Problem Set1

Problem Set1 - answer Problem 12 If e is normally...

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Problem Set 1 Problem 1 There are four conventional ways to describe classical linear regression equations, and one of them is given as below. y t = β 1 + x 2 t β 2 + x 3 t β 3 + e t , e t iid (0 2 ) What are the other three representations? Problem 2 What is the meaning of β 2 ? Problem 3 What is the diﬀerence between a regression equation and its estimated regression equation? Give an example. Problem 4 List the classical assumptions. Problem 5 Under the classical assumptions, derive the OLS estimator ˆ β and its variance V ar ( ˆ β ) . Problem 6 Show that ˆ β is an unbiased estimator. Problem 7 State the Gauss-Markov theorem. Prove it when the number of regressors is one. Problem 8 Suggest an unbiased estimator of σ 2 implied by the OLS estimator ˆ β , and prove its unbiasedness. Problem 9 What is the estimated variance of ˆ β, \ V ar ± ˆ β ² ? What is the diﬀerence between \ V ar ± ˆ β ² and V ar ( ˆ β ) . 1

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Problem 10 What happens to V ar ( ˆ β ) if the sample size increases? What is the intuition behind your answer? Problem 11 What happens to V ar ( ˆ β ) if σ 2 decreases? What is the intuition behind your
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Unformatted text preview: answer? Problem 12 If e is normally distributed and σ 2 is known, what is the distribution for ˆ β ? Problem 13 Show that ˆ Y and ˆ e are orthogonal, and explain its implication. Problem 14 Explain the variance decomposition of the estimated regression equation and de-rive R 2 . Problem 15 Distinguish R 2 and ¯ R 2 . What drawbacks are there to the use of each of them as the primary determinant of the overall quality of a regression? Problem 16 Explain how to construct a random variable W ∼ χ 2 ( n ) given mutually inde-pendent standard normally distributed random variables Z 1 ,Z n ,..,Z n . Problem 17 Explain how to construct a random variable V ∼ t ( n ) given mutually independent W ∼ χ 2 ( n ) and Z ∼ N (0 , 1) . Problem 18 Explain how to construct a random variable K ∼ F ( n 1 ,n 2 ) given mutually in-dependent W 1 ∼ χ 2 ( n 1 ) and W 2 ∼ χ 2 ( n 2 ) . 2...
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Problem Set1 - answer Problem 12 If e is normally...

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