Some Extrapolative Forecasting Methods
1. The Naïve Forecast
The simplest extrapolative forecasting method is the naïve forecast, which involves taking the actual value for the current period as the forecast for the next period. This method assumes that there is no pattern present in historical data. This is sometimes the case and the naïve forecast method is then the most accurate to use. Forecasts produced by this method can also be used as benchmarks to measure the accuracy of other forecasting methods when applying the statistical tests mentioned above.
2. Eyeball Forecasting
A time series is charted and a forecast is made by line of sight. Testing has shown this to be a surprisingly accurate method.
3. Moving Average
The simplest smoothing method is the moving average. The average can be taken over any number of periods. The fewer the number of periods the faster the forecast values will be to respond to changes in actual values. The greater the number of periods the slower the forecast will be to respond to changes and the less it will be influenced by random variations. Which period is appropriate will depend on the variability of the data.
It may be that in a data series there is a great deal of random variation and this needs to be smoothed out to prevent it corrupting the forecast. In this case a long period average would be used. In other circumstances it may be that it is felt that there is a rapidly changing pattern in the data and the forecast needs to respond quickly to changes. In that case a three period average may well be more appropriate. It should be noted that the moving average method is usually only used to forecast for one period in advance. The weakness with the method is that it attaches equal weight to all the periods taken into the averaging calculation.
4. Weighted Moving Average
One of the weaknesses of the simple moving average can be overcome by attaching weights to different past periods when performing the averaging calculation. This allows greater weight to be given to the most recent periods and less to earlier periods by varying the weights attached. Some experimentation is usually needed to determine what set of weights works best with a particular set of data.
5. Exponential Smoothing
Exponential smoothing employs the formula
Ft = Ft-1 + [Et-1*alpha]
This means that a forecast Ft is calculated by taking the previous periods forecast value Ft-1and adding to it some percentage [alpha] of the error Et-1 [difference between the previous periods actual value and the previous periods forecast value] in the last periods forecast. The value of alpha determines how much of the current error is taken into account when calculating the next periods forecast.
If the value of alpha is, say, 0.9 then nearly all the error will be taken into account and the forecast will be very responsive to changes in actual values. If the value of alpha is 0.1 then the forecast will respond much more slowly to changes in actual values.
The selection of an appropriate value of alpha will be a matter of trial and error [and the results of statistical tests of accuracy], and will depend very much upon the variability of the series being forecast. Selection of the alpha value can be done by calculating forecasting using several different values of alpha and then calculating several measures of error for each forecast. This method would be used for forecasting one period into the future. It does not work well with highly seasonal data.
Both the moving average and exponential smoothing methods work best with data, which exhibits a horizontal, though perhaps highly variable, data pattern. They work much less well with series that include a seasonal or trend pattern. If simple exponential smoothing is used on a series that includes a trend the forecast will always lag behind in recognising the trend. If a series is believed to contain randomness, a trend and seasonality then Winters exponential smoothing may be used.
Noise in the Data
Any trend that is present in a data set may be obscured by noise of various kinds. For example, the data set may contain the residual influences of some expired event. Or it may contain the affects of unusual or exceptional events. Almost certainly there will be at least some, and perhaps a great deal of, random variation affecting the data. The problems can be illustrated by taking the example of a company, which has sales records for eight previous periods and wishes to use this data to forecast the next period’s sales. At the end of period four the company dropped a product line, which was contributing 20% of sales volume. In period five part of its factory suffered a fire, which affected its sales for two periods. It has also suffered three brief component supply interruptions during the eight periods, all of which affected production and sales. Finally its sales are subject to quite considerable variations about a mean due to a variety of random variations in demand from its customers. It must decide how it should deal with each of these factors.
The dropped product line can be dealt with by removing the sales for that line from the first four periods values. The effect of a new product line could have been dealt with by the same kind of adjustment; by taking the sales values out of the mainstream data set and processing them separately. The effects of the fire could be dealt with by adjusting the values for periods five and six to what they would have otherwise have been. The component supply problems might require a different kind of treatment. If they seem to occur fairly frequently their effect should be left in the data set. No preliminary adjustment is required for the normal random variations.
The second problem that has to be dealt with is that there are many different kinds of patterns and each requires a different approach. The simplest case is the horizontal pattern where the data is subject to some random variations form period to period but overall is not increasing of decreasing over time.
A slightly more complicated case arises when the data is subject to some random variations from period to period and there is also and under lying increase or decrease over time. This underlying change may follow a linear or a non-linear pattern. If the pattern in the data set is a power or exponential trend it requires a different treatment to a linear trend.
A seasonal pattern occurs where a series varies according to the seasons of the year. A cyclical pattern is similar to a seasonal pattern but the cycle is determined by some factor other than the seasons. The cycle may be short or may be of several years’ duration. In a rapidly changing environment it is sometimes difficult to detect a cyclical pattern and even more difficult to forecast with confidence that the cycle will continue.
Statistical Testing
Statistical testing can be conducted to help identify which extrapolative forecasting method will work best with the available data. Testing is necessary in order that inappropriate methods can be avoided and effort concentrated on those methods, which produce the most accurate forecasts. We can carry out testing to select the best method by producing ex post forecasts for past periods and comparing the forecast values with the actual values for those periods. When the best method has been found it is used to make ex ante forecasts.
Testing can also be used to refine the accuracy of the technique being employed since many methods have internal parameters, which can be varied [e.g. weights in weighted moving average, or the value of alpha in exponential smoothing]. Experiment and testing is used to find the most suitable parameter values. The overall objective is to minimize the error [E] between forecast values [F] and actual values [A]. There are a number of different tests that are used for measuring forecasting accuracy.
- Mean Error [ME] is simply the average of the error [difference between forecast and actual]. This measure is simple to compute but positive and negative errors tend to cancel one another out and conceal the true magnitude of E.
- Mean Absolute Error [MAE] is the average of the absolute [ignoring sign] error. This test overcomes the weakness of ME and provides a better measure of error.
- Mean Squared Error [MSE] is the average of the square of the absolute errors. This measure emphasises substantial deviations from actual since an error of 2 will be squared to 4 but an error of 4 will be squared to 16. This measure would be particularly useful when minor errors could be tolerated, but it was particularly important to avoid forecasting methods, which produced large errors. For example, a company, which used sales forecasting to set production levels and only maintained small finished goods stocks.
- Mean percentage error [MPE] is calculated by taking the average of absolute error as a percentage of actual. This statistic allows easy comparisons to be made between different forecasting methods, or different applications of the same method.
Common Sense and Visualization
Finally, forecasts cannot be accepted unthinkingly but have to be subject to common sense verification. Visualization is also an important part of time series forecasting. Before using any mathematical forecasting method it is a good idea to chart the data, and then study the chart to try and understand the data patterns. Sometimes it will be very clear what kinds of trends are present in the data, and what forecasting methods should be used.