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Unformatted text preview: Introduction to Econometrics Professor Alexei Onatski Problem Set 5 – October 21 Problem 1) [ Stock and Watson5.4 ] (a) − 3.13 + 1.47 × 16 = $20.39 per hour (b) The wage is expected to increase from $14.51 to $17.45 or by $2.94 per hour. (c) The increase in wages for college education is β 1 × 4. Thus, the counselor’s assertion is that β 1 = 10/4 = 2.50. The t-statistic for this null hypothesis is 1.47 2.50 14.71, − 0.07 t = − which has a p-value of 0.00. Thus, the counselor’s assertion can be rejected at the 1% significance level. A 95% confidence for β 1 × 4 is 4 × (1.47 ± 1.97 × 0.07) or $5.33 ≤ Gain ≤ $6.43. Problem 2) [ Stock and Watson5.13 ] (a) Yes. The Gauss-Markov Theorem (Key Concept 5.5) says if (Y i , X i ) satisfy the assumptions in Key Concept 4.3 and if errors are homoskedastic then the OLS estimator is the Best Linear conditionally Unbiased Estimator. (b) Yes – See (a). (c) They would be unchanged. The Gauss-Markov Theorem does not require that the errors be normally distributed; it only requires homoskedasticity. (d) (a) is unchanged; (b) is no longer true as the errors are not conditionally homoskedastic. If the errors are heteroskedastic, GLS estimators have a smaller variance than the OLS estimators. Problem 3 ) a ) Consider the model: Y i = & + & 1 X i + u i ; (1) where X i is a binary variable. Assume that heteroskedasticity is known and has the following form: & V ar ( u i j X i = 1) = 4 V ar ( u i j X i = & 1) = 1 (2) From previous exercises (see problems 5 : 10 and 5 : 11 from the book), we know that the OLS estimator for this model will give us the following estimates: ( b & OLS + b & OLS 1 = Y m b & OLS & b & OLS 1 = Y w !...
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- Fall '09
- Econometrics, GLS, ui jxi