1. QUALITATIVE AND LIMITED DEPENDENT VARIABLE MODELS
Chapter 5
QUALITATIVE AND LIMITED DEPENDENT VARIABLE MODELS
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2. Qualitative Response Regression Models
The regressand or response variable (Y) is qualitative in nature, for example, the labor force participation (LFP), whether an adult is in the labor force or not, it acquires YES or NO decision.
The regressand can take only two values, 1 if the person is in the labor force and 0 if he or she is not. It is a binary or dichotomous variable.
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3. Two approaches to developing a probability model:
In models where Y is qualitative, the objective is to find the probability of something happening. The models are often known as probability models.
Two approaches to developing a probability model:
The Logit Model
The Probit model
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4. Logit Model
Consider a home ownership example that explain the relationship between home ownership and income. X is income and Y =1 means the family owns a house.
(1)
For ease of exposition, Equation (1) is written as:
(2)
where
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5. Then Equation (2) can be written as: (3)
If Pi is the probability of owning a house, the probability of not owning a house (1- Pi) is
Then Equation (2) can be written as:
(3)
is odds ratio, which is the ratio of the probability that a family will own a house to the probability that it will not own a house.
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6. Take the natural log of Equation (3), it becomes:
(4)
L, the log of the odds ratio, is not only linear in X but also linear in parameters. L is called logit.
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7. Features of Logit Model
As P goes from 0 to 1, the logit L goes from - to +.
Although L is linear in X, the probabilities themselves are not.
As many regressors can be added.
If L is positive, when the value of the regressors increase, the odds that the regressand equals 1 increases. If L is nagative, the odds that the regressand equals 1 decreases as the value of X increases.
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8. Features of Logit Model
β2 is the slope measures change in L for a unit change in X. Intercept β1 is the value of the log-odds in favour of Y=1 when X is 0.
Can directly measure Pi when β1 and β2 are given.
The log of the odds ratio is linearly related to Xi.
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9. Probit Model
Assume the home ownership decision of ith family to own a house or not depends on an unobservable utility index, Ii, that is determined by one or more explanatory variables (eg. Income Xi).
The larger the value of index Ii, the greater the probability of a family owning a house. Index Ii is expressed as:
(5)
where Xi is the income of the ith family.
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10. Assume the critical level of the index is , if Ii exceeds , the family will own a house.
Given the assumption of normality, the probability that is less than or equal to Ii can be computed from:
(6)
where means the probability that an event occurs given the value(s) of X, or explanatory, variable(s) and where Zi is the standard normal variable.
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11. To obtain Ii, β1 and β2, take the inverse of Equation (6):
The probability of owning a house is measured by the area of the standard normal curve from - to Ii.
To obtain Ii, β1 and β2, take the inverse of Equation (6):
Where F-1 is the inverse of the normal cumulative distribution function.
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12. 1 Pi
- 
+ 
In panel a we obtain from the ordinate the (cumulative) probability of owning a house given , whereas in panel b we obtain from the abscissa the value of Ii given the value of Pi.
- 
+  Pi 1
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