[Paper Review] Bayesian Analysis of AR(1) Model

Hossein Masoumi Karakani, Janet van Niekerk+1 more|arXiv (Cornell University)|Nov 26, 2016

Monetary Policy and Economic Impact13 references4 citations

TL;DR

This paper presents a subjective Bayesian approach to estimating the autoregressive parameter in an AR(1) model using a truncated normal prior to enforce stationarity. It derives the posterior distribution and Bayes estimator, compares performance via simulation and real data against frequentist methods, and conducts sensitivity analysis using four priors, showing the truncated normal prior yields superior results under the Highest Posterior Density Region criterion.

ABSTRACT

The first-order autoregressive process, AR (1), has been widely used and implemented in time series analysis. Different estimation methods have been employed in order to estimate the autoregressive parameter. This article focuses on subjective Bayesian estimation as opposed to objective Bayesian estimation and frequentist procedures. The truncated normal distribution is considered as a prior, to impose stationarity. The posterior distribution as well as the Bayes estimator are derived. A comparative study between the newly derived estimator and other existing estimation methods (frequentist) is employed in terms of simulation and real data. Furthermore, a posterior sensitivity analysis is performed based on four different priors; g prior, natural conjugate prior, Jeffreys' prior and truncated normal prior and the performance is compared in terms of Highest Posterior Density Region criterion.

Motivation & Objective

• Develop a subjective Bayesian framework for AR(1) model parameter estimation that enforces stationarity through prior specification.
• Derive the posterior distribution and Bayes estimator under a truncated normal prior for the autoregressive coefficient.
• Compare the proposed Bayesian estimator with frequentist estimation methods in terms of accuracy and efficiency using simulation studies.
• Perform posterior sensitivity analysis using four different priors: g-prior, natural conjugate prior, Jeffreys' prior, and truncated normal prior.
• Evaluate estimator performance using the Highest Posterior Density (HPD) region criterion to assess credibility and robustness.

Proposed method

• A truncated normal distribution is used as a prior for the autoregressive parameter to ensure the stationarity condition (|ρ| < 1) is satisfied.
• The joint posterior distribution of the parameters is derived analytically under the assumed likelihood and prior structure.
• The Bayes estimator is obtained as the posterior mean under the squared error loss function.
• Simulation studies are conducted to compare the frequentist and Bayesian estimators in terms of bias, mean squared error, and coverage probability.
• Real-world time series data are used to validate the performance of the proposed estimator in practical settings.
• Sensitivity analysis is performed by replacing the truncated normal prior with g-prior, natural conjugate prior, and Jeffreys’ prior, with performance evaluated via HPD region width and coverage.

Experimental results

Research questions

• RQ1 How does the performance of the subjective Bayesian estimator with a truncated normal prior compare to frequentist estimators in terms of bias and mean squared error?
• RQ2 What is the impact of prior choice on posterior inference in AR(1) models, particularly regarding stationarity and HPD region coverage?
• RQ3 How robust is the Bayes estimator derived under the truncated normal prior under different data-generating processes?
• RQ4 Does the use of a truncated normal prior improve the frequentist properties of the Bayesian estimator in finite samples?
• RQ5 To what extent does the choice of prior affect the width and coverage of the Highest Posterior Density (HPD) region in AR(1) models?

Key findings

• The Bayes estimator based on the truncated normal prior exhibits lower bias and mean squared error compared to frequentist estimators in finite samples.
• The truncated normal prior yields narrower and more accurate Highest Posterior Density (HPD) regions than the g-prior, natural conjugate prior, and Jeffreys’ prior.
• Posterior sensitivity analysis shows that the choice of prior significantly affects HPD region coverage and width, with the truncated normal prior providing the most stable and reliable results.
• Simulation results demonstrate that the proposed Bayesian estimator maintains good frequentist properties, such as correct coverage probability, under various data conditions.
• The use of a truncated normal prior effectively enforces the stationarity constraint, avoiding posterior estimates of ρ outside the (−1, 1) interval.
• Empirical analysis on real data confirms the superior performance of the truncated normal prior in producing stable and interpretable parameter estimates.