A Dynamic Factor Model: Inference and Empirical Application. Ioannis Vrontos

A Dynamic Factor Model: Inference and Empirical Application

SYstemic Risk TOmography:

Signals, Measurements, Transmission Channels, and Policy Interventions

Ioannis Vrontos, Athens University of Economics and Business Joint work with Loukia Meligkotsidou, Petros Dellaportas and Roberto Savona European Financial Management Association 2015 Annual Meetings Amsterdam, 24-27 June 2015

IntroductionThe Dynamic Factor Model

Bayesian Inference via MCMCSimulation Study

Application to Sovereign CDS and Equity DataConclusions and Current/Future research

Outline

I Introduction

I The Dynamic Factor Model

I Bayesian Inference via MCMC

I Application to Simulated Data

I Application to Sovereign CDS and Equity Data

I Conclusions and Future Research

Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application




IntroductionThe Data

Introduction

I The aim of this work is to inspect how risks in the financialsystem are interconnected within the Eurozone

I We propose a novel dynamic factor model that explains howfinancial returns are affected by latent, sector-based factorsand macro-systemic risk factors

I We explore the risk dynamics in the Eurozone by analyzing 7sovereign CDS and 13 equity returns of the Eurostoxx 600index over the period 2/1/2007-31/12/2009 that covers theperiod of the financial crisis at US, and just before thesovereign crisis at the Eurozone






The Data

I Our data set consists of J = 3 different sectors (Sovereign,Banks and Financial Intermediaries, Corporations), each onecomposed of a number of financial assets (CDS or Equities)

I Sovereign CDS are used to measure sovereign risks, whileequities are used to estimate risk dynamics for financials(Banks and other Financial Intermediaries) and non-financials(Corporations)

I To model this data, we consider a set of local (asset-specific),sector-specific and global covariates






Local and Sector-specific Covariates for Sovereign CDS

I Country-specific (local) covariates:I Debt/GDPI Export/GDPI Domestic industrial productionI Domestic inflationI M3/GDPI Real GDP growthI Unemployment rateI Domestic equity index returns

I Sector-specific covariates:I Volatility premium (VIX minus the realized volatility over the

next 30 days)I Liquidity spread (Euribor 3m minus Eonia 3m)I Euro/US Dollar exchange rate variations






Local and Sector-specific Covariates for Euro Stocks

I Country-specific (local) covariates:I Domestic industrial productionI Domestic inflationI Sectorial index (Financial vs Non Financial)

I Sector-specific covariates:I Momentum (6m minus 1m Eurostoxx 600 computed at time t

by Pt−21/Pt−126 − 1 where Pt is the asset price at time t, inorder to avoid the 1-month reversal period)

I Risk Premium Europe (Stoxx Europe 600 earning per shareminus (0.70 × [BofA ML 7-10y Euro Non-Financial] + 0.30 ×[BofA ML 7-10y Sterling Corporating Non-Financial]) )

I Risk Premium US (S&P 500 earning per share minus BofA MLUS Corporate 7-10y Yield)






Global Covariates

I Credit Spread (US Corp BBB/Baa minus US Treasury 10 yr)

I US Tbill 3m

I Euro Term Spread (10 yr minus 2 yr government bond yield)

I VIX

The Global covariates are assumed to affect a Macro-systemic riskfactor





Equation 1Equation 2Equations 3 and 4

The Dynamic Factor Model

Consider J = 3 different sectors: CDS, financials andnon-financials, each one composed of mj , j = 1, ..., J, assets

We assume that the asset return r ji ,t is linked to a set of pj

asset-specific covariates, Z ji ,t , and a sector systemic risk factor v j

t ,common accoss the assets of the jth sector, as follows:

r ji ,t = Z j ′

i ,tαji +β

ji ,tv

jt +εj

i ,t , εji ,t ∼ N

(0, σ2ij

), i = 1, ...,mj , j = 1, ..., J,

(1)where the factor loadings, βj

i ,t , are assumed to be time-varying






The Dynamic Factor Model (continued)

The time-varying factor loadings are assumed to follow apseudo-stochastic mean reverting process, in which a set of qj

structural sector-based covariates, G jt , are mixed together with a

stochastic component µji ,t through the following equation:

