Introduction to Econometrics Midterm Examination Fall 2005 Answer Key

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1 Introduction to Econometrics Midterm Examination Fall 2005 Answer Key Please answer all of the questions and show your work Clearly indicate your final answer to each question If you think a question is ambiguous, clearly state how you interpret it before providing an answer All question parts have equal weight, and there are 7 in total Be sure to write your name on your answer books! 1 Consider the following relationship: y i β 0 + β 1 x i + ε i, where x is an exogenous variable and i is a member of the population of interest Define the matrix 1 x 1 1 x 2 X, 1 x N and the vectors and write Y y 1 y 2 y N, ε ε 1 ε 2 ε N Y Xβ + ε, β β1 for the relationship between Y and X in a random sample of size N You can assume that E(ε X) 0 for all X and E(εε 0 X) σ 2 εi N, where I N istheidentitymatrixofdimensionn β 2, 1

2 You are given the following information from a random sample of size N 100 X 0 X , y i 40 yi X yi x i 60 (Recall that the inverse of a 2 2 matrix a b c d is equal to (ad bc) 1 d b c a Compute each estimate or say why you don t have enough information to do so: (a) The Ordinary Least Squares (OLS) estimate of β, ˆβ Answer: We already have been given X 0 X We find that the inverse of this matrix is (X 0 X) 1 ( ) ) Now Then X 0 y ˆβ ˆβ

3 (b) The estimated covariance matrix of ˆβ Answer: To compute this, we need an estimate of σ 2, the variance of the disturbance term To compute the unbiased estimator s 2 (N 2) 1 r2 i, we need to know ri 2 (y i ˆβ 0 ˆβ 1 x i ) 2 [yi 2 + ˆβ ˆβ 2 1x 2 i 2y iˆβ0 2y iˆβ1 x i +2ˆβ 0ˆβ1 x i ] (304) 2 +(236) (304)40 2(236)(60) + 2(304)(236) , so that s /(100 2) 126 Then the estimated covariance matrix of ˆβ is ˆΣˆβ 126 (X 0 X) Consider the linear relationship y i αx i, (1) where y i and x i are both expressed as deviations from their respective sample means (thus y i x i 0) In both of the questions that follow we assume that x i is measured without error in the sample information available to you (a) First consider the case where y i is measured with error, where the measured value of y is given by yi y i + u i, where u i is independently and identically distributed (iid) with mean 0 and variance σ 2 u The information available to you is yi, which is measured as a deviation from the sample, and x i,,,n Find unbiased estimators of α and σ 2 u Is your estimator of α best linear unbiased? Answer: After substitution, we have y i αx i + u i Since u i is iid, it is mean independent of x i Thus the OLS estimator of α, ˆα y i x i x2 i 3

4 is unbiased Since the disturbance term u is homoskedastic, the OLS estimator is BLU by the Gauss-Markov theorem An unbiased estimator for the variance of the measurement error is ˆσ 2 u (N 1) 1 N X (y i ˆαx i ) 2 (b) Now return to the specification given in (1) - in particular, assume that y i is measured without error Assume that individuals in the population differ in their value of α Say that α is iid with mean α and variance σ 2 α From the sample information {y i,x i } N, derive unbiased estimators for α and σ2 α (Hint: Recognize that we can always write α i α + ε i, where the ε i have mean 0 by construction) Answer: We can write y i αx i + ε i, where ε i (α i α)x i, which has mean 0 and conditional (on x) variance x 2 i σ2 α Then the OLS estimator bα y ix i x2 i is an unbiased estimator of the mean of α due to the mean independence property E(ε x) E((α α)x x) E((α α) x) x 0 since α is iid We can form an unbiased estimator of σ 2 α from ˆσ 2 α (y i bαx i ) 2 3 In class we discussed the linear probability model at length, in which a binary (ie, 0-1) variable d was the dependent variable In matrix notation, we wrote d Xβ + ε, where X was N K dimensional matrix, with N>>Kand rank K, d and ε were N 1 vectors and β was a K 1 vector of unknown regression coefficients 4 x 2 i

5 (a) Derive the distribution of ε i conditional on X i for this model and show that E(ε i X i )0 Answer: The disturbance can take two values in this case The probability distribution is d i ε i prob(ε i ) 1 1 X i β X i β 0 0 X i β (1 X i β) This follows since E(d X) P (d 1 X) Xβ It is easy to see that E(ε X) (1 X i β)(x i β) X i β(1 X i β) 0 (b) Derive the covariance matrix of the OLS estimator of the linear probability model (remember that the actual covariance matrix is a function of the true value of β and the sample X, and does not involve an estimate of β) Answer: We know that OLS is unbiased, and ˆβ β (X 0 X) 1 X 0 ε, so that the covariance matrix is given by E(ˆβ β)(ˆβ β) 0 X (X 0 X) 1 XE(εε 0 X)X(X 0 X) 1, where E(εε 0 X) (X 1 β)(1 X 1 β) (X 2 β)(1 X 2 β) (X N β)(1 X N β) (c) In practical situations we do not know β, so we must estimate the covariance matrix of the OLS estimator It is natural to replace the unknown β with ˆβ What numerical issues might arise when computing the estimated covariance matrix of ˆβ in this case? Answer: When we replace β in the matrix E(εε 0 X) with ˆβ, there is nothing to guarantee that X iˆβ (0, 1) for all i, i 1,,N In this case the estimated covariance matrix of the OLS estimator is not well-defined (ie, it is not positive definite, which is the basis requirement of a covariance matrix) 5