HW1_Solution (1) - Solution to Econ 140 Assignment 1...

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Solution to Econ 140 Assignment 1 Appendix C: C.1 (i) This is just a special case of what we covered in the text, with n = 4: E() = µ and Var() = σ 2 /4. (ii) E( W ) = E( Y 1 )/8 + E( Y 2 )/8 + E( Y 3 )/4 + E( Y 4 )/2 = µ [(1/8) + (1/8) + (1/4) + (1/2)] = µ (1 + 1 + 2 + 4)/8 = µ , which shows that W is unbiased. Because the Y i are independent, the covariance terms vanishes, we have Var( W ) = Var( Y 1 )/64 + Var( Y 2 )/64 + Var( Y 3 )/16 + Var( Y 4 )/4 = σ 2 [(1/64) + (1/64) + (4/64) + (16/64)] = σ 2 (22/64) = σ 2 (11/32). (iii) Because 11/32 > 8/32 = 1/4, Var( W ) > Var() for any σ 2 > 0, so is preferred to W because provides a more precise estimation. Appendix C: C.6 (i) Null H 0 : = μ 0 (ii) Alternative H 1 : < μ 0 (iii) Note this is a one-sided hypothesis, t statistic is = - =- . - × . =- . T y 01ns 32 8 01900 466 4 2 11 p-value is the probability of observing a t statistic T at least as far from 0 as your actual estimate =- T . 2 11 given that the null hypothesis is true. For one-sided hypothesis testing = - . = . p F 2 11 0 017 di normal(-2.11) Because . < = . < . 0 01 p 0 017 0 05 , we reject null hypothesis at 5% level, but fail to reject null hypothesis at 1% level. (iv) We have around 98% confidence to conclude that there is indeed a fall in liquor consumption statistically; practically we might infer that a tax liquor on liquor has decreased average liquor consumption in the state. (v) To infer the causality from the tax change to liquor consumption, we have to ensure there is no omitted variable bias. Otherwise, if there exists some variable which is both related to the tax level and the liquor consumption, the causality may not hold. For example, the tax on other
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