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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 Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
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
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
IntroductionThe Data
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
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
IntroductionThe Data
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
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
IntroductionThe Data
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)
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
IntroductionThe Data
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
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
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
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
Equation 1Equation 2Equations 3 and 4
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)
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
Equation 1Equation 2Equations 3 and 4
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
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
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]
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The Likelihood functionThe Joint Posterior distributionPrior SpecificationBayesian InferenceFull Conditional Posterior Distributions
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
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The Likelihood functionThe Joint Posterior distributionPrior SpecificationBayesian InferenceFull Conditional Posterior Distributions
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)
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The Likelihood functionThe Joint Posterior distributionPrior SpecificationBayesian InferenceFull Conditional Posterior Distributions
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)
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The Likelihood functionThe Joint Posterior distributionPrior SpecificationBayesian InferenceFull Conditional Posterior Distributions
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
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The Likelihood functionThe Joint Posterior distributionPrior SpecificationBayesian InferenceFull Conditional Posterior Distributions
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
i,t − βjic − ϕ
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
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The Likelihood functionThe Joint Posterior distributionPrior SpecificationBayesian InferenceFull Conditional Posterior Distributions
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
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The Likelihood functionThe Joint Posterior distributionPrior SpecificationBayesian InferenceFull Conditional Posterior Distributions
Full Conditional Posterior Distributions (continued)
ϕji |· ∝ exp
−1
2ψ−2ij
T∑t=1
(βj
i,t − βjic − ϕ
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
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The Likelihood functionThe Joint Posterior distributionPrior SpecificationBayesian InferenceFull Conditional Posterior Distributions
Full Conditional Posterior Distributions (continued)
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
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The Likelihood functionThe Joint Posterior distributionPrior SpecificationBayesian InferenceFull Conditional Posterior Distributions
Full Conditional Posterior Distributions (continued)
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
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The Likelihood functionThe Joint Posterior distributionPrior SpecificationBayesian InferenceFull Conditional Posterior Distributions
Full Conditional Posterior Distributions (continued)
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
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The Likelihood functionThe Joint Posterior distributionPrior SpecificationBayesian InferenceFull Conditional Posterior Distributions
Full Conditional Posterior Distributions (continued)
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
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
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
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
Simulation StudyAn Example of Simulated DataEvaluation StatisticEvaluation Figures for simulated factors
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 ,t˜N
(0, σ2ij
)βj
i ,t = βji ,t−1 + µj
i ,t , µji ,t˜N
(0, ψ2
ij
)v j
t = αj1Vt + ωj
t , ωjt˜N
(0, k21j = 1
)Vt = X
′tB + ut , ut˜N
(0, k22 = 1
)Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
Simulation StudyAn Example of Simulated DataEvaluation StatisticEvaluation Figures for simulated factors
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
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
Simulation StudyAn Example of Simulated DataEvaluation StatisticEvaluation Figures for simulated 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
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
