lecture 8 - COMBINATION OF FORECASTS As the old saying goes...

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1 COMBINATION OF FORECASTS As the old saying goes, “Don’t put all of your eggs in one basket les t you drop the basket and lose all of your eggs.” Suppose the head of a forecasting division of a company has two sources of forecasts for the company’s sales , one source being the forecasts generated by the division’s econometrics group using an econometric time series model and the other source being the aggregated forecasts of the regional sales managers of the company. Suppose that the forecast horizon is h = 1 and represent the one-step-ahead forecasts of the econometrics group made at time t by ) 1 ( 1 t f and the forecasts of the managers as ) 2 ( 1 t f . Now let us consider the statistical properties of the combination forecast ) 2 ( 1 ) 1 ( 1 ) ( 1 ) 1 ( t t c t f w wf f . (1) Suppose that both of the forecasting methods are unbiased in that the one-step- ahead forecast errors of the two methods, say ) 1 ( 1 1 ) 1 ( 1 t t t f y e and ) 2 ( 1 1 ) 2 ( 1 t t t f y e , have zero expectations. That is, 0 ) ( ) ( ) 2 ( 1 ) 1 ( 1 t t e E e E . Now consider the following theorem. Theorem 1 : Assume ) 1 ( 1 t f and ) 2 ( 1 t f are unbiased forecasts of 1 t y . Then the combination forecast ) ( 1 c t f is unbiased. That is, . 0 ) ( ) ( ) ( 1 1 ) ( 1 c t t c t f y E e E Proof: ) ) 1 ( ( ) ( ) ( ) 2 ( 1 ) 1 ( 1 1 ) ( 1 1 ) ( 1 t t t c t t c t f w wf y E f y E e E )) )( 1 ( ) ( ( ) 2 ( 1 1 ) 1 ( 1 1 t t t t f y w f y w E ) ( ) 1 ( ) ( ) 2 ( 1 1 ) 1 ( 1 1 t t t t f y E w f y wE . 0 0 ) 1 ( 0 ) ( ) 1 ( ) ( ) 2 ( 1 ) 1 ( 1 w w e E w e wE t t QED.
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