Nonlinear HAR Models and Nonlinear Least Squares: Asymptotic Properties

We empirically examine asymptotic properties of the nonlinear least 2 squares estimator for a nonlinear extension of the class of Heterogeneous Auto- 3 Regressive (HAR) models for realized covariance matrices, the Hadamard Expo4 nential HAR (HE–HAR). First, we confirm both consistent and normally distributed 5 OLS estimates for a classical multivariate HAR specification in vectorial form, used 6 as a benchmark. Then, we replicate the Monte Carlo experiment under different 7 specifications and distributional hypotheses for HE–HAR extensions as a sensitivity 8 check to verify the corresponding results’ robustness. The results establish the con9 vergence of regular HAR coefficients in most cases, while the asymptotic normality 10 of the NLS estimates is uniquely confirmed for the HE–vech-HAR specification 11 with log-transformed realized covariance series. The only persistent asymptotic bias 12 is evident for a HE parameter estimate.