Business Forecasting 1st Edition John E. Hanke - Solutions

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Suppose that ???????? = ???????? + ????????, where ???????? is a first-order autoregressive process (1 −????????)???????? = ???????? and ???????? is a white noise process with variance ????2???? . What model does the process ???????? follow? State your assumptions.
If ????1???? = (1 − ????1????)????1???? and ????2???? = (1 − ????2????)????2????, show that ????3???? = ????1???? + ????2???? may be written as ????3???? = (1 − ????3????)????3????, and derive an expression for ????3 and ????2????3 in terms of the parameters of the other two processes. State
If ????̄???? = ∑∞????=1 ????????????????+1−????, then for models (1) and (2) of Exercise 4.1, which are of the form (1 − ????)???????? = (1 − ????????)????????, ̄???????? is an exponentially weighted moving average.For these two models, by actual calculation, confirm that ????̄11,
Consider the IMA(0, 1, 1) model (1 − ????)???????? = (1 − ????)????????, where the ???????? are i.i.d.N(0, ????2????).(a) Derive the expected value and variance of ????????, ???? = 1, 2, … , assuming that the process starts at time ???? = 1 with ????0 = 10.(b) Derive the correlation
Using the inverted forms of each of the models in Exercise 4.1, obtain ????12, ????13, and????14, using only the values ????1, ????2, … , ????11 derived in Exercise 4.4 and ????12, ????13, and????14. Confirm that the values agree with those obtained in Exercise 4.4.
Consider the IMA(0, 2, 2) process with parameters ????1 = 0.8 and ????2 = −0.4.(a) Is the process invertible? If so, what is the expected pattern of the ???? weights?(b) Calculate and plot the first ten ???? weights for the original series ???????? and comment.(c) Calculate and plot the first ten
For the five models of Exercise 4.1, and using where appropriate the results there obtained,(a) Write each model in random shock form.(b) Write each model as a complementary function plus a particular integral in relation to an origin ???? = ???? − 3.(c) Write each model in inverted form.
For each of the models(1) (1 − ????)???????? = (1 − 0.5????)????????(2) (1 − ????)???????? = (1 − 0.2????)????????(3) (1 − 0.5????)(1 − ????)???????? = ????????(4) (1 − 0.2????)(1 − ????)???????? = ????????(5) (1 − 0.2????)(1 − ????)???????? = (1 − 0.5????)????????(a) Obtain
Calculate and plot the theoretical autocorrelation function and partial autocorrelation function for the AR(4) model specified in Exercise 3.11(b).
(a) Why is it important to factorize the autoregressive and moving average operators after fitting a model to an observed series?(b) It was shown by Jenkins (1975) that the number of mink skins ???????? traded annually between 1848 and 1909 in North Canada is adequately represented by the
(a) Plot the power spectrum ????(????) of the autoregressive process of Exercise 3.9, and show that it has a peak at a period that is close to the period in the autocorrelation function.(b) Graphically, or otherwise, estimate the proportion of the variance of the series in the frequency band
For the AR(2) process ????̃???? − 1.0????̃????−1 + 0.5????̃????−2 = ????????:(a) Calculate ????1.(b) Using ????0 and ????1 as starting values and the difference equation form for the autocorrelation function, calculate the values of ???????? for ???? = 2, … , 15.(c) Use the plotted
Consider the mixed ARMA(1,1) model ???????? − ????????????−1 = ???????? − ????????????−1, where −1
Consider the first-order autoregressive model ???????? = ????0 + ????????????−1 + ????????, where the constant ????0 is a function of the mean of the series.(a) Derive the autocovariances ???????? = ????([???????? − ????][????????−???? − ????]) for this series.(b) Calculate and plot the
(a) Write down the Yule--Walker equations for models (1) and (4) considered in Exercises 3.1 and 3.2.(b) Solve these equations to obtain ????1 and ????2 for the two models.(c) Obtain the partial autocorrelation function for the two models.
Classify each of the models (1) to (4) in Exercises 3.1 and 3.2 as a member of the class of ARMA(p, q) processes.
Calculate the first fifteen ???????? weights for each of the three models in Exercise 3.2 using the function ARMAtoMA in R. See help(ARMAtoMA) for details.
