Week 1

Question 1

What is the Linear Probability Model?

Answer 1

A model where you try to predict the probability of a binary variable using the regular OLS model.

Question 2

What is the disadvantage of using a Linear Probability Model?

Answer 2

It can estimate probabilities outside the [0, 1] range.

Question 3

What are two models we can use to predict probabilities of a binary variable? On what are they based?

Answer 3

We can choose the following two models:

1. Logit model, based on a standard logistic distribution (i.e. a student t distribution with 9 degrees of freedom)
2. Probit model, based on a standard normal distribution

Question 4

What is a Latent variable?

Answer 4

A latent variabale is a variable that is not directly observed but can be inferred. We can create the latent variable e.g. y_i* for y_i.

Question 5

Derive the probabilities for P(y_i = 1 | x_i) in a binary probiy/logit model for the case that y_i = 1_{{yi* > 0}}.

Answer 5

Note: in the image is G(.) the CDF for the standard logistic or standard normal distribution.

Question 6

Why can we assume e_i ~ N(0, 1) without loss of generality? (i.e. why do we not essume e_i ~ N(μ, σ²))

Answer 6

We can set: (see image).

This means that we do not identify any of the beta’s, mu’s or sigma’s. Even with infinitely many observations we could not observe any difference, thus we can set mu and sigma to 0 without loss of generality.

Question 7

What is the marginal effect of e.g. x₁ on the estimated value (in a binary probit/logit model)?

Answer 7

Question 8

What is the PEA?

Answer 8

Partial Effect at the Average, it is just the derivative of the probability function (relative to x) and then at the mean of x.

Question 9

What is the APE?

Answer 9

Average Partial Effect, calculated as follows:

Question 10

What are the advantages and disadvantages of the APE compared to the PEA?

Answer 10

Question 11

What is a disadvantage of both the PEA and APE?

Answer 11

Question 12

Derive the likelyhood and loglikelyhood of the binary probit and logit model.

Answer 12

Question 13

What is the z-statistic, how do we use it?

Answer 13

The z-statistic is similar to the t-statistic (but since we use MLE it is only asymptotically valid). The z-statistic = β₁/SE₁. The 95% confidence interval is it z-statistic +/- 1.96.

If 0 is not within this range we reject H₀: β_j = 0.

Question 14

How can we compare estimated values between the binary logit and probit models?

Answer 14

We could divide them, e.g.

Question 15

How can we compare parameters between the binary probit and logit models?

Answer 15

1. We can compare the β_i by multiplying it by the difference in marginal effect, this means that multiplying the β of the probit model by 1.5958.
2. We can compare the β_i by comparing the standard deviation of the logistic and standard normal distribution. Then you the β of the probit model by 1.8138.

Question 16

What is the AIC? And what is the SC?

Answer 16

They both measure the relative (i.e. which one is less bad) quality of models with estimated likelyhood. For both holds that lower is better.

Question 17

What is the McFadden (pseudo) R-squared?

Answer 17

Higher (i.e. closer to 1) is better. This model is not relative.

Question 18

What is the Percentage Correctly Predicted?

Answer 18

Higher is better (obv.)

Question 19

How is the McFadden (pseudo) R-squared increasing in the likelyhood?

Answer 19

Since we have that ln L₀ < 0, we have that:

1 - ln L / ln L₀ = 1 + (ln L /- ln L₀)

This means that L is increasing in the McFadden R-squared.