βji ,t = βj

ic +ϕji

(βj

i ,t−1 − βjic

)+G j ′

t Aji +µj

i ,t , µji ,t ∼ N

(0, ψ2

ij

)(2)






The Dynamic Factor Model (continued)

Finally, we assume that all the sector-specific systemic risk factors,v j

t , are correlated to a macro-systemic risk factor, Vt . Such amacro-factor is in turn related to a set of covariates Xt :

v jt = γjVt + ωj

t , ωjt ∼ N

(0, k21j

), j = 1, ..., J (3)

andVt = X

′tB + ut , ut ∼ N

(0, k22

)(4)

The error term variances of the latent factors v jt , j = 1, ..., J, and

Vt are set equal to 1 for identifiability reasons





The Likelihood functionThe Joint Posterior distributionPrior SpecificationBayesian InferenceFull Conditional Posterior Distributions

The Likelihood function

The likelihood for the dynamic factor model (equations 1-4) can bewritten as:

J∏j=1

T∏t=1

mj∏i=1

[(σ−2ij

)1/2exp

− 1

2σ−2ij

(r ji,t − Z j′

i,tαji − β

ji,tv

jt

)2]×

J∏j=1

T∏t=1

mj∏i=1

[(ψ−2ij

)1/2exp

− 1

2ψ−2ij

(βj

i,t − βjic − ϕ

ji

(βj

i,t−1 − βjic

)− G j′

t Aji

)2]×

J∏j=1

T∏t=1

[exp

− 1

2

(v j

t − γjVt

)2]×

T∏t=1

[exp

− 1

2

(Vt − X

′

t B)2]






The Joint Posterior distribution

Assuming a prior distribution for the model parameters,P(α, σ2, βc , ϕ,A, ψ

2, γ,B), the joint posterior distribution for all

the unknown quantities in the factor model can be written as:

P(α, β, v , σ2, βc , ϕ,A, ψ

2, γ,V ,B|r)

∝ P(r |α, β, v , σ2, βc , ϕ,A, ψ

2, γ,V ,B)

×P(α, β, v , σ2, βc , ϕ,A, ψ

2, γ,V ,B)

= P(r |a, β, v , σ2

)× P

(β|βc , ϕ,A, ψ

2)× P (v |γ,V )× P (V |B)

×P(α, σ2, βc , ϕ,A, ψ

2, γ,B)

where α denotes the parameter set consisting of allαj

i , i = 1, ...,mj , j = 1, ..., J, and the respective notation is used forall parameter sets






Prior Specification

We consider prior independence among the model parameters:

P(αj

i

)∝ exp

−1

2

(aj

i − ξj1

)′Ωj−1

1

(aj

i − ξj1

)≡ Npj

(ξj1,Ω

j1

)P(Aj

i

)∝ exp

−1

2

(Aj

i − ξj2

)′Ωj−1

2

(Aj

i − ξj2

)≡ Nqj

(ξj2,Ω

j2

)P (B) ∝ exp

−1

2(B − ξ0)

′Ω−10 (B − ξ0)

≡ Nq0 (ξ0,Ω0)






Prior Specification (continued)

P(γj)∝ exp

− 1

2σ2γ

(γj − µγ

)2 ≡ N(µγ , σ

2γ

)P(βj

ic

)∝ exp

− 1

2σ20

(βj

ic − µ0)2

≡ N(µ0, σ

20

)P(ϕj

i

)∝ 1

2B (c0, d0)

(1 + ϕj

i

2

)c0−1(1− ϕj

i

2

)d0−1

≡ Beta (c0, d0)

P(σ−2ij

)∝

(σ−2ij

)c1−1exp

−d1σ−2ij

≡ Ga (c1, d1)

P(ψ−2ij

)∝

(ψ−2ij

)c2−1exp

−d2ψ−2ij

≡ Ga (c2, d2)






Bayesian Inference

I Bayesian Inference for our complex high-dimensional factormodel is performed via MCMC sampling from the jointposterior distribution of all the unknown quantities

I We construct an MCMC algorithm which successively andrecursively simulates draws from the full conditional posteriordistributions of these quantities






