Granger Causality and Dynamic Structural Systems 1

Transcription

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

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6 2/35 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

7 3/35 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

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

9 5/35 Granger causality Notation subscript t denotes a variable at time t. superscript t denotes a variable s "t-history", (e.g., Y t fy 0, Y 1,..., Y t g). De nition 2.1: Granger non-causality Let fq 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.

10 6/35 Data generating process (DGP) Assumption A.1 (White and Kennedy, 2009) Let fd t, U t, W t, Y t, Z t ; t = 0, 1,...g be a stochastic process. Further, suppose that D t ( (D t 1, U t, W t, Z t ), Y t ( (Y t 1, D t, U t, W t, Z t ) where, for an unknown measurable k y 1 function q t, fy t g is structurally generated as Y t = q t (Y t 1, D t, Z t, U t ), t = 1, 2,...

11 7/35 Data generating process (DGP) Y t = q t (Y t 1, D t, Z t, U t ), t = 1, 2,... fd t, W t, Y t, Z t g observable; fu t g unobservable Interested in e ects of D t on Y t (time-series natural experiment) with Y t = (Y 0 1,t, Y 0 2,t )0, e ects of Y t 1 2 on Y 1,t (structural VAR)

12 8/35 Structural causality De nition 3.1 (Direct causality: structural VAR) Given A.1, for given t > 0, j 2 f1,..., k y g, and s, suppose (i) for all admissible values of y t 1 (s), d t, z t, and u t, ys t 1! q j,t (y t 1, d t, z t, u t ) is constant in ys t 1. Then Ys t 1 does not directly structurally cause Y j,t : Y t 1 s d 6) S Y j,t Else Ys t 1 directly structurally causes Y j,t : Ys t 1 Notation: d ) S Y j,t ys t 1 : sub-vector of y t 1 with elements indexed by non-empty set s f1,..., k y g f0,..., t 1g y t 1 : sub-vector of y t (s) 1 with elements of s excluded.

13 9/35 Structural causality De nition 3.1 (Direct causality: time-series natural experiment) Given A.1, for given t > 0, j 2 f1,..., k y g, and s, suppose that (ii) for all admissible values of y t 1, d t (s), zt, and u t, d t s! q j,t (y t 1, d t, z t, u t ) is constant in d t s. Then Ds t does not directly structurally cause Y jt : Ds t d ) S Y j,t Else D t s directly structurally causes Y j,t : D t s Notation: d 6) S Y j,t d t s : sub-vector of d t with elements indexed by non-empty set s f1,..., k d g f0,..., tg d t (s) : sub-vector of d t with the elements of s excluded

14 10/35 Structural causality Recursive substitution of yields Y t = q t (Y t 1, D t, Z t, U t ), t = 1, 2,... 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

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

16 12/35 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,... Proposition 4.1 Let A.1 and A.2(a) hold. If D t d 6)S Y t, t = 1, 2,..., then D does not G cause Y w.r.t. X.

17 13/35 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 1fq t (Y t 1, D t, Z t, u t ) < yg 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 t d 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 t d )S(Y t 1,X t ) Y t.

18 14/35 G-causality, conditional exogeneity, and direct causality Theorem 4.4 Let A.1 and A.2(a) hold. Then D t d 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.

19 15/35 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 fq 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.

20 16/35 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 τ1,..., X t+τ2 ) 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.

21 17/35 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 1fq t (Y t 1, D t, Z t, u t ) < yg df t (u t j Y t 1, X t ). d Then D t 6)S(Yt 1,X t ) Y t (direct non-causality σ(y t 1, X t ) w.p.1). Else D t d )S(Yt 1,X t ) Y t.

22 18/35 Finite-order G-causality and Markov structures Theorem 4.10 Let B.1 and B.2 hold. Then D t if and only if d 6)S(Yt 1,X t ) Y t, t = 1,..., T τ 2, 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.

23 3. Granger causality and structural VARs 19/35

24 20/35 G-causality and structural VARs Special case of A.1: structural VARs (set k d = 0) The DGP becomes Letting Y t (Y 0 1,t, Y 0 2,t )0, Y t = q t Y t 1, Z t, U t. Y 1,t = q 1,t Y1 t 1, Y2 t 1, Z t, U t Y 2,t = q 2,t (Y1 t 1, Y2 t 1, Z t, U t ).

25 21/35 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 τ1,..., X t+τ2 ). 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 Y t (Y1,t 0, Y 2,t 0 )0 and U t (U1,t 0, U0 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 1t j Y 1,t 1, X t, t = 1,..., T τ 2.

26 22/35 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 1fq 1,t (Y t 1, Z t, u 1,t ) < yg df 1,t (u 1,t j Y 1,t 1, X t ). d Then Y 2,t 1 6)S(Y1,t 1,X t ) Y 1,t (direct non-causality σ(y 1,t 1, X t ) w.p.1). Else Y 2,t 1 d )S(Y1,t 1,X t ) Y 1,t.

