The methodology of tests for martingale properties in return series is analyzed. Martingale results obtain frequently in finance. One case is focused on here, namely, rational price discovery. Price discovery is the process by which a market moves towards a new equilibrium after a major event. It is rational if price changes cannot be predicted from commonly available information. The price discovery process, however, cannot be assumed stationary. Hence, to avoid false inference in the presence of nonstationarities, event studies of field data have been advocating the use of cross-sectional information in the computation of test statistics. Under the martingale hypothesis, however, this inference strategy is shown to add little except if higher moments of the return series do not exist. On the contrary, the cross-sectional approach may even be invalid if there is cross-sectional heterogeneity in the price discovery process. The time series statistic of Patell (1976], originally suggested in the context of i.i.d. time series but cross-sectional heterosceclasticity, may be preferable. It will not provide valid inference either, if higher serial correlation coincides with higher volatility. Unfortunately, this appears to be the case in the dataset which is used in the paper to illustrate the methodological issues, namely, transaction price changes from experiments on continuous double auctions with stochastic private valuations
