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Financial classification models Part I: Discriminant
AnalysisAdvanced Financial Accounting IISchool of Business and
Economics atbo Akademi University
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ContentsThe classification problemClassification modelsCase:
Bankruptcy prediction of Spanish banksSome comments on hypothesis
testing References
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The classification problemIn a traditional classification
problem the main purpose is to assign one of k labels (or classes)
to each of n objects, in a way that is consistent with some
observed data, i.e. to determine the class of an observation based
on a set of variables known as predictors or input variablesTypical
classification problems in finance are for exampleFinancial
failure/bankruptcy predictionCredit risk rating
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Classification methodsThere are several statistical and
mathematical methods for solving the classification problem,
e.g.Discriminant analysisLogistic regressionRecursive partitioning
algorithm (RPA)Mathematical programmingLinear programming
modelsQuadratic programming modelsNeural network classifiersWe
begin by presenting the most common method, i.e. Discriminant
analysis Other methods later in part 2
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Discriminant analysisDiscriminant analysis is the most common
technique for classifying a set of observations into predefined
classesThe model is built based on a set of observations for which
the classes are knownThis set of observations is sometimes referred
to as the training set or estimation sample
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Discriminant Analysis Historical backgroundDiscriminant analysis
is concerned with the problem to assign or allocate an object (e.g.
a firm) to its correct populationThe statistical method originated
with Fisher (1936)The theoretical framework was derived by Anderson
(1951)The term discriminant analysis emerged from the research by
Kendall & Stuart (1968) and Lachenbruch & Mickey
(1968)Discriminant analysis was originally applied in accounting by
Altman (1968) using U.S. data and by Aatto Prihti (1975) on Finnish
data
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Discriminant analysis...Based on the training set, the technique
constructs a set of linear functions of the predictors, known as
discriminant functions, such that L = b1x1 + b2x2 + + bnxn + c,
where the b's are discriminant coefficients, the x's are the
input variables or predictors and c is a constant.Two types of
discriminant functions are discussed laterCanonical discriminant
functions (k-1)Fishers discriminant functions (k)
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Discriminant functionsThe discriminant functions are optimized
to provide a classification rule that minimizes the probability of
misclassificationSee figure on the next pageIn order to achieve
optimal performance, some statistical assumptions about the data
must be metEach group must be a sample from a multivariate normal
populationThe population covariance matrices must all be equalIn
practice the discriminant has been shown to perform fairly well
even though the assumptions on data are violated
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Distributions of the discriminant scores for two classes A
discriminant function is optimized to minimize the common area for
the distributions
Chart1
0.00443184840.0000000061
0.00514264090.0000000082
0.00595253240.000000011
0.00687276670.0000000148
0.00791545160.0000000198
0.00909356250.0000000264
0.01042093480.0000000351
0.01191224360.0000000467
0.01358296920.0000000618
0.01544934710.0000000817
0.01752830050.0000001077
0.01983735440.0000001416
0.02239453030.0000001857
0.02521821990.000000243
0.02832703770.0000003171
0.03173965180.0000004128
0.03547459280.0000005361
0.03955004160.0000006944
0.0439835960.0000008972
0.04879201860.0000011564
0.05399096650.0000014867
0.05959470610.0000019066
0.06561581480.000002439
0.07206487430.0000031122
0.07895015830.0000039613
0.08627731880.0000050295
0.09404907740.0000063698
0.10226492460.0000080472
0.11092083470.0000101409
0.12000900070.0000127473
0.12951759570.0000159837
0.13943056640.0000199918
0.14972746560.0000249425
0.16038332730.0000310414
0.1713685920.0000385352
0.18264908540.0000477186
0.1941860550.0000589431
0.20593626870.0000726259
0.2178521770.0000892617
0.22988214070.000109434
0.24197072450.0001338302
0.25405905650.0001632564
0.26608524990.0001986555
0.27798488610.0002411266
0.28969155280.0002919469
0.30113743220.0003525957
0.31225393340.0004247803
0.32297235970.000510465
0.33322460290.0006119019
0.3429438550.0007316645
0.35206532680.0008726827
0.36052696250.0010382813
0.36827014030.0012322192
0.37524034690.0014587308
0.38138781550.0017225689
0.38666811680.0020290481
0.3910426940.0023840882
0.39447933090.0027942584
0.39695254750.0032668191
0.39844391410.0038097621
0.39894228040.0044318484
0.39844391410.0051426409
0.39695254750.0059525324
0.39447933090.0068727667
0.3910426940.0079154516
0.38666811680.0090935625
0.38138781550.0104209348
0.37524034690.0119122436