27 23/35 G-causality and structural VARs Theorem 5.3 d Let C.1 and C.2 hold. Then Y 2,t 1 6)S(Y1,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.

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

29 25/35 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 j=1 ψ(y t 1 γ 0,j + S 0 t γ j )β j+1 + ε t. Test conditional independence with nonlinear transforms ψ y,1 (Y t ) = α 0 + ψ y,2 (Y t 1 )ρ 0 + ψ q (Q t ) 0 β 0 + S 0 t β 1 + r j=1 ψ(y t 1 γ 0,j + S 0 t γ j )β j+1 + η t.

30 5.Conditional exogeneity 26/35

31 27/35 The crucial role of conditional exogeneity Proposition 7.1 Given C.1, suppose Y 2,t 1 d 6)S Y 1,t, t = 1, 2,.... If C.2 (cond exog) does not hold, then for each t there exists q 1,t such that Y 1,t? Y 2,t 1 j Y 1,t 1, X t does not hold. Corollary 7.2 Given C.1 with Y 2,t 1 d 6)S Y 1,t, t = 1, 2,..., suppose that q 1,t is invertible in the sense that Y 1,t = q 1,t (Y 1,t 1, Z t, U 1,t ) implies the existence of ξ 1,t such that U 1,t = ξ 1,t (Y 1,t 1, Z t, Y 1,t ), t = 1, 2,... If C.2 fails, then Y 1,t? Y 2,t 1 j Y 1,t 1, X t fails, t = 1, 2,....

32 28/35 Separability and nite-order conditional exogeneity Proposition 7.3 Given C.1, suppose that E (Y 1,t ) < and q 1,t (Y t 1, Z t, U 1,t ) = ζ t (Y t 1, Z t ) + υ t (Y 1,t 1, Z t, U 1,t ), where ζ t and υ t are unknown measurable functions. Let If C.2 holds, then ε t Y 1,t E (Y 1,t jy t 1, X t ). ε t = υ t (Y 1,t 1, Z t, U 1,t ) E (υ t (Y 1,t 1, Z t, U 1,t ) j Y 1,t 1, X t ), E (ε t jy t 1, X t ) = E (ε t jy 1,t 1, X t ) = 0, and Y 2,t 1? ε t j Y 1,t 1, X t. ()

33 29/35 An indirect test for structural causality 1. Reject structural non-causality if either: (i) the conditional exogeneity test fails to reject and the G causality test rejects; or (ii) the conditional exogeneity test rejects and the G causality test fails to reject. 2. Fail to reject structural non-causality if the conditional exogeneity and Granger non-causality tests both fail to reject; 3. Make no decision as to structural non-causality if the conditional exogeneity and Granger non-causality tests both reject.

34 30/35 6. Applications Crude oil prices and gasoline prices (White and Kennedy, 2008) Monetary policy and industrial production (Angrist and Kuersteiner, 2004) Economic announcements and stock returns (Flannery and Protopapadakis, 2002)

35 31/35 Economic announcements and stock returns Decompose macro announcements into news and expected changes: Announcements: A t Announcement expectations: A e t Decomposition: A t A t 1 = (A t A e t ) + (A e t A t 1 ) = Z t + D t News: Z t = A t A e t Expected change: D t = A e t A t 1 A t : eight major macro announcements: (1) real GDP (advanced) (2) core CPI (3) core PPI (4) unemployment rate (5) new home sales (6) nonfarm payroll employment (7) consumer con dence (8) capacity utilization rate A e t : Money Market Service survey data.

36 32/35 Economic announcements and stock returns Y t : CRSP value-weighted NYSE-AMEX-NASDAQ index daily returns W t : drivers of D t and responses to unobservable causes (1) three month T-Bill yield (2) term structure premium (3) corporate bond premium (4) daily change in the Index of the Foreign Exchange Value of the Dollar (5) daily change in crude oil price These variables represent macro fundamentals. Covariates : X t = (Z t, W t ).

37 33/35 Economic announcements and stock returns Test retrospective conditional exogeneity (CE) by testing D t? ε t j Y t 1, X t where ε t = Y t E (Y t jd t, Y t 1, X t ) Test nite-order retrospective G non-causality (GN) by testing Y t? D t j Y t 1, X t

38 34/35 Economic announcements and stock returns Results Fail to reject GN for all τ = 0, 1,..., 8. Fail to reject CE for all τ = 0, 1,..., 8. Suggests no structural e ects of expected macro announcements on stock returns. Consistent with both weak market e ciency and absence of other distributional impacts. Non-retrospective GN and CE tests (conditioning on lags only) yield exhibit identical pattern.

39 35/35 7. Conclusions This paper Links G non-causality with structural non-causality Links G non-causality with conditional exogeneity Provides explicit guidance as to how to choose S so G non-causality gives structural insight Provides new tests of G non-causality, conditional exogeneity, and structural non-causality