For each of the models in Exercise 3.1, obtain:(a) The first four ???????? weights(b) The first four ???????? weights(c) The autocovariance generating function(d) The first four autocorrelations ????????(e) The variance of ????̃???? assuming that ????2???? = 1.0
For each of the models of Exercise 3.1 and also for the following models, state whether it is (a) stationary or (b) invertible.(4) ????̃???? − 1.5????̃????−1 + 0.6????̃????−2 = ????????(5) ????̃???? − ????̃????−1 = ???????? − 0.5????????−1 (6) ????̃???? − ????̃????−1 =
Write the following models in ???? notation:(1) ????̃???? − 0.5????̃????−1 = ????????(2) ????̃???? = ???????? − 1.3????????−1 + 0.4????????−2 (3) ????̃???? − 0.5????̃????−1 = ???????? − 1.3????????−1 + 0.4????????−2
Using the data of Exercise 2.1, calculate the periodogram for periods 36, 18, 12, 9, 36/5, and 6 and construct an analysis of variance table showing the mean squares associated with these periods and the residual mean square.
Simulate a time series of length ???? = 300 from the AR(2) model specified in Exercise 2.9 and plot the resulting series.(a) Estimate and plot the autocorrelation function for the simulated series. Compare the results with the theoretical autocorrelation function derived in Exercise 2.9.(b) Repeat
Calculate and plot the theoretical autocorrelation function and the spectral density function for the AR(2) process ???????? + 0.35????????−1 − 0.20????????−2 = ????????.
Calculate and plot the theoretical autocorrelation function and the spectral density function for the AR(1) process ???????? = 0.95????????−1 + ????????. (Hint: See the R code provided for Figure 2.9). Based on the results, how would you expect a time series generated from this model to fluctuate
Repeat the calculations in Exercise 2.6 for the annual sunspot series given as Series E in Part Five of this book. Use a square root transformation of the data in part (b) in Exercise 2.6.(Note: This series is also available for a slightly longer time period as series sunspot.year in the datasets
The annual sales of mink furs by a North American company during 1911--1950 are included as Series N in Part Five of this book. The series is also available at http://pages.stat.wisc.edu/ reinsel/bjr-data/.(a) Plot the time series using R. Calculate and plot the sample autocorrelation function of
On the assumption that ???????? = 0 for ???? > 2, obtain the following:(a) Approximate standard errors for ????1, ????2, and ????????,???? > 2.(b) The approximate correlation between ????4 and ????5.
Calculate ????0, ????1, ????2, ????3, ????1, ????2, ????3 for the series given in Exercise 2.1.Make a graph of????????, ???? = 0, 1, 2, 3.
Two stationary stochastic processes ????1???? and ????2???? have the following autocovariance functions:????1???? ∶ ????0 = 0.5, ????1 = 0.2, ???????? =0 (???? ≥2)????2???? ∶ ????0 = 2.30, ????1 = −1.43, ????2 = 0.30, ???????? =0 (???? ≥3)Calculate the autocovariance function of the
State whether or not a stationary stochastic process can have the following values of autocorrelations:(a) ????1 = 0.80, ????2 = 0.55, ???????? = 0, for ???? > 2 (b) ????1 = 0.80, ????2 = 0.28, ???????? = 0, for ???? > 2
The following are temperature measurements ???????? made every minute on a chemical reactor:200, 202, 208, 204, 204, 207, 207, 204, 202, 199, 201, 198, 200, 202, 203, 205, 207, 211, 204, 206, 203, 203, 201, 198, 200, 206, 207, 206, 200, 203, 203, 200, 200, 195, 202, 204.207, 206, 200(a) Plot the
Download a time series of your choosing from the Internet. Note that financial and economic time series are available from sources such as Google Finance and the Federal Reserve Economic Data (FRED) of Federal Reserve Bank in St. Louis, Missouri, while climate data is available from from NOAA’s
Monthly totals of international airline passengers (in thousands of passengers), January 1949--December 1960, are available as Series G in Part Five of this book. The data are also available as series AirPassengers in the R datasets package.(a) Load the dataset into R and examine the structure of
The dataset airquality in the R datasets package includes information on daily air quality measurements in New York, May to September 1973. The variables included are mean ozone levels at Roosevelt Island, solar radiation at Central Park, average wind speed at LaGuardia Airport, and maximum daily
2. Using Minitab or equivalent software, fit an 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?
1. Using Mary Beasley’s data in Table 13, construct a time series plot, and compute the sample autocorrelations and sample partial autocorrelations for the total billable visits, ; the differenced series, ; and the seasonally differenced series, . Given this information and the information in
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4. Do you have any reservations about using ARIMA modeling in this situation? Discuss.
forecasts.
