Quiz 9: The Box-Jenkins Arima Methodology

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?

Suppose the following time series model has been fit to historical data and found to be an adequate model. 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.

A time series model has been fit and checked with historical data yielding 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.

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

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

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

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)) TABLE P-7B Sample Autocorrelations for Original Data TABLE P-7C Autocorrelations for the First-Differenced Series TABLE P-7D Autocorrelations for the Second-Differenced Series

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

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

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

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

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

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

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

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

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

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.)

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.

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.

Use the Box-Jenkins methodology to model and forecast the quarterly sales of Wal-Mart stores shown in Table P-27 in Chapter 5.

Use the Box-Jenkins methodology to model and forecast the annual numbers of severe earthquakes shown in Table P-18 in Chapter 4.

Use the Box-Jenkins methodology to model and forecast the quarterly sales for The Gap shown in Table P-20 in Chapter 4.

Figure P-23 is a time series plot of the weekly number of Influenza A Positive cases in a region of Texas from the week of September 2, 2003, to the week of April 10, 2007. The number of cases peaks during the November-April period and is zero during the remaining weeks from late spring to early fall. Although medical data, this series shares many of the characteristics of the demand for certain spare parts. The number of flu cases and the demand for spare parts are "lumpy," with periods of activity followed by relatively long and variable periods of no activity. Can you think of any problems a series like the flu series might pose for ARIMA modeling? Can you think of a simple nonseasonal ARIMA model that might produce a reasonable forecast of the number of Influenza A Positive cases one week ahead? FIGURE P-23 Time Series Plot of Weekly Influenza A Positive Case

SOUTHWEST MEDICAL CENTER Mary Beasley has been disappointed in her forecasting efforts so far. She has tried Winters' smoothing (see Case 4-7), decomposition (see Case 5-8), and regression (see Case 8-9) with less than satisfactory results. Mary knows her time series of monthly total billable visits to Medical Oncology is nonstationary and seasonal. She knows from one of her Executive M.B.A. courses that ARIMA models can describe time series that are nonstationary and seasonal and decides to give ARIMA modeling a try. She has nothing to lose at this point. Following the prescribed procedure of looking at plots, autocorrelations, and partial autocorrelations of the original series and various differences, Mary generates these quantities for the original series (TotVisits), the differences of the series (DiffTotVisits), the seasonal differences of the series (Diff12TotVisits), and the series with one regular difference followed by one seasonal difference (DiffDiff12TotVisits). A plot of DiffDiff12TotVisits is shown in Figure 9-55. The corresponding autocorrelation function is given in Figure 9-56. Examining the time series plot in Figure 9-55, Mary sees that the differences appear to be stationary and vary about zero. The autocorrelation function in Figure 9-56 appears to have two significant negative autocorrelations at lags 1 and 12. This suggests a moving average parameter at lag 1 and a seasonal moving average parameter at lag 12 in an ARIMA model with one regular and one seasonal difference. Mary is ready to fit an ARIMA(0, 1, 1)(0, 1, 1) 12 model to her data. FIGURE 9-55 Time Series Plot of DiffDiff12TotVisits FIGURE 9-56 Sample Autocorrelations for DiffDiff12TotVisits Using Mary Beasley's data in Table 4-13, construct a time series plot, and compute the sample autocorrelations and sample partial autocorrelations for the total billable visits, Y ; the differenced series, ? Y ; and the seasonally differenced series, ? 12 Y. Given this information and the information in Figures 9-55 and 9-56, do you think Mary's initial model is reasonable? Discuss.

SOUTHWEST MEDICAL CENTER Mary Beasley has been disappointed in her forecasting efforts so far. She has tried Winters' smoothing (see Case 4-7), decomposition (see Case 5-8), and regression (see Case 8-9) with less than satisfactory results. Mary knows her time series of monthly total billable visits to Medical Oncology is nonstationary and seasonal. She knows from one of her Executive M.B.A. courses that ARIMA models can describe time series that are nonstationary and seasonal and decides to give ARIMA modeling a try. She has nothing to lose at this point. Following the prescribed procedure of looking at plots, autocorrelations, and partial autocorrelations of the original series and various differences, Mary generates these quantities for the original series (TotVisits), the differences of the series (DiffTotVisits), the seasonal differences of the series (Diff12TotVisits), and the series with one regular difference followed by one seasonal difference (DiffDiff12TotVisits). A plot of DiffDiff12TotVisits is shown in Figure 9-55. The corresponding autocorrelation function is given in Figure 9-56. Examining the time series plot in Figure 9-55, Mary sees that the differences appear to be stationary and vary about zero. The autocorrelation function in Figure 9-56 appears to have two significant negative autocorrelations at lags 1 and 12. This suggests a moving average parameter at lag 1 and a seasonal moving average parameter at lag 12 in an ARIMA model with one regular and one seasonal difference. Mary is ready to fit an ARIMA(0, 1, 1)(0, 1, 1) 12 model to her data. FIGURE 9-55 Time Series Plot of DiffDiff12TotVisits FIGURE 9-56 Sample Autocorrelations for DiffDiff12TotVisits Using Minitab or equivalent software, fit an ARIMA(0, 1, 1)(0, 1, 1) 12 model to Mary Beasley's data, and generate forecasts for the next 12 months. Examine the residuals to check for adequacy of fit. Should Mary be happy with the forecasts of total billable visits now?

SOUTHWEST MEDICAL CENTER Mary Beasley has been disappointed in her forecasting efforts so far. She has tried Winters' smoothing (see Case 4-7), decomposition (see Case 5-8), and regression (see Case 8-9) with less than satisfactory results. Mary knows her time series of monthly total billable visits to Medical Oncology is nonstationary and seasonal. She knows from one of her Executive M.B.A. courses that ARIMA models can describe time series that are nonstationary and seasonal and decides to give ARIMA modeling a try. She has nothing to lose at this point. Following the prescribed procedure of looking at plots, autocorrelations, and partial autocorrelations of the original series and various differences, Mary generates these quantities for the original series (TotVisits), the differences of the series (DiffTotVisits), the seasonal differences of the series (Diff12TotVisits), and the series with one regular difference followed by one seasonal difference (DiffDiff12TotVisits). A plot of DiffDiff12TotVisits is shown in Figure 9-55. The corresponding autocorrelation function is given in Figure 9-56. Examining the time series plot in Figure 9-55, Mary sees that the differences appear to be stationary and vary about zero. The autocorrelation function in Figure 9-56 appears to have two significant negative autocorrelations at lags 1 and 12. This suggests a moving average parameter at lag 1 and a seasonal moving average parameter at lag 12 in an ARIMA model with one regular and one seasonal difference. Mary is ready to fit an ARIMA(0, 1, 1)(0, 1, 1) 12 model to her data. FIGURE 9-55 Time Series Plot of DiffDiff12TotVisits FIGURE 9-56 Sample Autocorrelations for DiffDiff12TotVisits Looking at a time series plot of total billable visits to Medical Oncology, can you see any feature(s) of these data that might make modeling difficult?

RESTAURANT SALES This case refers to the sales data and situation for the restaurant discussed in Case 8-3. Jim Price has now completed a course in forecasting and is anxious to apply the Box-Jenkins methodology to the restaurant sales data. These data, shown in Table 9-17A, begin with the week ending Sunday, January 4, 1981, and continue through the week ending Sunday, December 26, 1982. Table 9-17B contains new data for the week ending January 2, 1983, through the week ending October 30, 1983. TABLE 9-17A Restaurant Sales: Old Data TABLE 9-17A ( Continued ) TABLE 9-17B Restaurant Sales: New Data \What is the appropriate Box-Jenkins model to use on the original data?

