Business Forecasting Study Set 1

Business

Quiz 8 :

The Box-Jenkins Arima Methodology

Quiz 8 :

The Box-Jenkins Arima Methodology

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For a sample of 100 observations of random data, calculate a 95% confidence interval for the autocorrelation coefficient at any lag. b. If all the autocorrelation coefficients are within their individual 95% confidence intervals and they show no particular pattern, what conclusion about the process can be drawn? c. If the first three autocorrelations are all positive and significantly different from zero and, in general, the autocorrelation pattern declines to zero, what conclusion can be drawn about the process? d. A process is observed quarterly. If the autocorrelations r 4 , r 8 , and r 12 are significantly different from zero, what conclusion can be drawn?
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Answer:

a.
Compute the 95% confidence interval for the autocorrelation coefficient at any lag.
Consider the confidence level is 95%, and
img .
The formula for confidence interval is given below:
img Here, the confidence level is 0.95.
For two tailed test,
img From Table B-2, the required
img value for 95% confidence level is 1.96. Thus,
img .
Substitute 1.96 for
img , 100 for n in the above confidence interval formula.
img Thus, the 95% confidence interval for the autocorrelation coefficient at any lag is
img .
b.
Conclusion:
Here, the series is random, because there is no particular pattern occurs in the plot and also all the autocorrelation coefficients lie within 95% confidence intervals.
c.
Conclusion:
The series may possibly stationary autoregressive process or non-stationary autoregressive process, because the inference depends on how rapid the autocorrelations decline to zero for the given time period.
d.
Conclusion:
Here, the series is seasonal with the period is 4, because the autocorrelations
img ,
img , and
img are significantly different from zero and also the process is quarterly.

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Suppose the following time series model has been fit to historical data and found to be an adequate model. img The first four observations are Y 1 = 32.5, Y 2 = 36.6, Y 3 = 33.3, and Y 4 = 31.9. Assuming ? 1 = 35 and ? 0 = 0, calculate forecasts for periods 5, 6, and 7 if period 4 is the forecast origin.
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Compute the forecasts for periods 5, 6, 7.
From the given information, the first four observations are
img ,
img ,
img and
img . Assume that
img and
img .
The model is given below:
img The forecast value for periods 1, 2, 3, and 4 are calculated as follows:
img The forecast value for period 5 is,
img Thus, the forecast value for period 5 is
img .
The forecast value for period 6 is,
img Thus, the forecast value for period 6 is
img .
The forecast value for period 7 is,
img Thus, the forecast value for period 7 is
img .

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A time series model has been fit and checked with historical data yielding img Suppose at time t = 60 the observation is Y 60 = 57. a. Determine forecasts for periods 61, 62, and 63 from origin 60. b. Suppose the observed value of Y 61 is 59. Update the forecasts for periods 62 and 63. c. Suppose the estimate for the variance of the error term is s 2 = 3.2. Compute a 95% prediction interval about the forecast for period 61.
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a.
Compute forecasts for periods 61, 62, and 63 from origin 60.
Assume that
img and
img .
The time series model is given below:
img The forecast value for period 61 is,
img Thus, the forecast value for period 61 is
img .
The forecast value for period 62 is,
img Thus, the forecast value for period 62 is
img .
The forecast value for period 63 is,
img Thus, the forecast value for period 63 is
img .
b.
Compute forecasts for periods 62 and 63.
Assume that
img and
img .
The time series model is given below:
img The forecast value for period 62 is,
img Thus, the forecast value for period 62 is
img .
The forecast value for period 63 is,
img Thus, the forecast value for period 63 is
img .
c.
Compute the 95% prediction interval about the forecast for period 61.
From part a.,
img .
The formula for prediction interval is given below:
img Substitute 75.65 for
img and 3.2 for
img in the above prediction interval formula.
img Thus, the 95% prediction interval about the forecast for period 61 is
img .

