Lecture notes 15 Limited dependent variable models

Question 1

How does a limited dependent variable regression model work?

Answer 1

Outcome Y is binary =1 or 0 and the regression estimates the probability given X

Question 2

What is the issue behind using a linear model to estimate LDV?

Answer 2

For extreme values of X it may mean probability is larger than 1 or negative probability.

Question 3

How can we address the issues of linear models not being suitable for probability

Answer 3

We can use a functional form eg
F(B0+B1X1)

where F
F= normal distribution phi
Logistic

Question 4

What are the binary outcome regression models?

Answer 4

-Linear probability model
-Probit model
-Logit model

Question 5

What is a sign of the function of a probit model? Logit model?

Answer 5

Probit has phi sign
Logit has open triangle sign.

Question 6

What is the CDF of
Probit
Logit

Answer 6

Probit = normal distribution CDF
Logit(u) = 1/ 1 + exp(-u)

Question 7

What are the issues with a linear probability model?

Answer 7

• Can be outside of [0,1]
• Only approximation
  -Violates homoscadacity and normality assumptions

Question 8

What are the advantages and disadvantages of probit model?

Answer 8

Prob always between 0 and 1

Cannot use OLS estimates but maximum likelhood estimates.

Likelhood of observing the data is as large as possible.

Question 9

What is the interpretation of the coefficient of the linear probability model?

Answer 9

A 1 unit increase of X causes a Beta 1 change in the probability

Question 10

How does interpreting coefficients of probit and logit functions work?

Answer 10

We need to take derivative of function with respect to X

dF(u)/ du F(B0+B1X1+ BK….)

First you take derivative of whole function (density of F) = PDF (Vary for probit and logit.
Then you take derivative of inside of function

Then you multiply both together

Question 11

What is the PDF of probit and logit?

Answer 11

Probit = 1/root 2 pi exp(-u^2/2)
Logit = exp(-u) / (1+ exp (-U))^2

Question 12

What is the symbol of the probit CDF and PDF?
What is the symbol of the logit CDF and PDF?

Answer 12

1. CDF is straight phi PDF is diaganol phi
2. CDF is open triangle, PDF is lamda

Question 13

With specific values how do you work out the derivative of a logit or probit

Answer 13

First you get the specific values and sub it into the whole function.

-Then with the value you get out of the whole function you sub this into the pdf
-You then times that by the coefficient derivative you took

Question 14

Write out the notation of when you take a derivative

Question 15

How do you interpret coefficient when in a LDV model when the regressor is discrete?

Answer 15

• You just subtract when the dummy is 0 and 1

First you evaluate the function for when it is equaled to the given numbers

Then you plug it into CDF

Then you subtract the two to find the probability difference

Question 16

What is the only type of hypothesis test done with LDV?
How do you do hypothesis test with LDV model?

Answer 16

Join significance
LR = 2[ log(Lu) - Log(Lr)] test statistic varies with chi squared with k degrees of freedom.

Then break down into a restricted and unrestricted log-liklehood models.

Question 17

How do you do goodness of fit with LDV model?

How do we do it?

Answer 17

There is no R^2

So you see how the estimated probability compares to the actual probability

-if estimated probability is close to one we should observe the. real outcome to be close to 1 as well.

Predicted Y=1 if the estimated probability is greater than 1/2

Predicted Y=0 if the estimated probability is less than 1/2

Then count how many times the predicted probability is equaled to the real outcome.

if the correct prediction is high we say the model fits well.

Question 18

What is the LDV goodness of fit compared to?

Answer 18

Compared to just taking what more observations were and then using that.

If this simplistic method is higher proportion than logistic method then the LDV is not suitable.