RESTAURANT SALES This case refers to the sales data and situation for the restaurant discussed in Case 8-3. Jim Price has now completed a course in forecasting and is anxious to apply the Box-Jenkins methodology to the restaurant sales data. These data, shown in Table 9-17A, begin with the week ending Sunday, January 4, 1981, and continue through the week ending Sunday, December 26, 1982. Table 9-17B contains new data for the week ending January 2, 1983, through the week ending October 30, 1983. TABLE 9-17A Restaurant Sales: Old Data TABLE 9-17A ( Continued ) TABLE 9-17B Restaurant Sales: New Data What are your forecasts for the first four weeks of January 1983?

RESTAURANT SALES This case refers to the sales data and situation for the restaurant discussed in Case 8-3. Jim Price has now completed a course in forecasting and is anxious to apply the Box-Jenkins methodology to the restaurant sales data. These data, shown in Table 9-17A, begin with the week ending Sunday, January 4, 1981, and continue through the week ending Sunday, December 26, 1982. Table 9-17B contains new data for the week ending January 2, 1983, through the week ending October 30, 1983. TABLE 9-17A Restaurant Sales: Old Data TABLE 9-17A ( Continued ) TABLE 9-17B Restaurant Sales: New Data How do these forecasts compare with actual sales?

RESTAURANT SALES This case refers to the sales data and situation for the restaurant discussed in Case 8-3. Jim Price has now completed a course in forecasting and is anxious to apply the Box-Jenkins methodology to the restaurant sales data. These data, shown in Table 9-17A, begin with the week ending Sunday, January 4, 1981, and continue through the week ending Sunday, December 26, 1982. Table 9-17B contains new data for the week ending January 2, 1983, through the week ending October 30, 1983. TABLE 9-17A Restaurant Sales: Old Data TABLE 9-17A ( Continued ) TABLE 9-17B Restaurant Sales: New Data How does your Box-Jenkins model compare to the regression models used in Chapter 8?

RESTAURANT SALES This case refers to the sales data and situation for the restaurant discussed in Case 8-3. Jim Price has now completed a course in forecasting and is anxious to apply the Box-Jenkins methodology to the restaurant sales data. These data, shown in Table 9-17A, begin with the week ending Sunday, January 4, 1981, and continue through the week ending Sunday, December 26, 1982. Table 9-17B contains new data for the week ending January 2, 1983, through the week ending October 30, 1983. TABLE 9-17A Restaurant Sales: Old Data TABLE 9-17A ( Continued ) TABLE 9-17B Restaurant Sales: New Data Would you use the same Box-Jenkins model if the new data were combined with the old data?

TUX John Mosby has decided to try the Box-Jenkins method for forecasting his monthly sales data. He understands that this procedure is more complex than the simpler methods he has been trying. He also knows that more accurate forecasts are possible using this advanced method. Also, he has access to the Minitab computer program that can fit ARIMA models. And since he already has the data collected and stored in his computer, he decides to give it a try. John decides to use his entire data set, consisting of 96 months of sales data. John knows that an ARIMA model can handle the seasonal structure in his time series as well as model the month-to-month correlation. John computed the sample autocorrelation function for his data in Case 3-2 (see Figure 3-25) and determined that his data were trending (nonstationary). He then computed the autocorrelation function for the first differences of his sales data. The result is shown in Figure 3-26. From this plot, John immediately noticed the significant spikes at lags 12 and 24, indicating seasonality and perhaps the need for a seasonal difference. The sample autocorrelation function for the data after a regular (first) difference and a seasonal difference is displayed in Figure 9-39. The sample partial autocorrelations for the differenced data are shown in Figure 9-40. FIGURE 9-39 Autocorrelations for Mr. Tux: Regular and Seasonally Differenced Series FIGURE 9-40 Partial Autocorrelations for Mr. Tux: Regular and Seasonally Differenced Series Discuss the problems, if any, of explaining the Box-Jenkins method to John's banker and others on his management team.

TUX John Mosby has decided to try the Box-Jenkins method for forecasting his monthly sales data. He understands that this procedure is more complex than the simpler methods he has been trying. He also knows that more accurate forecasts are possible using this advanced method. Also, he has access to the Minitab computer program that can fit ARIMA models. And since he already has the data collected and stored in his computer, he decides to give it a try. John decides to use his entire data set, consisting of 96 months of sales data. John knows that an ARIMA model can handle the seasonal structure in his time series as well as model the month-to-month correlation. John computed the sample autocorrelation function for his data in Case 3-2 (see Figure 3-25) and determined that his data were trending (nonstationary). He then computed the autocorrelation function for the first differences of his sales data. The result is shown in Figure 3-26. From this plot, John immediately noticed the significant spikes at lags 12 and 24, indicating seasonality and perhaps the need for a seasonal difference. The sample autocorrelation function for the data after a regular (first) difference and a seasonal difference is displayed in Figure 9-39. The sample partial autocorrelations for the differenced data are shown in Figure 9-40. FIGURE 9-39 Autocorrelations for Mr. Tux: Regular and Seasonally Differenced Series FIGURE 9-40 Partial Autocorrelations for Mr. Tux: Regular and Seasonally Differenced Series Given the autocorrelations in Figure 9-39 and the partial autocorrelations in Figure 9-40, what regular (nonseasonal) terms might John include in an ARIMA model for Mr. Tux sales? What seasonal terms might he include in the model?

TUX John Mosby has decided to try the Box-Jenkins method for forecasting his monthly sales data. He understands that this procedure is more complex than the simpler methods he has been trying. He also knows that more accurate forecasts are possible using this advanced method. Also, he has access to the Minitab computer program that can fit ARIMA models. And since he already has the data collected and stored in his computer, he decides to give it a try. John decides to use his entire data set, consisting of 96 months of sales data. John knows that an ARIMA model can handle the seasonal structure in his time series as well as model the month-to-month correlation. John computed the sample autocorrelation function for his data in Case 3-2 (see Figure 3-25) and determined that his data were trending (nonstationary). He then computed the autocorrelation function for the first differences of his sales data. The result is shown in Figure 3-26. From this plot, John immediately noticed the significant spikes at lags 12 and 24, indicating seasonality and perhaps the need for a seasonal difference. The sample autocorrelation function for the data after a regular (first) difference and a seasonal difference is displayed in Figure 9-39. The sample partial autocorrelations for the differenced data are shown in Figure 9-40. FIGURE 9-39 Autocorrelations for Mr. Tux: Regular and Seasonally Differenced Series FIGURE 9-40 Partial Autocorrelations for Mr. Tux: Regular and Seasonally Differenced Series Using a program for ARIMA modeling, fit and check an ARIMA model for Mr. Tux sales. Generate forecasts for the next 12 months.

CONSUMER CREDIT CONSULING The Consumer Credit Counseling (CCC) operation was described in Cases 1-2 and 3-3. The executive director, Marv Harnishfeger, concluded that the most important variable that CCC needed to forecast was the number of new clients that would be seen for the rest of 1993. Marv provided Dorothy Mercer with monthly data for the number of new clients seen by CCC for the period from January 1985 through March 1993 (see Case 3-3). Dorothy was very happy with your forecasting work so far. However, you are not completely satisfied and decide that now is the time to really impress her. You tell her that she should try one of the most powerful techniques used in forecasting, the Box-Jenkins methodology and ARIMA models. Dorothy has never heard of Box-Jenkins but is willing to have you give it a try. Write Dorothy a memo that explains the Box-Jenkins methodology.

