Bayesian Graphical Models for Structural Vector Autoregressive Processes - D. F. Ahelegbey, M....

Bayesian Graphical Models for Structural VectorAutoregressive Processes

SYstemic Risk TOmography:Signals, Measurements, Transmission Channels, and Policy Interventions

D. F. Ahelegbey, Ca’ Foscari University of Venice (Italy)M. Billio, Ca’ Foscari University of Venice (Italy)R. Casarin, Ca' Foscari University of Venice (Italy)

ENSAE. November 21, 2013.

Bayesian Graphical Models for Structural VectorAutoregressive Processes

Daniel Felix Ahelegbey, Monica Billio, Roberto Casarin

Department of Economics, Ca’ Foscari University of Venice, Italy

Networks in Economics and Finance,ENSAE-Paris Tech, November 18 - 22, 2013

Structural Vector Autoregressive (SVAR)

A standard SVAR of order p is of the form

Xt = B0Xt + B1Xt−1 + . . .+ BpXt−p + εt (1)

Xt = X1t ,X2

t , . . . ,Xnt is n dimensional time series

B0, . . . ,Bp are (n×n) matrices of coefficients, B0 is zero diagonalεt is (n×1) i.i.d disturbance, zero mean and diagonal matrix Σε

Equation (1) is not directly estimable. General approach is through reduced formand use of impulse response functions to recover B0.

VAR in reduced-form is given as:

Xt = A1Xt−1 + A2Xt−2 + . . .+ ApXt−p + ut (2)

Ai = (I−B0)−1Bi , ut = (I−B0)−1εt , A0 = (I−B0), Σu = A−10 Σε (A−1

0 )′.

Imposing restrictions on the structural form leads to a cost in the generalizationof the results.

Hard to provide convincing restrictions without relying on theories.

Ahelegbey, Billio, Casarin Bayesian Graphical VAR

Bayesian Graphical Models

Graphical Models can be informally defined as statistical models represented inthe form of a graph, where the nodes (vertices) represent the variables and theedges shows the interactions.

Advantages of Graphical Models

Represent graphically the logical implication of relationshipsSuitable representation of the causal relationships using directed edgesClarity of interpretation when analyzing complex interactions

Why Bayesian Graphical Models?

For n variables and p lags, the possible number of structures

C(n) =n∑i=1

(−1)i+1(

ni

)2i(n−i)C(n− i), L(p,n) = 2(pn2)

Structure learning is NP-hard

Bayesian Model Averaging accounts for structural uncertainties


Our Main Contributions

1 Develop a Bayesian graphical approach to over-parameterized problem in VAR

We collapsed the SVAR relationships into :(i) Multivariate Autoregressive (MAR) structure and(ii) Multivariate instantaneous (MIN) structure

We combine the MIN and MAR - Bayesian Graphical VAR (BGVAR)

2 Application of MCMC for graphical model - applying a modified and moreefficient version of (Giudici and Castelo, 2003) checking acyclic constraints

3 Applied Contributions

Modeling macroeconomic time series

Compare MIN with PC-algorithm, and MAR with C-GCCompare BGVAR with classical BVAR

Interconnectedness of the financial system

Compare MAR with pairwise and conditional Granger causalityLinkages between financial and non-financial super-sectors


Related Literature

Application of Graphical Models to SVAR estimation

Swanson and Granger (1997), Bessler and Lee (2002), Demiralp andHoover (2003)

Bayesian approach to graphical model determination

Madigan and York (1995), Giudici and Green (1999), Dawid and Lauritzen(2001)

Markov Chain Monte Carlo (MCMC) as standard inference for sampling thenetwork structure

Giudici and Green (1999), Grzegorczyk and Husmeier (2008)

Application of Networks to Systemic Risk Analysis

Hautsch et al. (2012), Billio et al. (2012), Diebold and Yilmaz (2013)


Graphical Models and Structural VAR

A typical structural VAR model order p is of the form:

Xt = B0Xt + B1Xt−1 + . . .+ BpXt−p + εt (3)

