Econ4261_HW3_key - ECON 4261 Introduction to Econometrics...

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ECON 4261: Introduction to Econometrics Fall 2009 Problem Set 3 – Answer Key Exercise 1 Given the following OLS estimates, C t = constant + 0 . 92 Y t + ˆ u 1 t C t = constant + 0 . 84 C t - 1 + ˆ u 2 t C t - 1 = constant + 0 . 78 Y t + ˆ u 3 t Y t = constant + 0 . 55 C t - 1 + ˆ u 4 t Calculate the OLS estimates of β 2 and β 3 in C t = β 1 + β 2 Y t + β 3 C t - 1 + u t Answer: First, working in the deviation form, observe that: c t y t y 2 t = 0 . 92 c t c t - 1 c 2 t - 1 = 0 . 84 y t c t - 1 y 2 t = 0 . 78 y t c t - 1 c 2 t - 1 = 0 . 55 Now, we write all of the summations in terms of y t c t - 1 , we obtain: X y 2 t = 1 0 . 78 X y t c t - 1 X c 2 t - 1 = 1 0 . 55 X y t c t - 1 X c t y t = 0 . 92 X y 2 t = 92 / 78 X y t c t - 1 X c t c t - 1 = 0 . 84 X c 2 t - 1 = 84 / 55 X y t c t - 1 Writing the equation we want to estimate in deviation form: c t = β 2 y t + β 3 c t - 1 + u t Using the normal equations in matrix form i.e. x 0 x ˆ β = x 0 y where ˆ β = ˆ β 2 ˆ β 3 1
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Then: ˆ β = y 2 t y t c t - 1 y t c t - 1 c 2 t - 1 - 1 c t y t c t c t - 1 = 1 / 0 . 78 1 1 1 / 0 . 55 - 1 92 / 78 84 / 55 = 0 . 464 0 . 585 Exercise 2 Prove that y 0 y where y = Ay with A = I - 1 n ii 0 , is the RSS when Y is regressed on x = i . ( i is a column vector of ones) Show also that the estimated coefficient of the regression is ¯ Y .
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