\[ \Pr (Y_i = 1) = \frac{{\exp \left( {\beta _0 + \beta _1 Year_t + \beta _2 X_i + \beta _{D1} D_i + \beta _{D2} D_i^2 + \beta _E E_i + \beta _T T_i + \beta _{TA} TA_i } \right)}}{{1 + \exp \left( {\beta _0 + \beta _1 Year_t + \beta _2 X_i + \beta _{D1} D_i + \beta _{D2} D_i^2 + \beta _E E_i + \beta _T T_i + \beta _{TA} TA_i } \right)}} \]
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\hat \beta _{(90)}=
Argmin_ß\[\in \]Reals
\[
\sum\limits_{i = 1}^N {\left[ {.90  1\left( {{\rm{Charges}}_i  X\beta < 0} \right)} \right]} \left[ {{\rm{Charges}}_{\rm{i}}  X\beta } \right]
\]
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The first graph shows the difference in Mt. Sinai's market share between the Target and Control Markets over the period 19942001. The target market share is about 6% higher relative to the control in 1996 compared to years before and after. Also, the target market share begins rising relative to the control group in 1999 and gets progressively larger, approaching 8% by 2000.
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The second graph shows NY Presbyterian's market share in the respective Target and Control markets over the period 19942001. Although both are trending upward over time from 1995 to 2000 (rising from about 7% to about 20%), there two series are very close and show no clear pattern of one being consistently above the other in share.
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This Venn diagram represents visually an overlap of two circular markets. Area inside a radius of T from the target market is white. Potential control markets reside in a second circle labeled 'A' outside the overlap region with the circle centered on T.
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\[ E(Y_{i,t} X_{i,t} {\rm{, Market = T, t = Post)}}  E(Y_i X_{i,t} {\rm{, Market = T, t = Pre)}}{\rm{.}}
\]
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\[ E(Y_{i,t} X_{i,t} {\rm{, Market = T, t = Post)}}  E(Y_i X_{i,t} {\rm{, Market = T, t = Pre)}}{\rm{.}}
\]
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\[ E(Y_{i,t} X_{i,t} {\rm{, Market = T, t = Post)}}  E(Y_i X_{i,t} {\rm{, Market = T, t = Pre)}}{\rm{.}}
\]
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\[ E(Y_{i,t} X_{i,t} {\rm{, Market = T, t = Post)}}  E(Y_i X_{i,t} {\rm{, Market = T, t = Pre)}}{\rm{.}}
\]
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\[ E(Y_{i,t} X_{i,t} {\rm{, Market = T, t = Post, TA = 1)}}  E(Y_i X_{i,t} {\rm{, Market = T, t = Pre, TA = 0)}}{\rm{.}} \]
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