Simulation StudyAn Example of Simulated DataEvaluation StatisticEvaluation Figures for simulated factors
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
M3: latent factor v2t
0 10 20 30 40 50 60 70 80 90 100−5
0
5
10
M3: latent factor v3t
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
Simulation StudyAn Example of Simulated DataEvaluation StatisticEvaluation Figures for simulated factors
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
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
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)
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
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
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
Traceplots for CDS’ αji ’s
0 10002000−0.2
00.2
a1ik
0 10002000−0.2
00.2
a1ik
0 10002000−0.5
00.5
a1ik
0 10002000−0.5
00.5
a1ik
0 10002000−0.5
00.5
a1ik
0 10002000−0.2
00.2
a1ik
0 10002000−0.5
00.5
a1ik
0 10002000−0.2−0.1
0
a1ik
0 10002000−101
a1ik
0 100020000
0.20.4
a1ik
0 10002000−0.5
00.5
a1ik
0 10002000−0.2
00.2
a1ik
0 10002000−0.5
00.5
a1ik
0 10002000−0.2
00.2
a1ik
0 10002000−0.5
00.5
a1ik
0 10002000−0.4−0.2
0
a1ik
0 10002000−0.5
00.5
a1ik
0 10002000−0.5
00.5
a1ik
0 10002000−101
a1ik
0 10002000−0.5
00.5
a1ik
0 10002000−101
a1ik
0 10002000−0.5
00.5
a1ik
0 10002000−101
a1ik
0 10002000−0.3−0.2−0.1
a1ik
0 10002000−0.2
00.2
a1ik
0 10002000−0.5
00.5
a1ik
0 10002000−0.5
00.5
a1ik
0 10002000−0.2
00.2
a1ik
0 10002000−0.5
00.5
a1ik
0 10002000−0.2
00.2
a1ik
0 10002000−0.5
00.5
a1ik
0 10002000−0.2−0.1
0
a1ik
0 10002000−101
a1ik
0 10002000−0.5
00.5
a1ik
0 10002000−0.1
00.1
a1ik
0 10002000−0.5
00.5
a1ik
0 10002000−0.5
00.5
a1ik
0 10002000−0.2
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a1ik
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a1ik
0 10002000−0.4−0.2
0
a1ik
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
Traceplots for Financial ’αji ’s
0 500 1000 1500 2000−0.2
00.2
a2ik
0 500 1000 1500 2000−0.2
00.2
a2ik
0 500 1000 1500 2000
0.70.80.9
a2ik
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00.05
a2ik
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a2ik
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a2ik
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a2ik
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a2ik
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a2ik
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a2ik
0 500 1000 1500 2000−0.1
00.1
a2ik
0 500 1000 1500 2000−0.1
00.1
a2ik
0 500 1000 1500 20000.70.80.9
a2ik
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
Traceplots for Non-Financial ’αji ’s
0 1000 2000−0.2
00.2
a3ik
0 1000 2000−0.2
00.2
a3ik
0 1000 2000
0.70.8
a3ik
0 1000 2000−0.2
00.2
a3ik
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a3ik
0 1000 20000.40.60.8
a3ik
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a3ik
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a3ik
0 1000 20000
0.20.4
a3ik
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a3ik
0 1000 20000.60.8
1
a3ik
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a3ik
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a3ik
0 1000 20000.70.80.9
a3ik
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a3ik
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a3ik
0 1000 20000
0.51
a3ik
0 1000 2000−0.2
00.2
a3ik
0 1000 2000−0.2
00.2
a3ik
0 1000 20000.40.60.8
a3ik
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
Traceplots for CDS: local covariates Ai
0 1000 2000−0.5
00.5
A1ik
0 1000 2000−0.5
00.5
A1ik
0 1000 2000−0.5
00.5
A1ik
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A1ik
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A1ik
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A1ik
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A1ik
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A1ik
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A1ik
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A1ik
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00.2
A1ik
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A1ik
0 1000 2000−0.5
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A1ik
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A1ik
0 1000 2000−0.5
00.5
A1ik
0 1000 2000−0.5
00.5
A1ik
0 1000 2000−0.2
00.2
A1ik
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
Traceplots for Financial: local covariates Ai
0 500 1000 1500 2000−0.5
00.5
A2ik
0 500 1000 1500 2000−0.5
00.5
A2ik
0 500 1000 1500 2000−0.5
00.5
A2ik
0 500 1000 1500 2000−0.2
00.2
A2ik
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A2ik
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00.2
A2ik
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00.2
A2ik
0 500 1000 1500 2000−0.2
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A2ik
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00.2
A2ik
0 500 1000 1500 2000−0.2
00.2
A2ik
0 500 1000 1500 2000−0.2
00.2
A2ik
0 500 1000 1500 2000−0.5
00.5
A2ik
0 500 1000 1500 2000−0.2
00.2
A2ik
0 500 1000 1500 2000−0.2
00.2
A2ik
0 500 1000 1500 2000−0.2
00.2
A2ik
0 500 1000 1500 2000−0.2
00.2
A2ik
0 500 1000 1500 2000−0.2
00.2
A2ik
0 500 1000 1500 2000−0.2
00.2
A2ik
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
Traceplots for Non-Financial: local covariates Ai
