risk.validation.migrationMatrixStabilityTest

Migration matrix stability test

Since R2026a

Syntax

hMMSTest = risk.validation.migrationMatrixStabilityTest(CountMatrix)

hMMSTest = risk.validation.migrationMatrixStabilityTest(CountMatrix,ConfidenceLevel=confidenceLevel)

[hMMSTest,Output] = risk.validation.migrationMatrixStabilityTest(___)

Description

hMMSTest = risk.validation.migrationMatrixStabilityTest(CountMatrix) returns the migration matrix stability (MMS) test result hMMSTest for a count matrix CountMatrix. The output is 1 if the test rejects the null hypothesis at the 95% confidence level, or 0 if it does not reject the null hypothesis.

hMMSTest = risk.validation.migrationMatrixStabilityTest(CountMatrix,ConfidenceLevel=confidenceLevel) specifies the confidence level for the MMS test.

[hMMSTest,Output] = risk.validation.migrationMatrixStabilityTest(___) also returns a structure Outputs that contains summary metrics.

example

Examples

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Apply Migration Matrix Stability Test to Probability of Default Data

Open Live Script

Perform a migration matrix stability test on the matrix Grades, which contains counts for probability of default (PD) grade migrations. The migration matrix stability test tests the null hypothesis that migration frequencies closer to the diagonal are greater than those further from the diagonal. The rows of Grades correspond to grades at the beginning of the observation period and the columns correspond to grades at the end.

Load the matrix.

load Grades.mat
Grades

Grades = 14×15

     1      11      21      31      41      51      61     71     81     91    101    111    121    131    141
    89     120      81      14       0       0       0      0      0      0      0      0      0      0      0
    24      81     114      81      15       0       0      0      0      0      0      0      0      0      0
     2      19      75     132      69      21       4      1      0      0      0      0      0      0      0
     0       0      19      89     100      66      32      1      0      0      0      0      0      0      0
     0       0       1      18      83     114      74     18      4      0      0      0      0      0      0
     0       0       0       4      27      61     107     73     13      0      0      0      0      0      0
     0       0       0       0       0      15      62    105     68     16      2      0      0      0      0
     0       0       0       0       0       1      10     84    102     75     14      1      0      0      0
     0       0       0       0       0       0       4     13     57    127     68     32      1      0      0
     0       0       0       0       0       0       0      1     14     85    122     83     26      0      0
     0       0       0       0       0       0       0      0      1     16     71    103     80     16      3
     0    1500    1400    1300    1200    1100    1000    900    800    700    600    500    400    300    200
     0       0       0       0       0       0       0      0      0      0      2     14     67    114    106
      ⋮

The element at position (3,5), for example, indicates that 15 customers migrated from rating grade 3 to rating grade 5.

Perform an MMS test using the counts in Grades.

[hMMSTest,Output] = risk.validation.migrationMatrixStabilityTest(Grades)

hMMSTest = 14×14

    NaN      1      1      0      0      0      0      0      0      0      0      0      0      0
      0    NaN      0      0      0    NaN    NaN    NaN    NaN    NaN    NaN    NaN    NaN    NaN
      0      0    NaN      0      0      0    NaN    NaN    NaN    NaN    NaN    NaN    NaN    NaN
      0      0      0    NaN      0      0      0      0      0    NaN    NaN    NaN    NaN    NaN
    NaN      0      0      0    NaN      0      0      0      0    NaN    NaN    NaN    NaN    NaN
    NaN      0      0      0      0    NaN      0      0      0      0    NaN    NaN    NaN    NaN
    NaN    NaN      0      0      0      0    NaN      0      0      0    NaN    NaN    NaN    NaN
    NaN    NaN    NaN    NaN      0      0      0    NaN      0      0      0      0    NaN    NaN
    NaN    NaN    NaN    NaN      0      0      0      0    NaN      0      0      0      0    NaN
    NaN    NaN    NaN    NaN    NaN      0      0      0      0    NaN      0      0      0      0
    NaN    NaN    NaN    NaN    NaN    NaN      0      0      0      0    NaN      0      0      0
    NaN    NaN    NaN    NaN    NaN    NaN    NaN      0      0      0      0    NaN      0      0
      0      1      1      1      1      1      1      1      1      1      1      1    NaN      0
    NaN    NaN    NaN    NaN    NaN    NaN    NaN    NaN    NaN      0      0      0      0    NaN
      ⋮

