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

(b) the characteristics of the time series;
(a) whether users want simply to forecast, or they also want to understand and in°uence the course of future events;
10.4 If you were to count, only once, from 0 to 100, how many times will you encounter the number eight?
Can we say how long it will be before it is interrupted by a recession?
10.3 The U.S. economy has been growing without interruption since May 1991. The longest previous expansion in the U.S.economy lasted for 105 months while the average post World War II expansion has been lasting a little more than 50 months• . What can one say about the current expansion?
who are paid to choose the best stocks could not beat the broad market averages."\This year's failure|the ninth in the past thirteen years, according to Lipper Analytical Services Inc.|brings to mind a famous quotation from A Random Walk Down Wall Street, by Burton Malkiel: `A blindfolded monkey
10.2 Glassman (1997) describes the reasons why various investment funds do not outperform the average of the market. Comment on the following quotes from Glassman's article:\For the third year in a row, fewer than one-fourth of U.S. stock mutual funds beat the Standard & Poor's 500 index. Once
10.1 In 1996 the giant ¯rm of Philips Electronics lost $350 million on revenues of $41 billion. For the last ¯ve years Philips has been having great trouble modernizing itself and turning to pro¯tability. During this time the various chief executive o±cers that passed through Philips were
4. The bottom line is that very few newsletters can \beat"the S&P 500. In addition, few can beat the market fore-casts derived from a statistical representation of publicly available information. There is no evidence that the letters can time the market (forecast direction). Consistent with mutual
3. Most of our tests focus on the ability of newsletters to call the direction of the market, that is, market timing.
2. Consistent with mutual funds studies, we ¯nd that poor performance is far more persistent than good performance.
1. Only 22.8% of newsletters have average returns higher than a passive portfolio of equity and cash with the same volatility. Indeed, some recommendations are remarkably poor. For example, the (once) high pro¯le Granville Mar-ket Letter-Traders has produced an average annual loss of 0.4% over the
9.4 In Johnson (1991), the following quote is said to have been made by Samuel Taylor in 1801. Comment on this quote.Samuel Taylor, of Fox & Taylor, °atly refused to believe that machinery could replace skilled craftsmen. He in-sisted that the existing method was perfect: \I have no hope of
9.3 Consider the price trends in computers and o±ce equipment.A computer whose speed was a fraction of a Mhz and which had only 8,000 words of RAM memory cost $10,000 in 1968($40,000 in 1997 prices). Similarly, a thermo-paper fax ma-chine, or black-and-white photocopier cost more than $1,000 a
9.2 Today one can buy a color photocopier which can also be used as a color computer printer, as a plain paper color fax machine and as a scanner. The price of this all-inclusive machine is under $1,000. Such a machine together with the powerful$3,000 computer mentioned above are all that one needs
(c) Today one can buy a PC running at 266Mhz and having 32 megabytes of RAM memory and 3 gigabytes of disk memory, plus many other powerful characteristics, for under $1,500. (In 1968 an electrical calculator which would only perform the four basic arithmetic functions used to cost $1,000{$4,000 in
(b) In 1997, an IBM computer program (Deep Blue) beat the world chess champion (Kasparov).
(a) There have been several computer programs developed recently that recognize continuous speech and turn it into written text in a computer word processing program.
9.1 Build a scenario about the future implications of the following new inventions:
(a) Cycles in copper prices and in cumulative random numbers; (b)Cycles in cumulative random numbers \0-200"; (c) Cycles in the S&P 500 and in cumulative random numbers; (d) Cycles in cumulative random numbers \0-400."
(d) Give two reasons why you might want to use the state space form of these models rather than the usual form.
(c) Show that Holt's method (Section 4/3/3) can be written in the following \error feedback form":Lt = Lt¡1 + bt¡1 + ®et bt = bt¡1 + ¯®et where et = Yt¡Lt¡1¡bt¡1. Use this result to ¯nd a state space form of Holt's method with state vector containing Lt and bt.
8.10 (a) Write an AR(3) model in state space form.(b) Write an MA(1) model in state space form. (Hint: Set F = 0.)
(c) In no more than half a page, discuss the di®erences between this model and that considered in Exercise 8.7.You should include mention of the assumptions in each model and explain which approach you think is most appropriate for these data.
(a) Let Yt denote the log of the average room rate and Xt denote the log of the CPI. Suppose these form a bivariate time series. Both series were di®erenced and a bivariate AR(12) model was ¯tted to the di®erenced data. The order of the model was chosen by minimizing the AIC statistic. Write
8.9 Consider the regression model ¯tted in Exercise 8.7 concern-ing the cost of tourist accommodation and the CPI.
(f) If the level of drug given varied from day-to-day, how could you modify your model to allow for this?
(e) Construct an ARIMA model ignoring the intervention and compare the forecasts with those obtained from your preferred intervention model. How much does the intervention a®ect the forecasts?
(d) Fit a new intervention model with a delayed response to the drug. Which model ¯ts the data better? Are the forecasts from the two models very di®erent?
(c) What does the model say about the e®ect of the drug?
(b) Fit an intervention model with a step function interven-tion to the series. Write out the model including the ARIMA model for the errors.
(a) Produce a time plot of the data showing where the intervention occurred.
