Problem Set # 1

Problem Set # 1

Master in Business and Quantitative Methods

Contents

Chapter1. Exercises on endogeneity of regressors

Chapter 2. Exercises on models with discrete dependent variables

1

0.1 Exercises on endogeneity of the regressors

Problem 1 Suppose that in the modely_i=X_iβ+u_i, (X_i, u_i) are an inde- pendent and identically distributed sequence of random vectors such thatX_i has a finite mean vector, µ_x, finite positive definite covariance matrix Σ_xx and finite fourth moments E[X_jX_kX_lX_m] = φ_jklm for all variables. How does the proof of consistency and asymptotic normality of βˆ change with these new hypotheses aboutX_i?

Problem 2 In the context of the instrumental variables we have shown that the least squares estimatorβˆ is biased and inconsistent. Nonetheless,βˆ does estimate something. Derive the asymptotic covariance matrix ofβ, and showˆ thatβˆ is asymptotically normally distributed.

Problem 3 Consider a model for the health of an individual:

healthi=β0+β1agei+β2weighti+β3heighti+β4malei+β5worki+β6exercisei+ui,

where health is some quantitative measure of the persons health, age, weight, height, and male are self-explanatory, work is weekly hours worked, and exercise is the hours of exercise per week.

a. Why might you be concerned about exercise being correlated with the error termu?

b. Suppose you can collect data on two additional variables, disthome and distwork, the distances from home and from work to the nearest health club or gym. Discuss whether these are likely to be uncorrelated with u.

Problem 4 Consider the model:

y^∗ =βx^∗+ε

Prove that when only x^∗ is measured with error, the squared correlation between y and x is less than that between y^∗ and x^∗. (Note the assumption thaty^∗=y). Does the same hold true if y^∗ is also measured with error?

Problem 5 The consumption function used in class is a very simple specifi- cation. One might wonder if the meager specification of the model could help

explain the finding in the Hausman test. The data set is called consump- tion.txt. Use these data to carry out the test in a more elaborate specification

c_t=β₁+β₂y_t+β₃i_t+β₄c_t−1+ε_t

wherec_tis the log of real consumption,y_tis the log of real disposable income, and i_t is the interest rate (90-day T bill rate).(Hint: use as instrument of y_t, y_t−1).

0.2 Exercises on models with discrete dependent variables

Problem 6 Evaluate the following statement: ’Estimation of a linear prob- ability model is more robust than probit or logit because the LPM does not assume homoskedasticity or a distributional assumption.’

Problem 7 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(z₁δ₁/(1 +γ₁²z₂²)^0.5).

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

d. If you have reason to believeρ₁>0, how would you estimateδ₁ along withρ₁?

Problem 8 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

y_i^∗ =−13.021

−2.641 + 2.826GP A_i

2.238 + 0.095T U CE_i

0.672 + 2.379P SI_i

2.234 logit where y^∗ is a non observable variable such that if y^∗_i >0, y_i = 1 (the grades of studentiimprove) and ify_i ≤0 (the grades of studentido 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 9 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?