Forecasting Methods And Applications 3rd Edition Spyros G. Makridakis, Steven C. Wheelwright, Rob J Hyndman - Solutions

(b) What can you learn from the ACF graph?
(a) Describe the time plot.
7.5 Figure 7-29 shows the data for Manufacturer's stocks of Evap-orated and Sweet Condensed Milk (case goods) for the period January 1971 through December 1980.
(f) Create a plot of the series with forecasts and prediction intervals for the next three periods shown.
(e) Forecast three times ahead by hand. Check your fore-casts with forecasts generated by the computer package.
(d) Fit the model using a computer package and examine the residuals. Is the model satisfactory?
(c) Write this model in terms of the backshift operator.
(b) Should you include a constant in the model? Explain.
(a) By studying appropriate graphs of the series, explain why an ARIMA(0,1,1) model seems appropriate.
7.4 Consider Table 7-11 which gives the number of strikes in the United States from 1951{1980.
(e) Generate data from an AR(2) model with Á1 = ¡0:8 andÁ2 = 0:3. Start with Y0 = Y¡1 = 0. Generate data from an MA(2) model with µ1 = ¡0:8 and µ2 = 0:3. Start with Z0 = Z¡1 = 0. Graph the two series and compare them.4737 5117 5091 3468 4320 3825 3673 3694 3708 3333 3367 3614 3362 3655 3963
(d) Generate data from an ARMA(1,1) model with Á1 = 0.6 and µ1 = ¡0:6. Start with Y0 = 0 and Z0 = 0.
(c) Produce a time plot for each series. What can you say about the di®erences between the two models?
(b) Generate data from an MA(1) model with µ1 = ¡0:6.Start with Z0 = 0.
(a) Using the normal random numbers of the table, generate data from an AR(1) model with Á1 = 0:6. Start with Y0 = 0.
7.3 The data below are from a white noise series with a standard normal distribution (mean zero and variance one). (Read left to right.)
7.2 A classic example of a non-stationary series is the daily closing IBM stock prices. Figure 7-28 shows the analysis of n = 369 daily closing prices for IBM stock. Explain how each plot shows the series is non-stationary and should be di®erenced.
(b) Why are the critical values at di®erent distances from the mean of zero? Why are the autocorrelations di®erent in each ¯gure when they each refer to white noise?
(a) Explain the di®erences among these ¯gures. Do they all indicate the data are white noise?
7-27 shows the ACFs for 360 random numbers and for 1,000 random numbers.
7.1 Figure 7-3 shows the ACF for 36 random numbers, and Figure
3. There may have been more than one plausible model identi¯ed, and we need a method to determine which of them is preferred.
2. The ACF and PACF provide some guidance on how to select pure AR or pure MA models. But mixture models are much harder to identify. Therefore it is normal to begin with either a pure AR or a pure MA model. Now it may be worth considering extending the selected model to a mixed ARMA model. ARMA
1. Some of the estimated parameters may have been insigni¯cant(their P-values may have been larger than 0.05). If so, a revised model with the insigni¯cant terms omitted may be considered.
3. Consider seasonal aspects. An examination of the ACF and PACF at the seasonal lags can help identify AR and MA models for the seasonal aspects of the data, but the indications are by no means as easy to ¯nd as in the case of the non-seasonal aspects. For quarterly data, the forecaster should
2. Consider non-seasonal aspects. An examination of the ACF and PACF of the stationary series obtained in Step 1 can reveal whether a MA or AR model is feasible.
1. Make the series stationary. An initial analysis of the raw data can quite readily show whether the time series is stationary in the mean and the variance. Di®erencing, (non-seasonal and/or seasonal) will usually take care of any non-stationarity in the mean. Logarithmic or power transformations
2. Determining the number of past values of Yt to include in equation (7.13) is not always straightforward.
1. In autoregression the basic assumption of independence of the error (residual) terms can easily be violated, since the explana-tory (right-hand side) variables in equation (7.13) usually have built-in dependence relationship.
2. If the plotted series shows no obvious change in the variance over time, then we say the series is stationary in the variance.
1. If a time series is plotted and there is no evidence of a change in the mean over time (e.g., Figure 7-6(a)), then we say the series is stationary in the mean.
4. illustrations of how the concepts, statistical tools, and notation can be combined to model and forecast a wide variety of time series.
3. de¯nition of some general notation (proposed by Box and Jenk-ins, 1970) for dealing with general ARIMA models;
2. description of the statistical tools that have proved useful in analyzing time series;
1. introduction of the various concepts useful in time series anal-ysis (and forecasting);
(b) Calculate the Durbin{Watson statistic and show that there is signi¯cant autocorrelation in the residuals.
(a) Fit the regression model Yt = a + bXt + et where Yt denotes sales, Xt denotes advertising, and et is the error.
6.7 A company which manufactures automotive parts wishes to model the e®ect of advertising on sales. The advertising expenditure each month and the sales volume each month for the last two years are given in Table 6-19.
(6.10) and (6.12) the twelfth-period was chosen as base.Rerun the regression model of equation (6.10) using some other period as the base. Then recompute the seasonal indices and compare them with those in part (a) above.
(c) The use of dummy variables requires that some time period is regarded as \base period" (the period for which all the dummy variables have zero value). In equations
(b) Repeat this procedure using equation (6.12) and compare the two sets of seasonal indices.
(a) From equation (6.10) compute a set of seasonal indices by examining the constant term in the regression equation when just one dummy variable at a time is set equal to 1, with all others set to 0. Finally, set all dummy variables to 0 and examine the constant term.
