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QUALITATIVE AND LIMITED DEPENDENT VARIABLE MODELS

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QUALITATIVE AND LIMITED DEPENDENT VARIABLE MODELS Powered By Docstoc
					QUALITATIVE AND LIMITED
 DEPENDENT VARIABLE
       MODELS
• Model to describe choice behavior
• Dependent variables that are limited, that
  is the range of values is constrained

• Or

• The values are not completely observable
e.g.
• a worker decides to drive to work or not

• a high school graduate decides to go to
  college or not

• a household decides to purchase a house
  or rent
• why are some loan applications accepted
  and others not?
Qualitative Choice Models:
Binary choice models
Occurs when an individual is making a
  choice.
Models with Binary Dependent Variables
• If the dependent variable assumes only 2
  values
1 if the outcome is chosen and 0 if not
• For these models, least squares
  estimation methods are not the best
  choices.

• In this case OLS is both biased and
  inconsistent
• OLS suffers from heteroscedasticity
  problem
• Instead, maximum likelihood estimation
  (MLE) is the usual method used
• MLE of probit (or logit) discrete choice
  model
• Slopes can only be estimated up to a
  scale factor
• But coefficient signs and t-values have the
  usual interpretation
              An example
• Y = Affairs = 1 if individual has had an
  affair
• X1 = Dummy for male
• X2 = Years of marriage
• X3 = Dummy for kids in the marriage
• X4 = Dummy for religion
• X5 = Years of education
• X6 = Dummy for Happy
Logit (MLE) estimate gives

Y = 1.29 + 0.25X1 + 0.05X2 + 0.44X3 -
  0.89X4 + 0.01X5 -0.87X6

X2, X4, X6 have p-values < 5%
Logit coefficients do not directly measure
 marginal effects so its hard to interpret
 them.
However, we can interpret the signs.
               Odds ratio
• Marginal effects on the odds ratio.
• Suppose the coefficient on Happy under
  the odds ratio is 0.42, how do we interpret
  this number.
• Happy is a dummy variable  if a person
  switches from an unhappy relationship to a
  happy one, the odds ratio would be 42% of
  what it was before.
Suppose an individual had an odds ratio of
   4.
i.e. P(Y=1) = 4/5 and P(Y=0) = 1/5  there’s
   an 80% chance that an individual will have
   an affair.
If the individual’s marriage becomes Happy,
   then the odds ratio becomes 42% higher
   as before.
4*0.42 = 1.68 
There is a 63% chance the individual will
   have an affair
The linear probability model

Suppose we wish to explain an individual’s
 choice between driving to work (private
 transportation) and taking the bus (public
 transportation)

Individual’s choice can be represented by a
  dummy variable
Y = 1 if individual drives to work
  = 0 if takes bus

We can collect a random sample of workers then
 the outcome Y will have a probability function




where p = probability that y=1.

E(y) = p
Explanatory variable = X
Difference between time by bus and time by
 car

Expectation:
As x increases, an individual would be more
 inclined to drive to work
We expect a positive relationship between x
 and p (the probability to drive to work)
The linear regression model = the linear
  probability model
=
Y= E(y) + e
Y=p+e
Y = Bo + B1X + e

This model is heteroscedastic, because the
 variance of the error term varies from one
 observation to another.
• If we use OLS to estimate the model,
  we’d obtain
Consider what happens when you use this
    model to predict behavior:
If you substitute different values of x in the
    the equation, you might obtain values of
    phat that are

1. Less than 0 (negative)
2. Greater than 1

Values that do not make sense as
   probabilities
• Generally, in a linear regression model the
  slope coefficient (if positive) suggests that
  the increase in x will have a constant
  effect on y

• But in probability models (binary
  dependent variable model), the constant
  rate of increase is impossible because
     0≥ p ≤1
To overcome this problem we use nonlinear
 probit and logit models.

In E-views, choose Objects, New Object,
  Equation and select the Binary Estimation
  option.
Specify your equation in the equation box.

E-views uses Maximum Likelihood estimator
• The estimates produces by a probit or logit
  model that look like slopes and intercepts
  in the output are actually standardized
  slopes and intercepts, known only up to a
  scale factor

• So the size of these coefficients is not
  viewed in the same way (as the change in
  Y for a one-unit change in X). Instead, we
  focus on the sign and the statistical
  significance of these coefficients, in
  particular, the "slopes.“
• If a "slope" is positive, it means that an
  increase in the corresponding X variable
  increases the latent propensity to choose
  the 1 alternative, thereby also increasing
  the predicted probability of choosing the 1
  alternative. (Analogously for negative
  slope coefficients.)
• If the fitted probability is greater than 0.5,
  we predict the individual will choose 1. If it
  is less than 0.5, we predict they will
  choose 0.
• The asymptotic t-ratios allow us to test
  zero hypotheses about these "slopes." If
  the t-ratios exceed (roughly) 2 in absolute
  value, we can reject the hypothesis that
  the relevant X variable has no effect on
  fitted choice probabilities.
       Missing Observations
• Collect data on Sleep and Age
• All data on Sleep but 20% of Age is
  missing
• How do you use all the data to show the
  effect of Age on Sleep?
     Non-Experimental Data
• Non-experimental data can sometimes
  make it very difficult to draw policy
  implications from regression analysis
                  GUN CONTROL

• Suppose your sample consists of households that have
  been victimized by robbery. The dependent variable
  takes a value of 1 if a household member is shot during
  the robbery and 0 otherwise. One of your explanatory
  variables is a dummy variable equal to 1 if there is a
  handgun present in the house, 0 otherwise. When a
  handgun is present in a household, an occupant of that
  house is much more likely to be shot in the process of a
  robbery than when no handgun is present. Therefore, to
  minimize injury and loss of life from robbery incidents,
  private ownership of handguns should be banned.
  Evaluate this policy proposal and the "evidence" upon
  which it is premised
• Briefly describe the nature of the true
  "experiment" that would allow an
  unambiguous determination of the effect of
  handgun presence on robbery shootings
  via a regression like this.
      LEGALIZATION OF MARIJUANA:
• Suppose you have a random sample of at-risk 18-year-
  olds. The dependent variable is the number of times
  each teenager has used heroin. Among the explanatory
  variables is a dummy variable that takes a value of 1 if
  the subject experimented with marijuana prior to age 13,
  and 0 otherwise. You find that the coefficient on this
  dummy variable is positive and strongly statistically
  significant. Therefore, we should not legalize marijuana
  use (which would make it much more accessible to pre-
  teens) since this will lead to widespread use of heroin.
  Evaluate this policy proposal and the "evidence" upon
  which it is premised
• Briefly describe the nature of the true
  "experiment" that would allow an
  unambiguous determination of the effect of
  pre-teen marijuana use on subsequent
  heroin use via a regression like this

				
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