Spillover Dynamics for Systemic Risk Measurement Using Spatial Financial Time Series Models. Andre Lucas. Amsterdam - June, 25 2015. European Financial Management Association 2015 Annual Meetings.

1. Spillover DynamicsSpillover DynamicsSpillover DynamicsSpillover Dynamics forforforfor
SystemicSystemicSystemicSystemic RiskRiskRiskRisk MeasurementMeasurementMeasurementMeasurement
Using Spatial Financial TimeUsing Spatial Financial TimeUsing Spatial Financial TimeUsing Spatial Financial Time
Series ModelsSeries ModelsSeries ModelsSeries Models
SYstemic Risk TOmography:
Signals, Measurements, Transmission Channels, and
Policy Interventions
Francisco Blasques (a,b)
Siem Jan Koopman(a,b,c)
Andre Lucas(a,b,d)
Julia Schaumburg(a,b)
(a)VU University Amsterdam (b)Tinbergen Institute (c)CREATES (d)Duisenberg School of Finance
EFMA Nyenrode
June 2015

2. Introduction 3
This project: four main research questions
What is the effectiveness of non-standard monetary policy on
markets’ perceptions of sovereign credit risk interconnectedness?
Can international debt interconnections be identified empirically as
possible transmission channels for systemic risk using economic
distances in a spatial analysis?
Is the economic significance of these channels stable or does it vary
over time and with economic conditions?
What are the empirical cross-sectional interactions as well as
country-specific and Europe-wide credit risk factors for European
sovereigns?
Spatial GAS

3. Introduction 4
Systemic risk: multi-faceted
Direct interconnectedness via cross-exposures
Common asset exposures and fire-sales commonalities
Global imbalances
Liability vulnerability (stable funding ratios, CoCos, etc.)
Sovereign and financial sector feedback loops
Shadow banking feedback
Real effects . . . ?
Spatial GAS

4. Introduction 5
Systemic sovereign credit risk
Systemic risk: Breakdown risk of
the financial system induced by the
interdependence of its constituents.
European sovereign debt since 2009:
Strong increases and comovements of credit spreads.
Financial interconnectedness across borders due to mutual
borrowing and lending
+ bailout engagements.
⇒ Spillovers of shocks between member states.
⇒ Unstable environment: need for time-varying parameter models and
fat tails.
Spatial GAS

5. Introduction 6
Four European sovereign CDS spreads
Spatial GAS

6. Introduction 7
Other ‘goodies’ in this paper
New parsimonious econometric model for overall time-varying
strength of cross-sectional spillovers in credit spreads (systemic risk).
⇒ Useful for flexible monitoring of policy measure effects.
Dynamic spatial dependence model with time-varying parameters,
accounting for typical data properties in finance (fat-tails,
time-varying volatility): score driven models
Econometric theory: asymptotic and finite sample properties of the
ML estimator of this ’spatial score driven model’
Spatial GAS

7. Introduction 8
Main findings
Spill-over strength particularly down since the OMT
announcements and implementation by the ECB;
. . . earlier LTRO activity only caused temporary effects in
spill-over strength
Relating spill-over strength to cross-exposures of the
financial sectors improves the model’s fit.
Spill-over is an important channel, but the strength of this
channel varies over time.
Control variables have little impact on CDS spread
changess: ok signs, but lack of significance.
Robust to a number of variations in the specification.
Spatial GAS

