A Four-class model formulation

DISCOVERING CREDIT CARDHOLDERS’ BEHAVIOR 269 Given a set of r variables about the cardholders A = A 1 , . . . , A i , let A t = A i 1 , . . . , A ir be the development sample of data for the variables, where i = 1, . . . , n and n is the sample size. We try to determine the coefficients of the variables, denoted by X = x 1 , . . . , x r , boundary b 1 to separate G 1 from G 2 , G 3 , and G 4 , boundary b 3 to separate G 4 from G 1 , G 2 , and G 3 , boundary b 2 to separate G 2 and G 3 . This separation can be represented by: A i X ≤ b 1 , A i ∈ G 1 b 1 ≤ A i X ≤ b 2 , A i ∈ G 2 b 2 ≤ A i X ≤ b 3 , A i ∈ G 3 , b 3 ≤ A i

X, A

i ∈ G 4 Similar to two-class and three-class models, we apply two measurements for better separations. Let α 1 i be the overlapping degree with respect of A i within G 1 and G 2 , α 2 i be the overlapping degree with respect of A i within G 2 and G 3 , and α 3 i be the overlapping degree with respect of A i , within G 3 and G 4 . Let β 1 i be the distance from A i within G 1 and G 2 to its adjusted boundaries A i X = b 1 − α 1 ∗ , and A i X = b 1 + α 1 ∗ , β 2 i be the distance from A i within G 2 and G 3 to its adjusted boundaries A i X = b 2 − α 2 ∗ , and A i X = b 2 + α 2 ∗ , and β 3 i be the distance from A i within G 3 and G 4 to its adjusted boundaries A i X = b 3 − α 3 i , and A i X = b 3 + α 3 i . We want to reach the maximization of β 1 i , β 2 i , and β 3 i and the minimization of α 1 i α 2 i and α 3 i simultaneously. After putting α 1 i , α 2 i , α 3 i , β 1 i , β 2 i , and β 3 i into the above four-class separation, we have: M4 Minimize i α 1 i + α 2 i + α 3 i and Maximize i β 1 i + β 2 i + β 3 i Subject to: G 1 : A i X = b 1 + α 1 i − β 1 i , A ∈ G 1 , Bankrupt charge-off G 2 : A i X = b 1 + α 1 i − β 1 i , A i X = b 2 + α 2 i − β 2 i , A i ∈ G 2, Non-bankrupt charge-off G 3 : A i X = b 2 + α 2 i − β 2 i , A i X = b 3 − α 3 i + β 3 i , Ai ∈ G 3, Delinquent G 4 : A i X = b 3 − α 3 i + β 3 i , A i ∈ G 4 Current b 1 + α 1 i ≤ b 2 − α 2 i , b 2 + α 2 i ≤ b 3 − α 3 i where A i , are given, X, b 1 , b 2 , b 3 are unrestricted, and α 1 i , α 2 i , α 3 i , β 1 i , β 2 i , and β 3 i ≥ 0. The constraints b 1 + α 1 i ≤ b 2 − α 2 i and b 2 + α 2 i ≤ b 3 − α 3 i guarantee the existence of four groups by enforcing b 1 lower than b 2 , and b 2 lower than b 3 . Then we apply the compromise solution approach Shi, 1989, 2001 to reform the model. We assume the ideal value of − i α 1 i be α 1 ∗ 0, − i α 2 i be α 2 ∗ 0, − i α 3 i be α 3 ∗ 0, and the ideal value of i β 1 i be β 1 ∗ 0, i β 2 i be β 2 ∗ 0, i β 3 i be β 3 ∗ 0. 270 KOU ET AL. Then, the four-group model M4 is transformed as: M5 Minimize d − α 1 + d + α 1 + d − β 1 + d + β 1 + d − α 2 + d + α 2 + d − β 2 + d + β 2 + d − α 3 + d + α 3 + d − β 3 + d + β 3 Subject to: α 1 ∗ + i α 1 i = d − α 1 − d + α 1 , β 1 ∗ − i β 1 i = d − β 1 − d + β 1 , α 2 ∗ + i α 2 i = d − α 2 − d + α 2 , β 2 ∗ − i β 2 i = d − β 2 − d + β 2 , α 3 ∗ + i α 3 i = d − α 1 − d + α 3 , β 3 ∗ − i β 3 i = d − β 3 − d + β 3 , b 1 + α 1 i ≤ b 2 − α 2 i , b 2 + α 2 i ≤ b 3 − α 3 i , G 1 : A i X = b 1 + α 1 i − β 1 i , A i = G 1 Bankrupt charge-off G 2 : A i X = b 1 − α 1 i + β 1 i , A i X = b 2 + α 2 i − β 2 i , A i ∈ G 2, Non-bankrupt charge-off G 3 : A i X = b 2 + α 2 i − β 2 i , A i X = b 3 − α 3 i + β 3 i , A i ∈ G 3, Delinquent G 4 : A i X = b 3 − α 3 i + β 3 i , A i ∈ G 4 , Current Where A i , are given, b 1 ≤ b 2 ≤ b 3 , X, b 1 , b 2 , b 3 , are unrestricted, and α 1 i , α 2 i , α 3 i , β 1 i , β 2 i and β 3 i ≥ 0. A SAS-based algorithm of this four-class model is proposed as follows: Algorithm 1. Step 1 . Use ReadCHD.sas to convert both the training and verifying data into SAS data sets. Step 2 . Use GroupDef.sas to divide the observations within the training data sets into four groups: G 1 , G 2 , G 3 , and G 4 . Step 3 . Use 4GModel.sas to perform the separation task on the training data. Here, PROC LP in SAS is called to calculate the M5 model for the best solution of the four- class separation given the values of control parameters α 1 ∗ , β 1 ∗ , α 2 ∗ , β 2 ∗ , α 3 ∗ , β 3 ∗ , b 1 , b 2 , b 3 . Step 4 . Use Score.sas to produce the graphical representations of the training results. Step 3–4 will not terminate until the best training result is found. Step 5 . Use Predict, sas to mine the four classes from the verifying data set.

5. Empirical study and managerial significance of four-class models