# Tobit

## Description

Create and analyze a `Tobit`

model object to calculate
the exposure at default (EAD) using this workflow:

Use

`fitEADModel`

to create a`Tobit`

model object.Use

`predict`

to predict the EAD.Use

`modelDiscrimination`

to return AUROC and ROC data. You can plot the results using`modelDiscriminationPlot`

.Use

`modelCalibration`

to return the R-squared, RMSE, correlation, and sample mean error of predicted and observed EAD data. You can plot the results using`modelCalibrationPlot`

.

## 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,
`TobitEADModel`

= fitEADModel(___,`Name=Value`

)```
eadModel =
fitEADModel(EADData,ModelType,PredictorVars={'UtilizationRate','Age','Marriage'},ConversionMeasure="ccf",DrawnVar='Drawn',LimitVar='Limit',ResponseVar='EAD')
```

creates an `eadModel`

object using a
`Tobit`

model type.

### Input Arguments

`data`

— Data for exposure at default

table

Data for exposure at 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`

**Name-Value Arguments**

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.

**Example: **```
eadModel =
fitEADModel(EADData,ModelType,PredictorVars={'UtilizationRate','Age','Marriage'},ConversionMeasure="ccf",DrawnVar='Drawn',LimitVar='Limit',ResponseVar='EAD')
```

`ModelID`

— User-defined model ID

`"Tobit"`

(default) | string | character vector

User-defined model ID, specified as `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
`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
`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 `ResponseVar`

and a string or character vector. The response variable contains the
EAD data and must be a numeric variable. By default,
`ResponseVar`

is set to the last column.

**Data Types: **`string`

| `char`

`LimitVar`

— Limit variable

string | character vector

Limit variable, specified as `LimitVar`

and a
string or character vector. `LimitVar`

indicates
which column in `data`

contains the limit amount.
The limit amount value in the `data`

must be a
positive numeric value. The limit depends on the loan. If its a
credit card, the limit is the credit limit, and if this is a
mortgage limit it is the initial loan amount. In general,
`LimitVar`

is the maximum amount that can be
borrowed.

**Note**

`LimitVar`

is required when
`ConversionMeasure`

is
`'ccf'`

or `'lcf'`

. For
more information on CCF and LCF, see Conversion Measure Options.

**Data Types: **`string`

| `char`

`DrawnVar`

— Drawn variable

string | character vector

Drawn variable, specified as `DrawnVar`

and a
string or character vector. `DrawnVar`

is the
balance on the account at the time of observation, prior to default
and EAD is the balance at the time of default.
`DrawnVar`

indicates which column in
`data`

contains the drawn amount. The drawn
variable value in the `data`

can be a positive or
negative numeric value.

**Note**

`DrawnVar`

is required when
`ConversionMeasure`

is
`'ccf'`

.

If the `ConversionMeasure`

is
`'lcf'`

, `DrawnVar`

is not
required. In this case, `DrawnVar`

is set to
`""`

.

For more information on CCF, see Conversion Measure Options.

**Data Types: **`string`

| `char`

`ConversionMeasure`

— Conversion measure for EAD response values

`"ccf"`

(default) | character vector with value of `'ccf'`

or
`'lcf'`

| string with value of `"ccf"`

or
`"lcf"`

Response transform, specified as
`ConversionMeasure`

and a character vector or string.

`"ccf"`

— Credit conversion factor (CCF) is the portion of the undrawn amount that will be converted into credit. The undrawn amount is the limit minus the drawn amount. The EAD thus becomes the drawn amount plus the CCF times the limit minus the drawn amount (`EAD = Drawn + CCF*(Limit - Drawn)`

).**Note**A

`Tobit`

model with`"ccf"`

can be unstable.`"lcf"`

— Limit conversion factor (LCF) is a fraction of the limit representing the total exposure. The EAD is then defined as the LCF times the limit (`EAD = LCF*Limit`

).

For more information on CCF and LCF, see Conversion Measure Options.

**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 `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
`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
`RightLimit`

and a scalar numeric between
`0`

and `1`

.

**Data Types: **`double`

`SolverOptions`

— `optimoptions`

object

object

Options for fitting, specified as
`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`

## 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`

`LimitVar`

— Limit variable

string

Limit variable, returned as a string.

**Data Types: **`string`

`DrawnVar`

— Drawn variable

string

Drawn variable, returned as a string.

**Data Types: **`string`

`ConversionMeasure`

— Conversion measure for EAD response values

`"ccf"`

(default) | string with value of `"ccf"`

or
`"lcf"`

Response transform, 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`

## Object Functions

`predict` | Predict exposure at 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 EADs |

`modelCalibrationPlot` | Scatter plot of predicted and observed EADs |

## Examples

### Create Tobit EAD Model

This example shows how to use `fitEADModel`

to create a `Tobit`

model for exposure at default (EAD).