0.36827014030.0135829692
0.36052696250.0154493471
0.35206532680.0175283005
0.3429438550.0198373544
0.33322460290.0223945303
0.32297235970.0252182199
0.31225393340.0283270377
0.30113743220.0317396518
0.28969155280.0354745928
0.27798488610.0395500416
0.26608524990.043983596
0.25405905650.0487920186
0.24197072450.0539909665
0.22988214070.0595947061
0.2178521770.0656158148
0.20593626870.0720648743
0.1941860550.0789501583
0.18264908540.0862773188
0.1713685920.0940490774
0.16038332730.1022649246
0.14972746560.1109208347
0.13943056640.1200090007
0.12951759570.1295175957
0.12000900070.1394305664
0.11092083470.1497274656
0.10226492460.1603833273
0.09404907740.171368592
0.08627731880.1826490854
0.07895015830.194186055
0.07206487430.2059362687
0.06561581480.217852177
0.05959470610.2298821407
0.05399096650.2419707245
0.04879201860.2540590565
0.0439835960.2660852499
0.03955004160.2779848861
0.03547459280.2896915528
0.03173965180.3011374322
0.02832703770.3122539334
0.02521821990.3229723597
0.02239453030.3332246029
0.01983735440.342943855
0.01752830050.3520653268
0.01544934710.3605269625
0.01358296920.3682701403
0.01191224360.3752403469
0.01042093480.3813878155
0.00909356250.3866681168
0.00791545160.391042694
0.00687276670.3944793309
0.00595253240.3969525475
0.00514264090.3984439141
0.00443184840.3989422804
0.00380976210.3984439141
0.00326681910.3969525475
0.00279425840.3944793309
0.00238408820.391042694
0.00202904810.3866681168
0.00172256890.3813878155
0.00145873080.3752403469
0.00123221920.3682701403
0.00103828130.3605269625
0.00087268270.3520653268
0.00073166450.342943855
0.00061190190.3332246029
0.0005104650.3229723597
0.00042478030.3122539334
0.00035259570.3011374322
0.00029194690.2896915528
0.00024112660.2779848861
0.00019865550.2660852499
0.00016325640.2540590565
0.00013383020.2419707245
0.0001094340.2298821407
0.00008926170.217852177
0.00007262590.2059362687
0.00005894310.194186055
0.00004771860.1826490854
0.00003853520.171368592
0.00003104140.1603833273
0.00002494250.1497274656
0.00001999180.1394305664
0.00001598370.1295175957
0.00001274730.1200090007
0.00001014090.1109208347
0.00000804720.1022649246
0.00000636980.0940490774
0.00000502950.0862773188
0.00000396130.0789501583
0.00000311220.0720648743
0.0000024390.0656158148
0.00000190660.0595947061
0.00000148670.0539909665
0.00000115640.0487920186
0.00000089720.043983596
0.00000069440.0395500416
0.00000053610.0354745928
0.00000041280.0317396518
0.00000031710.0283270377
0.0000002430.0252182199
0.00000018570.0223945303
0.00000014160.0198373544
0.00000010770.0175283005
0.00000008170.0154493471
0.00000006180.0135829692
0.00000004670.0119122436
0.00000003510.0104209348
0.00000002640.0090935625
0.00000001980.0079154516
0.00000001480.0068727667
0.0000000110.0059525324
0.00000000820.0051426409
0.00000000610.0044318484
Class 1
Class 2
Score
Normal
mean03
stdev11
ScoreClass 1Class 2
-30.00443184840.0000000061
-2.950.00514264090.0000000082
-2.90.00595253240.000000011
-2.850.00687276670.0000000148
-2.80.00791545160.0000000198
-2.750.00909356250.0000000264
-2.70.01042093480.0000000351
-2.650.01191224360.0000000467
-2.60.01358296920.0000000618
-2.550.01544934710.0000000817
-2.50.01752830050.0000001077
-2.450.01983735440.0000001416
-2.40.02239453030.0000001857
-2.350.02521821990.000000243
-2.30.02832703770.0000003171
-2.250.03173965180.0000004128
-2.20.03547459280.0000005361
-2.150.03955004160.0000006944
-2.10.0439835960.0000008972
-2.050.04879201860.0000011564
-20.05399096650.0000014867
-1.950.05959470610.0000019066
-1.90.06561581480.000002439
-1.850.07206487430.0000031122
-1.80.07895015830.0000039613
-1.750.08627731880.0000050295
-1.70.09404907740.0000063698
-1.650.10226492460.0000080472
-1.60.11092083470.0000101409
-1.550.12000900070.0000127473
-1.50.12951759570.0000159837
-1.450.13943056640.0000199918
-1.40.14972746560.0000249425
-1.350.16038332730.0000310414
-1.30.1713685920.0000385352
-1.250.18264908540.0000477186
-1.20.1941860550.0000589431
-1.150.20593626870.0000726259
-1.10.2178521770.0000892617
-1.050.22988214070.000109434
-10.24197072450.0001338302
-0.950.25405905650.0001632564
-0.90.26608524990.0001986555
-0.850.27798488610.0002411266
-0.80.28969155280.0002919469
-0.750.30113743220.0003525957
-0.70.31225393340.0004247803
-0.650.32297235970.000510465
-0.60.33322460290.0006119019
-0.550.3429438550.0007316645
-0.50.35206532680.0008726827
-0.450.36052696250.0010382813
-0.40.36827014030.0012322192
-0.350.37524034690.0014587308
-0.30.38138781550.0017225689
-0.250.38666811680.0020290481
-0.20.3910426940.0023840882
-0.150.39447933090.0027942584
-0.10.39695254750.0032668191
-0.050.39844391410.0038097621
00.39894228040.0044318484
0.050.39844391410.0051426409
0.10.39695254750.0059525324
0.150.39447933090.0068727667
0.20.3910426940.0079154516
0.250.38666811680.0090935625
0.30.38138781550.0104209348
0.350.37524034690.0119122436
0.40.36827014030.0135829692
0.450.36052696250.0154493471
0.50.35206532680.0175283005
0.550.3429438550.0198373544
0.60.33322460290.0223945303
0.650.32297235970.0252182199
0.70.31225393340.0283270377
0.750.30113743220.0317396518
0.80.28969155280.0354745928
0.850.27798488610.0395500416
0.90.26608524990.043983596
0.950.25405905650.0487920186
10.24197072450.0539909665
1.050.22988214070.0595947061
