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Use the Ljung-Box test to test the following hypothesis: H :ρ 0 1= 0 . Copy and paste the test results below including the R command used to perform the test. State your conclusion (level of significance .05) based on the results. (5 points) b.Use auto.arima to select the “best” model for tsloans. The model is of the form arima(p,d,q), where p = _____________ d = _____________ q = _____________ (1 point each) Express the above model using lag operator notation. (4 points) c.Fit the model to the data. Give graphical evidence that the model residuals are uncorrelated. Copy and paste the appropriate graph below. (3 points) Perform the appropriate hypothesis test to determine whether it is plausible to conclude that the residuals came from a white noise process. Copy and paste the test results below, including the R command used to perform the test. State your conclusion (level of significance .05) based on the results. (5 points) Assess the normality of the model residuals graphically. Copy and paste the appropriate graph below. (3 points) Perform the appropriate test to determine whether it is plausible that the residuals come from a normal distribution. Copy and paste the test results below, including the R command used to perform the test. State your conclusion (level of significance .05) based on the results. (5 points) d.Give the forecast for the number of loan applications 9 weeks in the future. Include the 95% prediction limits for your forecast. (4 points) e. You need to assess the forecast accuracy of your model. Fit the model to 1 y y2 , ,,y70 . Use the results to forecast the values for weeks 71 – 104. Complete the table below. (5 points) t et (1) (4 decimal places) 100 101 102 103 104 The mean squared one-step-ahead forecast error is: MSE = __________________________ (3 points) 2.The datafile Beer, available in the Data folder on Bb, contains monthly U.S. beer production over an 8- year period. Import the data into R, and format it as the time series tsbeer. a. Fit an arima(1,1,0)(0,1,1)[12] model to the data. Store the model information in the R object fit. Copy and paste the output containing the model coefficients and their standard errors below. (5 points) t 12 t−12 β ε L −1 et Mt = (1 − L) t (1 − L12 )Y , then Mt =1a Mt +ε + where: a1 = _______________ 1 β = _______________ (1 point each) b.Let ()()1211ttMLLY=−− . Calculate []9696EM . Your answer should be a numerical value rounded to three decimal places. Show your work below. (5 points) c.Show that []ˆ9697961968584EMyyyy+=−−+ . (5 points) d.Use the fact that [][].969796963687EMEM=−⋅ to calculate ˆ961y+ . Round your answer to three decimal places. Show your work below. (5 points) 3.The datafile Coal, available in the Data folder on Bb, contains annual U.S. coal production over an 57- year period. Import the data into R, and format it as the time series tscoal. Use Holt’s Method (.60α=,.20β=) to smooth the coal production time series. Answer the following questions. Round your answers to 5 decimal places. a. 2L= _______________ (3 points) b. 2T= _______________ (3 points) c. 43y= ________________ (2 points) d.ˆ35y= ________________(2 points) e.()281e= _________________(3 points) f.ˆ574y+= ___________________(3 points) g.The upper 80% prediction limit for 59Y equals ___________________.(3 points) h.The mean squared one-step-ahead forecast error for the 57 coal production observations equals ____________________ . (3 points) i.The values of the smoothing parameters that minimize the sum of squared one-step-ahead forecast errors are α= _________________ (2 points each) β= _________________ 4.Additive Holt-Winters Method was used to smooth quarterly grocery spending (millions of dollars) over an 11-year period. Information about the smoothing parameters and coefficients is given below. Smoothing parameters: alpha: 0.70 beta : 0.04 gamma: 0.80 Coefficients: a 2550.37806 b 11.04454 s1 205.28605 s2 12.69910 s3 -78.89692 s4 -78.67618 SSE = 101,917.1 Use the above information to answer the following questions. Round your answers to 5 decimal places. a. ˆ443y+= __________________ (4 points) Show your calculation below. b.Suppose that the grocery spending for quarter 45 is 2584.4, give a 95% prediction interval for 46Y that uses the new information. (8 points) Show your calculation below. 5.The datafile Retail, available in the Data folder on Bb, contains monthly retail sales of recreational goods over a 185-month period. Import the data into R, and format it as the time series tsret. a.Create a time-series plot of the data. Copy and paste the plot in the space below. (3 points) b.Select the appropriate exponential smoothing method to smooth the data. Use R to smooth the data with the optimal smoothing parameters. Describe the method that you used. Be specific. (4 points) c.Create a plot that adds forecasts 12 months into the future to the time series plot of Part a. Copy and paste the plot in the space below. (6 points) d.Give 95% prediction limits for retail sales in month 190. (4 points)

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