**Load EAD Data**

Load the EAD data.

```
load EADData.mat
head(EADData)
```

UtilizationRate Age Marriage Limit Drawn EAD _______________ ___ ___________ __________ __________ __________ 0.24359 25 not married 44776 10907 44740 0.96946 44 not married 2.1405e+05 2.0751e+05 40678 0 40 married 1.6581e+05 0 1.6567e+05 0.53242 38 not married 1.7375e+05 92506 1593.5 0.2583 30 not married 26258 6782.5 54.175 0.17039 54 married 1.7357e+05 29575 576.69 0.18586 27 not married 19590 3641 998.49 0.85372 42 not married 2.0712e+05 1.7682e+05 1.6454e+05

rng('default'); NumObs = height(EADData); c = cvpartition(NumObs,'HoldOut',0.4); TrainingInd = training(c); TestInd = test(c);

**Select Model Type**

Select a model type for `Tobit`

or `Regression`

.

`ModelType = "Tobit";`

**Select Conversion Measure**

Select a conversion measure for the EAD response values.

`ConversionMeasure = "LCF";`

**Create Tobit EAD Model**

Use `fitEADModel`

to create a `Tobit`

model using the `EADData`

.

eadModel = fitEADModel(EADData,ModelType,PredictorVars={'UtilizationRate','Age','Marriage'}, ... ConversionMeasure=ConversionMeasure,DrawnVar="Drawn",LimitVar="Limit",ResponseVar="EAD"); disp(eadModel);

Tobit with properties: CensoringSide: "both" LeftLimit: 0 RightLimit: 1 ModelID: "Tobit" Description: "" UnderlyingModel: [1x1 risk.internal.credit.TobitModel] PredictorVars: ["UtilizationRate" "Age" "Marriage"] ResponseVar: "EAD" LimitVar: "Limit" DrawnVar: "Drawn" ConversionMeasure: "lcf"

Display the underlying model. The underlying model's response variable is the transformation of the EAD response data. Use the `'LimitVar'`

and `'DrawnVar'`

name-value arguments to modify the transformation.

disp(eadModel.UnderlyingModel);

Tobit regression model: EAD_lcf = max(0,min(Y*,1)) Y* ~ 1 + UtilizationRate + Age + Marriage Estimated coefficients: Estimate SE tStat pValue __________ __________ ________ ________ (Intercept) 0.22735 0.026172 8.6868 0 UtilizationRate 0.47364 0.016412 28.859 0 Age -0.0013929 0.00063689 -2.1871 0.028789 Marriage_not married -0.0068879 0.012314 -0.55936 0.57594 (Sigma) 0.36419 0.00388 93.864 0 Number of observations: 4378 Number of left-censored observations: 0 Number of uncensored observations: 4377 Number of right-censored observations: 1 Log-likelihood: -1791.06

**Predict EAD**

EAD prediction operates on the underlying compact statistical model and then transforms the predicted values back to the EAD scale. You can specify the `predict`

function with different options for the `'ModelLevel'`

name-vale argument.

predictedEAD = predict(eadModel,EADData(TestInd,:),ModelLevel="ead"); predictedConversion = predict(eadModel,EADData(TestInd,:),ModelLevel="ConversionMeasure");

**Validate EAD Model**

For model validation, use `modelDiscrimination`

, `modelDiscriminationPlot`

, `modelCalibration`

, and `modelCalibrationPlot`

.

Use `modelDiscrimination`

and then `modelDiscriminationPlot`

to plot the ROC curve.

ModelLevel = "ConversionMeasure"; [DiscMeasure1,DiscData1] = modelDiscrimination(eadModel,EADData(TestInd,:),ModelLevel=ModelLevel); modelDiscriminationPlot(eadModel,EADData(TestInd, :),ModelLevel=ModelLevel,SegmentBy="Marriage");

Use `modelCalibration`

and then `modelCalibrationPlot`

to show a scatter plot of the predictions.

```
YData = "Observed";
[CalMeasure1,CalData1] = modelCalibration(eadModel,EADData(TestInd,:),ModelLevel=ModelLevel);
modelCalibrationPlot(eadModel,EADData(TestInd,:),ModelLevel=ModelLevel,YData=YData);
```

Plot a histogram of observed with respect to the predicted EAD.

figure; histogram(CalData1.Observed); hold on; histogram(CalData1.(('Predicted_' + ModelType))); legend('Observed','Predicted');

## More About

### Exposure at Default Tobit Models

The exposure at default (EAD) Tobit models fit a Tobit model to EAD 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. Using `ModelLevel`

, you can set the
Tobit model level to `EAD`

, `CCF`

, or
`LCF`

conversion measures. The `EAD`

model
level does not have any range, the `CCF`

conversion measure has a
range of `-Inf`

to `1`

, and the
`LCF`

conversion measure is `0`

to
`1`

. A distribution of response values where there is a high
frequency of observations at the limits is consistent with the model
assumptions.

