In **extrapolative forecasting** we predict the future by extrapolating a historical trend. What has happened in the past determines what is forecast for the future [with other forecasting methods, such as exploratory forecasting, this need not be so. For example, with exploratory forecasting we can explore revolutionary, as well as evolutionary, scenarios]. In some circumstances it is right to use** extrapolative forecasting**. In other cases different approaches might be more suitable. It is not an appropriate approach to use in a new product/ new business situation, or in situations where circumstances have radically changed, and the past is no guide to the future.

Any time series [a series of numbers recording past events] will have been produced by the interaction of a number of variables. For example, a time series of a company’s past profits will have been produced by a complex process, which involves an interaction between multiple revenue streams and numerous costs. **Extrapolative forecasting** would not require the forecaster to identify these individual variables, or the ways in which they interacted. The underlying process [variables and their interaction], which generated profits, would be treated as a black box [not analyzed in detail].

To produce a profit forecast scenario using exploratory [what if] methods it would be necessary to analyze and model the total sales and costs process; including cost and revenue behavior and the relationship between all the cost and revenue variables involved in the process. To produce a forecast profit scenario by **extrapolative forecasting** it is only necessary to obtain a series of past profit figures and extend them to produce a forecast figure.

This usually makes the **extrapolative forecasting approach** quicker and cheaper; it also makes it more risky. Since the underlying processes producing the forecast are not being modeled there is a danger of overlooking some fundamental change in the process, which makes it unwise to assume any continuity of events.

There are a number of different **extrapolative forecasting methods**, ranging form the simple to the very complex. One of the simplest methods is eyeball extrapolation. This involves producing a chart of past values and then forecasting by line of sight. Other simple methods include weighted average and exponential smoothing. Research has shown that the more complex methods are not necessarily superior to the simpler methods in terms of forecasting accuracy. Often, the simpler methods are just as reliable as the more complex. There is no single best method. Because different time series will contain different kinds of trends [linear, exponential, logarithmic etc], and different forms of noise [see below] forecasters need to carry out statistical tests [see below] to identify the method that will work best with their data and suit their particular requirements [e.g. how far in to the future do they need to forecast – some methods only work for one period ahead].

**When To Use **

The extrapolative forecasting approach can be employed when there is a set of historical data, which is believed to contain a trend, and it is believed that this trend can be isolated and used to forecast future events. For example, if a company has accurate records of its sales for the past six periods, and believes that there has been a trend in its past sales, and, also believes that this trend will continue into the future. In these circumstances it may be able to use extrapolative forecasting methods to help it to forecast its sales for future periods. Before extrapolative forecasting methods can be used a number of conditions have to be satisfied.

**Consistency In The Environment **

There needs to be a reasonable expectation that the conditions that have prevailed in the past will continue to hold true for the period being forecast. This is not an absolute requirement. If there has been some minor change in the environment a judgmental adjustment can be made to the values produced by forecast calculations. However, whenever there have been major changes in the environment, such as a competitor introducing a radical new product, extrapolative forecasting methods become less appropriate. If a manager is intending to make a decision with radical consequences extrapolative forecasting methods are of little value since the effect of the decision would be to change the environment.

**Data Availability**

Reliable quantitative historical data has to be available. Extrapolative forecasting methods would not be appropriate for start up, or new product launches where relevant historical data is not available. Nor would they be appropriate where the historical data is unreliable, perhaps because it has not been recorded in a consistent manner over the relevant time period. Most accounting and business systems contain copious amounts of data organized on a time basis, but one has to be certain that information is both reliable and in a usable form.

**Data Comparability**

Data relating to several past periods has to be reduced to a common basis. This can be a problem when forecasting something like future sales. Usually what one is trying to do is forecast future sales volumes and the temptation is to do this using sales values as a surrogate for volumes since the accounting system will have recorded values, not physical volumes. The danger in doing that is that price inflation will have affected sales values and £5m of sales in 2010 does not represent the same volume of physical sales as £5m in 2000. In a situation like that, if sales values are to be used for forecasting purposes the values have to be inflation adjusted. If possible physical volumes should be used instead of monetary units but there are unfortunately many instances where this is not feasible. Another problem in data comparability arises when a company sells a range of products or services and that mix has changed over time. Here again it may be necessary to forecast each component in the mix separately.

**Cost**

The benefit expected from the forecast should exceed the costs of collecting and processing the data. A supermarket might well want to forecast the short term demand for each of its many thousands of product lines but could only do so if it could easily collect data on past sales at point of sale via some method such a bar code reading by lasers at the checkout, and process the data using a fairly simple and computationally undemanding extrapolative forecasting technique.

**Simplicity**

As a general rule management will use information it understands. Someone who is responsible for a decision will usually not base that decision on a technique, which they cannot understand. There is a wide range of forecasting techniques. Ranging from simple ones, which are readily intelligible to most people, to others, which require a PhD in mathematics to understand. If it becomes obvious that the only appropriate forecasting technique is mathematically complex you have to ask yourself firstly whether you can competently employ it, and secondly whether the client will understand and accept the results of the calculations.

**Some Extrapolative Forecasting Methods**

**1. The Naive Forecast**

The simplest extrapolative forecasting method is the naive 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 naive 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

**F _{t} = F_{t-1} + [E_{t-1}*alpha]**

This means that a forecast **F _{t }**is calculated by taking the previous periods forecast value

**F**and adding to it some percentage [alpha] of the error

_{t-1}**E**[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.

_{t-1}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 recognizing 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.