STATISTICS AND ECONOMETRICS

STATISTICS AND ECONOMETRICS

Problem Set IV

Fall 2011

Jose G. Montalvo Universitat Pompeu Fabra

PART A

1. The le wages.dta contains the data of Blackburn and Newmark (1992)

"Unobserved ability, e ciency wages and interindustry wage", Quarterly Journal of Economics, 107, 1421-1436. The data are organized as follows (obs=935):

wage: monthly earnings hours: average weekly hours IQ: IQ score

KWW: knowledge of world work score educ: years of education

exper: years of work experience tenure: years with current employer age: age in years

married: =1 if married black: =1 if black

south: =1 if live in south urban: =1 if live in SMSA sibs: number of siblings brthord: birth order

meduc: mother's education feduc: father's education

a) Consider the following speci cation

ln(wage_i) = ₁+ ₂EDU C_i+ ₃EXP ER_i+ ₄EDU C_i EXP ER_i+others_i+u_i where others include tenure, married, south, black and urban. What is the interpretation of ₂? Why is this coe cient so important? Estimate the model and discuss the results. Does the estimated coe cient of ₃ makes sense?

b) Write down the return for another year of experience for someone with 12 years of education and 10 years of experience using simple calculus. What is your estimate for another year of experience?

c) Reparametrize the model so that the coe cient on EXPER is the return to another year of experience starting at EXPER=10 and educ=12.

Call the coe cient of the new variable : Estimate the model and obtain a 90% coe cient interval for :

d) Drop the cross product term (experience by education) and add the variables IQ. Why do you want to include this variable?. What happen with the return to education? Explain.

e) Include KWW in the regression in d). Comment on the individual statistical signi cance of educ and exper. Are these three varaibles jointly signi cant? Explain.

f) Run the regression in e) but without IQ and KWW. Is the return to education higher or lower than before? Explain.

g) Estimate the return to education, using the model in e), at exper=10 and obtain a 95% con dence interval.

2*. (Only for advance students. Alternative to 1.) Hausman, Hall and Griliches (Econometrica, 1984) suggest that the Poisson distribution approx- imates relatively well the distribution of patents granted to a rm in any given year (n_it). The speci cation used is

P r(n_it) =

nit it e ^it

n_it! (1)

where is the conditional expectation of the distribution. The speci ca- tion of this conditional distribution is

E(n_itj X^it) = _it; n_it = exp( +

j=5X

j=0

jln(R&D_{t j}) + ₆T + u_it) (2) where T represents a time trend.

(a) Derive the log-likelihood function of a sample of N rms over T periods for this Poisson speci cation. Derive a estimator for the information matrix.

Is the likelihood function globally concave? Why?

(b) The le PATENTS.PRN contains the data used by Hall, Griliches and Hausman ("Patents and R&D: Is there a lag," International Economic

Review, 1986. You can nd this paper in the website of the course) and extend the original sample of 128 rms used by Hall, Hausman and Griliches (1984). The structure of the data is as follows: rm id, industry number, scienti c sector dummy, ln(capital in 1972), number, ln(R&D) (10 numbers, from 1970 up to 1979), number of patents (10 numbers, from 1970 up to 1979). There are 346 rms in this le. Using this data set estimate a Poisson model with the speci cation presented above.

(c) Use the likelihood ratio principle to test the joint hypothesis that

1 = ₂ = ₃ = ₄ = ₅ = 0.

PART B

3. The Poisson distribution is a good description of the probability of a rare disease when we are random sampling from a large number of inhabitants or the probability of getting a patent in a large random sample of rms. The discrete function is given by

f (X; ) = e ^X

X! ; X = 0; 1; 2; 3; :::

For a random sample from this Poisson distribution a) Obtain the MLE estimator for :

b) Show that your estimator is unbiased and consistent.

c) Derive the Cramer-Rao lower bound for any unbiased estimator of : Show that your ML estimator attains that bound.

c) Obtain the likelihood ratio test for testing H₀ : = 2 versus H₁ : 6= 2:

Derive also the Wald and the Lagrange Multiplier test statitistic for the same null hypothesis.

4. Consider the random variables x₁ and x₂, which are bivariate normal with x₁ N (0; ₁²); x₂ N (0; ₂²); and correlation : Show that the ex- pectation of x₁ conditional on x₂ is linear on x₂: Calculate the variance of x₁ conditional on x₂: How are these results modi ed if the means of x₁ and x₂ are ₁ and ₂? What is the importance of these result? (Hint: notice that we have justi ed linear regression as an approximation to a potentially non-linear conditional expectation).

5*. (Only for advance students. Alternative to 4) Let's assume that X is a random vector normally distributed,

X =

"

x₁ x₂

#

E(X) =

"

1 2

#

V CV (X) =

"

11 12

21 22

#

(3)

If you feel more confortable working with scalars then convert the matrices in scalars and solve the questions below.

(a) Show that there is a non-singular transformation, Z = CX, such that Z can be partioned into two subvectors, one corresponding to x₂and the other distributed independently of x₂. Hint: Use the following transformation

z₁ = x₁ x₂ (4)

z₂ = x₂ (5)

Calculate E(Z) and V CV (Z).

(b) Calculate the joint density function of (x₁; x₂) and the conditional density of x₁ given x₂, f (x₁ j x²). Interpret the expectation and the variance- covariance matrix of this conditional density in terms of a regression.

(c) Explain the importance of the previous results in the context of sto- chastic regressors and normality. Explain the relevance of the linearity of the conditional expectation under this conditions in terms of the interpretation of linear regression.

6. Consider the following speci cation for heteroskedasticity

b

u²_i = ₁+ ₂Z_2i+ ₃Z_3i+ ::: + _sZ_si+ v_i

a) Describe a F tets for the null hypothesis that all the parameters except the constant are equal to 0 (H₀ : ₂ = ₃ = ::: = _s = 0;no heteroskedastic- ity) using the sum of squares residual approach. Explain what is the form of the sum of squares residuals without constrains and with constrains.

b) When we discuss in class the issue of asymptotic properties one of the results implied that the asympotitc ² test could be written as the product of the degrees of freedom of the numerator of the corresponding F by the value of the F (rF). Use that result to obtain the asymptotic distribution of the F test in section a.

c) Explain why var(u^d2_i) =var(v^d_i): Explain why the var(v^d_i) =

Pb^v2i

N s

d) Show that var(u^d2_i) = _N^{1 P}(u_b²_i u_b²_i): Use the method of moments to explain the appropriatness of the calculation.

e) In the expresion of the chi-square of b) you got in the denominator var(vdi): Use the equality in c) and the de nition of the estimated variance of u²_i to show that the chi-square distribution in b) can be written also a N R²:

f) Under what conditions the result in e) can be called White's test?

7. Consider the following SURE system

Y1 = X1 1+ u1 (6)

Y₂ = X_{2 2}+ u₂ (7)

where X₁ and X₂ are non-stochastic matrices. Assuming that Eu₁ = 0 and Eu₂ = 0. The variance-covariance matrix of the perturbations is characterized by the following conditions: E(u1u⁰₁) = 11I; E(u2u⁰₂) =

22I; E(u₁u⁰₂) = ₁₂I where the values of _ij are known. Besides, the X's are orthogonal, X₁⁰X₂ = 0.

a) Obtain the SURE estimators and show their relationship with the least squares estimator. Interpret.

b) Compare the variance-covariance of the LS estimator with the variance- covariance of the SURE estimator.