Workshop 12 Solutions - Workshop 12 Solutions THE...

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Workshop 12 Solutions T HE U NIVERSITY OF S YDNEY Q UANTITATIVE B USINESS A NALYSIS Prepared by: Scott Liu Workshop Aims 1. Use residual plots to investigate the properties of the assumptions about the error term. 2. Use regression outputs for prediction and to form interval estimates for the mean and for individual observations. 3. Conduct a Durbin-Watson test for auto-correlation.
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BUSS1020 Quantitative Business Analysis Workshop 12 Solutions Workshop 12 Learning the Basics Question 1 (BLS Q33, p. 543) The residuals for 15 consecutive time periods are as follows: Time Period Residual Time Period Residual 1 + 4 9 + 6 2 - 6 10 - 3 3 - 1 11 + 1 4 - 5 12 + 3 5 + 2 13 0 6 + 5 14 - 4 7 - 2 15 - 7 8 + 7 (a) Plot the residuals over time. What conclusion can you reach about the pattern of the residuals over time? Solution Looking at the graph, the residuals display an ostensible cyclic pattern. It may suggest that there exists autocorrelation and therefore a Durbin-Watson statistic should be taken (see part (b)). (b) Compute the Durbin-Watson statistic. At the 0.05 level of significance, is there evidence of positive autocorrelation among the residuals? Solution Stating the null and alternative hypotheses: H 0 : There is no autocorrelation H 1 : Autocorrelation is present Given the critical value using the table: d L = 1.08 and d U = 1.36: After doing calculations on Excel: D = n X i = 2 ( e i - e i - 1 ) 2 n X i = 1 e 2 i = 465 280 = 1.6607 > 1.36 = d U , Hence do not reject H 0 at 5% level of significance. University of Sydney Page 2
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BUSS1020 Quantitative Business Analysis Workshop 12 Solutions (c) Based on (a) and (b), what conclusions can you reach about the autocorrelation of the residuals? Solution Since from (b) the null hypothesis, H 0 was not rejected, there is insufficient evidence to say that there is autocorrelation of the residuals. Question 2 (BLS Q35, p. 543) What is the relationship between the price of crude oil and the price you pay at the pump for gasoline? The data in the excel file Oil & Gasoline . xlsx contains the price ($) for a barrel of crude oil (Cushing, Oklahoma, spot price) and a gallon of gasoline (U.S. average conventional spot price) for 181 weeks, ending June 14 2013. (a) Construct a scatter plot with the price of oil on the horizontal axis and the price of gasoline on the vertical axis. (b) Use the least-squares method to develop a simple linear regression equation to predict the price of a gallon of gasoline using the price of a barrel of crude oil as the independent variable. [ Hint : Just use Data Analysis ] Solution Let Y be the price of a gallon of gasoline and X be the price of barrel of crude oil. ˆ Y i = - 0.6047 + 0.0360 X i University of Sydney Page 3
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BUSS1020 Quantitative Business Analysis Workshop 12 Solutions (c) Interpret the meaning of the slope, b 1 , in this problem.
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