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Now we are ready to construct an estimate of the covariance of both newly derived time series, each of which has those polynomial trends removed interval by interval:. If both time series are the same, we obtain an estimate of the detrended variance in the k -th interval from partitioning with the scale s.

The definition of the family of fluctuation functions given by Eq. In such a case, we obtain, as it should be, the q -th order fluctuation function, which follows from the multifractal detrended fluctuation analysis MFDFA for a single time series [ 42 ].

The family of fluctuation functions of order q , defined by Eq. Therefore, one can define a q -dependent detrended cross-correlation q DCCA coefficient using a family of such fluctuation functions [ 41 ]:. The definition of the cross-correlation given by Eq.

The parameter q helps to identify the range of detrended fluctuation amplitudes corresponding to the most significant correlations for these two time series [ 41 ]. Hence, by choosing a range of values in q we may filter out correlation coefficient for either small or large fluctuations. Based on the methodology described, we will investigate multiscale properties for cross-correlations among time series corresponding to currency exchange rates for the whole period — as well as for some sub-periods.

We also apply standard methods of MATLAB source codes validation and surrogate data checking against artefacts or robustness of nonlinear correlations within our data sets [ 41 ]. Although in our study we focus on different signatures and statistical properties of multivariate time series with respect to triangular arbitrage, one may envisage a broader picture of such analysis, whereby one would like to uncover a specific kind of cross-correlations in these time series which would help us to detect underlying interconnections useful for the system behavior prediction in future.

In this subsection, we will present a general picture for the financial time series dynamics with an emphasis on events which have an impact on the Forex market. Color online Currency indexes as defined by Eq. For a reference, some important global political and economic events have been indicated over the timescale.

The currency index given by Eq. With such a single averaged characteristics, one may have a general overview of a global temporal behavior and performance of any currency in the Forex market. In Fig. For this plot, we take logarithmic returns arising from average bid and ask exchange rates. In the figure, we have indicated some political or economic events on the timescale with dotted vertical line which in principle could have impact on the Forex market performance during that period of time.

These labels may serve as an intuitive explanation of features observed on the curves related to the currency indexes. In general, one observes significant variations of considered currency indexes over the period of 8 years. In the following, we will explore in some more details statistical properties of the Forex market data in the vicinity of these events.

We will discuss to what extent our proposed statistical analysis corroborates these features, when looking from the hindsight with the help of historical data from the Forex market. Another interesting feature of the Forex market is related to the cumulative distribution of absolute logarithmic returns:.

Each solid line of different color demonstrates the tail behavior for the corresponding currency. The inset shows these distributions when a short period of an extreme volatility in currency exchange rates has been removed. This removed period a half an hour corresponds to the wake of the SNB intervention on January 15, We note that all currency exchange rates with the Swiss franc CHF as base currency yield higher probability of larger absolute logarithmic returns than other exchange rates.

These outliers could be attributed to two instances of the SNB interventions in and In order to demonstrate the origin of these deviations, we remove from our data sets a period of a half an hour in the morning on January 15, , when a significant volatility of currencies exchange rates has been observed in the wake of the SNB intervention [ 9 ]. The resultant tails of the probability distributions are shown in the inset of Fig.

Hence without this single, short-termed event on the market, the tails of the distributions approximately follow the inverse-cubic behavior. We already know that the outliers of the cumulative distributions document an increased level of larger fluctuations in absolute log-returns. This means also a higher chance to encounter fluctuations yielding larger returns. The question arises to what extent these fluctuations are cross-correlated among exchange rates.

Such cross-correlations at least between two exchange rate time series would offer a potential opportunity of triangular arbitrage. The graph illustrates the cross-correlation fluctuation functions scaling over the range of timescales s from 5 min up to 2 weeks for different q -coefficients.

We verify indeed that the scaling according to Eq. Additionally, for the reference both insets in Fig. This average is defined as follows [ 53 ]:. It is worth noting that in this shown example, the scaling of the fluctuation functions for the case of the triangular relation among two exchange rates the top panel in Fig. Two examples of the cross-correlation between two series of returns for exchange rates are shown in Fig. From the data shown in Fig. This seems to be expected as in the former case there is a common base currency JPY.

In such a way, a pair of returns is intrinsically correlated by JPY currency performance due to the triangular constraint in the exchange rates. This is an example of cross-correlations among 3 currencies. In this case, the cross-correlations are in the triangular relation the top panel of Fig. In the case shown in the bottom panel of Fig. On the bottom-left and bottom-right panels, the results are unsorted making an easier task to identify particularly high cross-correlations shown with labels for each of the both cases, the triangular and non-triangular relationship.

The magnitude of this cross-correlation measure is weakly dependent on the timescale and only slightly grows with time. Its growth is more pronounced for larger fluctuations cf. Let us investigate in some more detail the cross-correlations between relatively small and large fluctuations of two exchange rate return series.

Note that the cross-correlation pairs are grouped into two classes. One class of exchange rate pairs in black, left top and bottom panels which pertain to the triangular relation and the second class, where cross-correlated pairs are outside the triangular relation in red, right top and bottom panels. This gives an idea about the range of obtained values of cross-correlation coefficient distributions for the currency pairs which are in or out the triangular relation.

