Topic 1: Dynamics of Credit Ratings, continued Empirical data show - - PowerPoint PPT Presentation

Topic 1: Dynamics of Credit Ratings, continued

• Empirical data show that observed rating transition frequen-

cies vary from year to year.

• Some variation would be expected because of sampling vari-

ability, but we must consider the possibility that the process is not homogeneous over time.

• A careful look at the results for a time-homogeneous shows

that some carry over to the inhomogeneous case.

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• We now write, for t < u,

pi,j(t, u) = P[X(u) = j|X(t) = i], and P (t, u) as the matrix with these as entries.

• These matrices still satisfy Chapman-Kolmogorov equations:

if s < t < u, then

P (s, u) = P (s, t)P (t, u).

• In discrete time, one-step transition matrices are also still

the key, because repeated use of the Chapman-Kolmogorov equations gives

P (s, u) = P (s, s + 1)P (s + 1, s + 2) . . . P (u − 1, u).

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• Write P t = P (t, t + 1), the one-year transition matrix for the

the time step t → t + 1.

• For the entries in P t, write

pi,j,t = P[X(t + 1) = j|X(t) = i].

• Our problem is to model the dependence of pi,j,t on t.
• We assume that we have some covariates zt, and that pi,j,t

depends on t only through these covariates; it also depends

• n a parameter vector θ:

pi,j,t = pi,j(zt; θ).

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Likelihood Inference

• Suppose that we have rating histories for a number of issuers,

and wish to estimate θ.

• In inference problems generally, the likelihood function is of-

ten the starting point.

• Suppose that we have rating histories for N issuers for T + 1

time steps: Rn(t) = rating of nth issuer at time step t, n = 1, 2, . . . N; t = 0, 1, . . . , T.

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• For

the nth issuer, the probability

• f

being in rating R(t + 1) at time t + 1 given being in state R(t) at time t is pRn(t),Rn(t+1)(zt; θ).

• By the Markov property, the probability of the history for the

nth issuer is therefore

T−1

• t=0

pRn(t),Rn(t+1)(zt; θ) .

• If we assume that transitions for different issuers are inde-

pendent, given the covariates zt, the likelihood function is just the product of these issuer-specific probabilities: L(θ) =

N

• n=1

T−1

• t=0

pRn(t),Rn(t+1)(zt; θ) .

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• Suppose that Ni,j(t) issuers make the transition from state i

at time t to state j at time t + 1.

• Then L(θ) contains Ni,j(t) factors equal to pi,j(zt; θ).
• So we can also write

L(θ) =

• i,j

T−1

• t=0

pi,j(zt; θ)Ni,j(t) .

• Since the likelihood depends on the transition histories only

through the tables of transition frequencies Ni,j(t), these are sufficient statistics.

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• Suppose that

θ =

  

θ1 θ2

. . .

  

and that pi,j(zt; θ) depends only on the corresponding block

θi.

• Then we can also write

L(θ) =

• i

 

T−1

• t=0
• j

pi,j(zt; θi)Ni,j(t)

 

=

• i

Li(θi) , say.

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• So if we estimate θ by maximizing the likelihood, we can just

maximize each term Li(θi) individually.

• But

Li(θi) =

T−1

• t=0
• j

pi,j(zt; θi)Ni,j(t) is the same as the likelihood for a multinomial situation: – for t = 0, 1, . . . , T − 1, Ni(t) =

j Ni,j(t) issuers are ran-

domly assigned new ratings j; – probability of rating j is pi,j(zt; θi).

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• So we can use methods for modeling and estimating the

probabilities in multinomial situations to model and estimate the Markov Chain transition probabilities.

• The most widely used method for binomial data is logistic

regression.

• The simplest generalization of binomial logistic regression to

the multinomial case (with ordered categories such as credit ratings) is cumulative logistic regression.

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Regression Models for Probabilities

• Simplest case: a binary response X and a single covariate z.
• Example: Space shuttle O-ring failures:

X =

  

1 if O-ring shows “thermal distress” on launch;

• therwise;

z = ambient temperature at launch time.

• We want to express P[X = 1] as a function of z.

