Time Series Analysis Forecasting And Control 5th Edition George E. P. Box, Gwilym M. Jenkins, Gregory C. Reinsel, Greta M. Ljung - Solutions

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What will the forecast cost?
What data are available, and will the data be sufficient to generate the needed forecast?
What level of detail or aggregation is required, and what is the proper time horizon?
Who will use the forecast, and what are their specific requirements?
Why is a forecast needed?
Will there be a recession? If so, when will it begin, how severe will it be, and when will it end?
What is a year-by-year prediction for the total loan balance of our bank over the next 10 years?
What factors can we identify that will help explain the variability in monthly unit sales?
How many units might we sell in an effort to recover our fixed investment in production equipment?
What revenue might the state government expect over the next 2-year period?
If we increase our advertising budget by 10%, how will sales be affected?
Consider (Box and MacGregor, 1976) a system for which the process transfer function is gB and the noise model is (1 − ????)???????? = (1 − ????????)???????? so that the error ???????? at the output satisfies(1 − ????)???????? = ????(1 − ????)????????−1+ + (1 − ????????)????????Suppose
A pilot feedback control scheme, based on the following disturbance and transfer function models:∇???????? = ????????(1 − ????????)Y???? = ????0????????−1+ − ????1????????−2+was operated, leading to a series of adjustments ???????? and errors ????????. It was believed that the noise model
In a treatment plant for industrial waste, the strength ???????? of the influent is measured every 30 minutes and can be represented by the model ∇???????? = (1 − 0.5????)????????. In the absence of control, the strength of the effluent ???????? is related to that of the influent ????????by an
Given the following combinations of disturbance and transfer function models:(1) ∇???????? = (1 − 0.7????)????????(1 − 0.4????)Y???? = 5.0????????−1+(2) ∇???????? = (1 − 0.5????)????????(1 − 1.2???? + 0.4????2)Y???? = (20 − 8.5)????????−1+(3) ∇2???????? = (1 − 0.9???? +
In a chemical process, 30 successive values of viscosity ???????? that occurred during a period when the control variable (gas rate) ???????? was held fixed at its standard reference origin were recorded as follows:Time Viscosities 1--10 92 92 96 96 96 98 98 100 100 94 11--20 98 88 88 88 96 96 92
Express the model for the nonstationary bivariate process ???????? given in Exercise 14.5 in an error correction form, similar to equation (14.8.2), as ???? ???? = ????????????−1 +???????? − ????????????−1, where ???? ???? = (1 − ????)????????. Determine the structure (and the ranks) of the
Verify that any VAR(????) model ???????? = ∑????????=1 ????????????????−???? + ???????? can be expressed equivalently in the error correction form of equation (14.8.2) as???? ???? = ????????????−1 + ∑????−1????=1 ????∗???????? ????−???? + ????????, where ???? ???? = ???????? −
Suppose that a three-dimensional VARMA process ???????? has Kronecker indices ????1 = 3, ????2 = 1, and ????3 = 2.(a) Write the form of the coefficient matrices ????#???? and ????#???? in the echelon canonical ARMA model structure of equation (14.7.1) for this process.(b) For this process
Suppose ????1, … , ????????, with ???? = 60, is a sample from a bivariate VAR(1) process, with sample covariance matrices obtained as????̂(0) = [1.0 1.0 1.0 2.0]????̂(1) = [0.6 0.4 0.7 1.2]????̂(2) = [0.30 0.10 0.42 0.64](a) Obtain the corresponding estimated correlation matrices ????̂(0),
Consider the simple transfer function model(1 − ????)????1???? = ????1???? − ????????1,????−1 ????2???? = ????????1???? + ????2????where ????1???? and ????2???? are independent white noise processes.(a) Determine the univariate ARIMA model for ????2????, and note that ????2???? is
For a bivariate VAR(2) model ???????? = ????1????????−1 + ????2????????−2 + ????????, with????1 =[1.5 −0.6 0.3 0.2]????2 =[−0.5 0.3 0.7 −0.2]???? =[4 1 1 2](a) Verify that this model is stationary based on the nature of the roots of det{???? −????1???? − ????2????2}=0. (Note that you
Consider the bivariate VMA(1) process ???????? = (???? − ????????)????????, with???? =[ 0.4 0.3−0.5 0.8]???? =[4 1 1 2](a) Find the lag 0 and lag 1 autocorrelations and cross-correlations of ????????; that is, find the matrices ????(0), ????(1), and ????(0), ????(1).(b) Find the individual
Daily air quality measurements in New York, May--September 1973, are available in the data file ‘‘airquality’’ in the R datasets package. The file provides data on four air quality variables, including the solar radiation measured from 8 a.m. to 12 noon at Central Park. The solar radiation
A time series defined as ???????? = 1000 log10(????????), where ???????? is the price of hogs recorded annually by the U.S. Census of Agriculture over the period 1867--1948, was considered in Exercise 6.6.(a) Estimate the parameters of the model identified for this series. Perform diagnostic check
Figure 13.2 shows the residuals before and after an outlier adjustment for the temperature data in Series C. Construct a similar graph for the viscosity data in Series D.
