A C T A U N I V E R S I T A T I S L O D Z I E N S I S
F O L IA O E C O N O M IC A 194, 2005
J a n Ż ó ł t o w s k i *
A PPLICATIO N OF PROBIT M O D E L S
AND SELECTED D ISCRIM IN ATION ANA LY SIS M E T H O D S FOR CREDIT D ECISION EVALUATIO N
Abstract
R etail b a n k in g deals w ith servicing co n su m er credits and it c o n stitu tes one o f the m ajo r b a n k in g activities. A cu sto m er applying fo r the c redit fills in th e a p p lic atio n w hich is basis to ev alu ated o f his creditw orthiness.
T h e p a p e r co nsiders the p ro b lem o f evalu atio n to w hich o f th e tw o g ro u p s th e p erson ap p ly in g for a cred it should be assigned to: a) those w ho possess the cred itw o rth in ess; b) those w ho d o n o t possess the creditw orthiness. It analyses th e possib ility o f ap plying the p r o b it m o d els a n d th e d isc rim in a tio n analysis m e th o d s using th e q u a d r a tic an d linear d iscrim in atio n fu n ctio n . A n e v alu atio n o f the correctness o f the classification based o n the real d a ta fro m a com m ercial ban k is con d u cted .
Key words: B ayes d iscrim in atio n m eth o d s, q u a d ra tic d iscrim in atio n fu n ctio n , classification fu n c tio n , p ro b it m odel.
I. IN T R O D U C T IO N
A m o n g various types o f activities perform ed by banks, retail banking is one, which deals with the issue o f consum er credits. E ach b ank acts according to previously established regulations regarding credit g ra n tin g and repaying. A client applying for a consum er credit fills o u t a credit app lication , which constitutes a basis for the client’s creditw orthiness ev alu atio n. D a ta from the credit application are processed into scoring, which allows to assign the ap p lican t to one o f the tw o groups: a) able to repay a credit, b) unable to rep ay a credit.
T h erefore, a problem arises w hether, and if so, how we can predict w hich of the tw o groups the credit applicant will be assigned to, based on the statistical d a ta p ertaining to credit gran ting and on the inform ation a b o u t the client. A lso, how can we establish which values o f th e socio- cconom ic clicnt characteristics assure an a p p ro p ria te scoring level?
A credit decision m ade by a bank can be described by a binary variable: Y _ [1, when a credit was granted
jo , when a credit was n o t granted ^ Regression models are comm only used in the causality relationship analysis. O ne o f them is the follow ing linear regression m odel:
Yt = x (7a + £r, for t = 1 ,..., T, (2) where: x, is a vector o f exogenous variables, a - a vector o f param eters, £, - an e rro r term with the expected value o f 0.
Let us consider a case, in which the endogenous variable Y is binary with p ro b ab ility d istrib u tio n function given by:
p (Tr = 1) = n v P(Y t = 0) = 1 — я, and я , е ( 0 , 1). (3) H ence E ( Y t) = n t. M oreover, based on the assum ption s and the m odel specification E (Y () = x /a . 1 he existence o f a binary endogenous variable in the regression m odel causes a particu lar in terp re tatio n o f th e theoretical values Ý, = x / á obtained from m odel (2). Specifically, they are n o t unbiased estim ators o f probabilities Р(У ( = 1 ) = я (, if E (a) = a . As a result, it is necessary to select a m etho d, which while estim ating the param eters of m odel (2) satisfies the following condition: Ý, = x,r a e ( 0 , 1). A p ro b it m odel is one o f such m ethods. A fter having estim ated its p aram eters, one can estim ate the probability P(Y, = 1) also for o ther values o f the exogenous variables.