1.10.2178521770.0656158148
1.150.20593626870.0720648743
1.20.1941860550.0789501583
1.250.18264908540.0862773188
1.30.1713685920.0940490774
1.350.16038332730.1022649246
1.40.14972746560.1109208347
1.450.13943056640.1200090007
1.50.12951759570.1295175957
1.550.12000900070.1394305664
1.60.11092083470.1497274656
1.650.10226492460.1603833273
1.70.09404907740.171368592
1.750.08627731880.1826490854
1.80.07895015830.194186055
1.850.07206487430.2059362687
1.90.06561581480.217852177
1.950.05959470610.2298821407
20.05399096650.2419707245
2.050.04879201860.2540590565
2.10.0439835960.2660852499
2.150.03955004160.2779848861
2.20.03547459280.2896915528
2.250.03173965180.3011374322
2.30.02832703770.3122539334
2.350.02521821990.3229723597
2.40.02239453030.3332246029
2.450.01983735440.342943855
2.50.01752830050.3520653268
2.550.01544934710.3605269625
2.60.01358296920.3682701403
2.650.01191224360.3752403469
2.70.01042093480.3813878155
2.750.00909356250.3866681168
2.80.00791545160.391042694
2.850.00687276670.3944793309
2.90.00595253240.3969525475
2.950.00514264090.3984439141
30.00443184840.3989422804
3.050.00380976210.3984439141
3.10.00326681910.3969525475
3.150.00279425840.3944793309
3.20.00238408820.391042694
3.250.00202904810.3866681168
3.30.00172256890.3813878155
3.350.00145873080.3752403469
3.40.00123221920.3682701403
3.450.00103828130.3605269625
3.50.00087268270.3520653268
3.550.00073166450.342943855
3.60.00061190190.3332246029
3.650.0005104650.3229723597
3.70.00042478030.3122539334
3.750.00035259570.3011374322
3.80.00029194690.2896915528
3.850.00024112660.2779848861
3.90.00019865550.2660852499
3.950.00016325640.2540590565
40.00013383020.2419707245
4.050.0001094340.2298821407
4.10.00008926170.217852177
4.150.00007262590.2059362687
4.20.00005894310.194186055
4.250.00004771860.1826490854
4.30.00003853520.171368592
4.350.00003104140.1603833273
4.40.00002494250.1497274656
4.450.00001999180.1394305664
4.50.00001598370.1295175957
4.550.00001274730.1200090007
4.60.00001014090.1109208347
4.650.00000804720.1022649246
4.70.00000636980.0940490774
4.750.00000502950.0862773188
4.80.00000396130.0789501583
4.850.00000311220.0720648743
4.90.0000024390.0656158148
4.950.00000190660.0595947061
50.00000148670.0539909665
5.050.00000115640.0487920186
5.10.00000089720.043983596
5.150.00000069440.0395500416
5.20.00000053610.0354745928
5.250.00000041280.0317396518
5.30.00000031710.0283270377
5.350.0000002430.0252182199
5.40.00000018570.0223945303
5.450.00000014160.0198373544
5.50.00000010770.0175283005
5.550.00000008170.0154493471
5.60.00000006180.0135829692
5.650.00000004670.0119122436
5.70.00000003510.0104209348
5.750.00000002640.0090935625
5.80.00000001980.0079154516
5.850.00000001480.0068727667
5.90.0000000110.0059525324
5.950.00000000820.0051426409
60.00000000610.0044318484
Normal
Class 1
Class 2
Score
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Canonical discriminant functionsA canonical discriminant
function is a linear combination of the discriminating variables
which are formed to satisfy certain conditionsThe coefficients for
the first function are derived so that the group means on the
function are as different as possibleThe coefficients for the
second function are derived to maximize the difference between
group means under the added condition that values on the second
function are not correlated with the values on the first functionA
third function is defined in a similar way having coefficients
which maximize the group differences while being uncorrelated with
the previous function and so onThe maximum number on unique
functions is Min(Groups 1, No of discriminating variables)
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Fishers Discriminant functionsThe discriminant functions are
used to predict the class of a new observation with unknown
classFor a k class problem, k discriminant functions are
constructedGiven a new observation, all the k discriminant
functions are evaluated and the observation is assigned to class i
if the i:th discriminant function has the highest value
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Interpretation of the Fishers discriminant function
coefficientsThe discriminant functions are used to compute the
discriminant score for a case in which the original discriminating
variables are in standard formThe discriminant score is computed by
multiplying each discriminating variable by its corresponding
coefficient and adding together these productsThere will be a
separate score for each case on each functionThe coefficients have
been derived in such a way that the discriminant scores produced
are in standard formAny single score represents the number of
standard deviations the case is away from the mean for all cases on
the given discriminant function
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Interpretation of the Fishers discriminant function
coefficientsThe standardized discriminant function coefficients are
of great analytical importanceWhen the sign is ignored, each
coefficient represents the relative contribution of its associated
variable for that functionThe sign denotes whether the variable is
making a positive or negative contributionThe interpretation is
