ARE106SU09HW2KEY

ARE106SU09HW2KEY - 1 University of California Davis Managerial Economic(ARE 106 Summer 2009 Instructor John H Constantine KEY Problem Set#2 Due

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1 University of California, Davis Managerial Economic(ARE) 106, Summer 2009 Instructor: John H. Constantine KEY —Problem Set #2: Due MONDAY, August 17, 2009 Problem 1 (A table is provided at the end of the assignment to calculate the numbers) : The following data give the daily price (x, ($/unit)) and consumption (y) of some commodity Z; the data were collected at nine shops in Davis that sell Z for a one week period. The objective is to estimate the demand for Z. Obs x y 1 51 0 2 31 5 3 15 8 4 91 2 5 14 6 6 41 8 7 32 2 8 61 4 9 81 1 (a) Plot the data. (b) The PRF for this model is: y i = β 1 + β 2 x i + e i . Use the method of least-squares to estimate (by hand) the sample regression function. You must provide numeric estimates for the following variables: (i) β 1 (ii) β 2 (iii) σ 2 (iv) var( β 2 ) (v) s.e.( β 2 ) (c) State the SRF for the given sample above. Plug in the parameter estimates from part (b) into this equation . y = 19.61 – 0.90 x .
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2 (d) What is the economic interpretation of both b 1 and b 2 , the sample estimates of β 1 and β 2 , respectively. The equation gives an estimate for the underlying model that generates the given data. “b 1 ” is the intercept of the estimated equation. It is also the predicted value when x i is equal to 0. “b 2 is the slope of the estimated equation. It is the average marginal response of y i to the change of x i . i.e., if x i increases one unit, y i will ON AVERAGE increase (b 1 > 0) or decrease (b 1 > 0) by b 1 units. Our estimate of β 1 and β 2 1 2 = – 0.90, respectively. The important parameter estimate is b 2 . We expect that, on average, for each additional one dollar increase in the price of Z that quantity demanded will fall by 0.90 units. (d) Clearly distinguish the terms var(b 2 ) and σ 2 and state what your estimates mean. Var(b 2 ) is the variance of b 2 . b 2 is random since different samples will give us different estimates of b 2 , i.e., we get a different b 2 from each sample. Var(b 2 ) is estimated as 0.83. σ 2 is the variance of the error term (e). It is a measure of dispersion of the data around the estimated regression line. ^ 2 σ = 134.05. (e) Plot the data points given in the table and draw in your estimated regression function. Chart is given on the following page. (f) Suppose x is 9. What is the predicted y (i.e., y ), as determined by your regression equation? Does this coincide with observation 4, where x = 9 and y = 12? Why or why not? y = 19.61 – 0.91(9) = 11.42. The actual y value for observation 4 is 12. No, the prediction does not coincide with observation 4. The predicted values are the prediction of E( y|x ), not the actual datum y i . (g) The estimated linear regression line always passes through the mean values x and y (assuming we do not suppress the intercept, which we have not done here.) Verify that this estimated regression line passes through the orgin.
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This note was uploaded on 11/19/2009 for the course ARE ARE106 taught by Professor Constantine during the Spring '09 term at UC Davis.

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ARE106SU09HW2KEY - 1 University of California Davis Managerial Economic(ARE 106 Summer 2009 Instructor John H Constantine KEY Problem Set#2 Due

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