T h e problem o f a bank, decision prediction analysed abo ve can also be considered as classification issue. A pop u latio n П o f credit applicants can be divided into tw o su b-popu lation s П 0 i H j. A ssigning an applicant to the sub -p o p u latio n П 0 is equivalent to denying a credit, while assigning him or her to the sub-p o p u latio n corresponds to g ran tin g a credit. A ban k decision is m ad e after the analysis of the client’s ability to repay the credit. T h e assignm ent to one o f the tw o described above groups is based on values o f m statistical characteristics describing client’s socio econom ic situatio n. A vector x e R m will represent them . T he space o f values
o f the characteristics can be divided (based on their values f o r the elements o f the learning set) into two disjoint regions and X x = R"' v X 0. A situation in which vector x belongs to the region X 0 is eq uivalent to assigning a credit ap plican t to the sub-p o p u latio n H 0.
T his study exam ines an application o f b o th appro ach es in the prediction o f credit g ran tin g decisions based on the exam ple o f a b ranch o f a certain bank.
L et’s assum e th a t we have a large sam ple (obtained from an independent sam pling) and th a t we divide the set o f observations in to M subsets. F o r each o f the subsets we can derive the frequency o f th e v ariable Y taking a value o f one. Let each k-lh subset (i = 1,2, ....,M ) w ith nk elem ents have mk n u m b er o f ones. T hen the em pirical probab ility can be co m puted as frequency W e assum e th a t with the accuracy o f th e e rro r ek, it is equal to the theoretical prob ab ility л к, which can be interp reted as the value o f the cum ulative d istrib u tio n function o f a certain d istrib u tio n , i.e.:
II. P R O B IT M O D E L S 1 71 к = F (x J a). T herefore: (4) where: H ence, (5)
1 M o d els with discrete exogenous variable are discussed b y Ja ju g a in c h ap ter 8 o f w orks by S. B artosiew icz (1990).
A fte r having expanded the function F 1 into T ay lo r series a b o u t the point n k we o b tain the follow ing m odel:
F - ^ W a -И * , (6)
where:
J 2 (n x _ *>0 ~ 4 '/ ( x j o t ) ' w’ / 2 (x f <x)ni
M odel (6) is called a p ro b it m o d el2 an d it is a m odel, in which the error term is heteroscedastic. Such a m odel can be estim ated w ith the generalised least squares m eth o d o r with the m axim um likelihood m etho d.
U I. S E L E C T E D B A Y ES D IS C R IM IN A T IO N M E T H O D S
A selection o f the discrim ination m ethod based on the theory o f statistical decision functions and a procedure in the case, in which th ere exist two sets o f elem ents П 0 and 111, depend on the in fo rm atio n regarding the prior p rob abilities p 0 and p j o f a certain elem ent belonging to a p articu lar set and o f the d istrib u tio n o f the variables X = [ X t , X 2, ..., X J T characterising the elem ents o f the p o p u la tio n 3. A pplying Bayes classification rule, we can choose one o f th e alternative decisions regarding w hether th e elem ent belongs to a certain sub-populatio n.
Let,
/i(x ) = (2л) 2(det £;) exp — 2 (x ~ Ч х - Л ) (7)
be the prob ab ility density function o f the ran d o m variable X, when the analysed clem ent O e ll, dla ŕ = 0 ,1 .
S*(x) = p J f a ) , i = 0, 1 can be used as the classification fun ctio n provided th a t the loss is co n stan t when an elem ent is m isclassified. M o re th an one
2 In terestin g exam ples o f the ap p licatio n o f p ro b it analysis can be fo u n d fo r exam ple in p u b licatio n by: W iśniew ski (1986), P ru sk a (2001).
particular classification function can be chosen, sincc the classification will not be altered, w hen function Sf(x), is replaced with:
Sj(x) = g(ST (x)), (8)
w here g is any increasing function.
S f(x ) = In(Pi/,(x)) m ay be applied as the classification function, i.e.
