Nonlinear HAR Models and Nonlinear Least Squares: Asymptotic Properties

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