Refer to Exercise 12.9.A bivariate transfer function--noise model was given for these series in Section 12.5.3.(a) Use the results from Exercise 12.9 to justify the choice of transfer function model. Derive preliminary estimates of the parameters in this model.(b) Justify the choice of the noise
A bivariate time series consisting of sales data and a leading indicator is listed as Series M in Part Five of this book. The series is also available as ‘‘BJsales’’ in the datasets package of R.(a) Plot the two time series using R.(b) Calculate and plot the autocorrelation and partial
Consider the transfer function--noise model fitted to the gas furnace data in (12.4.1)and (12.4.2). Note that the estimate of ????2 is very close to zero. Re-estimate the parameters of this model setting ????2 equal to zero. Describe the resulting impact on the estimate of the residual variance and
Refer to Exercise 12.6.Build (identify, estimate, and check) a transfer function--noise model that uses the GDP series ???????? as input to help explain variations in the logged unemployment series ????????.
Quarterly measurements of unemployment and the gross domestic product (GDP)in the United Kingdom over the period 1955--1969 are included in Series P in Part Five of this book; see also http://pages.stat.wisc.edu/ reinsel/bjr-data/.(a) Plot the two time series using R.(b) Calculate and plot the
Consider the regression model???????? = ????1????1,???? + ????2????2,???? + ????????where ???????? is a nonstationary error term following an IMA(0, 1, 1) process ∇???????? =???????? − ????????????−1. Show that the regression model may be rewritten in the form???????? − ????̄????−1 =
After estimating a prewhitening transformation ????−1???? (????)????????(????)???????? = ???????? for an input series ???????? and then computing the transformed output ???????? = ????−1???? (????)????????(????)????????, crosscorrelations ????????????(????) were obtained as follows:????
If two series may be represented in ????-weight form as???????? = ????????(????)???????? ???????? = ????????(????)????�(a) Show that their cross-covariance generating function ????????????(????) = ∑∞????=−∞????????????(????)????????is given by ????2 ????????????(????)????????(????).(b)
Estimate of the cross-correlation function at lags −1, 0, and +1 for the following series of five pairs of observations:???? 12 3 4 5???????? 11 7 8 12 14???????? 7 10 6 7 10
(a) Calculate and plot the response of the two-input system???????? = 10 + 6 1−0.7????????1,????−1 +8 1−0.5????????2,????−2 to the orthogonal and randomized input sequences shown below.???? ????1???? ????2???? ???? ????1???? ????2????00 0 5 1 −1 1 −1 1 61 1 2 1 −1 7 −1 −1 3 −1
Express the models of Exercise 11.1 in ∇ form.
Use equation (11.2.8) to obtain the impulse weights ???????? for each of the models of Exercise 11.1, and check that they are the same as the impulse response obtained in Exercise 11.2(a).
For each of the models of Exercise 11.1, calculate from the difference equation and plot the responses to:(a) A unit impulse (0, 1, 0, 0, 0, 0, …) applied at time ???? = 0(b) A unit step (0, 1, 1, 1, 1, 1, …) applied at time ???? = 0(c) A ramp input (0, 1, 2, 3, 4, 5, …) applied at time ????
