158 L. Grilli, M.A. Russo, and A. Sfrecola
in long-term bonds, so the expectations about the market evolution are so similar that
the behaviour of long-term bond prices does not reflect any difference in theperceived
valueof such assets [3].
The analysis shows that the number of corrected predictions is dependent on the
parameterM; it is not the same for the parameterηsince the number of corrected
predictions remains quite constant (we report it in Fig. 3).
The neural network approach has shown the presence of a valueMfor which the
prediction power is maximum and this is a signal in the direction that the time series
is “pseudo-random” and agents useMas the time window. This can be interpreted
in terms of lack of EMH, but this is only partially true since the neural network
cannot predict in a significant way (in terms of number of corrected predictions)
the financial time series considered and this can indicate that these time series are
randomly generated and so these markets are efficient. This result is not surprising
since all the markets considered in this paper present a huge number of transactions
and huge volumes, and information provided to agents is immediately and easily
available. The neural network can predict, for these time series, slightly more than
50%, which is the expected value of corrected predictions in cases where choices are
randomly made, which is a signal in the direction that these markets fulfil the EMH
and results in other directions can be considered “anomalies”. A comparative analysis
reveals that the Fixed Income Markets seems to the least efficient since the number
of predictions is maximal.
3 Conclusions
In [10] the authors show that in an MG framework, a neural network that use a Hebbian
algorithm can predict almost every minority decision in the case in which the sequence
of minority decisions follows a “pseudo”-random distribution. The neural network
can capture the “periodicity” of the time series and then predict it. On the other hand
they show that the prediction power is not so good when the time series is randomly
generated. In this paper we consider financial time series from U.S. Fixed Income
Market, S&P500, DJ Eurostoxx 50, Dow Jones, Mibtel and Nikkei 225. If agents
make satisfactory choices instead of optimal ones, they are inductive in the sense that
they learn from experience and MG is a very good model for inductive behaviour of
financial agents. If financial time series are generated by some learning procedure,
then we can consider financial time series as “pseudo”-random time series and in this
case the prediction of neural networks is appreciable. So we consider the financial
time series from the Minority Game point of view and then we apply a neural network
with learning algorithm in order to analyse its prediction power as a measure of market
efficiency.
We show that the case of U.S. Treasury Bond seems to be the most interesting
since the time window of the past minorities considered by the investor isM=32,
which is very high with respect to other markets, and for this time series the neural
network can predict about 60% of entries. This is a signal in the direction that the
Fixed Income Market is more predictable as a consequence of features that make