Introduction to Econometrics Final Examination Fall 2006 Answer Sheet

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1 Introduction to Econometrics Final Examination Fall 2006 Answer Sheet Please answer all of the questions and show your work. If you think a question is ambiguous, clearly state how you interpret it before providing an answer. All question parts have a value of 8 points.. A latent variable y is related to a (scalar) covariate x by y i β 0 + β x i + ε i, where ε i is independently and identically distributed as a Uniform random variable on the interval [ a, a],.a>0. Recall that the probability density function (p.d.f.) of ε in this case is f(ε) for a [ a, a], 2a and the cumulative distribution function is F (ε) (ε + a)/(2a) for a [ a, a]. Assume that you have access to a random sample of N observations, {y i,x i } N i, where ½ if y y i i > 0 0 if yi 0. You estimate the linear regression model (y i.5) δ 0 + δ x i + w i by Ordinary Least Squares. Denote the OLS estimates of δ by ˆδ. (a) In the linear regression model, is E(ˆδ) δ? Answer: E(y Xβ) P (y X) P (Xβ + ε>0).5+ Xβ 2a E(y.5 X) Xβ 2a. This implies that E(w X) 0, so that OLS is an unbiased estimator of δ, where δ β/(2a).

2 (b) If (a) is true, is the OLS estimator the Best Linear Unbiased Estimator of δ? Answer: For OLS to be BLUE requires the disturbances to be i.i.d. (0,σ 2 ). The variance of P (y X) (Xβ)( Xβ)/4a 2, which is obviously not i.i.d. Thus OLS is not BLUE. (c) We aren t really interested in the parameters δ. Determine whether Eˆδ β. If not, what additional information would you need to be able to define an unbiased estimator of β (in addition to ˆδ )? Answer: Since we have an unbised estimator of ˆδ, and since β (2a)δ, knowledge of a is sufficient to allow us to form an unbiased estimator of β. E(2aˆδ )2aE(ˆδ )2a β 2a β. 2

3 2. Consider the panel data regression model where y it βx it + ε it ε it ρ i ε it + u it, and where u it is i.i.d. with mean 0 and variance σ 2 u, and where ρ i < for all i, and the ρ i are unknown. You have access to a random sample of observations from the population for N individuals and 2 time periods, {y it,x it } i,...,n;t,2. We assume that x it is a scalar for simplicity, and that all variables are measured as deviations from their means. (a) Using both time periods of data, define the OLS estimator of β. Determine whether the OLS estimator is unbiased. Answer: Stack the X observerations by time period, and call the matrix X, and similarly stack the y and the disturbances ε. Then ˆβ (X 0 X) X 0 y. The ε process is strictly exogenous, meaning that E(ε X) 0. OLS is unbiased. (b) Is it possible to form a Feasible Generalized Least Squares estimator for β? Why or why not? Answer: The covariance matrix of ε is σ 2 ρ σ 2 σ 2 2 ρ 2 σ E(εε 0 X) σ 2 N ρ N σ 2 N ρ σ 2 σ 2, ρ 2 σ 2 2 σ ρ N σ 2 N σ 2 N where σ 2 i σ2 u/( ρ 2 i ). The covariance matrix depends on N +parameters, ρ,...,ρ N and σ 2 u. Since each estimate of ρ i is based only on 2 pieces of information, as N we will not be able to consistently estimate ρ i (this would be possible only if T as well). Therefore we cannot consistently estimate the covariance matrix of the disturbances, and cannot form a consistent FGLS estimator.

4 (c) Is it possible to form a FGLS estimator for β if ρ i ρ for all i? If not, why not? If so, define the estimator. Answer: Yes. Now the covariance matrix depends on only two unknown parameters. Consistent estimators for the common variance term and the common covariance term are given by where r it is the OLS residual. (2N) 2X NX t i X N N r i r i2,. A population of unemployed job searchers has completed spell lengths that are uniformly distributed on the interval [0, a], that is, the probability density function and the cumulative distribution functions are: i r 2 it f(t) a,t (0,a], α>0. 0 if t 0 t F (t) a if 0 <t a. if t>a You have access to administrative records that record the length of time an individual was unemployed if the individual was unemployed for at least months. For all unemployment spells lasting less than months, no information is available, not even the number of individuals experiencing such an unemployment spell. There are N cases in the administrative records. (a) Can the administrative records be used to consistently estimate a? Answer this by writing down the log likelihood function for the administrative records. Answer: The maximum likelihood estimator of a is given by the maximum spell length observed in the sample. Since we do not observe spells less than months, but do observe spells greater than three months, the missing information has no informative value. Thus â max{t i } N i. This is biased, but consistent estimator of a. (b) Say that the administrative agency won t release the individual unemployment spell information to you. Instead, they are willing to tell you that the mean duration of unemployment in their administrative data base is 8. Can this 4

5 information be used to form a consistent estimator of a? If so, provide the estimator. If not, why not? Answer: Z a /a E(t t >) t p(t >) dt. Now p(t >) a a, so E(t t > ) a Z a Z a /a t (a )/a dt tdt t 2 a a 2.5 a (a2 9).5 (a +)(a ) a.5(a +). Then since we have a consistent estimator of E(t t >), a consistent estimator of a is â 2E(t t d>) 6. 5

6 4. A cross-sectional population regression model is given by y i βx i + ε i, where y i and x i are expressed as deviations from their respective means, and ε i is independently and identically distributed (i.i.d.) with mean 0 and variance σ 2 ε. You have access to a random sample of observations on {y i,x i }N i. The variable x i is a noisy measure of x i, that is, x i x i + u i, where u i is i.i.d. with mean 0 and variance σ 2 u > 0. (a) Say that you form the estimator Show that Answer: ˆδ i y ix i i (x i )2. plim ˆδ 6 β. plim ˆδ plim N i (βx i + ε i )(x i + u i ) plim N i (x i + u i ) 2 βσ 2 x σ 2 x + σ 2 6 β. u (b) Say that you have access to another variable {z i } N i, which is also expressed as a deviation from its mean, that has the following properties in the population: E(zx) 6 0 E(εz) 0. WIth this variable, can you define a consistent estimator of β? If so, define it, and provide a short proof of consistency. Answer: Define i β z iy i i z ix i Now plim β plim N i (βx i + ε i )z i ) plim N i (x i + u i )z i βcov(x, z) cov(x, z)+cov(u, z). Thus this estimator is consistent if cov(u, z) 0and is not if this is not the case. 6

7 5. You want to estimate the time series model where y t β 0 + β x t + ε t, ε t u t + αu t, and u i is independently and identically distributed with mean 0 and variance σ 2 u for all t. You have access to a sample of observations {y t,x t } T t. (a) Let x X.. x T, β (β 0 β ) 0, and y (y... y T ) 0. Is the OLS estimator ˆβ (X 0 X) X 0 y an unbiased estimator of β? Answer: Yes, the ε process is strictly exogenous with mean 0. (b) Could you implement a Feasible GLS estimator for β for this problem? If so, describe the estimator in detail. Answer: Yes. The covariance matrix is given by ( + α)σ 2 u ασ 2 u 0 0 ασ 2 u ( + α)σ 2 u ασ 2 u 0 E(εε 0 X) 0 ασ u ( + α)σ 2 u ασ 2 u 0 0 ασ 2 u ( + α)σ 2 u This is characterized by two unknown parameters. The terms on the diagonal of the matrix can be consistently estimated by the average squared residual. The terms on the off-diagonal by the average residual crossproduct for residuals separatedbyperiod. 7