Forecast Friday Topic: Procedures for Combining Forecasts

(Forty-first in a series)

We have gone through a series of different forecasting approaches over the last several months. Many times, companies will have multiple forecasts generated for the same item, usually generated by different people across the enterprise, often using different methodologies, assumptions, and data collection processes, and typically for different business problems. Rarely is one forecasting method or forecast superior to another, especially over time. Hence, many companies will opt to combine the forecasts they generate into a composite forecast.

Considerable empirical evidence suggests that combining forecasts works very well in practice. If all the forecasts generated by the alternative approaches are unbiased, then that lack of bias carries over into the composite forecast, a desirable outcome to have.

Two common procedures for combining forecasts include simple averaging and assigning weights inversely proportional to the sum of squares error. We will discuss both procedures in this post.

Simple Average

The quickest, easiest way to combine forecasts is to simply take the forecasts generated by each method and average them. With a simple average, each forecasting method is given equal weight. So, if you are presented with the following five forecasts:

You’ll get the average of $83,000 as your composite forecast.

The simplicity and quickness of this procedure is its main advantage. However, the chief drawback is if information is known that individual methods consistently predict superiorly or inferiorly, that information is disregarded in the combination. Moreover, look at the wide variation in the forecasts above. The forecasts range from $50,000 to $120,000. Clearly, one or more of these methods’ forecasts will be way off. While the combination of forecasts can dampen the impact of forecast error, the outliers can easily skew the composite forecast. If you suspect one or more forecasts may be inferior to the others, you may just choose to exclude them and apply simple averaging to the forecasts for which you have some reasonable degree of confidence.

Assigning Weights in (Inverse) Proportion to Sum of Squared Errors

If you know the past performance of individual forecasting methods available to you, and you need to combine multiple forecasts, it’s likely you will want to assign greater weights to those forecast methods that have performed best. You will also want to allow the weighting scheme to adapt over time, since the relative performance of forecasting methods can change. One way to do that would be to assign weights to each forecast in based on their inverse proportion to the sum of squared forecast errors.

Let’s assume you have 12 months of sales data, actual (Xt), and three forecasting methods, each generating a forecast for each month (f1t, f2t, and f2t). Each of those three methods have also generated forecasts for month 13, which you are trying to predict. The table below shows these 12 months of actuals and forecasts, along with each method’s forecasts for month 13:

How much weight do you give each forecast? Calculate the sum squared error for each:

To get the weight of the one forecast method, you need to divide the sum of the other two methods’ squared errors by the total sum of the squared errors for all three methods, and then divide by 2 (3 methods minus 1). You must then do the same for the other two methods, in order to get the weights to sum to 1. Hence, the weights are as follows:

 

Notice that the higher weights are given to the forecast methods with the lowest sum of squared error. So, since each method generated a forecast for month 13, our composite forecast would be:

Hence, we would estimate approximately 795 as our composite forecast for month 13. When we obtain month 13’s actual sales, we would repeat this process for sum of squared errors from months 1-13 for each individual forecast, reassign the weights, and then apply them to each method’s forecasts for month 14. Also, notice the fraction ½ at the beginning of each weight equation. The denominator depends on the number of weights we are generating. In this case, we are generating three weights, so our denominator is (3-1)=2. If we would have used four methods, each weight equation above would have been one-third; and if we had only two methods, there would be no fraction, because it would be one.

Regression-Based Weights – Another Procedure

Another way to assign weights would be with regression, but that’s beyond the scope of this post. While the weighting approach above is simple, it’s also ad hoc. Regression-based weights can be much more theoretically correct. However, in most cases, you will not have many months of forecasts for estimating regression parameters. Also, you run the risk of autocorrelated errors, most certainly for forecasts beyond one step ahead. More information on regression-based weights can be found in Newbold & Bos, Introductory Business & Economic Forecasting, Second Edition, pp. 504-508.

Next Forecast Friday Topic: Effectiveness of Combining Forecasts

Next week, we’ll take a look at the effectiveness of combining forecasts, with a look at the empirical evidence that has been accumulated.

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