modelAccuracy

Compute RMSE of predicted and observed PDs on grouped data

Syntax

[AccMeasure,AccData] = modelAccuracy(pdModel,data,GroupBy)

AccMeasure,AccData = modelAccuracy(___,Name,Value)

Description

[AccMeasure,AccData] = modelAccuracy(pdModel,data,GroupBy) computes the root mean squared error (RMSE) of the observed compared to the predicted probabilities of default (PD). GroupBy is required and can be any column in the data input (not necessarily a model variable). The model Accuracy function computes the observed PD as the default rate of each group and the predicted PD as the average PD for each group. modelAccuracy supports comparison against a reference model.

AccMeasure,AccData = modelAccuracy(___,Name,Value) specifies options using one or more name-value pair arguments in addition to the input arguments in the previous syntax.

Examples

Compute Model Accuracy for Logistic Lifetime PD Model

This example shows how to use fitLifetimePDModel to fit data with a Logistic model and then use modelAccuracy to compute the root mean squared error (RMSE) of the observed probabilities of default (PDs) with respect to the predicted PDs.

Load Data

Load the credit portfolio data.

load RetailCreditPanelData.mat
disp(head(data))

    ID    ScoreGroup    YOB    Default    Year
    __    __________    ___    _______    ____

    1      Low Risk      1        0       1997
    1      Low Risk      2        0       1998
    1      Low Risk      3        0       1999
    1      Low Risk      4        0       2000
    1      Low Risk      5        0       2001
    1      Low Risk      6        0       2002
    1      Low Risk      7        0       2003
    1      Low Risk      8        0       2004

disp(head(dataMacro))

    Year     GDP     Market
    ____    _____    ______

    1997     2.72      7.61
    1998     3.57     26.24
    1999     2.86      18.1
    2000     2.43      3.19
    2001     1.26    -10.51
    2002    -0.59    -22.95
    2003     0.63      2.78
    2004     1.85      9.48

Join the two data components into a single data set.

data = join(data,dataMacro);
disp(head(data))

    ID    ScoreGroup    YOB    Default    Year     GDP     Market
    __    __________    ___    _______    ____    _____    ______

    1      Low Risk      1        0       1997     2.72      7.61
    1      Low Risk      2        0       1998     3.57     26.24
    1      Low Risk      3        0       1999     2.86      18.1
    1      Low Risk      4        0       2000     2.43      3.19
    1      Low Risk      5        0       2001     1.26    -10.51
    1      Low Risk      6        0       2002    -0.59    -22.95
    1      Low Risk      7        0       2003     0.63      2.78
    1      Low Risk      8        0       2004     1.85      9.48

Partition Data

Separate the data into training and test partitions.

nIDs = max(data.ID);
uniqueIDs = unique(data.ID);

rng('default'); % For reproducibility
c = cvpartition(nIDs,'HoldOut',0.4);

TrainIDInd = training(c);
TestIDInd = test(c);

TrainDataInd = ismember(data.ID,uniqueIDs(TrainIDInd));
TestDataInd = ismember(data.ID,uniqueIDs(TestIDInd));

Create Logistic Lifetime PD Model

Use fitLifetimePDModel to create a Logistic model using the training data.

pdModel = fitLifetimePDModel(data(TrainDataInd,:),"Logistic",...
    'AgeVar','YOB',...
    'IDVar','ID',...
    'LoanVars','ScoreGroup',...
    'MacroVars',{'GDP','Market'},...
    'ResponseVar','Default');
 disp(pdModel)

  Logistic with properties:

        ModelID: "Logistic"
    Description: ""
          Model: [1x1 classreg.regr.CompactGeneralizedLinearModel]
          IDVar: "ID"
         AgeVar: "YOB"
       LoanVars: "ScoreGroup"
      MacroVars: ["GDP"    "Market"]
    ResponseVar: "Default"

Display the underlying model.

disp(pdModel.Model)

Compact generalized linear regression model:
    logit(Default) ~ 1 + ScoreGroup + YOB + GDP + Market
    Distribution = Binomial

Estimated Coefficients:
                               Estimate        SE         tStat       pValue   
                              __________    _________    _______    ___________

    (Intercept)                  -2.7422      0.10136    -27.054     3.408e-161
    ScoreGroup_Medium Risk      -0.68968     0.037286    -18.497     2.1894e-76
    ScoreGroup_Low Risk          -1.2587     0.045451    -27.693    8.4736e-169
    YOB                         -0.30894     0.013587    -22.738    1.8738e-114
    GDP                         -0.11111     0.039673    -2.8006      0.0051008
    Market                    -0.0083659    0.0028358    -2.9502      0.0031761


388097 observations, 388091 error degrees of freedom
Dispersion: 1
Chi^2-statistic vs. constant model: 1.85e+03, p-value = 0

Compute Model Accuracy

Model accuracy measures how accurate the predicted probabilities of default are. For example, if the model predicts a 10% PD for a group, does the group end up showing an approximate 10% default rate, or is the eventual rate much higher or lower? While model discrimination measures the risk ranking only, model accuracy measures the accuracy of the predicted risk levels.

modelAccuracy computes the root mean squared error (RMSE) of the observed PDs with respect to the predicted PDs. A grouping variable is required and it can be any column in the data input (not necessarily a model variable). The modelAccuracy function computes the observed PD as the default rate of each group and the predicted PD as the average PD for each group.

