Logit Model

Question 1

Why do we use a Logit model

Answer 1

• Probit model estimation is numerically complicated
• The logistical distribution is more suitable, giving rise to the logit model

Question 2

If L is a logistic random variable, what is it’s probability density function

Answer 2

• λ(l) = e^-l / (1+e^-l)^2
• for -∞ < l < ∞

Question 3

What is the cumulative function of a logistic random variable

Answer 3

• Λ(l) = P(L <= l) = 1 / 1 + e^-l = e^l / 1 + e^l

Question 4

How can we model a LPM using the Logistic function

Answer 4

• P(y = 1) = pi = 1 / 1 + e^-(b0 + b1xi) = e^(b0+b1xi) / 1 + e^(b0 + b1xi)
• P(y = 0) = 1 - pi = e^-(b0+b1xi) / 1 + e^-(b0+b1xi) = 1 / 1 + e^(b0+b1xi)

Question 5

How are the coefficients for the logit model interpreted

Answer 5

• We use the marginal effects

Question 6

How is the marginal effect for a continuous x calculated

Answer 6

• Take the partial derivative of pi with respect to x
• Set b0 + b1x = xb

Question 7

How is the marginal effect for a discrete x calculated

Answer 7

• The change in probability is computed

Question 8

What is another way to combat the limits of p between [0,1]

Answer 8

• The probability p at the LHS can be transformed so there are no restrictions

Question 9

What is an odds ratio

Answer 9

• The ratio of the probability of favourable to unfavourable cases
• odds = p / 1 - p

Question 10

How does an odds ratio work

Answer 10

• The odds are larger than one when the probability of a favourable case is higher than the probability of an unfavourable case
• Odds can take any positive value, so they have no ceiling

Question 11

How can we transform the odds ration to remove the floor restriction

Answer 11

• We can take logarithms to obtain the log-odds/logit
• ν = logit(p) = ln(p / 1 - p)

Question 12

How is negative and positive logit represented with odds

Answer 12

• odds < 1 = negative logit
• odds > 1 = positive logit

Question 13

How are odds ratios interpreted

Answer 13

• The odds of y are multipled by the OR