S*(x) = ~y 1п(2я) - 2( d e t L j ) - ^ ( x -ц / Е Г4 * - f t) + l n Pi- (9)
Since the first elem ent in the form ula (9) is c o n sta n t with respect to i we can ignore it and the equivalent classification fu nction is as follows:
St(x) = - 2 (x - ц,)т E f 4 x - fij) - \ (det Zj) + ln p„ for i = 0, 1. (10)
T he function (10) contains a q u a d ra tic form o f a vector ( х - ц , ) , and as a result it is called a q u ad ratic classification function. Its value for a given x depends up o n th e p rio r probability p, and upo n the p aram eters ol the d istrib u tio n o f (i; and £ (.
A pplying Bayes classification rule with respect to a p rio r d istrib utio n (Po, Pl), we include an observation x in the p o p u latio n П ,, for w hich the classification function S,(x) takes the biggest value for i = 0, 1. C lassification regions are determ ined using the Bayes rule and tak e the follow ing form:
X 0 = { x :S 0( x ) ^ S 1(x)}. (11)
о
T he inequality in form ula (11) can be substituted with the following equivalent inequality:
(S 0(x) - ln p 0) - S L(x) - ln p j) > ln (12) Po
D enoting the left-hand side o f the inequality (12) by S0i(x ) and tak in g into consid eratio n fo rm u la (10) we receive the follow ing function:
S o i(x ) = !,[(x - ц ^ Е Г Ч х - щ ) - (x - И о № ( х - fi0) + b £ | l (13)
w hich is independent o f the prio r p robability p, and called a q u ad ratic discrim inatio n function.
Q uasi-B ayesian estim ato r is a consistent estim ato r o f th e q u ad ratic disc rim in atio n function (13). It is obtained based on the n o rm al d istribu tio n prob ab ility density function estim ator o f the follow ing form (sec: K rzyśko,
1990: 53):
V M = y ln t1 +
- у ln [• +
D° ^ + ln c~*
(14) where: Г (— ) N ‘ 2 c 0 , |X J c i — ~Tkt2— T\ — ---- i --- an d phm — = l n — —» 'x »l D ?(x) = (x — Xj)T £ f x(x — Xj), for i = 0 ,1 . (15) Statistics from the sam ple are usually used as estim ators o f the param eters in the form ula:1 Nl л 1 N<
Á = x i = дг I х . , l ~ m T ^ = Z (*u - x i)(x u - X (). (16)
iVi J= 1 / V ; — J !
E m ploying th e estim ato r Ś0i(x ) o f a q u ad ratic discrim in atio n function given by the fo rm u la (15), we assign an observation x to the su b -p o p u latio n П 0 according to the Bayes classification rule when S01(x) 55 l n 1,1 w here p 0 and
Po p x are p rio r p ro babilities estim ators.
A lso, in th e d iscrim in atio n analysis one co nsiders th e p rob lem o f a redu ction o f the num ber o f variables characterising elem ents subject to classification. T h e set o f the original variables X u X 2, ..., X m is divided into disjoint subsets and a new variable, called a discrim ination variable, is assigned to each o f the subsets. T h e d iscrim ination variable constitutes a linear co m b in atio n o f the variables contained in a p artic u la r subset. S earching for the discrim ination variables, one should aim at U t , U 2, ..., Ur w hich are n o t m utu ally correlated, which have unit variances and m axim ise the selected d istrib u tio n m easu re4.
Let us assume, ju st like we did previously, th at П 0, П , are sub-populations o f the general p o p u latio n П and th a t x = [x „ x 2, ..., x J T, w hose d istribu tio n
is m u ltiv a ria te n o rm a l, is th e re a lisa tio n o f a ra n d o m vecto r X = [X ,, X 2, ..., X J 7 in the sam ple.
