Tobit
Description
Create and analyze a Tobit
model object to calculate loss
given default (LGD) using this workflow:
Use
fitLGDModel
to create aTobit
model object.Use
predict
to predict the LGD.Use
modelDiscrimination
to return AUROC and ROC data. You can plot the results usingmodelDiscriminationPlot
.Use
modelCalibration
to return the R-squared, RMSE, correlation, and sample mean error of predicted and observed LGD data. You can plot the results usingmodelCalibrationPlot
.
Creation
Description
specifies options using one or more name-value arguments in addition to the
input arguments in the previous syntax. The optional name-value arguments
set the model object properties. For example,
TobitLGDModel
= fitLGDModel(___,Name,Value
)lgdModel = fitLGDModel(data,'tobit',PredictorVars={'LTV' 'Age'
'Type'},ResponseVar="LGD",CensoringSide="left",LeftLimit=1e-4,WeightsVar="Weights")
creates a lgdModel
object using a
Tobit
model type.
Input Arguments
data
— Data for loss given default
table
Data for loss given default, specified as a table.
Data Types: table
ModelType
— Model type
string with value "Tobit"
| character vector with value 'Tobit'
Model type, specified as a string with the value of
"Tobit"
or a character vector with the value of
'Tobit'
.
Data Types: char
| string
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
Example: lgdModel = fitLGDModel(data,'tobit',PredictorVars={'LTV'
'Age'
'Type'},ResponseVar="LGD",CensoringSide="left",LeftLimit=1e-4)
ModelID
— User-defined model ID
"Tobit"
(default) | string | character vector
User-defined model ID, specified as the comma-separated pair
consisting of 'ModelID'
and a string or character
vector. The software uses the ModelID
text to
format outputs and is expected to be short.
Data Types: string
| char
Description
— User-defined description for model
""
(default) | string | character vector
User-defined description for model, specified as the
comma-separated pair consisting of 'Description'
and a string or character vector.
Data Types: string
| char
PredictorVars
— Predictor variables
all columns of data
except for ResponseVar
(default) | string array | cell array of character vectors
Predictor variables, specified as the comma-separated pair
consisting of 'PredictorVars'
and a string array
or cell array of character vectors. PredictorVars
indicates which columns in the data
input
contain the predictor information. By default,
PredictorVars
is set to all the columns in
the data
input except for
ResponseVar
.
Data Types: string
| cell
ResponseVar
— Response variable
last column of data
(default) | string | character vector
Response variable, specified as the comma-separated pair
consisting of 'ResponseVar'
and a string or
character vector. The response variable contains the LGD data and
must be a numeric variable. An LGD value of 0
indicates no loss (full recovery), 1
indicates
total loss (no recovery), and values between 0
and 1
indicate a partial loss. By default,
ResponseVar
is set to the last column.
Data Types: string
| char
CensoringSide
— Censoring side
"both"
(default) | character vector with value of 'left'
, 'right'
, or 'both'
| string with value of "left"
, "right"
, or "both"
Censoring side, specified as the comma-separated pair consisting
of 'CensoringSide'
and a character vector or
string. CensoringSide
indicates whether the
desired Tobit model is left-censored, right-censored, or censored on
both sides.
Data Types: string
| char
LeftLimit
— Left-censoring limit
0
(default) | numeric between 0
and 1
Left-censoring limit, specified as the comma-separated pair
consisting of 'LeftLimit'
and a scalar numeric
between 0
and 1
.
Data Types: double
RightLimit
— Right-censoring limit
1
(default) | numeric between 0
and 1
Right-censoring limit, specified as the comma-separated pair
consisting of 'RightLimit'
and a scalar numeric
between 0
and 1
.
Data Types: double
SolverOptions
— optimoptions
object
object
Options for fitting, specified as the comma-separated pair
consisting of 'SolverOptions'
and an
optimoptions
object that is created using
optimoptions
from
Optimization Toolbox™. The defaults for the optimoptions
object are:
"Display"
—"none"
"Algorithm"
—"sqp"
"MaxFunctionEvaluations"
—500
✕ Number of model coefficients"MaxIterations"
— The number of Tobit model coefficients is determined at run time, it depends on the number of predictors and the number of categories in the categorical predictors.
Note
When using optimoptions
with a Tobit
model, specify the SolverName
as
fmincon
.
Data Types: object
WeightsVar
— Column name containing weights
""
(default) | string scalar
Column name of the input table containing weights, specified as a string scalar.
