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Unformatted text preview: CHAPTER 9 BOXJENKINS (ARIMA) METHODOLOGY ANSWERS TO PROBLEMS AND CASES 1. a. 0 .196 b. Series is random c. Series could be a stationary autoregressive process or series could be nonstationary. Interpretation depends on how fast the autocorrelations decay to 0. d. Seasonal series with period of 4 2. t Y t t Y e t 1 32.5 35.000 2.500 2 36.6 34.375 2.225 3 33.3 36.306 3.006 4 31.9 33.581 1.681 Y 5 = 35 + .25(1.681)  .3(3.006) = 35.482 Y 6 = 35 + .25(0)  .3(1.681) = 35.504 Y 7 = 35 3. a. 61 Y = 75.65 62 Y = 84.04 63 Y = 87.82 b. 62 Y = 76.55 63 Y = 84.45 c. 75.65 23.2 4. a. Model Autocorrelations Partial Autocorrelations AR die out cut off MA cut off die out ARIMA die out die out 5. a. MA(2) b. AR(1) c. ARIMA(1,0,1) 6. a. Model is not adequate. 173 b. Q = 44.3 df = 11 = .05 Reject H if 2 > 19.675 Since Q = 44.3 > 19.675, reject H 0 and conclude model is not adequate. Also, there is a significant residual autocorrelation at lag 2. Add a MA term to the model at lag 2 and fit an ARIMA(1,1,2) model. 7. a. Autocorrelations of original series fail to die out, suggesting that demand is nonstationary. Autocorrelations for first differences of demand, do die out (cut off relative to standard error limits) suggesting series of first differences is stationary. Low lag autocorrelations of series of second differences increase in magnitude, suggesting second differencing is too much. A plot of the demand series shows the series is increasing linearly in time with almost a perfect (deterministic) straight line pattern. In fact, a straight line time trend fit to the demand data represents the data well as shown in the plot below. If an ARIMA model is fit to the demand data, the autocorrelations and plots of the original series and the series of first differences, suggest an ARIMA(0,1,1) model with a constant term might be good starting point. The first order moving average term is suggested by the significant autocorrelation at lag 1 for the first differenced series. b. The Minitab output from fitting an ARIMA(0,1,1) model with a constant is shown below. 174 The least squares estimate of the constant term, .7127, is virtually the same as The least squares slope coefficient in the straight line fit shown in part a. Also, The first order moving average coefficient is essentially 1. These two results are consistent with a straight line time trend regression model for the original data. Suppose t Y is demand in time period t . The straight line time trend regression model is: t t t Y + + = 1 . Thus 1 1 1 ) 1 ( + + = t t t Y and 1 1 1 + = t t t t Y Y . The latter is an ARIMA(0,1,1) model with a constant term (the slope coefficient in the straight line model) and a first order moving average coefficient of 1....
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 Spring '08
 DR.STEPHANIES.PANEHADEN
 Management

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