Spillover Dynamics for Systemic Risk Measurement Using Spatial Financial Time Series 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) Julia Schaumburg (a,b) (a)VU University Amsterdam (b)Tinbergen Institute (c)CREATES

ESEM Toulouse, August 25-29, 2014

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.

Introduction 3

Measuring systemic sovereign credit risk

Systemic risk: Breakdown risk of thefinancial system, induced by theinterdependence of its constituents.

European sovereign debt since 2009:

I Strong increases and comovements of credit spreads.

I 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 andfat tails.

Spillover Dynamics

Introduction 3

Measuring systemic sovereign credit risk

Systemic risk: Breakdown risk of thefinancial system, induced by theinterdependence of its constituents.

European sovereign debt since 2009:

I Strong increases and comovements of credit spreads.

I 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 andfat tails.

Spillover Dynamics

Introduction 3

Measuring systemic sovereign credit risk

Systemic risk: Breakdown risk of thefinancial system, induced by theinterdependence of its constituents.

European sovereign debt since 2009:

I Strong increases and comovements of credit spreads.

I 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 andfat tails.

Spillover Dynamics

Introduction 4

This project

I New parsimonious model for overall time-varying strength ofcross-sectional spillovers in credit spreads (systemic risk).⇒ Useful for flexible monitoring of policy measure effects.

I Extension of widely used spatial lag model to generalizedautoregressive score (GAS) dynamics and fat tails in financial data.

I Asymptotic theory and assessment of finite sample performance ofthis ’Spatial GAS model’.

Spillover Dynamics

Introduction 5

European sovereign systemic risk 2009-2014

Mario Draghi: „Whatever it takes“

Ireland bailed out Help offer to Greece

First LTRO Second LTRO

ESM inaugurated

Greece : record deficit

New supervisory authority

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Introduction 6

Some related literature

I Systemic risk in sovereign credit markets:

. Ang/Longstaff (2013), Lucas/Schwaab/Zhang (2013),

Ait-Sahalia/Laeven/Pelizzon (2014), 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).

I 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).

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Spatial GAS model 7

Spatial lag model for panel data

yi,t = ρt

n∑j=1

wijyj,t +K∑

k=1

xik,tβk + ei,t , ei,t ∼ tν(0, σ2)

where

I |ρt | < 1 is time-varying spatial dependence parameter,

I wij , j = 1, ..., n, are nonstochastic spatial weights adding up to one with wii = 0,

I xik,t , k = 1, ...,K are individual-specific regressors,

I βk , k = 1, ...,K , σ2 and ν are unknown coefficients.

Matrix notation:

yt = ρt Wyt︸︷︷︸’spatial lag’

+Xtβ + et or

yt = ZtXtβ + Ztet , with Zt = (In − ρtW )−1.

⇒ Model is highly nonlinear and captures feedback.

Spillover Dynamics

Spatial GAS model 7

Spatial lag model for panel data

yi,t = ρt

n∑j=1

wijyj,t +K∑

k=1

xik,tβk + ei,t , ei,t ∼ tν(0, σ2)

where

I |ρt | < 1 is time-varying spatial dependence parameter,

I wij , j = 1, ..., n, are nonstochastic spatial weights adding up to one with wii = 0,

I xik,t , k = 1, ...,K are individual-specific regressors,

I βk , k = 1, ...,K , σ2 and ν are unknown coefficients.

Matrix notation:

yt = ρt Wyt︸︷︷︸’spatial lag’

+Xtβ + et or

yt = ZtXtβ + Ztet , with Zt = (In − ρtW )−1.

⇒ Model is highly nonlinear and captures feedback.

Spillover Dynamics

Spatial GAS model 8

GAS dynamics for ρt

I Reparameterization: ρt = h(ft) = tanh(ft).

I ft is assumed to follow a dynamic process,

ft+1 = ω + ast + bft ,

where ω, a, b are unknown parameters.

I We specify st as the first derivative (“score”) of the predictive likelihoodw.r.t. ft (Creal/Koopman/Lucas, 2013).

I Model can be estimated straightforwardly by maximum likelihood (ML).

I 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).

Spillover Dynamics

Spatial GAS model 9

Score

Score for Spatial GAS model with normal errors:

st =

((1 + n

ν)y ′tW

′Σ−1(yt − h(ft)Wyt − Xtβ)

1 + 1ν

(yt − h(ft)Wyt − Xtβ)′Σ−1(yt − h(ft)Wyt − Xtβ)− tr(ZtW )

)· h′(ft)

Spillover Dynamics

Spatial GAS model 10

Score

Score for Spatial GAS model with t-errors:

st =

((1 + n

ν)y ′tW

′Σ−1(yt − h(ft)Wyt − Xtβ)

1 + 1ν

(yt − h(ft)Wyt − Xtβ)′Σ−1(yt − h(ft)Wyt − Xtβ)− tr(ZtW )

)· h′(ft)

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Theory 11

Theory for Spatial GAS model

I Extension of theoretical results on GAS models inBlasques/Koopman/Lucas (2014a, 2014b).

