The analysis of statistical correlation between the most developed countries based on chosen macroeconomy parameters is performed. The main aim of this analysis is to investigate similarities in the development pattern. The following set of countries is considered: Austria (AUT), Belgium (BEL), Canada (CAN), Denmark (DNK), Finland (FIN), France (FRA), Germany\footnote{Germany is considered as a one country. To have a record before consolidation the data are constructed as a sum of GDP of both German countries.} (DEU), Greece (GRC), Ireland (IRL), Italy (ITA), Japan (JPN), the Netherlands (NLD), Norway (NOR), Portugal (PRT), Spain (ESP), Sweden (SWE), Switzerland (CHE), United Kingdom (GBR) and USA. The countries are described by theirs Gross Domestic Product (GDP), since in most countries GDP is considered as an official parameter of the economic situation. GDP is usually defined as a sum of all final goods and services produced in the country equal to total consumer, investment and government spending, plus the value of exports, minus the value of imports [1]. Additionally in order to define a reference country an artificial "All" country is constructed. GDP of "All" country is defined as a sum of GDP of all 19 countries. So the GDP increment of "All" can be considered as a reference level of development.
The GDP values for each of these countries are first normalised to their 1990 value given in US dollars as published by the Groningen Growth and Development Center on their web page [2]. The data cover the period between 1950 and 2003, i.e. 54 points for each country.
The GDP yearly increment given by Eq.(1) is considered and its statistical properties are presented
Δ GDP(t) = [GDP(t)  GDP(t1)]/[GDP(t1)]. (1)
A distance matrix is calculated, where the distance is defined as
d(i,j)_{(t,T)} = {[1 corr_{(t,T)} (c_{i},c_{j})]/2}^{1/2}. (2)
The correlation function corr_{(t,T)} is
corr_{(t,T)} (c_{i},c_{j}) = (<c_{i} c_{j}>_{(t,T)}  <c_{i}>_{(t,T)}<c_{j}>_{(t,T)})/[(<c_{i}^{2}>_{(t,T)}  <c_{i}>^{2}_{(t,T)})(<c_{j}^{2}>_{(t,T)}  <c_{j}>^{2}_{(t,T)})]^{1/2},
where c_{i} denotes the time series of increments of GDP for the i^{th} country, and <c_{i}>_{(t,T)} is the average of yearly GDP increments in the time window (t,t+T) of size T. This definition gives the distance, which is decreasing while correlations are increasing, so the shortest distance is between the countries with the highest correlation GDP increments. Eq.(2) maps the linear space of the series L_{n} of the length n onto the interval [0,1]:
d: L_{n} x L_{n} >[0,1],
where the distance d takes the value 0 for correlated time series and 1 for anticorrelated series.
The distances between country are illustrated for a few (5y, 15y, 25y, 35y, 45y) time windows, where y denotes years. It means that correlations between such countries are measured within very short, short, medium, long and very long time windows.
In order to obtain some quantitative information on the country correlations, we have looked for clusters or structures formation. Classical way to search for cluster is find subgraphs with high clustering coefficient [3]. The alternative way is to build a well defined structure e.g. Minimal Spanning Tree (MST) and look for a structures repeating in consecutive time windows or to find a set of nodes connected by short links. For the sake of simplicity minimal length path algorithm (MLP), which is a 1D modification of the MST algorithm is used. This algorithm emphasizes the strongest correlation between entities with the constraint that the item is attached only once to the network. This results in a lack of loops in the ''tree''. The construction of more elaborate networks is left for further studies. Two different graphs: the unidirectional (with a given initial point) and bidirectional minimal length paths (UMLP and BMLP respectively) are constructed, as a function of time and for moving time windows of various sizes. The size of time window is constant during the displacement.
The UMLP and BMLP algorithms are defined as follows:
[UMLP]: The algorithm begins with choosing an initial point of the chain. Here the initial point is the "All" country. Next the shortest connection (in terms of the distance definition  Eq.(2) is looked for between the initial point and the other possible neighbours. The closest possible one is selected and attached to the initial point. One searches next for the entity closest to the previously attached one, and repeat the process.
[BMLP]: The algorithm begins with searching for the pair of countries which has the shortest distance between them. Then these countries become the root of a chain. In the next step the closet country for both ends of the chain is searched. Being selected it is attached to the appropriate end. Next a search is made for the closest neighbour of the new ends of the chain. Being selected, the entity is attached, a.s.o.
Considering different time windows a sort of critical correlation time has been found. It is pointed out that the size of the time window for which the correlations are well seen should not be shorter then 15y, but the most appropriate is 25y time window. This means that on the level of global economy correlations are well seen in the medium length time window and cooperations between countries form a stable relationship. In the case of medium and long time window formation of clusters, understood as a set of countries with highly correlated GDP increment is observed.
The properties of UMLP and BMLP algorithms are compared and it is found that BMLP algorithm is more sensitive to searching for a clustering patterns among considered entities, while UMLP is suitable for ranking countries (companies) and could be useful in solving portfolio problems.
A new method for estimating a realistic minimal time window to observe correlations in macroeconomy is thus suggested. This method could be also applied to a stock market analysis. The mean distances analysis is expected to be useful in estimating the shortest time window in analyzing correlations on a stock market as well. In such a case it should be compared to moving average windows.
Bibliography
[1] http://www.investorwords.com/2153/GDP.html
[2] http://www.ggdc.net/indexdseries.html#top
[3] R. Albert, A.L. Barab'asi: Statistical Mechanics of Complex Networks, Rev. Mod. Phys. 47 (2004) p.74
