The Journal of finance From this it follows that for all lags, the serial covariances between lagged values of a“ fair game” variable are zero.Thus, observations of a“ fair game” ariable are linearly independent. o But the fair game del does not necessarily imply that the serial ovariances of one-period returns are zero. In the weak form tests of this model the“ fair game” variable is t=r,t-E(ftr3t-1,r3t-2,…).(C.in.9) But the covariance between, for example, r]t and rj, t+1 is [f3. t+1-E(3t+1)] [F3t-E(Fjt [rjt-E(st)] [E(f, +1 r3)-E(f,t+1)]f(rit)drjt and(9)does not imply that E(Ft+1rst)=E(j,t+1): In the "fair gam fficient markets model, the deviation of the return for t+ 1 from its condi- tional expectation is a"fair game"variable, but the conditional expectation itself can depend on the return observed for t. 1 In the random walk literature, this problem is not recognized since it is assumed that the expected return (and indeed the entire distribution of eturns) is stationary through time. In practice, this implies estimating serial covariances by taking cross products of deviations of observed returns from the overall sample mean return. It is somewhat fortuitous, then, that this pro- cedure, which represents a rather gross approximation from the viewpoint of the general expected return efficient markets model, does not seem to greatly affect the results of the covariance tests at least for common stocks But the integral in brackets is just E(x++1l which by the "fair game"assumption is 0, so that a process. A "fair game "also rules out many types of non linear dependence ments similar to those above, it can be shown that if x is a"fair game "E(x,++1.. x++1)=0 for all t, which is not implied by E(x+t+r)=0 for all t. For example, consider a three-period case where x must be either# 1. Suppose the process is x++2=sign(x, *++1), i.e, x++ If probabilities are uniformly distributed across events E(x1+2x1+1)=E(x1+21x)=E(x+1x)=E(x+2)=E(x+1)=E(1)=0, so that all pairwise serial covariances are zero. But the process is not a "fair game, since E(x++21x++1+)+0, and knowledge of (x++1, x,) can be used as the basis of a simple"system' 11. For example, suppose the level of one-period returns follows a martingale so that E differe nce of nets hs win s cncsirel ated) 12. The reason is probably that for stocks, changes in equilibrium e d returns for the