Granger Causality and Dynamic Structural Systems 1 Halbert White and - PowerPoint PPT Presentation

Granger Causality and Dynamic Structural Systems 1 Halbert White and Xun Lu Department of Economics, University of California, San Diego December 10, 2009 1 forthcoming, Journal of Financial Econometrics 1/35

Objective Relate Granger causality to a notion of structural causality � Granger ( G ) causality Granger, 1969 and Granger and Newbold, 1986 � Structural causality White and Kennedy, 2008 and White and Chalak "Settable Systems," JMLR 2009 2/35

Outline 1. Granger causality, a dynamic DGP and structural causality 2. Granger causality and time-series natural experiments 3. Granger causality and structural vector autoregressions (VARs) 4. Testing …nite-order Granger causality 5. Conditional exogeneity 6. Applications 7. Conclusions 3/35

1. Granger causality, a dynamic DGP and structural causality 4/35

Granger causality Notation � subscript t denotes a variable at time t . � superscript t denotes a variable’s " t -history", (e.g., Y t � f Y 0 , Y 1 , ..., Y t g ). De…nition 2.1: Granger non-causality Let f Q t , S t , Y t g be a sequence of random vectors. Suppose that Y t ? Q t j Y t � 1 , S t t = 1 , 2 , ... . Then Q does not G � cause Y w.r.t. S . Else Q G � causes Y w.r.t. S . 5/35

Data generating process (DGP) Assumption A.1 (White and Kennedy, 2009) Let f D t , U t , W t , Y t , Z t ; t = 0 , 1 , ... g be a stochastic process. Further, suppose that ( D t � 1 , U t , W t , Z t ) , D t ( ( Y t � 1 , D t , U t , W t , Z t ) Y t ( where, for an unknown measurable k y � 1 function q t , f Y t g is structurally generated as Y t = q t ( Y t � 1 , D t , Z t , U t ) , t = 1 , 2 , .... 6/35

Data generating process (DGP) � Y t = q t ( Y t � 1 , D t , Z t , U t ) , t = 1 , 2 , .... � f D t , W t , Y t , Z t g observable; f U t g unobservable � Interested in � e¤ects of D t on Y t (time-series natural experiment) 2 , t ) 0 , e¤ects of Y t � 1 � with Y t = ( Y 0 1 , t , Y 0 on Y 1 , t (structural 2 VAR) 7/35

Structural causality De…nition 3.1 (Direct causality: structural VAR) Given A.1, for given t > 0 , j 2 f 1 , ..., k y g , and s , suppose ( i ) for all admissible values of y t � 1 ( s ) , d t , z t , and u t , y t � 1 ! q j , t ( y t � 1 , d t , z t , u t ) is constant in y t � 1 . s s Then Y t � 1 does not directly structurally cause Y j , t : s d Y t � 1 6) S Y j , t s d Else Y t � 1 directly structurally causes Y j , t : Y t � 1 ) S Y j , t s s Notation: : sub-vector of y t � 1 with elements indexed by non-empty � y t � 1 s set s � f 1 , ..., k y g � f 0 , ..., t � 1 g ( s ) : sub-vector of y t � 1 with elements of s excluded. � y t � 1 8/35

Structural causality De…nition 3.1 (Direct causality: time-series natural experiment) Given A.1, for given t > 0 , j 2 f 1 , ..., k y g , and s , suppose that ( ii ) for all admissible values of y t � 1 , d t ( s ) , z t , and u t , d t s ! q j , t ( y t � 1 , d t , z t , u t ) is constant in d t s . d Then D t s does not directly structurally cause Y jt : D t 6) S Y j , t s d Else D t s directly structurally causes Y j , t : D t ) S Y j , t s Notation: s : sub-vector of d t with elements indexed by non-empty set � d t s � f 1 , ..., k d g � f 0 , ..., t g ( s ) : sub-vector of d t with the elements of s excluded � d t 9/35

