Conditional variance estimation in regression models with long memory

Abstract

This paper studies asymptotic properties of a nonparametric kernel estimator of the conditional variance in a random design model with parametric mean and heteroscedastic errors, for a class of long memory errors and predictors. We establish small and large bandwidths asymptotics, which show a different behaviour compared to that of kernel estimators of the conditional mean. We distinguish between an oracle case (i.e., where the errors are directly observed) and a non-oracle case (where the errors are replaced with residuals), and show non-equivalence between the oracle and non-oracle case. We also discuss a practical problem of bandwidth choice. Theoretical results are justified by simulation studies. We apply our theory to DJA and FTSE indices.

On Local Trigonometric Regression Under Dependence

“…Related results in the context of parametric regression can be found in Yajima (1991). In a broader context, similar phenomena have also been observed in random design regression (Künsch et al, 1993;Koul et al, 2004;Guo and Koul, 2008;Kulik and Wichelhaus, 2012) and in the behavior of estimated wavelet coefficients (Roueff and Sachs, 2011).…”

Detecting long-range dependence for time-varying linear models

“…i,n . Furthermore, our proof extends that of Proposition 1 and 2 in Kulik and Wichelhaus (2012) in a non-trivial way, since they focus on the partial sum assuming i.i.d. covariates {x i,n } and i.i.d errors {e i,n }.…”

“…covariates {x i,n } and i.i.d errors {e i,n }. Kulik and Wichelhaus (2012) utilized their Propositions 1 and 2 to estimate the conditional variance in the heteroscedastic model (1.1).…”

“…Condition (E4) assumes strict exogeneity of covariates, which is general and has been frequently imposed in the literature. Our setting includes the heteroscedastic errors considered in He and Zhu (2003) and Kulik and Wichelhaus (2012) where they considered Hi (x 1,n , ..., x i,n ) of the form s(x i,n ), where s(•) is an unknown smooth function. Moreover, we extend the assumption 1(b) of Cavaliere et al (2017) to allow heteroscedasticity of errors induced by covariates.…”

“…The time-varying regression model (1.1) with time series errors has attracted enormous attention, see for instance Hoover et al (1998), Fan and Zhang (2000), Huang et al (2004), Zhou and Wu (2010), Cai (2007) and Chen et al (2018) where the errors are assumed to be SRD, and Kulik and Wichelhaus (2012), Ferreira et al (2013), Ferreira et al (2018) where the LRD errors are considered. The aforementioned research reveals that nonparametric estimators of the time-varying coefficient β(•) possess distinct properties under the two scenarios, d = 0 and 0 < d < 1/2.…”