Robust Estimation Procedure for Autoregressive Models with Heterogeneity
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Robust Estimation Procedure for Autoregressive Models with Heterogeneity A. Callens1
· Y.-G. Wang2 · L. Fu3 · B. Liquet1
Received: 7 January 2020 / Accepted: 24 August 2020 © Springer Nature Switzerland AG 2020
Abstract In environmental studies, regression analysis is frequently performed. The classical approach is the ordinary least squares method which consists in minimizing the sum of the squares of the residuals. However, this method relies on strong assumptions that are not always satisfied. In environmental data, the response variable often contains outliers and errors can be heteroscedastic. This can have significant effects on parameter estimation. To solve this problem, the weighted Mestimation was developed. It assumes a parametric function for the variance, and, estimates alternately and robustly, mean and variance parameters. However, this method is limited to the independent errors case, and is not applicable to time series data. Therefore, we introduce a new estimation procedure which adapts the weighted M-estimation to environmental time series data, while selecting optimal value for the tuning parameter present in the M-estimation. We compare the efficiency of our procedure on simulated data to other usual regression methods. Our estimation procedure outperforms the other methods providing estimates with lower biases and mean squared errors. Finally, we illustrate the proposed method using an air quality dataset from Beijing. This method has been implemented in the R package RlmDataDriven. Keywords Heteroscedasticy · Model selection · Robust estimation · Temporal correlations
1 Introduction In environmental modelling, pure homoscedasticity is uncommon. For example, residuals of hydrological [8], or air pollution models [27], are usually heteroscedastic. Ignoring this problem and performing an ordinary least squares method would result in regression parameters with biased covariance matrix and hence would lead to erroneous inference. Electronic supplementary material The online version of this article (https://doi.org/10.1007/s10666-020-09730-w) contains supplementary material, which is available to authorized users. A. Callens
[email protected] 1
Laboratoire de Math´ematiques et de leurs Applications de Pau, Universit´e de Pau et des Pays de l’Adour, UMR CNRS 5142, E2S-UPPA, Pau, France
2
ARC Centre of Excellence for Mathematical and Statistical Frontiers and School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
3
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, China
One method to deal with this biased covariance matrix is to use the White estimator [26] which provides a heteroscedasticity consistent covariance matrix. In case of a time series, the Newey-West estimator [18] can provide heteroscedasticity and an autocorrelation-consistent covariance matrix. Both of these methods account for heteroscedasticity but do not give information on the variability of the data generation process. Another approach to cope with h
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