In part one, 'Thinking ahead', moving averages were used to identify the underlying trend and seasonal variations in a time series.

Sales data was given for four years, and forecasts for a fifth year were required.

The starting point for these forecasts is to extend the trend line (extrapolate into year five). There are a number of ways of doing this.

Since the trend has been plotted on a graph, it can be extended by eye (by placing a ruler on the graph, fitting the trend as closely as possible). This method is reliant on the accuracy of the graph and is subjective, especially if the trend is not linear.

Alternatively, the extrapolation can be done using linear regression. Here, a line of best fit is expressed as a linear equation (y=a+bx). Determination of such an equation is not required at Intermediate or Technician level, but use of a given equation may be required at Technician level.

The method we will use here is more of a common-sense approach, where it is assumed that the average movement in the trend in the past will be repeated in the future.

Consider the final table produced in article one Thinking ahead. Between year one, quarter three and year four, quarter two (a jump of 11 quarters), the centred trend has increased from 58.75 to 75 - a quarterly increase of 1.477 (75-58.75 = 16.25, divided by 11). If it is assumed that this trend will continue, the relevant (forecast) figures are as follows:

Note that it is necessary to ‘predict’ figures for the second half of year four, even though actual figures are available. This is because the centred trend stopped at quarter two of this year. Averaging helps to eliminate variation, but some detail from the original data is lost.

However, extrapolated trend points alone do not provide useful forecasts. The original data shows that some quarters tend to exceed the trend, and others fall below it. Forecasts must be adjusted for seasonal variations.

Again, it is assumed that past patterns will be repeated. Actual seasonal variations were calculated last month. These are analysed to produce an average variation for each quarter. The adjustment is made because the average seasonal variations should sum to zero. The difference is spread evenly over the quarters to force this equality, as below:

Seasonal Variations

The adjusted average seasonal variations are now combined with the extrapolated trend to produce forecast sales figures:

Forecast Sales: Year Five

The extended trend and forecast sales are then often superimposed on the graph, to see how they fit with the pattern of the original data - see illustration on this page).

Time series analysis using moving averages is a useful technique, but has its limitations. The forecasts above ignore random fluctuations. If these exist there is little one can do about it, but they will limit accuracy.

Consider year two, quarter three. The centred trend is 66.5 and the adjusted average variation is (7.042), so the model predicts sales for quarter three of 59.458. However, actual sales were 60, giving a random variation of 0.542. This means little but, if done for all available data, will suggest how reliable the forecast is likely to be (the higher the random variations, the less reliable the forecast).

Extrapolation of the trend line has limitations. It is assumed that a linear extrapolation is appropriate, but even if the trend has tended to be linear in the past, changes in market conditions, government policy, advertising strategy or other external factors may change the pattern. In addition, many sectors of the economy experience cyclical variations (like seasonal variations but over several years),which limit the effectiveness of forecasts.

Finally, consider inflation. Changing prices can distort a time series. One way to account for inflation might be to use index numbers before using moving averages to forecast sales in real terms, then re-inflate forecasts to anticipated price levels.

Andrew Harrington teaches at Chelmsford College

AT, January 2001, page 32