Financial time series and neural networks in a minority game context 155The main point is that we suppose that players make their decisionsaccording to
a learning rule, as a consequence they follow an inductive behaviour and this affects
the time series of the minority decisions that is not simply a random sequence of
−1and+1. If this is the case the time series can be defined “pseudo-random” as a
consequence of the periodicity derived from the generating rule. This periodicity is not
due to the presence of a “trend” which is buried under noise but it is a consequence of
the inductive behaviour of players and this is the reason why classical techniques such
as simple autocorrelation analysis do not give us information, by definition, on the
learning procedure. On the contrary, the neural network with an appropriate learning
algorithm can capture such “regularities” very well and consequently can predict the
time series as shown in [10]. The main result presented in [10] is that a neural network
with an appropriate learning algorithm can predict “pseudo-random” time series very
well whatever the learning algorithm. On the other hand the neural network is not
able to predict a randomly generated time series. As a consequence, if we apply a
similar analysis to financial time series in the MG context presented before, we can
test for EMH since bad results in terms of prediction power of the neural network can
suggest that EMH is fulfilled and time series are randomly generated. If this is not the
case and the prediction power is remarkable then the time series is “pseudo-random”
as a consequence of inductive behaviour of the players. The neural network approach
also reveals the time window of past decisions that players are considering in order
to make their choice. As we will see, it is dependent on the market we consider.
2 Neural network and financial time series
The main issue of this paper is to determine the predictability of financial time se-
ries taking into account the imperfection of the market as a consequence of agents’
behaviour. In [10] it is shown that, when players make their decisions according to
some learning rule, then the time series of the minority decisions is not simply a
random sequence of−1and+1. The time series generated with learning algorithms
can be defined as “pseudo-random” time series. The reason is that, by construction, it
presents a sort of periodicity derived from the generating algorithm. This periodicity
is not evident directly from the time series but a neural network with an appropriate
learning algorithm can capture such “regularities” and consequently can predict the
time series. The authors show that, for three artificial sequences of minority deci-
sions generated according to different algorithms, the prediction power of the neural
network is very high.
In this paper we suppose thateach player, in order to make her decision, is provided
with a neural network. We consider time series from U.S. Fixed Income Market,
S&P500, DJ Eurostoxx 50, Dow Jones, Mibtel and Nikkei 225 (all the time series
from Jan 2003 to Jan 2008, daily prices, data from the Italian Stock Exchange).
Following the motivations presented in [10], in this paper we consider a neural
network that uses the Hebbian Learning algorithm to update the vector of weights.
The neural network is able to adjust its parameters, at eachround of the game, and
so the perceptron is trying to learn the minority decisions. IfSindicates the minority