Structural causality � Recursive substitution of Y t = q t ( Y t � 1 , D t , Z t , U t ) , t = 1 , 2 , .... yields Y t = r t ( Y 0 , D t , Z t , U t ) , t = 1 , 2 , ..., De…nition 3.2 (Total causality: time-series natural experiment) Given A.1, suppose for all admissible values of y 0 , z t , and u t , d t ! r t ( y 0 , d t , z t , u t ) is constant in d t . Then D t does not structurally cause Y t : D t 6) S Y t Else D t structurally causes Y t : D t ) S Y t 10/35

2. Granger causality and time-series natural experiments 11/35

G-causality, conditional exogeneity, and direct causality � Let X t � ( W t , Z t ) , t = 0 , 1 , ... . Assumption A.2 ( a ) (conditional exogeneity) D t ? U t j Y t � 1 , X t , t = 1 , 2 , .... d Proposition 4.1 Let A.1 and A.2 ( a ) hold. If D t 6) S Y t , t = 1 , 2 , ... , then D does not G � cause Y w.r.t. X . 12/35

G-causality, conditional exogeneity, and direct causality De…nition 4.3 Suppose A.1 holds and that for each y 2 supp ( Y t ) there exists a measurable mapping ( y t � 1 , x t ) ! f t , y ( y t � 1 , x t ) such that w . p . 1 Z 1 f q t ( Y t � 1 , D t , Z t , u t ) < y g dF t ( u t j Y t � 1 , X t ) = f t , y ( Y t � 1 , X t ) Then D t does not directly cause Y t w.p.1 w.r.t. ( Y t � 1 , X t ) : d D t 6) S ( Y t � 1 , X t ) Y t . Else D t directly causes Y t with pos. prob. w.r.t. ( Y t � 1 , X t ) : d D t ) S ( Y t � 1 , X t ) Y t . 13/35

G-causality, conditional exogeneity, and direct causality Theorem 4.4 d Let A.1 and A.2 ( a ) hold. Then D t 6) S ( Y t � 1 , X t ) Y t , t = 1 , 2 , ..., if and only if D does not G � cause Y w.r.t. X. 14/35

Finite-order G-causality and Markov structures Notation: …nite histories Y t � 1 � ( Y t � ` , ..., Y t � 1 ) and Q t � ( Q t � k , ..., Q t ) . De…nition 4.8 Let f Q t , S t , Y t g be a sequence of random variables, and let k � 0 and ` � 1 be given …nite integers. Suppose Y t ? Q t j Y t � 1 , S t , t = 1 , 2 , ... Then Q does not …nite-order G � cause Y w.r.t. S. Else Q …nite-order G � causes Y w.r.t. S . 15/35

Finite-order G-causality and Markov structures Notation: …nite histories D t � ( D t � k , ..., D t ) , Z t � ( Z t � m , ..., Z t ) , X t � ( X t � τ 1 , ..., X t + τ 2 ) Assumption B.1 A.1 holds, and for k , ` , m 2 N , ` � 1 , Y t = q t ( Y t � 1 , D t , Z t , U t ) , t = 1 , 2 , .... Assumption B.2 For k , ` , and m as in B.1 and for τ 1 � m , τ 2 � 0, suppose D t ? U t j Y t � 1 , X t , t = 1 , ..., T � τ 2 . 16/35

Finite-order G-causality and Markov structures De…nition 4.9 Suppose B.1 holds and that for given τ 1 � m , τ 2 � 0 and for each y 2 supp ( Y t ) there exists a σ ( Y t � 1 , X t ) � measurable version of Z 1 f q t ( Y t � 1 , D t , Z t , u t ) < y g dF t ( u t j Y t � 1 , X t ) . d Then D t 6) S ( Y t � 1 , X t ) Y t ( direct non-causality � σ ( Y t � 1 , X t ) w . p . 1). d ) S ( Y t � 1 , X t ) Y t . Else D t 17/35

Finite-order G-causality and Markov structures Theorem 4.10 d Let B.1 and B.2 hold. Then D t 6) S ( Y t � 1 , X t ) Y t , t = 1 , ..., T � τ 2 , if and only if Y t ? D t j Y t � 1 , X t , t = 1 , ..., T � τ 2 , i.e., D does not …nite-order G � cause Y w.r.t. X . 18/35