In the following transfer function models, ???????? is the methane gas feed rate to a gas furnace, measured in cubic feet per minute, and ???????? the percent carbon dioxide in the outlet gas:(1) ???????? = 10 + 25 1−0.7????????????−1(2) ???????? = 10 + 22 − 12.5????1−0.85????????????−2
Measurements of the annual flow of the river Nile at Aswan from 1871 to 1970 are provided as series ‘‘Nile’’ in the R datasets package; type help(Nile) for details.(a) Plot the data along with the ACF and PACF of the series. Fit an appropriate ARIMA model to this series and comment.(b)
Consider the annual sunspot series referred to as Series E in this text. The series is also available for a slightly longer time period as series ‘‘sunspot.year’’ in the R datasets package.(a) Plot the time series and fit an AR(3) model to the series.(b) Use the procedure described by
Consider the first-order bilinear model ???????? = ????????????−1 + ???????? − ????????????−1????????−1, where the ????????are independent variates with mean 0 and variance ????2????. Assume the process {????????} is stationary, which involves the condition that ????2 + ????2????????2 < 1,
Assume that {????????} is a stationary, zero mean, Gaussian process with autocovariance function ????????(????) and autocorrelation function ????????(????). Use the property that for zero mean Gaussian variates, ????[????????????????+????????????+????????????+????] =
Suppose that a time series of stock returns {????????} can be represented using an ARCH(1)-M process ???????? = ????????2???? + ????????, ???????? = ????????????????, and ????2???? = ????0 + ????1????2????−1, where the ????????are iid Normal(0, 1).(a) Derive the conditional and unconditional mean
Derive the five-step-ahead forecast of the conditional variance ????2???? from a time originℎ for the GARCH(1, 1) process. Repeat the derivation for a GARCH(2, 1) process.
Consider the GARCH(1, 1) model ???????? = ????????????????, where the ???????? are iid random variables with mean 0 and variance 1, and ????2???? = ????0 + ????1????2????−1 + ????1????2????−1. Show that the unconditional variance of ???????? equals var[????????] = ????0∕[1 − (????1 +
Daily closing prices of four major European stock indices are available for the period 1991--1998 in the file ‘‘EuStockMarkets’’ in the R datasets package; see help(EuStockMarkets) for details.(a) Select two series and plot the data using R. Are there any unusual features worth noting?
Download from the Internet the daily stock prices of a company of your choosing.(a) Plot the data using the graphics capabilities in R. Are there any unusual features worth noting? Perform a statistical test to determine the presence of a unit root in the series.(b) Compute and plot the series of
Monthly averages of hourly ozone readingsin downtown Los Angeles for the period from January 1955 to December 1972 are included as Series R in Part 5 of this book;see also http://pages.stat.wisc.edu/reinsel/bjr-data/.(a) Plot the time series and comment.(b) Develop a suitable time model for this
Suppose the quarterly seasonal process {????????} is represented as ???????? = ???????? + ????2????, where???????? follows a ‘‘seasonal random walk’’ model (1 − ????4)???????? = ????0 + ????1????, and ????1???? and????2???? are independent white noise processes with variances ????2????1
Consider the time series model ???????? = ????0 + ???????? where ???????? follows the AR(1) model???????? = ????????????−1 + ????????. Assume that a series of length ???? is available for analysis.(a) Assuming that the parameter ???? is known, derive the generalized least-squares estimator of the
A time series representing the total monthly electricity generated in the United States(in millions of kilowatt-hours) for the period January 1970 to December 2005 is available as series ‘electricity’ in the R TSA package.(a) Plot the series and comment. Is a variance stabilizing transformation
Monthly Mauna Loa atmospheric CO2 concentration readings for the period 1959--1997 are available as series ‘co2’ in the R datasets package.(a) Plot the time series and comment on the pattern in the data.(b) Examine the autocorrelation structure and develop a suitable time series model for this
