HW05_answers_Econ140A_2011Fall

HW05_answers_Econ140A_2011Fall - Economics 140A Fall 2011...

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Economics 140A Fall 2011 Professor Startz Sketched Answers to Homework 5 1. a. The formula for the OLS variance under heteroskedasticity is ±²(³ ̂ ) ´ ∑ µ · ¸ · ¹∑ ¸ · º · In this case that’s just ±²(³ ̂ ) ´ µ » · ¸ · ¼ ½ ¶¾» ¿ µ · · ¸ · ¼ ¶¾¼ ½ À» (∑ ¸ · ¼ ½ ¶¾» ¿ ∑ ¸ · ¼ ¶¾¼ ½ À» ) · b. The GLS estimator requires dividing each observation by its corresponding standard deviation. ³ ÁÂà ´ Ä µ » · Å ¸ ¼ ½ ¶¾» ¿ Ä µ · · Å ¸ ¼ ¶¾¼ ½ À» Ä µ » · ¸ · ¼ ½ ¶¾» ¿ Ä µ · · ¸ · ¼ ¶¾¼ ½ À» 2. a. ³ ̃ ´ Ä Æ Å ¸ ´ Ä Æ ∑ Ç ³¸ ¿ È ¸ É ´ ³ ¿ Ä Æ ∑ Ê È ¸ ËÌ Í(³ ̃ ) ´ ³ ¿ Ä Æ ∑ Π͹Ⱥ ¸ Ï ´ ³ b. ÐÑÒ(³ ̃ ) ´ Ó Ê(³ ̃ Ô ³) · Ë ´ Ó ÇÊ » Æ Õ È ¸ Ë · É ´ Ê » Æ Ë · Ó ÇÊÕ È ¸ Ë · ÉÌ ´ Ê » Æ Ë · Õ µ · ¸ · ¸ · ´ µ · Æ c. Note that Ö × Ø × ´ ³ Ø × Ø × ¿ Ù × Ø × is the same as Ö × Ø × ´ ³ ¿ Ù × Ø × , so is just OLS on this equation. Note that this is a true equation, the RHS variable (1.0) is fixed, the errors have mean zero, are uncorrelated, and Í ÇÊ Ù × Ø × Ë · É ´ Ê » Ø × Ë · Í¹È · º ´ Ê » Ø × Ë · µ · ¸ · ´ µ · , so the errors are homoskedastic. All the Gauss-Markov assumptions hold for ³ ̃ , so ³ ̃ is BLUE.
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page 2 3. I began by generating data with the EViews commands: wfcreate u 1000 rndseed series x=nrnd series epsilon=nrnd series y=2*x+epsilon series xuhoh = x + nrnd (1) The two regressions gave me Dependent Variable: Y Method: Least Squares Date: 11/19/03 Time: 15:54 Sample: 1 1000 Included observations: 1000 Variable Coefficient Std. Error t-Statistic Prob. X
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HW05_answers_Econ140A_2011Fall - Economics 140A Fall 2011...

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