There are two main methods of forecasting, although both are primarily concerned with short-term forecasts because the assumptions mentioned previously will break down gradually for periods of longer than about a year.
(a) Moving Averages Method
This method involves extending the moving average trend line drawn on the graph of the time series. The trend line is extended by assuming that the gradient remains the same as that calculated from the data. The further forward you extend it, the more
unreliable becomes the forecast.
When you have read the required trend value from the graph, the appropriate seasonal fluctuation is added to this and allowance is made for the residual variation. For
example, consider the Premium Bond sales shown in Figure 9.7. On this figure the moving average trend line stops at the final quarter of 19x4. If this line is extrapolated with the same gradient to the first quarter of 19x5 then:
19x5 1st Qtr: Trend = 750
This is multiplied by the seasonal variation as it is a multiplicative model,
i.e. 750 × 149 = 1,118, and the residual variation which varied by as much as 18 per cent is added to this. Therefore the final short-term estimate for the sales of Premium Bonds for the first quarter of 19x5 is £1,118,000 £201,000.
Although fairly easy to calculate, this forecast, like all others, must be treated with caution, because it is based on the value of the trend calculated for the final quarter of 19x4, so if this happens to be an especially high or low value then it would influence the trend, and thus the forecast, considerably.
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(b) Least Squares Method
If the line of best fit (y = a + bx) is used as the trend line and drawn on a time series graph, it can be extended to give an estimate of the trend. Preferably the required value of x can be substituted in the equation to give the trend value. The seasonal fluctuation and residual variations must be added as in the moving averages method. Using the results of the earlier example involving days lost through sickness at a factory, the trend line was:
y = 21.92 + 1.46x
where x took all the integer values between 1 and 20.
Now suppose we want to estimate the number of days lost in the first quarter of 19x5, i.e. when x = 21. The value of the trend would be:
Y = 21.92 + 1.46 × 21 Y = 52.58
Y = 53 days, rounded to whole days.
(This result could also be read from the graph in Figure 9.6.)
To this must be added, as it is an additive model, the seasonal fluctuation for a first quarter, which was about 11 days, making a total of 64 days. The residual variation for this series was a maximum of 5 days. Therefore the forecast for days lost through sickness for the first quarter of 19x5 is between 59 and 69 days.
This forecast again is not entirely reliable, as the trend is depicted by one straight line of a fixed gradient. It is a useful method for short-term forecasting, although like the previous method it becomes more unreliable the further the forecast is extended into the future.
There are no hard and fast rules to adopt when it comes to choosing a forecast method. Do not think that the more complicated the method the better the forecast. It is often the case that the simpler, more easily understood methods produce better forecasts, especially when you consider the amount of effort expended in making them. Remember that, whatever the method used for the forecast, it is only an educated guess as to future values.
150 Time Series Analysis
D. THE Z CHART
We will conclude this study unit with a short description of a particular type of chart which plots a time series, called a Z chart. It is basically a means of showing three sets of data relating to the performance of an organisation over time. The three sets of data are plotted on the same chart and should be kept up-to-date. The graphs are:
(a) The plot of the current data, be it monthly, quarterly or daily. (b) The cumulative plot of the current data.
(c) The moving total plot of the data.
The Z chart is often used to keep senior management informed of business developments. As an example, we will plot a Z chart for the sales of Premium Bonds in 19x2 using the data of the table below with the sales broken down into months. The table also shows the
cumulative monthly sales and the moving annual totals. Note that the scale used for (a) is shown on the right of the chart and is twice that used for (b) and (c) so that the fluctuations in monthly sales show up more clearly. This is a device often used so that the chart is not too large.
Year Month Sales Cumulative Sales Moving Annual Total 1,240 19x2 Jan 150 150 1,290 Feb 350 500 1,460 Mar 300 800 1,640 Apr 100 900 1,670 May 150 1,050 1,730 June 150 1,200 1,830 July 120 1,320 1,890 Aug 120 1,440 1,940 Sept 100 1,540 1,990 Oct 300 1,840 2,140 Nov 400 2,240 2,340 Dec 200 2,440 2,440
These totals are presented in Figure 9.8. It is called a Z chart because the position of the three graphs on the chart resembles the letter Z.
This is a useful chart because management can see at a glance how production is progressing from one month to the next. It is also possible to compare the current year's performance with a set target or with the same periods in previous years.
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152 Time Series Analysis
SUMMARY
In this study unit we have discussed the main models used to analyse time series. We began by identifying the various factors into which a time series may be divided in order to use these models, and went on to show how to separate a time series into these constituent factors. This is an important subject and you should particularly note the following points:
Set out all calculations systematically in tables.
The layout of the table used for calculation of centred moving averages is very important for all models.
You must learn thoroughly the method of calculating and adjusting seasonal variations for all models.
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