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Fill in the missing information in Table P-4, indicating whether the theoretical autocorrelations and partial autocorrelations die out or cut off for these models. TABLE P-4 img
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Given the graphs in Figure P-5 of the sample autocorrelations and the sample partial autocorrelations, tentatively identify an ARIMA model from each pair of graphs. FIGURE P-5 Sample Autocorrelations and Sample Partial Autocorrelations img
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An ARIMA(1, 1, 0) model (AR(1) model for first differences) is fit to observations of a time series. The first 12 residual autocorrelations are shown in Figure P-6. The model was fit with a constant. a. Based on an inspection of the residual autocorrelations, does the model appear to be adequate? Why or why not? b. If you decide the model is not adequate, indicate how it might be modified to eliminate any inadequacy. FIGURE P-6 Autocorrelations for the Residuals of an ARIMA(1, 1, 0) Model img
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Chips Bakery has been having trouble forecasting the demand for its special highfiber bread and would like your assistance. The data for the weekly demand and the autocorrelations of the original data and various differences of the data are shown in Tables P-7A-D. a. Inspect the autocorrelation plots and suggest a tentative model for these data. How did you decide on this model? ( Note: Do not difference more than necessary. Too much differencing is indicated if low lag sample autocorrelations tend to increase in magnitude.) b. Using a computer program for ARIMA modeling, fit and check your identified model with the bread demand data. c. Write down the equation for forecasting the demand for high-fiber bread for period 53. d. Using a computer program for ARIMA modeling, forecast the demand for high-fiber bread for the next four periods from forecast origin 52. Also, construct 95% prediction intervals. TABLE P-7A Weekly Sales Demand for High-Fiber Bread (in 1,000s)) img TABLE P-7B Sample Autocorrelations for Original Data img TABLE P-7C Autocorrelations for the First-Differenced Series img TABLE P-7D Autocorrelations for the Second-Differenced Series img
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Table P-8 contains a time series with 126 observations. Using a computer program for ARIMA modeling, obtain a plot of the data, the sample autocorrelations, and the sample partial autocorrelations. Develop an appropriate ARIMA model, and generate forecasts for the next three time periods from forecast origin t = 126. TABLE P-8 img img
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Table P-9 contains a time series of 80 observations. Using a computer program for ARIMA modeling, obtain a plot of the data, the sample autocorrelations, and the sample partial autocorrelations. Develop an appropriate ARIMA model, and generate forecasts for the next three time periods from forecast origin t = 80. TABLE P-9 img
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Table P-10 contains a time series of 80 observations. Using a computer program for ARIMA modeling, obtain a plot of the data, the sample autocorrelations, and the sample partial autocorrelations. Develop an appropriate ARIMA model, and generate forecasts for the next three time periods from forecast origin t = 80. TABLE P-10 img
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Table P-11 contains a time series of 96 monthly observations. Using a computer program for ARIMA modeling, obtain a plot of the data, the sample autocorrelations, and the sample partial autocorrelations. Develop an appropriate ARIMA model, and generate forecasts for the next 12 time periods from forecast origin. TABLE P-11 img
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The data in Table P-12 are weekly prices for IBM stock. a. Using a program for ARIMA modeling, obtain a plot of the data, the sample autocorrelations, and the sample partial autocorrelations. Use this information to tentatively identify an appropriate ARIMA model for this series. b. Is the IBM series stationary? What correction would you recommend if the series is nonstationary? c. Fit an ARIMA model to the IBM series. Interpret the result. Are successive changes random? d. Perform diagnostic checks to determine the adequacy of your fitted model. e. After a satisfactory model has been found, forecast the IBM stock price for the first week of January of the next year. How does your forecast differ from the naive forecast, which says that the forecast for the first week of January is the price for the last week in December (current price)? TABLE P-12 img
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The data in Table P-13 are closing stock quotations for the DEF Corporation for 150 days. Determine the appropriate ARIMA model, and forecast the stock price five days ahead from forecast origin t = 145. Compare your forecasts with the actual prices using the MAPE. How accurate are your forecasts? TABLE P-13 img img
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The data in Table P-14 are weekly automobile accident counts for the years 2006 and 2007 in Havana County. Determine the appropriate ARIMA model, and forecast accidents for the 91st week. Comment on the accuracy of your forecast. TABLE P-14 img
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Table P-15 gives the 120 monthly observations on the price in cents per bushel of corn in Omaha, Nebraska. Determine the best ARIMA model for these data. Generate forecasts of the price of corn for the next 12 months. Comment on the pattern of the forecasts. TABLE P-15 img
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Use the Box-Jenkins methodology to model and forecast the monthly sales of the Cavanaugh Company given in Table P-14 in Chapter 5. ( Hint: Consider a log transformation before modeling these data.) TABLE P-14 img
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Use the Box-Jenkins methodology to model and forecast the quarterly sales of Disney Company given in Table P-16 in Chapter 5. ( Hint: Consider a log transformation before modeling these data.)
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Use the Box-Jenkins methodology to model and forecast the monthly gasoline demand of Yukong Oil Company shown in Table P-17 in Chapter 5.
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Use the Box-Jenkins methodology to model and forecast the monthly numbers of employed men 16 years of age and older in the United States shown in Table P-25 in Chapter 5.
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Use the Box-Jenkins methodology to model and forecast the quarterly sales of Wal-Mart stores shown in Table P-27 in Chapter 5.
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