Introduction SVAR GMs Application to SVAR Recent Developments Graphical Causal Models for Time Series Econometrics: Some Recent Developments and Applications Alessio Moneta Max Planck Institute of Economics, Jena 10 December 2009 MAX-PLANCK-GESELLSCHAFT Mini Symposium: Causality and Time Series Analysis NIPS 2009, Vancouver Moneta Causal Search in TS Econometrics
Introduction SVAR GMs Application to SVAR Recent Developments Motivations Overview Scope ⊲ Application of methods of causal search to the problem of finding the appropriate causal order for the Structural Vector Autoregressive models (SVAR). Moneta Causal Search in TS Econometrics
Introduction SVAR GMs Application to SVAR Recent Developments Motivations Overview Overview ⊲ VAR and SVAR model ⊲ Causal search methods: graphical models ⊲ Application to the linear/Gaussian setting ⊲ Extensions: • Nonparametric setting • Non-Gaussian case: application of a method based on ICA Moneta Causal Search in TS Econometrics
Introduction SVAR GMs Application to SVAR Recent Developments Motivations Overview Overview ⊲ VAR and SVAR model ⊲ Causal search methods: graphical models ⊲ Application to the linear/Gaussian setting ⊲ Extensions: • Nonparametric setting • Non-Gaussian case: application of a method based on ICA Moneta Causal Search in TS Econometrics
Introduction SVAR GMs Application to SVAR Recent Developments VAR Identification Problem VAR vs. SVAR model Basic VAR model (reduced-form): Y t = A 1 Y t − 1 + . . . + A p Y t − p + u t . (1) • Y t : ( y t 1 , . . . , y tk ) ′ ; • A j ( j = 1, . . . , p ) are k × k coefficient matrices; • u t is the vector white noise process; • E ( u t u ′ t ) = Σ u . Moneta Causal Search in TS Econometrics
Introduction SVAR GMs Application to SVAR Recent Developments VAR Identification Problem VAR vs. SVAR model Basic VAR model (reduced-form): Y t = A 1 Y t − 1 + . . . + A p Y t − p + u t . (1) • Y t : ( y t 1 , . . . , y tk ) ′ ; • A j ( j = 1, . . . , p ) are k × k coefficient matrices; • u t is the vector white noise process; • E ( u t u ′ t ) = Σ u . Moneta Causal Search in TS Econometrics
Introduction SVAR GMs Application to SVAR Recent Developments VAR Identification Problem VAR vs. SVAR model Basic VAR model (reduced-form): Y t = A 1 Y t − 1 + . . . + A p Y t − p + u t . (1) • Y t : ( y t 1 , . . . , y tk ) ′ ; • A j ( j = 1, . . . , p ) are k × k coefficient matrices; • u t is the vector white noise process; • E ( u t u ′ t ) = Σ u . Moneta Causal Search in TS Econometrics
Introduction SVAR GMs Application to SVAR Recent Developments VAR Identification Problem VAR vs. SVAR model Basic VAR model (reduced-form): Y t = A 1 Y t − 1 + . . . + A p Y t − p + u t . (1) • Y t : ( y t 1 , . . . , y tk ) ′ ; • A j ( j = 1, . . . , p ) are k × k coefficient matrices; • u t is the vector white noise process; • E ( u t u ′ t ) = Σ u . Moneta Causal Search in TS Econometrics
Introduction SVAR GMs Application to SVAR Recent Developments VAR Identification Problem VAR vs. SVAR model Wold representation (in case of stationarity): ∞ ∑ Y t = Φ j u t − j , (2) j = 0 j where Φ j = ∑ i = 1 Φ j − i A i But for any nonsingular k × k matrix P we get: ∞ ∞ Φ j PP − 1 u t − j = ∑ ∑ Y t = Ψ j ε t − j , (3) j = 0 j = 0 where ε t − j = P − 1 u t − j and Ψ j = Φ j P ( j = 0, 1, 2, ... ) . Moneta Causal Search in TS Econometrics
Introduction SVAR GMs Application to SVAR Recent Developments VAR Identification Problem VAR vs. SVAR model Wold representation (in case of stationarity): ∞ ∑ Y t = Φ j u t − j , (2) j = 0 j where Φ j = ∑ i = 1 Φ j − i A i But for any nonsingular k × k matrix P we get: ∞ ∞ Φ j PP − 1 u t − j = ∑ ∑ Y t = Ψ j ε t − j , (3) j = 0 j = 0 where ε t − j = P − 1 u t − j and Ψ j = Φ j P ( j = 0, 1, 2, ... ) . Moneta Causal Search in TS Econometrics
Introduction SVAR GMs Application to SVAR Recent Developments VAR Identification Problem VAR vs. SVAR model If we premultiply equation (1) by P − 1 we get P − 1 Y t = P − 1 A 1 Y t − 1 + . . . + P − 1 A p Y t − p + P − 1 u t . (4) SVAR model (structural form): Γ 0 Y t = Γ 1 Y t − 1 + . . . + Γ p Y t − p + ε t , (5) where Γ 0 = P − 1 , Γ j = P − 1 A j ( j = 1, . . . , p ) . ⊲ choice of P ( Γ 0 ) based on information about the contemporaneous causal structure. Moneta Causal Search in TS Econometrics
