# Seasonal Indexes adding to zero

In the textbook Forecasting: principles and practice by Hyndman and Athana­sopou­los, in the Classical Decomposition (Sec 6.3), in step 3 of the additive decomposition algorithm, the authors state that the seasonal indexes have to be adjusted to ensure that they add to zero. Why is that?

Any help would be appreciated.

• Suppose you investigate the yield of four different seeds of corn. The yield $y_{ij}$of the $j$ plot of corn sown with the $i$ kind of corn is modelled as: $y_{ij} = \alpha + \beta_i + \epsilon_{ij}$. The coefficients $\beta_i$ ($i=1,\ldots,4$) measure the effect of the different kinds of corn, and $\alpha$ the average yield. Clearly, it doesn't make sense to have all $\beta_i$ positive, for then we would subtract part of their value and include it in $\alpha$: rather, it makes sense to impose that they add up to zero. Same with the seasonal coefficients. – F. Tusell Feb 23 '15 at 17:00