318 ECONOMETRICA entire group the results listed above have been reduced to effective annual rates, and each has been given a weight to conform with the length of the record analysed. After these adjustments, we are enabled to conclude that the average forecasting agency fell approximately 4 per cent per annum below a record representing the average of all performances achievable by pure chance. This would seem to indicate that, in general, these stock market forecasters failed to accomplish their objective. The most that can be said in extenuation is that the long-continued decline in securities has been, naturally, a handicap to a group which, taking warning from the experience of Cassandra,usu- ally seems constrained to look on the bright side During the 41 year period under analysis the number of weeks in which the stock market declined almost exactly equalled the number of weeks in which ad vances were recorded, and the total amount of the declines consider ably exceeded the total amount of the advances. Yet we recorded during this period 2035 bullish, 804 bearish, and 479 doubtful fore- casts. Further we note that in 1928, the only year the market showed a net gain, the excess of bullish over bearish forecasts was smaller than in any succeeding year. Taking a glaring example, in the rising market of 1928 the ratio of bullish to bearish forecasts was only four to three. In 1931, when the market declined 54 per cent, there were sixteen bullish forecasts to every three bearish STATISTICAL INTERPRETATIONS OF RESULTS In an attempt to illuminate the problem of whether the records of all these forecasters lay within the limits of pure chance, we compiled 24 records, identical with those of the 24 forecasters as to the total period covered, but having purely fortuitous advices applied to random intervals within these perids. For example, to compile a purely chance record to compare with the actual record of a forecaster whose opera- tions covered 230 weeks from January 1, 1928, to June 1, 1932, we first determined the average number of changes of advice for such a period which was 33. Cards numbered from 1 to 229 were shuffled, drawn, re- shuffled, drawn, in all 33 times. Thus 33 random dates were selected as of which forecasts were to be changed. The investment policies which were to apply to the intervals between those dates were derived in similar fortuitous fashion, by drawing 33 times from nine cards on each of which a different one of the nine possible investment policies It only remained to relate these random advices to a stock market index, cumulate the results, relate them as we had done with the records of the actual forecasters to the average of all chances for the period, and subtract 100. Thus we had a list of 24 purely chance fore-