Improving Heterogeneous Agent Models by Avoiding Explicit Discretizations of Stiff Equations

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Baumann, Michael Heinrich ; Baumann, Michaela ; Grüne, Lars ; Herz, Bernhard:
Improving Heterogeneous Agent Models by Avoiding Explicit Discretizations of Stiff Equations.
Bayreuth , 2020 . - 38 S.
DOI: https://doi.org/10.15495/EPub_UBT_00004718

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Abstract

We consider a standard heterogeneous agent model that is widely used to analyze price developments in financial markets. The model is linear in log-prices and, in its basic setting, populated by fundamentalists and chartists. These fundamentalists are typically believed to stabilize markets by bringing asset prices back to their fundamental values. However, we illustrate that in this type of model, this does not necessarily hold as - unintended and so far over-looked - instabilities might occur. As the number of fundamentalists increases and exceeds a specific threshold, oscillations occur whose amplitude might even grow exponentially over time.

We show that this instability phenomenon is due to a "hidden" explicit discretization of a stiff ordinary differential equation contained in the model. Replacing this explicit discretization by an implicit one removes this artifact, bringing the model's prediction in line with standard theory.
We extend our analysis and simulate markets with evolutionary rules, i.e., replicator dynamics, for the explicit as well as the implicit model. Overall, we find that our analytical results carry over to the extended model. Models based on explicit discretization are likely to overrate price instabilities and, in particular, bubbles and crashes and imply biased results in the empirical application of heterogeneous agent models.