# Forecasting a series with changes in growth rate

I have a particular time series (demand in units for a particular product category), which I forecast each year (specifically, at/for Christmas, although the data runs all year round.

Historically, I have used Holt-Winters to forecast for this category. However, in Autumn and Winter of 2015, the slope began to progressively decline, while still remaining positive. I.e., the average ratio of demand yt to yt-365 was lower than it previously had been. This meant that forecasts which were generated using Holt-Winters for Christmas, in late Autumn, were biased upwards because the parameters were optimised using SSE minimisation over a period for most of which the "growth rate" had been higher than it actually was over the target period.

Subsequently, beginning in late Winter/Spring of this year, the slope has progressively started to turn back up again, back to historical norms. I thus have a trend or beta which has dipped downwards, then back upwards, again over time (while continually remaining positive).

I don't know how to accommodate for this when attempting to forecast for Christmas 2016. Is there a particular model which is best for generating forecasts in this situation? Or is this simply a "known unknown" which will make it difficult to generate accurate forecasts, and there's nothing I can do about it?

In practical terms, if I continue using Holt-Winters (and of course this may not be the best model to use), which period should I use for minimising the SSE? A twelve-month period backwards from today? A different period? Presumably, the value of the coefficient estimated for the slope parameter will be significantly affected by this choice, and by the apparent changes in the slope over time.

• Your time series model does not know any more than what you feed it in your data series. Do you expect this Fall/Winter dip to repeat this year? Is that caused by a one-time event? You need to bring domain knowledge to bear on this. One possibility would be to incorporate exogenous predictors (econ growth, etc). – horaceT Sep 19 '16 at 17:32
• Hi HoraceT, thanks for the comment. The answer to your question, do I expect it to be repeated, is: "I don't know", although I can hypothesise. Supposing I knew, though, do you have any advice on how, in practical terms, to incorporate exogenous parameters? I don't think the Holt-Winters model allows for this, but are there any alternative models which you would recommend? Any choice would have to preserve a seasonal component, as the demand has two very clear intra-year seasonal cycles (which collectively can be effectively captured with a single seasonality component/parameter). – Statsanalyst Sep 19 '16 at 17:42
• @Statanalyst Not sure what package/lib/functions you use for Holt-Winters. In R, the arima function has xreg to allow external covariates. – horaceT Sep 19 '16 at 17:53
• @horaceT, external variables are difficult to incorporate in exponential smoothing models in general; see Rob J. Hyndman's blog post "Exponential smoothing and regressors". There he recommends using regression with ARMA errors instead. – Richard Hardy Sep 19 '16 at 19:24