Represented in the form of a graphical model with a one-to-one correspondencebetween the matrices of coefficients and a directed acyclic graph (DAG):

X jt−s → X i

t ⇐⇒ Bs (i , j) 6= 0 0≤ s ≤ p (4)

For Graphical approach to SVAR, we define the following notations:

Bs = (Gs Φs ) 0≤ s ≤ p (5)

Bs is n×n matrix of structural coefficientsGs is n×n matrix of causal structure, gij,s = 1 ⇐⇒ X j

t−s → X it

Φs is n×n matrix of coefficients, bij,s = φij,s if gij,s = 1() is the Hadamard product

From the above definition, equation (3) can be expressed in terms of (5) as:

Xt = (G0 Φ0)Xt + (G1 Φ1)Xt−1 + . . .+ (Gp Φp)Xt−p + εt (6)


Efficient Inference Scheme

We assume the marginal prior distribution on gij is a Bernoulli, therefore we havea uniform prior over all DAGs

Following (Geiger and Heckerman, 1994), we assume parameter independenceand modularity. This allows for marginalizing out the parameters to estimate thedependence structure.

Assume variables are samples from a multivariate normal distribution

Assume normal-Wishart prior (Geiger and Heckerman, 1994)

Score DAGs with Bayesian Gaussian equivalent (BGe) metric.

Search involves Addition or Removal of an edge at each iteration

Modify the proposal by (Giudici and Castelo, 2003) for acyclic constraints incontemporaneous dependence

Use the standard acceptance and rejection criterion for MCMC


Efficient Model Inference Scheme

1 Sample the structural relationships: (G0,G+|Data), G+ = (G1,G2, . . . ,Gp)

Following Friedman et al. (1998), we collapse the structural relationship inmultivariate autoregressive (MAR) - G+, and multivariate instantaneous (MIN)dependence - G0

The first step of the sampler is:

1.1 Sample MAR: (G+|Data), : X it |Xt−p ∼N (µ+,Σ+)

1.2 Sample MIN: (G0|Data), : X it |Xt\i ∼N (µ0,Σ0)

2 Sample the reduced-form parameters: (A+,Σa,0|G0,G+,Data)

Ga,+ = (In−G0)−1G+ Ga,0 = (In−G0)

A+,A0 is the parameters corresponding to Ga,+ and Ga,0αg ,+ and αg ,0 as the vectorized form of A+,A0The reduced-form Graphical VAR is represented as:

y = Zg ,+ (αg ,+)′+ vg ,0 (7)

vg ,0 ∼N (0,Ωg ,0), Ω = Σa,0⊗ I, Σa,0 = A−10 (A−1

0 )′ A0 = In− (G0 Φ0)


Modeling Macroeconomic Time Series

Data : 20 US-macroeconomic variables from 1959:Q1 to 2008:Q4 (medium VARdataset of Koop, 2013),

We estimate the model with lag length (p = 1) chosen according to the BICcriterion.

The dependent variables are (7)

Real gross domestic product (Y )Consumer price index (Π)Federal funds rate (R)Money stock - M2 (M)Real personal consumption expenditure (C)Industrial production index (IP)Unemployment rate (U)

The explanatory variables consists of the first lag of of the dependent variablesand the remaining 13 variables are considered as predictor variables

We use a moving window with an initial in-sample from 1960:Q1 to 1984:Q4

Forecast the next four quaters using a step-ahead forecast.


Comparing PC Algorithm and Multivariate Instantaneous

Ut

IPt

Yt Πt Rt

Ct Mt

(a) PC

Ut

IPt

Yt Πt Rt

Ct Mt

(b) MIN

Figure : The PC and MIN from 1960:Q1 - 2008:Q4.


Comparing Lagged dependence from 1960:Q1 to 2008:Q4.