CONSUMER CREDIT CONSULING The Consumer Credit Counseling (CCC) operation was described in Cases 1-2 and 3-3. The executive director, Marv Harnishfeger, concluded that the most important variable that CCC needed to forecast was the number of new clients that would be seen for the rest of 1993. Marv provided Dorothy Mercer with monthly data for the number of new clients seen by CCC for the period from January 1985 through March 1993 (see Case 3-3). Dorothy was very happy with your forecasting work so far. However, you are not completely satisfied and decide that now is the time to really impress her. You tell her that she should try one of the most powerful techniques used in forecasting, the Box-Jenkins methodology and ARIMA models. Dorothy has never heard of Box-Jenkins but is willing to have you give it a try. Develop an ARIMA model using the Box-Jenkins methodology, and forecast the monthly number of new clients for the rest of 1993.

CONSUMER CREDIT CONSULING The Consumer Credit Counseling (CCC) operation was described in Cases 1-2 and 3-3. The executive director, Marv Harnishfeger, concluded that the most important variable that CCC needed to forecast was the number of new clients that would be seen for the rest of 1993. Marv provided Dorothy Mercer with monthly data for the number of new clients seen by CCC for the period from January 1985 through March 1993 (see Case 3-3). Dorothy was very happy with your forecasting work so far. However, you are not completely satisfied and decide that now is the time to really impress her. You tell her that she should try one of the most powerful techniques used in forecasting, the Box-Jenkins methodology and ARIMA models. Dorothy has never heard of Box-Jenkins but is willing to have you give it a try. Write Dorothy a second memo that summarizes the results of this analysis.

THE LYDIA E. PINKHAM MEDICINE COMPANY The Lydia E. Pinkham Medicine Company was a family-owned concern whose income was derived largely from the sale of Lydia Pinkham's Vegetable Compound. Perhaps students today could use some of the compound to relieve stress; unfortunately, it is no longer sold. Lydia Pinkham's picture was on the label, and the compound was marketed to women. Ads for the compound included this invitation: "write freely and fully to Mrs. Pinkham, at Lynn, Mass., and secure the advice which she offers free of charge to all women. This is the advice that has brought sunshine into many homes which nervousness and irritability had nearly wrecked." In fact, the company ensured that a female employee answered every letter. Women did write to Mrs. Pinkham. Their claims included this one: "Without [Lydia Pinkham's Vegetable Compound] I would by this time have been dead or, worse, insane.... I had given up on myself; as I had tried so many things, I believed nothing would ever do me any good. But, thanks to your medicine, I am now well and strong; in fact, another person entirely." This testimonial and others were reproduced in print ads for the compound. The unique nature of the company-one dominant product that accounted for most of the company's sales, no sales staff, and a large proportion of sales revenues invested in advertising-and the availability of data on both sales and advertising led the Committee on Price Determination of the National Bureau of Economic Research (NBER) in 1943 to recommend that the data be subjected to thorough analysis. The research was not undertaken for several years (Palda 1964). Analysts have studied the data using causal models that include the advertising data and other economic variables (similar to those presented in Chapter 8). However, several researchers have suggested that Box-Jenkins approaches using only the sales data provide comparable, or even superior, predictions when compared with the causal approaches (see, e.g., Kyle 1978). The sales data are appealing to study for two reasons: 1. The product itself was unchanged for the span of the data; that is, there are no shifts in the series caused by changes in the product. 2. There was no change in the sales force over the span of the data, and the proportion of revenues spent on advertising was fairly constant. Thus, there are no shifts in the data caused by special promotions or other marketing phenomena. Typically, actual data are not this "clean" in terms of product and marketing continuity. The task at hand, then, is to determine which Box-Jenkins (ARIMA) model is the "best" for these data. The model will be developed using the 1907-1948 data and tested using the 1949-1960 data shown in Table 9-18. MODEL IDENTIFICATION A computer program capable of ARIMA modeling was used to examine the data for 1907 through 1948; the data for 1949 through 1960 are used to examine the forecasting ability of the selected model. Preliminary tests suggest that the data are stationary (that is, there is no apparent trend), so differencing is not employed. After examining the autocorrelations and partial autocorrelations, it was determined that an AR model was most appropriate for the data. (The autocorrelations (ACF) and partial autocorrelations (PACF) for 10 periods are given in Table 9-19.) The autocorrelations and partial autocorrelations seemed to be consistent with those of an AR(2) process. To verify the order p of the AR component, Akaike's information criterion ( AIC ) (see Equation 9.7) was used with autoregressive models of orders, p = 2, and 3. The AIC leads to the choice of an AR(2) model for the Lydia Pinkham data. TABLE 9-18 Lydia E. Pinkham Medicine Data TABLE 9-19 MODEL ESTIMATION AND TESTING OF MODEL ADEQUACY A computer program was used to estimate the parameters (including a constant term) of the AR(2) model using the data for 1907 through 1948. Using the estimated parameters, the resulting model is where the numbers in parentheses beneath the autoregressive coefficients are their estimated standard deviations. Each autoregressive coefficient is significantly different from zero for any reasonable significance level. The residual autocorrelations were all small, and each was within its 95% error limits. The Ljung-Box chi-square statistics had p -values of.63,.21, and.64 for groups of lags m = 24, and 36, respectively. The AR(2) model appears to be adequate for the Lydia Pinkham sales data. FORECASTING WITH THE MODEL The final step in the analysis of the AR(2) model for these data is to forecast the values for 1949 through 960 one period ahead. (For example, data through 1958 are used in computing the forecast for 1959.) The forecasting equation is and the one-step-ahead forecasts and the forecast errors are shown in Table 9-20. Table 9-20 In addition to the one-step-ahead forecasts, some accuracy measures were computed. The forecasts from the AR(2) model have a mean absolute percentage error ( MAPE ) of 6.9% and a mean absolute deviation ( MAD ) of $112 (thousands of current dollars).These figures compare favorably to the accuracy measures from the causal models developed by other researchers. SUMMARY AND CONCLUSIONS A parsimonious (smallest number of parameters) AR(2) model has been fit to the Lydia Pinkham data for the years 1907 through 1948.This model has produced fairly accurate one-step-ahead forecasts for the years 1949 through 1960. The product itself was unchanged for the span of the data; that is, there are no shifts in the series caused by changes in the product.