I Nonstandard due to nonlinearity of the model, particularly in thecase of Spatial GAS-t specification.

I Conditions:

. moment conditions;

. b + a ∂st∂ftis contracting on average.

I Result: strong consistency and asymptotic normality of MLestimator.

I Also: Optimality results (see paper).

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Simulation 12

Simulation results (n = 9, T = 500)

0 100 200 300 400 500

0.0

0.4

0.8

Sine, dense W, t−errorsrh

o.t

0 100 200 300 400 500

0.0

0.2

0.4

0.6

0.8

1.0

Step, dense W, t−errors

rho.

t

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Application 13

Systemic risk in European credit spreads:Data

I Daily log changes in CDS spreads from February 2, 2009 - May 12,2014 (1375 observations).

I 8 European countries: Belgium, France, Germany, Ireland, Italy,Netherlands, Portugal, Spain.

I Country-specific covariates (lags):

. returns from leading stock indices,

. changes in 10-year government bond yields.

I Europe-wide control variables (lags):

. term spread: difference between three-month Euribor and EONIA,

. change in volatility index VSTOXX.

Spillover Dynamics

Application 14

Five European sovereign CDS spreads

2009 2010 2011 2012 2013 2014

200

400

600

800

1000

1200

spre

ad (

bp)

IrelandSpainBelgiumFranceGermany

average correlation of log changes = 0.65

Spillover Dynamics

Application 15

Spatial weights matrix

I Idea: Sovereign credit risk spreads are (partly) driven by cross-border debtinterconnections of financial sectors (see, e.g. Korte/Steffen (2013),Kallestrup et al. (2013)).

I Intuition: European banks are not required to hold capital buffers againstEU member states’ debt (’zero risk weight’).

I If sovereign credit risk materializes, banks become undercapitalized, sothat bailouts by domestic governments are likely, affecting their creditquality.

I 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.

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Application 16

Empirical model specifications

model mean equation errors et ∼

(0, σ2In) (0,Σt)

Static spatial yt = ρWyt + Xtβ + et N, t

Sp. GAS yt = h(f ρt )Wyt + Xtβ + et N, t t

Sp. GAS+mean fct. yt = ZtXtβ + λf λt + Ztet t

Benchmark yt = Xtβ + λf λt + et t

Spillover Dynamics

Application 17

Model fit comparison

Static spatial Time-varying spatial

et ∼ N(0, σ2In) tν(0, σ2In) N(0, σ2In) tν(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) (+mean f.+tv.volas) (+mean f.+tv.volas)

logL -24175.70 -24156.96 -26936.15

AICc 48389.97 48375.30 53927.42

Spillover Dynamics

Application 18

Parameter estimates

I Spatial dependence is high and significant.

I Spatial GAS parameters:

. High persistence of dynamic factors reflected by largeestimates for b.

. Estimates for score impact parameters a are small butsignificant.

I Estimates for β have expected signs.

I Mean factor loadings:

. Positive for Ireland, Portugal, Spain.

. Negative for Belgium, France, Germany, Netherlands, Italy.

Spillover Dynamics

Application 19

Different choices of W

Candidates (all row-normalized):

I Raw exposure data (constant): Wraw

I Raw exposure data (updated quarterly): Wdyn

I Three categories of exposure amounts (high, medium, low): Wcat

I 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.

Spillover Dynamics

Application 19

Different choices of W

Candidates (all row-normalized):

I Raw exposure data (constant): Wraw

I Raw exposure data (updated quarterly): Wdyn

I Three categories of exposure amounts (high, medium, low): Wcat

I 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.

Spillover Dynamics

Application 20

Spillover strength 2009-2014

Mario Draghi: „Whatever it takes“

Ireland bailed out Help offer to Greece

First LTRO Second LTRO

ESM inaugurated

Greece : record deficit

New supervisory authority

Spillover Dynamics

Conclusions 21

Conclusions

I Spatial model with dynamic spillover strength and fat tails isnew, and it works (theory, simulation, empirics).

I European sovereign CDS spreads are strongly spatiallydependent.

I Decrease of systemic risk from mid-2012 onwards; possiblydue to EU governments’ and ECB’s bailout measures.

I Best model: Time-varying spatial dependence based ont-distributed errors, time-varying volatilities, additional meanfactor, and categorical spatial weights.

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Thank you.

