Efficient Estimation for Models With Nonlinear Heteroscedasticity

Abstract

We study efficient estimation for models with nonlinear heteroscedasticity. In two-step quantile regression for heteroscedastic models, motivated by several undesirable issues caused by the preliminary estimator, we propose an efficient estimator by constrainedly weighting information across quantiles. When the weights are optimally chosen under certain constraints, the new estimator can simultaneously eliminate the effect of preliminary estimator as well as achieve good estimation efficiency. When compared to the Cramér-Rao lower bound, the relative efficiency loss of the new estimator has a conservative upper bound, regardless of the model design structure. The upper bound is close to zero for practical situations. In particular, the new estimator can asymptotically achieve the optimal Cramér-Rao lower bound if the noise has either a symmetric density or the asymmetric Laplace density. Monte Carlo studies show that the proposed method has substantial efficiency gain over existing ones. In an empirical application to GDP and inflation rate modeling, the proposed method has better prediction performance than existing methods.

Keywords:

• Cramér-Rao bound
• Efficient estimation
• Nonlinear heteroscedasticity
• Quantile regression

Acknowledgments

We are grateful to an associate editor and a referee for their constructive comments. Part of this work is based on Zhanxiong Xu’s PhD dissertation at Penn State University.