−0

−0

0

−0

0

0

−0

−0

−1

−1

0

0

−1

1

−0

1

−0

0

−1

−0

0

−0

0

0

−0

−0

0

−0

0

0

0

0

0

1

−1

0

−0

−0

−0

−0

0

−0

−0

−0

−0

0

−0

−1

0

0

0

1

−0

−0

0

−0

−0

−0

0

0

0

−0

−0

−0

−0

−0

0

0

−0

0

−0

0

0

0

0

−0

1

0

0

0

−0

0

0

−0

0

−0

0

−0

−0

0

−0

−0

1

0

−0

−0

0

0

0

0

−0

0

0

−0

1

0

0

0

−1

−0

−0

0

0

0

1

−0

0

1

−1

0

0

1

−1

−0

1

−0

0

0

−1

−0

0

−0

0

0

−0

0

−0

0

0

−1

Y_t−

1P

i_t−

1R

_t−

1M

_t−

1C

_t−

1IP

_t−

1U

_t−

1M

P_t−

1N

B_t−

1R

T_t−

1C

U_t−

1H

S_t−

1P

P_t−

1P

C_t−

1H

E_t−

1M

1_t−

1S

P_t−

1IR

_t−

1E

R_t−

1E

N_t−

1

Y_t

Pi_t

R_t

M_t

C_t

IP_t

U_t

(a) C-GC

−0

−0

0

−0

0

0

−0

−0

−1

−1

0

0

−0

1

−1

1

−0

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0

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1

0

1

0

0

1

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0

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1

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1

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1

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1

1

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0

−1

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0

−0

0

0

−0

0

−0

0

0

−1

Y_t−

1P

i_t−

1R

_t−

1M

_t−

1C

_t−

1IP

_t−

1U

_t−

1M

P_t−

1N

B_t−

1R

T_t−

1C

U_t−

1H

S_t−

1P

P_t−

1P

C_t−

1H

E_t−

1M

1_t−

1S

P_t−

1IR

_t−

1E

R_t−

1E

N_t−

1

Y_t

Pi_t

R_t

M_t

C_t

IP_t

U_t

(b) MAR


Comparing BIC Scores

−1620

−1600

−1580

−1560

−1540

−1520

−1500

60:Q

1−84:Q

4

64:Q

1−88:Q

4

68:Q

1−92:Q

4

72:Q

1−96:Q

4

76:Q

1−00:Q

4

80:Q

1−04:Q

4

84:Q

1−08:Q

4

PC

MIN

(c) PC and MIN BIC

−1700

−1680

−1660

−1640

−1620

−1600

60:Q

1−84:Q

4

64:Q

1−88:Q

4

68:Q

1−92:Q

4

72:Q

1−96:Q

4

76:Q

1−00:Q

4

80:Q

1−04:Q

4

84:Q

1−08:Q

4

C−GC

MAR

(d) C-GC and MAR BIC


Comparing BVAR and BGVAR

−0.15

0.00

0.07

−0.27

0.07

0.08

0.02

−0.10

−0.80

−0.34

0.34

0.18

0.32

−0.10

−0.01

−0.03

0.32

0.15

−0.12

−0.03

−0.08

−0.06

0.01

0.18

−0.29

−0.20

0.19

−0.19

0.32

0.11

0.22

0.11

−0.11

0.32

−0.19

0.20

−0.14

−0.02

−0.08

0.03

0.43

−0.23

0.19

−0.34

−0.19

−0.01

−0.13

−0.23

0.07

−0.05

−0.06

0.06

0.23

−0.19

−0.14

0.11

−0.22

−0.31

0.05

0.34

−0.03

−0.12

−0.06

0.40

−0.38

−0.17

−0.11

0.29

−0.03

0.01

−0.33

0.17

0.08

−0.02

−0.07

−0.06

0.28

−0.03

0.11

0.08

−0.19

0.27

−0.08

0.01

0.16

0.04

0.24

−0.42

−0.04

−0.16

0.08

−0.06

0.61

0.06

−0.03

−0.36

−0.17

0.09

0.02

0.09

−0.04

0.01

0.21

0.07

0.03

−0.09

0.05

−0.08

−0.16

−0.01

−0.07

0.15

0.24

0.05

0.11

−0.20

0.16

0.27

−0.14

0.02

0.12

0.37

−0.48

−0.14

0.23

−0.12

0.05

0.00

−0.08

−0.05

0.00

−0.12

0.11

0.62

−0.29

−0.11

0.28

0.37

−0.12

−0.50

Y_t−

1P

i_t−

1R

_t−

1M

_t−

1C

_t−

1IP

_t−

1U

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1M

P_t−

1N

B_t−

1R

T_t−

1C

U_t−

1H

S_t−

1P

P_t−

1P

C_t−

1H

E_t−

1M

1_t−

1S

P_t−

1IR

_t−

1E

R_t−

1E

N_t−

1

Y_t

Pi_t

R_t

M_t

C_t

IP_t

U_t

(e) BVAR

−0.08

0.03

