1. 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

2. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
Outline
Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Application to Simulated Data
Application to Sovereign CDS and Equity Data
Conclusions and Future Research
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

3. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
Introduction
The Data
Introduction
The aim of this work is to inspect how risks in the financial
system are interconnected within the Eurozone
We propose a novel dynamic factor model that explains how
financial returns are affected by latent, sector-based factors
and macro-systemic risk factors
We explore the risk dynamics in the Eurozone by analyzing 7
sovereign CDS and 13 equity returns of the Eurostoxx 600
index over the period 2/1/2007-31/12/2009 that covers the
period of the financial crisis at US, and just before the
sovereign crisis at the Eurozone
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

4. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
Introduction
The Data
The Data
Our data set consists of J = 3 different sectors (Sovereign,
Banks and Financial Intermediaries, Corporations), each one
composed of a number of financial assets (CDS or Equities)
Sovereign CDS are used to measure sovereign risks, while
equities are used to estimate risk dynamics for financials
(Banks and other Financial Intermediaries) and non-financials
(Corporations)
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

5. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
Introduction
The Data
Local and Sector-specific Covariates for Sovereign CDS
Country-specific (local) covariates:
Debt/GDP
Export/GDP
Domestic industrial production
Domestic inflation
M3/GDP
Real GDP growth
Unemployment rate
Domestic equity index returns
Sector-specific covariates:
Volatility premium (VIX minus the realized volatility over the
next 30 days)
Liquidity spread (Euribor 3m minus Eonia 3m)
Euro/US Dollar exchange rate variations
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

6. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
Introduction
The Data
Local and Sector-specific Covariates for Euro Stocks
Country-specific (local) covariates:
Domestic industrial production
Domestic inflation
Sectorial index (Financial vs Non Financial)
Sector-specific covariates:
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, in
order to avoid the 1-month reversal period)
Risk Premium Europe (Stoxx Europe 600 earning per share
minus (0.70 × [BofA ML 7-10y Euro Non-Financial] + 0.30 ×
[BofA ML 7-10y Sterling Corporating Non-Financial]) )
Risk Premium US (S&P 500 earning per share minus BofA ML
US Corporate 7-10y Yield)
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

7. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
Introduction
The Data
Global Covariates
Credit Spread (US Corp BBB/Baa minus US Treasury 10 yr)
US Tbill 3m
Euro Term Spread (10 yr minus 2 yr government bond yield)
VIX
The Global covariates are assumed to affect a Macro-systemic risk
factor
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

8. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
Equation 1
Equation 2
Equations 3 and 4
The Dynamic Factor Model
Consider J = 3 different sectors: CDS, financials and
non-financials, each one composed of mj , j = 1, ..., J, assets
We assume that the asset return rj
i,t is linked to a set of pj
asset-specific covariates, Zj
i,t, and a sector systemic risk factor vj
t ,
common accoss the assets of the jth sector, as follows:
rj
i,t = Zj
i,tαj
i +βj
i,tvj
t +εj
i,t, εj
i,t ∼ N 0, σ2
ij , 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

9. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
Equation 1
Equation 2
Equations 3 and 4
The Dynamic Factor Model (continued)
The time-varying factor loadings are assumed to follow a
pseudo-stochastic mean reverting process, in which a set of qj
structural sector-based covariates, Gj
t , are mixed together with a
stochastic component µj
i,t through the following equation:
βj
i,t = βj
ic +ϕj
i βj
i,t−1 − βj
ic +Gj
t Aj
i +µj
i,t, µj
i,t ∼ N 0, ψ2
ij (2)
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

10. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
Equation 1
Equation 2
Equations 3 and 4
The Dynamic Factor Model (continued)
Finally, we assume that all the sector-specific systemic risk factors,
vj
t , are correlated to a macro-systemic risk factor, Vt. Such a
macro-factor is in turn related to a set of covariates Xt:
vj
t = γj
Vt + ωj
t, ωj
t ∼ N 0, k2
1j , j = 1, ..., J (3)
and
Vt = Xt B + ut, ut ∼ N 0, k2
2 (4)
The error term variances of the latent factors vj
t , 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

11. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The Likelihood function
The Joint Posterior distribution
Prior Specification
Bayesian Inference
Full Conditional Posterior Distributions
The Likelihood function
The likelihood for the dynamic factor model (equations 1-4) can be
written as:
J
j=1
T
t=1
mj
i=1
σ−2
ij
1/2
exp −1
2 σ−2
ij rj
i,t − Zj
i,tαj
i − βj
i,tvj
t
2
×
J
j=1
T
t=1
mj
i=1
ψ−2
ij
1/2
exp −1
2 ψ−2
ij βj
i,t − βj
ic − ϕj
i βj
i,t−1 − βj
ic − Gj
t Aj
i
2
×
J
j=1
T
t=1
exp −1
2 vj
t − γj
Vt
2
×
T
t=1
exp −1
2 Vt − Xt B
2
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

12. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The Likelihood function
The Joint Posterior distribution
Prior Specification
Bayesian Inference
Full Conditional Posterior Distributions
The Joint Posterior distribution
Assuming a prior distribution for the model parameters,
P α, σ2, βc, ϕ, A, ψ2, 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, ψ2
, γ, V , B|r
∝ P r|α, β, v, σ2
, βc, ϕ, A, ψ2
, γ, V , B
×P α, β, v, σ2
, βc, ϕ, A, ψ2
, γ, V , B
= P r|a, β, v, σ2
× P β|βc, ϕ, A, ψ2
× P (v|γ, V ) × P (V |B)
×P α, σ2
, βc, ϕ, A, ψ2
, γ, B
where α denotes the parameter set consisting of all
αj
i , i = 1, ..., mj , j = 1, ..., J, and the respective notation is used for
all parameter sets
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

13. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The Likelihood function
The Joint Posterior distribution
Prior Specification
Bayesian Inference
Full Conditional Posterior Distributions
Prior Specification
We consider prior independence among the model parameters:
P αj
i ∝ exp −
1
2
aj
i − ξj
1 Ωj−1
1 aj
i − ξj
1 ≡ Npj ξj
1, Ωj
1
P Aj
i ∝ exp −
1
2
Aj
i − ξj
2 Ωj−1
2 Aj
i − ξj
2 ≡ Nqj ξj
2, Ωj
2
P (B) ∝ exp −
1
2
(B − ξ0) Ω−1
0 (B − ξ0) ≡ Nq0 (ξ0, Ω0)
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

14. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The Likelihood function
The Joint Posterior distribution
Prior Specification
Bayesian Inference
Full Conditional Posterior Distributions
Prior Specification (continued)
P γj
∝ exp −
1
2σ2
γ
γj
− µγ
2
≡ N µγ, σ2
γ
P βj
ic ∝ exp −
1
2σ2
0
βj
ic − µ0
2
≡ N µ0, σ2
0
P ϕj
i ∝
1
2B (c0, d0)
1 + ϕj
i
2
c0−1
1 − ϕj
i
2
d0−1
≡ Beta (c0, d0)
P σ−2
ij ∝ σ−2
ij
c1−1
exp −d1σ−2
ij ≡ Ga (c1, d1)
P ψ−2
ij ∝ ψ−2
ij
c2−1
exp −d2ψ−2
ij ≡ Ga (c2, d2)
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

15. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The Likelihood function
The Joint Posterior distribution
Prior Specification
Bayesian Inference
Full Conditional Posterior Distributions
Bayesian Inference
Bayesian Inference for our complex high-dimensional factor
model is performed via MCMC sampling from the joint
posterior distribution of all the unknown quantities
We construct an MCMC algorithm which successively and
recursively simulates draws from the full conditional posterior
distributions of these quantities
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

16. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The Likelihood function
The Joint Posterior distribution
Prior Specification
Bayesian Inference
Full Conditional Posterior Distributions
Full Conditional Posterior Distributions
σ−2
ij |· ∼ Ga
T
2
+ c1,
1
2
T
t=1
rj
i,t − Zj
i,tαj
i − βj
i,tvj
t
2
+ d1
ψ−2
ij |· ∼ Ga
T
2
+ c2,
1
2
T
t=1
βj
i,t − βj
ic − ϕj
i βj
i,t−1 − βj
ic − Gj
t Aj
i
2
+ d2
vj
t |· ∼ N




γj
Vt +
mj
i=1
σ−2
ij βj
i,t rj
i,t − Zj
i,tαj
i
1 +
mj
i=1
σ−2
ij βj
i,t
2
,
1
1 +
mj
i=1
σ−2
ij βj
i,t
2




Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

17. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The Likelihood function
The Joint Posterior distribution
Prior Specification
Bayesian Inference
Full Conditional Posterior Distributions
Full Conditional Posterior Distributions (continued)
αj
i |· ∼ Npj
ξj
1∗, Ωj
1∗
where ξj
1∗ = Ωj
1∗ Ωj−1
1 ξj
1 + σ−2
ij Zj
i,t rj
i − diag(vj
)βj
i
and Ωj
1∗ = Ωj−1
1 + σ−2
ij Zj
i Zj
i
−1
βj
ic |· ∼ N µ∗
0, (σ∗
0 )
2
where µ∗
0 = (σ∗
0 )
2
σ−2
0 µ0 + ψ−2
ij 1 − ϕj
i
T
t=1
βj
i,t − ϕj
i βj
i,t−1 − Gj
t Aj
i
and (σ∗
0 )
2
= Tψ−2
ij 1 − ϕj
i
2
+ σ−2
0
−1
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

18. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The Likelihood function
The Joint Posterior distribution
Prior Specification
Bayesian Inference
Full Conditional Posterior Distributions
Full Conditional Posterior Distributions (continued)
ϕj
i |· ∝ exp −
1
2
ψ−2
ij
T
t=1
βj
i,t − βj
ic − ϕj
i βj
i,t−1 − βj
ic − Gj
t Aj
i
2
×
1 + ϕj
i
2
c0−1
1 − ϕj
i
2
d0−1
Aj
i |· ∼ Nqi
ξj
2∗, Ωj
2∗
where ξj
2∗ = Ωj
2∗ Ωj−1
2 ξj
2 + ψ−2
ij Gj
βj
i − βj
ic − ϕj
i βj
i,−1 − βj
ic
and Ωj
2∗ = Ωj−1
2 + ψ−2
ij Gj
Gj
−1
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

19. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The Likelihood function
The Joint Posterior distribution
Prior Specification
Bayesian Inference
Full Conditional Posterior Distributions
Full Conditional Posterior Distributions (continued)
γj
|· ∼ N





σ−2
γ µγ +
T
t=1
vj
t Vt
σ−2
γ +
T
t=1
V 2
t
,
1
σ−2
γ +
T
t=1
V 2
t





Vt|· ∼ N





Xt B +
J
j=1
γj vj
t
1 +
J
j=1
γj2
,
1
1 +
J
j=1
γj2





B|· ∼ Nq0 (ξ∗
0, Ω∗
0)
where ξ∗
0 = Ω∗
0 X V + Ω−1
0 ξ0 , and Ω∗
0 = X X + Ω−1
0
−1
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

20. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The Likelihood function
The Joint Posterior distribution
Prior Specification
Bayesian Inference
Full Conditional Posterior Distributions
Full Conditional Posterior Distributions (continued)
In order to draw the vector βj
i = βj
i,1, ..., βj
i,T jointly, from its full
conditional posterior, we rewrite equation (1) in the form:
rj
i = Zj
i αj
i + diag vj
βj
i + εj
i , εj
i ∼ NT 0T , σ2
ij IT
where
rj
i is a T × 1 vector
Zj
i is a T × pj design matrix
diag vj
is a T × T diagonal matrix with elements vj
1, vj
2,..., vj
T in
the diagonal
εj
i is a T × 1 vector of innovations
IT is the T × T identity matrix
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

21. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The Likelihood function
The Joint Posterior distribution
Prior Specification
Bayesian Inference
Full Conditional Posterior Distributions
Full Conditional Posterior Distributions (continued)
and equation (2) in the form:
βj
i = βj
ic 1T + ϕj
i S1βj
i − βj
ic 1T + Gj
Aj
i + µj
i , µj
i ∼ NT 0T , ψ2
ij IT
where
1T is a T × 1 vector of ones
S1 is a T × T matrix of the form S1 =







0 0 . . . 0 0
1 0 . . . 0 0
0 1 . . . 0 0
...
...
...
...
...
0 0 . . . 1 0







Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

22. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The Likelihood function
The Joint Posterior distribution
Prior Specification
Bayesian Inference
Full Conditional Posterior Distributions
Full Conditional Posterior Distributions (continued)
Then
βj
i |· ∼ NT ξ∗
ij , Ω∗
ij
where
ξ∗
ij = Ω∗
ij σ−2
ij diag vj
rj
i − Zj
i αj
i
+ Ω∗
ij ψ−2
ij IT − ϕj
i S1 βj
ic 1T − ϕj
i βj
ic 1T + Gj
Aj
i
Ω∗
ij = σ−2
ij diag vj
diag vj
+ ψ−2
ij IT − ϕj
i S1 IT − ϕj
i S1
−1
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

23. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
Simulation Study
An Example of Simulated Data
Evaluation Statistic
Evaluation Figures for simulated factors
Simulation Study
The aim of the simulation study is to assess the performance
of the Bayesian methodology, to estimate the model
parameters, the latent factors, the time-varying betas, as well
as the macro-systemic risk factor
We have considered different sample sizes, namely T = 100 ,
T = 200 and T = 500, however we present results only for
T = 100, for reasons of space. The accuracy of our
estimation approach increases as the sample size increases
We have simulate data for J = 3 sectors, with m1 = 2,
m2 = 3 and m3 = 5
We have run the MCMC algorithm for 10, 000 iterations
discarding the first 1, 000 draws as burn-in
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

24. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
Simulation Study
An Example of Simulated Data
Evaluation Statistic
Evaluation Figures for simulated factors
An Example of Simulated Data
As an example, we simulate data from the following dynamic
factor model:
rj
i,t = αj
i + βj
i,tvj
t + εj
i,t, εj
i,t˜N 0, σ2
ij
βj
i,t = βj
i,t−1 + µj
i,t, µj
i,t˜N 0, ψ2
ij
vj
t = αj
1Vt + ωj
t, ωj
t˜N 0, k2
1j = 1
Vt = Xt B + ut, ut˜N 0, k2
2 = 1
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

25. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
Simulation Study
An Example of Simulated Data
Evaluation Statistic
Evaluation Figures for simulated factors
Evaluation Statistic
Evaluate the performance of the latent factor estimates by
using 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 denotes
the true simulated factor
This statistic takes values between zero and one, and
therefore, values of SSF0 close to one indicate good
approximation of the true latent factors
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

26. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
Simulation Study
An Example of Simulated Data
Evaluation Statistic
Evaluation Figures for simulated factors
Evaluation Statistic
The values of the SSF0 statistic for all three estimated sector
latent factors, vj
t , and the macro-systemic risk factor, Vt, are
above 0.95
The values of the SSF0 statistic for the estimated
time-varying betas parameters, βj
i,t, vary between 0.96 and
0.997
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

27. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
Simulation Study
An Example of Simulated Data
Evaluation Statistic
Evaluation Figures for simulated factors
Posterior means for the latent factors vj
t , 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 v1
t
0 10 20 30 40 50 60 70 80 90 100
−5
0
5
10
M3: latent factor v2
t
0 10 20 30 40 50 60 70 80 90 100
−5
0
5
10
M3: latent factor v3
t
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

28. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
Simulation Study
An Example of Simulated Data
Evaluation Statistic
Evaluation 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

29. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
The data
In our empirical analysis we used daily returns for 7 sovereign
CDS of the Eurozone and 13 equity returns of the Eurostoxx
600 index
CDS: Germany, Greece, Spain, France, Ireland, Italy and
Portugal
Financial assets: Banca Popolare Di Milano (Italy), Bank of
Ireland (Ireland), Deutsche Bank (Germany), Banco Comercial
Portugues (Portugal), Banco Popular Espanol (Spain), BNP
Paribas (France)
Non-Financial assets: Fiat (Italy), Hellenic Telecommunication
(Greece), C&C Group (Ireland), Bayer (Germany), EDP
Energas (Portugal), Endesa (Spain), Air France (France)
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

30. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
The MCMC algorithm
We run the MCMC algorithm for 60, 000 iterations using a
burn-in period of 20, 000 and estimate the model parameters
and the latent factors based on 2, 000 sample values taken
every 20 iterations from the last 40, 000 iterations of the
MCMC algorithm
The trace plots provide evidence of convergence of the
MCMC algorithm
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

31. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
Traceplots for CDS’ αj
i ’s
0 10002000
−0.2
0
0.2
a1
ik
0 10002000
−0.2
0
0.2
a1
ik
0 10002000
−0.5
0
0.5
a1
ik
0 10002000
−0.5
0
0.5
a1
ik
0 10002000
−0.5
0
0.5
a1
ik
0 10002000
−0.2
0
0.2
a1
ik
0 10002000
−0.5
0
0.5
a1
ik
0 10002000
−0.2
−0.1
0
a1
ik
0 10002000
−1
0
1
a1
ik
0 10002000
0
0.2
0.4
a1
ik
0 10002000
−0.5
0
0.5
a1
ik
0 10002000
−0.2
0
0.2
a1
ik
0 10002000
−0.5
0
0.5
a1
ik
0 10002000
−0.2
0
0.2
a1
ik
0 10002000
−0.5
0
0.5
a1
ik
0 10002000
−0.4
−0.2
0
a1
ik
0 10002000
−0.5
0
0.5
a1
ik
0 10002000
−0.5
0
0.5
a1
ik
0 10002000
−1
0
1
a1
ik
0 10002000
−0.5
0
0.5
a1
ik
0 10002000
−1
0
1
a1
ik
0 10002000
−0.5
0
0.5
a1
ik
0 10002000
−1
0
1
a1
ik
0 10002000
−0.3
−0.2
−0.1
a1
ik
0 10002000
−0.2
0
0.2
a1
ik
0 10002000
−0.5
0
0.5
a1
ik
0 10002000
−0.5
0
0.5
a1
ik
0 10002000
−0.2
0
0.2
a1
ik
0 10002000
−0.5
0
0.5
a1
ik
0 10002000
−0.2
0
0.2
a1
ik
0 10002000
−0.5
0
0.5
a1
ik
0 10002000
−0.2
−0.1
0
a1
ik
0 10002000
−1
0
1
a1
ik
0 10002000
−0.5
0
0.5
a1
ik
0 10002000
−0.1
0
0.1
a1
ik
0 10002000
−0.5
0
0.5
a1
ik
0 10002000
−0.5
0
0.5
a1
ik
0 10002000
−0.2
0
0.2
a1
ik
0 10002000
−1
0
1
a1
ik
0 10002000
−0.2
0
0.2
a1
ik
0 10002000
−0.5
0
0.5
a1
ik
0 10002000
−0.5
0
0.5
a1
ik
0 10002000
−0.5
0
0.5
a1
ik
0 10002000
−0.5
0
0.5
a1
ik
0 10002000
−1
0
1
a1
ik
0 10002000
−0.5
0
0.5
a1
ik
0 10002000
−0.5
0
0.5
a1
ik
0 10002000
−0.3
−0.2
−0.1
a1
ik
0 10002000
−0.5
0
0.5
a1
ik
0 10002000
−0.2
0
0.2
a1
ik
0 10002000
−0.2
0
0.2
a1
ik
0 10002000
−0.5
0
0.5
a1
ik
0 10002000
−0.2
0
0.2
a1
ik
0 10002000
−0.2
0
0.2
a1
ik
0 10002000
−0.5
0
0.5
a1
ik
0 10002000
−0.4
−0.2
0
a1
ik
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

32. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
Traceplots for Financial ’αj
i ’s
0 500 1000 1500 2000
−0.2
0
0.2
a2
ik
0 500 1000 1500 2000
−0.2
0
0.2
a2
ik
0 500 1000 1500 2000
0.7
0.8
0.9
a2
ik
0 500 1000 1500 2000
−0.05
0
0.05
a2
ik
0 500 1000 1500 2000
−0.05
0
0.05
a2
ik
0 500 1000 1500 2000
0.7
0.8
0.9
a2
ik
0 500 1000 1500 2000
−0.2
0
0.2
a2
ik
0 500 1000 1500 2000
−0.2
0
0.2
a2
ik
0 500 1000 1500 2000
0.7
0.8
a2
ik
0 500 1000 1500 2000
−0.1
0
0.1
a2
ik
0 500 1000 1500 2000
−0.2
0
0.2
a2
ik
0 500 1000 1500 2000
0.5
1
1.5
a2
ik
0 500 1000 1500 2000
−0.1
0
0.1
a2
ik
0 500 1000 1500 2000
−0.1
0
0.1
a2
ik
0 500 1000 1500 2000
0.7
0.8
0.9
a2
ik
0 500 1000 1500 2000
−0.1
0
0.1
a2
ik
0 500 1000 1500 2000
−0.1
0
0.1
a2
ik
0 500 1000 1500 2000
0.7
0.8
0.9
a2
ik
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

33. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
Traceplots for Non-Financial ’αj
i ’s
0 1000 2000
−0.2
0
0.2
a3
ik
0 1000 2000
−0.2
0
0.2
a3
ik
0 1000 2000
0.7
0.8
a3
ik
0 1000 2000
−0.2
0
0.2
a3
ik
0 1000 2000
−0.2
0
0.2
a3
ik
0 1000 2000
0.4
0.6
0.8
a3
ik
0 1000 2000
−0.2
0
0.2
a3
ik
0 1000 2000
−0.2
0
0.2
a3
ik
0 1000 2000
0
0.2
0.4
a3
ik
0 1000 2000
−0.5
0
0.5
a3
ik
0 1000 2000
−0.5
0
0.5
a3
ik
0 1000 2000
0.6
0.8
1
a3
ik
0 1000 2000
−0.2
0
0.2
a3
ik
0 1000 2000
−0.2
0
0.2
a3
ik
0 1000 2000
0.70.80.9
a3
ik
0 1000 2000
−0.1
0
0.1
a3
ik
0 1000 2000
−0.2
0
0.2
a3
ik
0 1000 2000
0
0.5
1
a3
ik
0 1000 2000
−0.2
0
0.2
a3
ik
0 1000 2000
−0.2
0
0.2
a3
ik
0 1000 2000
0.4
0.6
0.8
a3
ik
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

34. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
Traceplots for CDS: local covariates Ai
0 1000 2000
−0.5
0
0.5
A1
ik
0 1000 2000
−0.5
0
0.5
A1
ik
0 1000 2000
−0.5
0
0.5
A1
ik
0 1000 2000
−0.5
0
0.5
A1
ik
0 1000 2000
−0.5
0
0.5
A1
ik
0 1000 2000
−0.2
0
0.2
A1
ik
0 1000 2000
−0.5
0
0.5
A1
ik
0 1000 2000
−0.5
0
0.5
A1
ik
0 1000 2000
−0.5
0
0.5
A1
ik
0 1000 2000
−0.2
0
0.2
A1
ik
0 1000 2000
−0.5
0
0.5
A1
ik
0 1000 2000
−0.5
0
0.5
A1
ik
0 1000 2000
−0.5
0
0.5
A1
ik
0 1000 2000
−0.5
0
0.5
A1
ik
0 1000 2000
−0.2
0
0.2
A1
ik
0 1000 2000
−0.5
0
0.5
A1
ik
0 1000 2000
−0.5
0
0.5
A1
ik
0 1000 2000
−0.5
0
0.5
A1
ik
0 1000 2000
−0.5
0
0.5
A1
ik
0 1000 2000
−0.5
0
0.5
A1
ik
0 1000 2000
−0.2
0
0.2
A1
ik
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

35. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
Traceplots for Financial: local covariates Ai
0 500 1000 1500 2000
−0.5
0
0.5
A2
ik
0 500 1000 1500 2000
−0.5
0
0.5
A2
ik
0 500 1000 1500 2000
−0.5
0
0.5
A2
ik
0 500 1000 1500 2000
−0.2
0
0.2
A2
ik
0 500 1000 1500 2000
−0.2
0
0.2
A2
ik
0 500 1000 1500 2000
−0.2
0
0.2
A2
ik
0 500 1000 1500 2000
−0.2
0
0.2
A2
ik
0 500 1000 1500 2000
−0.2
0
0.2
A2
ik
0 500 1000 1500 2000
−0.2
0
0.2
A2
ik
0 500 1000 1500 2000
−0.2
0
0.2
A2
ik
0 500 1000 1500 2000
−0.2
0
0.2
A2
ik
0 500 1000 1500 2000
−0.5
0
0.5
A2
ik
0 500 1000 1500 2000
−0.2
0
0.2
A2
ik
0 500 1000 1500 2000
−0.2
0
0.2
A2
ik
0 500 1000 1500 2000
−0.2
0
0.2
A2
ik
0 500 1000 1500 2000
−0.2
0
0.2
A2
ik
0 500 1000 1500 2000
−0.2
0
0.2
A2
ik
0 500 1000 1500 2000
−0.2
0
0.2
A2
ik
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

36. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
Traceplots for Non-Financial: local covariates Ai
0 500 1000 1500 2000
−0.5
0
0.5
A3
ik
0 500 1000 1500 2000
−0.5
0
0.5
A3
ik
0 500 1000 1500 2000
−0.5
0
0.5
A3
ik
0 500 1000 1500 2000
−0.5
0
0.5
A3
ik
0 500 1000 1500 2000
−0.5
0
0.5
A3
ik
0 500 1000 1500 2000
−0.5
0
0.5
A3
ik
0 500 1000 1500 2000
−0.5
0
0.5
A3
ik
0 500 1000 1500 2000
−0.5
0
0.5
A3
ik
0 500 1000 1500 2000
−0.5
0
0.5
A3
ik
0 500 1000 1500 2000
−0.5
0
0.5
A3
ik
0 500 1000 1500 2000
−0.5
0
0.5
A3
ik
0 500 1000 1500 2000
−0.5
0
0.5
A3
ik
0 500 1000 1500 2000
−0.5
0
0.5
A3
ik
0 500 1000 1500 2000
−0.5
0
0.5
A3
ik
0 500 1000 1500 2000
−0.5
0
0.5
A3
ik
0 500 1000 1500 2000
−0.5
0
0.5
A3
ik
0 500 1000 1500 2000
−0.5
0
0.5
A3
ik
0 500 1000 1500 2000
−0.5
0
0.5
A3
ik
0 500 1000 1500 2000
−0.5
0
0.5
A3
ik
0 500 1000 1500 2000
−0.5
0
0.5
A3
ik
0 500 1000 1500 2000
−0.5
0
0.5
A3
ik
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

37. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
Traceplots for Error variance parameters (Sigmas)
0 10002000
0
0.01
0.02
0.03
σ2
ij
0 10002000
0.05
0.1
0.15
0.2
0.25
σ2
ij
0 10002000
0
0.05
0.1
0.15
0.2
σ2
ij
0 10002000
0
0.01
0.02
0.03
0.04
σ2
ij
0 10002000
0
0.05
0.1
σ2
ij
0 10002000
0
0.05
0.1
0.15
0.2
σ2
ij
0 10002000
0
0.02
0.04
0.06
0.08
σ2
ij
0 10002000
0.05
0.1
0.15
0.2
0.25
σ2
ij
0 10002000
0
0.005
0.01
0.015
σ2
ij
0 10002000
0
0.01
0.02
0.03
0.04
σ2
ij
0 10002000
0.05
0.1
0.15
0.2
σ2
ij
0 10002000
0.02
0.04
0.06
0.08
σ2
ij
0 10002000
0.005
0.01
0.015
0.02
0.025
σ2
ij
0 10002000
0
0.05
0.1
0.15
0.2
σ2
ij
0 10002000
0.1
0.2
0.3
0.4
0.5
σ2
ij
0 10002000
0
0.1
0.2
0.3
0.4
σ2
ij
0 10002000
0
0.1
0.2
0.3
0.4
σ2
ij
0 10002000
0.05
0.1
0.15
0.2
0.25
σ2
ij
0 10002000
0
0.05
0.1
0.15
0.2
σ2
ij
0 10002000
0.1
0.2
0.3
0.4
0.5
σ2
ij
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

38. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
Traceplots for Error variance parameters (Psis)
0 10002000
0.8
1
1.2
1.4
1.6
ψ2
ij
0 10002000
0
0.5
1
1.5
ψ2
ij
0 10002000
0.5
1
1.5
ψ2
ij
0 10002000
0
0.5
1
1.5
ψ2
ij
0 10002000
0.8
1
1.2
1.4
ψ2
ij
0 10002000
0
0.5
1
1.5
ψ2
ij
0 10002000
0
0.5
1
1.5
ψ2
ij
0 10002000
0.2
0.4
0.6
0.8
ψ2
ij
0 10002000
0.05
0.1
0.15
0.2
0.25
ψ2
ij
0 10002000
0.2
0.25
0.3
0.35
0.4
ψ2
ij
0 10002000
0.1
0.2
0.3
0.4
ψ2
ij
0 10002000
0.1
0.2
0.3
0.4
ψ2
ij
0 10002000
0.1
0.15
0.2
0.25
ψ2
ij
0 10002000
0.2
0.4
0.6
0.8
1
ψ2
ij
0 10002000
0
0.2
0.4
0.6
0.8
ψ2
ij
0 10002000
0
0.5
1
1.5
ψ2
ij
0 10002000
0.2
0.4
0.6
0.8
1
ψ2
ij
0 10002000
0.1
0.2
0.3
0.4
0.5
ψ2
ij
0 10002000
0.2
0.4
0.6
0.8
1
ψ2
ij
0 10002000
0.2
0.4
0.6
0.8
1
ψ2
ij
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

39. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
Traceplots for Latent factors at time t
0 500 1000 1500 2000
−1
0
1
0 500 1000 1500 2000
0
1
2
3
0 500 1000 1500 2000
−1
0
1
2
0 500 1000 1500 2000
0
1
2
3
0 500 1000 1500 2000
0
1
2
3
0 500 1000 1500 2000
0
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

40. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments 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 2000
0
2
4
6
0 500 1000 1500 2000
−6
−4
−2
0
0 500 1000 1500 2000
0
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

41. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
CDS Local Covariates
CDS sector: The estimated parameters a1
i of the local covariates
and their corresponding standard errors
DebtGDP ExpGDP Indprod Inflation m3GDP realGDP Unempl. Mrk.
acds
GER -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)
acds
GR 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)
acds
SP -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)
acds
FR -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)
acds
IRE 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)
acds
IT -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)
acds
PT 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

42. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
Comments on the results
Discuss the results of the estimated parameters of the
country-specific (local) covariates appeared in the first equation of
the proposed model. We observe that:
The α1
i estimated parameters for the domestic equity index
(Mrk) are negative and most of them seem to be statistically
significant, indicating that a decrease in the equity index leads
to an increase of the CDS values (negatively correlated)
With the term ’statistically significant’ we mean that the
posterior distribution of the respective parameter does not
contain the value of zero, or the probability around zero is
very small
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

43. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
Comments on the results
The returns of the CDS of Greece and Portugal are affected
by the local covariates of Debt/GDP (0.19 and 0.14
respectively), indicating that an increase of the Debt/GDP
leads to an increase of the CDS returns. This is reasonable,
since an increase of the ratio of debt over the GDP indicates
that the government debt is more than what the country
produces and leads to an increase in the CDS price
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

44. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
Comments on the results
The CDS returns of Germany and France are negatively
related to the Export/GDP covariate, while CDS of Greece
and Ireland are positive related to Export/GDP
An increase of Industrial production in Greece and Spain leads
to increased CDS returns
Positive relation exist between Inflation and CDS returns of
Germany and Ireland, and between m3GDP and CDS of
Germany, Spain, Ireland and Italy
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

45. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
Financial and NonFinancial Local Covariates
Financial and Non-Financial sector: The estimated parameters a2
i
and a3
i of the local covariates and their corresponding standard
errors
Indprod Inflation Sectorial Index Indprod Inflation Sectorial Index
aFin
IT -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)
aFin
IRL -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)
aFin
GER -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)
aFin
PT 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)
aFin
SP 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)
aFin
FR -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)
aNFin
FR -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

46. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
Comments on the results
The estimated parameters for the sectorial index
(Financial/Non-Financial) are positive and all of them seem to
be statistically significant indicating that an increase in the
sectorial index leads to an increase of their returns and prices
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

47. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
CDS Sector Covariates
CDS sector: The estimated parameters A1
i of the sector covariates
and their corresponding standard errors
Vpremium Liquidity Spread Exchange Rate
Acds
GER 0.12 0.13 -0.08
(0.05) (0.05) (0.05)
Acds
GR 0.06 0.08 -0.03
(0.05) (0.05) (0.04)
Acds
SP 0.04 0.03 -0.07
(0.05) (0.06) (0.04)
Acds
FR 0.04 0.09 -0.12
(0.04) (0.05) (0.04)
Acds
IRE 0.08 0.03 0.01
(0.05) (0.05) (0.04)
Acds
IT 0.07 0.10 -0.08
(0.04) (0.05) (0.04)
Acds
PT 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

48. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
Financial and NonFinancial Sector Covariates
Financial and Non-Financial sector: The estimated parameters A2
i
and A3
i of the sector covariates and their corresponding standard
errors
Momentum RP − Europe RP − US Momentum RP − Europe RP − US
AFin
IT 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)
AFin
IRL -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)
AFin
GER 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)
AFin
PT -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)
AFin
SP -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)
AFin
FR -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)
ANFin
FR 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

49. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
Comments on the results
The structural sector-based covariates, Gj
t , affect the beta
parameters in time, since some parameters A1
i of the
Vpremium (for Germany) of the Liquidity Spread (for
Germany and France) and of Exchange Rate (for France)
seem to be statistically significant
The beta of the asset of Italy for the Financial sector seems to
be affected by the Risk Premium of Europe and US, while the
betas of Financial sector of Ireland is affected by momentum
covariate. For the assets of Non-Financial sector only the
betas of Italy and Spain seem to be affected by the
momentum and RP-US, respectively.
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

50. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
Common Covariates
The estimated parameters Bj of the common covariates and their
corresponding standard errors
Credit Spread Tbill Term Spread VIX
CDS Bcds
cr.sp Bcds
tbill Bcds
ter.sp. Bcds
vix
Estimates 0.09 0.15 -0.01 0.05
Std.Error (0.10) (0.07) (0.07) (0.08)
Financials BFin
cr.sp BFin
tbill BFin
ter.sp. BFin
vix
Estimates -0.21 -0.13 -0.03 0.09
Std.Error (0.10) (0.08) (0.08) (0.08)
Non-Financial BNFin
cr.sp BNFin
tbill BNFin
ter.sp. BNFin
vix
Estimates 0.02 0.01 -0.05 0.09
Std.Error (0.13) (0.09) (0.08) (0.10)
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

51. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
Autoregressive Parameters
The estimated autoregressiver parameters ϕj
i and their
corresponding standard errors [in the time-varying betas equation]
CDS ϕcds
GER ϕcds
GR ϕcds
SP ϕcds
FR ϕcds
IRE ϕcds
IT ϕcds
PO
Estimates -0.15 -0.04 0.02 -0.12 0.11 -0.03 -0.01
Std.Error (0.05) (0.06) (0.07) (0.05) (0.05) (0.06) (0.05)
Financials ϕFin
IT ϕFin
IRL ϕFin
GER ϕFin
PT ϕFin
SP ϕFin
FR
Estimates 0.29 0.14 -0.04 0.09 -0.01 0.09
Std.Error (0.08) (0.05) (0.06) (0.09) (0.08) (0.06)
Non − Financial ϕNFin
IT ϕNFin
GR,t ϕNFin
IRL ϕNFin
GER ϕNFin
PT ϕNFin
SP ϕNFin
FR
Estimates -0.18 0.14 -0.28 -0.32 -0.27 -0.16 0.14
Std.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

52. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
Constant Parameters
The estimated parameters βj
ic and their corresponding standard
errors [constants in the time-varying betas equation]
CDS βcds
GER βcds
GR βcds
SP βcds
FR βcds
IRE βcds
IT βcds
PO
Estimates -0.05 0.11 0.11 0.07 0.04 0.11 0.09
Std.Error (0.04) (0.04) (0.05) (0.04) (0.05) (0.04) (0.04)
Financials βFin
IT βFin
IRL βFin
GER βFin
PT βFin
SP βFin
FR
Estimates -0.19 0.05 0.03 0.06 -0.04 0.004
Std.Error (0.06) (0.02) (0.03) (0.05) (0.03) (0.03)
Non − Financial βNFin
IT βNFin
GR,t βNFin
IRL βNFin
GER βNFin
PT βNFin
SP βNFin
FR
Estimates -0.11 -0.04 0.004 0.02 -0.04 0.04 -0.02
Std.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

53. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
Error Variance Parameters
The estimated error variance parameters σ2
ij and their
corresponding standard errors
CDS σ2
GER σ2
GR σ2
SP σ2
FR σ2
IRL σ2
IT σ2
PT
Estimates 0.009 0.094 0.111 0.014 0.028 0.076 0.036
Std.Error (0.003) (0.015) (0.021) (0.004) (0.009) (0.014) (0.009)
Financials σ2
IT σ2
IRL σ2
GER σ2
PT σ2
SP σ2
FR
Estimates 0.15 0.008 0.017 0.089 0.049 0.014
Std.Error (0.02) (0.002) (0.004) (0.013) (0.007) (0.003)
Non − Financial σ2
IT σ2
GR σ2
IRL σ2
GER σ2
PT σ2
SP σ2
FR
Estimates 0.09 0.31 0.17 0.19 0.11 0.04 0.21
Std.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

54. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
Error Variance Parameters
The estimated error variance parameters ψ2
ij and their
corresponding standard errors [in the time-varying betas equation]
CDS ψ2
GER ψ2
GR ψ2
SP ψ2
FR ψ2
IRL ψ2
IT ψ2
PT
Estimates 1.06 0.73 0.78 0.83 0.99 0.74 0.75
Std.Error (0.09) (0.07) (0.08) (0.07) (0.10) (0.07) (0.07)
Financials ψ2
IT ψ2
IRL ψ2
GER ψ2
PT ψ2
SP ψ2
FR
Estimates 0.34 0.14 0.28 0.18 0.19 0.17
Std.Error (0.05) (0.01) (0.03) (0.03) (0.02) (0.02)
Non − Financial ψ2
IT ψ2
GR ψ2
IRL ψ2
GER ψ2
PT ψ2
SP ψ2
FR
Estimates 0.45 0.23 0.69 0.42 0.21 0.57 0.48
Std.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

55. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
CDS sector: Median and 5-th, 95-th percentiles for Time-varying
betas 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

56. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
Financial sector: Median and 5-th, 95-th percentiles for
Time-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

57. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
Non-Financial sector: Median and 5-th, 95-th percentiles for
Time-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

58. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments 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

59. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
Financial sector: Scatterplots of latent factor vs the time-varying
betas
−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

60. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
The data
Some convergence plots
Parameter Estimates
Comments on the results
Non-Financial sector: Scatterplots of latent factor vs the
time-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

61. Introduction
The Dynamic Factor Model
Bayesian Inference via MCMC
Simulation Study
Application to Sovereign CDS and Equity Data
Conclusions and Current/Future research
Conclusions and Current/Future research
Conclusions and Current/Future research
We present a dynamic factor model that explains how
financial returns are affected by latent, sector-based factors
and macro-systemic risk factors
Model selection exercise
Correlation among the different sectors
Time-varying variances/covariances
Ioannis Vrontos, Athens University of Economics and Business DFM: Inference and Empirical Application

62. 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.