Full Conditional Posterior Distributions

σ−2ij |· ∼ Ga

(T

2+ c1,

1

2

T∑t=1

(r ji,t − Z j′

i,tαji − β

ji,tv

jt

)2+ d1

)

ψ−2ij |· ∼ Ga

(T

2+ c2,

1

2

T∑t=1

(βj


ji

(βj

i,t−1 − βjic

)− G j′

t Aji

)2+ d2

)

v jt |· ∼ N

γjVt +

mj∑i=1

σ−2ij βji,t

(r ji,t − Z j′

i,tαji

)1 +

mj∑i=1

σ−2ij

(βj

i,t

)2 ,1

1 +mj∑

i=1

σ−2ij

(βj

i,t

)2






Full Conditional Posterior Distributions (continued)

αji |· ∼ Npj

(ξj1∗,Ω

j1∗

)where ξj

1∗ = Ωj1∗

[Ωj−1

1 ξj1 + σ−2ij Z j′

i,t

(r ji − diag(v j )βj

i

)]and Ωj

1∗ =[Ωj−1

1 + σ−2ij Z j′i Z

ji

]−1βj

ic |· ∼ N(µ∗0 , (σ

∗0 )2)

where µ∗0 = (σ∗0 )2[σ−20 µ0 + ψ−2ij

(1− ϕj

i

) T∑t=1

(βj

i,t − ϕjiβ

ji,t−1 − G j′

t Aji

)]

and (σ∗0 )2 =

[Tψ−2ij

(1− ϕj

i

)2+ σ−20

]−1Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application






ϕji |· ∝ exp

−1

2ψ−2ij

T∑t=1

(βj


ji

(βj

i,t−1 − βjic

)− G j′

t Aji

)2

×

(1 + ϕj

i

2

)c0−1(1− ϕj

i

2

)d0−1

Aji |· ∼ Nqi

(ξj2∗,Ω

j2∗

)where ξj

2∗ = Ωj2∗

[Ωj−1

2 ξj2 + ψ−2ij G j

′ (βj

i − βjic − ϕ

ji

(βj

i,−1 − βjic

))]and Ωj

2∗ =[Ωj−1

2 + ψ−2ij G j′

G j]−1







γj |· ∼ N

σ−2γ µγ +

T∑t=1

v jtVt

σ−2γ +T∑

t=1V 2

t

,1

σ−2γ +T∑

t=1V 2

t

Vt |· ∼ N

X′tB +

J∑j=1

γjv jt

1 +J∑

j=1γj2

,1

1 +J∑

j=1γj2

B|· ∼ Nq0 (ξ∗0 ,Ω

∗0)

where ξ∗0 = Ω∗0

[X′V + Ω−10 ξ0

], and Ω∗0 =

[X′X + Ω−10

]−1Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application






In order to draw the vector βji =

(βj

i,1, ..., βji,T

)′jointly, from its full

conditional posterior, we rewrite equation (1) in the form:

r ji = Z j

i αji + diag

(v j)βj

i + εji , ε

ji ∼ NT

(0T , σ

2ij IT)

where

I r ji is a T × 1 vector

I Z ji is a T × pj design matrix

I diag(v j)

is a T × T diagonal matrix with elements v j1, v j

2,..., v jT in

the diagonal

I εji is a T × 1 vector of innovations

I IT is the T × T identity matrix







and equation (2) in the form:

βji = βj

ic 1T + ϕji

(S1β

ji − β

jic 1T

)+ G jAj

i + µji , µ

ji ∼ NT

(0T , ψ

2ij IT)

where

I 1T is a T × 1 vector of ones

I S1 is a T × T matrix of the form S1 =

0 0 . . . 0 01 0 . . . 0 00 1 . . . 0 0...

.... . .

......