Output = struct with fields:
             RejectTest: [14×14 double]
           PValueMatrix: [14×14 double]
    TestStatisticMatrix: [14×14 double]
          CriticalValue: -1.6449
        MigrationMatrix: [14×15 double]

The matrix hMMSTest contains the results of the MMS test. Above the main diagonal, an value of 1 indicates enough evidence exists to reject the null hypothesis that the corresponding probability in the migration matrix is significantly smaller than the probability in the same row of the previous column. Below the main diagonal, a value of 1 indicates enough evidence exists to reject the null hypothesis that the corresponding probability in the migration matrix is significantly smaller than the probability in the same row of the following column. NaN entries indicate that the test statistic is NaN.

The Output structure contains values that the migrationMatrixTest function uses to perform the hypothesis test, including p-values, test statistics, and the migration matrix.

Display the migration matrix.

Output.MigrationMatrix

ans = 14×15

    0.0009    0.0103    0.0197    0.0291    0.0385    0.0479    0.0573    0.0667    0.0761    0.0854    0.0948    0.1042    0.1136    0.1230    0.1324
    0.2928    0.3947    0.2664    0.0461         0         0         0         0         0         0         0         0         0         0         0
    0.0762    0.2571    0.3619    0.2571    0.0476         0         0         0         0         0         0         0         0         0         0
    0.0062    0.0588    0.2322    0.4087    0.2136    0.0650    0.0124    0.0031         0         0         0         0         0         0         0
         0         0    0.0619    0.2899    0.3257    0.2150    0.1042    0.0033         0         0         0         0         0         0         0
         0         0    0.0032    0.0577    0.2660    0.3654    0.2372    0.0577    0.0128         0         0         0         0         0         0
         0         0         0    0.0140    0.0947    0.2140    0.3754    0.2561    0.0456         0         0         0         0         0         0
         0         0         0         0         0    0.0560    0.2313    0.3918    0.2537    0.0597    0.0075         0         0         0         0
         0         0         0         0         0    0.0035    0.0348    0.2927    0.3554    0.2613    0.0488    0.0035         0         0         0
         0         0         0         0         0         0    0.0132    0.0430    0.1887    0.4205    0.2252    0.1060    0.0033         0         0
         0         0         0         0         0         0         0    0.0030    0.0423    0.2568    0.3686    0.2508    0.0785         0         0
         0         0         0         0         0         0         0         0    0.0034    0.0552    0.2448    0.3552    0.2759    0.0552    0.0103
         0    0.1261    0.1176    0.1092    0.1008    0.0924    0.0840    0.0756    0.0672    0.0588    0.0504    0.0420    0.0336    0.0252    0.0168
         0         0         0         0         0         0         0         0         0         0    0.0066    0.0462    0.2211    0.3762    0.3498
      ⋮

The migration matrix contains probabilities for customer migration from one state to another. The software uses these probabilities to calculate the test statistics for the hypothesis tests. For more information about the MMS test, see More About.

Input Arguments

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`CountMatrix` — Count matrix
wide matrix of nonnegative values | wide matrix of nonnegative values

Count matrix, specified as a wide matrix of nonnegative values. The elements of the matrix are counts for rating grade migrations and status migrations. The rating grades and statuses can belong to any ordinal rating system. Examples of ordinal rating systems include credit rating grades, loss given default (LGD) rating grades, probability of default (PD) rating grades, and mortgage delinquency states.

The rows of CountMatrix correspond to the grade at the beginning of the observation period and the columns correspond to the grade at the end. In other words, the element at position i,j represents the number of customers whose grade was i at the start of the observation period and j at then end of the observation period.

Data Types: single | double

`confidenceLevel` — Confidence level
`0.95` (default) | numeric scalar in the range (0,1)

Confidence level of the hypothesis test, specified as a numeric scalar in the range (0,1).

