Questions and Answers of Forecasting Predictive Analytics

8.1 Consider the problem in Exercise 6.7. We ¯tted a regression model Yt = a + bXt + Nt where Yt denotes sales, Xt denotes advertising and showed that Nt had signi¯cant autocorrelation.
5. Forecasts for the model are produced.
4. The parameters of the state space model are estimated.
3. The best of the revised models is then approximated by a state space model with fewer parameters.
2. It tries to improve the ¯t of the selected AR model by adding moving average terms and removing some of the autoregressive terms.
1. It ¯ts a sequence of multivariate autoregressive models for lags multivariate 0 to 10. For each model, the AIC is calculated and the model autoregression with the smallest AIC value is selected
4. the variation due to estimating the ARIMA part of the model.Monthly housing starts (thousands of units), construction contracts (millions of dollars), and average new home mortgage rates from
3. the variation due to estimating the regression part of the model;
2. the variation due to the error in forecasting the explanatory variables (where necessary);
1. the variation due to the error series Nt;
5. Check that the et residual series looks like white noise.
4. Re¯t the entire model using the new ARMA model for the errors.
3. If the errors now appear stationary, identify an appro-priate ARMA model for the error series, Nt.
2. If the errors from the regression appear to be non-stationary, and di®erencing appears appropriate, then di®erence the forecast variable and all explanatory vari-ables. Then ¯t the model using
1. Fit the regression model with a proxy AR(1) or AR(2)model for errors.
2. The standard errors of the coe±cients are incorrect when there are autocorrelations in the errors. They are most likely too small. This also invalidates the t-tests and F-test and predic-tion
1. The resulting estimates are no longer the best way to com-pute the coe±cients as they do not take account of the time-relationships in the data.
(d) What would your next step be if you were trying to develop an ARIMA model for this time series? Can you identify a model on the basis of Figures 7-34 and 7-35?Would you want to do some more
(c) What does the one large partial autocorrelation in Figure 7-35 suggest?
(b) What can you say about trend in the time series?
(a) What can you say about seasonality of the data?
(e) In general, when there is a reasonably long time series such as this one, and there is a clear long-term cycle(shown by plotting a 12-month moving average, for in-stance) what should the
(d) For the last 96 months (1963 through 1970) use the model obtained in (b) above to forecast the next 12 months ahead. Compare your forecast with the actual ¯gures given below.Year J F M A M J J A
(c) For the ¯rst 96 months (1955 through 1962) use the ARIMA model obtained in (b) above to forecast the next 12 months ahead. How do these forecasts relate to the actuals?
(b) Split the data set into two parts, the ¯rst eight years(96 months) and the second eight years (96 months) and do Box-Jenkins identi¯cation, estimation, and diagnostic testing for each part
(a) How consistent is the seasonal pattern? Examine this question using several di®erent techniques, including de-composition methods (Chapter 4) and autocorrelations for lags up to 36 or 48.
7.9 Table 7-13 shows monthly employment ¯gures for the motion picture industry (SIC Code 78) for 192 months from Jan. 1955 through Dec. 1970. This period covers the declining months due to the
(f) Forecast the next 24 months of generation of electric-ity by the U.S. electric industry. See if you can get the latest ¯gures from your library (or on the web at www.eia.doe.gov) to check on the
(e) Estimate the parameters of your best model and do diagnostic testing on the residuals. Do the residuals resemble white noise? If not, try to ¯nd another ARIMA model which ¯ts better.
(d) Identify a couple of ARIMA models that might be useful in describing the time series. Which of your models is the best according to their AIC values?
(c) Are the data stationary? If not, ¯nd an appropriate di®erencing which yields stationary data.
(b) Do the data need transforming? If so, ¯nd a suitable transformation.
(a) Examine the 12-month moving average of this series to see what kind of trend is involved.
7.8 Table 7-12 shows the total net generation of electricity (in billion kilowatt hours) by the U.S. electric industry (monthly for the period 1985{1996). In general there are two peaks per year: in
(c) The last ¯ve values of the series are given below:t (year) 1964 1965 1966 1967 1968 Yt (million tons net) 467 512 534 552 545
(b) Explain why this model was chosen.
7.7 Figure 7-33 shows the annual bituminous coal production in the United States from 1920 to 1968. You decide to ¯t the following model to the series:Yt = c + Á1Yt¡1 + Á2Yt¡2 + Á3Yt¡3 +
(c) The last ¯ve values of the series are given below:Year 1935 1936 1937 1938 1939 Millions of sheep 1648 1665 1627 1791 1797 Given the estimated parameters are Á1 = 0:42, Á2 =¡0:20, and Á3 =
(b) By examining Figure 7-32, explain why this model is appropriate.
(a) What sort of ARIMA model is this (i.e., what are p, d, and q)?
7.6 The sheep population of England andWales from 1867{1939 is graphed in Figure 7-31. Assume you decide to ¯t the following model:Yt = Yt¡1+Á1(Yt¡1¡Yt¡2)+Á2(Yt¡2¡Yt¡3)+Á3(Yt¡3¡Yt¡4)+et
(e) Write the model in terms of the backshift operator, and then without using the backshift operator.
(d) Figure 7-30 shows an analysis of the di®erenced data(1 ¡ B)(1 ¡ B12)Yt|that is, a ¯rst-order non-seasonal di®erencing (d = 1) and a ¯rst-order seasonal di®erencing(D = 1). What model do
(c) What can you learn from the PACF graph?
(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
(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
(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
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
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
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
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
(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
(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
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
(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
(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
(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

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