Let Al t A2, —Д г be the largest ro o ts o f the equation:
det(B — I W ) = 0, (17)
and I , , I 2 )...,I , vectors o f length 1 satisfying the follow ing m atrix equation:
( B - I j W ) í = 0, (18) respectively fo r j = 1 ,2 ,..., r where W = (W 0 + W 1), В = N 0(x0 - x )(x 0 - X)T + N f a - x )(x x - X)T, _ = JV0 Xq+ N ^ j x n0 + n T
T h e d iscrim inatory variable Üj can be estim ated from the sam ple as:
17 j = t jx . (1 9)
D en o tin g by 0 = [ Üt , Ü 2, Ü r]T and v, = [ í t j í 2 J ... j i r ]T x ; for i = 0 ,1 we o b tain the follow ing form o f the classification function:
Si ( 0 ) = - ^ ( 0 - vf)T(C - V,.) + ln p , (20)
O b servation x is assigned to the sub -p o p u latio n П 0, w hen S0(C ) > ^ ( Ü ) .
IV. E M P IR IC A L E X A M P L E
In his o r h e r credit application a client provides basic d a ta (such as p ersonal in fo rm atio n , address, net incom e, ad d itio n al sources o f incom e, housing and o th er stable m onthly expenses, p otential o bligation s to serve in the arm y) and supplem ental d a ta (regarding his o r her h ousing situatio n, m arital statu s, nu m b er o f m em bers o f the househ old, type o f em ployer and years w orked for th a t em ployer, finally regarding the n um b er o f credits tak en or guaranteed).
T h e d a ta contained in the ap plication are tran sfo rm ed into scoring, which constitutes a basis for assigning the applicant to one o f the tw o groups:
1) with the ability to repay a credit; 2) w ith o u t the ability to repay.
T h e second g ro u p is som etim es divided into tw o sections: ap p licants who will be denied a credit and those who will be fu rth e r considered in the credit decision after having supplied an add itio nal collateral.
D a ta conccrning received credit applications and ban k decisions ab o u t g ran tin g or denying a crcdit over a period o f six consecutive m o n th s in 2001 were gathered in one o f the branches o f a com m ercial bank. It was established at th a t tim e th at, as a general rule, a credit was denied if a client has n o t fulfilled his arm y obligations. T herefo re, all the applications in which this was the case were removed and as a result a set o f 239 ob serv atio n s was obtained.
T hose ap plications were divided into tw o groups. T h e first g ro up was created from the applications received d u rin g the first 5 m o n th s and was treated as a learning set. T his g roup consisted o f 203 ap plicatio ns (including 131 cases followed by a negative decision - crcdit d enial, and 72 cases followed by a positive decision). Applications received in June (36 applications, including 24 cases followed by a negative decision) m ad e up the second g ro u p (which was treated as the exam ined set), which was used to predict a credit decision. T h is enabled us to evaluate the accuracy (fitness) o f the applied m ethods.
Based on the applications th e follow ing variables ch aracterisin g a crcdit ap p lican t were singled out:
1) quantitative variables:
X y - prim ary m onthly net incomc [in PLZ], X 2 - supplem ental m onthly net incom e [in PLZ], X 3 - stable m onth ly housing expenses [in PLZ], X 4 - o th er stable m onthly expenses [in PLZ], X s - n u m b er o f household m em bers,
X 6 - period w orked w ith the cu rren t em ployer [in years], X 7 - n u m b er o f tak en or guaranteed credits,
X B - m onthly incom e o f the co-ap plicant if there is one. 2) qualitative variables:
X 9 - variable specifying whether the applicant rents/ow ns an apartm ent (a house) ( X g = 1, when a client rents o r owns an ap a rtm en t (a house) and X 9 = 0 otherw ise),
X l0 variable specifying the ap p lica n t’s m arital statu s ( X 10 = 1, if the applican t is m arried and A'lo = 0 if the app licant is single),
Xyy - variable specifying the applican t’s em ploym ent statu s ( I u = 1, if the applicant w orks for a governm ental com pany, public ad m inistration ,
ow ns a p ro p rie to rsh ip o r is a p a rtn e r in a p a rtn e rsh ip an d X u = 0 otherw ise).
A crcdit application decision m ade by a ban k can be described by a binary variable:
T he a m o u n t o f credit requested in the application [in PLZ] is an additional variable: У2.