0 500 1000 1500 2000−0.5
00.5
A3ik
0 500 1000 1500 2000−0.5
00.5
A3ik
0 500 1000 1500 2000−0.5
00.5
A3ik
0 500 1000 1500 2000−0.5
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A3ik
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A3ik
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A3ik
0 500 1000 1500 2000−0.5
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A3ik
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A3ik
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A3ik
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A3ik
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A3ik
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A3ik
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A3ik
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A3ik
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A3ik
0 500 1000 1500 2000−0.5
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A3ik
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00.5
A3ik
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A3ik
0 500 1000 1500 2000−0.5
00.5
A3ik
0 500 1000 1500 2000−0.5
00.5
A3ik
0 500 1000 1500 2000−0.5
00.5
A3ik
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
Traceplots for Error variance parameters (Sigmas)
0 100020000
0.01
0.02
0.03
σ2ij
0 100020000.05
0.1
0.15
0.2
0.25
σ2ij
0 100020000
0.05
0.1
0.15
0.2
σ2ij
0 100020000
0.01
0.02
0.03
0.04
σ2ij
0 100020000
0.05
0.1
σ2ij
0 100020000
0.05
0.1
0.15
0.2
σ2ij
0 100020000
0.02
0.04
0.06
0.08
σ2ij
0 100020000.05
0.1
0.15
0.2
0.25
σ2ij
0 100020000
0.005
0.01
0.015
σ2ij
0 100020000
0.01
0.02
0.03
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σ2ij
0 100020000.05
0.1
0.15
0.2
σ2ij
0 100020000.02
0.04
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0.08
σ2ij
0 100020000.005
0.01
0.015
0.02
0.025
σ2ij
0 100020000
0.05
0.1
0.15
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σ2ij
0 100020000.1
0.2
0.3
0.4
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σ2ij
0 100020000
0.1
0.2
0.3
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σ2ij
0 100020000
0.1
0.2
0.3
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σ2ij
0 100020000.05
0.1
0.15
0.2
0.25
σ2ij
0 100020000
0.05
0.1
0.15
0.2
σ2ij
0 100020000.1
0.2
0.3
0.4
0.5
σ2ij
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
Traceplots for Error variance parameters (Psis)
0 100020000.8
1
1.2
1.4
1.6
ψ2ij
0 100020000
0.5
1
1.5
ψ2ij
0 100020000.5
1
1.5
ψ2ij
0 100020000
0.5
1
1.5
ψ2ij
0 10002000
0.8
1
1.2
1.4
ψ2ij
0 100020000
0.5
1
1.5
ψ2ij
0 100020000
0.5
1
1.5
ψ2ij
0 100020000.2
0.4
0.6
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ψ2ij
0 100020000.05
0.1
0.15
0.2
0.25
ψ2ij
0 100020000.2
0.25
0.3
0.35
0.4
ψ2ij
0 100020000.1
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0.3
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ψ2ij
0 100020000.1
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0.3
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ψ2ij
0 100020000.1
0.15
0.2
0.25
ψ2ij
0 100020000.2
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0.8
1
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0 100020000
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0 100020000
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1
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ψ2ij
0 100020000.2
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1
ψ2ij
0 100020000.1
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ψ2ij
0 100020000.2
0.4
0.6
0.8
1
ψ2ij
0 100020000.2
0.4
0.6
0.8
1
ψ2ij
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
Traceplots for Latent factors at time t
0 500 1000 1500 2000−1
0
1
0 500 1000 1500 20000
1
2
3
0 500 1000 1500 2000−1
0
1
2
0 500 1000 1500 20000
1
2
3
0 500 1000 1500 20000
1
2
3
0 500 1000 1500 20000
1
2
3
0 500 1000 1500 2000−2
0
2
0 500 1000 1500 2000−2
0
2
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
Traceplots for Time-varying betas at time t
0 500 1000 1500 2000−5
0
5
0 500 1000 1500 2000−6
−4
−2
0
0 500 1000 1500 2000−5
0
5
0 500 1000 1500 20000
2
4
6
0 500 1000 1500 2000−6
−4
−2
0
0 500 1000 1500 20000
2
4
0 500 1000 1500 2000−5
0
5
0 500 1000 1500 2000−5
0
5
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
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)
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
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
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
Comments on the results
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
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
Comments on the results
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
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
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)
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
Comments on the results
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
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
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)
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
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)
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
Comments on the results
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.