Note
The default value (""
) results in a weight
of 1
for each row in data
.
All weight values in data
must be
nonnegative.
For an example using WeightsVar
, see Create Weighted LGD Model.
Properties
ModelID
— User-defined model ID
Tobit
(default) | string
User-defined model ID, returned as a string.
Data Types: string
Description
— User-defined description
""
(default) | string
User-defined description, returned as a string.
Data Types: string
UnderlyingModel
— Underlying statistical model
compact linear model
This property is read-only.
Underlying statistical model, returned as a compact linear model object.
The compact version of the underlying regression model is an instance of the
classreg.regr.CompactLinearModel
class. For more
information, see fitlm
and CompactLinearModel
.
Data Types: CompactLinearModel
PredictorVars
— Predictor variables
all columns of data
except for the ResponseVar
(default) | string array
Predictor variables, returned as a string array.
Data Types: string
ResponseVar
— Response variable
last column of data
(default) | string
Response variable, returned as a string.
Data Types: string
CensoringSide
— Censoring side
"both"
(default) | string with value of "left"
, "right"
, or "both"
This property is read-only.
Censoring side, returned as a string.
Data Types: string
LeftLimit
— Left-censoring limit
0
(default) | numeric between 0
and 1
This property is read-only.
Left-censoring limit, returned as a scalar numeric between
0
and 1
.
Data Types: double
RightLimit
— Right-censoring limit
1
(default) | numeric between 0
and 1
This property is read-only.
Right-censoring limit, returned as a scalar numeric between
0
and 1
.
Data Types: double
WeightsVar
— Column name containing weights
""
(default) | string scalar
Column name of the input table containing weights, returned as a string
scalar. This property is also used to determine the weights variable for
validation data when you use the modelDiscrimination
or modelCalibration
functions.
Object Functions
predict | Predict loss given default |
modelDiscrimination | Compute AUROC and ROC data |
modelDiscriminationPlot | Plot ROC curve |
modelCalibration | Compute R-square, RMSE, correlation, and sample mean error of predicted and observed LGDs |
modelCalibrationPlot | Scatter plot of predicted and observed LGDs |
Examples
Create Tobit LGD Model
This example shows how to use fitLGDModel
to create a Tobit
model for loss given default (LGD).
Load LGD Data
Load the LGD data.
load LGDData.mat
head(data)
LTV Age Type LGD _______ _______ ___________ _________ 0.89101 0.39716 residential 0.032659 0.70176 2.0939 residential 0.43564 0.72078 2.7948 residential 0.0064766 0.37013 1.237 residential 0.007947 0.36492 2.5818 residential 0 0.796 1.5957 residential 0.14572 0.60203 1.1599 residential 0.025688 0.92005 0.50253 investment 0.063182
rng('default'); NumObs = height(data); c = cvpartition(NumObs,'HoldOut',0.4); TrainingInd = training(c); TestInd = test(c);
Create Tobit
LGD Model
Use fitLGDModel
to create a Tobit
model using the TrainingInd
data.
lgdModel = fitLGDModel(data(TrainingInd,:),'Tobit',... 'ModelID','Example Tobit',... 'PredictorVars',{'LTV' 'Age' 'Type'},... 'ResponseVar','LGD',... 'CensoringSide','left',... 'LeftLimit',1e-4); disp(lgdModel)
Tobit with properties: CensoringSide: "left" LeftLimit: 1.0000e-04 RightLimit: 1 Weights: [0x1 double] ModelID: "Example Tobit" Description: "" UnderlyingModel: [1x1 risk.internal.credit.TobitModel] PredictorVars: ["LTV" "Age" "Type"] ResponseVar: "LGD" WeightsVar: ""
Display the underlying model. The underlying model is a left-censored Tobit model. Use the 'CensoringSide'
argument and the 'LeftLimit'
and
'RightLimit'
arguments to modify the underlying Tobit model.
disp(lgdModel.UnderlyingModel)
Tobit regression model, left-censored: LGD = max(0.0001,Y*) Y* ~ 1 + LTV + Age + Type Estimated coefficients: Estimate SE tStat pValue ________ _________ _______ __________ (Intercept) 0.057356 0.026585 2.1575 0.031083 LTV 0.2003 0.030596 6.5464 7.3912e-11 Age -0.09405 0.0072999 -12.884 0 Type_investment 0.10071 0.017922 5.6193 2.1732e-08 (Sigma) 0.28833 0.0055224 52.211 0 Number of observations: 2093 Number of left-censored observations: 547 Number of uncensored observations: 1546 Number of right-censored observations: 0 Log-likelihood: -638.353
Predict LGD
For Tobit models, use predict
to calculate the predicted LGD value, which is the unconditional expected value of the response, given the predictor values.