−0.00

−0.19

0.16

0.19

−0.06

0.00

−0.78

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.37

0.00

0.00

0.00

0.00

0.00

0.00

0.00

−0.07

0.00

0.00

0.00

0.27

0.06

0.25

0.00

−0.21

0.21

−0.11

0.05

−0.05

0.04

0.00

−0.03

0.34

−0.14

0.00

0.00

−0.19

0.00

0.00

0.00

0.07

0.00

0.00

0.00

0.00

−0.12

0.00

0.00

−0.22

−0.46

0.00

0.03

−0.08

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0.00

0.00

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0.08

0.00

0.19

0.00

−0.16

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.63

0.00

0.00

−0.23

0.00

0.00

0.00

0.08

0.00

0.00

0.16

0.00

0.00

0.00

0.00

0.00

−0.30

0.00

0.00

0.00

0.18

0.05

0.04

0.00

0.19

0.21

−0.04

0.05

0.10

0.24

−0.31

−0.09

0.15

−0.06

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.24

0.00

0.02

0.00

0.26

0.00

−0.39

Y_t−

1P

i_t−

1R

_t−

1M

_t−

1C

_t−

1IP

_t−

1U

_t−

1M

P_t−

1N

B_t−

1R

T_t−

1C

U_t−

1H

S_t−

1P

P_t−

1P

C_t−

1H

E_t−

1M

1_t−

1S

P_t−

1IR

_t−

1E

R_t−

1E

N_t−

1

Y_t

Pi_t

R_t

M_t

C_t

IP_t

U_t

(f) BGVAR


Comparing Forecast Performance of BVAR and BGVAR

−2

0

2

4 D

Mij t

1984:Q

4

1988:Q

4

1992:Q

4

1996:Q

4

2000:Q

4

2004:Q

4

2008:Q

4

Yt

Pit

Rt

Mt

Ct

IPt

Ut

Figure : Evolution of the Diebold and Mariano statistic of the BGVAR andBVAR from 1985:Q1 - 2008:Q4.


Interconnectedness of the Financial System

Data : 19 super-sectors of Euro Stoxx 600 from January 2001 to August 2013,obtained from Datastream

Financial : Banks, Insurance, Financial Services and Real Estate

Non-Financial: Construction & Materials, Industrial Goods & Services,Automobiles & Parts, Food & Beverage, Personal & Household Goods,Retail, Media, Travel & Leisure, Chemicals, Basic Resources, Oil & Gas,Telecommunications, Health Care, Technology, Utilities

We estimate the model with lag length (p = 1) chosen according to the BICcriterion.

Compare MAR with pairwise (P-GC) and conditional Granger causality (C-GC)

Percentage of all possible links among the super-sectors

Linkages between Financial and Non-financial super-sectors

Following Billio et al. (2012), we use a 36-month moving window with an initialin-sample from January 2001 to December 2013


Percentage of all possible linkages among the super-sectors

0

0.1

0.2

0.3

0.4

0.5

0.6

Jan01−D

ec0

3

Jul0

1−Ju

n04

Jan02−D

ec0

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Jul0

2−Ju

n05

Jan03−D

ec0

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Jul0

3−Ju

n06

Jan04−D

ec0

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5−Ju

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Jan06−D

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Jan07−D

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Jul0

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n10

Jan08−D

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Jul0

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n11

Jan09−D

ec1

1

Jul0

9−Ju

n12

Jan10−D

ec1

2

Jul1

0−Ju

n13

P−GC

C−GC

MAR

Figure : Evolution of total links of the P-GC, C-GC and MAR among thesuper-sectors of Euro Stoxx 600, from January 2001 to August 2013.

