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1. The time series plot for 30 observations of a time series is given below. The least squares regression line has been added to the plot. Its equation is given in the upper right-hand corner. a. Complete the following table. Show your work below the table. Round your answer to 3 decimal places. Use discount factors α β = = .30 . (10 points) t t y t L t T ˆ t t 1 y y = −  0 1 35 2 36 27 103 98.947 2.637 28 100 101.109 2.495 29 106 30 107 107.024 2.708 y = 2.4747x + 32.375 20 30 40 50 60 70 80 90 100 110 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 b. Give each of the following. Round your answers to 3 decimal places. Show your calculations. (3 points each) ˆ30 1 y + = _____________ ˆ30 3 y + = ____________ e28 ( ) 1 = ______________ 26 y = _____________ c. Given 31 y = 112 , calculate ˆ31 1 y + , the one-step-ahead forecast for 32 Y . Round your answer to 3 decimal places. Show your work. (5 points) d. Given that the mean squared (one-step-ahead) forecast error is MSE = 31 10303 . , use the forecast of part c to create a 95% prediction interval for 32 Y . Round your prediction limits to 3 decimal places. Show your work. (5 points) 2. In the following table, t y denotes the first-order exponential smoother with λ = .25 . Mt denotes the 3-span moving average smoother. a. Calculate ˆ50 1 y + using the 3-span moving average. Show your work. (3 points) b. Calculate ˆ50 1 y + using first-order exponential smoothing with λ = .25 . Round your answer to 3 decimal places. Show your work. (4 points) 3. Given the following time series data with T = 3. The data are to be smoothed using first-order exponential smoothing with 0 1 y y  = , a. Calculate the mean squared one-step-ahead forecast error ( ) ˆ 2 1 T t t t MSE y y = = − ∑ if λ = .40 . Show your work. (5 points) t t y Mt r y 47 43.9 48 48.6 47.28 49 63.8 50 43.6 t t y 1 30 2 36 3 28 b. Use calculus to find the value of λ that minimizes the MSE. Show your work. (5 points) 4. Express the following process using lag operator notation. (4 points) . . 12 1 20 4 6 t t ttt Y YY ε ε = + − ++ −− − 5. Express the following process without lag operator notation. (5 points) ( )( ) ( ) 1 3 1 6 25 1 3 .. . t t − − = ++ L LY L ε 6. Let 1 D denote the first non-seasonal difference, and let 1 Ds denote the first seasonal difference of period s. t 35 36 37 38 39 40 t y 129 133 128 137 135 141 For the above time series data, find (3points each) a. ( ) 1 D y37 = b. ( ) 1 D y 3 39 = c. ( ( )) 1 1 DDy 4 40 = 7. Suppose that the following model was fitted to a set of time series data containing T = 45 observations. . . 1 15 25 7325 t t t Y Y ε =+ + − {ε t} WN(0 12 36 , . ) . 45 y = 69 2 a. Give a 95% prediction interval for 47 Y . (5 points) b. Give a 95% prediction interval for 250 Y . (5 points) 8. A time series consists of 100 observations. Holt’s method is used to obtain the one-step, two-steps, and three-steps-ahead forecasts. Given that . 100 L = 68 25 and ˆ 100 2 y 73 + = and calculate ˆ 100 3 y + . 9. Suppose that the following model was fitted to a set of time series data containing T = 75 observations. .. . 1 2 15 2 46 23 t t tt Y YY ε =+ + + − − { } ( ) 0 10 82 , . ε t WN . 74 y = 56 33 . 75 y = 59 48 Give a 95% prediction interval for 78 Y . 10. Suppose that the following model was fitted to a set of time series data containing T = 55 observations. .. . 1 1 8 75 57 43 t tt t Y Y ε ε = + +− − − { } ( ) 0 42 44 , . ε t WN t t y ˆ t y 52 20.48 19.26 53 12.80 19.72 054 18.59 19.99 55 16.03 20.11 a. Give the two-steps-ahead forecast. (4 points) b. Give a 95% prediction interval for Y58

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