THE LYDIA E. PINKHAM MEDICINE COMPANY The Lydia E. Pinkham Medicine Company was a family-owned concern whose income was derived largely from the sale of Lydia Pinkham's Vegetable Compound. Perhaps students today could use some of the compound to relieve stress; unfortunately, it is no longer sold. Lydia Pinkham's picture was on the label, and the compound was marketed to women. Ads for the compound included this invitation: "write freely and fully to Mrs. Pinkham, at Lynn, Mass., and secure the advice which she offers free of charge to all women. This is the advice that has brought sunshine into many homes which nervousness and irritability had nearly wrecked." In fact, the company ensured that a female employee answered every letter. Women did write to Mrs. Pinkham. Their claims included this one: "Without [Lydia Pinkham's Vegetable Compound] I would by this time have been dead or, worse, insane.... I had given up on myself; as I had tried so many things, I believed nothing would ever do me any good. But, thanks to your medicine, I am now well and strong; in fact, another person entirely." This testimonial and others were reproduced in print ads for the compound. The unique nature of the company-one dominant product that accounted for most of the company's sales, no sales staff, and a large proportion of sales revenues invested in advertising-and the availability of data on both sales and advertising led the Committee on Price Determination of the National Bureau of Economic Research (NBER) in 1943 to recommend that the data be subjected to thorough analysis. The research was not undertaken for several years (Palda 1964). Analysts have studied the data using causal models that include the advertising data and other economic variables (similar to those presented in Chapter 8). However, several researchers have suggested that Box-Jenkins approaches using only the sales data provide comparable, or even superior, predictions when compared with the causal approaches (see, e.g., Kyle 1978). The sales data are appealing to study for two reasons: 1. The product itself was unchanged for the span of the data; that is, there are no shifts in the series caused by changes in the product. 2. There was no change in the sales force over the span of the data, and the proportion of revenues spent on advertising was fairly constant. Thus, there are no shifts in the data caused by special promotions or other marketing phenomena. Typically, actual data are not this "clean" in terms of product and marketing continuity. The task at hand, then, is to determine which Box-Jenkins (ARIMA) model is the "best" for these data. The model will be developed using the 1907-1948 data and tested using the 1949-1960 data shown in Table 9-18. MODEL IDENTIFICATION A computer program capable of ARIMA modeling was used to examine the data for 1907 through 1948; the data for 1949 through 1960 are used to examine the forecasting ability of the selected model. Preliminary tests suggest that the data are stationary (that is, there is no apparent trend), so differencing is not employed. After examining the autocorrelations and partial autocorrelations, it was determined that an AR model was most appropriate for the data. (The autocorrelations (ACF) and partial autocorrelations (PACF) for 10 periods are given in Table 9-19.) The autocorrelations and partial autocorrelations seemed to be consistent with those of an AR(2) process. To verify the order p of the AR component, Akaike's information criterion ( AIC ) (see Equation 9.7) was used with autoregressive models of orders, p = 2, and 3. The AIC leads to the choice of an AR(2) model for the Lydia Pinkham data. TABLE 9-18 Lydia E. Pinkham Medicine Data TABLE 9-19 MODEL ESTIMATION AND TESTING OF MODEL ADEQUACY A computer program was used to estimate the parameters (including a constant term) of the AR(2) model using the data for 1907 through 1948. Using the estimated parameters, the resulting model is where the numbers in parentheses beneath the autoregressive coefficients are their estimated standard deviations. Each autoregressive coefficient is significantly different from zero for any reasonable significance level. The residual autocorrelations were all small, and each was within its 95% error limits. The Ljung-Box chi-square statistics had p -values of.63,.21, and.64 for groups of lags m = 24, and 36, respectively. The AR(2) model appears to be adequate for the Lydia Pinkham sales data. FORECASTING WITH THE MODEL The final step in the analysis of the AR(2) model for these data is to forecast the values for 1949 through 960 one period ahead. (For example, data through 1958 are used in computing the forecast for 1959.) The forecasting equation is and the one-step-ahead forecasts and the forecast errors are shown in Table 9-20. Table 9-20 In addition to the one-step-ahead forecasts, some accuracy measures were computed. The forecasts from the AR(2) model have a mean absolute percentage error ( MAPE ) of 6.9% and a mean absolute deviation ( MAD ) of $112 (thousands of current dollars).These figures compare favorably to the accuracy measures from the causal models developed by other researchers. SUMMARY AND CONCLUSIONS A parsimonious (smallest number of parameters) AR(2) model has been fit to the Lydia Pinkham data for the years 1907 through 1948.This model has produced fairly accurate one-step-ahead forecasts for the years 1949 through 1960. There is some evidence that the Lydia Pinkham data may be nonstationary. For example, the sample autocorrelations tend to be large (persist) for several lags. Difference the data. Construct a time series plot of the differences. Using Minitab or a similar program, fit an ARIMA(1, 1, 0) model to annual sales for 1907 to 1948. Generate one-step-ahead forecasts for the years 1949 to 1960. Which model, AR(2) or ARIMA(1,1,0), do you think is better for the Lydia Pinkham data?

THE LYDIA E. PINKHAM MEDICINE COMPANY The Lydia E. Pinkham Medicine Company was a family-owned concern whose income was derived largely from the sale of Lydia Pinkham's Vegetable Compound. Perhaps students today could use some of the compound to relieve stress; unfortunately, it is no longer sold. Lydia Pinkham's picture was on the label, and the compound was marketed to women. Ads for the compound included this invitation: "write freely and fully to Mrs. Pinkham, at Lynn, Mass., and secure the advice which she offers free of charge to all women. This is the advice that has brought sunshine into many homes which nervousness and irritability had nearly wrecked." In fact, the company ensured that a female employee answered every letter. Women did write to Mrs. Pinkham. Their claims included this one: "Without [Lydia Pinkham's Vegetable Compound] I would by this time have been dead or, worse, insane.... I had given up on myself; as I had tried so many things, I believed nothing would ever do me any good. But, thanks to your medicine, I am now well and strong; in fact, another person entirely." This testimonial and others were reproduced in print ads for the compound. The unique nature of the company-one dominant product that accounted for most of the company's sales, no sales staff, and a large proportion of sales revenues invested in advertising-and the availability of data on both sales and advertising led the Committee on Price Determination of the National Bureau of Economic Research (NBER) in 1943 to recommend that the data be subjected to thorough analysis. The research was not undertaken for several years (Palda 1964). Analysts have studied the data using causal models that include the advertising data and other economic variables (similar to those presented in Chapter 8). However, several researchers have suggested that Box-Jenkins approaches using only the sales data provide comparable, or even superior, predictions when compared with the causal approaches (see, e.g., Kyle 1978). The sales data are appealing to study for two reasons: 1. The product itself was unchanged for the span of the data; that is, there are no shifts in the series caused by changes in the product. 2. There was no change in the sales force over the span of the data, and the proportion of revenues spent on advertising was fairly constant. Thus, there are no shifts in the data caused by special promotions or other marketing phenomena. Typically, actual data are not this "clean" in terms of product and marketing continuity. The task at hand, then, is to determine which Box-Jenkins (ARIMA) model is the "best" for these data. The model will be developed using the 1907-1948 data and tested using the 1949-1960 data shown in Table 9-18. MODEL IDENTIFICATION A computer program capable of ARIMA modeling was used to examine the data for 1907 through 1948; the data for 1949 through 1960 are used to examine the forecasting ability of the selected model. Preliminary tests suggest that the data are stationary (that is, there is no apparent trend), so differencing is not employed. After examining the autocorrelations and partial autocorrelations, it was determined that an AR model was most appropriate for the data. (The autocorrelations (ACF) and partial autocorrelations (PACF) for 10 periods are given in Table 9-19.) The autocorrelations and partial autocorrelations seemed to be consistent with those of an AR(2) process. To verify the order p of the AR component, Akaike's information criterion ( AIC ) (see Equation 9.7) was used with autoregressive models of orders, p = 2, and 3. The AIC leads to the choice of an AR(2) model for the Lydia Pinkham data. TABLE 9-18 Lydia E. Pinkham Medicine Data TABLE 9-19 MODEL ESTIMATION AND TESTING OF MODEL ADEQUACY A computer program was used to estimate the parameters (including a constant term) of the AR(2) model using the data for 1907 through 1948. Using the estimated parameters, the resulting model is where the numbers in parentheses beneath the autoregressive coefficients are their estimated standard deviations. Each autoregressive coefficient is significantly different from zero for any reasonable significance level. The residual autocorrelations were all small, and each was within its 95% error limits. The Ljung-Box chi-square statistics had p -values of.63,.21, and.64 for groups of lags m = 24, and 36, respectively. The AR(2) model appears to be adequate for the Lydia Pinkham sales data. FORECASTING WITH THE MODEL The final step in the analysis of the AR(2) model for these data is to forecast the values for 1949 through 960 one period ahead. (For example, data through 1958 are used in computing the forecast for 1959.) The forecasting equation is and the one-step-ahead forecasts and the forecast errors are shown in Table 9-20. Table 9-20 In addition to the one-step-ahead forecasts, some accuracy measures were computed. The forecasts from the AR(2) model have a mean absolute percentage error ( MAPE ) of 6.9% and a mean absolute deviation ( MAD ) of $112 (thousands of current dollars).These figures compare favorably to the accuracy measures from the causal models developed by other researchers. SUMMARY AND CONCLUSIONS A parsimonious (smallest number of parameters) AR(2) model has been fit to the Lydia Pinkham data for the years 1907 through 1948.This model has produced fairly accurate one-step-ahead forecasts for the years 1949 through 1960. The Lydia Pinkham data are interesting due to the unique (unchanging) nature of the product and marketing for the 54-year period represented. What factors might affect annual sales data for automobiles and copper over this same period? Why? For further reading on the Lydia E. Pinkham Medicine Company and Lydia Pinkham, see Sarah Stage, Female Complaints: Lydia Pinkham and the Business of Women's Medicine (New York: Norton, 1979).