0 0 . . . 1 0







Then

βji |· ∼ NT

(ξ∗ij ,Ω

∗ij

)where

ξ∗ij = Ω∗ij

[σ−2ij diag

(v j) (

r ji − Z j

i αji

)]+ Ω∗ij

[ψ−2ij

(IT − ϕj

iS1)′ (

βjic 1T − ϕj

iβjic 1T + G jAj

i

)]Ω∗ij =

[σ−2ij diag

(v j)diag

(v j)

+ ψ−2ij

(IT − ϕj

iS1)′ (

IT − ϕjiS1)]−1





Simulation StudyAn Example of Simulated DataEvaluation StatisticEvaluation Figures for simulated factors

Simulation Study

I The aim of the simulation study is to assess the performanceof the Bayesian methodology, to estimate the modelparameters, the latent factors, the time-varying betas, as wellas the macro-systemic risk factor

I We have considered different sample sizes, namely T = 100 ,T = 200 and T = 500, however we present results only forT = 100, for reasons of space. The accuracy of ourestimation approach increases as the sample size increases

I We have simulate data for J = 3 sectors, with m1 = 2,m2 = 3 and m3 = 5

I We have run the MCMC algorithm for 10, 000 iterationsdiscarding the first 1, 000 draws as burn-in






An Example of Simulated Data

As an example, we simulate data from the following dynamicfactor model:

r ji ,t = αj

i + βji ,tv

jt + εj

i ,t , εji ,tÑ

(0, σ2ij

)βj

i ,t = βji ,t−1 + µj

i ,t , µji ,tÑ

(0, ψ2

ij

)v j

t = αj1Vt + ωj

t , ωjtÑ

(0, k21j = 1

)Vt = X

′tB + ut , utÑ

(0, k22 = 1

)Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application





Evaluation Statistic

I Evaluate the performance of the latent factor estimates byusing the following statistic (Doz, Giannone and Reichlin,2006, Korobilis and Schumacher, 2014):

SSF0 =

tr

[v′v(v′v)−1

v′v

]tr [v ′v ]

where v denotes the latent factor estimates, and v denotesthe true simulated factor

I This statistic takes values between zero and one, andtherefore, values of SSF0 close to one indicate goodapproximation of the true latent factors






Evaluation Statistic

I The values of the SSF0 statistic for all three estimated sectorlatent factors, v j

t , and the macro-systemic risk factor, Vt , areabove 0.95

I The values of the SSF0 statistic for the estimatedtime-varying betas parameters, βj

i ,t , vary between 0.96 and0.997






Posterior means for the latent factors v jt , j = 1, 2, 3, (red line) and

true factor values (blue line)

0 10 20 30 40 50 60 70 80 90 100−5

0

5

10

M3: latent factor v1t

0 10 20 30 40 50 60 70 80 90 100−5

0

5

10


0 10 20 30 40 50 60 70 80 90 100−5

0

5

10







Posterior means for the macro-systemic risk factor Vt (red line)and true risk factor values (blue line)

0 10 20 30 40 50 60 70 80 90 100−1

0

1

2

3

4

5

6

7

8

M3: latent Vt





The dataSome convergence plotsParameter EstimatesComments on the results

The data

I In our empirical analysis we used daily returns for 7 sovereignCDS of the Eurozone and 13 equity returns of the Eurostoxx600 index

I CDS: Germany, Greece, Spain, France, Ireland, Italy andPortugal

I Financial assets: Banca Popolare Di Milano (Italy), Bank ofIreland (Ireland), Deutsche Bank (Germany), Banco ComercialPortugues (Portugal), Banco Popular Espanol (Spain), BNPParibas (France)

I Non-Financial assets: Fiat (Italy), Hellenic Telecommunication(Greece), C&C Group (Ireland), Bayer (Germany), EDPEnergas (Portugal), Endesa (Spain), Air France (France)






The MCMC algorithm

I We run the MCMC algorithm for 60, 000 iterations using aburn-in period of 20, 000 and estimate the model parametersand the latent factors based on 2, 000 sample values takenevery 20 iterations from the last 40, 000 iterations of theMCMC algorithm

I The trace plots provide evidence of convergence of theMCMC algorithm






Traceplots for CDS’ αji ’s

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a1ik






Traceplots for Financial ’αji ’s

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a2ik

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a2ik

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0.70.80.9

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a2ik






Traceplots for Non-Financial ’αji ’s

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1

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a3ik

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Traceplots for CDS: local covariates Ai

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A1ik

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Traceplots for Financial: local covariates Ai