1 Pre-2005 - Aftermath of scandals, Enron and Worldcom2 2007-2009 - Recent financial crisis3 2010-2013 - The European sovereign crisis.


Linkage Between Financial and Non-Financial Super-sectors

0

0.1

0.2

0.3

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0.5

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Jan01−D

ec0

3

Jul0

1−Ju

n04

Jan02−D

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Jul0

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n05

Jan03−D

ec0

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Jul0

3−Ju

n06

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ec0

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Jul0

4−Ju

n07

Jan05−D

ec0

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Jul0

5−Ju

n08

Jan06−D

ec0

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Jul0

6−Ju

n09

Jan07−D

ec0

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Jul0

7−Ju

n10

Jan08−D

ec1

0

Jul0

8−Ju

n11

Jan09−D

ec1

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Jul0

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n12

Jan10−D

ec1

2

Jul1

0−Ju

n13

P−GC

C−GC

MAR

(a) Financial → Non Financial

0

0.1

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Jan01−D

ec0

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Jul0

1−Ju

n04

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Jul0

2−Ju

n05

Jan03−D

ec0

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Jul0

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ec0

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ec0

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7−Ju

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Jan08−D

ec1

0

Jul0

8−Ju

n11

Jan09−D

ec1

1

Jul0

9−Ju

n12

Jan10−D

ec1

2

Jul1

0−Ju

n13

P−GC

C−GC

MAR

(b) Non Financial → Financial


Comparing BIC Scores

−1400

−1300

−1200

−1100

−1000

−900

−800

Jan01−D

ec0

3

Jul0

1−Ju

n04

Jan02−D

ec0

4

Jul0

2−Ju

n05

Jan03−D

ec0

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Jul0

3−Ju

n06

Jan04−D

ec0

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Jul0

4−Ju

n07

Jan05−D

ec0

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Jul0

5−Ju

n08

Jan06−D

ec0

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Jul0

6−Ju

n09

Jan07−D

ec0

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Jul0

7−Ju

n10

Jan08−D

ec1

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Jul0

8−Ju

n11

Jan09−D

ec1

1

Jul0

9−Ju

n12

Jan10−D

ec1

2

Jul1

0−Ju

n13

P−GC

C−GC

MAR

Figure : BIC scores of P-GC, C-GC and MAR for linkages among thesuper-sectors of Euro Stoxx 600.


Conclusion: Relevance of Results

Our graphical approach allows for inference of the lagged dependence andcontemporaneous structure from the data providing a natural way to insertrestriction in the VAR dynamics

The BGVAR is more parsimonious than the BVAR and the forecast performs aresignificantly equivalent.

Able to learn from the data the structure and sign restrictions on thecontemporaneous relationships

Not only are financial institutions highly interconnected before and during crisisperiods (Billio et al 2012)

Financial and Non-financial institutions are also highly interconnected duringsuch periods

Pairwise Granger causality (P-GC) overestimates links and Conditional Granger(C-GC) suffers due to over-fitting.

The MAR deliver satisfactory results on the causal structures by accounting forcausal uncertainties when applying Bayesian model averaging.


Thank you !for your attention

Research supported by funding from the European Union, Seventh FrameworkProgramme FP7/2007-2013 under grant agreement SYRTO-SSH-2012-320270


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

Bayesian Graphical Models for Structural Vector Autoregressive Processes - D. F. Ahelegbey, M....

Economy & Finance

Transcript of Bayesian Graphical Models for Structural Vector Autoregressive Processes - D. F. Ahelegbey, M....