8. Introduction 9
Related literature (partial and incomplete)
Systemic risk in sovereign credit markets:
Ang/Longstaff (2013), Lucas/Schwaab/Zhang (2013),
Aretzki/Candelon/Sy (2011), Kalbaska/Gatkowski (2012), De Santis
(2012), Caporin et al. (2014), Korte/Steffen (2013),
Kallestrup/Lando/Murgoci (2013), Beetsma et al. (2013, 2014).
Spatial econometrics:
General: Cliff/Ord (1973), Anselin (1988), Cressie (1993), LeSage/Pace
(2009), Ord (1975), Lee (2004), Elhorst (2003);
Panel data: Kelejian/Prucha (2010), Yu/de Jong/Lee (2008, 2012),
Baltagi et al. (2007, 2013), Kapoor/Kelejian/Prucha (2007);
Empirical finance: Keiler/Eder (2013), Fernandez (2011),
Asgarian/Hess/Liu (2013), Arnold/Stahlberg/Wied (2013), Wied (2012),
Denbee/Julliard/Li/Yuan (2013), Saldias (2013).
Spatial GAS

9. Spatial lag model 10
Modeling framework
Spatial GAS

10. Spatial lag model 11
Basic spatial lag model
Let y denote a vector of observations of a dependent variable for n units.
A basic spatial lag model of order one is given by
y = ρWy
’spatial lag’
+Xβ + e, e ∼ N(0, σ2
In), (1)
where
W is a nonstochastic (n × n) matrix of spatial weights with rows adding
up to one and with zeros on the main diagonal,
X is a (n × k)-matrix of covariates,
|ρ| < 1, σ2
> 0, and β = (β1, ..., βk ) are unknown coefficients.
Model (1) for observation i:
yi = ρ
n
j=1
wij yj +
K
k=1
xik βk + ei (2)
Spatial GAS

11. Spatial lag model 12
Spatial spillovers (LeSage/Pace (2009))
Rewriting model (1) as
y = (In − ρW )−1
Xβ + (In − ρW )−1
e (3)
and expanding the inverse matrix as a power series yields
y = Xβ + ρWXβ + ρ2
W 2
Xβ + · · · + e + ρWe + ρ2
W 2
e + · · ·
Implications:
The model is nonlinear in ρ.
Each unit with a neighbor is its own second-order neighbor.
The model can be interpreted as a structural VAR model with restricted
parameters.
Spatial GAS

12. Spatial lag model 13
Spatial models in empirical finance
Spatial lag models: Keiler/Eder (2013), Fernandez (2011),
Asgarian/Hess/Liu (2013), Arnold/Stahlberg/Wied (2013),
Wied (2012).
Spatial error models: Denbee/Julliard/Li/Yuan (2013),
Saldias (2013).
CDS application: Unrealistic to assume that systemic
sovereign credit risk is static over time.
So far, no model for time-varying spatial dependence
parameter in the literature.
Spatial GAS

13. Spatial GAS 14
Dynamic spatial dependence
Idea: Let the strength of spillovers ρ change over time.
GAS-SAR model for panel data, i = 1, ..., n, and t = 1, ..., T:
yt = ρt Wyt + Xt β + et , et ∼ pe (0, Σ), or
yt = Zt Xt β + Zt et ,
where Zt = (In − ρt W )−1
, and pe corresponds to the error distribution,
e.g. pe = N or pe = tν , with covariance matrix Σ.
The model can be estimated by maximizing
=
T
t=1
t =
T
t=1
(ln pe (yt − ρt Wyt − Xt β; λ) + ln |(In − ρt W )|) , (4)
where λ is a vector of variance parameters.
Ensure that ln |(In − ρt W )| exists: ρt = h(ft ) = γ tanh(ft ), γ < 1.
Spatial GAS

14. Spatial GAS 15
GAS dynamics for ρt
Reparamerization: ρt = h(ft ) = tanh(ft ).
ft is assumed to follow a dynamic process,
ft+1 = ω + ast + bft ,
where ω, a, b are unknown parameters.
We specify st as the first derivative (“score”) of the predictive likelihood
w.r.t. ft (Creal/Koopman/Lucas, 2013).
Model can be estimated straightforwardly by maximum likelihood (ML).
For theory and empirics on different GAS/DCS models, see also, e.g.,
Creal/Koopman/Lucas (2011), Harvey (2013), Harvey/Luati (2014),
Blasques/Koopman/Lucas (2012, 2014a, 2014b).
Spatial GAS

