At Christmas I will enjoy a glass or two of white wine and I suspect I will not be unusual in that regard. The figure below plots data from Her Majesty's Revenue and Customs (HMRC) on the volume of wine on which excise duty has been paid and shows that wine has become a much more popular drink in the UK over the last 20 years or so.

In contrast, the volume of beer production in the UK has fallen by around 20 percent over the last ten years. The figures do not give the level of consumption but rather the volume of beer and wine on which excise duty has been paid, but I’ll use them in the following as a rough guide to changing consumer tastes. In addition to long-run trends, the figures from HMRC show that there is significant monthly variation in the amount of excise duty collected. Unsurprisingly, the amount of duty collected on wine rises markedly each year around Christmas but, in many years (for example 2002 and 2003), there is also a spike in the level of duty collected in April which seems to be due to suppliers trying to shift stock prior to the Budget. The volume of beer production shows less seasonal variation in comparison to the excise duty collected for wine, but in each year there is always a marked drop in beer production in February and March.

**Data from https://www.uktradeinfo.com/statistics/pages/taxanddutybulletins.aspx **

Drinks companies clearly have an interest in trying to forecast the level of future demand. In statistics, forecasting is part of time series analysis. A time series is simply a set of observations taken over a period of time. In time series analysis, one aim is to understand how the observations vary over time as a function of their own past values rather than as a function of other explanatory variables. An understanding of how past values influence the current value opens the way to forecasting future values using current values.

There are two main approaches to modelling time series. In the state space approach to modelling time series the key idea is to view the observations (Y_{t}) as being a function of an unobserved or latent process termed the state process (α_{t}). The state represents the different components of the time series such as trends and seasonals that we want to model. The figure below illustrates the basic structure of a state space model. The observation at any time, Y_{t}, depends only on the state at that time, α_{t}, while the dynamics of the state process are such that αt depends only on α_{t-1}. The beauty of the model for forecasting is that we can predict the future observation Y_{t+1} using only the state at time t, α_{t}, rather than the entire previous history of the process.

The alternative approach to modelling time series is the autoregressive integrated moving average model (ARIMA) of Box and Jenkins. The ARIMA model first eliminates the trend and seasonal components of the time series by differencing, for example, if a series has a time trend, Y_{t} = α + βt, then taking the first difference of the series removes the trend: Y_{t} – Y_{t-1} = α + βt – α – β(t-1) = β. The resulting series is then modelled as a function of its own past values.

It might be helpful to illustrate the state space approach with an example and for this purpose I’ll use the above monthly series on the volume of UK beer production. The state space model I have used includes only two states, the first represents the underlying trend in the time series and the second models monthly seasonal variation. Our main interest is in the underlying trend and there are two methods available for estimating this. We can either estimate the current value of the trend as new data arrive each month (termed filtering) or we can retrospectively estimate the entire tend given all the observed data (termed smoothing). The figure below shows estimates of the filtered and the smoothed trend for the beer series. As expected, the smoothed estimate of the trend (based on all the data) shows much less variation than the filtered estimate (based only on the past data at time t). In comparison to the smoothed trend, the filtered trend also tends to lag the changes in the series since it does not take account of current and future observations. For example, the filtered trend takes some time to record the fall in beer volumes from 2005 onwards while the smoothed trend adjusts much more rapidly.

The motivation for time-series analysis is often to forecast future observations and the figure below shows forecasts of the level of beer production for the next two years. The most likely forecast of future beer production is given by the solid black line while the grey lines allow us to gauge the uncertainty associated with the predictions. The figure suggests that beer volumes are anticipated to remain at roughly their current level for the next two years although the level of uncertainty associated with the predictions does not rule out a continued fall in the level of production.

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## Comments

## P

Nice little article, but I really felt it ended rather abruptly.

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