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• We could use a linear model:

P[X = 1] = β0 + β1z, but extreme values of z would give “probabilities” either < 0

• r > 1, both impossible.
• The “odds ratio”

P[X = 1] P[X = 0] = P[X = 1] 1 − P[X = 1] can be any positive value, and its logarithm log

• P[X = 1]

1 − P[X = 1]

• can be any real value.

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• So the model

log

• P[X = 1]

1 − P[X = 1]

• = β0 + β1z

always implies a probability between 0 and 1 (exclusive).

• Solving for P[X = 1] gives

P[X = 1] = eβ0+β1z 1 + eβ0+β1z = 1 1 + e−(β0+β1z).

• This is called a logistic regression model. It is an example of

a generalized linear model.

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• Often we have more than one covariate; the model general-

izes to P[X = 1] = 1 1 + e−(β0+β1z1+···+βkzk) = 1 1 + e−zTβ where

z =

    

1 z1 . . . zk

    

and

β =

    

β0 β1 . . . βk

     .

• SAS proc logistic will fit this model.

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• What if X takes more values? Say 0, 1, . . . , N − 1 for N > 2.
• Two different generalizations for different situations:

– If X is an ordinal scale, use cumulative (or ordinal) logistic regression. – If X is a nominal (that is, unordered) scale, use multino- mial logistic regression.

• Credit ratings are an ordinal scale, so cumulative logistic

regression is the natural choice, but multinomial logistic re- gression could also be used.

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Cumulative Logistic Regression

• For j = 0, 1, . . . , N − 1, write

Xj =

  

X ≤ j 1 X > j.

• Xj is binary, so we can write a logistic model:

log

• P[Xj = 1]

1 − P[Xj = 1]

• = zTβj.
• We have assumed that the same covariates z are relevant to

each Xj, but we must allow the parameters βj to depend on j.

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• In the conventional cumulative logistic regression, only the

intercept depends on j:

βj =

    

β0,j β1 . . . βk

     .

• In terms of X, we have

log

• P[X > j]

P[X ≤ j]

• = zTβj

= β0,j + β1z1 + · · · + βkzk.

• SAS proc logistic will fit this model.

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Motivation for Cumulative Logistic Regression model

• Suppose that X is determined by an unobserved continuous

variable ξ.

• For example, a bond issuer’s credit rating is determined by

its underlying financial strength, which does not have to fall into neat categories.

• X results from categorizing ξ at N − 1 cut-points x0 < x1 <

· · · < xN−2: X = j ⇐ ⇒ xj−1 < ξ ≤ xj where we take x−1 = −∞ and xN−1 = +∞.

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• Now suppose that ξ is related to the covariates by

ξ = β1z1 + · · · + βkzk + ǫ where ǫ has cumulative distribution function F(·).

• Then

P[X ≤ j] = P

• ξ ≤ xj
• = P
• ǫ ≤ xj − β1z1 − · · · − βkzk
• = F
• xj − β1z1 − · · · − βkzk
• .

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• If ǫ has the logistic distribution, that is

F(x) = 1 1 + e−x, then log

• 1 − F(x)

F(x)

• = −x

and log

• P[X > j]

P[X ≤ j]

• = log

  

1 − F

• xj − β1z1 − · · · − βkzk
• F
• xj − β1z1 − · · · − βkzk
• 

 

= −xj + β1z1 + · · · + βkzk.

• If we write β0,j = −xj, this is the same as the cumulative

logistic regression model.

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• So fitting the cumulative logistic regression model is essen-

tially the same as estimating the cut-points xj and the re- gression coefficients β1, . . . , βk.

• If ǫ is assumed to have the standard normal distribution in-

stead of the logistic distribution, we have the cumulative probit regression model.

• SAS proc logistic will fit this model.

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Multinomial Logistic Regression model

• For nominal data, the cumulative probabilities modeled in

the cumulative logistic regression model have no meaning.

• In the Multinomial Logistic Regression model, also known as

the Generalized Logistic Regression model, the ratios of the individual probabilities are modeled.

• Choose a reference (or base) category j0. Then for j = j0,

log

• P[X = j]

P[X = j0]

• = zTβj.

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• In this model, all elements of βj depend on j:

βj =

    

β0,j β1,j . . . βk,j

     .

• Note that the model depends on the choice of the base cat-

egory only cosmetically–fitted probabilities are the same, re- gardless of the choice.

• SAS proc logistic will fit this model, using the link = glogit
• ption on the model statement.

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