Time Series Analysis And Its Applications With R Examples 3rd Edition Robert H. Shumway, David S. Stoffer - Solutions

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Show that Cov(Y) = E[(Y −μ )(Y −μ )].
Show that the form (11.3.2) simplifies to the form (11.3.1) for simple linear regression.
Repeat Exercise 9.12.6 on the body fat data with special emphasis on diagnostics and model selection.
Reanalyze the pollution data of Exercise 9.12.5. Examine residuals and influential observations. Explore the use of various model selection methods.
Reanalyze the Chapman data of Exercise 9.12.4. Examine residuals and influential observations. Explore the use of the various model selection methods.
Reanalyze the Prater data of Atkinson (1985) and Hader and Grandage (1958)from Exercise 9.12.3. Examine residuals and influential observations. Explore the use of the various model selection methods.
Consider the information in Table 10.21 on stepwise regression for the wood data of Exercise 9.12.2.(a) What is being given in the rows labeled x1, x2, x3, and x4?What is being given in the rows labeled t?(b) Is this table for forward selection, backwards elimination, stepwise regression, or some
Consider the information in Table 10.20 on best subset regression for the wood data of Exercise 9.12.2.(a) In order, what are the three best models as measured by the Cp criterion?(b) What is the mean squared error for the model with variables x1, x3, and x4?(c) In order, what are the three best
Consider the information given in Table 10.19 on diagnostic statistics for the wood data of Exercise 9.12.2.(a) Are there any outliers in the predictor variables? Why are these considered outliers?(b) Are there any outliers in the dependent variable? If so, why are these considered outliers?(c)
Reconsider the advertising data of Exercise 9.12.1.(a) Are there any high-leverage points?Why or why not?(b) Test whether each case is an outlier using an overall significance level no greater than α = 0.05.Completely state the appropriate reference distribution.(c) Discuss the importance of
Go to http://lib.stat.cmu.edu/datasets/ and click on “bodyfat.”There are data for 15 variables along with a description of the data.(a) Using the body density measurements as a dependent variable, perform a multiple regression using all of the other variables except body fat as predictor
Table 9.4 contains a subset of the pollution data analyzed by McDonald and Schwing (1973). The data are from various years in the early 1960s. They relate air pollution to mortality rates for various standard metropolitan statistical areas in the United States. The dependent variable y is the total
Analyze the Chapman data of Example 9.8.1. Find a good model for predicting weight from the other variables.
Atkinson (1985) and Hader and Grandage (1958) have presented Prater’s data on gasoline. The variables are y, the percentage of gasoline obtained from crude oil; x1, the crude oil gravity oAPI; x2, crude oil vapor pressure measured in lbs/in2; x3, the temperature, in oF, at which 10% of the crude
The information below relates y, a second measurement on wood volume, to x1, a first measurement on wood volume, x2, the number of trees, x3, the average age of trees, and x4, the average volume per tree. Note that x4 = x1/x2. Some of the information has not been reported, so that you can figure it
Younger (1979, p. 533) presents data from a sample of 12 discount department stores that advertise on television, radio, and in the newspapers. The variables x1, x2, and x3 represent the respective amounts of money spent on these advertising activities during a certain month while y gives the
Cubic Splines.Table 8.6: Axial stiffness index data.Plate ASI Plate ASI Plate ASI Plate ASI Plate ASI 4 309.2 6 402.1 8 392.4 10 346.7 12 407.4 4 409.5 6 347.2 8 366.2 10 452.9 12 441.8 4 311.0 6 361.0 8 351.0 10 461.4 12 419.9 4 326.5 6 404.5 8 357.1 10 433.1 12 410.7 4 316.8 6 331.0 8 409.9 10
Consider fitting quadraticmodels yi =γ0+γ1xi+γ2x2i+εi and yi =β0+β1(xi−¯ x·)+β2(xi− ¯ x·)2 +εi. Show that γ2 =β2, γ1 =β1−2β2 ¯ x·, and γ0 =β0−β1 ¯ x·+β2 ¯ x2· .