analogous to the interpretation of beta weights in multiple
regressionAs in factor analysis, the coefficients can be used to
name the functions by identifying the dominant characteristics they
measure
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Variable selection: Analyzing group differencesAlthough the
variables are interrelated and the multivariate statistical
techniques such as discriminant analysis incorporate these
dependencies, it is often helpful to begin analyzing the
differences between groups by examining univariate statisticsThe
first step is to compare the group means of the predictor
variablesA significant inequality in group means indicates the
predictor variables ability to separate between the groupsThe
significance test for the equality of the group means is an F-test
with 1 and n-g degrees of freedomIf the observed significance level
is less than 0.05, the hypothesis of equal group means is
rejected
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Analyzing group differences: Wilks LambdaAnother statistic used
to analyze the univariate equality of group means is Wilks Lambda,
sometimes called the U-statisticLambda is the ratio of the
within-groups sum of squares to the total sum of squaresLambda has
values between 0 and 1A lambda of 1 occurs when all observed group
means are equalValues close to 0 occur when within-groups
variability is small compared to total variabilityLarge values of
lambda indicate that group means do not appear to be different
while small values indicate that group means do appear to be
different
-
Multivariate Wilks Lambda statisticIn the case of several
variables {X1, X2,...,Xp}, the total variability is expressed by
the total cross product matrix TThe sum of cross-product matrix T
is decomposed into the within-group sum of cross- product matrix W
and the between-group sum of cross-product matrix B such thatT = W
+ B W = T - B
-
Multivariate Wilks Lambda statistic...For the set of the X
variables, the multivariate global Wilks Lambda is defined as
Lp = |W| / |W + B| = |W| / |T| ~ L(p,m,n) where|W| = the
determinant of the within-group SSCP matrix|B| = the determinant of
the between-groups SSCP matrix|T| = the determinant of the total
sum of cross product matrixL(p,m,n) = Wilks Lambda distributionFor
large m, Bartlett's (1954) approximation allows Wilks' lambda to be
approximated by a Chi-square distribution
-
Variable selection: Correlations between predictor variables
Since interdependencies among the variables affect most
multivariate analyses, it is worth examining the correlation matrix
of the predictor variablesIncluding highly correlated variables in
the analysis should be avoided as correlations between variables
affect the magnitude and the signs of the coefficientsIf correlated
variables are included in the analysis, care should be exercised
when interpreting the individual coefficients
-
Case: Bankruptcy prediction in the Spanish banking
sectorReference: Olmeda, Ignacio and Fernndez, Eugenio: "Hybrid
classifiers for financial multicriteria decision making: The case
of bankruptcy prediction", Computational Economics 10, 1997,
317-335.Sample: 66 Spanish banks37 survivors29 failedSample was
divided in two sub-samplesEstimation sample, 34 banks, for
estimating the model parametersHoldout sample, 32 banks, for
validating the results
-
Case: Bankruptcy prediction in the Spanish banking sectorInput
variablesCurrent assets/Total assets(Current assets-Cash)/Total
assetsCurrent assets/LoansReserves/LoansNet income/Total assetsNet
income/Total equity capitalNet income/LoansCost of sales/SalesCash
flow/Loans
-
Empirical resultsAnalyzing the total set of 66 observationsGroup
statistics comparing the group meansTesting for the equality of
group meansCorrelation matrixClassification with different
methodsEstimating classification models using the estimation sample
of 34 observationsChecking the validity of the models by
classifying the holdout sample of 32 observations
-
Group statistics
Class 0 N=37Class 1 N=29Total
N=66MeanSt.devMeanSt.devMeanSt.devCA/TA,410,114,370,108,393,112(CA-Cash)/TA,268,089,264,092,266,089CA/Loans,423,144,390,117,409,133Reserves/Loans,038,054,016,012,028,043NI/TA,008,005-,003,019,003,014NI/TEC,167,082-,032,419,079,299NI/Loans,008,005-,003,020,003,015CofS/Sales,828,062,957,188,885,147CF/Loans,018,029,004,012,012,024
-
Tests of equality of group meansNo significant difference in
group meansInsignificant difference
Wilks
LambdaFdf1df2Sig.CA/TA,9692,072164,155(CA-Cash)/TA1,000,027164,871CA/Loans,985,981164,326Reserves/Loans,9324,667164,034NI/TA,86410,041164,002NI/TEC,8898,011164,006NI/Loans,86310,149164,002CofS/Sales,80515,463164,000CF/Loans,9185,713164,020
-
F(1,64)-distribution5% critical value = 3.99CA/TA 2.072CA/Loans
0.981(CA-Cash)/TA 0.027
Chart2
0
0.1251351043
0.0510893915
0.0387023349
0.032214465
0.0280230808
0.0250152815
0.0227140657
0.0208755276
0.0193599885
0.0180808215
0.016981014
0.0160212729
0.0151734999
0.0144169858
0.0137360775
0.0131186881
0.0125553111
0.0120383498
0.0115616477
0.011120154
0.0107096786
0.0103267113
0.009968285
0.0096318718
0.0093153016
0.0090166992
0.0087344342
0.0084670809
0.0082133861
0.007972243
0.0077426693
0.0075237899
