Bayesian Graphical Models for Structural Vector Autoregressive - PowerPoint PPT Presentation

Bayesian Graphical Models for Structural Vector Autoregressive Processes Daniel Ahelegbey, Monica Billio, and Roberto Cassin (2014) Presented by: Jacob Warren March 21, 2015 Presented by: Jacob Warren Bayesian Graphical Models for Structural Vector Autoregressive Processes March 21, 2015 1 / 1

Background This paper is about combining information from Dynamic Networks to inform the causal structure of Structural Vector Autoregressions The paper discusses using networks for estimating both structural and autoregressive coefficients, but my focus will be on the structural components (the autoregressive ones are easier) Presented by: Jacob Warren Bayesian Graphical Models for Structural Vector Autoregressive Processes March 21, 2015 2 / 1

Background: What is a SVAR? Consider the structural VAR(L) process: Y t = B 0 Y t + B 1 Y t − 1 + · · · + B L Y t − L + ε t Where ε t ∼ N (0 , I p ), and B 0 has 0s on the diagonal Presented by: Jacob Warren Bayesian Graphical Models for Structural Vector Autoregressive Processes March 21, 2015 3 / 1

Background: What is a SVAR? Consider the structural VAR(L) process: Y t = B 0 Y t + B 1 Y t − 1 + · · · + B L Y t − L + ε t Where ε t ∼ N (0 , I p ), and B 0 has 0s on the diagonal Rewriting the SVAR into a reduced-form VAR: Y t = A − 1 0 B 1 Y t − 1 + A − 1 0 B 2 Y t − 2 + . . . A − 1 0 B L Y t − L + A − 1 0 ε t where A 0 = I − B 0 The problem is that the structural parameter, A 0 is not identified. Presented by: Jacob Warren Bayesian Graphical Models for Structural Vector Autoregressive Processes March 21, 2015 3 / 1

Background: Identification Issue Y t = A − 1 0 B 1 Y t − 1 + A − 1 0 B 2 Y t − 2 + . . . A − 1 0 B L Y t − L + A − 1 0 ε t Observe that A − 1 0 A − 1 ′ is the covariance of the error term. 0 Presented by: Jacob Warren Bayesian Graphical Models for Structural Vector Autoregressive Processes March 21, 2015 4 / 1

Background: Identification Issue Y t = A − 1 0 B 1 Y t − 1 + A − 1 0 B 2 Y t − 2 + . . . A − 1 0 B L Y t − L + A − 1 0 ε t Observe that A − 1 0 A − 1 ′ is the covariance of the error term. 0 But, for any orthogonal matrix Q (so QQ ′ = I ), A − 1 0 A − 1 ′ = A − 1 0 QQ ′ A − 1 ′ = ( A − 1 0 Q )( A 0 Q ) ′ 0 0 So the structural parameters, A 0 are not identified ◮ Requires making some identification assumptions to perform structural analysis Presented by: Jacob Warren Bayesian Graphical Models for Structural Vector Autoregressive Processes March 21, 2015 4 / 1

Background: Directed Acyclic Graphs A Directed Acyclic Graph is a collection G = {V , E} , where V is the set of vertices and E is the set of edges. For example: Figure 1 : DAG from Grzegorcyzk (2001) In this graph, node A is the parent of B and C. B and C are child nodes of A. D is the child node of both B and C, and E is the child node of E. A graph is said to be acyclic if no node is a descendant of itself Presented by: Jacob Warren Bayesian Graphical Models for Structural Vector Autoregressive Processes March 21, 2015 5 / 1

Connecting DAG and SVAR Y t = B 0 Y t + B 1 Y t − 1 + · · · + B L Y t − L + ε t Then, there is a one-to-one relationship between the regression matrices and Directed Acyclic Graphs, given as: X j t − s → X i t ⇐ ⇒ B s ( i , j ) � = 0 Where X j t means that X j t − s → X i t − s ”causes” in some way, X i t . Presented by: Jacob Warren Bayesian Graphical Models for Structural Vector Autoregressive Processes March 21, 2015 6 / 1

Connecting DAG and SVAR Y t = B 0 Y t + B 1 Y t − 1 + · · · + B L Y t − L + ε t Then, there is a one-to-one relationship between the regression matrices and Directed Acyclic Graphs, given as: X j t − s → X i t ⇐ ⇒ B s ( i , j ) � = 0 Where X j t means that X j t − s → X i t − s ”causes” in some way, X i t . ◮ Note that despite the directed aspect, the assumption of causality is not totally innocuous ◮ Think about a stock market ascent that may lead to an increase in GDP. The stock market may not cause GDP to increase, it could merely lead it in time. ◮ For more information, see Dawid 2008 This is similar to the notion of Granger Causality (with the exception of contemporaneous causation and conditional independence) Presented by: Jacob Warren Bayesian Graphical Models for Structural Vector Autoregressive Processes March 21, 2015 6 / 1

DAG and Cholesky One identification of the orthogonal shocks in the VAR is to use a Cholesky Decomposition PP ′ = A − 1 0 A − 1 ′ 0 Acyclicality implies a specific ordering of variables For example, if X 1 → X 2 , then let X 1 be the first variable in the system, and X 2 the second Presented by: Jacob Warren Bayesian Graphical Models for Structural Vector Autoregressive Processes March 21, 2015 7 / 1

