Individual and Combination Forecasts

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Five methods were chosen as the individual forecasting methods. These are the Box-Jenkin, state-space, exponential smoothing, direct smoothing and decomposition methods. All these methods are widely used in forecasting seasonal data. Our empirical example deals with a set of seasonal data.

Combination of forecasts in time series started with the seminal work of Bates and Granger (1969). It has had considerable impact on the forecasting literature for the past 30 years. See, for example, Aksu and Gunter (1992), Bordley (1982), Bunn (1985), DeGroot and Feinberg (1983), Gupta and Wilton (1987), Öller (1978), and Winkler and Makridakis (1983). A survey of this literature was provided by Clemen (1989), with an update in Clemen (1993).

In this study, three forecast combination methods will be compared.

1. Simple average (SA): Let X_{i, t-1} (1) (i = 1, 2, ..., M) be the one-step ahead forecasts of M individual methods. This entails M series of forecast errors , (i = 1, 2, ..., M) where . Then the combined forecast, denoted as X_{c, t-1}(1), is equal to i.e., the average of the individual forecasts. Note that an equivalent formulation of the combined forecast is

   

   (11) 

   

   where for all i and t.

The next two methods were proposed by Granger, see Granger and Newbold (1986). Suppose we are at time n - 1. Then we have

2. Granger 1(G1): With reference to equation (11),

   (12) 

   

   Here ν is usually taken to be 1, 3, 6, 9 and 12 in Granger and Newbold [15]. Intuitively we see this formula will allow the weighting to change so as to capture the changes in the relative performance of the different methods.

3. Granger 2 (G2): This is similar to Granger 1, but we introduce smoothing for the λ weights.

   (13) 

   

   Winkler and Markridakis (1983) applied the same combining rules to 10 forecasting methods used on 1,001 different time series. They found techniques based on G1 and G2 to be superior to other combinations.

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