predictedLGD = predict(lgdModel,data(TestInd,:))
predictedLGD = 1394×1
0.0871
0.1228
0.3181
0.0926
0.1654
0.2215
0.2347
0.0102
0.1576
0.1969
⋮
Validate LGD Model
Use modelDiscriminationPlot
to plot the ROC curve.
modelDiscriminationPlot(lgdModel,data(TestInd,:))
Use modelCalibrationPlot
to show a scatter plot of the predictions.
modelCalibrationPlot(lgdModel,data(TestInd,:))
More About
Loss Given Default Tobit Models
The loss given default (LGD) Tobit models fit a Tobit model to LGD data.
Tobit models are “censored” regression models. Tobit models assume that the response variable can be observed only within certain limits, and no value outside the limits can be observed. In the case of LGD models, the limits are typically 0 (total recovery or cure) and 1 (total loss). A distribution of response values where there is a high frequency of observations at the limits is consistent with the model assumptions. For LGD models, it is common to have distributions with a high proportion of cures, or high proportion of total losses, or both.
The Tobit model combines the following two formulas:
where
Y is the observed response variable, the observed LGD data for an LGD model.
L is the left limit, the lower bound for the response values, typically
0
for LGD models.R is the right limit, the upper bound for the response values, typically
1
for LGD models.Y* is a latent, unobserved variable.
βj is the coefficient of the jth predictor (or the intercept for j =
0
).σ is the standard deviation of the error term.
ε is the error term, assumed to follow a standard normal distribution.
The first formula above is written using min
and
max
operators and is equivalent to
The standard deviation of the error is explicitly indicated in the formulas. Unlike traditional regression least-squares estimation, where the standard deviation of the error can be inferred from the residuals, for Tobit models the estimation is via maximum likelihood and the standard deviation needs to be handled explicitly during the estimation. If there are p predictor variables, the Tobit model estimates p+2 coefficients, namely, one coefficient for each predictor, plus an intercept, plus a standard deviation.
Three censoring side options are supported in the Tobit LGD models with the
CensoringSide
name-value argument:
'both'
— This is the default option, with censoring on both sides. The estimation uses left and right limits.'left'
— The left-censored version of the model has no right limit (or R = ∞). The relationship between Y and Y* is Y =max
{L,Y* }.'right'
— The right-censored version of the model has no left limit (or L = -∞). The relationship between Y and Y* is Y =min
{Y*,R}.
The parameters of the Tobit model are estimated using maximum likelihood. For observation i = 1,…,n, the likelihood function is
where
(x;m,s) is the cumulative normal distribution with mean m and standard deviation s.
(x;m,s) is the normal density function with mean m and standard deviation s.
This likelihood function is for models censored on both sides. For left-censored models, the right limit has no effect, and the likelihood function has two cases only (R = ∞); likewise for right-censored models (L = -∞).
The log-likelihood function is the sum of the logarithm of the likelihood functions for individual observations
The parameters are estimated by maximizing the log-likelihood function. The only constraint is that the σ parameter must be positive.
To predict an LGD value, Tobit LGD models return the unconditional expected value of the response, given the predictor values
The expression for the expected value can be separated into the cases
Using the previous expression and the properties of the (truncated) normal distribution, it follows that
where
This expression applies to the models censored on both sides. For models censored on one side only, the corresponding expressions can be derived from here. For example, for left-censored models, let the R limit in the expression above go to infinity, and the resulting expression is
Similarly, for right-censored models, the L limit is decreased to minus infinity to get
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.
Version History
Introduced in R2021aR2024a: Added WeightsVar
name-value argument for Tobit
model
The Tobit
model supports a WeightsVar
name-value argument for observation weights.
R2023a: modelAccuracy
object function is renamed to modelCalibration
function
The modelAccuracy
object function is renamed to
modelCalibration
function. The use of
modelAccuracy
is discouraged, use modelCalibration
instead.
R2023a: modelAccuracyPlot
object function is renamed to modelCalibrationPlot
function
The modelAccuracyPlot
object function is renamed to
modelCalibrationPlot
function. The use of
modelAccuracyPlot
is discouraged, use modelCalibrationPlot
instead.
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