Difference between revisions of "Cohort Estimator"
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== Definition == | == Definition == | ||
− | The '''Cohort estimator''' is a simple ''frequentist'' estimation of multi-state transitions. The estimator can be used to derive the transition probability matrix of a Markov process with a finite number of states | + | The '''Cohort estimator''' is a simple ''frequentist'' estimation of multi-state transitions. The estimator can be used to derive the transition probability matrix of a [[Markov Chain]] process with a finite number of states |
== Single Period Cohort Estimator == | == Single Period Cohort Estimator == | ||
− | The position in state space for an entity <math>i</math> at discrete time <math>l</math> is a random variable <math>R^i_l</math> taking values in the state space S. We assume a finite [[State Space | state space]] <math>S ={0, \dots ,D}</math> | + | The position in state space for an entity <math>i</math> at discrete time <math>l</math> is a random variable <math>R^i_l</math> taking values in the state space <math>S</math>. We assume a finite [[State Space | state space]] <math>S ={0, \dots , D}</math> |
− | Let <math>C^{mn}_k</math> be the number (count) of entities at state m at time k-1 and at state n at k. The directly estimated transition probability is: | + | Let <math>C^{mn}_k</math> be the number (count) of entities at state <math>m</math> at time <math>k-1</math> and at state <math>n</math> at <math>k</math>. The directly estimated transition probability is: |
:<math> | :<math> | ||
− | T^{mn}_{k} = \frac{C^{mn}}{ \sum_{n=0}^{D} C^{mn}} = \frac{C^{mn}}{ N^{m}_{k}} | + | T^{mn}_{k} = \frac{C^{mn}}{ \sum_{n=0}^{D} C^{mn}} = \frac{C^{mn}}{N^{m}_{k}} |
</math> | </math> | ||
The estimator reflects that the probability of transition from m to n is the observed count number <math>C^{mn}_k</math> of all entities that migrated from m to n as a fraction of the count of all entities whose rating was m at k-1, that is, <math>\sum_{n=0}^{D} C^{mn}_k</math>, irrespective of where they migrated to. The denominator includes all entities that did not migrate <math>C^{nn}_k</math> | The estimator reflects that the probability of transition from m to n is the observed count number <math>C^{mn}_k</math> of all entities that migrated from m to n as a fraction of the count of all entities whose rating was m at k-1, that is, <math>\sum_{n=0}^{D} C^{mn}_k</math>, irrespective of where they migrated to. The denominator includes all entities that did not migrate <math>C^{nn}_k</math> | ||
− | State changes which occur within the period [k-1,k] are | + | State changes which occur within the period [k-1,k] are ignored. The time resolution (the number of cohorts) must be chosen so that there is as little ignored transition history as possible. |
== Multi-Period Cohort Estimator == | == Multi-Period Cohort Estimator == | ||
− | + | A multi-period estimator (without the assumption of time-homogeneity) is also straight-forward: | |
− | A multi-period estimator without the assumption of time-homogeneity is also straight-forward: | ||
:<math> | :<math> | ||
− | T^{mn}_{kl} = \frac{C^{mn}_{kl}}{ \sum_{n=0}^{D} C^{mn}_{kl}} | + | T^{mn}_{kl} = \frac{C^{mn}_{kl}}{\sum_{n=0}^{D} C^{mn}_{kl}} |
</math> | </math> | ||
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== Issues and Challenges == | == Issues and Challenges == | ||
− | |||
* The cohort estimator gives zero probability to a migration event that is not present in the data. E.g. in the presence of (right) censoring where we do not know what happens to the firm after the sample window closes (e.g. does it default right away or does it live on until the present) <ref>Measurement and Estimation of Credit Migration Matrices by Til Schuermann Yusuf Jafry, 2003</ref> | * The cohort estimator gives zero probability to a migration event that is not present in the data. E.g. in the presence of (right) censoring where we do not know what happens to the firm after the sample window closes (e.g. does it default right away or does it live on until the present) <ref>Measurement and Estimation of Credit Migration Matrices by Til Schuermann Yusuf Jafry, 2003</ref> | ||
* Left truncation where firms only enter sample if they have either survived long enough or have received a rating. | * Left truncation where firms only enter sample if they have either survived long enough or have received a rating. | ||
+ | |||
+ | == See Also == | ||
+ | * [[Aalen-Johansen Estimator]] | ||
== References == | == References == |
Revision as of 15:07, 30 April 2021
Contents
Definition
The Cohort estimator is a simple frequentist estimation of multi-state transitions. The estimator can be used to derive the transition probability matrix of a Markov Chain process with a finite number of states
Single Period Cohort Estimator
The position in state space for an entity at discrete time is a random variable taking values in the state space . We assume a finite state space
Let be the number (count) of entities at state at time and at state at . The directly estimated transition probability is:
The estimator reflects that the probability of transition from m to n is the observed count number of all entities that migrated from m to n as a fraction of the count of all entities whose rating was m at k-1, that is, , irrespective of where they migrated to. The denominator includes all entities that did not migrate
State changes which occur within the period [k-1,k] are ignored. The time resolution (the number of cohorts) must be chosen so that there is as little ignored transition history as possible.
Multi-Period Cohort Estimator
A multi-period estimator (without the assumption of time-homogeneity) is also straight-forward:
where denotes the migration count of the period [k,l].
Confidence Intervals
Confidence intervals for the cohort estimator can be estimated using the multinomial proportions method[1]
Issues and Challenges
- The cohort estimator gives zero probability to a migration event that is not present in the data. E.g. in the presence of (right) censoring where we do not know what happens to the firm after the sample window closes (e.g. does it default right away or does it live on until the present) [2]
- Left truncation where firms only enter sample if they have either survived long enough or have received a rating.