Probit Model

Question 1

What is the first steps in moving from a LPM to a probit model

Answer 1

• We allow p to vary, depending on x

Question 2

How do we keep p within the [0,1] interval

Answer 2

• We need a non-linear relationship between x and p

Question 3

What type of curve do we need for the interval of p to be between [0,1]

Answer 3

• An S-shaped curve
• The slope is not constant as in the LPM

Question 4

What defines the probit model

Answer 4

• The choice of a standard normal distribution for the relationship between x and p

Question 4

Why cant the coefficients be used to find the magnitude of the effect for a probit model

Answer 4

• The x’s are a non linear function of p
• Interpretation is done through the examination of marginal effects of a one-unit change in x on the probability that y=1

Question 5

Write down how a probit model is defined

Answer 5

• pi = P[Z <= b0 + b1xi]
• pi = Φ(b0 + b1xi)
• pi = integral of 1/(2π)^1/2 * e^-t^2 / 2 dt between b0+b1xi and -∞

Question 6

How to we find the partial effect of a continuous variable in a probit model

Answer 6

• We need to take the partial derivative with respect to that variable
• ∂Pr(y=1|x1,…,xk) / ∂xk = ∂Φ(b0h + b1h * x1 + … + bkh * xk) / ∂xk = φ(b0h + b1h * x1 + … + bkh * xk) * bkh

Question 7

What is the difference between φ and Φ

Answer 7

• φ is the normal density function of Φ
• The derivative of the cumulative distribution function Φ is the density function φ

Question 8

How is the marginal effect of the variable determined

Answer 8

• Determined by the sign of bkh, since the pdf part of the marginal effect is always positive

Question 9

How is computing the partial effect of a discrete variable different to a continuous variable

Answer 9

• The first derivative cannot be used
• We must find the discrete change in probability

Question 10

How do we find the discrete change in probability for a discrete random variable

Answer 10

• Subract P(y=1 | x=1) from P(y=1 | x=0)
• Φ(bh0 + bh1 * 1 + bh2 * x2 + … + bhk * xk) - Φ(bh0 + bh1 * 0 + b2h * x2 + … + bhk * xk)