15. Spatial GAS 16
Score
Score for Spatial GAS model with normal errors:
εt = yt − ρt Wyt − Xt β
st = wt · yt W Σ−1
εt − tr(Zt W ) · h (ft )
wt = 1
Spatial GAS

16. Spatial GAS 17
Score
Score for Spatial GAS model with t-errors:
εt = yt − ρt Wyt − Xt β
st = wt · yt W Σ−1
εt − tr(Zt W ) · h (ft )
wt =
1 + n
ν
1+ 1
ν
εt Σ−1εt
Spatial GAS

17. Theory 18
Theory for Spatial GAS model
Extension of theoretical results on GAS models in
Blasques/Koopman/Lucas (2014a, 2014b):
Nonstandard due to nonlinearity of the model, particularly in the
case of Spatial GAS-t specification.
Conditions:
moment conditions;
b + a∂st
∂ft
is contracting on average.
Result: strong consistency and asymptotic normality of ML
estimator.
Also: Optimality results (see paper).
Spatial GAS

18. Theory 19
Asymptotic theory: Assumptions
Assumption
Let θ = (ω, a, b, β, λ), and Θ ⊂ R3+dβ +dλ
is a compact set. Assume that
1. the scaled score has Nf finite moments:
sup(λ,β)∈Λ×B E|s(f , yt, Xt; β, λ)|Nf
< ∞,
2. the contraction condition for the GAS update holds:
sup(f ,y,X,β,λ)∈R×Y×X×B×Λ |b + a ∂s(f ,y,X;β,λ)
∂f | < 1
3. Z, Z−1
, h, and log pe have bounded derivatives.
Spatial GAS

19. Theory 20
Asymptotic theory: Results
Theorem
(Consistency)
Let {yt }t∈Z and {Xt }t∈Z be stationary and ergodic sequences satisfying
E|yt |Ny
< ∞ and E|Xt |Nx
< ∞ for some Ny > 0 and Nx > 0. Furthermore, let
θ0 ∈ int(Θ) be the unique maximizer of ∞(θ) on Θ. Assume additionally that
Assumption holds. Then the MLE satisfies θT (f1)
a.s.
→ θ0 as T → ∞ for every
initialization value f1.
(Asymptotic Normality)
Under the above assumptions and some additional moment conditions,
√
T(θT (f1) − θ0)
d
→ N 0, I−1
(θ0)J (θ0)I−1
(θ0) as T → ∞,
where J (θ0) := E t (θ0) t (θ0) is the mean outer product of gradients and
I(θ0) := E t (θ0) is the Fisher information matrix.
Spatial GAS

20. Simulation 21
Simulation results (n = 9, T = 500)
0 100 200 300 400 500
0.00.40.8
Sine, dense W, t−errorsrho.t
0 100 200 300 400 500
0.00.20.40.60.81.0
Step, dense W, t−errors
rho.t
Spatial GAS

21. Simulation 22
Simulation results I
0 100 200 300 400 500
0.860.900.94
Constant, dense W, t−errors
rho.t
0 100 200 300 400 500
0.00.20.40.60.81.0
Sine, dense W, t−errors
rho.t
0 100 200 300 400 500
0.00.20.40.60.81.0
Fast sine, dense W, t−errors
rho.t
0 100 200 300 400 500
0.00.20.40.60.81.0
Step, dense W, t−errors
rho.t
0 100 200 300 400 500
0.00.20.40.60.81.0
Ramp, dense W, t−errors
rho.t
Spatial GAS

22. Simulation 23
Simulation: Consistency check
Simulate from GAS model, check whether parameters are
estimated consistently.
DGP:
yt = ZtXtβ + Ztet, et ∼ i.i.d.N(0, σ2
In).
Parameters: ω = 0.05, a = 0.05, b = 0.8, β = 1.5, and
σ2 = 2.
Sample sizes: N = 9, T = { 500, 1000, 2000} .
500 replications.
Spatial GAS

