Latent Variable Nonparametric Cointegrating Regression

Abstract

This article studies the asymptotic properties of empirical nonparametric regressions that partially misspecify the relationships between nonstationary variables. In particular, we analyze nonparametric kernel regressions in which a potential nonlinear cointegrating regression is misspecified through the use of a proxy regressor in place of the true regressor. Such models occur in linear and nonlinear regressions where the regressor suffers from measurement error or where the true regressor is a latent or filtered variable as in mixed-data-sampling. The treatment allows for endogenous regressors as the latent variable and proxy variables that cointegrate asymptotically with the true latent variable, including correctly specified as well as misspecified systems, and is therefore intermediate between nonlinear nonparametric cointegrating regression and completely spurious nonparametric nonstationary regression. The results relate to recent work on dynamic misspecification in nonparametric nonstationary systems and the limit theory accommodates regressor variables with autoregressive roots that are local to unity and whose errors are driven by long memory and short memory innovations, thereby encompassing applications with a wide range of economic and financial time series. Some implications for forecasting under misspecification are also examined.

Weighted Nonlinear Regression With Nonstationary Time Series

“…Then, for any cn → ∞ satisfying cn/n → 0, we have where ĝ1(s, x) = Eg1(s, x, Vm). This is essentially the same as in the proof of (A.20) with i = 2 in Wang, et al (2021) and hence the details are omitted.…”

“…In the latter papers, the IVX method has been generalized to multi-regression (linear) models with nonstationary time series and the proposed method has been used to test the episodic predictability in stock returns. More recently, for a simple nonlinear in variables cointegrating regression model, the locally trimmed lease squares was introduced in Hu, Kasparis and Wang (2021) and Kasparis and Phillips (2020) has investigated the model with single covariate heavy tailed regressor.…”

“…These developments have provided a framework of econometric estimation and inference for a wide class of nonlinear, nonstationary relationships. See, for instance, Wang and Phillips (2009a, 2009b, 2016, Duffy (2016Duffy ( , 2020, Wang, Phillips and Kasparis (2021), Chang, Park and Phillips (2001), Bae and de Jong (2007), Kim and Kim (2012), Gao and Phillips (2013), Dong, et al (2016), Dong and Linton (2018), Lin, Tu and Yao (2020), Wang (2021), together with the references cited therein.…”

Least Squares Estimation for Nonlinear Regression Models With Heteroscedasticity

“…As in Theorem 3.1, we may prove Theorem 3.2 by checking the conditions of Theorem 2.1. Full details will be given in Wang (2020).…”

“…Since then, many authors have contributed to parametric, nonparametric, and semi-parametric estimation and inference theory in this field. We only refer to Chang, Park, and Phillips (2001), Chang and Park (2003), Bae and De Jong (2007), Phillips (2009a,b, 2016), Kim and Kim (2012), Gao and Phillips (2013), Chan and Wang (2015), Dong, Gao, and Tjötheim (2016), Dong and Linton (2018), and Wang et al (2020) together with the references cited therein. Although there have been significant developments in the last few decades, these existing studies do not consider the impact of conditional heteroscedasticity on the estimation and inference theory.…”