The General Equivalence of Granger and Sims Causality

Linear predictor definitions of causality are not adequate for discrete data. The paper
extends the Granger and Sims definitions by using conditional independence instead of
linear predictors. The extended definition of "y does not cause x" is that x is independent
of pasty conditional on past x. This is stronger than the strict exogeneity condition that y
be independent of future x conditional on current and past x. Under a weak regularity
condition, however, if y is independent of future x conditional on current and past x and
pasty, then y does not cause x .

1. INTRODUCTION
LET [(x,, y,), t = . . . ,- 1,0,1, . . . ] be a collection of random variables on a
common probability space-a stochastic process. Granger [6] defined "y does
not cause x" as follows: the (minimum mean square error) linear predictor of
x,+ based on x,, xt-, , . . . ,y,, y,- ,,. . . is identical to the linear predictor based
on x,, x,-, ,. . . alone.* Sims [lo] defined x to be strictly exogenous relative toy
if the linear predictor of y, based on . . .,xt-, ,x,, x,, ,, . . . is identical to the
linear predictor based on x,,xt-,, . . . . Sims [lo] showed that these two definitions
are equivalent.3 This is a beautiful result; we would like to know whether it
still holds if linear predictors are replaced by a more general form of dependence.
In applying these ideas to longitudinal data on individuals, the prevalence of
qualitative variables argues for considering models based on the entire conditional
distribution instead of looking only at linear predictors. Suppose that yit is
zero or one, indicating, for example, whether or not individual i was employed in
period t . We observe (xi', yil, . . . ,xi,, y,,) for i = 1, . . . ,N individuals, and we
regard these vectors as independent and identically distributed (i.i.d.) observations
from the joint distribution of (x,, y,, . . . ,x,, y,). Let t = 1 be the first
period of the individual's (economic) life. Consider the following specification for
the conditional probability that yi, equals one:
P(yit= IXil,"' XiT,ci)= P(yil= 1 /xi,, ,xif,~i),

where c is a latent variable that represents unmeasured characteristics of the
individual; c is assumed to be constant over the sample period.
If c is independent of the x's, then, dropping the i subscripts, we have
so that x is strictly exogenous. However, if P(c < u I x,, . . .,x,) # P(c 2 u),
then in general P(c 5 u 1 x,, . . . ,x,) # P(c 2 u I x,, . . . ,x,); a latent variable
that is constant over time is generally related to all of the x,'s if it is related to
any of them. In that case
Hence the failure of strict exogeneity indicates that the latent variable is not
independent of the measured x's. Is there an extension of Granger's definition
of "y does not cause x" that will imply P(y, = 1 1 x,, . . . ,x,) = P(y, =
l l x , , . . .,x,)?