0.1 Exercises on models with discrete dependent variables

Part B of Problem Set # 1

Master in Business and Quantitative Methods

Contents

Chapter2. Exercises on models with discrete dependent variables.

1

0.1 Exercises on models with discrete dependent variables

Problem 1 Consider the following model:

y_i^∗=β¹+β²Di+ui

and

yi= 1 if y^∗_i >0 yi= 0 if y^∗_i ≤0,

whereDiis a dummy variable anduiis the error term that follows a standard normal distribution.

In a sample of n observations of D and y, the number of observations in each of the four combinations below is A, B, C and D such that:

Di = 0 Di= 1

yi= 0 A B

yi= 1 C D

a. Write the log likelihood and obtain the system of first order conditions that allows to obtain the maximum likelihood estimators of β1 and β2

in function of A, B, C and D.

b. Characterize the solution of the previous system and comment the re- sults in the following cases:

1. A=B=C=D;

2. A=D=0;

Problem 2 Consider the probit model:

P(y = 1|z, q) = Φ(z₁δ₁+γ1z2q),

where q is independent of z and distributed as Normal(0,1); the vector z is observed but the scalar q is not.

a. Find the partial effect of z² on the response probability, namely, δP(y= 1|z, q)

δz2

b. Show that P(y= 1|z) = Φ(z₁δ

1/(1 +γ²1z2²)^0.5).

c. Defineρ1≡γ1² . How would you test H0:ρ1= 0?

d. If you have reason to believeρ1>0, how would you estimate δ1 along withρ¹?

Problem 3 Spector y Mazzeo (1980) analyzed the performance of a new learning system called PSI (Personalizad System of Instruction), and ob- tained by ML the following results:

ˆ

yi∗ =−13.021

−2.641 + 2.826GP Ai 2.238

+ 0.095T U CEi 0.672

+ 2.379P SIi 2.234

logit

where y^∗ is a non observable variable such that if y^∗_i >0, yi = 1 (the grades of studentiimprove) and ifyi ≤0(the grades of student ido not improve), GPA is the average grade, TUCE is the grade obtained in a pretest and PSI=1 if the student was exposed to the new learning system and 0 otherwise.

a. Explain how the estimates were obtained.

b. Obtain the marginal effects of each explanatory variable on the proba- bility of grade improvement.

c. Determine the probability of improvement when:

i. GPA=4,TUCE=20 and PSI=1 ii. GPA=4,TUCE=20 and PSI=0 Comment the results.

Problem 4 Letpatentsbe the number of patents applied for by a firm dur- ing a given year. Assume that the conditional expectation of patents given salesand RD is

E(patents|sales, RD) =exp(β0+β1log(sales) +β2RD+β3RD²), where sales is annual firm sales and RD is total spending on research and development over the past 10 years.

1. How would you estimate the βj? Justify your answer by discussing the nature of patents.

2. How do you interpret β1?

3. Find the partial effect of RD on E(patents|sales, RD).

Problem 5 Use the data in SMOKE.RAW to answer this question.

a. Use a poisson regression model to explaincigs, the number of cigarettes smoked per day. Use as explanatory variableslog(cigpric),log(income), restaurn,white, educ, age, and age². Are the price and income sig- nificant variables?