Output Arguments

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`hMMSTest` — Hypothesis test results
numeric matrix containing values of `1`, `0`, or `NaN`

Hypothesis test results, returned as a numeric matrix. Each element in hMMSTest corresponds to the element in CountMatrix at the same position.

A value of 1 rejects the null hypothesis at the specified confidence level.
A value of 0 fails to reject the null hypothesis at the specified confidence level.
A value of NaN indicates that the corresponding test statistic is NaN.

`Output` — Output metrics
structure

Output metrics, returned as a structure the following fields:

RejectTest — Numeric matrix indicating whether each null hypothesis was rejected. Each element in RejectTest corresponds to the element in CountMatrix at the same position. This field represents the same values as hMMSTest.
PValueMatrix — p-values for the hypothesis tests returned as a matrix of values in the range [0,1]. Each element in pValue corresponds to the element in CountMatrix at the same position. A small value indicates that the null hypothesis might not be valid.
TestStatisticMatrix — Test statistic values for the hypothesis tests, returned as a numeric matrix. Each element in TestStatistic corresponds to the element in CountMatrix at the same position.
CriticalValue — Critical value for the hypothesis tests returned as a numeric scalar.
MigrationMatrix — A numeric matrix containing migration probabilities. The element at position i,j is the probability of migrating from state i to state j.

The test statistic can be undefined for some values of i and j. In this case, the software returns 0 in the corresponding positions of hMMSTest and RejectTest, and NaN in the corresponding positions of PValue, TestStatistic, and CriticalValueMatrix.

For more information about the MMS test and its corresponding statistics, see More About.

More About

collapse all

Migration Matrix Stability Test

The migration matrix stability test [1] tests the null hypothesis that migration frequencies closer to the diagonal are greater than those further from the diagonal.

For a model that uses K grades, the test statistic z_i,j for an upper diagonal element is

$z_{i, j} = \frac{p_{i, j - 1} - p_{i, j}}{\sqrt{\frac{p_{i, j} (1 - p_{i, j}) + p_{i, j - 1} (1 - p_{i, j - 1}) + 2 p_{i, j} p_{i, j - 1}}{N_{i}}}},$

where i<j≤K, p_i,j is the migration frequency of customers from state i to state j, and N_i,j is the number of customers that migrated from state i to state j. The test statistic for a lower diagonal element is

$z_{i, j} = \frac{p_{i, j + 1} - p_{i, j}}{\sqrt{\frac{p_{i, j} (1 - p_{i, j}) + p_{i, j + 1} (1 - p_{i, j + 1}) + 2 p_{i, j} p_{i, j + 1}}{N_{i}}}},$

where 1≤j<i. The corresponding p-values are equal to $F (z_{i, j})$ , where F is the cumulative distribution function (cdf) for the standard normal distribution.

For more information about the MMS test, see Section 2.5.5.2 in [1].

References

[1] European Central Bank, “Instructions for reporting the validation results of internal models.” February, 2019. https://www.bankingsupervision.europa.eu/activities/internal_models/shared/pdf/instructions_validation_reporting_credit_risk.en.pdf.

Version History

Introduced in R2026a

risk.validation.migrationMatrixStabilityTest

Syntax

Description

Examples

Apply Migration Matrix Stability Test to Probability of Default Data

Input Arguments

CountMatrix — Count matrix wide matrix of nonnegative values | wide matrix of nonnegative values

confidenceLevel — Confidence level 0.95 (default) | numeric scalar in the range (0,1)

Output Arguments

hMMSTest — Hypothesis test results numeric matrix containing values of 1, 0, or NaN

Output — Output metrics structure

More About

Migration Matrix Stability Test

References

Version History

See Also

`CountMatrix` — Count matrix
wide matrix of nonnegative values | wide matrix of nonnegative values

`confidenceLevel` — Confidence level
`0.95` (default) | numeric scalar in the range (0,1)

`hMMSTest` — Hypothesis test results
numeric matrix containing values of `1`, `0`, or `NaN`

`Output` — Output metrics
structure