T w o new variables were derived:
X 12 - net discretionary incom e (a sum o f p rim ary and supplem ental net incom e afte r deducting stable m onth ly expenses, X 12 = (X j + X 2) —
X i2 - d isposable gross incom e (the sum o f the net incom e o f the applicant and the co-applicant X 13 = X l2 + X 8).
L et us consider th e problem o f predicting, which o f th e tw o groups a client will be assigned to based on the decisions m ad e in the learning set and on the d a ta regarding the new client. We will utilise p ro b it m odels and Bayes discrim ination analysis m eth o d to exam ine this problem .
In ord er to co m p are results o f the client classification obtained with different m eth o d s described above, we had to select variables, w hich can be em ployed by all m ethods. In particular, norm al d istrib u tio n o f all utilised variables was assum ed in Bayes discrim ination. W e verified th a t X l2 and
T herefore, basic variables used in all exam ined m odels were: w1 = l n X 12 and w2 = ln A '13- l n У2.
T hree types o f p ro b it m odel were analysed: 1, when a credit was granted 0, when a credit was n o k t granted
( * 2 + * 4 » , and 2 (21) Ф 1 = ßo + ß l Wl + ß z w2 + РзХд + t]2, (22) Ф “ 1 = ľ o + ľ l Wl + ľ 2 W2 + ľ 3 * 9 + ľ 4 * 1 0 + '/ 3 . (23)
where:
w, = ln J ŕ 12 - logarithm o f net discretionary incom e, X
w2 = In 13 - logarithm o f gross disposable incom e an d credit am o un t, ^2
X 9 - binary variable accepting the value o f 1 if an ap plicant rents or ow ns an ap a rtm en t (house),
X 10 - binary variable accepting the value o f 1 if an ap plican t is m arried. A fter an application o f a p ro b it analysis and estim a tio n 5 o f ap p ro p riate p ro b it m odels param eters (based on the d a ta from the learning set) we have obtained the follow ing results:
Table 1. A ccuracy o f credit ap p licatio n s classification based on p ro b it m odels
M odel O bserved
value У,
R esults for the learning set R esu lts fo r th e exam ined set
Predicted value У, % accu rate class. P redicted value У1 % accurate class. 0 1 0 1 (21) 0 117 14 89.3 19 5 79.2 1 20 52 72.2 3 9 75.0 (22) 0 121 10 92.4 22 2 91.7 1 11 61 84.7 2 10 83.3 (23) 0 124 7 94.7 20 4 83.3 1 7 65 90.3 1 11 91.7 S ource: A u th o r’s c o m p u tatio n s.
W hile analysing the results, we note th a t for the learning set, the percentage o f accurate classification based on m odel (21) is relatively high (89% and 72% ). H ow ever, classification based on p ro b it m odels (22) and (23) is m ore accurate (the num ber o f correctly predicted credit decisions increases). M odels “ perform b e tte r” in term s o f identifying the cases o f cred it denial in the learning sam ple. T h e percentage o f an accurate p rediction o f a credit denial is 17 points higher th a n the percentage o f an accurate p rediction o f a credit grantin g decision (for m odel (21)). F o r the exam ined sam ple (u nfortu nately, no t very num erous) the general situation regarding the accuracy o f prediction is sim ilar.
U sing the estim ato r S0t(x) o f a q u ad ratic discrim ination function given by the fo rm u la (14), observation x is assigned to su b -p o p u latio n П 0(У, = 0), according to the Bayes classification rule if S01( x ) ^ l n „ .