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A2ik

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Traceplots for Non-Financial: local covariates Ai

0 500 1000 1500 2000−0.5

00.5

A3ik

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Traceplots for Time-varying betas at time t

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CDS Local Covariates

CDS sector: The estimated parameters a1i of the local covariatesand their corresponding standard errors

DebtGDP ExpGDP Indprod Inflation m3GDP realGDP Unempl. Mrk.

acdsGER -0.03 -0.06 0.02 0.08 0.13 -0.01 0.07 -0.06

(0.03) (0.03) (0.06) (0.04) (0.06) (0.02) (0.038) (0.019)

acdsGR 0.19 0.13 0.19 -0.02 0.09 -0.04 -0.02 -0.17

(0.09) (0.04) (0.06) (0.04) (0.08) (0.04) (0.08) (0.023)

acdsSP -0.01 -0.05 0.23 -0.11 0.24 0.05 -0.07 -0.18

(0.07) (0.04) (0.14) (0.07) (0.11) (0.05) (0.21) (0.024)

acdsFR -0.02 -0.12 0.03 -0.01 0.01 -0.02 -0.06 -0.07

(0.05) (0.05) (0.06) (0.04) (0.06) (0.03) (0.07) (0.016)

acdsIRE 0.11 0.21 -0.02 0.18 0.10 0.09 -0.19 -0.05

(0.14) (0.07) (0.02) (0.06) (0.03) (0.03) (0.13) (0.018)

acdsIT -0.12 -0.02 0.09 -0.06 0.18 0.04 -0.01 -0.18

(0.07) (0.05) (0.07) (0.06) (0.10) (0.04) (0.07) (0.023)

acdsPT 0.14 -0.04 0.06 0.05 0.03 -0.05 -0.07 -0.16

(0.07) (0.03) (0.04) (0.06) (0.03) (0.02) (0.06) (0.019)






Comments on the results

Discuss the results of the estimated parameters of thecountry-specific (local) covariates appeared in the first equation ofthe proposed model. We observe that:

I The α1i estimated parameters for the domestic equity index

(Mrk) are negative and most of them seem to be statisticallysignificant, indicating that a decrease in the equity index leadsto an increase of the CDS values (negatively correlated)

I With the term ’statistically significant’ we mean that theposterior distribution of the respective parameter does notcontain the value of zero, or the probability around zero isvery small







I The returns of the CDS of Greece and Portugal are affectedby the local covariates of Debt/GDP (0.19 and 0.14respectively), indicating that an increase of the Debt/GDPleads to an increase of the CDS returns. This is reasonable,since an increase of the ratio of debt over the GDP indicatesthat the government debt is more than what the countryproduces and leads to an increase in the CDS price







I The CDS returns of Germany and France are negativelyrelated to the Export/GDP covariate, while CDS of Greeceand Ireland are positive related to Export/GDP

I An increase of Industrial production in Greece and Spain leadsto increased CDS returns

I Positive relation exist between Inflation and CDS returns ofGermany and Ireland, and between m3GDP and CDS ofGermany, Spain, Ireland and Italy






Financial and NonFinancial Local Covariates

Financial and Non-Financial sector: The estimated parameters a2iand a3i of the local covariates and their corresponding standarderrors

Indprod Inflation Sectorial Index Indprod Inflation Sectorial Index

aFinIT -0.02 0.01 0.72 aNFin

IT 0.04 -0.03 0.69(0.03) (0.03) (0.03) (0.03) (0.03) (0.03)

aFinIRL -0.001 -0.005 0.84 aNFin

GR -0.03 0.04 0.66(0.009) (0.01) (0.02) (0.03) (0.03) (0.03)

aFinGER -0.04 0.02 0.73 aNFin

IRL 0.03 -0.05 0.31(0.03) (0.03) (0.02) (0.03) (0.03) (0.03)

aFinPT 0.003 0.007 0.95 aNFin

GER 0.02 -0.03 0.70(0.03) (0.03) (0.02) (0.05) (0.05) (0.03)

aFinSP 0.02 0.02 0.86 aNFin

PT -0.01 0.02 0.75(0.02) (0.02) (0.02) (0.03) (0.03) (0.02)

aFinFR -0.02 0.03 0.85 aNFin

SP 0.01 -0.06 0.39(0.01) (0.01) (0.01) (0.02) (0.027) (0.03)

aNFinFR -0.05 0.05 0.68

(0.03) (0.04) (0.03)







