Spatial GAS Models for Systemic Risk Measurement - Blasques F., Koopman S.J., Lucas A., Schaumburg J. January, 9 2014

1. Spatial GAS Models for
Systemic Risk Measurement
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
Workshop on Dynamic Models driven by the Score of
Predictive Likelihoods
La Laguna, January 9-11, 2014

2. 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.

3. Introduction 3
Introduction
Strong increases and comovements of sovereign credit spreads since
the beginning of the European debt crisis in 2009.
Common currency area: Mutual borrowing and lending leads to
financial interconnectedness across borders.
Shocks that affect the credit quality of a member country are likely
to spill over to the other members, possibly creating feedback loops
⇒ Systemic risk.
Suitable models should capture complex correlation dynamics and
feedback effects, but be empirically tractable and intuitively
interpretable.
Spatial GAS

4. Introduction 4
European sovereign credit spread dynamics:
Some related literature
Contagion/comovement of sovereign credit spreads:
Kalbaska/Gatkowski (2012), Caporin et al. (2013), Aretzki et
al. (2011), Lucas/Schwaab/Zhang (2013), Ang/Longstaff
(2013), Metiu (2012), Favero (2013), De Santis (2012),
Constancio (2012).
Sovereign credit spreads vs. banks’ aggregate foreign
exposures: Kallestrup et al. (2013), Korte/Steffen (2013),
Beetsma et al. (2013).
Spatial GAS

5. Introduction 5
This project: New dynamic spatial model for
sovereign credit spreads
Joint model for European sovereign credit spreads, accounting for
cross-sectional interactions of units as well as country-specific, and
Europe-wide credit risk factors.
Transmission channels are defined explicitly as economic distances
in a spatial weights matrix of international debt interconnections.
Single measure of the degree of comovement, the spatial
dependence parameter, follows a generalized autoregressive score
(GAS) process.
Asymptotic and finite sample properties of the ML estimator of this
’Spatial GAS model’.
Spatial GAS

6. Outline 6
Outline
1. Introduction
2. Basic spatial lag model
3. Spatial lag model with GAS dynamics
4. Consistency of the Spatial GAS model
5. Simulation
6. Application: European CDS dynamics
7. Conclusions, Outlook
Spatial GAS

7. Spatial lag model 7
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

8. Spatial lag model 8
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.
Spatial GAS

9. Spatial lag model 9
Some related literature: Spatial econometrics
Cliff/Ord (1973), Anselin (1988), Cressie (1993), LeSage/Pace (2009);
Spatial panel models: Lee/Yu (2010a), Anselin/Le Gallo/Jayet (2008),
Kelejian/Prucha (2010), Kukenova/Monteiro (2008);
Spatial lag panel models:
Fixed effects: Yu/de Jong/Lee (2008, 2012), Lee/Yu (2010b,
2010c, 2012);
Random effects: Baltagi et al. (2007, 2013),
Kapoor/Kelejian/Prucha (2007), Mutl/Pfaffermayr (2011);
Maximum likelihood estimation of spatial lag models: Ord (1975), Lee
(2004), Hillier/Martellosio (2013);
Spatial error models: y = Xβ + e, e = We + u
e.g. Anselin/Bera (1998), Kelejian/Robinson (1995), Anselin/Moreno
(2003), Chudik/Pesaran (2013).
Spatial GAS

10. Spatial lag model 10
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).
! So far, no model for time-varying spatial dependence
parameter in the literature (t.t.b.o.o.k.).
Spatial GAS

11. Spatial GAS 11
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 = ρtWyt + Xtβ + et, et ∼ pe(0, Σ), or
yt = ZtXtβ + Ztet,
where Zt = (In − ρtW )−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 − ρtWyt − Xtβ; λ) + ln |(In − ρtW )|) ,
(4)
where λ is a vector of variance parameters.
Spatial GAS

12. Spatial GAS 12
GAS dynamics for ρt
To ensure that ln |(In − ρtW )| exists, we use ρt = h(ft) = tanh(ft).
ft is assumed to follow a GAS(1,1) process, see Creal et al. (2011,
2013), and Harvey (2013):
ft+1 = ω + ast + bft, (5)
where ω, a, b are unknown parameters, and st is the scaled score of
the log likelihood function,
st = St t. (6)
For simplicity, we use the unity matrix as scaling function, i.e.
St = 1.
Spatial GAS

