# How to model market share or a fraction of anything that sums to 100%?

Suppose we have sales of 3 products A, B and C which market share always sum up to 100%. How to model market share of product A using market shares of B and C? So that we will know that if share of A grows 1 percentage point then A has to steal that growth in some proportion form products B and C.

A means increase of share = A on Tuesday - A on Monday, and so on with B, C.

A = 0*A + b1*B + c1*C
B = a2*A + 0*B + c2*C
C = a3*A + b3*B + 0*C
b1+c1=-1
a2+c2=-1
a3+b3=-1

How to estimate it? Is there any other method how to estimate structure which sums to 100% by the parts of the structure? Regression doesn't work here.

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It seems you've made an edit without being logged in, thus it is showed as an anonymous one. Next time, please check it out before sending -- this way it will be applied right away. –  mbq Jul 7 '12 at 21:37
To alternatives I can think of, 1) estimate a model for the sales of A,B,C and calculate the shares afterward. 2) estimate a model for the shares of A and B and then calculate the shares of C by definition. –  John Jul 7 '12 at 21:45
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## 2 Answers

You did not really describe what you mean with model share, i.e. what the purpose of the modelling is, but the hts R package and hierarchical time series provide a way to create forecasts of time series which sum up to 100%, etc. Prof. Hyndman has at least two publications on using them: Hierarchical forecasts for Australian domestic tourism and Optimal combination forecasts for hierarchical time series. Besides documentation of the package I don't know other sources about hierarchical time series than the papers and the references therein.

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I give some explanations: –  Przemyslaw Remin Jul 9 '12 at 21:14
The purporse of the model is to give answer how much growth of A is gained from B and how much from C. A(B,C) = ? If we estimate model A = a*B + b*C without constant then both parameters a and b equal -1. But empiricaly I know that product A grew from 0% share to 30% share. It "stole" 75% of this 30% share growth from product B and 25% from product C. But I have no idea how to estimate it. The purpose of the model is explanation, not prediction. –  Przemyslaw Remin Jul 9 '12 at 21:25
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Apart from the hierarchical approach already mentioned, an alternative is to model the three series as compositional time series. See the following papers for recent surveys of the literature.

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I have access only to abstract. What is the concept of the compositional and hierarchical approach to time series? –  Przemyslaw Remin Jul 11 '12 at 11:10
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