Beineke and Suddarth (1979) and Devore (1991, p. 380) consider data on roof supports involving trusses that use light-gauge metal connector plates. Their dependent variable is an axial stiffness index (ASI) measured in kips per inch. The predictor variable is the length of the light-gauge metal
Jensen (1977) and Weisberg (1985, p. 101) considered data on the outside diameter of crank pins that were produced in an industrial process. The diameters of batches of crank pins were measured on various days; if the industrial process is “under control” the diameters should not depend on the
Based on the height and weight data given in Table 8.4, fit a simple linear regression of weight on height for these data and check the assumptions. Give a 99% confidence interval for the mean weight of people with a 72-inch height. Test for lack of fit of the simple linear regression model.Table
Use two methods other than fitting polynomial models to test for lack of fit in Exercise 8.7.1
In Exercise 7.4.2 we considered data on the relationship between farm sizes and the acreage in corn. Fit the linear, quadratic, cubic, and quartic polynomial models to the logs of the acreages in corn. Find the model that fits best. Check the assumptions for this model.
Dixon andMassey (1969) presented data on the relationship between IQ scores and results on an achievement test in a general science course. Table 8.3 contains a subset of the data.Fit the simple linear regression model of achievement on IQ and the quadratic model of achievement on IQ and IQ
Repeat Exercise 7.4.2 but instead of using the number of acres of corn as the dependent variable, use the proportion of acreage in corn as the dependent variable. Compare the results to those given earlier.
Snedecor and Cochran (1967, Section 6.18) presented data obtained in 1942 from South Dakota on the relationship between the size of farms (in acres) and the number of acres planted in corn. The data are given in Table 7.4.Plot the data. Fit a simple linear regression to the data. Examine the
Using the complete data of Exercise 6.11.2, test the need for a transformation of the simple linear regression model. Repeat the test after eliminating any outliers. Compare the results.
6.11.6. Use part (4) of Proposition 1.2.11 to show that Cov( ¯ y·, ˆβ1) = 0 whenever Var(εi) =σ 2 for all i and Cov(εi,ε j) = 0 for all i = j. Hint: write out ¯ y· and ˆβ1 in terms of the yis.
6.11.5. Working–Hotelling confidence bands are a method for getting confidence intervals for every point on a line with a guaranteed simultaneous coverage. The method is essentially the same as Scheff´e’s method for simultaneous confidence intervals discussed in Section 13.3.For estimating the
6.11.4. Exercise 6.11.3 involves the calibration of a measuring instrument. Often, calibration curves are used in reverse, i.e., we would use the calorimeter to measure a melting point y and use the regression equation to give a point estimate of x. If a new substance has a measured melting point
6.11.3. Mulrow et al. (1988) presented data on the calibration of a differential scanning calorimeter. The melting temperatures of mercury and naphthalene are known to be 234.16 Table 6.9: Angle between the plane of the equator and the plane of rotation about the Sun.Year Angle Year Angle Year
6.11.2. Stigler (1986, p. 6) reported data from Cassini (1740) on the angle between the plane of the equator and the plane of the Earth’s revolution about the Sun. The data are given in Table 6.9. The years −229 and −139 indicate 230 B.C. and 140 B.C. respectively. The angles are listed as
6.11.1. Draper and Smith (1966, p. 41) considered data on the relationship between the age of truck tractors (in years) and the cost (in dollars) of maintaining them over a six-month period. The data are given in Table 6.8. Plot cost versus age and fit a regression of cost on age. Give 95%
5.7.8. Stigler (1986, p. 208) reports data from the Edinburgh Medical and Surgical Journal (1817) on the relationship between heights and chest circumferences for Scottish militia men. Measurements were made in inches. We concern ourselves with two groups of men, those with 39-inch chests and those
5.7.7. Table 5.20 contains additional data from Lazerwitz (1961). These consist of a breakdown of the Protestants in Table 5.6 but with the addition of fourmore occupational categories.The additional categories are E, semiskilled; F, unskilled; G, farmers; H, no occupation. Analyze the data with an
5.7.6. Quetelet (1842) and Stigler (1986, p. 175) report data on conviction rates in the French Courts of Assize (Law Courts) from 1825 to 1830. The data are given in Table 5.19. Test whether the conviction rate is the same for each year. Use α = 0.05. (Hint: Table 5.19 is written in a nonstandard
5.7.5. Snedecor and Cochran (1967) report data from an unpublished report by E.W. Lindstrom. The data concern the results of cross-breeding two types of corn (maize). In 1301 crosses of two types of plants, 773 green, 231 golden, 238 green-golden, and 59 golden-greenstriped plants were obtained. If
5.7.4. Use the data in Table 5.2 to test whether the probability of a birth in each month is the number of days in the month divided by 365. Thus the null probability for January is 31/365 and the null probability for February is 28/365.
