Week 5

Question 1

Linear Probability Model (LPM)

Answer 1

P(yi = 1) = pi = E(yi) = xi’β

Question 2

What are the problems with LPM?

Answer 2

1) Distribution of error terms not normal (discontinuous)
2) Heteroskedasticity
3) OLS is unbiased and consistent, but inefficient
4) OLS ignores restriction 0<=pi<=1 s.t. fitted values can lay outside [0,1]

Question 3

What is the transformation from linear to non-linear model for p?

Answer 3

P(yi = 1) = F(xi’β)

Question 4

Logit model

Answer 4

εi ~ LOG(0,1)

F(xi’β) = 1 / (1 + exp(-xi’β))

Question 5

Probit model

Answer 5

εi ~ N(0,1)

F(xi’β) = Φ(xi’β)

Question 6

What is the difference between the logit and probit model?

Answer 6

β(logit) ≈ 1.8β(probit)

σ(logit) = π/3

Question 7

How can the parameters of logit/probit models be estimated?

Answer 7

ML: yi~Bernoulli(pi)

f(yi) = 1 - pi, if yi = 0
= pi , if yi = 1

L(β) = f(y1, ..., yn) = Πf(yi) = Πpi^(yi)(1-pi)^(1-yi)
ℓ(β) = Σyilog(pi) + (1-yi)log(1-pi)

Question 8

What is the ML estimate’s distribution and properties?

Answer 8

β^ ~ N(β, V^)

V^ = (Σpi(1-pi)xixi’)^(-1)

Consistent and asymptotically normal

Question 9

What is identification?

Answer 9

A vector of parameters is identified by a given data set, model, and estimation method if, for that data set, the estimation method provides a unique way to estimate the parameters in the model.

Unless there is collinearity, parameters in the linear model are always identified

Identification issues commonly arise in non-linear models; if not identified it is impossible to pin down the estimates of the true value -> impose restrictions

Question 10

What are the 2 identification issues?

Answer 10

1) The variance σ^2 of the latent variable is unidentified

2) The threshold T is unidentified

Question 11

What is the marginal effect of the logit, profit and LPM model?

Answer 11

LPM: βj
Logit: P(yi=1)P(yi=0)βj
Profit = Φ(xi’β)βj

Question 12

Odds Ratio

Answer 12

P(yi=1) / P(yi=0)

Describes the importance of x in terms of determining the probability of outcome 1 in terms of the probability of outcome 0

Question 13

Odds Ratio - Logit

Answer 13

P(yi=1)/P(yi=0) = exp(xi’β)

0 -> two outcomes equally likely
+ -> outcome 1 more likely
- -> outcome 0 more likely

Question 14

If all variables are demeaned, what does the sign of β denote?

Answer 14

Average preferred choice

Question 15

Diagnostics of logit/probit models

Answer 15

Residuals: ei = yi - F(xi’β) = yi - pi^

Standardized residuals: ei = (y - pi^) / sqrt(pi^(1-pi^)

McFadden R^2: 1 - ℓ(β^)/ℓ(β0^) => 0<=R^2<=1 (β0^ is model with only intercept)
larger value => more that the regressors explain outcome

Question 16

Hit Rate

Answer 16

h = p00 + p11 (when realization = prediction)

Question 17

How to test whether predictions of model are better than “random”?

Answer 17

q = p1^2 + (1-p1)^2

h -q/sqrt(q(1-q)/n) ~ N(0,1)

Stat large => predictions better than random