DataSetChoice = "Training";
 if DataSetChoice=="Training"
    Ind = TrainDataInd;
else
   Ind = TestDataInd;
 end

GroupingVar = "YOB";
[AccMeasure,AccData] = modelAccuracy(pdModel,data(Ind,:),GroupingVar,'DataID',DataSetChoice);
disp(AccMeasure) 

                                            RMSE   
                                          _________

    Logistic, grouped by YOB, Training    0.0004142

disp(AccData) 

     ModelID      YOB       PD    
    __________    ___    _________

    "Observed"     1      0.017421
    "Observed"     2      0.012305
    "Observed"     3      0.011382
    "Observed"     4      0.010741
    "Observed"     5       0.00809
    "Observed"     6     0.0066747
    "Observed"     7     0.0032198
    "Observed"     8     0.0018757
    "Logistic"     1      0.017185
    "Logistic"     2      0.012791
    "Logistic"     3       0.01131
    "Logistic"     4      0.010615
    "Logistic"     5     0.0083982
    "Logistic"     6     0.0058744
    "Logistic"     7     0.0035872
    "Logistic"     8     0.0023689

Visualize the model accuracy.

AccDataUnstacked = unstack(AccData,"PD","ModelID");
plot(AccDataUnstacked.(GroupingVar),AccDataUnstacked.(pdModel.ModelID),'-o')
hold on
plot(AccDataUnstacked.(GroupingVar),AccDataUnstacked.Observed,'*')
hold off
title(strcat(AccMeasure.Properties.RowNames,", RMSE = ",num2str(AccMeasure.RMSE)))
xlabel(GroupingVar)
ylabel('PD')
legend(pdModel.ModelID,"Observed")
grid on

You can use more than one variable for grouping. For this example, group by the variables YOB and ScoreGroup.

[AccMeasure,AccData] = modelAccuracy(pdModel,data(Ind,:),["YOB","ScoreGroup"],'DataID',DataSetChoice);
disp(AccMeasure)

                                                         RMSE   
                                                      __________

    Logistic, grouped by YOB, ScoreGroup, Training    0.00066239

disp(tail(AccData))

     ModelID      YOB    ScoreGroup        PD    
    __________    ___    ___________    _________

    "Logistic"     6     Medium Risk    0.0050296
    "Logistic"     6     Low Risk       0.0028847
    "Logistic"     7     High Risk      0.0061491
    "Logistic"     7     Medium Risk    0.0030905
    "Logistic"     7     Low Risk       0.0017571
    "Logistic"     8     High Risk      0.0040759
    "Logistic"     8     Medium Risk    0.0020492
    "Logistic"     8     Low Risk       0.0011611

Now visualize the two grouping variables.

AccDataUnstacked = unstack(AccData,"PD","ModelID");
ScoreGroupLevels = categories(AccDataUnstacked.ScoreGroup);
c = cell(3,1);
figure;
hold on
for ii=1:length(ScoreGroupLevels)
   IndScore = AccDataUnstacked.ScoreGroup==ScoreGroupLevels{ii};
   h = plot(AccDataUnstacked.YOB(IndScore),AccDataUnstacked.(pdModel.ModelID)(IndScore),'-o');
   c{ii} = h.Color;
end
Legend1 = strcat(ScoreGroupLevels,", predicted");
for ii=1:length(ScoreGroupLevels)
   IndScore = AccDataUnstacked.ScoreGroup==ScoreGroupLevels{ii};
   plot(AccDataUnstacked.YOB(IndScore),AccDataUnstacked.Observed(IndScore),'*','Color',c{ii})
end
Legend2 = strcat(ScoreGroupLevels,", observed");
hold off

title(strcat(AccMeasure.Properties.RowNames,", RMSE = ",num2str(AccMeasure.RMSE)))
xlabel('YOB')
ylabel('PD')
legend([Legend1;Legend2])
grid on

Input Arguments

pdModel — Probability of default model

Logistic object | Probit object

Probability of default model, specified as a Logistic or Probit object previously created using fitLifetimePDModel.

Note

The 'ModelID' property of the pdModel object is used as the identifier or tag for pdModel.