0.007314822
0.0071150628
0.0069238789
0.006740698
0.0065650013
0.0063963165
0.0062342139
0.0060783001
0.0059282147
0.0057836267
0.0056442317
0.0055097485
0.0053799175
0.0052544982
0.0051332676
0.0050160182
0.0049025571
0.0047927044
0.0046862922
0.0045831633
0.0044831707
0.0043861764
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F(1,64)-distribution
normality test
Test sheet for visual checking of empirical data
Copy your own data hereHistograms:
Title:Original dataStandardized data
Data
Data:standardized
Basic statistics0.0000
0.0000
Mean:0.00000.0000
St.dev:1.00000.0000
Skewness:0.00000.0000
Kurtosis:0.00000.0000
Q1-0.50000.0000
Median0.00000.0000
Q30.50000.0000
n1000.0000
0.0000
Basic statistics for the0.0000
standardized data0.0000
0.0000
Mean:0.00000.0000
St.dev:1.00000.0000
Skewness:0.00000.0000
Kurtosis:0.00000.0000
Q1-0.50000.0000
Median0.00000.0000
Q30.50000.0000
n1000.0000
0.0000ObservedExpectedObservedExpected
0.0000Class lowfrequencyfrequencyClass lowfrequencyfrequency
0.000000
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0.0000-1.6701.670.10-2.0001.080.05
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0.0000-1.1703.400.20-1.4003.000.15
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0.0000-0.3306.350.38-0.4007.380.37
0.0000-0.1706.620.39-0.2007.840.39
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0.00002.5000.290.023.0000.090.00
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0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
normality test
&A
Page &P
Observed
Expected
Chi Square
Observed
Expected
F-distribution
x-increment:0.9
n10(degrees of freedom)
CDFpdf
010
0.90.99989409520.0001059048
1.80.99765587750.0022382178
2.70.9876298490.0100260284
3.60.963593340.0240365091
4.50.92198589510.0416074449
5.40.86290786740.0590780276
6.30.78946028810.0734475793
7.20.70643845970.0830218284
8.10.6190696180.0873688418
90.53210359050.0869660275
9.90.449310090.0827935005
10.80.37331077130.0759993186
11.70.30563601870.0676747526
12.60.2469037330.0587322857
13.50.19704337510.0498603579
14.40.15551561570.0415277594
15.30.12150116970.0340144461
16.20.09404851720.0274526524
17.10.07218031430.0218682029
180.05496364150.0172166728
18.90.04155137250.013412269
19.80.03120212140.0103492511
20.70.02328538080.0079167405
21.60.01727720440.0060081765
22.50.01275047340.004526731
23.40.00936261740.003387856
24.30.00684269710.0025199203
25.20.0049790280.0018636691
26.10.0036079940.001371034
270.00260434030.0010036538
27.90.0018730010.0007313393
28.80.00134238240.0005306185
29.70.00095894770.0003834347
30.60.00068292010.0002760275
31.50.00048492150.0001979987
32.40.00034337150.00014155
33.30.00024249820.0001008733
34.20.00017082840.0000716698
35.10.00012005240.000050776
360.00008417610.0000358763
36.90.00005889240.0000252837
37.80.00004111730.0000177751
38.70.000028650.0000124674
39.60.00001992480.0000087251
40.50.00001383150.0000060933
41.40.00000958480.0000042467
42.30.00000663080.000002954
43.20.00000457980.000002051
44.10.00000315830.0000014215
450.00000217470.0000009836
45.90.00000149530.0000006794
46.80.00000102670.0000004686
47.70.00000070410.0000003227
48.60.00000048220.0000002219
49.50.00000032980.0000001524
50.40.00000022530.0000001045
51.30.00000015370.0000000716
52.20.00000010480.0000000489
53.10.00000007130.0000000334
540.00000004850.0000000228
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F-distribution
Chi Square(n)-distribution
t-distribution
n1:1
n2:64
CDFpdf
010
0.0250.87486489570.1251351043
0.050.82377550410.0510893915
0.0750.78507316920.0387023349
0.10.75285870420.032214465
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0.5750.4510622630.009968285
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0.6250.43211508950.0093153016
0.650.42309839030.0090166992
0.6750.41436395610.0087344342
0.70.40589687520.0084670809
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0.750.38971124610.007972243
0.7750.38196857690.0077426693
0.80.3744447870.0075237899
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0.9250.3397853240.0065650013
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0.9750.32715479350.0062342139
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1.2250.27252441330.0049025571
1.250.26773170880.0047927044
1.2750.26304541660.0046862922
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1.3250.25397908260.0044831707
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1.4750.22902064090.0039419082
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1.5250.22137900250.0037812015
1.550.21767488650.003704116
1.5750.21404578750.003629099
1.60.21048971410.0035560734
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1.650.20358904120.003415707
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1.70.19695833690.0032824737
1.7250.19373996020.0032183767
1.750.19058407760.0031558826
1.7750.18748914080.0030949369
1.80.1844536530.0030354878
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1.850.17855528330.0029208838
1.8750.17568964680.0028656365
1.90.17287794610.0028117007
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2.2750.13639610020.0021355581
2.30.13429812270.0020979775
2.3250.13223693180.0020611909
2.350.13021175490.0020251769
2.3750.12822183960.0019899153
