67
where 0(B) and (B) are invertible polynomials in the lag operator B.
The terms ufc and vfc represent the white noise processes (innovations)
obtained from of X and Y, respectively. The polynomials 9(3) and ( S)
may be viewed as filters which are identified and estimated by using the
Box-Jenkins approach. The sample cross correlations between ut and vt
{r^(k)} provide a means by which the properties of interdependency
(causality) between X and Y can be assessed. In addition, tests of
unidirectional causality can be performed using the chi-square Haugh-
Pierce statistic. These inferences regarding the direction of price
determination are vital for specification of the transfer function.
Dynamic Shock Model
Having determined a lead/lag structure; e.g. Xfc leads Yt, Haugh and
Box (1977) show that it is possible to express Yt as a distributed lag
on X^. as
Yt S(B)Xt + at
where 5(B) is some polynomial of Xt and afc is an error process. The
weights on the terms of the polynomial 5(B) are referred to as the
impulse response parameters. These parameters characterize the response
of Yt to changes in the "input" Xfc, net of the "masking effect" of the
stationary white noise process at. To identify the order of the poly
nomial 5(B) connecting Yt and X^., a model must first be identified that
connects the innovations ut and vt. This procedure will make use of the
sample residual cross correlations r(k) f where k is the order of lag,
to arrive at a dynamic shock model given as
v V(B)u + Y(B)a
t t t