WAVELET-BASED BAYESIAN ESTIMATION OF PARTIALLY LINEAR REGRESSION MODELSWITH LONG MEMORY ERRORS.

Area of Science:

• Statistics
• Econometrics
• Time Series Analysis

Background:

• Partially linear regression models are widely used in statistical modeling.
• Long memory processes introduce complex dependencies in error terms, posing estimation challenges.
• Existing methods may struggle with the dense variance-covariance matrices typical of long memory errors.

Purpose of the Study:

• To develop a robust Bayesian procedure for estimating partially linear regression models with long memory errors.
• To address the computational complexity arising from the variance-covariance matrix of long memory errors.
• To provide a unified framework for simultaneous estimation of parametric and nonparametric components.

Main Methods:

• A wavelet-based Bayesian approach utilizing discrete wavelet transforms.
• Simplification of the long memory error variance-covariance matrix through wavelet decomposition.
• Fully Bayesian inference implemented via a Metropolis-Hastings algorithm within a Gibbs sampler.
• Evaluation of the method's performance using simulated datasets.

Main Results:

• The proposed wavelet-based Bayesian method successfully estimates both model parameters and the nonparametric function.
• Discrete wavelet transforms effectively simplify the complex variance-covariance structure of long memory errors.
• The Metropolis-within-Gibbs sampler achieves efficient fully Bayesian inference.
• Demonstrated performance on simulated data and a real-world application.

Conclusions:

• The wavelet-based Bayesian procedure offers an effective solution for partially linear regression with long memory errors.
• This method provides a computationally efficient and statistically sound approach to complex time series analysis.
• The technique is applicable to benchmark datasets in the long memory literature, such as Northern Hemisphere temperature data.