2.40.12626645350.0019553861
2.4250.12434488320.0019215703
2.450.12245643380.0018884494
2.4750.12060042830.0018560055
2.50.11877620680.0018242215
2.5250.11698312630.0017930805
2.550.11522055980.0017625665
2.5750.1134878960.0017326638
2.60.11178453860.0017033573
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2.6750.10684456070.0016188707
2.70.10525275380.001591807
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2.7750.10063450470.0015137298
2.80.09914580320.0014887015
2.8250.09768165050.0014641527
2.850.09624157840.0014400721
2.8750.09482512940.001416449
2.90.09343185650.0013932729
2.9250.09206131740.0013705391
2.950.09071308750.0013482298
2.9750.08938676260.001326325
30.08808192470.0013048379
3.0250.0867981750.0012837497
3.050.08553512360.0012630515
3.0750.08429238890.0012427347
3.10.08306959790.001222791
3.1250.08186638570.0012032122
3.150.08068239520.0011839905
3.1750.07951727720.001165118
3.20.07837068970.0011465874
3.2250.07724229850.0011283913
3.250.07613177590.0011105226
3.2750.07503880160.0010929743
3.30.07396306190.0010757397
3.3250.07290424960.0010588123
3.350.07186206410.0010421856
3.3750.07083621080.00102585333.9909236883
3.40.06982640140.0010098094
3.4250.06883235340.000994048
3.450.06785379020.0009785632
3.4750.06689044080.0009633494
3.50.0659420340.0009484069
3.5250.06500832150.0009337125
3.550.06408904230.0009192792
3.5750.06318394670.0009050956
3.60.062292790.0008911567
3.6250.06141533220.0008774578
3.650.06055133830.0008639939
3.6750.05970057780.0008507605
3.70.05886282470.0008377531
3.7250.05803785760.0008249671
3.750.05722545940.0008123983
3.7750.0564254170.0008000424
3.80.05563752170.0007878953
3.8250.05486156870.000775953
3.850.05409735720.0007642115
3.8750.05334469030.0007526669
3.90.05260337470.0007413156
3.9250.0518732210.0007301538
3.950.05115404310.0007191778
3.9750.05044565880.0007083843
40.04974788910.0006977697
4.0250.04906055840.0006873307
4.050.04838349440.000677064
4.0750.04771652810.0006669663
4.10.04705949350.0006570346
4.1250.04641222780.0006472657
4.150.04577457120.0006376566
4.1750.04514636680.0006282044
4.20.04452746060.0006189062
4.2250.04391769810.0006097625
4.250.04331693780.0006007603
4.2750.04272503060.0005919073
4.30.04214183330.0005831973
4.3250.04156720550.0005746278
4.350.04100100930.0005661962
4.3750.04044310930.0005579
4.40.03989337250.0005497368
4.4250.03935166840.0005417041
4.450.03881786870.0005337997
4.4750.03829184740.0005260213
4.50.03777348070.0005183666
&A
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t-distribution
F(1,64)-distribution
n100
tails2(if tails = 1, tdist() returns the 1-tailed
distribution)
CDFpdf
-30.00340791530.0004784311
-2.950.00395767390.0005497586
-2.90.0045881680.0006304942
-2.850.0053098390.000721671
-2.80.00613424150.0008244025
-2.750.00707412480.0009398833
-2.70.00814351480.00106939
-2.650.00935779490.0012142802
-2.60.01073378310.0013759882
-2.550.01228981020.001556027
-2.50.01404578850.0017559783
-2.450.0160232770.0019774885
-2.40.01824553690.0022222599
-2.350.02073757970.0024920428
-2.30.0235261950.0027886153
-2.250.02663997810.0031137831
-2.20.03010932890.0034693508
-2.150.03396643770.0038571088
-2.10.03824525390.0042788163
-2.050.04298141320.0047361593
-20.04821217370.0052307604
-1.950.05397629990.0057641262
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-1.650.10208062760.0098413005
-1.60.11275331380.0106726862
-1.550.12430064860.0115473348
-1.50.13676505820.0124644096
-1.450.15018772690.0134226688
-1.40.16460817460.0144204477
-1.350.18006380710.0154556325
-1.30.19658946340.0165256563
-1.250.21421694890.0176274855
-1.20.23297456730.0187576183
-1.150.25288665490.0199120876
-1.10.27397312620.0210864713
-1.050.29624903430.0222759081
-10.31972415580.0234751215
-0.950.34440260550.0246784497
-0.90.3702824890.0258798834
-0.850.39735559840.0270731094
-0.80.42560716060.0282515623
-0.750.45501564040.0294084798
-0.70.48555260650.030536966
-0.650.51718266390.0316300574
-0.60.54986345760.0326807937
-0.550.58354574860.033682291
-0.50.61817356590.0346278173
-0.450.65368443370.0355108678
-0.40.69000967440.0363252407
-0.350.72707478440.0370651099
-0.30.76479988030.037725096
-0.250.80310021220.0383003319
-0.20.84188673620.038786524
-0.150.88106674280.0391800066
-0.10.92054453110.0394777883
-0.050.96022212170.0396775906
010.0397778783
0.050.96022212170.0397778783
0.10.92054453110.0396775906
0.150.88106674280.0394777883
0.20.84188673620.0391800066
0.250.80310021220.038786524
0.30.76479988030.0383003319
0.350.72707478440.037725096
0.40.69000967440.0370651099
0.450.65368443370.0363252407
0.50.61817356590.0355108678
0.550.58354574860.0346278173
0.60.54986345760.033682291
0.650.51718266390.0326807937
0.70.48555260650.0316300574
0.750.45501564040.030536966
0.80.42560716060.0294084798
0.850.39735559840.0282515623
0.90.3702824890.0270731094
0.950.34440260550.0258798834
10.31972415580.0246784497
1.050.29624903430.0234751215
1.10.27397312620.0222759081
1.150.25288665490.0210864713
1.20.23297456730.0199120876
1.250.21421694890.0187576183
1.30.19658946340.0176274855
1.350.18006380710.0165256563
1.40.16460817460.0154556325
1.450.15018772690.0144204477
1.50.13676505820.0134226688
1.550.12430064860.0124644096
1.60.11275331380.0115473348