T h e m ean values, variances and a covariance o f variables w ,, w2 for both su b -p o p u latio n s were derived from the learning set (including 203 observations from the first 5 m o n th s o f 2001) an d x = wU) = [w ^ w ^ ]7 was substituted in form u la (14). As a result, wc have arrived a t the following form o f an estim ato r o f a discrim ination function for an i-th observation:
Śoi(vyU)) = 36 ln [1 + 0.01389 D f(w 0))] — 65.5 ln [1 + 0.00763 Dg(wü))] - 1.0376 (24) where: Do(wU)) = 11.6419(wV> — 6 .71 18)2 4.78541(w^> + 1.1622)2 + - 4.5013(wi" - 6 .7 1 1 8 )(w ^ )+ 1.1622) D?(w“ >) = 10.8785(wV> - 7.2581)2 + 12.1320(wj/> + 0.5310)2 + - 7.2689(wV> - 7.2581 -I- 0.5310)
F o r each elem ent j o f the learning set (j = 1 ,2 ,..., 203) and th e exam ined set (j = 204, ...,2 3 9 ) (crcdit applicant), the values o f a discrim ination function s ot(w 0)) w ere c o m p u te d 6 based on th e e stim a te d elem en ts o f vecto r
wVn
■ 1 hen credit applications were assigned to su b -p o p u latio n П 0, nam ely to the set o f applications followed by a credit denial, when
Soi(wU))5= - 0 .5 9 8 5
and to su b -p o p u latio n П х o f applications follow ed by a credit granting decision otherw ise. T h e follow ing classifications o f credit ap p lication s have been received:
T abic 2. A ccuracy o f credit ap p licatio n s classification based o n the value o f the q u a d ra tic discrim ination fu n ctio n estim ato r
R esu lts fo r th e learning set R esults fo r th e exam ined set
O bserved Predicted value P redicted value
value У, % accu rate
1 % accu rate class. class. 0 1 0 1 0 117 14 89.3 19 5 79.2 1 14 58 80.6 2 10 83.3 Source: A u th o r’s co m p u ta tio n s.
W hile analysing the results we n o te th a t, the percentage o f accurate classifications for the learning set is 89% and 80% . T he classification obtained w ith the estim ato r o f a q u ad ratic discrim ination function (24) “ perform ed b e tte r” in predicting credit denial. T he percentage o f accurate prediction o f credit denial is 9 points higher th an th a t o f a cred it g ra n tin g decision. F o r the exam ined sam ple (u nfortunately not very n um erou s) the general situ atio n regarding the accuracy o f prediction is sim ilar. H ow ever, the percentage o f a correct prediction o f a credit g ran tin g decision increased (by 3 points) and the percentage o f a correct prediction o f credit denial decreased (by 10 points).
V ariables used in the discrim ination were (just like above) variables wx, w2. H av ing com puted their m ean values, variances an d their covariance for b o th su b -p o p u latio n s we received: x x, £ 0, W 0, x 1( £ 1; W „ and then derived for the entire learning set x, W , B.
In o rd e r to approxim ate a discrim ination variable from the sam ple ú, Л = т а х { Я 1,Д2} was introduced, where Я,,Я2 symbolise ro ots o f the quadratic equatio n (17), and was estim ated as 2 = 0.934984. Vector Í satisfying equation (18) tu rn ed o u t to have the follow ing elem ents I = [0.8805 0.4741]T.
T herefore, the follow ing linear com bination o f the variables Wj, w2 is a discrim in ation variable form the sample:
ü = 0.8805 wt + 0.4741 w2. (25)
H aving com puted co n stan ts v1 (for the su b-po pulatio n I I 0) and v2 (for the su b -p o p u latio n П ,) we received the explicit form s o f estim ators o f both classification fu n c tio n s7.
S 0(ü) = - ^ (0.8805 wx + 0.4741 w2 - 5.3581)2 + ln p 0,
S ^ ü ) = - *(0.8805 w i + 0.4741 w2 - 6.139)2 + l n p x.
O n th e ir basis th e follow ing cred it a p p lic a tio n s classific atio n was o b ta in e d 8:
7 L et us n o te th a t the d iscrim in atio n fu n ctio n e stim a to r derived fro m b o th classification fu n ctio n s w ould take the follow ing form : S 01(u) = — 0.6877*v, —0.3702w 2 + 4 .5 9 6 3 .