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
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)
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
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)
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
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)
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
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
Estimates 0.09 0.31 0.17 0.19 0.11 0.04 0.21Std.Error (0.02) (0.043) (0.04) (0.03) (0.02) (0.01) (0.03)
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
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
Estimates 1.06 0.73 0.78 0.83 0.99 0.74 0.75Std.Error (0.09) (0.07) (0.08) (0.07) (0.10) (0.07) (0.07)
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
Estimates 0.45 0.23 0.69 0.42 0.21 0.57 0.48Std.Error (0.06) (0.06) (0.10) (0.06) (0.03) (0.06) (0.07)
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
CDS sector: Median and 5-th, 95-th percentiles for Time-varyingbetas across time
0 200 400 600 800−5
0
5
0 200 400 600 800−4
−2
0
2
4
0 200 400 600 800−4
−2
0
2
4
0 200 400 600 800−5
0
5
0 200 400 600 800−10
−5
0
5
0 200 400 600 800−4
−2
0
2
4
0 200 400 600 800−4
−2
0
2
4
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
Financial sector: Median and 5-th, 95-th percentiles forTime-varying betas across time
0 200 400 600 800−4
−2
0
2
4
0 200 400 600 800−2
−1
0
1
2
0 200 400 600 800−2
−1
0
1
2
0 200 400 600 800−2
−1
0
1
2
0 200 400 600 800−2
−1
0
1
2
0 200 400 600 800−2
−1
0
1
2
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
Non-Financial sector: Median and 5-th, 95-th percentiles forTime-varying betas across time
0 200 400 600 800−4
−2
0
2
4
0 200 400 600 800−2
−1
0
1
2
0 200 400 600 800−4
−2
0
2
4
0 200 400 600 800−4
−2
0
2
4
0 200 400 600 800−2
−1
0
1
2
0 200 400 600 800−5
0
5
0 200 400 600 800−4
−2
0
2
4
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
CDS sector: Scatterplots of latent factor vs the time-varying betas
−4 −2 0 2 4−4
−2
0
2
4
−4 −2 0 2 4−4
−2
0
2
4
−4 −2 0 2 4−4
−2
0
2
4
−4 −2 0 2 4−4
−2
0
2
4
−5 0 5−4
−2
0
2
4
−4 −2 0 2 4−4
−2
0
2
4
−2 0 2 4−4
−2
0
2
4
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
Financial sector: Scatterplots of latent factor vs the time-varyingbetas
−2 −1 0 1 2−4
−2
0
2
4
−1.5 −1 −0.5 0 0.5 1 1.5−4
−2
0
2
4
−1.5 −1 −0.5 0 0.5 1 1.5−4
−2
0
2
4
−1.5 −1 −0.5 0 0.5 1 1.5−4
−2
0
2
4
−1.5 −1 −0.5 0 0.5 1 1.5−4
−2
0
2
4
−1.5 −1 −0.5 0 0.5 1 1.5−4
−2
0
2
4
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
The dataSome convergence plotsParameter EstimatesComments on the results
Non-Financial sector: Scatterplots of latent factor vs thetime-varying betas
−4 −2 0 2−4
−2
0
2
4
−1 0 1 2−4
−2
0
2
4
−4 −2 0 2 4−4
−2
0
2
4
−4 −2 0 2 4−4
−2
0
2
4
−2 −1 0 1 2−4
−2
0
2
4
−4 −2 0 2−4
−2
0
2
4
−2 −1 0 1 2−4
−2
0
2
4
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
IntroductionThe Dynamic Factor Model
Bayesian Inference via MCMCSimulation Study
Application to Sovereign CDS and Equity DataConclusions and Current/Future research
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
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application
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.