Towards a Computationally Tractable Maximum Entropy Principle for Nonstationary Financial Time Series

Statistical analysis of financial time series of equity returns can be hindered by various unobserved factors, resulting in a nonstationarity of the overall problem. Parametric methods approach the problem by restricting it to a certain (stationary) distribution class through various assumptions, which often result in a model misspecification when the underlying process does not belong to this predefined parametric class. Nonparametric methods are more general but can lead to ill-posed problems and computationally expensive numerical schemes. This paper presents a nonparametric methodology addressing these issues in a computationally tractable way by combining such key concepts as the maximum entropy principle for nonparametric density estimation and a Lasso regularization technique for numerical identification of redundant parameters and persistent latent regimes. In the context of volatility modeling, the presented approach identifies an a priori unknown nonparametric persistent regime transition process switching between distinct local nonparametric i.i.d. volatility distribution regimes with maximum entropy. Using historical return data for an equity index, we demonstrate that despite viewing the data as serially independent conditional on the latent regimes (serial dependence is introduced only via the stochastic latent regime), our methodology leads to identification of robust models that are superior to considered conditional heteroskedasticity models when compared to the model log-likelihood, Akaike, and Bayesian information criteria for the analyzed data.