CITY OF COLLEGE STATION The City of College Station relies on monthly sales tax revenue to fund operations. The sales tax revenue generated accounts for approximately 44% of the total General Fund budget. It is important to be able to forecast the amount of sales tax revenue the city will receive each month from the state of Texas so that anticipated expenditures can be matched with projected revenue. Charles Lemore, an analyst for the city, was asked to develop a forecasting model for monthly sales tax revenue. Charles has recently completed a forecasting course at the local university and decides to try the Box-Jenkins methodology. He begins by looking at the data and the sample autocorrelation and sample partial autocorrelation functions. The monthly sales tax revenues (in thousands of dollars) for the period from January 1991 to November 1999 are listed in Table 9-21. A time series plot of revenues is given in Figure 9-41. A strong seasonal component is evident in a plot of the series and in the sample autocorrelations and sample partial autocorrelations shown in Figures 9-42 and 9-43, respectively. The autocorrelations at multiples of lag 3 suggest that there is a within-year quarterly seasonal pattern (the second monthly payment to the city within any quarter is relatively large) as well as an annual pattern. In addition, the series contains a clear upward trend. Charles decides to difference the revenue series with respect to the seasonal period S = 12. The sample autocorrelations for the seasonally differenced series are shown in Figure 9-44. A plot of the seasonally differenced series (not shown) appears to vary about a constant level of about 50. Charles is ready to fit and check an ARIMA model for the sales tax revenue series, but he needs your help. He would also like to generate forecasts for the next 12 months, once he has an adequate model. FIGURE 9-41 City of College Station Sales Tax Revenues FIGURE 9-42 Sample Autocorrelations for Sales Tax Revenues TABLE 9-21 FIGURE 9-43 Sample Partial Autocorrelations for Sales Tax Revenues FIGURE 9-44 Sample Autocorrelations for the Seasonally Differenced Sales Tax Revenues Develop an ARIMA model for sales tax revenue using the Box-Jenkins methodology.

CITY OF COLLEGE STATION The City of College Station relies on monthly sales tax revenue to fund operations. The sales tax revenue generated accounts for approximately 44% of the total General Fund budget. It is important to be able to forecast the amount of sales tax revenue the city will receive each month from the state of Texas so that anticipated expenditures can be matched with projected revenue. Charles Lemore, an analyst for the city, was asked to develop a forecasting model for monthly sales tax revenue. Charles has recently completed a forecasting course at the local university and decides to try the Box-Jenkins methodology. He begins by looking at the data and the sample autocorrelation and sample partial autocorrelation functions. The monthly sales tax revenues (in thousands of dollars) for the period from January 1991 to November 1999 are listed in Table 9-21. A time series plot of revenues is given in Figure 9-41. A strong seasonal component is evident in a plot of the series and in the sample autocorrelations and sample partial autocorrelations shown in Figures 9-42 and 9-43, respectively. The autocorrelations at multiples of lag 3 suggest that there is a within-year quarterly seasonal pattern (the second monthly payment to the city within any quarter is relatively large) as well as an annual pattern. In addition, the series contains a clear upward trend. Charles decides to difference the revenue series with respect to the seasonal period S = 12. The sample autocorrelations for the seasonally differenced series are shown in Figure 9-44. A plot of the seasonally differenced series (not shown) appears to vary about a constant level of about 50. Charles is ready to fit and check an ARIMA model for the sales tax revenue series, but he needs your help. He would also like to generate forecasts for the next 12 months, once he has an adequate model. FIGURE 9-41 City of College Station Sales Tax Revenues FIGURE 9-42 Sample Autocorrelations for Sales Tax Revenues TABLE 9-21 FIGURE 9-43 Sample Partial Autocorrelations for Sales Tax Revenues FIGURE 9-44 Sample Autocorrelations for the Seasonally Differenced Sales Tax Revenues Using your model, generate forecasts of revenue for the next 12 months. Append these forecasts to the end of the series, and plot the results. Are you happy with the pattern of the forecasts?

CITY OF COLLEGE STATION The City of College Station relies on monthly sales tax revenue to fund operations. The sales tax revenue generated accounts for approximately 44% of the total General Fund budget. It is important to be able to forecast the amount of sales tax revenue the city will receive each month from the state of Texas so that anticipated expenditures can be matched with projected revenue. Charles Lemore, an analyst for the city, was asked to develop a forecasting model for monthly sales tax revenue. Charles has recently completed a forecasting course at the local university and decides to try the Box-Jenkins methodology. He begins by looking at the data and the sample autocorrelation and sample partial autocorrelation functions. The monthly sales tax revenues (in thousands of dollars) for the period from January 1991 to November 1999 are listed in Table 9-21. A time series plot of revenues is given in Figure 9-41. A strong seasonal component is evident in a plot of the series and in the sample autocorrelations and sample partial autocorrelations shown in Figures 9-42 and 9-43, respectively. The autocorrelations at multiples of lag 3 suggest that there is a within-year quarterly seasonal pattern (the second monthly payment to the city within any quarter is relatively large) as well as an annual pattern. In addition, the series contains a clear upward trend. Charles decides to difference the revenue series with respect to the seasonal period S = 12. The sample autocorrelations for the seasonally differenced series are shown in Figure 9-44. A plot of the seasonally differenced series (not shown) appears to vary about a constant level of about 50. Charles is ready to fit and check an ARIMA model for the sales tax revenue series, but he needs your help. He would also like to generate forecasts for the next 12 months, once he has an adequate model. FIGURE 9-41 City of College Station Sales Tax Revenues FIGURE 9-42 Sample Autocorrelations for Sales Tax Revenues TABLE 9-21 FIGURE 9-43 Sample Partial Autocorrelations for Sales Tax Revenues FIGURE 9-44 Sample Autocorrelations for the Seasonally Differenced Sales Tax Revenues Write a brief memo summarizing your findings.

