Question

17.3

Consider a binary response variable y and an explanatory variable x. The following table contains the parameter estimates of the linear probability model (LPM) and the logit model, with the associated p-values shown in parentheses.

Variable	LPM	Logit
Constant	−0.69	−6.30
	(0.04)	(0.06)
x	0.05	0.15
	(0.07)	(0.06)

a.	Test for the significance of the intercept and the slope coefficients at a 5% level in both models.

Coefficients	LPM	Logit
Intercept	(Click to select)Not significantSignificant	(Click to select)Not significantSignificant
Slope	(Click to select)SignificantNot significant	(Click to select)Not significantSignificant

b.	What is the predicted probability implied by the linear probability model for x = 21 and x = 37? (Round intermediate calculations to 4 decimal places and final answers to 2 decimal places.)

	y-hat
x = 21
x = 37

c.	What is the predicted probability implied by the logit model for x = 21 and x = 37? (Round intermediate calculations to 4 decimal places and final answers to 2 decimal places.)

	y-hat
x = 21
x = 37

Answer 1

a.)

siginificance level, = 0.05

Null Hypotheses, Ho: Not significant

Alternate Hypotheses, Ha: Significant

For LPM Model:

P(Intercept) = 0.04 which is less than 0.05

Hence Null Hypotheses is rejected which means Intercept is Significant

P(Slope) = 0.07 which is not less than 0.05

Hence Null Hypotheses is not rejected which means Slope is not Significant

For LOGIT Model:

P(Intercept) = 0.06 which is more than 0.05

Hence Null Hypotheses is not rejected which means Intercept is not Significant

P(Slope) = 0.06 which is not less than 0.05

Hence Null Hypotheses is not rejected which means Slope is not Significant

b.)

LPM is given as

p = -0.69 + 0.05*x

-> For x = 21

p = -0.69 + 1.05

p =0.36

-> For x = 37

p = -0.69 + 37 * 0.05

p = 1.16

c.)

In case of logit model, we get

ln (p / (1-p)) = -6.30 + 0.15 * x

So, For x = 21

ln (p / (1-p)) = -6.30 + 0.15 * 21

ln (p / (1-p)) = 0.36

=> p / (1-p) = exp(0.36)

=> p / (1-p) = 1.433

=> p = 1.433 - 1.433*p

=>p = 1.433 / 2.433

=> p = 0.5889 = 0.59

For x = 37

ln (p / (1-p)) = -6.30 + 0.15 * 37

ln (p / (1-p)) = 1.16

=> p / (1-p) = exp(1.16)

=> p / (1-p) = 3.189

=> p = 3.189 - 3.189 * p

=>p = 3.189 / 4.189

=> p = 0.7612 = 0.76