Fig. 3 shows the predictions of the future of the US S&P 500 index obtained by applying the so-called 'zero-phase' Weierstrass-type function, which is another child of our general theory of imitation and herding between investors. As for the previous figures, our theory takes into account the competition between positive feedback (self-fulfilling sentiment), negative feedbacks (contrariant behavior and fundamental/value analysis) and inertia (everything takes time to adjust). This 'zero-phase' Weierstrass-type function adds one additional ingredient: it attempts to capture the existence of 'critical' points within the anti-bubble, corresponding to accelerating waves of imitation within the large scale unraveling of the herding anti-bubble. The continuous black line is the forward prediction using all the data from Aug. 9, 2000 to Oct. 17, 2003, while the dashed black line is the retroactive prediction using the data from Aug. 9, 2000 to Aug. 24, 2002. Both lines are reconstructed and extrapolated from the fits to a six-term zero-phase Weierstrass-type function. We also present the two previous fits (red lines) performed on Aug. 24, 2002 (shown in Fig. 1) for comparison. The blue dots show the daily price evolution from Aug. 9, 2000 to Oct. 17, 2003. The large (respectively small) ticks in the abscissa correspond to January 1st (respectively to the first day of each quarter) of each year.
The development observed in the update on June, 19, 2003 is further confirmed: the fit obtained here is very close to that on June 19, 2003. The 'zero-phase' Weierstrass-type function, which up to May 18, 2003 selected a series of downward critical crashes, is now selecting as the dominant critical points the bullish accelerations. The formula is thus deciphering the coexistence of two sets of critical points: (i) the crashes previously recognized which have punctuated the descent in the last three years and (ii) the bursts of upward accelerating rallies. This formula is however not rich enough in its present version to capture these two sets simultaneously and has to choose between the two, as a result of their relative strengths. This new twist does not change fundamentally our prediction of a drastic turn towards a systematic downward trajectory till the summer of 2004.