UPS AIR FINANCE DIVISION The Air Finance Division is one of several divisions within United Parcel Service Financial Corporation, a wholly owned subsidiary of United Parcel Service (UPS). The Air Finance Division has the responsibility of providing financial services to the company as a whole with respect to all jet aircraft acquisitions. In addition, this division provides financing to independent outside entities for aviation purchases as a separate and distinct operation. Historically, the non-UPS financing segment of the Air Finance Division has provided a higher rate of return and greater fee income than the UPS segment; however, it is more costly to pursue. Funding forecasts for the non-UPS market segment of the Air Finance Division have been highly subjective and not as reliable as the forecasts for other segments. Table 9-22 lists 10 years of monthly funding requirements for the non-UPS Air Finance Division segment. These data were collected from management reports during the period from January 1989 through December 1998 and are the month-end figures in millions of dollars. The data in Table 9-22 are plotted in Figure 9-45. The funding requirement time series has a strong seasonal component along with a general upward trend. The sample autocorrelations and sample partial autocorrelations are shown in Figures 9-46 and 9-47, respectively. The Box-Jenkins methodology will be used to develop an ARIMA model for forecasting future funding requirements for the non-UPS segment of the Air Finance Division. Because of the nonstationary character of the time series, the autocorrelations and partial autocorrelations for various differences of the series are examined. The autocorrelation function for the series with one regular difference and one seasonal difference of order S = 12 is shown in Figure 9-48. That is, these autocorrelations are computed for the differences There is clearly a significant spike at lag 12 in the sample autocorrelations shown in Figure 9-48. This suggests an ARIMA model for the W's (the process with a regular difference and a seasonal difference) that has a moving average term at the seasonal lag 12. Consequently, an ARIMA(0, 1, 0)(0, 1, 1) 12 model might be a good initial choice for the funding requirements data. TABLE 9-22 FIGURE 9-45 Funding Requirements for Non-UPS Segment of Air Finance Division FIGURE 9-46 Sample Autocorrelations for Funding Requirements FIGURE 9-47 Sample Partial Autocorrelations for Funding Requirements Using Minitab or equivalent software, fit an ARIMA(0, 1, 0)(0, 1, 1 ) 12 model to the data in Table 9-22. Do you think a constant term is required in the model? Explain.

UPS AIR FINANCE DIVISION The Air Finance Division is one of several divisions within United Parcel Service Financial Corporation, a wholly owned subsidiary of United Parcel Service (UPS). The Air Finance Division has the responsibility of providing financial services to the company as a whole with respect to all jet aircraft acquisitions. In addition, this division provides financing to independent outside entities for aviation purchases as a separate and distinct operation. Historically, the non-UPS financing segment of the Air Finance Division has provided a higher rate of return and greater fee income than the UPS segment; however, it is more costly to pursue. Funding forecasts for the non-UPS market segment of the Air Finance Division have been highly subjective and not as reliable as the forecasts for other segments. Table 9-22 lists 10 years of monthly funding requirements for the non-UPS Air Finance Division segment. These data were collected from management reports during the period from January 1989 through December 1998 and are the month-end figures in millions of dollars. The data in Table 9-22 are plotted in Figure 9-45. The funding requirement time series has a strong seasonal component along with a general upward trend. The sample autocorrelations and sample partial autocorrelations are shown in Figures 9-46 and 9-47, respectively. The Box-Jenkins methodology will be used to develop an ARIMA model for forecasting future funding requirements for the non-UPS segment of the Air Finance Division. Because of the nonstationary character of the time series, the autocorrelations and partial autocorrelations for various differences of the series are examined. The autocorrelation function for the series with one regular difference and one seasonal difference of order S = 12 is shown in Figure 9-48. That is, these autocorrelations are computed for the differences There is clearly a significant spike at lag 12 in the sample autocorrelations shown in Figure 9-48. This suggests an ARIMA model for the W's (the process with a regular difference and a seasonal difference) that has a moving average term at the seasonal lag 12. Consequently, an ARIMA(0, 1, 0)(0, 1, 1) 12 model might be a good initial choice for the funding requirements data. TABLE 9-22 FIGURE 9-45 Funding Requirements for Non-UPS Segment of Air Finance Division FIGURE 9-46 Sample Autocorrelations for Funding Requirements FIGURE 9-47 Sample Partial Autocorrelations for Funding Requirements Is the model suggested in Question 1 adequate? Discuss with reference to residual plots, residual autocorrelations, and the Ljung-Box chi-square statistics. If the model is not adequate, modify and refit the initial model until you feel you have achieved a satisfactory model.

UPS AIR FINANCE DIVISION The Air Finance Division is one of several divisions within United Parcel Service Financial Corporation, a wholly owned subsidiary of United Parcel Service (UPS). The Air Finance Division has the responsibility of providing financial services to the company as a whole with respect to all jet aircraft acquisitions. In addition, this division provides financing to independent outside entities for aviation purchases as a separate and distinct operation. Historically, the non-UPS financing segment of the Air Finance Division has provided a higher rate of return and greater fee income than the UPS segment; however, it is more costly to pursue. Funding forecasts for the non-UPS market segment of the Air Finance Division have been highly subjective and not as reliable as the forecasts for other segments. Table 9-22 lists 10 years of monthly funding requirements for the non-UPS Air Finance Division segment. These data were collected from management reports during the period from January 1989 through December 1998 and are the month-end figures in millions of dollars. The data in Table 9-22 are plotted in Figure 9-45. The funding requirement time series has a strong seasonal component along with a general upward trend. The sample autocorrelations and sample partial autocorrelations are shown in Figures 9-46 and 9-47, respectively. The Box-Jenkins methodology will be used to develop an ARIMA model for forecasting future funding requirements for the non-UPS segment of the Air Finance Division. Because of the nonstationary character of the time series, the autocorrelations and partial autocorrelations for various differences of the series are examined. The autocorrelation function for the series with one regular difference and one seasonal difference of order S = 12 is shown in Figure 9-48. That is, these autocorrelations are computed for the differences There is clearly a significant spike at lag 12 in the sample autocorrelations shown in Figure 9-48. This suggests an ARIMA model for the W's (the process with a regular difference and a seasonal difference) that has a moving average term at the seasonal lag 12. Consequently, an ARIMA(0, 1, 0)(0, 1, 1) 12 model might be a good initial choice for the funding requirements data. TABLE 9-22 FIGURE 9-45 Funding Requirements for Non-UPS Segment of Air Finance Division FIGURE 9-46 Sample Autocorrelations for Funding Requirements FIGURE 9-47 Sample Partial Autocorrelations for Funding Requirements Using the model you have developed in Question 2, generate forecasts for the funding requirements for the next 12 months.

UPS AIR FINANCE DIVISION The Air Finance Division is one of several divisions within United Parcel Service Financial Corporation, a wholly owned subsidiary of United Parcel Service (UPS). The Air Finance Division has the responsibility of providing financial services to the company as a whole with respect to all jet aircraft acquisitions. In addition, this division provides financing to independent outside entities for aviation purchases as a separate and distinct operation. Historically, the non-UPS financing segment of the Air Finance Division has provided a higher rate of return and greater fee income than the UPS segment; however, it is more costly to pursue. Funding forecasts for the non-UPS market segment of the Air Finance Division have been highly subjective and not as reliable as the forecasts for other segments. Table 9-22 lists 10 years of monthly funding requirements for the non-UPS Air Finance Division segment. These data were collected from management reports during the period from January 1989 through December 1998 and are the month-end figures in millions of dollars. The data in Table 9-22 are plotted in Figure 9-45. The funding requirement time series has a strong seasonal component along with a general upward trend. The sample autocorrelations and sample partial autocorrelations are shown in Figures 9-46 and 9-47, respectively. The Box-Jenkins methodology will be used to develop an ARIMA model for forecasting future funding requirements for the non-UPS segment of the Air Finance Division. Because of the nonstationary character of the time series, the autocorrelations and partial autocorrelations for various differences of the series are examined. The autocorrelation function for the series with one regular difference and one seasonal difference of order S = 12 is shown in Figure 9-48. That is, these autocorrelations are computed for the differences There is clearly a significant spike at lag 12 in the sample autocorrelations shown in Figure 9-48. This suggests an ARIMA model for the W's (the process with a regular difference and a seasonal difference) that has a moving average term at the seasonal lag 12. Consequently, an ARIMA(0, 1, 0)(0, 1, 1) 12 model might be a good initial choice for the funding requirements data. TABLE 9-22 FIGURE 9-45 Funding Requirements for Non-UPS Segment of Air Finance Division FIGURE 9-46 Sample Autocorrelations for Funding Requirements FIGURE 9-47 Sample Partial Autocorrelations for Funding Requirements Write a report summarizing your findings. Include in your report a plot of the original series and the forecasts.