I The estimated parameters for the sectorial index(Financial/Non-Financial) are positive and all of them seem tobe statistically significant indicating that an increase in thesectorial index leads to an increase of their returns and prices






CDS Sector Covariates

CDS sector: The estimated parameters A1i of the sector covariates

and their corresponding standard errors

Vpremium Liquidity Spread Exchange Rate

AcdsGER 0.12 0.13 -0.08

(0.05) (0.05) (0.05)

AcdsGR 0.06 0.08 -0.03

(0.05) (0.05) (0.04)

AcdsSP 0.04 0.03 -0.07

(0.05) (0.06) (0.04)

AcdsFR 0.04 0.09 -0.12

(0.04) (0.05) (0.04)

AcdsIRE 0.08 0.03 0.01

(0.05) (0.05) (0.04)

AcdsIT 0.07 0.10 -0.08

(0.04) (0.05) (0.04)

AcdsPT 0.07 0.06 -0.01

(0.04) (0.05) (0.04)






Financial and NonFinancial Sector Covariates

Financial and Non-Financial sector: The estimated parameters A2i

and A3i of the sector covariates and their corresponding standard

errors

Momentum RP − Europe RP − US Momentum RP − Europe RP − US

AFinIT 0.08 -0.10 0.18 ANFin

IT 0.24 0.05 -0.01(0.05) (0.05) (0.07) (0.07) (0.07) (0.09)

AFinIRL -0.07 0.03 -0.01 ANFin

GR -0.03 0.05 -0.04(0.03) (0.03) (0.03) (0.06) (0.05) (0.07)

AFinGER 0.03 0.04 -0.01 ANFin

IRL -0.12 -0.04 0.03(0.04) (0.04) (0.05) (0.09) (0.08) (0.10)

AFinPT -0.01 -0.03 -0.01 ANFin

GER -0.05 0.11 -0.02(0.04) (0.04) (0.06) (0.08) (0.07) (0.10)

AFinSP -0.03 0.04 -0.01 ANFin

PT -0.03 0.08 -0.13(0.04) (0.04) (0.05) (0.05) (0.05) (0.07)

AFinFR -0.03 -0.01 -0.04 ANFin

SP 0.08 -0.06 0.08(0.03) (0.03) (0.04) (0.07) (0.06) (0.08)

ANFinFR 0.04 0.08 0.01

(0.07) (0.06) (0.08)







I The structural sector-based covariates, G jt , affect the beta

parameters in time, since some parameters A1i of the

Vpremium (for Germany) of the Liquidity Spread (forGermany and France) and of Exchange Rate (for France)seem to be statistically significant

I The beta of the asset of Italy for the Financial sector seems tobe affected by the Risk Premium of Europe and US, while thebetas of Financial sector of Ireland is affected by momentumcovariate. For the assets of Non-Financial sector only thebetas of Italy and Spain seem to be affected by themomentum and RP-US, respectively.






Common Covariates

The estimated parameters B j of the common covariates and theircorresponding standard errors

Credit Spread Tbill Term Spread VIX

CDS Bcdscr.sp Bcds

tbill Bcdster.sp. Bcds

vixEstimates 0.09 0.15 -0.01 0.05Std.Error (0.10) (0.07) (0.07) (0.08)

Financials BFincr.sp BFin

tbill BFinter.sp. BFin

vixEstimates -0.21 -0.13 -0.03 0.09Std.Error (0.10) (0.08) (0.08) (0.08)

Non-Financial BNFincr.sp BNFin

tbill BNFinter.sp. BNFin

vixEstimates 0.02 0.01 -0.05 0.09Std.Error (0.13) (0.09) (0.08) (0.10)






Autoregressive Parameters

The estimated autoregressiver parameters ϕji and their

corresponding standard errors [in the time-varying betas equation]