The black dotted horizontal line on the top-left panel shows the average cross-correlation of different pairs pertaining to the triangular relation. The value of that overall average is about 0. It indicates a possibility of observing stronger correlations in exchange rates among four currencies in comparison with what we would expect on average in the case of exchange rates linked with the triangular relations.

This somewhat unexpected result could be ascribed to mechanisms coupling economies of these two countries. Hence, from our study it follows that indeed some cross-correlations of the pairs, which are not linked by a common currency and are traded on the Forex market, may reach that overall average cross-correlation of exchange rates with a common base.

This seems to be a surprising conclusion, since typically we would expect stronger correlations between explicitly correlated two series by means of a common, base currency rather than in a case where there is no such common base. However, we have to appreciate the fact that cross-correlations between any pair of exchange rates will have some impact on the cross-correlations of other pairs through mutual connections arising from different combinations of currencies being exchanged.

Nonetheless, the small fluctuations in logarithmic returns would be difficult to use in viable trading strategies, mainly due to finite spreads in bid and ask rates. From the practical point of view, correlations of large fluctuations seem to be more promising in finding and exploiting arbitrage opportunities. The order of currency pairs in the bottom panels is kept the same as in the top panels.

For the case of the larger fluctuations, the level of overall average of cross-correlations is marked again with horizontal black dotted line at a value of 0. The cross-correlations of the large fluctuations are therefore approximately two times smaller than in the case of small correlations. In the case of cross-correlations outside the triangular relations, the strong cross-correlations arise when we take AUD as base currency on the one side and NZD on the other.

This is in agreement with previous findings [ 25 , 41 , 54 ] for other financial instruments like stocks and commodities. The time evolution of averaged cross-correlations over currency pairs with the common base for large fluctuations will be discussed later cf. As we have already seen, studying quantitative levels of cross-correlations may uncover some less obvious connections among currencies than just the explicit link through a common base currency.

In order to uncover a hierarchy of currencies in terms of logarithmic returns from exchange rates, one may consider cross-correlation coefficients as a measure of the distance d i , j between different exchange rate pairs. We define the distance d i , j similarly to the definition introduced by [ 55 ], but instead of correlation coefficient, we use q -dependent cross-correlation coefficient [ 41 , 56 ].

Thus, the distance for agglomerative hierarchical trees takes the following form:. It is worth noting that to the best of our knowledge, it is the first ever such use of the cross-correlation q -coefficient as a way to induce measure for creating a hierarchy tree a dendrogram.

As a result of adopting the distance given by Eq. An interesting observation follows that Australlian AUD and New Zealand NZD dollars are strongly correlated—they appear together in the same clusters of exchange rates for both the small and large fluctuations. This indicates a possibility of building strong cross-correlations between exchange rate pairs which do not have the same common base.

Such findings are important when designing the trading strategies, both for optimizing portfolio and for its hedging. We would like to stress the fact that our method is not limited only to time series from the Forex and it may well be applied to the signals in a form of time series arising in other fields of research and applications. Color online Dendrograms corresponding to Fig. We have looked already into the cross-correlations within fluctuation magnitude domain. Let us now investigate the cross-correlations in the time domain.

The results are shown in Fig. For each q value, we consider 4 values of s corresponding to 5 min, 1 h, 24 h and 1 week. We also show the overall average for the currency exchange rate pairs complying to the triangular relation the black dotted line and for currency exchange rate pairs which are not bounded by the triangular relation red dotted line.

The most striking feature when comparing the small and large fluctuation cross-correlations over different timescales is that in the former case little is happening over different timescales considered. The plots indicate nearly static cross-correlations, almost independent on the timescale for the small fluctuations.

Specifically, the overall average denoted by the black dotted horizontal line grows from a value which is less than 0. The growth of the overall average of cross-correlation is even more convincing for the class of pairs of currency exchange rates which are not in a strict triangular relation. What is more, for the shortest timescale shown here, the difference between cross-correlations for pairs that are in the triangular relations and those that are not, is the biggest.

The cross-correlation for currency exchange pairs outside the triangular relation in the case of large fluctuations in logarithmic return rates grows in time, which indicates propagation of correlations in time. This explains why averaged cross-correlations for such currency pairs may be unexpectedly high cf. This gives us some idea about the information propagation time through the Forex market, which is the time needed to reflect the maximum average cross-correlation between any pair of exchange currency rates.

As we have already mentioned above, in the Forex market all currency rates are connected through mutual exchange rate mechanism. However, in some cases the inherently stronger correlations e. This time lag could be regarded as an estimate for the time duration of window of opportunity to execute an arbitrage opportunity.

The results presented in Fig 8 show how these averages change in consecutive years. The result is consistent for a range of timescales s taken in our approach. It is interesting to note that this effect becomes more pronounced when longer timescales are implemented 24 h, 1 week. A similar conclusion is valid when considering GBP or JPY taken as the base currency—corresponding curves have a maximum in This could mean that the sudden overnight increase in the rates by the Bank of Canada in did not have longer lasting effect and was only causing very short term effect.

