**Authors**: Ivan Svetunkov, John E. Boylan

**Journal**: IMA Journal of Management Mathematics

**Abstract**: Exponential smoothing in state space form (ETS) is a popular forecasting technique, widely used in research and practice. While the additive error ETS models have been well studied, the multiplicative error ones have received much less attention in forecasting literature. Still, these models can be useful in cases, when one deals with positive data, because they are supposed to work in such situations. Unfortunately, the classical assumption of normality for the error term might break this property and lead to non-positive forecasts on positive data. In order to address this issue we propose using Log-Normal, Gamma and Inverse Gaussian distributions, which are defined for positive values only. We demonstrate what happens with ETS(M,*,*) models in this case, discuss conditional moments of ETS with these distribution and show that they are more natural for the models than the Normal one. We conduct the simulation experiments in order to study the bias introduced by point forecasts in these models and then compare the models with different distributions. We finish the paper with an example of application, showing how pure multiplicative ETS with a positive distribution works.

**DOI**: 10.1093/imaman/dpad028.

# About the paper

**DISCLAIMER**: This is quite a technical paper focusing on solving a small problem of the ETS model that would allow using it in specific non-standard situations. It acts as a building block for the iETS paper. But the latter does not work without this paper, so while it seems small, it is an important brick in the wall.

The conventional ETS works great for regular demand, where the volume of the data is high. In that case, a forecaster can decide which of the 30 models to select for the data, not worrying too much about the assumption of normality for the error term and about forecast trajectories from the selected model. The situation changes when one needs to work with the positive low volume data. One would think that pure multiplicative ETS should work fine in that case, however, due to the normality assumption, the model might produce negative prediction intervals and in some situations even point forecasts. Trying to fix this issue, we considered several distributions for the error term in the multiplicative error ETS:

- \( 1 + \epsilon_t \sim \mathcal{N}\left(1, \sigma^2\right) \) – the conventional assumption of Normality;
- \( 1 + \epsilon_t \sim \mathcal{IG}\left(1, \sigma^2\right) \) – the error term follows the Inverse Gaussian distribution with the expectation of one and the variance of \(\sigma^2\);
- \( 1 + \epsilon_t \sim \mathrm{log}\mathcal{N} \left(-\frac{\sigma^2}{2}, \sigma^2 \right) \) – the error term follows the Log-Normal distribution with the location of \(-\frac{\sigma^2}{2}\) and the scale of \( \sigma^2 \);
- \( 1 + \epsilon_t \sim \Gamma\left(\sigma^{-2}, \sigma^2\right) \) – the error term follows the Gamma distribution with the shape parameter \(\sigma^{-2}\) and the scale \( \sigma^2 \).

The restrictions imposed on the parameters of distributions above are necessary to make sure that the expectation of the error term \(1 + \epsilon_t \) is zero. If it isn’t then the ETS model would need to be modified to cater for the non-zero mean, otherwise the model will produce incorrect forecasts.

In the paper, we show how ETS works with these assumptions, what forecasting trajectories it produces and how it can be estimated. We also demonstrate that the distribution selection can be easily automated using AIC. All these aspects of the model are already implemented and supported in the `adam()`

function from the `smooth`

package in R (read more here and here).

# Story of the paper

John Boylan and I started working on this paper after getting a rejection from the IJF for the other paper of ours, “iETS: State space model for intermittent demand forecasting“. The rejection showed us that we need to take a completely new look at the paper, and it became apparent that the pure multiplicative ETS is not well studied in the literature. At the same time, its discussion would be outside of the scope of the original paper, so, we decided to write a separate one, focusing on the non-intermittent, but low volume demand.

We needed to discuss two points in the paper, which were then used in the iETS one:

- The conventional ETS assumes that the demand follows Normal distribution. In case of low volume demand this assumption may lead to negative forecasts, which makes the model inappropriate;
- Point forecasts from multiplicative ETS models does not coincide with the conditional expectations. Hyndman et al. (2008) discuss this in their book, but not to the extent we needed. We thought that lots of people do not understand the implications of this, so we added that discussion to the paper.

The paper was written over the period of 2021 – 2023 and was ready in Spring 2023. John and I discussed it several times, and we agreed to have a final look at it in May 2023 before submitting it to IMA Journal of Management Mathematics. When I found out that John was ill, I decided not to wait for his comments further and just submitted it. The paper went through a couple of rounds, changed its name to reflect concerns of one of reviewers (the new name is objectively better than the old one) and was accepted for publication in November 2023. This is the last paper that John and I wrote together.