A Hessian Regularized Nonlinear Time Series Model

We introduce a nonparametric nonlinear time series model. The novel idea is to fit a model via penalization, where the penalty term is an unbiased estimator of the integrated Hessian of the underlying function. The underlying model assumption is very general: it has Hessian almost everywhere in its domain. Numerical experiments demonstrate that our model has better predictive power: if the underlying model complies with an existing parametric/semiparametric form (e.g., a threshold autoregressive model (TAR), an additive autoregressive model (AAR), or a functional coefficient autoregressive model (FAR)), our model performs comparably; if the underlying model does not comply with any preexisting form, our model outperforms in nearly all simulations. We name our model a Hessian regularized nonlinear model for time series (HRM). We conjecture on theoretical properties and use simulations to verify. Our method can be viewed as a way to generalize splines to high dimensions (when the number of variates is more than three), under which an analogous analytical derivation cannot work due to the curse of dimensionality. Supplemental materials are provided, and will help readers reproduce all results in the article.