Journal of Econometrics 23 (1983) 331-335. North-Holland

PIERCE AND HAUGH ON CHARACTERIZATIONS OF CAUSALITY

A Re-examination*

Lewis EVANS Victoria University, Wellington, New Zealand

Graeme WELLS New Zealand Institute of Economic Research, Wellington, New Zealand

Received August 1982, final version received June 1983

This paper amends the set of equivalent necessary and sufficient conditions under which y does not cause x, provided by Pierce and Haugh (1977).

1. Introduction

Pierce and Haugh (1977) have provided a set of equivalent conditions under which a variable y does not cause x. This paper argues that these conditions appear to have been established in isolation, in that consistency between the AR and MA representations requires modification of this theorem. The role of instantaneous causality in achieving this consistency is discussed.

2. Manipulations

Pierce and Haugh (1977), PH hereafter, use the following equivalent characterizations of the bivariate process (x,, y,):

where Y(B) is a matrix of lag polynomials, (a,,b,) is a vector white noise

*The paper has benefited from the helpful comments of an unknown referee.

03044076/83/$3.00 0 1983, Elsevier Science Publishers B.V. (North-Holland)

332 L.

sequence satisfying

Evans and G. Wells, Pierce and Haugh on causality

at E bt =O,

0

C (positive definite), t=s, E[(~~)bs~b,)]={o, rfs,

(2)

(3)

(4)

(5)

In terms of these alternatives, the relevant part of Theorem 4.2 of PH is:

y does not cause x if and only if the following equivalent conditions hold:

(1) Y,,(B) [equivalently, O,,(B)] can be chosen zero.

(2) O,,(B) is either 0 or a constant.

(3) Y,,(B) is either 0 or proportional to YI,(B).

(5) j?(B) is either 0 or a constant.

(6) H(B) is either 0 or proportional to A(B).

Our amendments to this theorem concern the nature of the equivalences between the conditions of the theorem. Consistency between the AR and MA parameterisations (1) and (2) requires that the ‘proportionality or constant’ parts of Theorem 4.2 be eliminated.

A route to this conclusion with respect to conditions (3) and (6) is provided by a consideration of

YY,,(B) = -H(B)/det n(B).

L. Evans and G. Wells, Pierce and Haugh on causality 333

Since detLI(B) is a polynomial in B, we cannot have ‘Y,,(B) and H(B) equal to non-zero constants. Similarly,

or

H(B)=0 and Y’,,(B)= Yy,,

H(B) = HO and Y’,,(B) =O,

are also ruled out.

Thus, for conditions (6) and (3) to be consistent we require H(B)= !Yr,(B) =0 when y does not cause x. It is obvious from the foregoing and the facts

o(B)=[~~) CyBJ Y(B) and rr(~)=n(s)[~(:~~ G(i)_1]

that any one of H(B), Y,,(B), O,,(B) and P(B) equal to zero implies that the others are zero. We conclude that PH’s Theorem 4.2 does not list equivalent parameterisations of y not causing x in the various representations of the sequence (x,, y,) and that it should be replaced with a Theorem 4.2’ in which y does not cause x if and only if the conditions

(2’) @1,(B)=&

(5’) P(B) = 0,

(6’) H(B) = 0.

and the conditions (4) and (7) hold.

3. Discussion

An implication of Theorem 4.2’ is that when the model is identified by normalising so that in the AR form C’=Ea,b, =0 instantaneous causality and y causing x are both ruled out by (6’).l The difficulty arises because the model is not identified in the absence of such constraints. Consider alternative representations of the AR form. Since Basmann’s (1965) work it has been known that two of an infinite number of observationally equivalent

IA” = Do = 1 and Co denotes the first element of C(L).

334 L. Evans and G. Wells, Pierce and Haugh on causality

representations are

and

(6b)

where the contemporaneous matrix II: (II!) is upper (lower) triangular, Ea.+..b*, =Eattbtt =0 and where (6a) and (6b) are non-singular linear transforms of (2) and hence of each other. In (6a) we have the cited restriction as CO, = Ea*,b*, =O. Although it is tempting to conclude that y does not cause x when n,(B) is lower triangular (6a) will not yield a lower triangular MA matrix polynomial if Hi #O.’ This is the content of the previous section and it is obvious from the MA form

where for any matrix G, [G]‘rZ. It is clear from (7) that n*(B) lower

triangular does not imply that Y,(B) and/or p,(B)= Y,(B)- !Pi will be lower triangular as no, is upper triangular when HO, is not zero. This lack of correspondence between the AR and MA forms arises because if y contemporaneously causes x then the innovation in y,, namely b*z, will influence xt even though a,, and b,, are uncorrelated. Because this influence is present in each period x, will be determined by the history of both a,, and b,, even when if,(B) is lower triangular. In the first row of (6a) yt reflects past values of the b,, innovation but there is no such variable in the MA form as (a,,, b,,) IS serially uncorrelated. In consequence, the dependence of x on b, must be manifest through the MA matrix. In general, the MA coefficients on past innovations will be an amalgam of lagged as well as contemporaneous AR matrices.

It is straightforward to demonstrate - see Sargent (1979, p. 284) - that n?(B) is lower triangular iff Y+(B) is lower triangular. Here the system has been normalised so that y does not contemporaneously cause x. We conclude

‘Note that n,(B) lower triangular would be in accord with y not causing x in that E(x, 1 yt, {y,_ ,}, {x1- ,}) = E(x, 1 y,, {x, _ ,}). But this equality does not imply that y does not cause x in the MA representation. We consider that Theorem 4.2’ is closer to the germ of Granger’s (1969) basic proposition that causality requires time. An appealing way to implement this proposition would be to restrict attention to y causing x and to evaluate if the prediction of x, from past information is not improved by knowledge of past y. that is, determine if E(x, 1 (yt _ 1}, {x, _ ,}) = E(x, 1 {x, _ ,}). This equality is implied by condition (6’).

L. Evans and G. Wells, Pierce and Haugh on causality 335

that this normalisation is required so that R+(B) = H+(L) - Hv (where Hy = 0) equal to zero accords with the various conditions of Theorem 4.2’.

The unidentified system can be normalised so that the AR and MA representations yield the same conclusion regarding Granger causality of x by y. We cannot extend this normalisation practice to the circumstance where y does not cause x and there is no feedback. In this case the presence of instantaneous causality means that a diagonal n(B) matrix will not map into a diagonal Y(B) matrix unless instantaneous causality is absent. So if we wish to parameterise y not causing x jointly with x not causing y in a way which is consistent with AR and MA representations (1) and (2), the constraint of no instantaneous causality is required.

References

Basmann, R.L., 1965, A note on the statistical testability of explicit causal chains against the class of ‘interdependent’ models, Journal of the American Statistical Association 60, 108(r 1093.

Granger, C.W.J., 1969, Investigating causal relations by econometric models and cross-spectral methods, Econometrica 37, 424438.

Pierce, D.A. and L.D. Haugh, 1977, Causality in temporal systems: Characterizations and a survey, Journal of Econometrics, 265-293.

Sargent, Thomas J., 1979, Macroeconomic theory (Academic Press, New York).