5.7.3. Pauling (1971) reports data on the incidence of colds among French skiers who where given either ascorbic acid or a placebo. Of 139 people given ascorbic acid, 17 developed colds. Of 140 people given the placebo, 31 developed colds. Do these data suggest that the proportion of people who get
5.7.2. In France in 1827, 6929 people were accused in the courts of assize and 4236 were convicted. In 1828, 7396 people were accused and 4551 were convicted. Give a 95% confidence interval for the proportion of people convicted in 1827. At the 0.01 level, test the null model Table 5.19: French
5.7.1. Reiss et al. (1975) and Fienberg (1980) reported that 29 of 52 virgin female undergraduate university students who used a birth control clinic thought that extramarital sex is not always wrong. Give a 99% confidence interval for the population proportion of virgin undergraduate university
4.5.17. Using the data of Example 4.2.1 give a 95% prediction interval for the difference in total points between a new female and a new male. This was not discussed earlier so it requires a deeper understanding of Section 3.5.Table 4.15: Dead adult flies.Medium A B C D E F G 0 units 423 326 246
4.5.16. Snedecor (1945b) presented data on a pesticide spray. The treatments were the number of units of active ingredient contained in the spray. Several different sources for breeding mediums were used and each spray was applied on each distinct breeding medium. The data consisted of numbers of
4.5.15. Snedecor and Haber (1946) presented data on cutting dates of asparagus.On two plots of land, asparagus was grown every year from 1929 to 1938. On the first plot the asparagus was cut on June 1, while on the second plot the asparagus was cut on June 15. Note that growing conditions will vary
4.5.14. Snedecor and Cochran (1967) presented data on the number of acres planted in corn for two sizes of farms. Size was measured in acres. Some of the data are given in Table 4.13.Are the data two independent samples or a paired comparison? Is the assumption of equal variances reasonable? Test
4.5.13. Garner (1956) presented data on the tensile strength of fabrics. Here we consider a subset of the data. The complete data and a more extensive discussion of the experimental procedure are given in Exercise 11.5.2. The experiment involved testing fabric strengths on different machines. Eight
4.5.12. Bethea et al. (1985) reported data on the peel-strengths of adhesives. Some of the data are presented in Table 4.11. Give an approximate P value for testing no difference between adhesives, a 95% confidence interval for the difference between mean peel-strengths, and a 95%prediction
4.5.11. Mandel (1972) reported stress test data on elongation for a certain type of rubber. Four pieces of rubber sent to one laboratory yielded a sample mean and variance of 56.50 and 5.66, respectively. Four different pieces of rubber sent to another laboratory yielded a sample mean and variance
4.5.10. Let y1 ∼ N(89,4) and y2 ∼ N(90,16). Show that Pr[y1 ≥ 87] > Pr[y2 ≥ 87], so that the population with the lower mean has a higher probability of exceeding 87. Recall that(y1 −89)/√4 ∼ N(0,1) with a similar result for y2 so that both probabilities can be rewritten in terms of a
4.5.9. Reanalyze the data of Example 4.1.1 after deleting the one possible outlier.Does the analysis change much? If so, how?
