The asymptotic behavior of the sample paths of two popular statistics that test market efficiency are investigated when markets learn to have rational expectations. Two cases are investigated, where, should markets start out at a rational expectations equilibrium, both statistics would asymptotically generate standard Brownian motions. In a first case, where agents are Bayesian and payoffs exogenous, the statistics have 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 considered, where payoffs are in part endogenous, yet agents consider them to be drawn from a stationary, exogenous distribution, which they attempt to learn in a frequentist way. In that case, one statistic behaves as 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 negative bias. The behavior of the two statistics in the second case matches remarkably well the empirical results in an investigation of the prices of six foreign currency contracts over the period 1973–1990.
Rational expectations equilibrium with transaction costs in financial markets