Local Markov Property A graph is said to be posses the Local Markov Property if N � P ( X 1 , X 2 , . . . X n ) = P ( X i | pa ( X i )) i =1 where pa ( X i ) is the set of parent nodes for node X i Presented by: Jacob Warren Bayesian Graphical Models for Structural Vector Autoregressive Processes March 21, 2015 8 / 1

Local Markov Property A graph is said to be posses the Local Markov Property if N � P ( X 1 , X 2 , . . . X n ) = P ( X i | pa ( X i )) i =1 where pa ( X i ) is the set of parent nodes for node X i In the example above, the full likelihood of the graph can be simplified into: P ( G ) = P ( A ) P ( B | A ) P ( C | A ) P ( D |{ B , C } ) P ( E | D ) Presented by: Jacob Warren Bayesian Graphical Models for Structural Vector Autoregressive Processes March 21, 2015 8 / 1

Estimation Define B ⋆ s = G s ◦ Φ s , 0 ≤ s ≤ p , where ◦ is the Hadamarad element-wise matrix multiplication, and Φ s are the reduced form parameters G s are connectivity matrices that indicates dependence Presented by: Jacob Warren Bayesian Graphical Models for Structural Vector Autoregressive Processes March 21, 2015 9 / 1

Estimation Define B ⋆ s = G s ◦ Φ s , 0 ≤ s ≤ p , where ◦ is the Hadamarad element-wise matrix multiplication, and Φ s are the reduced form parameters G s are connectivity matrices that indicates dependence Thus, the reduced form parameters of the VAR can be written as: A 0 = I − G 0 ◦ Φ 0 A i = ( I − G 0 ◦ Φ 0 ) − 1 ( G i ◦ Φ i ) Presented by: Jacob Warren Bayesian Graphical Models for Structural Vector Autoregressive Processes March 21, 2015 9 / 1

Estimation: Bayesian Paradigm (priors) Y t = B ⋆ 0 Y t + B ⋆ 1 Y t − 1 + · · · + B ⋆ L Y t − L + ε t , ε t ∼ N (0 , I ) Likelihood: ◮ The data matrix, X ∼ N (0 , Σ x ) Prior: ◮ Define the probability distribution over a graph as: P ( G , Θ) = P ( G ) P (Θ |G ) Where G is the set of graph structures (nodes, edges and directions), and Θ is the set of parameters. ◮ P ( G ) ∝ 1 ◮ B i are distributed normally ◮ Conditional on a complete graph, P (Σ | G ) ∼ IW Presented by: Jacob Warren Bayesian Graphical Models for Structural Vector Autoregressive Processes March 21, 2015 10 / 1

Estimation: Bayesian Paradigm (priors) Y t = B ⋆ 0 Y t + B ⋆ 1 Y t − 1 + · · · + B ⋆ L Y t − L + ε t , ε t ∼ N (0 , I ) Likelihood: ◮ The data matrix, X ∼ N (0 , Σ x ) Prior: ◮ Define the probability distribution over a graph as: P ( G , Θ) = P ( G ) P (Θ |G ) Where G is the set of graph structures (nodes, edges and directions), and Θ is the set of parameters. ◮ P ( G ) ∝ 1 ◮ B i are distributed normally ◮ Conditional on a complete graph, P (Σ | G ) ∼ IW Note: This seems a little strange, since they have not specified a prior on the covariance except conditional on a complete graph. If the graph is not complete, the covariance will not be IW distributed Presented by: Jacob Warren Bayesian Graphical Models for Structural Vector Autoregressive Processes March 21, 2015 10 / 1

Marginal Likelihood The marginal likelihood can be factorized � P ( X| G ) = P ( X| G , Σ x ) P (Σ x | G ) d Σ x Under the assumed likelihood/priors, the marginal likelihood has a closed form They estimate a Multivariate-Normal-Inverse-Wishart process and a Minnesota Prior process Presented by: Jacob Warren Bayesian Graphical Models for Structural Vector Autoregressive Processes March 21, 2015 11 / 1

Model Inference Since everything has closed forms, they implement the following Gibbs Sampler: Sample the graph from the conditional posterior using Metropolis 1 Hastings Sample the reduced form parameters directly from their posterior 2 Presented by: Jacob Warren Bayesian Graphical Models for Structural Vector Autoregressive Processes March 21, 2015 12 / 1

Simulations They compare their scheme to a competitor (the PC algorithm) They find that their model does about 10% better for a small-scale VAR (n=5), but comparably well in a larger system (n=10) Presented by: Jacob Warren Bayesian Graphical Models for Structural Vector Autoregressive Processes March 21, 2015 13 / 1

Simulations They compare their scheme to a competitor (the PC algorithm) They find that their model does about 10% better for a small-scale VAR (n=5), but comparably well in a larger system (n=10) An additional advantage of their model over the PC algorithm is that they can do prediction Presented by: Jacob Warren Bayesian Graphical Models for Structural Vector Autoregressive Processes March 21, 2015 13 / 1