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Periodic Time Series Forecasting with Bayesian Model Averaging

R code used in: Vosseler, A., Weber, E. (2018): Forecasting seasonal time series data: a Bayesian model averaging approach

Overview

This repository provides R implementations for Bayesian estimation and forecasting of Periodic Autoregressive (PAR) models with potential structural breaks. The methodology uses Gibbs sampling for posterior inference and Bayesian Model Averaging (BMA) to account for model uncertainty, including uncertainty about the presence and timing of structural breaks.

Key Features

  • Bayesian estimation of PAR(p) models for quarterly (S=4) and monthly (S=12) time series
  • Structural break detection with unknown break date
  • Gibbs sampling for posterior inference
  • Bayesian Model Averaging across models with/without breaks
  • Marginal posterior computation for break date estimation
  • Forecast generation with predictive density estimation
  • Statistical tests for forecast evaluation

Repository Structure

Main Estimation Scripts

File Description
par4_estimation_gibbs_B_with_BMA_marg.R Complete estimation and forecasting pipeline for quarterly PAR(p) models with BMA
par12_estimation_gibbs_with_BMA_marg.R Complete estimation and forecasting pipeline for monthly PAR(p) models with BMA

Core Functions

File Description
bayes.par.R Bayesian PAR model setup for quarterly data (S=4). Creates design matrices, computes OLS estimates, and information criteria (AIC, BIC, HQ, FPE)
bayes.par12.R Bayesian PAR model setup for monthly data (S=12). Analogous to bayes.par.R
gibbs4.R Gibbs sampler for quarterly PAR(p) models. Draws from posterior distributions of coefficients, variance, and break dates
gibbs12.R Gibbs sampler for monthly PAR(p) models. Analogous to gibbs4.R

Marginal Posterior for Break Date

File Description
margpostTb.R Computes marginal posterior density of break date $T_b$ for quarterly models. Includes HPD interval computation
margpostTb12.R Computes marginal posterior density of break date $T_b$ for monthly models

Design Matrix Construction

File Description
design.R Constructs seasonal dummy design matrices with mean or effect coding
design.break.R Constructs design matrices accommodating structural breaks at time $T_b$
strbr.comp.R Comprehensive structural break component builder for various deterministic specifications
mats01.R Constructs $\Omega_0$ and $\Omega_1$ matrices for the companion form representation of PAR models

Simulation Functions

File Description
par_simulation4.R Simulates quarterly PAR(1), PAR(2), PAR(3), PAR(4), and broken PAR processes. Includes unit root restrictions
par_simulation12.R Simulates monthly PAR(1), PAR(2), and broken PAR processes

Forecasting Functions

File Description
forec.sarima.R Forecasting using SARIMA models with seasonal dummies (benchmark comparison)
forec.sdummy.R Forecasting using seasonal dummy regression models

Forecast Evaluation

File Description
predictive_tests.R Statistical tests for comparing predictive accuracy: Sign test, Wilcoxon test, and Bayesian posterior probability of equal predictive ability
signtest.R Implementation of the sign test for paired comparisons with confidence intervals

Dependencies

The code requires the following R packages:

install.packages(c("MASS", "Bolstad", "TSA", "pear"))

Usage Example

# Source simulation functions
source("utils/par_simulation4.R")

# Simulate a quarterly PAR(1) process
y <- par1(N = 200, mu = c(1, 1, 1, 1), beta = 0, 
          phi = c(0.87, 1.2, 0.95, 0.85), start = c(1976, 2))$y

# Source Gibbs sampler
gibbs4 <- dget("utils/gibbs4.R")

# Run Gibbs sampling with structural break detection
results <- gibbs4(y = y, p = 1, n.ahead = 8, M = 5000, burnin = 1000,
                  sbreak = TRUE, in.samp.fc = TRUE, 
                  sdeter = "const", tau = 3)

Model Specification

The PAR(p) model is specified as:

$$y_t = \sum_{s=1}^{S} D_{s,t} \left( \mu_s + \sum_{j=1}^{p} \phi_{s,j} y_{t-j} \right) + \varepsilon_t$$

where:

  • $S$ = number of seasons (4 for quarterly, 12 for monthly)
  • $D_{s,t}$ = seasonal dummy for season $s$
  • $\mu_s$ = season-specific intercept
  • $\phi_{s,j}$ = season-specific autoregressive coefficients
  • $\varepsilon_t \sim N(0, \sigma^2)$

Citation

If you use this code, please cite:

@article{vosseler2018forecasting,
  title={Forecasting seasonal time series data: A Bayesian model averaging approach},
  author={Vosseler, Alexander and Weber, Enzo},
  journal={Computational Statistics},
  volume={33},
  pages={1733--1765},
  year={2018},
  publisher={Springer}
}

License

This project is provided for academic and research purposes.