Introduction SVAR GMs Application to SVAR Recent Developments VAR Identification Problem VAR vs. SVAR model If we premultiply equation (1) by P − 1 we get P − 1 Y t = P − 1 A 1 Y t − 1 + . . . + P − 1 A p Y t − p + P − 1 u t . (4) SVAR model (structural form): Γ 0 Y t = Γ 1 Y t − 1 + . . . + Γ p Y t − p + ε t , (5) where Γ 0 = P − 1 , Γ j = P − 1 A j ( j = 1, . . . , p ) . ⊲ choice of P ( Γ 0 ) based on information about the contemporaneous causal structure. Moneta Causal Search in TS Econometrics
Introduction SVAR GMs Application to SVAR Recent Developments VAR Identification Problem VAR vs. SVAR model If we premultiply equation (1) by P − 1 we get P − 1 Y t = P − 1 A 1 Y t − 1 + . . . + P − 1 A p Y t − p + P − 1 u t . (4) SVAR model (structural form): Γ 0 Y t = Γ 1 Y t − 1 + . . . + Γ p Y t − p + ε t , (5) where Γ 0 = P − 1 , Γ j = P − 1 A j ( j = 1, . . . , p ) . ⊲ choice of P ( Γ 0 ) based on information about the contemporaneous causal structure. Moneta Causal Search in TS Econometrics
Introduction SVAR GMs Application to SVAR Recent Developments VAR Identification Problem VAR vs. SVAR model ⊲ choice of P ( Γ 0 ) in the literature: • Choleski decomposition such that: P is lower diagonal and Ω = E ( ε t ε ′ t ) = I k . • a priori (theoretical, institutional) zero-restrictions; ⊲ Our proposal: inferring P ( Γ 0 ) starting from the estimated residuals ˆ u t • conditional independence relations − → causal relationships (graphical models) • independent component analysis (in case of non-Gaussianity) Moneta Causal Search in TS Econometrics
Introduction SVAR GMs Application to SVAR Recent Developments VAR Identification Problem VAR vs. SVAR model ⊲ choice of P ( Γ 0 ) in the literature: • Choleski decomposition such that: P is lower diagonal and Ω = E ( ε t ε ′ t ) = I k . • a priori (theoretical, institutional) zero-restrictions; ⊲ Our proposal: inferring P ( Γ 0 ) starting from the estimated residuals ˆ u t • conditional independence relations − → causal relationships (graphical models) • independent component analysis (in case of non-Gaussianity) Moneta Causal Search in TS Econometrics
Introduction SVAR GMs Application to SVAR Recent Developments Causal Search Algorithm Graphical models Graphs have two functions: ⊲ Representation of causal structures ⊲ Representation of causal independence relations Moneta Causal Search in TS Econometrics
Introduction SVAR GMs Application to SVAR Recent Developments Causal Search Algorithm Graphical models Representation of causal structures: ⊲ edge: causal influence • Undirected edges: X — Y (ambiguous causal influence between X and Y ) • Directed edges: X − → Y , X ← − Y • Bi-directed edges: X ← → Y ⊲ DAGs: Directed acyclic graphs (only directed edges). Moneta Causal Search in TS Econometrics
Introduction SVAR GMs Application to SVAR Recent Developments Causal Search Algorithm Graphical models Representation of conditional independence relations: ⊲ edge: statistical dependence • X ⊥ ⊥ Y | ∅ : no edge between X and Y • X ⊥ ⊥ / Y : X — Y ; X − → Y ; X ← − Y • X ⊥ ⊥ Z | Y : X − → Y − → Z ; X ← − Y ← − Z ; X ← − Y − → Z • X ⊥ ⊥ / Z | Y : X − → Y ← − Z Moneta Causal Search in TS Econometrics
Introduction SVAR GMs Application to SVAR Recent Developments Causal Search Algorithm Graphical models Rules of Inference: ⊲ edge: from conditional independence �→ causal relationships • DAGs among V 1 , . . . , V k • Causal Markov Condition: conditioned on its parents every node is independent of its nondescendants ; or: conditioned on its direct causes every variable is independent of its non-effects • Faithfulness Condition: every conditional independence relation among V 1 , . . . , V k is entailed by the Causal Markov Condition Moneta Causal Search in TS Econometrics
Introduction SVAR GMs Application to SVAR Recent Developments Causal Search Algorithm Search algorithm: PC, SGS, modified PC algorithm (Spirtes, Glymour and Scheines 2000): • Input: conditional independence tests • Output: set of Markov equivalent DAGs ⊲ Start: complete undirected graph among V 1 , . . . , V k ⊲ First step: elimination of edges whenever ⊥ ⊥ ⊲ Second step: statistical orientation of edges • search for unshielded colliders : X − → Y ← − Z ⊲ Third step: logical orientation of edges • X − → Y — Z �→ X − → Y − → Z • orient edges in order to avoid cycles Moneta Causal Search in TS Econometrics
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