85. Stochastic Change-Point Models of Asset Returns and Their Volatilities

We begin with an overview of frequentist and Bayesian approaches to incorporating change points in time series models of asset returns and their volatilities. It has been found in many empirical studies of stock returns and exchange rate that ignoring the possibilities of parameter changes yields time series models with long memory, such as unit-root nonstationarity and high volatility persistence. We therefore focus on the ARX-GARCH model and introduce two timescales, using the “short” timescale to define GARCH dynamics and the “long” timescale to incorporate parameter jumps. This leads to a Bayesian change-point ARX-GARCH model, whose unknown parameters may undergo occasional changes at unspecified times and can be estimated by explicit recursive formulas when the hyperparameters of the Bayesian model are specified. We describe efficient estimators of the hyperparameters of the Bayesian model leading to empirical Bayes estimators of the piecewise constant parameters with relatively low computational complexity. We also show how the computationally tractable empirical Bayes approach can be applied to the frequentist problem of partitioning the time series into segments under sparsity assumptions on the change points.