AAA WASHINGTON An overview of AAA Washington was provided in Case 5-5 when students were asked to prepare a time series decomposition of the emergency road service calls received by the club over five years. The time series decomposition performed in Case 5-5 showed that the pattern Michael DeCoria had observed in emergency road service call volume was probably cyclical in nature. Michael would like to be able to predict the cyclical effect on emergency road service call volume for future years. In Case 6-6, four linear regression models using the total number of emergency road service calls as the dependent variable and the unemployment rate, temperature, rainfall, and number of members as the four independent variables were investigated. The temperature variable was transformed by subtracting 65 degrees from the average monthly temperature values. A nonlinear relationship was then researched. In Case 7-2, a multiple regression model was developed. Variables such as the rainfall, number of members, exponentially transformed average monthly temperature, and unemployment rate lagged 11 months were tested. In Case 8-6, the multiple regression model developed in Case 7-2 was checked for serial correlation, and a model was recommended to Michael that was believed to be most appropriate for predicting the cyclical nature of emergency road service call volume. Michael is still not satisfied. The model presented in Case 8-6 explained only about 71 percent of the variance of the emergency road service call volume variable. He has asked you to try the Box-Jenkins methodology to forecast the road call volume data presented in Table 9-23. You decide to divide the data set into two sections. The values from May 1988 through December 1992 will be the init.ia.1-ization or fitting section, and the first four months of 1993 will be the test or forecasting section. FIGURE 9-48 Sample Autocorrelations for the Regularly Differenced and Seasonally Differenced Funding Requirements TABLE 9-23 Emergency Road Service Call Volume by Month Develop an ARIMA model for the emergency road service call volume data. Write Michael a memo summarizing your findings.

WEB RETAILER In Case 4-6, Pat Niebuhr and his team were interested in developing alternative forecasts for total orders and contacts per order (CPO) as part of a staffing plan project for the contact centers of his employer, a large web retailer. Monthly forecasts of total orders and CPO were supplied by the finance department for Pat's use, but since he had some historical data available, he thought he would try to generate alternative forecasts using smoothing methods. He could then compare his forecasts with those supplied by the finance department. However, the appropriate variable for planning purposes is Contacts = Orders × CPO and Pat was convinced, since the data were available, that forecasting contacts directly would be better than generating a contacts forecast by multiplying an orders forecast by a CPO forecast. Pat had some training in Box-Jenkins methods and thought he should give ARIMA modeling a try. But Pat recognized that he didn't have very many observations and wondered if ARIMA modeling was even feasible. The monthly contacts data are given in Table 9-24. A plot of the contacts time series is shown in Figure 9-49. Even though he had only about two years' worth of data, Pat concluded from the time series plot that the number of contacts is seasonal. Pat was unsure about stationarity. The sample autocorrelations for the contacts series are shown in Figure 9-50. There is a significant autocorrelation at lag 1, but the autocorrelation coefficients are not slow to die out like those for a nonstationary series. Because the time series plot seems to show a slight downward trend, Pat decides to difference the contacts series and look at a plot and the autocorrelations of the differences. The results are shown in Figures 9-51 and 9-52. The contact differences appear to vary about zero, and the autocorrelation function shows no significant autocorrelation, although the error bands are fairly wide because of the small sample size. There is a reasonably large (almost significant) autocorrelation at lag 12, and since the time series plot of contacts shows a seasonal pattern, Pat decided to try a seasonal model for contacts. TABLE 9-24 Contacts for Web Retailer, June 2001-June 2003 FIGURE 9-49 Monthly Number of Contacts for a Web Retailer FIGURE 9-50 Sample Autocorrelation Function for Contacts FIGURE 9-51 Time Series Plot of the Differences of Contacts Using Minitab or equivalent software, fit an ARIMA(0, 1, 0)(0, 0, 1) 12 model to the data in Table 9-24. Why does this model seem to be a reasonable initial choice? Is there another candidate model that you might consider?

WEB RETAILER In Case 4-6, Pat Niebuhr and his team were interested in developing alternative forecasts for total orders and contacts per order (CPO) as part of a staffing plan project for the contact centers of his employer, a large web retailer. Monthly forecasts of total orders and CPO were supplied by the finance department for Pat's use, but since he had some historical data available, he thought he would try to generate alternative forecasts using smoothing methods. He could then compare his forecasts with those supplied by the finance department. However, the appropriate variable for planning purposes is Contacts = Orders × CPO and Pat was convinced, since the data were available, that forecasting contacts directly would be better than generating a contacts forecast by multiplying an orders forecast by a CPO forecast. Pat had some training in Box-Jenkins methods and thought he should give ARIMA modeling a try. But Pat recognized that he didn't have very many observations and wondered if ARIMA modeling was even feasible. The monthly contacts data are given in Table 9-24. A plot of the contacts time series is shown in Figure 9-49. Even though he had only about two years' worth of data, Pat concluded from the time series plot that the number of contacts is seasonal. Pat was unsure about stationarity. The sample autocorrelations for the contacts series are shown in Figure 9-50. There is a significant autocorrelation at lag 1, but the autocorrelation coefficients are not slow to die out like those for a nonstationary series. Because the time series plot seems to show a slight downward trend, Pat decides to difference the contacts series and look at a plot and the autocorrelations of the differences. The results are shown in Figures 9-51 and 9-52. The contact differences appear to vary about zero, and the autocorrelation function shows no significant autocorrelation, although the error bands are fairly wide because of the small sample size. There is a reasonably large (almost significant) autocorrelation at lag 12, and since the time series plot of contacts shows a seasonal pattern, Pat decided to try a seasonal model for contacts. TABLE 9-24 Contacts for Web Retailer, June 2001-June 2003 FIGURE 9-49 Monthly Number of Contacts for a Web Retailer FIGURE 9-50 Sample Autocorrelation Function for Contacts FIGURE 9-51 Time Series Plot of the Differences of Contacts Is the model suggested in Question 1 adequate? Discuss with reference to residual plots, residual autocorrelations, and the Ljung-Box chi-square statistics. If the model is not adequate, modify and refit the initial model until you feel you have achieved a satisfactory model.

WEB RETAILER In Case 4-6, Pat Niebuhr and his team were interested in developing alternative forecasts for total orders and contacts per order (CPO) as part of a staffing plan project for the contact centers of his employer, a large web retailer. Monthly forecasts of total orders and CPO were supplied by the finance department for Pat's use, but since he had some historical data available, he thought he would try to generate alternative forecasts using smoothing methods. He could then compare his forecasts with those supplied by the finance department. However, the appropriate variable for planning purposes is Contacts = Orders × CPO and Pat was convinced, since the data were available, that forecasting contacts directly would be better than generating a contacts forecast by multiplying an orders forecast by a CPO forecast. Pat had some training in Box-Jenkins methods and thought he should give ARIMA modeling a try. But Pat recognized that he didn't have very many observations and wondered if ARIMA modeling was even feasible. The monthly contacts data are given in Table 9-24. A plot of the contacts time series is shown in Figure 9-49. Even though he had only about two years' worth of data, Pat concluded from the time series plot that the number of contacts is seasonal. Pat was unsure about stationarity. The sample autocorrelations for the contacts series are shown in Figure 9-50. There is a significant autocorrelation at lag 1, but the autocorrelation coefficients are not slow to die out like those for a nonstationary series. Because the time series plot seems to show a slight downward trend, Pat decides to difference the contacts series and look at a plot and the autocorrelations of the differences. The results are shown in Figures 9-51 and 9-52. The contact differences appear to vary about zero, and the autocorrelation function shows no significant autocorrelation, although the error bands are fairly wide because of the small sample size. There is a reasonably large (almost significant) autocorrelation at lag 12, and since the time series plot of contacts shows a seasonal pattern, Pat decided to try a seasonal model for contacts. TABLE 9-24 Contacts for Web Retailer, June 2001-June 2003 FIGURE 9-49 Monthly Number of Contacts for a Web Retailer FIGURE 9-50 Sample Autocorrelation Function for Contacts FIGURE 9-51 Time Series Plot of the Differences of Contacts Using the model you have developed in Question 2, generate forecasts of contacts for the next 12 months. Include the forecasts on a plot of the original contacts series. Write a brief report commenting on the reasonableness of the forecasts.