T able 3. A ccuracy o f credit ap p licatio n s classification based on the value o f the d iscrim in atio n variable
Scoring P i
O bserved v alu e У,
R esults fo r the learning set R esu lts for th e exam ined set
Predicted value У, % accurate class. Predicted value У, % accurate class. 0 1 0 1 0.50 0 110 21 84.0 19 5 79.2 1 13 59 81.9 2 10 83.3 0.55 0 121 10 92.4 20 4 83.3 1 28 44 61.1 3 9 75.0 0.60 0 130 1 99.2 23 1 95.8 1 44 28 38.9 9 3 25.0 Source: A u th o r’s c o m p u ta tio n s.
C lassification obtained with the discrim ination variable ü leads to sim ilar conclusions as the one obtained with properly co nstru cted estim ato r o f the q u a d ra tic discrim ination function S0x, if one assum es a p rio r probability o f 0.5. T h e estim ation o f this probab ility obtained from the frequency o f a credit denial decision in the learning sam ple am o u n ted to 0.64. If we tak e values higher th an 0.5 for p 0 we observe an increase in the percentage o f correct classification o f the credit denial decision for b o th sam ples (over 90% ). H ow ever, this increase is accom panied by a rapid decrease in the correctly classified credit granting decisions.
V. F IN A L C O N C L U S IO N S
T h e results o btained from the p ro b it m odel (utilising the sam e variables as Bayes m ethods) are sim ilar to the ones received from Bayes discrim ination, alth o u g h the percentage o f correct classification is slightly low er in the p ro b it m odel. T h e results provided by the extended m odels (22) and (23) are better as the percentage o f correctly classified, b o th accepted and denied, credit ap plications increases. In conclusion, add itio nal exogenous variables are relevant fo r the process o f accurate classification. I t w ould be interesting to utilise th e sam e variables in Bayes analysis. H ow ever do ing so is not trivial since wc assumed th at the variables used in this m odel are continuously d istrib u ted , while add itional variables are binary.
R E F E R E N C E S
B artosiew icz S. et al. (1990), Estym acja m odeli ekonom etrycznych, P W E , W arszaw a. Ja ju g a K . (1990), S ta ty sty czn a teoria rozpoznawania obrazów, P W N , W arszaw a. K rzy śk o M . (1990), A naliza dyskrym inacyjna, W yd. N aukow o-T echniczne, W arszaw a. K rzy śk o M . (1998), S ta ty sty k a m atem atyczna, W yd. N a u k . U A M , P oznań.
P ru sk a K . (2001), M odele probitow e i logitow e w program ach n au czan ia studiów ekonom icznych, [in]: M e to d y analizy cech jakościow ych w procesie podejm ow ania decyzji (w orking papers), W yd. U Ł , Ł ó d ź, 89-98. J a n Ż ó ł t o w s k i Z A S T O S O W A N IE M O D E L I P R O B IT O W Y C H I W Y B R A N Y C H M E T O D A N A L IZ Y D Y S K R Y M IN A C Y JN E I D O P R Z E W ID Y W A N IA D E C Y Z J I K R E D Y T O W E J Streszczenie
O b słu g a k red y tó w k onsum pcyjnych jest jed n y m z ro d zajó w działalności b an k ó w . Z d olność k re d y to w a k lien ta je s t ocen ian a n a podstaw ie złożonego przez niego w niosku.
W pracy ro zw ażan y jest problem przew idyw ania, d o k tó rej z d w ó c h g ru p klientów , p osiadających zdolność kred y to w ą lub nie (w ocenie ban k u ), zostanie zaliczona o so b a ubiegająca się o k red y t. A n alizo w an e są tu m ożliwości zasto so w an ia m odeli p ro b ito w y ch o raz m etod analizy dyskrym inacyjnej w ykorzystujących k w a d rato w ą funkcję d y sk ry m in acy jn ą i zm ienną d y sk ry m in acy jn ą z p ró b y . P rzep ro w ad zo n a jest także ocena p o p raw n o ści klasyfikacji danych z pew nego b an k u .
