- Regression models can be used to predict outcomes and understand which factors influence them. Examples given include predicting India's energy consumption based on GDP growth and the probability of a customer defaulting on a loan. - Simple and multiple regression models define the dependent variable Y and identify independent variables X to estimate relationships and interpret results. - Non-linear probability models like logistic (logit) and probit models are better suited when the dependent variable is dichotomous like default/no default. These transform the probability in a non-linear way compared to the linear probability model (LPM).

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Regression Models

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Regression
 Predictive Modeling
 Which Factors Explain?
 Regressive vs. Correlation
Examples:
 What will be India’s Energy Consumption as GDP grows by 6.5%?
 What is the probability that a customer will default on housing loan
 How many fatal road accidents will you have in Delhi if the traffic
volume increases by 10%?

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Simple vs multiple Regression
Define Y
Identify X
Estimate
Interpret
uxxxy kk   ....22110

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Non Linear Probability Models
 ktiit Xfp  ,

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Linear Probability Model (LPM)
uxxxp kk   ....22110 …….
where,
p
kxxx ,...,, 10
is the probability of default
kare the explanatory variables
.
yRegress an indicator variable on kxxx ,...,, 10
y is a dichotomous variable with possible values




defaultednothasfirmtheif0
defaultedhasfirmtheif1
y

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Problems with LPM
 Goodness of Fit
 Improbable Probability Estimates
 Linear Incremental Effect of variables on Default Probability

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Goodness of Fit

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 Improbable Probability Estimates
 Linear Incremental Effect of variables on Default
Probability
Other Problems with LPM

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How should it look like?

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Non Linear Probability Models
 Linear vs Non Linear Regression
 Logit Model
 Probit Model

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Logistic Model (Logit Models)





 k
j
ij
k
j
ij
e
e
P
1
0
1
0
1



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Linear Transformation




k
j
ij
i
i
i
P
P
L
1
0
1
ln 
 Log-Odds Ratio

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How does the probability change?
)1( PP
dx
dP
j
j
 

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Estimation and Interpretation
  

01
1
ii y
i
y
i PP
Maximum Likelihood Technique
Likelihood function
.Choose j to maximize

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Goodness of Fit: Concordant Analysis/Specificity vs Sensitivity
Estimated
Equation
Actual  Won Lost Total
Predicted
Won 160 61 221
Lost 133 1345 1478
Total 293 1406 1699
Correct 160 1345 1505
% Correct 54.6 95.7 88.6
% Incorrect 45.4 4.3 11.4
Constant
Probability
Won Lost Total
0 0 0
293 1406 1699
293 1406 1699
0 1406 1406
0.0 100.0 82.8
100.0 0.0 17.2
Sensitivity 54.61%
Specificity 95.66%
Positive predictive value 72.40%
Negative predictive value 91.00%

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Probit Model

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Logit versus Probit