WEB RETAILER In Case 4-6, Pat Niebuhr and his team were interested in developing alternative forecasts for total orders and contacts per order (CPO) as part of a staffing plan project for the contact centers of his employer, a large web retailer. Monthly forecasts of total orders and CPO were supplied by the finance department for Pat's use, but since he had some historical data available, he thought he would try to generate alternative forecasts using smoothing methods. He could then compare his forecasts with those supplied by the finance department. However, the appropriate variable for planning purposes is Contacts = Orders × CPO and Pat was convinced, since the data were available, that forecasting contacts directly would be better than generating a contacts forecast by multiplying an orders forecast by a CPO forecast. Pat had some training in Box-Jenkins methods and thought he should give ARIMA modeling a try. But Pat recognized that he didn't have very many observations and wondered if ARIMA modeling was even feasible. The monthly contacts data are given in Table 9-24. A plot of the contacts time series is shown in Figure 9-49. Even though he had only about two years' worth of data, Pat concluded from the time series plot that the number of contacts is seasonal. Pat was unsure about stationarity. The sample autocorrelations for the contacts series are shown in Figure 9-50. There is a significant autocorrelation at lag 1, but the autocorrelation coefficients are not slow to die out like those for a nonstationary series. Because the time series plot seems to show a slight downward trend, Pat decides to difference the contacts series and look at a plot and the autocorrelations of the differences. The results are shown in Figures 9-51 and 9-52. The contact differences appear to vary about zero, and the autocorrelation function shows no significant autocorrelation, although the error bands are fairly wide because of the small sample size. There is a reasonably large (almost significant) autocorrelation at lag 12, and since the time series plot of contacts shows a seasonal pattern, Pat decided to try a seasonal model for contacts. TABLE 9-24 Contacts for Web Retailer, June 2001-June 2003 FIGURE 9-49 Monthly Number of Contacts for a Web Retailer FIGURE 9-50 Sample Autocorrelation Function for Contacts FIGURE 9-51 Time Series Plot of the Differences of Contacts Do you have any reservations about using ARIMA modeling in this situation? Discuss.

SURTIDO COOKIES Jame Luna's efforts to model and forecast Surtido Cookies monthly sales have been documented in Cases 3-5,4-8,5-7, and 8-8. A time series plot of sales and the corresponding autocorrelations were considered in Case 4-8. In Case 5-7, a decomposition analysis of sales confirmed the existence of a trend component and a seasonal component. A multiple regression model with a time trend and seasonal dummy variables was considered in Case 8-8. At a recent forecasting seminar, Jame encountered ARIMA modeling. He learned that to identify an appropriate ARIMA model, it is usually a good idea to examine plots of the time series and its various differences. Moreover, the sample autocorrelations and sample partial autocorrelations for the series and its differences provide useful model identification information. Jame decides to try to fit an ARIMA model to his data and generate forecasts. Since Surtido Cookies sales are seasonal, S = 12, Jame constructed plots and autocorrelation functions for the original series, the differences of the original series (DiffSales), the seasonal differences of the series (Diff12Sales), and the series with one regular difference followed by one seasonal difference (DiffDiff12 Sales). A time series plot and the sample autocorrelation function for Diff12Sales are shown in Figures 9-53 and 9-54, respectively. Looking at this information, Jame has selected an initial ARIMA model but needs your help. FIGURE 9-53 Time Series Plot of Seasonally Differenced Surtido Cookies Sales FIGURE 9-54 Sample Autocorrelations for the Seasonally Differenced Cookie Sales Using Minitab or equivalent software, fit an ARIMA(0, 0, 0)(0, 1, 1) 12 model to the data in Table 3-12. Why does this model seem to be a reasonable initial choice for Jame? Looking at the plots and autocorrelations of the original series and differences other than Diff12Sales, is there another candidate model that you might consider?

SURTIDO COOKIES Jame Luna's efforts to model and forecast Surtido Cookies monthly sales have been documented in Cases 3-5,4-8,5-7, and 8-8. A time series plot of sales and the corresponding autocorrelations were considered in Case 4-8. In Case 5-7, a decomposition analysis of sales confirmed the existence of a trend component and a seasonal component. A multiple regression model with a time trend and seasonal dummy variables was considered in Case 8-8. At a recent forecasting seminar, Jame encountered ARIMA modeling. He learned that to identify an appropriate ARIMA model, it is usually a good idea to examine plots of the time series and its various differences. Moreover, the sample autocorrelations and sample partial autocorrelations for the series and its differences provide useful model identification information. Jame decides to try to fit an ARIMA model to his data and generate forecasts. Since Surtido Cookies sales are seasonal, S = 12, Jame constructed plots and autocorrelation functions for the original series, the differences of the original series (DiffSales), the seasonal differences of the series (Diff12Sales), and the series with one regular difference followed by one seasonal difference (DiffDiff12 Sales). A time series plot and the sample autocorrelation function for Diff12Sales are shown in Figures 9-53 and 9-54, respectively. Looking at this information, Jame has selected an initial ARIMA model but needs your help. FIGURE 9-53 Time Series Plot of Seasonally Differenced Surtido Cookies Sales FIGURE 9-54 Sample Autocorrelations for the Seasonally Differenced Cookie Sales Is the model suggested in Question 1 adequate? Discuss with reference to residual plots, residual autocorrelations, and the Ljung-Box chi-square statistics. If the model is not adequate, modify and refit the initial model until you feel you have achieved a satisfactory model.

SURTIDO COOKIES Jame Luna's efforts to model and forecast Surtido Cookies monthly sales have been documented in Cases 3-5,4-8,5-7, and 8-8. A time series plot of sales and the corresponding autocorrelations were considered in Case 4-8. In Case 5-7, a decomposition analysis of sales confirmed the existence of a trend component and a seasonal component. A multiple regression model with a time trend and seasonal dummy variables was considered in Case 8-8. At a recent forecasting seminar, Jame encountered ARIMA modeling. He learned that to identify an appropriate ARIMA model, it is usually a good idea to examine plots of the time series and its various differences. Moreover, the sample autocorrelations and sample partial autocorrelations for the series and its differences provide useful model identification information. Jame decides to try to fit an ARIMA model to his data and generate forecasts. Since Surtido Cookies sales are seasonal, S = 12, Jame constructed plots and autocorrelation functions for the original series, the differences of the original series (DiffSales), the seasonal differences of the series (Diff12Sales), and the series with one regular difference followed by one seasonal difference (DiffDiff12 Sales). A time series plot and the sample autocorrelation function for Diff12Sales are shown in Figures 9-53 and 9-54, respectively. Looking at this information, Jame has selected an initial ARIMA model but needs your help. FIGURE 9-53 Time Series Plot of Seasonally Differenced Surtido Cookies Sales FIGURE 9-54 Sample Autocorrelations for the Seasonally Differenced Cookie Sales Using the model you have developed in Question 2, generate forecasts of cookie sales for the next 12 months. Include the forecasts on a plot of the original Surtido Cookies sales series. Write a brief report describing the nature of the forecasts and whether they seem reasonable.