High-dimensional Holt-Winters trend model: Fast estimation and prediction

Abstract

Prediction and trend estimation are vital tasks for managers in business and industry. When the number of series increases and multiple trends arise, the estimation of trend co-movement gains importance. Indeed, it makes it possible to extract joint dynamics, measure correlations between shocks, and improve prediction performance. Unfortunately, when the model dimension increases, estimating multiple trends becomes extremely challenging, since the number of parameters tends to explode. This paper provides a closed-form result that helps simplify the prediction and estimation of the multivariate smooth-trend model, one of the state-space representations of Holt-Winters celebrated recursions. The result has several practical consequences. Indeed, we suggest a simple method that quickly maximizes the likelihood even for high-dimensional models. A Monte Carlo simulation shows the computational advantages provided by these results compared with the standard maximum likelihood approach. In addition, an empirical analysis focusing on forecasting retail sales data in the USA shows the potential of the proposed technique.

Keywords:

• Forecasting
• time series
• econometrics

Notes

1 All the results as well as the numerical proofs have been derived using Mathematica 9 by Wolfram (see Mathematica (2012)). The codes are available upon request.