آزمون ثبات ضد حباب در بازار سهام ایالات متحده سال 2000

Since August 2000, the stock market in the USA as well as most other western markets have depreciated almost in synchrony according to complex patterns of drops and local rebounds. In (Quantitative Finance 2 (2002) 468), we have proposed to describe this phenomenon using the concept of a log-periodic power law antibubble, characterizing behavioral herding between investors leading to a competition between positive and negative feedbacks in the pricing process. A monthly prediction for the future evolution of the US S&P 500 index has been issued, monitored and updated in (http://www.ess.ucla.edu/faculty/sornette/prediction/index.asp#prediction), which is still running as the article goes to press. Here, we test the possible existence of a regime switching in the US S&P 500 antibubble. First, we find some evidence that the antibubble has exhibited a transition in log-periodicity described by a so-called second-order log-periodicity. Second, we develop a battery of tests to detect a possible end of the antibubble of the first order which suggest that the antibubble was alive in August 2003 but has ended in the USA, when expressed in the local US dollar currency. Our tests provide quantitative measures to diagnose the end of an antibubble. Such diagnostic is not instantaneous and requires from three to six months within the new regime before assessing its existence with confidence. From the perspective of foreign investors in their currencies (S&P 500 denominated in British pound or in euro) or when expressed in gold so as to correct for an arguably artificial US$ valuation associated with the Federal Reserve interest rate and monetary policy, we find that the S&P 500 antibubble is still alive and running its course. Similar analyses performed on the major European stock markets (CAC 40 of France, DAX of Germany, and FTSE 100 of United Kingdom) show that the antibubble is also present and continuing there.

In 1999, in order to describe the evolution of the Japanese stock market since its all-time high in December 1989, Johansen and Sornette introduced the concept of an “antibubble” as a counterpart of a bubble resulting from the same herding behavior and characterized by log-periodic power-law (LPPL) structures but with decelerating (rather than accelerating) oscillations [1]. The term “antibubble” is inspired by the concept of “antiparticle” in physics. Just as an antiparticle is identical to its sister particle except that it carries exactly opposite charges and destroys its sister particle upon encounters, an antibubble is both the same and the opposite of a bubble; it is the same because similar herding patterns occur, but with a mostly bearish versus bullish slant. Some antibubbles can also describe increasing markets over long times, although a bearish phase is more commonly recognized in the markets [2], [3], [4] and [5]. In August 2002, we detected the existence of a clear signature of an antibubble in the relaxation of the US S&P 500 index since August 2000 with high statistical significance, in the form of strong log-periodic components [6]. Similarly to the prediction offered in Ref. [1] for the evolution of the Nikkei index which was later evaluated in Ref. [7], we presented a prediction for the future evolution of the US S&P 500 index [4] and [6]. This prediction has been monitored and updated once a month at the URL [8]. Accompanying the US stock markets, the antibubble regime since 2000 seems to be a world-wide phenomenon in the major western stock market [2]. These works on antibubbles extend a large amount of theoretical and empirical work on LPPL bubbles which often end in crashes or strong corrections (see [9], [10], [11] and [12] and references therein). In this context, Roehner has investigated the resilience pattern around large price peaks [13] and has found strong negative correlations between stock market crash-recovery and interest rate spread [14]. In contrast to a LPPL bubble whose end is automatically described by one of the parameters, the critical time tctc, the LPPL formulation of an antibubble does not say anything a priori about its duration. For prediction purpose, the agonizing question is whether the detection of an antibubble pattern ensures its continuation in the future and for how long. In the case of the Japanese antibubble studied in depth [1] and [7], the detection was performed in early January 1999, corresponding to 9 years since the birth of the antibubble. The prediction issued in early January 1999 turned out to be followed subsequently ex-ante by the Nikkei index over more than two years. However, in January 1999, it was hard a priori to assess for how long the theory would be a correct predictor of the future evolution of the Nikkei index. Here, we address this question of the detection of a change of regime from an antibubble phase to something else. For this purpose, the present situation is perhaps more favorable than for the Nikkei in January 1999 for the following reason. As we said above, in January 1999, the antibubble has been unfolding itself already for 9 years. It was found necessary to extend the LPPL theory from a first-order log-periodic formula to a second-order and then to a third-order formula. It should be stressed that the first-order formula is embedded as a special case of the second-order formula which is itself embedded as a special case of the third-order formula. The logic of this succession of formulas is that they represent successive improvement to describe the market price at time intervals further and further away from the early development of the antibubble. The larger is the order of the formula, the larger is the time interval over which the theory applies. The prediction issued in January 1999 was performed based on the third-order LPPL formula. In contrast, the analysis of the US S&P 500 antibubble has been performed much earlier after about only 2.5 years since its inception in August 2000 [6]. Due to this relatively short time span, it was found that the first-order formula was sufficient to describe the empirical data, while the second-order (and a fortiori the third-order) formula was not needed as it did not lead to any statistically significant improvement. We thus concluded that the S&P 500 index had not yet entered into the second phase in which the angular log-frequency may start its shift to another value, as did the 1990 Nikkei antibubble after about 2.5 years. However, this situation offers the possibility for tracking a possible future change of regime from the first-order to the second-order formula. This is the first purpose of this paper. Using data garnered over ten additional months, we show that one can start to detect the occurrence of such a change of regime. Adding an additional year of data confirms further this conclusion, as we shall show. The statistical tests described below give the probability to reject the hypothesis that the market has not entered the second phase in which the angular log-frequency is shifting to another value. These results suggest the possibility that, indeed, we have entered a cross-over regime in log-frequency shift. The improved second-order log-periodic formula has implications in the prediction of the future drops of the markets. The second purpose of our paper is to develop a battery of tests to gain a better understanding of which scenario might be the most likely to unfold: is the antibubble likely to continue and is the market when expressed in one of the major foreign currencies or in gold to drop further? Or will the stock market transit to another regime, perhaps rebound to develop a new bullish regime? Or even worse (from the point of view of our model): is it possible that the US and European stock market has already entered a regime different from that described by the LPPL antibubble and that we have not yet taken this into account in our updates presented in Ref. [8]? The present paper provides the theoretical basis and the statistical anchor underlying the monthly prediction updates which are available at the URL [8].