In order to identify promising arbitrage opportunities e. In view of the above findings where we have already identified an important role of the large fluctuations, a question arises to what extent even briefly occurring in time such extreme events fluctuations in currency exchange returns may influence the detrended cross-correlations.

It is interesting to see how these extreme events manifest themselves as far as cross-correlations are concerned. These exchange rates exhibit substantial volatility during considered years. The dashed line corresponds to the cross-correlation results with rejected periods of time with large volatility and existence of triangular arbitrage opportunities. The periods of extreme variation of exchange rates are shown in the corresponding insets of Fig.

The insets show that in fact the exchange rates compared red and black curves were changing so rapidly that they could not follow each other. In such a way, the possible arbitrage opportunities have arisen. Finally, let us investigate closely these brief in time periods of arbitrage opportunities we have identified by our data analysis.

Color online Deviations from the triangular relations. In , existed a big arbitrage opportunity CHF , moderate arbitrage opportunity GBP in and no such opportunity in In this case, we use ask and bid prices for exchange rates instead of averaged ones in order to show this in more details.

All events indicated by values greater than 0 in fact could potentially offer triangular arbitrage opportunities. The top panel shows an example of potentially significant arbitrage opportunity which is related to the SNB intervention in and fluctuations in the CHF exchange rates. The middle panel of Fig. Finally, the bottom panel illustrates rather weak chance of exploiting triangular arbitrage opportunity—there is only one very brief in time instance when in theory this might be possible.

The arbitrage opportunities are very closely related to large fluctuations which tend to be more pronounced in the longer timescales s. This is the case for exchange rates related to CHF and GBP, and this is precisely what opens windows of opportunities for the triangular arbitrage. We have investigated currency exchange rates cross-correlations within the basket of 8 major currencies. Distributions of 10 s historical logarithmic exchange rate returns follow approximately the inverse cubic power-law behavior when the brief period of trading on January 15, , in the wake of the SNB intervention is excluded from the exchange rate data sets.

The tails of the cumulative distributions of the high-frequency intra-day quotes exhibit non-Gaussian distribution of the rare events by means of the so-called fat tails large fluctuations. This clearly documents that large fluctuations in the logarithmic rate returns occur more frequently than one may expect from the Gaussian distribution. We have found that on average the cross-correlations of exchange rates for currencies in the triangular relationship are stronger than cross-correlations between exchange rates for currencies outside the triangular relationship.

Such dendrograms may have important applications related to hedging, risk optimization, and diversification of the currency portfolio in the Forex market. Such abrupt changes of cross-correlations combined with the presence of relatively large fluctuations may signal potential triangular arbitrage opportunities.

Finally, our conjecture is that during significant events e. Such events and the resultant opportunities indeed have been identified in the historical trading data for the period — The evidence we have shown clearly indicates that the multifractal cross-correlation methodology should contribute significantly to predictive modeling of temporal and multiscale patterns in time series analysis.

We believe that our present study, where we consider currencies interaction through their mutual exchange rates and the dynamics of the rates adjustment to a new conditions due to a sudden event, may encourage future research in studying the information propagation through complex networks of interacting entities. This in turn may have some consequences for design of new smart learning methods for neural networks and a general computational intelligence in predicting a future behavior of complex systems.

For example, since we have demonstrated feasibility of financial time series analysis against favorable patterns, we may expect future advancement in computer algorithms for financial engineering when trading tick-by-tick data are available in real time. Google Scholar. Accessed 29 March Rickles, D. In: Hooker, C.

Philosophy of Complex Systems. Handbook of Philosophy of Science, vol. North Holland Ghashghaie, S. Nature , — Vandewalle, N. B 4 2 , — C 9 5 , — Basnarkov, L. Physica A , Boilard, J. Physica A , — Yang, Y. Han, C.

Financial Econ. Aiba, Y. Fenn, D. Finance 12 8 , — New J. Cui, Z. Finance Buchanan, M. Guida, T. Wiley, New York Moews, B. Expert Syst. Ghosh, I. Soft Comput. Miller, T. Chen-hua, S. Fan, Q. Physica A , 17—27 I have done an event study and the result is the graph below. The x-axis shows dummy variables before and after event. Tags: None. Marcos Almeida. Unfortunately, I cannot see thr graph in your message.

That said, I was also in doubt whether it is a survival or time-series analysis. Comment Post Cancel. There is the graph, hopefully the attachment uploaded this time. Attached Files. Phil Bromiley. I am not sure if Marcos is going to follow up on his reply. In case he doesn't, let me add my 2 cents worth.

First, you're more likely to get a useful answer if you follow the FAQ on asking questions - provide Stata code in code delimiters, readable Stata output, and sample data using dataex. In your case, it looks to me like you have a very strong time trend.

Declare a dataset to be time-series data. Stata is continually being updated, and Stata users are always writing new commands. To ensure. The Time-Series Reference Manual organizes the commands alphabetically, making it easy to find individual command entries if you know the name of the. A detrend involves removing the effects of trend from a data set to show only the differences in values from the trend; it allows cyclical and other.