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One challenge of economics is that the systems treated by these sciences have no perfect metronome in time and no perfect spatial architecture—crystalline or otherwise. Nonetheless, as if by magic, out of nothing but one finds remarkably fine-tuned processes in time. To understand this “miracle,” one might consider placing aside the human tendency to see the universe as a machine. Instead, one might address the challenge of uncovering how, through randomness (albeit, as we shall see, strongly correlated randomness), one can arrive at many temporal patterns in economics. Inspired by principles developed by statistical physics over the past 50 years—scale invariance and universality—we review some recent applications of correlated randomness to economics.

Complex systems comprise a large number of interacting elements, whose dynamics is not always a priori known. In these cases — in order to uncover their key features — we have to turn to empirical methods, one of which was recently introduced by Menezes and Barabási. It is based on the observation that for the activity () of the constituents there is a power law relationship between the standard deviation and the mean value: ∝ <>. For stock market trading activity (traded value), good scaling over 5 orders of magnitude with the exponent = 0.72 was observed. The origin of this non-trivial scaling can be traced back to a proportionality between the rate of trades <> and their mean sizes <>. One finds <> ∝ <> for the ∼ 1000 largest companies of New York Stock Exchange. Model independent calculations show that these two types of scaling can be mapped onto each other, with an agreement between the error bars. Finally, there is a continuous increase in if we look at fluctuations on an increasing time scale up to 20 days.

Analysis of tick data of yen-dollar exchange using random walk methods has showed that there exists a characteristic time scale approximately at 10 minutes. Accordingly, for time scales shorter than 10 minutes the market exhibits anti-persistence, meaning that it self-organizes so that to restore a given tendency. For time scales longer than 10 minutes the market approaches a behavior appropriate to pure Brownian motion. This property is explored here to elucidate the predictability of this type of data. We find that improvement in predictability is possible provided that the data are not “contaminated” with noise.

We firstly introduce an optimal moving average for Yen-Dollar tick data that makes the residual term to be an independent noise. This noise separation is realized for weight functions decaying nearly exponentially with characteristic time about 30 seconds. We further introduce another moving average applied to the optimal moving average in order to elucidate underlying force acting on the optimal moving average. It is found that for certain time scale we can actually estimate potential force that satisfies a simple scaling relation with respect to the time scale of moving average.

In foreign exchange markets monotonic rate changes can be observed in time scale of order of an hour on the days that governmental interventions took place. We estimate the starting time of an intervention using this characteristic behavior of the exchange rates. We find that big amount of interventions can shift the averaged rate about 1 yen per 1 dollar in an hour, and the rate change distribution becomes asymmetric for a few hours.

We study here the behavior of the eigenvalues of the covariance matrices of returns for emerging and mature markets at times of crises. Our results appear to indicate that mature markets respond to crashes differently to emerging ones and that emerging markets take longer to recover than mature markets. In addition, the results appear to indicate that the eigenvalue gives additional information on market movement and that a study of the behavior of the other eigenvalues may provide insight on crash dynamics.

We study the statistics of the return intervals between two consecutive return losses below a threshold −, in various stocks, currencies and commodities. We find the probability distribution function (pdf) of scales with the mean return interval in a quite universal way, which may enable us to extrapolate rare events from the behavior of more frequent events with better statistics. The functional form of the pdf shows deviation from a simple exponential behavior, suggesting memory effects in losses. The memory shows up strongly in the conditional mean loss return intervals which depend significantly on the previous interval. This dependence can be used to improve the estimate of the risk level.

Recurrence Plot (RP) and Recurrence Quantification Analysis (RQA) are signal numerical analysis methodologies able to work with non linear dynamical systems and non stationarity. Moreover they well evidence changes in the states of a dynamical system. It is shown that RP and RQA detect the critical regime in financial indices (in analogy with phase transitions) before a bubble bursts, whence allowing to estimate the bubble initial time. The analysis is made on NASDAQ daily closing price between Jan. 1998 and Nov. 2003. The NASDAQ bubble initial time has been estimated to be on Oct. 19, 1999.

We introduce an autoregressive-type model with self-modulation effects for a foreign exchange rate by separating the foreign exchange rate into a moving average rate and an uncorrelated noise. From this model we indicate that traders are mainly using strategies with weighted feedbacks of the past rates in the exchange market. These feedbacks are responsible for a power law distribution and characteristic autocorrelations of rate changes.

We introduce a weighted-moving-average analysis for the tick-by-tick data of yen-dollar exchange market: price, transaction interval and volatility. The weights are determined automatically for given data by applying the Yule-Walker formula for autoregressive model. Although the data are non-stationary the resulting moving average gives a quite nice property that the deviation around the moving-average becomes a white noise.