financial classification models – part i: discriminant analysis advanced financial accounting ii...
TRANSCRIPT
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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
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0.05399096650.0000014867
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0.16038332730.0000310414
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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
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0.01983735440.342943855
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0.01191224360.3752403469
0.01042093480.3813878155
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0.00791545160.391042694
0.00687276670.3944793309
0.00595253240.3969525475
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0.00326681910.3969525475
0.00279425840.3944793309
0.00238408820.391042694
0.00202904810.3866681168
0.00172256890.3813878155
0.00145873080.3752403469
0.00123221920.3682701403
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0.00061190190.3332246029
0.0005104650.3229723597
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0.00029194690.2896915528
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Class 1
Class 2
Score
Normal
mean03
stdev11
ScoreClass 1Class 2
-30.00443184840.0000000061
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-0.650.32297235970.000510465
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-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
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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
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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
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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
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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
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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
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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
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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
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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
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F(1,64)-distribution5% critical value = 3.99CA/TA 2.072CA/Loans 0.981(CA-Cash)/TA 0.027
Chart2
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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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0.0000-1.1703.400.20-1.4003.000.15
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0.0000-0.1706.620.39-0.2007.840.39
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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
0.1250.72483562340.0280230808
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1.2250.27252441330.0049025571
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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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2.7750.10063450470.0015137298
2.80.09914580320.0014887015
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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
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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
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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
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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
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-2.30.0235261950.0027886153
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-2.20.03010932890.0034693508
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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
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t(n)-distribution
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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
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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
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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
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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
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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
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Confusion matrix Classification results
Predicted classSurvivedFailedTrue classSurvived28975,7 %24,3 %Failed42513,8 %86,2 %
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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