALass = arg min 
β
β̂Enet
i
i=1
!2
T
X
yt −
X
t=1
βi xit
+λ
i
K
X

wi |βi |
i=1
where wi = 1/β̂iν , β̂i is OLS or ridge, and ν > 0


!2
T
K
X
X
X
= arg min 
yt −
βi xit
+λ
α|βi | + (1 − α)βi2 
β

β̂AEnet = arg min 
β
t=1
i
T
X
X
t=1
i=1
!2
yt −
i
βi xit
+λ
K
X

2
αwi |βi | + (1 − α)βi 
i=1
where wi = 1/β̂iν , β̂i is OLS or ridge, and ν > 0.
9/9