3. Granger causality and structural VARs 19/35

G-causality and structural VARs � Special case of A.1: structural VARs (set k d = 0 ) � The DGP becomes � Y t � 1 , Z t , U t � Y t = q t . � Letting Y t � ( Y 0 1 , t , Y 0 2 , t ) 0 , � , Z t , U t � Y t � 1 , Y t � 1 Y 1 , t = q 1 , t 1 2 q 2 , t ( Y t � 1 , Y t � 1 , Z t , U t ) . = Y 2 , t 1 2 20/35

G-causality and structural VARs Notation: Y 1 , t � 1 � ( Y 1 , t � ` , ..., Y 1 , t � 1 ) , Y 2 , t � 1 � ( Y 2 , t � ` , ..., Y 2 , t � 1 ) , Z t � ( Z t � m , ..., Z t ) , and X t � ( X t � τ 1 , ..., X t + τ 2 ) . Assumption C.1 A.1 holds, and for ` , m , 2 N , ` � 1 , suppose that Y t = q t ( Y t � 1 , Z t , U t ) , t = 1 , 2 , ... , such that, with 2 , t ) 0 and U t � ( U 0 Y t � ( Y 0 1 , t , Y 0 1 , t , U 0 2 , t ) 0 , Y 1 , t = q 1 , t ( Y t � 1 , Z t , U 1 , t ) Y 2 , t = q 2 , t ( Y t � 1 , Z t , U 2 , t ) . Assumption C.2 For ` and m as in C.1 and for τ 1 � m , τ 2 � 0, suppose that Y 2 , t � 1 ? U 1 t j Y 1 , t � 1 , X t , t = 1 , ..., T � τ 2 . 21/35

G-causality and structural VARs De…nition 5.2 Suppose C.1 holds and that for given τ 1 � m , τ 2 � 0 and for each y 2 supp ( Y 1 , t ) there exists a σ ( Y 1 , t � 1 , X t ) � measurable version of Z 1 f q 1 , t ( Y t � 1 , Z t , u 1 , t ) < y g dF 1 , t ( u 1 , t j Y 1 , t � 1 , X t ) . d Then Y 2 , t � 1 6) S ( Y 1 , t � 1 , X t ) Y 1 , t ( direct non-causality � σ ( Y 1 , t � 1 , X t ) w . p . 1). d Else Y 2 , t � 1 ) S ( Y 1 , t � 1 , X t ) Y 1 , t . 22/35

G-causality and structural VARs Theorem 5.3 d Let C.1 and C.2 hold. Then Y 2 , t � 1 6) S ( Y 1 , t � 1 , X t ) Y 1 , t , t = 1 , ..., T � τ 2 , if and only if Y 1 , t ? Y 2 , t � 1 j Y 1 , t � 1 , X t , t = 1 , ..., T � τ 2 , i.e., Y 2 does not …nite-order G � cause Y 1 w.r.t. X . 23/35

4. Testing …nite-order Granger causality 24/35

Testing …nite-order Granger causality Test : Y t ? Q t j Y t � 1 , S t . � Test conditional mean independence with linear regression Y t = α 0 + Y t � 1 ρ 0 + Q 0 t β 0 + S 0 t β 1 + ε t . � Test conditional mean independence with neural nets Y t = α 0 + Y t � 1 ρ 0 + Q 0 t β 0 + S 0 t β 1 r ψ ( Y t � 1 γ 0 , j + S 0 ∑ + t γ j ) β j + 1 + ε t . j = 1 � Test conditional independence with nonlinear transforms α 0 + ψ y , 2 ( Y t � 1 ) ρ 0 + ψ q ( Q t ) 0 β 0 + S 0 ψ y , 1 ( Y t ) = t β 1 r ∑ ψ ( Y t � 1 γ 0 , j + S 0 + t γ j ) β j + 1 + η t . j = 1 25/35

5.Conditional exogeneity 26/35