Ebook Forecasting Financial Time-Series using Artificial Market Models
Judging from the literature, in particular the wide range of popular finance books, the possibility of predicting future movements in financial markets ranges from significant (see, for example, the many books on chartism) to impossible (see, for example, [14]). Another scenario of course does exist that financial markets may neither be predictable or unpredictable all the time, but may instead have periods where they are predictable (i.e. non-random) and periods where they are not (i.e. random). Evidence for such ‘pockets of predictability’ were found several years ago, by Johnson et al. [10]. A similar study was reported subsequently by Sornette et al. [2]. However, a formal report of a theoretical framework for identifying such periods of predictability has not appeared in the literature to date.
The rationale behind our initial proposal to predict financial markets using artificial market models, is as follows. Financial markets produce time-series, as does any dynamical system which is evolving in time, such as the ambient air-temperature, or the electrical signals generated by heart rhythms. In principle, one could use any time-series analysis technique to build up a picture of these statistical fluctuations and variations - and then use this technique to attempt some form of prediction, either on the long or short time-scale. One example would be to use a multivariate analysis of the prices themselves in order to build up an estimate of the parameters in the multivariate expansion, and then run this model forwards. However, such a multivariate model may not bear a relation to any physical representation of the market itself. Instead, imagine that we are able to identify an artificial market model which seems to produce the aggregate statistical behavior (i.e. the stylized facts) observed in the financial market itself. It now has the additional advantage that it also mimics the microscopic structure of the market, i.e. it contains populations of artificial traders who use strategies in order to make decisions based on available information, and will adapt their behavior according to past successes or failures. All other things being equal, we believe that such a model may be intrinsically ‘better’ than a purely numerical multivariate one - and may even be preferable to many more sophisticated models such as certain neural network approaches, which also may not be correctly capturing a realistic representation of the microscopic details of a physical market. The question then arises as to whether such an artificial market model could be ‘trained’ on the real market in order to produce reliable forecasts.
Although in principle one could attempt to train any artificial market model on a given financial time-series, each of the model parameters will need to be estimated - and if the model has too many parameters, this will become practically impossible as the model will become over-determined. For this reason, our own (and Sornette et al.’s [2]) attempts have focused on training a minimal artificial market model. We had already shown in [10] that a minimal model could be built from a binary multi-agent game in which agents are allowed to not participate if they were not sufficiently confident.
Here we focus on the basic Minority Game where all agents trade at every time-step, since we are interested in describing in detail the parameter estimation process as opposed to creating the best possible market prediction model. In particular, there is no reason to expect that (i) the Minority Game’s pay-off structure whereby only the minority group gets rewarded, or (ii) the Minority Game’s traditional feature whereby all agents have the same memory time-scale m, are either realistic or optimal. For a more complete discussion of suitable pay-off structures in artificial financial markets, see Chapter 4 in [9] (also see [8]). For the present discussion, we retain both these features since they do not affect the formalism presented - however we note that recent work by Mitman et al. [16] and subsequent work by Guo [7] have shown that allowing agents to have multiple memory values does indeed lead to improved performance both overall and for specific individual traders.
Binary Agent Resource games, such as the Minority Game and its many extensions, have a number of real-world applications in addition to financial markets. For example, they are well-suited to studying traffic flow in the fairly common situation of drivers having to choose repeatedly between two particular routes from work to home. In these examples, and in particular the financial market setting, one would like to predict how the system will evolve in time given the current and past states. In the context of the artificial market corresponding to the Minority Game and its generalizations, this ends up being a parameter estimation problem. In particular, it comes down to estimating the composition of the heterogeneous multi-trader population, and specifically how this population of traders is distributed across the strategy space. Here we investigate the use of iterative numerical optimization schemes to estimate these quantities, in particular the population’s composition across the strategy space - we will then use these estimates to make forecasts on the time-series we are analyzing. Along with these forecasts, we also need to find a covariance matrix in order to determine the certainty with which we believe our forecast is correct. Such a covariance matrix is important for a number of reasons including risk analysis.
Given the forecast and its associated covariance matrix, we will also need to decide whether to use the forecast or throw it away based on the covariance matrix (which represents the expected errors on the forecast). We discuss this point, and in so doing we will see that the system can fall into ‘pockets of predictability’ during which the system becomes predictable over some significant time-window. In the rest of this paper, we discuss these ideas and apply them to simulated and real financial market data, in order to identify pockets of predictability based on the model.
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