Tutorial 6 - Binary Dependent Variables

Question 1

What are different approaches to estimate binary dependent variables?

Answer 1

• linear probability,
• logit,
• probit
• complementary loglog

Question 2

What is a Linear Probability model?

Answer 2

• can be estimate by OLS (only variance is different)
• Var(ϵ_i|x_i) = xœ_i‘β (1 − xœ_i‘β)
• > error term is heteroscedastic, adjust standard errors
• estimated values, ^y, can take any value, not restricted to [0; 1]-interval (not what we want to have)
• linear probability is still often used due to the easy interpretation and the flexibility of the linear model

Question 3

What are latent variables?

Answer 3

latent variables are variables that are not directly observed but are rather inferred (through a mathematical model) from other variables

Question 4

How does the latent variable model look like?

Answer 4

• y* latent variable, continuos and unobserved variable driving the dependent variable
• binary outcome variable y
  
  • y = 1 if y* > 0
  • y = 0 if y* ≤ 0
• eg. preferred working time in hours for working full time or ability to cover credit in Euro

Question 5

What is the probability for dependent variable y to be one under latent model?

Answer 5

Question 6

Which question does F(x) answer?

Answer 6

F(x) answers the question which share of distribution (described by f(t) ) is smaller (or equal) to the value x

Question 7

What is the distribution function for probit?

Answer 7

see below with where Φ(x) is the distribution function of the standard normal distribution -> probit

Question 8

What is the distribution function for logit?

Answer 8

see below with Λ(x) = standard logistic function

Question 9

What is complementary log-log model?

Answer 9

• Third alternative to logistic regression and probit analysis for binary response variables.
• Frequently used when the probability of an event is very small or very large -> has advantages for cases with average probabilities close to zero or one
• Unlike logit and probit, the complementary log-log function is asymmetrical.

Question 10

What is the distribution function for complementary log-log model?

Answer 10

extreme value distribution function:

Question 11

What type of estimator do linear probability, logit, probit and complementary loglog have?

Answer 11

Question 12

What are the distribution functions for linear probability, logit, probit and complementary loglog?

Answer 12

Question 13

What are the two ways of interpreting probabilistic models?

Answer 13

• average marginal effect
• marginal effect evaluated at average

Question 14

Which Tests for Goodness of Fit can you use for models with binary dependent variables?

Answer 14

• Pearson’s test
• Hosmer-Lemeshow test

(same test statistic below, but different groups)

Question 15

How can you apply Person’s test for goodness of fit?

Answer 15

1. form m groups according to covariates:
   
   • n_j: number of observation in group j
   • Y_j: number of observations being one,
   • ^p_j: predicted probability of being one
2. Sum of squared Pearson’s residuals (group residuals) approximately Χ²_M−K distributed

Question 16

How can you apply Hosmer-Lemeshow test for goodness of fit for models with binary dependent variables?

Answer 16

1. form m groups according to predicted probability ^p (typically ten equally large groups, with 0.0-0.1, 0.1-0.2,…):
   
   • nj: number of observation in group j
   • Yj: number of observations being one,
   • ^pj: predicted probability of being one
2. Test statistic is Χ²_M−K distributed

Question 17

How do you calculate the Receiver Operating Characteristic-curve?

Answer 17

Question 18

What is sensitivity?

Answer 18

Question 19

What is specificity?

Answer 19

Question 20

What does the ROC-curve plot?

Answer 20

Question 21

How would you interpret the coefficient below in a linear probability model?

lm(default ~ age + sex + income + children, data=data)

…

sex estimate: - 0.057 P-value: 0.047

Answer 21

probability of default changes by -6 percentage points for males, keeping everything else constant. it is significant at 5% level.

Question 22

What is the Wald-test?

Answer 22

Joint test of significance: no explanatory variable (except the intercept β₀) has explanatory power:

Question 23

What is the Wald-test test statistic?

Answer 23

with J degrees of freedom and J equals the number of restrictions (four in this example)