Methods of Hyperparameter Estimation in Time-Varying Regression Models with Application to Dynamic Style Analysis of Investment Portfolios

The problem of estimating time-varying regression inevitably concerns the necessity to choose the appropriate level of model volatility – ranging from the full stationarity of instant regression models to their absolute independence of each other. In the stationary case the number of regression coefficients, constituting the model parameter to be estimated, equals that of regressors, whereas the absence of any smoothness assumptions augments the dimension of the unknown vector by the factor of the time-series length. We consider here a family of continuously nested a priori probability distributions matching the specificity of time-varying data models, in which the dimension of the parameter is fixed, but the freedom of its values is softly constrained by a family of continuously nested a priori probability distributions, which contains a number of hyperparameters. The aim of this paper is threefold. First, in accordance with the specificity of the time-varying regression, we modify three commonly adopted methods of estimating hyperparameters in data models, namely, Leave-One-Out Cross Validation, Evidence Maximization and Hypothetical Cross Validation. Second, we experimentally compare these methods on both simulated and real-world data. Third, on the basis of the proposed technique we develop a new approach to the problem of detecting the hidden dynamics of an investment portfolio in respect to certain market or economic factors.