The asymptotic behavior of the sample paths of two popular statistics
that test market efficiency are investigated when markets learn to hav
e rational expectations. Two cases are investigated, where, should mar
kets start out at a rational expectations equilibrium, both statistics
would asymptotically generate standard Brownian motions. In a first c
ase, where agents are Bayesian and payoffs exogenous, the statistics h
ave identical sample paths, but they are not standard Brownian motions
. Whereas the finite-dimensional distributions are Gaussian, there may
be a bias if agents' initial beliefs differ. A second case is conside
red, where payoffs are in part endogenous, yet agents consider them to
be drawn from a stationary, exogenous distribution, which they attemp
t to learn in a frequentist way. In that case, one statistic behaves a
s if the economy were at a rational expectations equilibrium from the
beginning on. The other statistic has sample paths with substantially
non-Gaussian finite-dimensional distributions. Moreover, there is a ne
gative bias. The behavior of the two statistics in the second case mat
ches remarkably well the empirical results in an investigation of the
prices of six foreign currency contracts over the period 1973-1990.