Functional Coefficient Regression Models for Non‐linear Time Series: A Polynomial Spline Approach

Abstract

ABSTRACT. We propose a global smoothing method based on polynomial splines for the estimation of functional coefficient regression models for non-linear time series. Consistency and rate of convergence results are given to support the proposed estimation method. Methods for automatic selection of the threshold variable and significant variables (or lags) are discussed. The estimated model is used to produce multi-step-ahead forecasts, including interval forecasts and density forecasts. The methodology is illustrated by simulations and two real data examples.

Optimal Uniform Convergence Rates and Asymptotic Normality for Series Estimators Under Weak Dependence and Weak Conditions

“…This approach is not without precedent in the nonparametric analysis of nonlinear time series models. For example, Huang and Shen (2004) use a similar approach to trim extreme observations in nonparametric functional coefficient regression models, following Tjøstheim and Auestad (1994). More generally, we can consider an expanding sequence of compact nonempty sets D n ⊂ X with D n ⊆ D n+1 for all n and set w n (x) = {x ∈ D n } for all n. For example, if X = R d we could take D n = {x ∈ R d : x p ≤ r n } where 0 < r n ≤ r n+1 < ∞ for all n. This approach is similar to excluding functions far from the support of the data when performing series LS estimation with a compactly-supported wavelet basis for…”

Modeling Nonlinear Vector Time Series Data

“…Once it is estimated, one may consider making simultaneous inference about parameters and using the residuals to study the structure of the volatility matrix. This model is a generalization of vector autoregressive models [1], threshold models [3] and functional-coefficient models [7][8][9][10]. Even for onedimensional settings with k = 1, model (5) includes important predictive regression models in econometrics, such as the linear predictive models with nonstationary predictors [11][12][13] and functional-coefficient models for nonstationary time series data [14].…”

A New Forecasting Model for USD/CNY Exchange Rate

“…Besides the above local linear estimation method, Huang and Shen (2004) proposed a global smoothing method based on a polynomial spline to estimate the functional-coefficient model. One appealing feature of their method is that the functional coefficients can depend on different smoothing variables.…”