4.5.8. Table 4.10 presents data given by Shewhart (1939, p. 118) on various atomic weights as reported in 1931 and again in 1936. Analyze the data. Check your assumptions.Table 4.9: Burt’s IQ data.Case y1 y2 Case y1 y2 Case y1 y2 1 82 82 10 93 82 19 97 87 2 80 90 11 95 97 20 87 93 3 88 91 12 88
4.5.7. Burt (1966) and Weisberg (1985) presented data on IQ scores for identical twins that were raised apart, one by foster parents and one by the genetic parents. Variable y1 is the IQ score for a twin raised by foster parents, while y2 is the corresponding IQ score for the twin raised by the
4.5.6. Koopmans (1987) gave the data in Table 4.8 on verbal ability test scores for 8 year-olds and 10 year-olds. Test whether the two groups have the same mean with α = 0.01 and give a 95% confidence interval for the difference in means. Give a 95% prediction interval for a new 10 year old. Check
4.5.5. Jolicoeur and Mosimann (1960) gave data on turtle shell lengths. The data for females and males are given in Table 4.7. Explore the need for a transformation. Test whether there is a difference in lengths using α = 0.01. Give a 95% confidence interval for the difference in lengths.
4.5.4. Use the data of Exercise 4.5.3 to test whether the fitness scores for farm boys are more or less variable than fitness scores for town boys.Table 4.6: Physical fitness of male high school seniors.Town 12.7 16.9 7.6 2.4 6.2 9.9 Boys 14.2 7.9 11.3 6.4 6.1 10.6 12.6 16.0 8.3 9.1 15.3 14.8 2.1
4.5.3. Conover (1971, p. 226) considered data on the physical fitness of male seniors in a particular high school. The seniors were divided into two groups based on whether they lived on a farm or in town. The results in Table 4.6 are from a physical fitness test administered to the students. High
4.5.2. Box (1950) also considered data on rats given Thyroxin and a control group of rats. The weight gains are given in Table 4.5. Give a 95% confidence interval for the difference in weight gains between the Thyroxin group and the control group. Give the P value for a test of whether the control
4.5.1. Box (1950) gave data on the weights of rats that were given the drug Thiouracil.The rats were measured at the start of the experiment and at the end of the experiment. The data are given in Table 4.4. Give a 99% confidence interval for the difference in weights between the finish and the
3.10.11. Given that y1, . . . ,yn are independent with E(yi) = μi and σ 2 = Var(yi) =E[yi −μi]2, give intuitive justifications for why both ˆ σ 2 ≡ Σni=1(yi − ˆ yi)2/n and MSE ≡ Σni=1(yi −ˆ yi)2/dfE are reasonable estimates of σ 2. Recall that ˆ yi is an estimate of μi.
3.10.10. Suppose we have three random variables y, y1, and y2 and let α be a number between 0 and 1. Show that if y =α y1 +(1−α )y2 and if E(y) = E(y2) =θ then E(y1) =θ .
3.10.9. With reference to Exercise 2.8.2, give the approximate number of observations necessary to estimate the mean rat weight to within 1 unit with 95% confidence. How large a sample is needed to get a reasonable test of H0 : μ = 55 when μ = 54 using Cox’s rule?
3.10.8. With reference to Exercise 2.8.3, give the approximate number of observations necessary to estimate the mean of FORM to within 0.5 units with 95% confidence. How large a sample is needed to get a reasonable test of H0 : μ = 20 when μ = 20.2 using Cox’s rule?
3.10.7. With reference to Exercise 2.8.3, give the approximate number of observations necessary to get a 99% confidence for the mean of K that has a length of 60. How large a sample is needed to get a reasonable test of H0 : μ = 1200 when μ = 1190 using Cox’s rule? What is the number when μ =
3.10.6. With reference to Exercise 2.8.3, give the approximate number of observations necessary to estimate the mean of BX to within 0.01 units with 99% confidence. How large a sample is needed to get a reasonable test of H0 : μ = 10 when μ = 11 using Cox’s rule?