1.650.10208062760.0106726862
1.70.09223932710.0098413005
1.750.08318568320.0090536439
1.80.07487589270.0083097905
1.850.06726632940.0076095634
1.90.06031390930.00695242
1.950.05397629990.0063376094
20.04821217370.0057641262
2.050.04298141320.0052307604
2.10.03824525390.0047361593
2.150.03396643770.0042788163
2.20.03010932890.0038571088
2.250.02663997810.0034693508
2.30.0235261950.0031137831
2.350.02073757970.0027886153
2.40.01824553690.0024920428
2.450.0160232770.0022222599
2.50.01404578850.0019774885
2.550.01228981020.0017559783
2.60.01073378310.001556027
2.650.00935779490.0013759882
2.70.00814351480.0012142802
2.750.00707412480.00106939
2.80.00613424150.0009398833
2.850.0053098390.0008244025
2.90.0045881680.000721671
2.950.00395767390.0006304942
30.00340791530.0005497586
3.050.00292948420.0004784311
&A
Page &P
t(n)-distribution
-
Tests of equality of group meansThe tests of equality of the
group means indicate that for the three first predictor variables
there does not seem to be any significant difference in group
meansF-values < 3.99, the 5 % critical value for
F(1,64)Significance > 0.05 The result is confirmed by the Wilks
lambda values close to 1As the results indicate low univariate
discriminant power for these variables, some or all of them may be
excluded from analysis in order to get a parsimonious model
-
Pooled Within-Groups Correlation Matrix
CA/TA(C-C)/TACA/LoaRes/LoaNI/TANI/TECNI/LoaCS/SalCF/LoaCA/TA1,000(C-C)/TA,7601,000CA/Loa,917,6411,000Res/Loa,013-,230,0991,000NI/TA,038-,007,058,1741,000NI/TEC-,023-,016-,035,033,9561,000NI/Loa,048-,015,072,194,999,9471,000CS/Sal-,087-,147-,104-,288-,565-,419-,5701,000CF/Loa-,007-,013,014,116,223,181,225-,3721,000
-
Correlations between predictor variablesThe variables Current
assets/Total assets and Current assets/Loans are highly correlated
(Corr = 0,917)The variables explain the same variation in the
dataIncluding both the variables in the discriminant function does
not improve the explanation power but may lead to multicollinearity
problem in estimationOnly one of the variables should be selected
into the set of explanatory variablesFor the same reason, only one
of the variables Net income/Total assets, Net income/Total equity
capital and Net income/Loans should be selected
-
Summary of Canonical Discriminant Functionsa. First 1 canonical
discriminant functions were used in the analysis.EigenvaluesWilks
Lambda
FunctionEigenvalue% of VarianceCumulative %Canonical
Correlation1,417a100,0100,0,542
Test of Function(s)Wilks
LambdaChi-squaredfSig.1,70620,8998,007
-
Canonical Discriminant Function CoefficientsRelative
contribution of each variable to discriminant function
Function
1StandardizedUnstandardizedCA/TA-1,318-11,825(CA-Cash)/TA,6256,940CA/Loans,6124,601Reserves/Loans-,228-5,510NI/TA1,13485,998NI/TEC-1,264-4,456CofS/Sales,7805,884CF/Loans-,180-7,864Constant-3,957
-
Functions at group centroidsUnstandardized canonical
discriminant functions evaluated at group means
ClassFunction 10-,5631,718
-
Example of classifying an observation by the canonical
discrimiant functionDistance to group centroid for Group 1
(Failed), 0,718, smaller than distance to group centroid for Group
0 (Survived), -0,563 Classification to the closest group Failed
Obs.
1Coeff.ScoreConstant-3,957-3,957CA/TA0.4611-11,825-5,453CA_Cash/TA0.38376,9402,663CA/Loans0.48944,6012,252Res/Loans0.0077-5,510-0,042NI/TA0.005785,9980,490NI/TEC0.0996-4,456-0,444CofS/Sales0.87995,8845,177CF/Loans0.0092-7,864-0,072Total
Score0,614
-
Fishers discriminant function coefficients
SurvivedFailedConstant -66,485 -71,653CA/TA 15,352,207CA_Cash/TA
82,320912,208CA/Loans -29,866-23,973Res/Loans 81,189
74,071NI/TA2006,853 2116.987NI/TEC-65,300-71,007CofS/Sales
126,771134,307CF/Loans185,726175,654
-
Example on classifying an observation by Fishers discriminant
functionsLarger score Classification: Failed
Obs. 1SurvivedScoreFailedScoreConstant -66,485
-66,485-71,653-71,653CA/TA0.4611
15,3527,079,2070,095CA_Cash/TA0.3837
82,32031,586912,20834,997CA/Loans0.4894
-29,866-14,616-23,973-11,732Res/Loans0.0077 81,189
0,62574,0710,570NI/TA0.00572006,853
11,4392116.98712,067NI/TEC0.0996-65,300-6,054-71,007-7,072CofS/Sales0.8799
126,771111,546134,307118,177CF/Loans0.0092185,7261,709175,6541,616Total
Score76,37877,064
-
Confusion matrix Classification results
Predicted classSurvivedFailedTrue classSurvived28975,7 %24,3
%Failed42513,8 %86,2 %
-
Summary of classifications with different classification
methods(Estimation sample)
Case Presentation
The bankruptcy prediction data by Olmeda and Fernandez
(1997)
Reference:Olmeda, Ignacio and Fernndez, Eugenio: "Hybrid
classifiers for financial
Sample66 Spanish banks
VariablesSurvived (0)/Failed (1)Class
Current assets/Total assetsCA/TA
(Current assets-Cash)/Total assets(CA-Cash)/TA
Current assets/LoansCA/Loans
Reserves/LoansReserves/Loans
Net income/Total assetsNI/TA
Net incomne/Total equity capitalNI/TEC
Net income/LoansNI/Loans
Cost of sales/SalesCofS/Sales
Cash flow/LoansCF/Loans
MethodsRecursive Partitioning AlgorithmRPA
Multivariate Linear Discriminant AnalysisMDA
Multivariate Quadratic Discriminant AnalysisMDA-Q
MDA-W
Logistic RegressionLogR
Linear Programming model by Freed & GloverLP