Quarterly earnings per share of the U.S. company Johnson & Johnson are available for the period 1960--1980 as series ’JohnsonJohnson’ in the R datasets package.(a) Plot the time series using the graphics capabilities in R.(b) Determine a variance stabilizing transformation for the series.(c)
Consider the airline series analyzed earlier in this chapter. We have seen that the logarithm of the series is well represented by the multiplicative model ???????? = (1 −????????)(1 − Θ12????12)????????(a) Compute and plot the 36-step-ahead forecasts and associated ±2 forecast error limits
Thompson and Tiao (1971) have shown that the outward station movements of telephones (logged data) in Wisconsin are well represented by the model(1 − 0.5????3)(1 − ????12)???????? = (1 − 0.2????9 − 0.3????12 − 0.2????13)????????Obtain and plot the autocorrelation function of ???????? = (1
The monthly oxidant averages in parts per hundred million in Azusa from January 1969 to December 1972 were as follows:Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.1969 2.1 2.6 4.1 3.9 6.7 5.1 7.8 9.3 7.5 4.1 2.9 2.6 1970 2.0 3.2 3.7 4.5 6.1 6.5 8.7 9.1 8.1 4.9 3.6 2.0 1971 2.4 3.3 3.3
It has been shown (Tiao et al., 1975) that monthly averages for the (smog-producing)oxidant level in Azusa, California, may be represented by the model(1 − ????12)???????? = (1 + 0.2????)(1 − 0.9????12)???????? ????2???? = 1.0(a) Compute and plot the ???????? weights of this model.(b) Compute
Refer to the daily air quality measurements for New York, May to September 1973, analyzed in Problem 7.10 of Chapter 7. Perform diagnostic checks to determine the adequacy of the models fitted to average daily temperature and wind speed series.8.10 Repeat the analysis in Problem 8.9 by performing
Global mean surface temperature deviations (from the 1951--1980 average level) are available for the period 1880--2009 as series ’gtemp2’ in the astsa package in R.(a) Plot the data and comment. Are there any unusual features worth noting?(b) Perform a statistical analysis to determine a
Monthly sales, {????????}, of a company over a period of 150 months are provided as part of Series M in Part 5 of this book. This series is also available as series ???????? ????????????????????along with a related series ???????? ????????????????????.???????????????? in the datasets package in
Two time series models, AR(2) and AR(3), were fitted to the yearly time series of sunspot numbers for the period 1770--1869 in Chapter 7. The sunspot data are available for the slightly longer time period 1700--1988 as series ‘sunspot.year’ in the datasets package in R; type help(sunspot.year)
Suppose that a (0, 1, 1) model ∇???????? = (1 − ????????)????????, corresponding to the use of an exponentially weighted moving average forecast, with ???? arbitrarily chosen to be equal to 0.5, was used to forecast a series that was, in fact, well represented by the(0, 1, 2) model ∇????????
(a) Show that the variance of the sample mean ???? of ???? observations from a stationary AR(1) process (1 − ????????)????̃???? = ???????? is given by var[????] ≃ ????2????????(1 − ????)2(b) The yields from consecutive batches of a chemical process obtained under fairly uniform conditions of
A long series containing ???? = 326 observations was split into two halves and a(1, 1, 0) model (1 − ????????)∇???????? = ???????? identified, fitted, and checked for each half. If the estimates of the parameter ???? for the two halves are ????̂(1) = 0.5 and ????̂(2) = 0.7, is there any
The residuals from a model ∇???????? = (1 − 0.6????)???????? fitted to a series of ???? = 82 observations yielded the following residual autocorrelations:???? ????????(????̂) ???? ????????(????̂)1 0.39 6 −0.13 2 0.20 7 −0.05 3 0.09 8 0.06 4 0.04 9 0.11 5 0.09 10 0.02(a) Plot the residual
The following are the first 30 residuals obtained when a tentative model was fitted to a time series:???? Residuals 1--6 0.78 0.91 0.45 −0.78 −1.90 −2.10 7--12 −0.54 −1.05 0.68 −3.77 −1.40 −1.77 13--18 1.18 0.02 1.29 −1.30 −6.20 −1.89 19--24 0.95 1.49 1.08 0.80 2.02 1.25
Refer to the annual river flow measurements in the time series ‘Nile’ analyzed in Exercise 6.7.Estimate the parameters of the model or models identified for this time series and comment.