Data Types: object

data — Data

table

Data, specified as a NumRows-by-NumCols table with projected predictor values to make lifetime predictions. The predictor names and data types must be consistent with the underlying model.

Data Types: table

GroupBy — Name of column in data input used to group the data

string | character vector

Name of column in the data input used to group the data, specified as a string or character vector. GroupBy does not have to be a model variable name. For each group designated by GroupBy, the modelAccuracy function computes the observed default rates and average predicted PDs are computed to measure the RMSE.

Data Types: string | char

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: [AccMeasure,AccData] = modelAccuracy(pdModel,data(Ind,:),'GroupBy',["YOB","ScoreGroup"],'DataID',DataSetChoice)

'DataID' — Data set identifier

" " (default) | character vector | string

Data set identifier, specified as the comma-separated pair consisting of 'DataID' and a character vector or string. DataID is included in the modelAccuracy output for reporting purposes.

Data Types: char | string

'ReferencePD' — Conditional PD values predicted for data by reference model

[ ] (default) | character vector | string

Conditional PD values predicted for data by the reference model, specified as the comma-separated pair consisting of 'ReferencePD' and a NumRows-by-1 numeric vector. The functions reports the modelAccuracy output information for both the pdModel object and the reference model.

Data Types: char | string

'ReferenceID' — Identifier for reference model

" " (default) | character vector | string

Identifier for the reference model, specified as the comma-separated pair consisting of 'ReferenceID' and a character vector or string. ReferenceID is used in the modelAccuracy output for reporting purposes.

Data Types: char | string

Output Arguments

AccMeasure — RMSE values

table

Accuracy measure, returned as a table.

RMSE values, returned as a single-column 'RMSE' table. The table has one row if only the pdModel accuracy is measured and it has two rows if reference model information is given. The row names of AccMeasure report the model IDs, grouping variables, and data ID.

Note

The reported RMSE values depend on the grouping variable for the required GroupBy argument.

AccData — Observed and predicted PD values for each group

table

Accuracy data, returned as a table.

Observed and predicted PD values for each group, returned as a table. The reported observed PD values correspond to the observed default rate for each group. The reported predicted PD values are the average PD values predicted by the pdModel object for each group, and similarly for the reference model. The modelAccuracy function stacks the PD data, placing the observed values for all groups first, then the predicted PDs for the pdModel, and then the predicted PDs for the reference model, if given.

The column 'ModelID' identifies which rows correspond to the observed PD, pdModel, or reference model. The table also has one column for each grouping variable showing the unique combinations of grouping values. The last column of AccData is a 'PD' column with the PD data.

More About

Model Accuracy

Model accuracy measures the accuracy of the predicted probability of default (PD) values.

To measure model accuracy, also called model calibration, you must compare the predicted PD values to the observed default rates. For example, if a group of customers is predicted to have an average PD of 5%, then is the observed default rate for that group close to 5%?

The modelAccuracy function requires a grouping variable to compute average predicted PD values within each group and the average observed default rate also within each group. modelAccuracy uses the root mean squared error (RMSE) to measure the deviations between the observed and predicted values across groups. For example, the grouping variable could be the calendar year, so that rows corresponding to the same calendar year are grouped together. Then, for each year the software computes the observed default rate and the average predicted PD. The modelAccuracy function then applies the RMSE formula to obtain a single measure of the prediction error across all years in the sample.

Suppose there are N observations in the data set, and there are M groups G₁,…,G_M. The default rate for group G_i is

$$D{R}_i=\frac{D_i}{N_i}$$

where:

D_i is the number of defaults observed in group G_i.

N_i is the number of observations in group G_i.

The average predicted probability of default PD_i for group G_i is

$$P{D}_i=\frac{1}{N_i}{\displaystyle {\sum }_{j\in G_i}PD(j)}$$

where PD(j) is the probability of default for observation j. In other words, this is the average of the predicted PDs within group G_i.

Therefore, the RMSE is computed as

$$RMSE\text{}=\sqrt{{\displaystyle {\sum }_{i=1}^M\left(\frac{N_i}{N}\right){(D{R}_i-P{D}_i)}^2}}$$

The RMSE, as defined, depends on the selected grouping variable. For example, grouping by calendar year and grouping by years-on-books might result in different RSME values.

References

[1] Baesens, Bart, Daniel Roesch, and Harald Scheule. Credit Risk Analytics: Measurement Techniques, Applications, and Examples in SAS. Wiley, 2016.

[2] Bellini, Tiziano. IFRS 9 and CECL Credit Risk Modelling and Validation: A Practical Guide with Examples Worked in R and SAS. San Diego, CA: Elsevier, 2019.

[3] Breeden, Joseph. Living with CECL: The Modeling Dictionary. Santa Fe, NM: Prescient Models LLC, 2018.