Quadratic Programming model by Freed & GloverLP-Q
Linear Programming model by GochetLPG
Quadratic Programming model by GochetLPGQ
Supervised Kohonen Neural NetKohonen
Data
ClassCA/TA(CA-Cash)/TACA/LoansReserves/LoansNI/TANI/TECNI/LoansCofS/SalesCF/Loans
10.46110.38370.48940.00770.00570.09960.00610.87990.0092
10.47680.27620.50210.00840.00240.04790.00250.93630.0039
00.44970.26760.46060.01200.00550.23590.00570.80290.0141
10.28360.14470.30000.01710.00260.04800.00280.94280.0055
10.32620.22220.33690.01120.00140.04300.00140.96960.0023
00.33580.22110.34690.02830.01140.35660.01170.71880.0210
00.30720.25500.31870.02440.00180.05000.00190.91940.0065
00.34470.25700.35730.01490.00680.19260.00710.85100.0111
00.40740.27640.43970.01560.00430.05820.00480.88460.0095
00.37470.19700.39560.03210.00700.13240.00740.86040.0100
00.39300.20190.40620.02030.00850.26390.00880.86130.0113
10.37400.22050.39380.0012-0.0537-1.0671-0.05651.3713-0.0231
00.23890.15340.25340.29900.01710.29880.01820.69520.0204
00.55700.38620.60580.02640.01210.15050.01320.77110.0201
00.42980.30120.04430.01930.00630.21110.00650.80670.0144
00.31260.24190.32510.03210.00390.10080.00400.84050.0108
10.41790.32400.43230.00580.00400.11990.00410.89550.0081
10.28380.24490.30750.00640.00080.01070.00090.92830.0106
00.46080.34550.48060.03370.01290.31340.01350.79570.0172
10.21770.17220.22590.01890.00210.05660.00210.92490.0049
10.22800.14540.23400.0054-0.0026-0.1008-0.00261.1019-0.0012
00.35980.31710.37920.03060.00830.16300.00880.86510.0108
10.47520.38670.49600.02520.00180.04400.00190.95250.0045
10.57910.49420.60500.01620.00000.00000.00000.97350.0026
00.43850.30730.46280.04980.00760.14510.00800.83650.0119
00.39650.31160.41500.00440.00180.04020.00190.92910.0047
00.28990.14590.33200.08140.00960.07540.01090.74430.0104
00.40020.26910.41380.02330.00740.22470.00760.82920.0136
10.21840.17620.22870.00770.00000.00000.00000.88080.0078
10.32500.22320.33890.02690.00150.03670.00160.92820.0050
10.23670.17250.24350.00920.00220.08000.00230.95760.0030
10.43110.32840.45050.00430.00530.12390.00560.86020.0070
00.38450.21460.39500.02030.00500.18940.00520.83960.0128
00.33470.20090.34230.01050.00320.14570.00330.89000.0067
00.54850.37230.58510.03540.01840.29500.01970.71980.0307
10.30350.22530.31510.01170.00230.06220.00240.89960.0065
10.44110.33840.45540.01120.00080.02410.00080.93860.0046
00.36670.23660.37670.01740.00510.19200.00520.85430.0113
10.39230.27340.40830.00970.00000.00000.00000.97140.0021
10.30600.19490.31320.01010.00110.09340.00220.92580.0048
00.46480.31890.48250.02610.00670.18230.00700.88100.0096
00.47100.37070.52220.03410.00840.08540.00930.82790.0166
10.44810.23460.49080.02490.00040.00450.00040.99470.0004
10.25680.17160.27800.04610.00240.03140.00260.89960.0065
10.59680.48930.64980.02430.00260.03180.00280.69530.0032
00.45710.28710.47310.02340.00320.09450.00330.88370.0095
00.63120.08830.77690.19860.02260.12060.02780.70560.0282
10.32370.27540.34550.01000.00030.00400.00030.91440.0051
10.29800.24550.31010.01210.00360.09100.00370.92470.0037
00.44020.33850.45620.02290.00430.12220.00450.87700.0104
00.43220.27230.44530.01810.00480.16330.00500.87600.0093
00.22870.08940.23960.02510.01420.31460.01490.74970.0153
00.31300.23450.32460.00670.00410.11600.00430.85850.0010
00.33730.21140.35310.03480.00340.07610.00360.79330.1830
00.22250.15560.22860.01300.00300.11190.00310.87030.0071
00.45590.29310.49030.02900.00620.08780.00660.85680.0123
10.35770.28970.37530.0243-0.0822-1.7434-0.08621.2671-0.0182
10.29110.22030.29930.01010.02230.81090.02291.20400.0126
10.55830.24540.59470.0203-0.0135-0.0221-0.01441.3492-0.0144
10.37230.19720.38740.01420.00290.07340.00300.92650.0041
00.30590.20810.31440.01590.00300.11100.00310.90600.0061
10.46260.34900.50940.05670.00570.06210.00630.34780.0511
00.62280.45720.65930.04700.01240.22370.01310.75840.0237
00.50420.35320.52840.03450.01050.22980.01100.78220.0202
00.39350.26720.41030.02300.00820.19950.00850.79770.0155
00.76710.48910.80700.02120.00510.10290.00540.89710.0106
MCA-results
Summary over classifications (Holdout sample)
MethodCorrect classErrorsTotal numberPercents
SWNECorrectSWNE
RPA27233284.38%6.25%9.38%
MDA25433278.13%12.50%9.38%
MDA-Q20753262.50%21.88%15.63%
MDA-W25523278.13%15.63%6.25%
LogR28313287.50%9.38%3.13%
LP24533275.00%15.63%9.38%
LP-Q21743265.63%21.88%12.50%
LPG25433278.13%12.50%9.38%
LPGQ21743265.63%21.88%12.50%
Kohonen164123250.00%12.50%37.50%
Summary over classifications (Estimation sample)
MethodCorrect classErrorsTotal numberPercents
SWNECorrectSWNE
RPA30133488.24%2.94%8.82%
MDA30043488.24%0.00%11.76%
MDA-Q31033491.18%0.00%8.82%
MDA-W31033491.18%0.00%8.82%
LogR33013497.06%0.00%2.94%
LP28153482.35%2.94%14.71%
LP-Q340034100.00%0.00%0.00%
LPG33013497.06%0.00%2.94%
LPGQ340034100.00%0.00%0.00%
Kohonen24373470.59%8.82%20.59%
Coefficients for the classification models
=Constant
=CA/TA
=CA_Cash/TA
=CA/Loans
=Res/Loans
=NI/TA
=NI/TEC
=NI/Loans
=CofS/Sales
=CF/Loans
Multiple Discriminant Analysis with Equal Prior
Probabilities
Class
0-758.24248.5889.800-18.031351.432-246563.2774.368223681.31499.65914625.844
1-758.80034.57223.506-16.947342.204-236546.7740.035214974.01505.54714245.368
Multiple Discriminant Analysis with Estimated Prior
Probabilities
Class
0-758.13148.5889.800-18.031351.432-24