Appendix 23

Model specifications (t-errors)I individual variance factors

. Σt = Σ(fσt ) = diag(exp(f σ1t ), ..., exp(f σnt ))

. fσt+1 = ωσ + asσt + bfσt , with

sσt =

− 1

2− ν+n

2·

1ν exp(fσ

t,1)·(yt,1−h(f

ρt )

∑nj=1 w1j yt,j−x′t,1β)2

1+ 1ν

(yt−h(fρt )Wyt−Xtβ)′Σ(fσt )−1(yt−h(f

ρt )Wyt−Xtβ)

...

− 12− ν+n

2·

1ν exp(fσt,n)

·(yt,n−h(fρt )

∑nj=1 wnj yt,j−x′t,nβ)2

1+ 1ν

(yt−h(fρt )Wyt−Xtβ)′Σ(fσt )−1(yt−h(f

ρt )Wyt−Xtβ)

I mean factor

. factor loadings: λ = (λ1, ..., λn)′

. f λt+1 = ωλ + aλsλ + bλf λt with

sλt =(1 + n

ν)(Z−1

t λ)′Σ−1et

1 + 1νe′tΣ−1et

, et = yt − h(f ρt )Wyt − Xtβ − Z−1t λf λt

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Appendix 24

Estimation results: Full model

ωλ -0.0012 ωσ1 Belgium 0.0426 ω 0.0307(0.0252) (0.0125) (0.0229)

Aλ 0.3494 ωσ2 France 0.0448 A 0.019(0.8937) (0.0142) (0.007)

Bλ 0.6891 ωσ3 Germany 0.0573 B 0.9636(0.1065) (0.0155) (0.0271)

λ1 Belgium -0.2776 ωσ4 Ireland 0.0301 const. -0.0621(0.2308) (0.01) (0.024)

λ2 France -0.2846 ωσ5 Italy 0.0471 VStoxx -0.0257(0.3137) (0.0136) (0.0157)

λ3 Germany -0.2029 ωσ6 Netherlands 0.0443 term sp. 0.0693(0.2811) (0.0132) (0.0705)

λ4 Ireland 0.405 ωσ7 Portugal 0.0524 stocks -0.102(0.6928) (0.0153) (0.0183)

λ5 Italy -0.1604 ωσ8 Spain 0.0591 yields 0.0173(0.2429) (0.016) (0.0026)

λ6 Netherlands -0.1891 Aσ 0.1826 λ0 3.1357(0.2519) (0.023) (0.1977)

λ7 Portugal 0.4614 Bσ 0.9479(0.8334) (0.0135)

λ8 Spain 0.0988 logLik -24156.96(0.3635) AICc 48375.3

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Appendix 25

Basic model: filtered GAS parameter

2009 2010 2011 2012 2013 2014

0.4

0.5

0.6

0.7

0.8

0.9

rho_

t

t−GAS modelnormal−GAS model

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Appendix 26

Filtered parameter: Full vs. basic model

2009 2010 2011 2012 2013 2014

0.4

0.5

0.6

0.7

0.8

0.9

rho_

t

basic t−GAS modelfull t−GAS model

I Neglecting heteroskedasticity and common mean dynamics leads toslightly biased filtered process.

Spillover Dynamics

Appendix 27

Residual diagnostics: Full model

Test for remaining autocorrelation and ARCH effects in standardized residualsfrom 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 0.07France 49.48 30.42 160.44 43.32∗ 0.66 -0.01Germany 62.61 19.49 142.70 53.78∗ 0.63 -0.07Ireland 129.89 17.53 302.23 87.11∗ 0.64 -0.07Italy 99.02 42.43∗ 102.13 150.88∗ 0.71 0.08Netherlands 55.69 33.29∗ 124.41 20.96 0.64 -0.05Portugal 167.91 32.56∗ 189.35 56.89∗ 0.65 0.03Spain 105.81 48.88∗ 253.68 154.42∗ 0.69 0.06∗Remaining effects at 5% level

Spillover Dynamics

Appendix 28

Simulation: Parameter tracking

I Data generating process:

yt = Ztet , et ∼ i .i .d .t5(0, In),

where Zt = (In − ρtW )−1, and t = 1, ..., 500.

I Weights matrix (row-normalized): cross-border debt of 9 Europeancountries (BIS data)

I Spatial dependence processes (Engle 2002):

1. Constant: ρt = 0.92. Sine: ρt = 0.5 + 0.4 cos(2πt/200)3. Fast sine: ρt = 0.5 + 0.4 cos(2πt/20)4. Step: ρt = 0.9− 0.5 ∗ I (t > T/2)5. Ramp: ρt = mod (t/200)

Spillover Dynamics