Consider the solar radiation series that is part of the New York airquality data file described in Problem 7.10.This series has a few missing values.(a) Impute suitable estimates of the missing values. (Note: A formal procedure for estimating missing values is described in Chapter 13, but is not
Daily air quality measurements in New York, from May to September 1973, are available in a file called ‘airquality’ in the R datasets package. The file provides data on four air quality variables: mean ozone levels at Roosevelt Island, solar radiation at Central Park, maximum daily temperature
Repeat the analysis in Exercise 7.8 by fitting (a) an AR(1) and (b) an ARMA(0, 1, 1)model to the chemical process viscosity readings in Series D.
The preliminary model identification performed in Chapter 6 suggested that either an ARIMA(1, 1, 0) or an ARIMA(0, 2, 2) model might be appropriate for the chemical process temperature readings in Series C. The series is available for download from http://pages.stat.wisc.edu/ reinsel/bjr-data/.(a)
For the ARIMA(1, 1, 1) model (1 − ????????)???????? = (1 − ????????)????????, where ???????? = ∇????????:(a) Write down the linearized form of the model.(b) Set out how you would start off the calculation of the conditional nonlinear least-squares algorithm with start values ???? = 0.5 and
(a) Problems were experienced in obtaining a satisfactory fit to a series, the last 16 values of which were recorded as follows:129, 135, 130, 130, 127, 126, 131, 152, 123, 124, 131, 132, 129, 127, 126, 124 Plot the series and suggest where the difficulty might lie.(b) In fitting a model of the
For the process ???????? = ???????? + (1 − ????????)???????? show that for long series the variance--covariance matrix of the maximum likelihood estimates ????̂????, ????̂ is approximately????−1 [(1 − ????)2????2???? 0 0 1− ????2]
Using the value of ????̂0 calculated in Exercise 7.2:(a) Show that the unconditional sum of squares ????(−0.5) is 143.4.(b) Show that for the (0, 1, 1) model, for large ????,????(????) = ????∗(????|0) − ????̂2 01 − ????2
Using the data in Exercise 7.1:(a) Show (using least-squares) that the value ????̂0 of ????0 that minimizes ????∗(−0.5|0) is????̂0 = (2)(0.50) + (4)(−0.25) + ⋯ + (−2)(0.0078)12 + 0.52 + ⋯ + 0.00782 = − ∑????????=0 ????????????0???? ∑????????=0 ????2????where ????0???? =
The following table shows calculations for an (unrealistically short) series ???????? for which the (0, 1, 1) model ???????? = ∇???????? = (1 − ????????)???????? is being considered with ???? = −0.5 and with an unknown starting value ????0???? ???????? ???????? = ∇???????? ???????? =
Download a time series of your choice from the Internet. Plot the time series and identify a suitable model for the series.
The file ‘‘EuStockMarkets’’ in the R datasets package contains the daily closing prices of four major European stock indices: Germany DAX (Ibis), Switzerland SMI, France CAC, and UK FTSE. The data are sampled in business time, so weekends and holidays are omitted.(a) Plot each of the four
Measurements of the annual flow of the river Nile at Ashwan from 1871 to 1970 are available as series ‘‘Nile’’ in the datasets package in R; type help(Nile) for details.(a) Plot the series and compute the ACF and PACF for the series.(b) Repeat the analysis in part (a) for the differenced
A time series defined by ???????? = 1000 log10(????????), where ???????? is the price of hogs recorded annually by the U.S. Census of Agriculture on January 1 for each of the 82 years, from 1867 to 1948 is listed as Series Q in the Collection of Time Series in Part Five. This is a well-known time
Quarterly UK unemployment rate (in thousands) is part of Series P analyzed in Exercise 6.4.Repeat parts (a) to (e) of Exercise 6.4 for this series.