13. Spatial GAS 13
Normally distributed error terms
Likelihood:
t = ln |Z−1
t | −
n
2
ln(2π) −
1
2
ln |Σ|
−
1
2
(yt − h(ft )Wyt − Xt β) Σ−1
(yt − h(ft )Wyt − Xt β)
Score:
t = yt W Σ−1
(yt − h(ft )Wyt − Xt β) − tr(Zt W ) · h (ft )
with Zt = (In − h(ft )W )−1
and h (ft ) = 1 − tanh2
(ft ).
Spatial GAS

14. Spatial GAS 14
t-distributed error terms
Likelihood:
t = ln |Z−1
t | + ln
Γ ν+n
2
|Σ|1/2(νπ)n/2Γ ν
2
+ −
ν + n
2
ln 1 +
(yt − h(ft )t Wyt − Xt β) Σ−1(yt − h(ft )Wyt − Xt β)
ν
.
Score:
t =
(1 + n
ν
)yt W Σ−1(yt − h(ft )Wyt − Xt β)
1 + 1
ν
(yt − h(ft )Wyt − Xt β) Σ−1(yt − h(ft )Wyt − Xt β)
− tr(Zt W ) · h (ft )
with h (ft ) = 1 − tanh2
(ft ).
Spatial GAS

15. Consistency 15
Consistency of the Spatial GAS estimator
Assumption
Let θ = (ω, a, b, β, λ), and Θ∗
⊂ R3+dβ +dλ
. Assume that
1. the scaled score has Nf finite moments:
supλ∈Λ,β∈B E|s(f , yt, Xt; β, λ)|Nf
< ∞,
2. the contraction condition for the GAS update holds:
sup(f ,y,X,θ)∈R×Y×X×Θ∗ |b + a ∂s(f ,y,X;λ)
∂f | < 1
3. Z, Z−1
, h, and log pe have bounded derivatives.
Spatial GAS

16. Consistency 16
Consistency of the Spatial GAS estimator
Theorem
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 and assume
that 1.-3. in Assumption hold.
Furthermore, let θ0 ∈ Θ be the unique maximizer of ∞(θ) on the
parameter space Θ ⊆ Θ∗
.
Then the MLE satisfies θT (f1)
a.s.
→ θ0 as T → ∞ for every initialization
value f1.
Spatial GAS

17. Simulation 17
Simulation: Spatial GAS model
Data generating process:
yt = Ztet, et ∼ i.i.d.N(0, In),
where Zt = (In − ρtW )−1
, and t = 1, ..., 500.
Weights matrices (row-normalized):
1. ’sparse’: neighborhood of 9 European countries (binary)
2. ’dense’: cross-border debt of 9 European countries (BIS data)
Spatial dependence processes (Engle 2002):
1. ’sine’ ρt = 0.5 + 0.4 cos(2πt/200)
2. ’step’ ρt = 0.9 − 0.5 ∗ I(t > T/2),
Spatial GAS

18. Simulation 18
Simulation results: DGP 1 (200 replications)
0 100 200 300 400 500
0.00.20.40.60.81.0
Sparse W, MSE=0.00842
rho.t
0 100 200 300 400 500
0.00.20.40.60.81.0
Dense W, MSE=0.01139
rho.t
Spatial GAS

19. Simulation 19
Simulation results: DGP 2 (200 replications)
0 100 200 300 400 500
0.00.20.40.60.81.0
Sparse W, MSE=0.00895
rho.t
0 100 200 300 400 500
0.00.20.40.60.81.0
Dense W, MSE=0.01088
rho.t
Spatial GAS

20. Application 20
Data
Daily relative CDS changes from November 7, 2008 - September 30,
2013 (1277 observations)
9 European countries: Belgium, France, Germany, Ireland, Italy,
Netherlands, Portugal, Spain, United Kingdom
Country-specific covariates (lags):
returns from leading stock indices
changes of 10-year government bond yields
Euro area-wide control variables (lags):
risk appetite: differences between implied volatility index VStoxx
and GARCH(1,1) volatility estimates of Eurostoxx 50
term spread: differences between three-month Euribor and EONIA
interbank interest rate: changes in three-month Euribor
Spatial GAS

