Lemma 1: In any covariance-stationary equilibrium, φ>0. That is,
momentum traders must rationally behave as trend-chasers.The lemma is proved in the appendix, but it is trivially easy to see why
φ=0 cannot be an equilibrium. Suppose to the contrary it is. Then prices
are given as in the all-newswatcher case in Eq. (1). And in this case,
cov(Pt+j−Pt, ∆Pt− 1 )>0, so that Eq. (7) tells us that φ>0, establishing a
contradiction.
We are now in a position to make some qualitative statements about the
dynamics of prices. First, let us consider the impulse response of prices to
news shocks. The thought experiment here is as follows. At time t, there is a
one-unit positive innovation εt+z− 1 that begins to diffuse among newswatch-
ers. There are no further news shocks from that point on. What does the
price path look like?
The answer can be seen by decomposing the price at any time into two
components: that attributable to newswatchers, and that attributable to
momentum traders. Newswatchers’ aggregate estimate of DTrises from
time tto time t+z−1, by which time they have completely incorporated
the news shock into their forecasts. Thus by time t+z−1, the price is just
right in the absence of any order flow from momentum traders. But with
φ>0, any positive news shock must generate an initially positive impulse
to momentum-trader order flow. Moreover, the cumulative order flow must
be increasing until at least time t+j, since none of the momentum trades
stimulated by the shock begin to be unwound until t+j+1. This sort of
reasoning leads to the following conclusions:
Proposition 1:In any covariance-stationary equilibrium, given a posi-
tive one-unit shock εt+z− 1 that first begins to diffuse among news-
watchers at time t: (1) there is always overreaction, in the sense that
the cumulative impulse response of prices peaks at a value that is
strictly greater than one; (2) if the momentum traders’ horizon jsat-
isfies j≥z− 1 , the cumulative impulse response peaks at t+j, and
then begins to decline, eventually converging to one; (3) if j<z−1,
the cumulative impulse response peaks no earlier than t+j, and
eventually converges to one.In addition to the impulse response function, it is also interesting to con-
sider the autocovariances of prices at various horizons. We can develop
some rough intuition about these autocovariances by considering the limit-
ing case where the risk tolerance of the momentum traders γgoes to infinity.
In this case, Eq. (7) implies that the equilibrium must have the property that
cov(Pt+j−Pt, ∆Pt− 1 )=0. Expanding this expression, we can write:
cov(∆Pt+ 1 , ∆Pt− 1 )+cov(∆Pt+ 2 , ∆Pt− 1 )+..... +cov(∆Pt+j, ∆Pt− 1 )= 0 (8)
A UNIFIED THEORY OF UNDERREACTION 511