Parametric quantile autoregressive moving average models with exogenous terms applied to Walmart sales data

TL;DR

This paper introduces a new class of quantile ARMAX models based on log-symmetric distributions, capable of modeling complex data features like bimodality, heavy tails, and heteroscedasticity, with applications to Walmart sales data.

Contribution

The paper develops a novel quantile ARMAX modeling framework using log-symmetric distributions and demonstrates its effectiveness through simulations and real-world sales data analysis.

Findings

Models effectively capture bimodal and heavy-tailed distributions.

Proposed estimation method shows accurate parameter retrieval.

Models outperform traditional approaches in sales forecasting.

Abstract

Parametric autoregressive moving average models with exogenous terms (ARMAX) have been widely used in the literature. Usually, these models consider a conditional mean or median dynamics, which limits the analysis. In this paper, we introduce a class of quantile ARMAX models based on log-symmetric distributions. This class is indexed by quantile and dispersion parameters. It not only accommodates the possibility to model bimodal and/or light/heavy-tailed distributed data but also accommodates heteroscedasticity. We estimate the model parameters by using the conditional maximum likelihood method. Furthermore, we carry out an extensive Monte Carlo simulation study to evaluate the performance of the proposed models and the estimation method in retrieving the true parameter values. Finally, the proposed class of models and the estimation method are applied to a dataset on the competition "M5 Forecasting - Accuracy" that corresponds to the daily sales history of several Walmart products. The results indicate that the proposed log-symmetric quantile ARMAX models have good performance in terms of model fitting and forecasting.