The Tobit model combines the following two formulas:

$$\begin{array}{l}Y=\mathrm{min}\left\{\mathrm{max}\left\{L,{Y}^{*}\right\},R\right\}\\ {Y}^{*}={\beta}_{0}+{\beta}_{1}{X}_{1}+\mathrm{...}+{\beta}_{p}{X}_{p}+\sigma \epsilon =X\beta +\sigma \epsilon \end{array}$$

where

*Y*is the observed response variable, the observed EAD data for an EAD model.*L*is the left limit, the lower bound for the response values, typically`0`

for EAD models.*R*is the right limit, the upper bound for the response values, typically`1`

for EAD models.*Y*^{*}is a latent, unobserved variable.β

_{j}is the coefficient of the*j*th 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

$$Y=\left\{\begin{array}{l}L\text{if}{Y}^{*}\le L\\ {Y}^{*}\text{if}L{Y}^{*}R\\ R\text{if}{Y}^{*}\ge R\end{array}\right\}$$

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 EAD models with the
`CensoringSide`

name-value argument:

`'both'`

— This option 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`

â¡{*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

$$LF(\beta ,\sigma |{X}_{i},{Y}_{i})=\left\{\begin{array}{l}\Phi (L;{X}_{i}\beta \text{,}\sigma \text{)if}{Y}_{i}\le L\\ \varphi ({Y}_{i}{\text{;X}}_{i}\beta \text{,}\sigma \text{)if}L{Y}_{i}R\\ 1-\Phi (R;{X}_{i}\beta ,\sigma )\text{if}{Y}_{i}\ge R\end{array}\right\}$$

where

Φ(

*x*;*m*,*s*) is the cumulative normal distribution with mean*m*and standard deviation*s*.φ(

*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

$$LLF(\beta ,\sigma |X,Y)={\displaystyle \sum _{i=1}^{n}\mathrm{log}(LF(}\beta ,\sigma |{X}_{i},{Y}_{i}))$$

The parameters are estimated by maximizing the log-likelihood function. The only constraint is that the Ïƒ parameter must be positive.

To predict an EAD value, Tobit EAD models return the unconditional expected value of the response, given the predictor values

$$EA{D}_{i}^{pred}=E\left[{Y}_{i}|{X}_{i}\right]$$

The expression for the expected value can be separated into the cases

$$\begin{array}{l}E\left[Y\right]=E\left[Y|Y=L\right]P(Y=L)\\ +E\left[Y|L<Y<R\right]P(L<Y<R)\\ +E\left[Y|Y=R\right]P(Y=R)\end{array}$$

Using the previous expression and the properties of the (truncated) normal distribution, it follows that

$$E\left[{Y}_{i}|{X}_{i}\right]=\Phi ({a}_{i})L+(\Phi ({b}_{i})-\Phi ({a}_{i}))({X}_{i}\beta +\sigma {\lambda}_{i})+(1-\Phi ({b}_{i}))R$$

where

$${a}_{i}=\frac{L-{X}_{i}\beta}{\sigma},{b}_{i}=\frac{R-{X}_{i}\beta}{\sigma},\text{and}{\lambda}_{i}=\frac{\varphi ({a}_{i})-\varphi ({b}_{i})}{\Phi ({b}_{i})-\Phi ({a}_{i})}$$

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

$$E\left[{Y}_{i}|{X}_{i}\right]=\Phi ({a}_{i})L+(1-\Phi ({a}_{i}))\left({X}_{i}\beta \text{+}\sigma \text{}\frac{\varphi ({a}_{i})}{1-\Phi ({a}_{i})}\right)$$

Similarly, for right-censored models, the *L* limit is
decreased to minus infinity to get

$$E\left[{Y}_{i}|{X}_{i}\right]=\Phi ({b}_{i})\left({X}_{i}\beta -\sigma \text{}\frac{\varphi ({b}_{i})}{\Phi ({b}_{i})}\right)+(1-\Phi ({b}_{i}))R$$

### Conversion Measure Options

You can relate the EAD to a scaling variable and derive
conversion measures like credit conversion factor (CCF) and limit conversion factor
(LCF) using the `'ccf'`

or `'lcf'`

options for the
`ConversionMeasure`

name-value argument.

The following table summarizes the supported transformations using the
`'ccf'`

or `'lcf'`

options for the
`ConversionMeasure`

name-value argument:

Measure | EAD Formula | Lower Bound | Upper Bound | Inverse Transformation |
---|---|---|---|---|

CCF | ```
EAD = Drawn + CCF Ã— (Limit -
Drawn)
``` | -Inf | 1 | `CCF = 1 - e` |

LCF | `EAD = LCF ⨉ Limit` | 0 | 1 | `LCF = e` |

## 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] Brown, Iain.
*Developing Credit Risk Models Using SAS Enterprise Miner and
SAS/STAT: Theory and Applications.* SAS Institute,
2014.

[4] Roesch, Daniel and Harald
Scheule. *Deep Credit Risk.* Independently published,
2020.

## Version History

**Introduced in R2021b**

### 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.

## See Also

### Functions

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