Binary responses

Question 1

what does the line B0 +B1X mean?

Linear regressor with a single regressor
Y = B0 +B1X+ u

Answer 1

The conditional mean of Y given X

E[Y|X]

Question 2

what does B1 mean?

Linear regressor with a single regressor
Y = B0 +B1X+ u

Answer 2

The partial affect of X on E[Y|X] (on conditional mean of Y given X)

^E[Y|X] / ^X

^ = average

Question 3

What hoes the predicted value ^Y mean?

^Y(estimation of Y)

Answer 3

A predicted conditional mean of Y given X

^E[Y|X] - ^b0 +^b1X

Question 4

what does the line B0 +B1X mean?

E[Y|X] - b0+b1X

Answer 4

P(Y=1|X)

Probability of Y +1 is conditional on X

Question 5

Linear probability model (advantages)

Answer 5

• Simple to estimate and to interpret

- inference is the same as for multiple regression (need heteroskeddasticity - robust standard errors)

Question 6

Linear probability model (disadvantages)

Answer 6

• A LPM says that the change in the predicted probability for a given change in X is the same for all values of X
• Also, LPM predicted probabilities can be < 0 or > 1!

These disadvantages can be addressed by using a nonlinear probability model: probit and logit regression

Question 7

a nonlinear functional form is used when we want:

Answer 7

1. P(Y=1|X) to be increasing in X for b1 >0,

2. 0

Question 8

Differences between linear and nonlinear functions (FORMULAS)

Answer 8

Linear: P (Y+1|X) = b0 + b1X

Non-linear: P (Y+1|X) = F(b0 + b1X)

F is a cumulative distribution function of a random variable

Question 9

Probit regression model

Answer 9

P (Y=1|X) = o| ( b0 + b1X)

o| is the cdf of standard normal distribution

b0 + b1X is the “z value” of the probit model

Question 10

Why use normal cdf? (probit regression)

Answer 10

• two properties are satisfied
• easy to use
• straightforward interpretation of “z value”
• a latent variable model with normal error u

Question 11

Probit regression model (formula

Answer 11

P (Y=1|X) = o| ( b0 + b1X)

F(x) = [ 1 / 1 + e^ (b0 + b1X) ]

Question 12

cdf of logistic distribution

Answer 12

• logit and probit use different probability functions, the coeficients (b’s) are different in logit and probit
• in practice, logit and probit are very similar - since empirical results typically don’t hinge on the logit/ probit choice, both tend to be used in practice

Question 13

P (Y=1|X) = o| ( b0 + b1X)

how can we estimate b0 and b1 ?

Answer 13

• Non linear least square

- Maximum likelihood estimation (MLE)

Question 14

what is the OLS estimator?

Answer 14

It is the coefficient values that minimize the squared residuals.

min E [ Y - ( b0 + b1X) ]^2

Question 15

Nonlinear lest square

Answer 15

• no explicit solution exist in general
• Solved numerically using the computer (specialized minimization algorithms)
• In practice, MLE is used more ( a more efficient estimator )

min E [ Y - o| ( b0 + b1X) ]^2

Question 16

likelihood function

Answer 16

• The conditional density of Y1,…,Yn given X1,…,Xn, treated as a function of the unknown parameter b0 and b1
• the probability that data was generated with the given parameter values

Question 17

Maximum likelihood estimator (MLE)

Answer 17

• The values of (b0, b1) that maximizes the likelihood function
• “the probability thqat data was generated with the parameter values of MLE is the largest.”
• MLE is consistent, asymptotically normal, and efficient ( with the minimum variance)