Kohonen Maps and TS Algorithms

Kohonen Maps and TS Algorithms

Marie-Thérèse Boyer-Xambeu (Université de Paris VII – LED, France), Ghislain Deleplace (Université de Paris 12 – ERUDITE, France), Patrice Gaubert (CNRS – LED, France) and Lucien Gillard (Université de Paris I – CES SAMOS, France)
Copyright: © 2009 |Pages: 8
DOI: 10.4018/978-1-59904-849-9.ch147
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Abstract

In the analysis of a temporal process, Kohonen maps may be used together with time-series (TS) algorithms. Previous research aimed at combining Kohonen algorithms and Markov switching models in order to suggest a periodization of the international bimetallism in the 19th century (Boyer-Xambeu, Deleplace, Gaubert, Gillard and Olteanu, 2006). This research was based on an economic study of the international monetary system ruling at this time in Europe, which combined three monetary zones: a gold-standard one, centred in London, a bimetallic one, centred in Paris, and a silver-standard one, centred in Hamburg (Boyer- Xambeu, Deleplace and Gillard, 2006). The three major financial centres of that system (London, Paris, and Hamburg, hence the label LPH used hereafter) were linked through arbitrage operations between markets for gold and silver and markets for foreign exchange located in those centres. Since two metals, gold and silver, acted as monetary standards in that system, it worked as an international bimetallism. Its growing integration during half a century (from 1821 to 1873) was reflected in the convergence of the observed levels of the relative price of gold to silver in London, Paris, and Hamburg. However, this integration process was subject to various changes, which can be understood as exogenous shocks disturbing that process. One such shock is vastly documented in the literature: the discovery of new gold mines in the United States and Australia, which led to a sudden decline in 1850 of the gold-silver price over all the markets in the world. This decline was not of the same magnitude everywhere, and therefore the spread between the London, Paris, and Hamburg gold-silver prices increased, stopping for a time the integration process of the system. This is what we will call a breaking in that process. The present paper aims at locating the major breakings occurring during the period of international bimetallism; a historical study could link them to special events, which operated as exogenous shocks on that system. The indicator of integration used is the spread between the highest and the lowest among the London, Paris, and Hamburg gold-silver prices. Three algorithms are combined to study this integration: a periodization obtained with the SOM algorithm is confronted to the estimation of a two-regime Markov switching model, in order to give an interpretation of the changes of regime; at the same time changepoints are identified over the whole period providing a more precise interpretation of these varying types of regulation. Section 2 summarizes the results obtained with the SOM algorithm to differentiate the sub-periods obtained using the whole available data. Section 3 presents the kind of model used and the results of its estimation using the new indicator, the spread computed at each period of quotation between the three relative prices of gold in silver. The sub-periods are confronted to the two regimes obtained and some evidence of a relation between the regime and the volatility of the spread is presented. Section 4 presents the technique used to identify change-points in the temporal process and some strong results of breaks in mean and in variance of the spread are obtained. They are interpreted in terms of monetary history as, for some of them, they are quite new in the literature of this domain. Some further directions of research are indicated in conclusion.
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The Sub-Periods Obtained With A Som Algorithm1

The Data

The relative prices of gold in silver are computed from the price of each metal observed, twice a week, in each of the three financial places, Paris, London and Hamburg (respectively, poa, lgs, and hoa), from the beginning of 1821 until the end of 1860. The same type of data is available for the exchange rates (Pound in Francs, Pound in Marks, Mark in Francs: respectively, lpv, hlv, and phv).

An observation is a set of twelve values, two quotations (Tuesday and Friday) for each of the six variables.

A computed variable has been added to emphasize the relation between the relative price of metals in Hamburg and the average level in Paris and London of this value (hpl).

Most of the time the quotations show rather small differences within a given week, but periods with important troubles, Paris in the late 1840s for instance, may be well separated from the more classical ones.

After the Kohonen classification using a grid of 25 nodes, a hierarchical ascending classification is used to produce a small number of macro classes, in this case 6 macro classes, corresponding to the main sub-periods. This latter classification is constructed with the code vectors obtained from the first process2.

Key Terms in this Chapter

Change-Point: Instant of time where the basic parameters of time series change (in mean and/or in variance); the series may be considered as a piecewise stationary process between two change-points

International Monetary System: A system linking the currencies of various countries, which ensures the stability of the exchange rates between them. Its working depends on the monetary rules adopted in each country and on international arbitrage (see that definition) between the foreign exchange markets. Historical examples are the gold-standard system (1873-1914) and the Bretton-Woods system (1944-1976). The paper studies some characteristics of another historical example: international bimetallism (see that definition).

International Arbitrage: Activity of traders in gold and silver and in foreign exchange, which consisted in comparing their prices in different places, and in moving the precious metals and the bills of exchange accordingly, in order to make a profit. Arbitrage and monetary rules were the two factors explaining the working of international bimetallism (see that definition).

Markov Switching Model: An autoregressive model where the process linking a present value to its lags is an hidden Markov chain defined by its transition matrix

SOM Algorithm: An unsupervised technique of classification (Kohonen,1984) combining adaptative learning and neighbourhood to construct a very stable classification, with a more simple interpretation (‘Kohonen maps’) than other techniques.

Gold-Silver Price: Ratio of the market price of gold to the market price of silver in one place. The stability of that ratio through time and the convergence of its levels in the various places constituting the international bimetallism (see that definition) are tests of the integration of that system.

International Bimetallism: An international monetary system (see that definition) which worked from 1821 to 1873. It was based on gold and silver acting as monetary standards, either together in the same country (like France) or separately in different countries (gold in England, silver in German and Northern states). The integration of that system was reflected in the stability and the convergence of the observed levels of the relative price of gold to silver (see that definition) in London, Paris, and Hamburg.

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