8.8 The data in Table 8-8 are the daily scores achieved by a schizophrenic patient on a test of perceptual speed. The pa-tient began receiving a powerful tranquilizer (chlorpromazine)on the sixty-¯rst day and continued receiving the drug for the rest of the sample period. It is expected that this
(e) Forecast the average price per room for the next twelve months using your ¯tted model. (Hint: You will need to¯nd forecasts of the CPI ¯gures ¯rst.)
(d) Follow the modeling procedure in Section 8/2 to ¯t a dynamic regression model. Explain your reasoning in arriving at the ¯nal model.
(c) Produce time series plots of both variables and explain why logarithms of both variables need to be taken before¯tting any models.
(b) Estimate the monthly CPI using the data in Table 8-7.
(a) Use the data in Table 8-6 to calculate the average cost of a night's accommodation in Victoria each month.
8.7 Table 8-6 gives the total monthly takings from accommodation and the total room nights occupied at hotels, motels, and guest houses in Victoria, Australia, between January 1980 and June 1995. Table 8-7 gives quarterly CPI values for the same period and the same region.
(b) Now generate data with the same input data from Table 8-5 and the following transfer functions.4. r = 1, s = 0, b = 1 with !0 = 2:0 and ±1 = 0:7.5. r = 0, s = 2, b = 0 with !0 = 1:2, !1 = ¡2:0,!2 = 0:8.Again, assume Nt is white noise with mean 0 and stan-dard deviation 1.
8.6 An input (explanatory) time series Xt is shown in Table 8-5.(a) Using equation (8.5), generate three output time series Yt corresponding to the three sets of transfer function weights below.v1 v2 v3 v4 v5 v6 v7 Set 1 0.2 0.4 0.3 0.1 0.0 0.0 0.0 Set 2 0.0 0.2 0.4 0.3 0.1 0.0 0.0 Set 3 0.0 0.0
8.5 Sketch the graph of the impulse response weights for the following transfer functions:(a) Yt = 2(1 ¡ 0:5B)B2Xt(b) Yt =3B 1 ¡ 0:7B Xt(c) Yt =1 ¡ 0:5B 1:2 ¡ 0:8B Xt(d) Yt =1 1 ¡ 1:1B + 0:5B2Xt.
(d) If the methane input was increased, how long would it take before the carbon dioxide emission is a®ected?
(c) What are the values of the coe±cients !0, !1, !2, ±1, ±2,µ1, µ2, Á1, and Á2?
(b) What sort of ARIMA model is used for the errors?
(a) What are the values ofb, r, and s for the transfer function?
8.4 Box, Jenkins, and Reinsell (1994) ¯t a dynamic regression model to data from a gas combustion chamber. The two variables of interest are the volume of methane entering the chamber (Xt in cubic feet per minute) and the percentage concentration of carbon dioxide emitted (Yt). Each variable is
(d) Explain why the Nt term should be modeled with an ARIMA model rather than modeling the data using a standard regression package. In your discussion, com-ment on the properties of the estimates, the validity of the standard regression results, and the importance of the Nt model in producing
c) Describe how this model could be used to forecast elec-tricity demand for the next 12 months.
Explain what the estimates of b1 and b2 tell us about electricity consumption.
(b) The estimated coe±cients are Parameter Estimate s.e. Z P-value b1 0.0077 0.0015 4.98 0.000 b2 0.0208 0.0023 9.23 0.000µ1 0.5830 0.0720 8.10 0.000Á12 ¡0:5373 0.0856 -6.27 0.000Á24 ¡0:4667 0.0862 -5.41 0.000
(a) What sort of ARIMA model is identi¯ed for Nt? Explain how the statistician would have arrived at this model.
8.3 Electricity consumption is often modeled as a function of temperature. Temperature is measured by daily heating de-grees and cooling degrees. Heating degrees is 65±F minus the average daily temperature when the daily average is below 65±F; otherwise it is zero. This provides a measure of our
(b) Using the ACF and PACF of the errors, identify and estimate an appropriate ARMA model for the error.Write down the full regression model and explain how you arrived at this model.
(b) Plot the ACF and PACF of the errors to verify that an AR(1) model for the errors is appropriate.8.2 (a) Fit a linear regression model with an AR(1) proxy model for error to the Lake Huron data given in Table 8-4 using the year as the explanatory variable.
(a) Re¯t the regression model with an AR(1) model for the errors. How much di®erence does the error model make to the estimated parameters?
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 for use in subsequent steps.
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 January 1983 to October 1989. Source: Survey of Current Business, U.S. Department of Commerce, 1990.
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 the same proxy model for errors, this time using di®erenced variables.
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 intervals.
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 analyses, and if so, what would they be?
(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 forecaster do? Use all the data?Use only the last so-many years? If the object is to forecast the next 12
(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 S O N D 1971 194.5 187.9 187.7 198.3 202.7 204.2 211.7 213.4 212.0 203.4 199.5 199.3
(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 separately. Is there any di®erence between the two identi¯ed models?
(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 advent of TV and then a recovery.
(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 accuracy of your fore-casts.
(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 mid-summer and mid-winter.
(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 + Á4Yt¡4 + et where Yt is the coal production in year t and et is a white noise series.
(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 = ¡0:30, give forecasts for the next three years (1940{1942).
(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 where et is a white noise series.
(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 these graphs suggest?
(c) What can you learn from the PACF graph?

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

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