6.6 Equations (6.10) and (6.12) give two regression models for the mutual savings bank data.
(c) Plot the data (Y against X) and join up the points according to their timing|that is, join the point for t = 1 to the point for t = 2, and so on. Note that the relationship between Y and X changes over time.
(b) Regress Y on X and t (time) and check the signi¯cance of the results.
(a) Regress Y on X and check the signi¯cance of the results.
6.5 The data set in Table 6-18 shows the dollar volume on the New York plus American Stock Exchange (as the explanatory variable X) and the dollar volume on the Boston Regional Exchange (as the forecast variable Y ).
(f) What would be the heat emitted for cement consisting of X1 = 10, X2 = 40, and X3 = 30? Give a 90% prediction interval.
(e) Which of the three components cause an increase in heat and which cause a decrease in heat? Which component has the greatest e®ect on the heat emitted?
(d) What proportion of the variation in Y is explained by the regression relationship?
(c) Plot the residuals against each of the explanatory vari-ables. Does the model appear satisfactory?
(b) Carry out an F-test for the regression model. What does the P-value mean?
(a) Regress Y against the three components and ¯nd con¯-dence intervals for each of the three coe±cients.
6.4 Table 6-17 shows the percentages by weight of three compo-nents in the cement mixture, and the heat emitted in calories per gram of cement.
(c) Compare your new forecasts with the actual D(EOM)values in Table 6-15 and compute the MAPE and other statistics to show the quality of the forecasts. How well do your new forecasts compare with those in Table 6-14?
(b) It was necessary to forecast (AAA) and (3-4) rates for future periods before it was possible to get forecasts for D(EOM). Holt's linear exponential smoothing method was used in Section 6/5/2 to do this. However, this is not necessarily the best choice and no attempt was made to optimize the
(a) Compare the forecasts with the actuals (Table 6-15) and determine the MAPE and other summary statistics for these forecasts.
6.3 The regression analysis resulting in equation (6.12) was used to forecast the change in end-of-month balance for a bank for the next six time periods (periods 54 through 59). See Section 6/5/2 for details. However, since we originally omitted the known values for D(EOM) for these periods, it is
(c) What is the correlation between P and P2? Does this suggest any general problem to be considered in dealing with polynomial regressions|especially of higher orders?
(b) For the quadratic regression compute prediction intervals for forecasts of consumption for various prices, for exam-ple, P = 20, 40, 60, 80, 100, and 120 cents per 1,000 cubic feet [using (6.13)].
(a) Now try what is known as a quadratic regression with^ C = b0 + b1P + b2P2:Compare this with the previous results. Check the ¹R 2value, the t values for the coe±cients, and consider which of the three models makes most sense.
6.2 The Texas natural gas data in Table 5-10 show gas consump-tion (C) and price (P). In Exercise 5.10, two regression models were ¯tted to these data: a linear regression of log C on P and a piecewise linear regression of C on P.
(e) What should be done next?
(d) Which coe±cients are signi¯cantly di®erent from zero(i.e., have P-value less than 0.05)?
(c) Is the overall regression signi¯cant?
(b) What would be the value of ¹R 2?
(a) How many observations were involved?
6.1 Table 6-16 presents some of the computer output from a regression analysis.
3. estimating in a simultaneous manner the parameters of all the equations;
2. determining the functional form (i.e., linear, exponential, loga-rithmic, etc.) of each of the equations;
1. determining which variables to include in each equation (spec-i¯cation);
19. National Personal Income
18. First Savings and Loan Index
17. U.S. Savings and Loan Index
16. U.S. Consumer Price Index
15. N.H. Mutual Savings Bank Savings
14. Massachusetts Mutual Savings Bank Savings
13. First Mutual Savings Bank Savings
12. U.S. Mutual Savings Bank Savings
11. First Negotiable CDs
10. U.S. Negotiable CDs
9. Rates for AAA Bonds
8. Rates for 3-4 Year Government Issues
7. Rates for Three-Month Bills
6. N.H. Personal Income
5. Massachusetts Personal Income
4. Northeast Personal Income
3. U.S. Personal Income
2. First Gross Demand Deposits
1. U.S. Gross Demand Deposits
² When there is a distinct fast-moving (zigzag) pattern|as in case (c)|the successive error di®erences tend to be large and the DW statistic will be large.
When there is a distinct slow-moving pattern to the errors|as in case (b)|successive error di®erences tend to be small and the DW statistic will be small.
4. normality of residuals.
3. homoscedasticity
2. independence of residuals
1. model form
Scatterplots of each combination of variables. The variable on the vertical axis is the variable named in that row; the variable on the horizontal axis is the variable named in that column. This scatterplot matrix is a very useful way of visualizing the relationships between each pair of variables.
(a) A time plot of the monthly change of end-of-month balances at a mutual savings bank. (b) A time plot of AAA bond rates. (c) A time plot of 3-4 year government bond issues. (d) A time plot of the monthly change in 3-4 year government bond issues. All series are shown over the period February
(h) Compute the standard errors for each of the forecasts, and 95% prediction intervals. [If using the second model, use n¡4 degrees of freedom in obtaining the multiplying factor t¤.] Make a graph of these prediction intervals and discuss their interpretation.
(g) For prices 40, 60, 80, 100, and 120 cents per 1,000 cubic feet, compute the forecasted per capita demand using the best model of the two above

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Forecasting Methods And Applications 3rd Edition Spyros G. Makridakis, Steven C. Wheelwright, Rob J Hyndman - Solutions

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