21. Application 21
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’). This can lead to
regulatory arbitrage incentives and
excessive issuing of of sovereign debt.
If sovereign credit risk materializes, banks become undercapitalized and
bailouts by domestic governments may be necessary, which in turn affects
their credit quality.
Entries of W : Row-standardized averages of quarterly across-the-border
debt exposures (Million US-$). Source: BIS homepage, Table 9B:
International bank claims, consolidated - immediate borrower basis.
Spatial GAS

22. Application 22
Estimation results
Model: yi,t = ρt
n
j=1 wij yj,t + 6
k=1 xt,k βk + ei,t , ei,t ∼ i.i.d.pe (0, σ2
)
x1: const., x2: risk app., x3: term spread, x4: interest rate, x5: stock ind., x6: yield
Static model GAS model
N tν N tν
ρ 0.6464 0.6123
ω 0.0101 0.0075
a 0.0091 0.0092
b 0.9852 0.9888
σ2 12.3124 2.8238 12.1195 2.8211
β1 0.1540 -0.0388 0.1001 -0.0661
β2 -1.0612 0.0038 -0.8996 0.0439
β3 0.4316 0.1337 0.3982 0.1642
β4 0.0642 0.0647 0.0648 0.0761
β5 -0.2260 -0.1267 -0.2123 -0.1253
β6 0.0902 0.0450 0.0916 0.0461
ν 2 2
logL -31287.93 -28688.66 -31151.42 -28622.93
AICc 62598.07 57399.54 62325.04 57268.07
Spatial GAS

23. Application 23
Estimation results: Categorical W
wij =



1 if 0 < wraw,ij ≤ Q0.25(Wraw )
2 if Q0.25(Wraw ) ≤ wraw,ij < Q0.5(Wraw )
3 if Q0.5(Wraw ) ≤ wraw,ij < Q0.75(Wraw )
4 if wraw,ij ≥ Q0.75(Wraw )
Static model GAS model
N tν N tν
ρ 0.6858 0.6622
ω 0.0096 0.0085
a 0.0096 0.0094
b 0.9874 0.9888
σ2
11.4287 2.6641 11.3410 2.6679
β1 0.0999 -0.0347 0.0418 -0.0591
β2 -0.7318 0.0307 -0.5728 0.0739
β3 0.3118 0.0887 0.3036 0.1135
β4 0.0497 0.0463 0.0527 0.0554
β5 -0.1997 -0.1133 -0.1903 -0.1128
β6 0.0837 0.0436 0.0840 0.0448
ν 2 2
logL -30926.16 -28372.33 -30826.08 -28326.93
AICc 61874.53 56766.87 61674.36 56675.23
Spatial GAS

24. Application 24
Filtered spatial parameter
2009 2010 2011 2012 2013
0.40.50.60.70.80.9
rho_t
Spatial GAS

25. Application 25
Estimation results: 2 sub-periods, t-errors, cat. W
Period 1: November 6, 2008 - March 30, 2012
Period 2: April 2, 2012 - September 30, 2013.
Static model GAS model
period 1 period 2 period 1 period 2
ρ 0.6960 0.5340
ω 0.1919 0.0144
a 0.0184 0.0113
b 0.7756 0.9741
σ 3.0030 2.1623 2.9964 2.1676
β1 0.0236 0.0899 0.0172 0.0844
β2 0.0695 -1.4151 0.0913 -1.5038
β3 -0.0232 0.8400 -0.0232 0.9779
β4 0.0488 0.0394 0.0463 0.0477
β5 -0.1089 -0.1493 -0.1070 -0.1499
β6 0.0476 0.0396 0.0487 0.0392
ν 2 2 2 2
AICc 40294.7 16289.52 40280.19 16277.72
Spatial GAS

26. Application 26
Conclusions
Spatial GAS model is new, and it works (theory, simulation).
European sovereign CDS spreads are spatially dependent.
Suitable spillover channel: debt interconnections.
Best model: Spatial GAS with t-distributed errors and
categorical spatial weights.
Some evidence for a level shift in spatial dependence after
Greek default (winter 2012).
Spatial GAS

27. Outlook 27
Outlook
Theory:
asymptotic normality of ML parameter estimator.
Simulation:
more DGPs for ρt,
t-distributed errors.
Sovereign CDS application:
check conditions implied by theory,
significance of covariates,
volatility clustering,
other choices of W .
Other application(s).
Spatial GAS