Estimating hidden transfers of market share Suppose we have yearly data representing the market share of three companies,
say A, B and C. In other words, we have observations:
$$
 A_t, \; B_t \;\; \text{and} \;\; C_t \;\; \text{where} \; \; A_t+B_t+C_t = 1  
$$
for $t = 1, \dots,T$.
Suppose that in year $t$ the market share of company A has changed by $\Delta A_t = A_t - A_{t-1}$. Is there any way of estimating how that change can be sub-divided into market share lost to or acquired from companies B and C? My actual problem includes 5 companies, but I guess that the solution shouldn't change too much.  
 A: Write your system explicitly for time $t$ as ("$L$" for "loss", as a positive quantity, and "$G$" for "gain")
$$ A_t - A_{t-1} = - L^A_{t} + G_{t}^{B\rightarrow A}+G_{t}^{C\rightarrow A}$$
$$ B_t - B_{t-1} = - L^B_{t} + G_{t-1}^{A\rightarrow B}+G_{t}^{C\rightarrow B}$$
$$ C_t - C_{t-1} = - L^C_{t} + G_{t}^{A\rightarrow C}+G_{t}^{B\rightarrow C}$$
The following three relations hold exactly:
$$  L^A_{t} = G_{t}^{A\rightarrow B} +  G_{t}^{A\rightarrow C} $$
$$  L^B_{t} = G_{t}^{B\rightarrow A} +  G_{t}^{B\rightarrow C} $$
$$  L^C_{t} = G_{t}^{C\rightarrow A} +  G_{t}^{C\rightarrow B} $$
If you substitute in the first three you obtain
$$ A_t - A_{t-1} = - G_{t}^{A\rightarrow B} -  G_{t}^{A\rightarrow C} + G_{t}^{B\rightarrow A}+G_{t}^{C\rightarrow A}$$
$$ B_t - B_{t-1} = - G_{t}^{B\rightarrow A} -  G_{t}^{B\rightarrow C} + G_{t}^{A\rightarrow B}+G_{t}^{C\rightarrow B}$$
$$ C_t - C_{t-1} = - G_{t}^{C\rightarrow A} -  G_{t}^{C\rightarrow B} + G_{t}^{A\rightarrow C}+G_{t}^{B\rightarrow C}$$
You have $6$ unknown quantities to estimate per time period. There is just not enough information to do that. So you need assumptions that will impose structure (=restrictions) on the situation, and will permit you to estimate something. What? Let's say you assume that there is a relatively stable "churn" from one company to another, as a linear function of their market share in the previous period. This assumption brings in a set of unknown coefficients to be estimated (which will then give you an estimate of "hidden transfers of market share"). Write $G_{t}^{A\rightarrow B} = a_bA_{t-1}$ (market share lost from $A$ to $B$ as a linear function of $A$'s market share in period $t-1$).
Your equations will become
$$ A_t - A_{t-1} = - a_bA_{t-1} -  a_cA_{t-1} + b_aB_{t-1}+c_aC_{t-1} $$
$$ B_t - B_{t-1} = - b_aB_{t-1} -  b_cB_{t-1} + a_bA_{t-1}+c_bC_{t-1}$$
$$ C_t - C_{t-1} = - c_aC_{t-1} -  c_bC_{t-1} + a_cA_{t-1}+ b_cB_{t-1}$$
We have turned a set of mathematical identities into a model. It is doubtful that this model will hold exactly for each $t$, so you should add a stochastic error term. Rearranging we obtain a first-order Vector Autoregression (VAR): 
$$ \left[ \begin{matrix}
A_t  \\
B_t  \\
C_t  \\
\end{matrix} \right] = \left [\begin{matrix}
1-a_b-a_c  & b_a & c_a \\
a_b & 1-b_a-b_c & c_b \\
a_c & b_c & 1-c_a-c_b \\
\end{matrix} \right]  \left[ \begin{matrix}
A_{t-1}  \\
B_{t-1}  \\
C_{t-1}  \\ 
\end{matrix} \right]+ \left[ \begin{matrix}
u^A_{t}  \\
u^B_{t}  \\
u^C_{t}  \\
\end{matrix} \right] $$
or, to homogenize notation,
$$ \left[ \begin{matrix}
A_t  \\
B_t  \\
C_t  \\
\end{matrix} \right] = \left [\begin{matrix}
\gamma_{11}  & \gamma_{12} & \gamma_{13} \\
\gamma_{21} & \gamma_{22} & \gamma_{23} \\
\gamma_{31} & \gamma_{32} & \gamma_{33} \\
\end{matrix} \right]  \left[ \begin{matrix}
A_{t-1}  \\
B_{t-1}  \\
C_{t-1}  \\ 
\end{matrix} \right]+ \left[ \begin{matrix}
u^A_{t}  \\
u^B_{t}  \\
u^C_{t}  \\
\end{matrix} \right] $$
subject to the equality restrictions
$$  \begin{matrix}
\gamma_{11} +  \gamma_{21} + \gamma_{31} =1 \\
\gamma_{12} +  \gamma_{22} + \gamma_{32} =1  \\
\gamma_{13} +  \gamma_{23} + \gamma_{33} =1  \\
\end{matrix} $$
So you have essentially $6$ unknown coefficients and a sample of $T-1$ observations (for each company).
Note that these restrictions imply the "add up to unity" restriction $A_t+B_t+C_t =1$ for each $t$, so this last one does not impose any additional structure on the unknown coefficients -but it does imply a relation between the error terms, namely that $u^A_{t} + u^B_{t}  +u^C_{t} =0$. Any additional assumptions on the three error terms should either come from knowledge of the specific real world phenomenon under study, and/or through a statistical specification search.
Then, an estimation for a hidden transfer of market share will be, for example
$$\hat G_{t}^{A\rightarrow B} = \hat \gamma_{21}A_{t-1}$$
etc.
Of course you may find that such a model does not fit your data sample well - for example you expect that all estimated coefficients should be positive and smaller than or equal to unity, but the estimation procedure may not give you that. But this is what we do: we come up with specification hypotheses and we test them against the data - "success" is never guaranteed. Then you should try to come up with a different model.
