Statistics

t W5: Discussion Answer completely and show all your work for Problem 12.10 (80%) Full Data Set: xt 2.052 2.026 2.002 1.949 1.942 1.887 1.986 2.053 2.102 2.113 2.058 2.060 2.035 2.080 2.102 2.150 yt 102.900 101.500 100.800 98.000 97.300 93.500 97.500 102.200 105.000 107.200 105.100 103.900 103.000 104.800 105.000 107.200 This discussion is moderated. Please respond to at least 2 other students after the initial post due date. Responses should include direct questions. (20%) Initial Post Due: Wednesday of Week 5, by 11:59 p.m., ET* Responses Due: Sunday of Week 5, by 11:59 p.m., ET Show your complete work to Problem 12.10. Discuss your solution with your classmates after the initial posting date. problem 12.10 *12.10. Refer to Microcomputer components Problem 12.9. The analyst has decided to employ regression model (12.1) and use the Cochrane-Orcutt procedure to fit the model. a. Obtain a point estimate of the autocorrelation parameter. How well does the approximate relationship (12.25) hold here between this point estimate and the Durbin-Watson test statistic? b. Use one iteration to obtain the estimates and of the regression coefficients and in transformed model (12.17) and state the estimated regression function. Also obtain and . c. Test whether any positive autocorrelation remains after the first iteration using α = .05. State the alternatives, decision rule, and conclusion. d. Restate the estimated regression function obtained in part (b) in terms of the original variables. Also obtain s{b0} and s{b1}. Compare the estimated regression coefficients obtained with the Cochrane-Orcutt procedure and their estimated standard deviations with those obtained with ordinary least squares in Problem 12.9a. e. On the basis of the results in parts (c) and (d), does the Cochrane-Orcutt procedure appear to have been effective here? f. The value of industry production in month 17 will be $2.210 million. Predict the value of the firm’s components used in month 17; employ a 95 percent prediction interval. Interpret your interval. g. Estimate β1 with a 95 percent confidence interval. Interpret your interval. problem 12.9 *12.10. Refer to Microcomputer components Problem 12.9. The analyst has decided to employ regression model (12.1) and use the Cochrane-Orcutt procedure to fit the model. a. Obtain a point estimate of the autocorrelation parameter. How well does the approximate relationship (12.25) hold here between this point estimate and the Durbin-Watson test statistic? b. Use one iteration to obtain the estimates and of the regression coefficients and in transformed model (12.17) and state the estimated regression function. Also obtain and . c. Test whether any positive autocorrelation remains after the first iteration using α = .05. State the alternatives, decision rule, and conclusion. d. Restate the estimated regression function obtained in part (b) in terms of the original variables. Also obtain s{b0} and s{b1}. Compare the estimated regression coefficients obtained with the Cochrane-Orcutt procedure and their estimated standard deviations with those obtained with ordinary least squares in Problem 12.9a. e. On the basis of the results in parts (c) and (d), does the Cochrane-Orcutt procedure appear to have been effective here? problem 12.9 *12.9. Microcomputer components. A staff analyst for a manufacturer of microcomputer components has compiled monthly data for the past 16 months on the value of industry production of processing units that use these components (X, in million dollars) and the value of the firm’s components used (Y, in thousand dollars). The analyst believes that a simple linear regression relation is appropriate but anticipates positive autocorrelation. The data follow: t: 1 2 3 … 14 15 16 Xt 2.052 2.026 2.002 … 2.080 2.102 2.150 Yt: 102.9 101.5 100.8 … 104.8 105.0 107.2 503 a. Fit a simple linear regression model by ordinary least squares and obtain the residuals. Also obtain s{b0} and s{b1}. b. Plot the residuals against time and explain whether you find any evidence of positive autocorrelation. c. Conduct a formal test for positive autocorrelation using α = .05. State the alternatives, decision rule, and conclusion. Is the residual analysis in part (b) in accord with the test results

Notes and comments

Yt = β0 + β1Xt + εt (12.1)
εt = ρεt−1 + ut
where:

ρ is a parameter such that |ρ| < 1

ut are independent N(0, σ2)

as follows:
Yt′=β0′+β1′Xt′+ut (12.17)
where:

Yt′ = Yt − ρYt−1

Xt′ = Xt − ρXt−1

β0′ = β0(1 − ρ)

β1′ = β1

D ≈ 2(1 − r) (12.25)
This relation indicates that the Durbin-Watson statistic ranges approximately between 0 and 4 since r takes on values between −1 and 1, and that D is approximately 2 when r = 0.