First, we have presented a general methodology to test for a cross-over or a shift in log-periodicity. Second, we have developed a battery of tests to detect a possible end of the antibubble. Our conclusion is that the antibubble was still probably alive in August 2003 but has ended since in the USA (i.e., when viewed from the view point of a US investor valuing in US dollars). More generally, our tests provide new quantitative measures to diagnose the end of an antibubble and this will be useful for future applications. We find that such diagnostic is not instantaneous and requires probably 3–6 months within the new regime before assessing its existence with confidence. We have also found that the antibubble is still continuing when viewed from the point of view of a European investor or alternatively from an investor valuing the US stock market with respect to gold. We attribute the discrepancy between our two conclusions to the depreciation of the US dollar in the last 2 years, which is linked to the Federal Reserve interest rate and monetary policy. This present paper follows several others [2], [4], [6] and [18] which were nucleated by noticing a similarity between the Nikkei antibubble that started in January 1990 and the present US antibubble that started in August 2000, when shifted approximately by 11 years. We conclude with a word of caution concerning this similitude between the two time series: the noted similarity should not lead to the belief that the S&P 500 index is bound to follow blindly in the future the path suggested by the Nikkei. In contrast with chartism or technical analysis, our approach is to develop a scientific understanding of these bubble-antibubble phases. The similitude between the Nikkei and US markets is part of the search for “universal” properties, that allow us to establish a theory (in short, a theory is a story of repeatable/reproducible occurrences). Using this theory then allows us to describe idiosyncratic behaviors, that is, deviations from one case to another, or in other words, the parts of the evolutions that are not universal. This is what should give us an hedge for predictions. This is why we have emphasized in previous works the similitude between the shifted Nikkei and the US stock markets [4]. However, after three-year evolution of the S&P 500 antibubble, the discrepancy between the Nikkei and S&P 500 antibubbles became detectable. The qualitative analogy is still there but, quantitatively, there are differences. Technically, after two years and a half after the top in December 31, 1989, we find that the Nikkei has started to shift to another antibubble regime while no such shift was detectable after the same time span since the start of the antibubble in the US. Only when using data up to the summer of 2003, we find suggestions of such a change of regime. In addition, the US markets have been characterized by much stronger crashes and rallies, modelled by the so-called zero-phase Weierstrass-type functions [18]. These two facts suggest that the herding forces are even stronger in the US and that investors react even more on hair-trigger to any “news.” The similarities between the shifted Nikkei and the S&P 500 are qualitative: bubble preceding antibubble, strong speculation and herding, similar fear and herding in the antibubble regime, some problems with bad loans or bad accounting, strong commitment from the central banks and governments to provide liquidity and cash... But there are differences and these differences can be detected already after three-year evolution of the S&P 500 antibubble and even more after 4 years. There are also interesting structural differences in the origin of the bubbles that preceded their antibubbles. Japan was (and still is) a surplus country, whose strong positive balance of payment led to “high-powered” money being poured in the country. This in turn powered speculation and price appreciation to sky-rocketing levels. The so-called bad loans dragging down the Japanese recovery came from this epoch when the high-powered money input was used by banks to provide loans amplified by the multiplier effect for purchases at prices often substantially larger than the real value. In contrast, the USA has become in the last decade a deficit country, accumulating an increasingly large negative balance of payment with the rest of the world. The bubble that developed in the 1990s was fuelled indirectly by the surplus dollars accumulated by foreign countries which were re-injected in the US in the hope of getting a reasonable return while avoiding the risk of appreciation of their own currencies [22]. The bubble had also a very strong endogenous component of self-reinforcing belief in a “new economy,” a characteristic that could be matched to the faith in the Japanese miracle underlying the Japanese bubble. Thus both the Japanese and USA markets are strongly linked to the behavior of international investors and central bankers and to the belief and confidence of investors, but the specifics of the herding and over-optimism have sometimes different origins. It remains to be seen if this will lead to appreciable differences in the evolution of the two antibubbles.