23. Simulation 24
Simulation results II
0.02 0.04 0.06 0.08 0.10
010203040506070
Density of estimates for ω, true value=0.05
N = 500 Bandwidth = 0.002946
Density
T=500
T=1000
T=2000
0.040 0.045 0.050 0.055 0.060
050150250350
Density of estimates for a, true value=0.0.05
N = 500 Bandwidth = 0.0006868
Density
T=500
T=1000
T=2000
0.70 0.75 0.80 0.85 0.90
010203040
Density of estimates for b, true value=0.8
N = 500 Bandwidth = 0.00627
Density
T=500
T=1000
T=2000
1.46 1.48 1.50 1.52 1.54
0102030405060
Density of estimates for β, true value=1.5
N = 500 Bandwidth = 0.003423
Density
T=500
T=1000
T=2000
1.85 1.90 1.95 2.00 2.05 2.10 2.15
05101520
Density of estimates for σ2
, true value=2
N = 500 Bandwidth = 0.01108
Density
T=500
T=1000
T=2000
Spatial GAS

24. Application 25
Systemic risk in European credit spreads:
Data
Daily log changes in CDS spreads from February 2, 2009 - May 12,
2014 (1375 observations).
8 European countries: Belgium, France, Germany, Ireland, Italy,
Netherlands, Portugal, Spain.
Country-specific covariates (lags):
returns from leading stock indices,
changes of 10-year government bond yields.
Europe-wide control variables (lags):
term spread: difference between three-month Euribor and EONIA,
interbank interest rate: change in three-month Euribor,
change in volatility index VSTOXX.
Spatial GAS

25. Application 26
Five European sovereign CDS spreads
2009 2010 2011 2012 2013 2014
20040060080010001200
spread(bp)
Ireland
Spain
Belgium
France
Germany
average correlation of log changes = 0.65
Spatial GAS

26. Application 27
Spatial weights matrix
Idea: Sovereign credit risk spreads are (partly) driven by cross-border debt
interconnections of the financial sector (see, e.g. Korte/Steffen (2013),
Kallestrup et al. (2013)).
Intuition: European banks are not required to hold capital buffers against
EU member states’ debt (’zero risk weight’).
If sovereign credit risk materializes, banks become undercapitalized, so
that bailouts by domestic governments are likely, affecting their credit
quality.
Entries of W : Three categories (high - medium - low) of cross-border
exposures in 2008.∗
∗Source: Bank for International Settlements statistics, Table 9B: International
bank claims, consolidated - immediate borrower basis.
Spatial GAS

27. Application 28
Empirical model specifications
model mean equation errors et ∼
(0, σ2
In) (0, Σt)
Static spatial yt = ρWyt + Xtβ + et N, t
Sp. GAS yt = h(f ρ
t )Wyt + Xtβ + et N, t t
Sp. GAS+mean fct. yt = ZtXtβ + λf λ
t + Ztet t t
Benchmark yt = Xtβ + λf λ
t + et t
Spatial GAS

28. Application 29
Model fit comparison
Static spatial Time-varying spatial
et ∼ N(0, σ2In) tλ(0, σ2In) N(0, σ2In) tλ(0, σ2In)
logL -26396.63 -24574.48 -26244.45 -24506.11
AICc 52807.35 49165.06 52507.03 49032.39
Time-varying spatial-t Benchmark-t
(+tv. volas) (+tv.volas) (+tv.volas)
(+mean f.) (+mean f.)
logL -24175.70 -24156.96 -26936.15
AICc 48389.97 48375.30 53927.42
Spatial GAS