3.10.5. The turtle shell data of Jolicoeur andMosimann (1960) given in Exercise 2.7.4 has a standard deviation of about 21.25. If we were to collect a new sample, how large should the sample size be in order to have a 95% confidence interval with a length of (about) four units?According to Cox’s
3.10.4. Consider that I am collecting (normally distributed) data with a variance of 4 and I want to test a null hypothesis of H0 : μ = 10. What sample size should I take according to Cox’s rule if I want a reasonable chance of rejecting H0 when μ = 13? What if I want a reasonable chance of
3.10.3. Identify the parameter, estimate, standard error of the estimate, and reference distribution for Exercise 2.8.4.
3.10.2. Identify the parameter, estimate, standard error of the estimate, and reference distribution for Exercise 2.8.2.
3.10.1. Identify the parameter, estimate, standard error of the estimate, and reference distribution for Exercise 2.8.1.
as your guide. Compare the optimal transformation near or at the zero frequency with the usual transformation yt = r2 t
For the NYSE returns, say, rt, analyzed in Chapter 5, Example 5.5:(a) Estimate the spectrum of the rt. Does the spectral estimate appear to support the hypothesis that the returns are white?(b) Examine the possibility of spectral power near the zero frequency for a transformation of the returns,
on BNRF1 of herpesvirus saimiri(the data file is bnrf1hvs), and compare the results with the results obtained for Epstein–Barr.
Repeat the analysis of Example
Verify, as stated in (7.181), the imaginary part of a k×k spectral matrix, fim(ω), is skew symmetric, and then show β0 fim yy (ω)β = 0 for a real k × 1 vector, β.
Extend the EM algorithm for classical factor analysis, (7.160)-(7.165), to the time series case of maximizing ln L B(ωj ), D(ωj ) in (7.176). Then, for the data used in Example 7.16, find the approximate maximum likelihood estimates of B(ωj ) and D(ωj ), and, consequently, Λt.Section 7.9
In the factor analysis model (7.154), let p = 3, q = 1, andΣxx =1 .4 .9.4 1 .7.9 .7 1 .Show there is a unique choice for B and D, but δ2 3 < 0, so the choice is not valid.
For this problem, consider the first three earthquake series (EQ1, EQ2, EQ3) listed in eqexp.(a) Estimate and compare the spectral density of the P component and then of the S component for each individual earthquake.(b) Estimate and compare the squared coherency between the P and S components of
Perform principal component analyses on the stimulus conditions (i)awake-heat and (ii) awake-shock, and compare your results to the results of Example 7.14. Use the data in fmri and average across subjects.
Assume the same additive signal plus noise representations as in the previous problem, except, the signal is now a random process with a zero mean and covariance matrix σ2 s I. Derive the comparable version of (7.115) as a quadratic detector, and characterize its performance under both hypotheses
The problem of detecting a signal in noise can be considered using the model xt = st + wt, t = 1, . . . , n, for p1(x) when a signal is present and the model xt = wt, t = 1, . . . , n, for p2(x) when no signal is present. Under multivariate normality, we might specialize even further by assuming
Suppose we have I = 2 groups and the models y1jt = µt + α1t + v1jt for the j = 1, . . . , N observations in group 1 and y2jt = µt + α2t + v2jt for the j = 1, . . . , N observations in group 2, with α1t + α2t = 0. Suppose we want to test equality of the two group means; i.e., yijt = µt +
For the random coefficient model, verify the expected mean square of the regression power component is E[SSR(ωk)] = E[Y ∗(ωk)Z(ωk)S−1 z (ωk)Z∗(ωk)Y (ωk)]= Lfβ(ωk)tr {Sz(ωk)} + Lqfv(ωk).Recall, the underlying frequency domain model is Y (ωk) = Z(ωk)B(ωk) + V (ωk), where B(ωk)
Consider the estimator (7.67) as applied in the context of the random coefficient model (7.65). Prove the filter coefficients for the minimum mean square estimator can be determined from (7.68) and the mean square covariance is given by (7.71).