3. 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
2. 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 ARIMA10, 1, 0210, 0, 1212 and refit the initial model until you feel you have achieved a satisfactory
1. Using Minitab or equivalent software, fit an model to the data in Table 24. Why does this model seem to be a reasonable initial choice? Is there another candidate model that you might consider?
4. Write a report summarizing your findings. Include in your report a plot of the original series and the forecasts.
3. Using the model you have developed in Question 2, generate forecasts for the funding requirements for the next 12 months.
2. 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 ARIMA10, 1, 0210, 1, 1212 and refit the initial model until you feel you have achieved a satisfactory
1. Using Minitab or equivalent software, fit an model to the data in Table 22. Do you think a constant term is required in the model? Explain.
3. Write a brief memo summarizing your findings.
2. 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?
1. Develop an ARIMA model for sales tax revenue using the Box-Jenkins methodology.
3. 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?
2. 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
1. After this analysis was completed, the figure for sales in 1961 became available: $1,426. What is the model’s forecast for 1961? If this year were added to the testing data set, what would the revised MAPE and MAD be?
3. Write Dorothy a second memo that summarizes the results of this analysis
2. Develop an ARIMA model using the BoxJenkins methodology, and forecast the monthly number of new clients for the rest of 1993.
1. Write Dorothy a memo that explains the BoxJenkins methodology.
3. Using a program for ARIMA modeling, fit and check an ARIMA model for Mr. Tux sales. Generate forecasts for the next 12 months.
2. Given the autocorrelations in Figure 39 and the partial autocorrelations in Figure 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?
1. Discuss the problems, if any, of explaining the Box-Jenkins method to John’s banker and others on his management team.
5. Would you use the same Box-Jenkins model if the new data were combined with the old data?
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3. How do these forecasts compare with actual sales?
2. What are your forecasts for the first four weeks of January 1983?
1. What is the appropriate Box-Jenkins model to use on the original data?
23. 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
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16. Use the Box-Jenkins methodology to model and forecast the monthly sales of the Cavanaugh Company given in Table P-14. (Hint: Consider a log transformation before modeling these data.)TABLE P-14 Period Observation Period Observation Period Observation 1 101 31 16 61 79 2 84 32 17 62 53 3 54 33
15. 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.
14. 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.
13. 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 TABLE P-13 Period DEF Period DEF Period DEF 1 136.0 16 136.0 31 140.0 2 132.8 17 138.4 32 144.8 3 130.4 18 137.6 33 140.0 4 128.8 19
12. 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
11. 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
10. 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
9. 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
8. 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
7. 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
6. 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?
5. 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.
4. 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.
3. A time series model has been fit and checked with historical data yielding Suppose at time the observation is .a. Determine forecasts for periods 61, 62, and 63 from origin 60.b. Suppose the observed value of is 59. Update the forecasts for periods 62 and 63.c. Suppose the estimate for the
2. Suppose the following time series model has been fit to historical data and found to be an adequate model. The first four observations are and Assuming and , calculate forecasts for periods 5, 6, and 7 if period 4 is the forecast origin.
1.a. 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
5. Which model, the dummy variable regression or the autoregression, do you prefer? Why?
4. Fit an autoregressive model to Jame’s data with sales lagged 12 months as the predictor variable.Is this model reasonable? Generate forecasts for the remaining seven months of 2003 using your autoregressive model.
3. Using Jame’s fitted model in Table 23, generate forecasts for the remaining seven months of 2003.
2. Are you happy with Jame’s regression model? What changes would you make, if any?
1. All the coefficients of the dummy variables in Jame’s regression are negative except that for Nov. Does this make sense? Explain.
4. What conditions might prompt Julie to reexamine her regression model or, perhaps, to look for another method of forecasting sales?
3. How might Julie’s model be used to determine future amounts spent on newspaper and TV advertising?
2. Assuming there are no additional important predictor variables, are you satisfied with Julie’s forecasting model? How would you “sell” the model to management (and Jackson Tilson)?
1. Julie has collected data on other variables that were not included in her multiple regression model. Should one or more of these other variables be included in her model? More generally, how can Julie be sure she has the “right” set of predictor variables?
3. Prepare a memo to Michael recommending the regression model you believe is more appropriate for predicting the cyclical nature of emergency road service call volume.
2. Is serial correlation a problem? If any coefficients are not significantly different from zero, try running a regression without these independent variables. Try experimenting with different powers for the exponential transformation.

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Business Forecasting 1st Edition John E. Hanke - Solutions

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