29. Application 30
Parameter estimates
Spatial dependence is high and significant.
Spatial GAS parameters:
High persistence of dynamic factors reflected by large
estimates for b.
Estimates for score impact parameters a are small but
significant.
Estimates for β have expected signs.
Mean factor loadings:
Positive for Ireland, Portugal, Spain.
Negative for Belgium, Italy, France, Germany, Netherlands.
Spatial GAS

30. Application 31
Estimation results: Full model
ωλ
-0.0012 ωσ
1 Belgium 0.0426 ω 0.0307
Aλ
0.3494 ωσ
2 France 0.0448 A 0.0190
Bλ
0.6891 ωσ
3 Germany 0.0573 B 0.9636
λ1 Belgium -0.2776 ωσ
4 Ireland 0.0301 const. -0.0621
λ2 France -0.2846 ωσ
5 Italy 0.0471 VStoxx -0.0257
λ3 Germany -0.2029 ωσ
6 Netherlands 0.0443 term sp. 0.0693
λ4 Ireland 0.4050 ωσ
7 Portugal 0.0524 stocks -0.1020
λ5 Italy -0.1604 ωσ
8 Spain 0.0591 yields 0.0173
λ6 Netherlands -0.1891 Aσ
0.1826 λ0 3.1357
λ7 Portugal 0.4614 Bσ
0.9479
λ8 Spain 0.0988 logLik -24156.96
AICc 48375.30
Spatial GAS

31. Application 32
Residual diagnostics: Full model
Test for remaining autocorrelation and ARCH effects in standardized residuals
from full model (Spatial GAS+volas+mean factor)
sovereign LB test stat. ARCH LM test stat. average cross-corr.
raw residuals raw residuals raw residuals
Belgium 108.64 15.93 169.91 25.53 0.70 0.07
France 49.48 30.42 160.44 43.32 0.66 -0.01
Germany 62.61 19.49 142.70 53.78 0.63 -0.07
Ireland 129.89 17.53 302.23 87.11 0.64 -0.07
Italy 99.02 42.43 102.13 150.88 0.71 0.08
Netherlands 55.69 33.29 124.41 20.96 0.64 -0.05
Portugal 167.91 32.56 189.35 56.89 0.65 0.03
Spain 105.81 48.88 253.68 154.42 0.69 0.06
Spatial GAS

32. Application 33
Different choices of W
Candidates (all row-normalized):
Raw exposure data (constant): Wraw
Raw exposure data (updated quarterly): Wdyn
Three categories of exposure amounts (high, medium, low): Wcat
Exposures standardized by GDP: Wgdp
Geographical neighborhood (binary, symmetric): Wgeo
Spatial GAS

33. Application 33
Different choices of W
Candidates (all row-normalized):
Raw exposure data (constant): Wraw
Raw exposure data (updated quarterly): Wdyn
Three categories of exposure amounts (high, medium, low): Wcat
Exposures standardized by GDP: Wgdp
Geographical neighborhood (binary, symmetric): Wgeo
Model fit comparison (only t-GAS model):
Wraw Wdyn Wcat Wgeo
logL -24745.56 -24679.44 -24506.11 -25556.85
Parameter estimates are robust.
Spatial GAS

34. Application 34
Spillover strength 2009-2014
Spatial GAS

35. Conclusions 35
Conclusions
Decrease of systemic risk from mid-2012 onwards; possibly
due to believable EU governments’ and ECB’s measures
European sovereign CDS spreads are strongly spatially
dependent via cross-exposure channel, but the channel’s
strength may vary over time
Spatial model with dynamic spillover strength and fat tails is
new, and it works (theory, simulation, empirics).
Best model: Time-varying spatial dependence based on
t-distributed errors, time-varying volatilities, additional mean
factor, and categorical spatial weights.
Spatial GAS

37. This project has received funding from the European Union’s
Seventh Framework Programme for research, technological
development and demonstration under grant agreement no° 320270
www.syrtoproject.eu
This document reflects only the author’s views.
The European Union is not liable for any use that may be made of the information contained therein.