Consider a linear model with mean value function µt and a signal αt delayed by an amount τj on each sensor, i.e., yjt = µt + αt−τj + vjt.Show the estimators (7.42) for the mean and the signal are the Fourier transforms of Mc(ω) = Y·(ω) − φ(ω)Bw(ω)1 − |φ(ω)|2 and whereφ(ω) = 1
Consider estimating the functionψt = X∞r=−∞a0 rβt−r by a linear filter estimator of the formψbt = X∞r=−∞a0 rβbt−r, where βbt is defined by (7.42). Show a sufficient condition for ψbt to be an unbiased estimator; i.e., E ψbt = ψt, is H(ω)Z(ω) = I for all ω. Similarly, show
Consider the complex regression model (7.28) in the form Y = XB + V , where Y = (Y1, Y2, . . . YL)0 denotes the observed DFTs after they have been reindexed and X = (X1, X2, . . . , XL)0 is a matrix containing the reindexed input vectors. The model is a complex regression model with Y = Y c−iY s,
Verify (7.18) and (7.19) for the mean-squared prediction error MSE in(7.11). Use the orthogonality principle, which implies MSE = E(yt − X∞r=−∞β0 rxt−r)ytand gives a set of equations involving the autocovariance functions. Then, use the spectral representations and Fourier transform
Prove f b in (7.6) maximizes the log likelihood (7.5) by minimizing the negative of the log likelihood Lln |f| + L tr {ffb −1}in the form LX iλi − ln λi − 1+ Lp + Lln |f b|, where the λi values correspond to the eigenvalues in a simultaneous diagonalization of the matrices f and ˆf;
Consider the complex Gaussian distribution for the random variable X = Xc − iXs, as defined in (7.1)-(7.3), where the argument ωk has been suppressed. Now, the 2p × 1 real random variable Z = (X0c, X0 s)0 has a multivariate normal distribution with density p(Z) = (2π)−p|Σ|−1/2 exp−1
Fit a stochastic volatility model to the returns of one (or more) of the four financial time series available in the R datasets package as EuStockMarkets.
Verify (6.200) and (6.207).Section 6.10
Verify (6.194) and (6.195).
Use the material presented in Example 6.21 to perform a Bayesian analysis of the model for the Johnson & Johnson data presented in Example 6.10.
Fit the switching model described in Example 6.15 to the growth rate of GNP. The data are in gnp and, in the notation of the example, yt is log-GNP and ∇yt is the growth rate. Use the code in Example 6.17 as a guide.Section 6.9
Repeat the bootstrap analysis of Example 6.13 on the entire three-month Treasury bills and rate of inflation data set of 110 observations. Do the conclusions of Example 6.13—that the dynamics of the data are best described in terms of a fixed, rather than stochastic, regression—still
Verify Property 6.6.Section 6.7
Verify Property 6.5.
Use Property 6.6 to complete the following exercises.(a) Write a univariate AR(1) model, yt = φyt−1 + vt, in state-space form.Verify your answer is indeed an AR(1).(b) Repeat (a) for an MA(1) model, yt = vt + θvt−1.(c) Write an IMA(1,1) model, yt = yt−1 + vt + θvt−1, in state-space form.
(a) Fit an AR(2) to the recruitment series, Rt in rec, and consider a lag-plot of the residuals from the fit versus the SOI series, St in soi, at various lags, St−h, for h = 0, 1, . . .. Use the lag-plot to argue that St−5 is reasonable to include as an exogenous variable.(b) Fit an ARX(2) to
Using Example 6.10 as a guide, fit a structural model to the Federal Reserve Board Production Index data and compare it with the model fit in Example 3.46.Section 6.6

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Time Series Analysis And Its Applications With R Examples 3rd Edition Robert H. Shumway, David S. Stoffer - Solutions

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