Semiparametric Ultra-High Dimensional Model Averaging of Nonlinear Dynamic Time Series

Abstract

We propose two semiparametric model averaging schemes for nonlinear dynamic time series regression models with a very large number of covariates including exogenous regressors and autoregressive lags. Our objective is to obtain more accurate estimates and forecasts of time series by using a large number of conditioning variables in a nonparametric way. In the first scheme, we introduce a Kernel Sure Independence Screening (KSIS) technique to screen out the regressors whose marginal regression (or auto-regression) functions do not make a significant contribution to estimating the joint multivariate regression function; we then propose a semiparametric penalized method of Model Averaging MArginal Regression (MAMAR) for the regressors and auto-regressors that survive the screening procedure, to further select the regressors that have significant effects on estimating the multivariate regression function and predicting the future values of the response variable. In the second scheme, we impose an approximate factor modelling structure on the ultra-high dimensional exogenous regressors and use the principal component analysis to estimate the latent common factors; we then apply the penalized MAMAR method to select the estimated common factors and the lags of the response variable that are significant. In each of the two schemes, we construct the optimal combination of the significant marginal regression and auto-regression functions. Asymptotic properties for these two schemes are derived under some regularity conditions. Numerical studies including both simulation and an empirical application to forecasting inflation are given to illustrate the proposed methodology.

A new semiparametric estimation approach for large dynamic covariance matrices with multiple conditioning variables

“…The moment condition (3.1) is similar to those in Bickel and Levina (2008a,b) and Chen and Leng (2016), and can be replaced by the weaker condition of E(|X ti | κ ) for κ > 2 sufficiently large if the dimension N diverges at a polynomial rate of T . The restriction of the conditioning variables U t having a compact support in Assumption 1(iii) is imposed mainly in order to facilitate the proofs of our asymptotic results and can be removed by using an appropriate truncation technique on U t (c.f., Remark 1 in Chen et al, 2018). The smoothness condition on the marginal regression functions in Assumption 2(i) is commonly used in kernel smoothing, and it is relevant to the asymptotic bias of the kernel estimators (c.f., Wand and Jones, 1995).…”

“…, p, on the right hand side of the equation for a ij,0 in (2.11). By Theorem 2(ii) in Chen et al (2018), under the sparsity assumption and some technical conditions, the zero optimal weights can be estimated exactly as zeros with probability approaching one. After obtaining b i,k and a ij,k , 0 k p, we can calculate the penalised estimates of the optimal MAMAR approximation to c 0 ij (u) and m 0 i (u) as…”

“…The MAMAR method provides an alternative way to estimate nonparametric joint mean regression with multiple regressors. The MAMAR approximation is introduced by Li, Linton and Lu (2015) in a semiparametric time series setting, and is applied to semiparametric dynamic portfolio choice by and further generalised to the ultra-high dimensional time series setting by Chen et al (2018). A similar idea is also used by Fan et al (2016) in high-dimensional classification.…”

MALMEM: Model Averaging in Linear Measurement Error Models

“…Hansen (2007), a seminal work on asymptotically optimal model averaging, selected the weights through minimizing the Mallows criterion, because of its unbiasedness (up to a constant) in estimating expected squared error. Other frequentist model averaging strategies include adaptive regression through mixing (Yang, 2001), jackknife model averaging (Hansen and Racine, 2012), heteroscedasticity robust model averaging (Liu and Okui, 2013), model averaging marginal regression (Chen et al, 2018;Li et al, 2015) and the plug-in method (Liu, 2015). Model averaging has also been extended to other contexts such as structural break models (Hansen, 2009), mixed effects models (Zhang et al, 2014), factor-augmented regression models (Cheng and Hansen, 2015), quantile regression models (Lu and Su, 2015), generalized linear models (Zhang et al, 2016) and missing data models (Fang et al, 2019;Zhang, 2013).…”

Variable screening for high dimensional time series

“…However, the dependence structure of longitudinal data is too restrictive to cover the type of dependence present in most time series. To the best of our knowledge there have been only two works, Chen et al (2017) and Yousuf (2018), dealing with the issue in a general stationary time series setting. The former work extended the nonparametric independence screening approach used for independent observations to the time series setting.…”