Semiparametric Regression Smoothing of Non-linear Time Series

In this paper, we consider using a semiparametric regression approach to modelling non-linear autoregressive time series. Based on a finite series approximation to non-parametric components, an adaptive selection procedure for the number of summands in the series approximation is proposed. Meanwhile, a large sample study is detailed and a small sample simulation for the Mackey–Glass system is presented to support the large sample study.     

Nonparametric kernel estimation of the impact of tax policy on the demand for private health insurance in Australia

This paper is motivated by our attempt to answer a policy question: how is private health insurance take‐up in Australia affected by the income threshold at which the Medicare Levy Surcharge (MLS) kicks in? We propose a new difference deconvolution kernel estimator for the location and size of regression discontinuities. We also propose a bootstrapping procedure for estimating the confidence interval for the estimated discontinuity. Performance of the estimator is evaluated by Monte Carlo simulations before it is applied to estimating the effect of the income threshold of MLS on the take‐up of private health insurance in Australia, using contaminated data.     

Semiparametric methods in nonlinear time series analysis: a selective review

Time series analysis is a tremendous research area in statistics and econometrics. In a previous review, the author was able to break down up 15 key areas of research interest in time series analysis. Nonetheless, the aim of the review in this current paper is not to cover a wide range of somewhat unrelated topics on the subject, but the key strategy of the review in this paper is to begin with a core the ‘curse of dimensionality’ in nonparametric time series analysis, and explore further in a metaphorical domino-effect fashion into other closely related areas in semiparametric methods in nonlinear time series analysis.     

Bayesian Bandwidth Estimation in Nonparametric Time-Varying Coefficient Models

Bandwidth plays an important role in determining the performance of nonparametric estimators, such as the local constant estimator. In this article, we propose a Bayesian approach to bandwidth estimation for local constant estimators of time-varying coefficients in time series models. We establish a large sample theory for the proposed bandwidth estimator and Bayesian estimators of the unknown parameters involved in the error density. A Monte Carlo simulation study shows that (i) the proposed Bayesian estimators for bandwidth and parameters in the error density have satisfactory finite sample performance; and (ii) our proposed Bayesian approach achieves better performance in estimating the bandwidths than the normal reference rule and cross-validation. Moreover, we apply our proposed Bayesian bandwidth estimation method for the time-varying coefficient models that explain Okun’s law and the relationship between consumption growth and income growth in the U.S. For each model, we also provide calibrated parametric forms of the time-varying coefficients. Supplementary materials for this article are available online.     

Adaptive estimation in partially linear autoregressive models

The authors consider a partially linear autoregressive model and construct kernel‐based estimates for both the parametric and nonparametric components. They propose an estimation procedure for the model and illustrate it through simulated and real data. Their work shows that the proposed estimation procedure not only has good asymptotic properties but also works well numerically. It also suggests that a partially linear autoregression is more appropriate than a completely nonparametric autoregression for some sets of data.     

Nonparametric Methods in Continuous Time Model Specification

Some popular parametric diffusion processes have been assumed as such underlying diffusion processes. This paper considers an important case where both the drift and volatility functions of the underlying diffusion process are unknown functions of the underlying process, and then proposes using two novel testing procedures for the parametric specification of both the drift and diffusion functions. The finite-sample properties of the proposed tests are assessed through using data generated from four popular parametric models. In our implementation, we suggest using a simulated critical value for each case in addition to the use of an asymptotic critical value. Our detailed studies show that there is little size distortion when using a simulated critical value while the proposed tests have some size distortions when using an asymptotic critical value in each case.     

Nonparametric Estimation and Forecasting for Time-Varying Coefficient Realized Volatility Models

This article introduces a new specification for the heterogenous autoregressive (HAR) model for the realized volatility of S&P 500 index returns. In this modeling framework, the coefficients of the HAR are allowed to be time-varying with unspecified functional forms. The local linear method with the cross-validation (CV) bandwidth selection is applied to estimate the time-varying coefficient HAR (TVC-HAR) model, and a bootstrap method is used to construct the point-wise confidence bands for the coefficient functions. Furthermore, the asymptotic distribution of the proposed local linear estimators of the TVC-HAR model is established under some mild conditions. The results of the simulation study show that the local linear estimator with CV bandwidth selection has favorable finite sample properties. The outcomes of the conditional predictive ability test indicate that the proposed nonparametric TVC-HAR model outperforms the parametric HAR and its extension to HAR with jumps and/or GARCH in terms of multi-step out-of-sample forecasting, in particular in the post-2003 crisis and 2007 global financial crisis (GFC) periods, during which financial market volatilities were unduly high.     

Local linear kernel estimation for discontinuous nonparametric regression functions

In this paper, we consider using a local linear (LL) smoothing method to estimate a class of discontinuous regression functions. We establish the asymptotic normality of the integrated square error (ISE) of a LL-type estimator and show that the ISE has an asymptotic rate of convergence as good as for smooth functions, and the asymptotic rate of convergence of the ISE of the LL estimator is better than that of the Nadaraya-Watson (NW) and the Gasser-Miiller (GM) estimators.     

ROBUST ESTIMATION IN PARAMETRIC TIME SERIES MODELS UNDER LONG- AND SHORT-RANGE-DEPENDENT STRUCTURES

This paper studies the asymptotic behaviour of an M-estimator of regression parameters in the linear model when the design variables are either stationary short-range dependent (SRD), α-mixing or long-range dependent (LRD), and the errors are LRD. The weak consistency and the asymptotic distributions of the M-estimator are established. We present some simulated examples to illustrate the efficiency of the proposed M-estimation method.