that some arbitrageurs eventually switch over from the momentum strat-
egy. Figure 14.4 illustrates by considering a situation where z=3, j=1,
where the contrarians trade at a lag that is c=2 periods greater than the
momentum traders, and where the risk tolerance takes on the value 1/0.3.
Given these parameter values, w=0.786. That is, 78.6 percent of traders
opt to play momentum strategies and the remaining 21.4 percent become
contrarians. The contrarians appear to have a modest stabilizing impact—
the impulse response function peaks at 1.197 when there are only momen-
tum traders, and this figure declines somewhat, to 1.146, when we allow
for contrarian strategies. Nevertheless, price dynamics are still remarkably
similar to what we have seen throughout. This underscores our key point:
Across a wide range of parameter values, allowing for contrarian strategies
need not alter the important qualitative features of our model.
A.2.arbitrageurs can run bivariate regressionsTo further relax our assumptions in the direction of rationality, we now ask
what happens if every arbitrageur becomes incrementally smarter, and can
condition on not one, but two lags of past prices. Said differently, instead of
A UNIFIED THEORY OF UNDERREACTION 519
Figure 14.4. Cumulative impulse response and contrarian trading. Cumulative im-
pulse responses for all-momentum trading equilibrium (w=1); the equilibrium in
which traders endogenously choose whether to follow either momentum or contrar-
ian strategies (w=0.786); and the equilibrium in which traders can optimally condi-
tion on both momentum and contrarian variables (“Both”). The other parameter
values are set as follows: the information diffusion parameter zis 3, the momentum
traders’ horizon jis 1, the volatility of news shocks is 1 and risk tolerance gamma is
1/0.3. The contrarians are assumed to trade based on returns from three periods ago.