Quarterly measurements of the gross domestic product (GDP) in the United Kingdom over the period 1955--1969 are included in Series P in Part Five of this book.(a) Calculate and plot the ACF and PACF of this series.(b) Repeat the analysis in part (a) for the first differences of the series.(c)
Consider the chemical process temperature readings referred to as Series C in this book.(a) Plot the original series and the series of first differences using R.(b) Use the R package to calculate and plot the ACF and PACF of this series. Repeat the calculation for the first and second differences
For the (2, 1, 0) process considered on line (5) of Exercise 6.1, the sample mean and variance of ???????? = ∇???????? are ???? = 0.23 and ????2???? = 0.25.If the series contains ???? = 101 observations,(a) show that a constant term needs to be included in the model,(b) express the model in the
Given the five identified models and the corresponding values of the estimated autocorrelations of ???????? = ∇???? ???????? in the following table:Identified Model???????? ???? Estimated Autocorrelations(1) 1 1 0 ????1 = 0.72(2) 0 1 1 ????1 = −0.41(3) 1 0 1 ????1 = 0.40, ????2 = 0.32(4) 0 2 2
For the model (1 − 0.6????)(1 − ????)???????? = (1 + 0.3????)????????, express explicitly in the statespace form of (5.5.2) and (5.5.3), and write out precisely the recursive relations of the Kalman filter for this model. Indicate how the (exact) forecasts ????̂????+????|???? and their
A time series representing a global mean land--ocean temperature index from 1880 to 2009 is available in a file called ‘‘gtemp’’ in the astsa package of R. The data are temperature deviations, measured in degree centigrades, from the 1951--1980 average temperature, as described by Shumway
Consider the annual Wolfer sunspot numbers for the period 1770--1869 listed as ¨Series E in Part Five of this text. The same series is available for the longer period 1700--1988 as "sunspot.year" in the datasets package of R. You can use either data set. Suppose that the series can be represented
Suppose that a quarterly economic time series is well represented by the model∇???????? = 0.5 + (1 − 1.0???? + 0.5????2)????????with ????2???? = 0.04.(a) Given ????48 = 130, ????47 = −0.3, ????48 = 0.2, calculate and plot the forecasts ????̂48(????)for ???? = 1, 2, … , 12.(b) Calculate and
For the model ∇???????? = ???????? − 1.1????????−1 + 0.28????????−2 of Exercise 5.2:(a) Write down expressions for the forecast errors ????????(1), ????????(2), … , ????????(6), from the same origin ????.(b) Calculate and plot the autocorrelations of the series of forecast errors
Using the data and forecasts of Exercise 5.2, and given the further observation????101 = 174:(a) Calculate the forecasts ????̂101(????) for ???? = 1, 2, … , 11 using the updating formula????̂????+1(????) = ????̂????(???? + 1) + ????????????????+1(b) Verify these forecasts using the difference
Suppose that the data of Exercise 5.2 represent monthly sales.(a) Calculate the minimum mean square error forecasts for quarterly sales for 1, 2, 3, 4 quarters ahead, using the data up to ???? = 100.(b) Calculate 80% probability limits for these forecasts.
The following observations represent values ????91, ????92, … , ????100 from a series fitted by the model ∇???????? = ???????? − 1.1????????−1 + 0.28????????−2:166, 172, 172, 169, 164, 168, 171, 167, 168, 172(a) Generate the forecasts ????̂100(????) for ???? = 1, 2, … , 12 and draw a
For the models(1) ????̃???? − 0.5????̃????−1 = ????????(2) ∇???????? = ???????? − 0.5????????−1(3) (1 − 0.6????)∇???????? = ????????write down the forecasts for lead times ???? = 1 and ???? = 2:(a) From the difference equation(b) In integrated form (using the ???????? weights)(c) As
Repeat the analysis in Exercise 11 for the Dow Jones Industrial Average, or for a time series of your own choosing.
Download the daily S& P 500 Index stock price values for the period January 2, 2014 to present from the Internet (e.g., http://research.stlouisfed.org).(a) Plot the series using R. Calculate and graph the autocorrelation and partial autocorrelation functions for this series. Does the series appear
(a) Simulate a time series of ???? = 200 observations from an IMA(0, 2, 2) model with parameters ????1 = 0.8 and ????2 = −0.4 using the arima.sim() function in R; type help(arima.sim) for details. Plot the resulting series and comment on its behavior.(b) Estimate and plot the autocorrelation

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Time Series Analysis Forecasting And Control 5th Edition George E. P. Box, Gwilym M. Jenkins, Gregory C. Reinsel, Greta M. Ljung - Solutions

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