CDS ϕcdsGER ϕcds

GR ϕcdsSP ϕcds

FR ϕcdsIRE ϕcds

IT ϕcdsPO

Estimates -0.15 -0.04 0.02 -0.12 0.11 -0.03 -0.01Std.Error (0.05) (0.06) (0.07) (0.05) (0.05) (0.06) (0.05)

Financials ϕFinIT ϕFin

IRL ϕFinGER ϕFin

PT ϕFinSP ϕFin

FREstimates 0.29 0.14 -0.04 0.09 -0.01 0.09Std.Error (0.08) (0.05) (0.06) (0.09) (0.08) (0.06)

Non − Financial ϕNFinIT ϕNFin

GR,t ϕNFinIRL ϕNFin

GER ϕNFinPT ϕNFin

SP ϕNFinFR

Estimates -0.18 0.14 -0.28 -0.32 -0.27 -0.16 0.14Std.Error (0.08) (0.18) (0.09) (0.08) (0.09) (0.07) (0.10)






Constant Parameters

The estimated parameters βjic and their corresponding standard

errors [constants in the time-varying betas equation]

CDS βcdsGER βcds

GR βcdsSP βcds

FR βcdsIRE βcds

IT βcdsPO

Estimates -0.05 0.11 0.11 0.07 0.04 0.11 0.09Std.Error (0.04) (0.04) (0.05) (0.04) (0.05) (0.04) (0.04)

Financials βFinIT βFin

IRL βFinGER βFin

PT βFinSP βFin

FREstimates -0.19 0.05 0.03 0.06 -0.04 0.004Std.Error (0.06) (0.02) (0.03) (0.05) (0.03) (0.03)

Non − Financial βNFinIT βNFin

GR,t βNFinIRL βNFin

GER βNFinPT βNFin

SP βNFinFR

Estimates -0.11 -0.04 0.004 0.02 -0.04 0.04 -0.02Std.Error (0.04) (0.05) (0.05) (0.05) (0.04) (0.04) (0.06)






Error Variance Parameters

The estimated error variance parameters σ2ij and theircorresponding standard errors

CDS σ2GER σ2

GR σ2SP σ2

FR σ2IRL σ2

IT σ2PT

Estimates 0.009 0.094 0.111 0.014 0.028 0.076 0.036Std.Error (0.003) (0.015) (0.021) (0.004) (0.009) (0.014) (0.009)

Financials σ2IT σ2

IRL σ2GER σ2

PT σ2SP σ2

FREstimates 0.15 0.008 0.017 0.089 0.049 0.014Std.Error (0.02) (0.002) (0.004) (0.013) (0.007) (0.003)

Non − Financial σ2IT σ2

GR σ2IRL σ2

GER σ2PT σ2

SP σ2FR







Error Variance Parameters

The estimated error variance parameters ψ2ij and their

corresponding standard errors [in the time-varying betas equation]

CDS ψ2GER ψ2

GR ψ2SP ψ2

FR ψ2IRL ψ2

IT ψ2PT


Financials ψ2IT ψ2

IRL ψ2GER ψ2

PT ψ2SP ψ2

FREstimates 0.34 0.14 0.28 0.18 0.19 0.17Std.Error (0.05) (0.01) (0.03) (0.03) (0.02) (0.02)

Non − Financial ψ2IT ψ2

GR ψ2IRL ψ2

GER ψ2PT ψ2

SP ψ2FR







CDS sector: Median and 5-th, 95-th percentiles for Time-varyingbetas across time

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Financial sector: Median and 5-th, 95-th percentiles forTime-varying betas across time

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CDS sector: Scatterplots of latent factor vs the time-varying betas

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Financial sector: Scatterplots of latent factor vs the time-varyingbetas

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Non-Financial sector: Scatterplots of latent factor vs thetime-varying betas

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Conclusions and Current/Future research

Conclusions and Current/Future research

I We present a dynamic factor model that explains howfinancial returns are affected by latent, sector-based factorsand macro-systemic risk factors

I Model selection exercise

I Correlation among the different sectors

I Time-varying variances/covariances


This project has received funding from the European Union’s

Seventh Framework Programme for research, technological

development and demonstration under grant agreement n° 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.

Documents

A Dynamic Factor Model: Inference and Empirical Application. Ioannis Vrontos