Time Series Analysis And Its Applications 4th Edition Robert H. Shumway, David S. Stoffer - Solutions

Unlock a comprehensive understanding of "Time Series Analysis And Its Applications 4th Edition" by Robert H. Shumway and David S. Stoffer with our expertly crafted solutions. Access an extensive range of solved problems and solutions PDF to enhance your learning experience. Our chapter solutions provide detailed, step-by-step answers, ensuring you grasp each concept thoroughly. Benefit from our online answers key and instructor manual, designed to assist both students and educators. Explore our test bank for diverse questions and answers, and download the solution manual for free to facilitate your studies with ease.

For the S&P500 weekly returns, say, rt, analyzed in Example 6.17(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, say,
Verify, as stated in (7.179), the imaginary part of a k ×k spectral matrix, f im(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, (7.158)-(7.163), to the time series case of maximizing ln L B(ωj), D (ωj)in (7.174). Then, for the data used in Example 7.15, find the approximate maximum likelihood estimates of B(ωj)and D (ωj), and, consequently, Λt
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 7.13. Use the data in fmri and average across subjects.
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,
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., yi jt = µt + vi
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 )]= L fβ(ωk )tr{Sz (ωk )} + Lq fv(ωk ).Recall, the underlying frequency domain model is Y(ωk ) = Z(ωk )B(ωk ) + V(ωk ), where
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 Mˆ (ω) =Y·(ω) − φ(ω)Bw(ω)1 − |φ(ω)|2 and Aˆ(ω)
Consider estimating the functionψt =Õ∞r=−∞a 0r βt−r by a linear filter estimator of the formψˆt =Õ∞r=−∞a 0r βˆt−r, where βˆt is defined by (7.42). Show a sufficient condition for ψˆt to be an unbiased estimator; i.e., E ψˆt = ψt, is H(ω)Z(ω) = I for all ω.
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 re-indexed and X = (X1, X2, . . ., XL)0 is a matrix containing the reindexed input vectors. The model is a complex regression model with Y = Yc − iYs, X
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 −Õ∞r=−∞β0 rxt−r )ytand gives a set of equations involving the autocovariance functions. Then, use the spectral representations and Fourier transform
Prove ˆf in (7.6) maximizes the log likelihood (7.5) by minimizing the negative of the log likelihood L ln | f | + L tr{ˆff−1}in the form LÕiλi − ln λi − 1+ Lp + L ln |ˆf |, 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 = (X 0c, X 0s)0 has a multivariate normal distribution with density p(Z) = (2π)−p|Σ|−1/2 exp−1 2(Z
Fit an AR(1) to the returns of the US GNP (gnp) using a Bayesian approach via MCMC?
Repeat Example 6.27 on the log of the Johnson & Johnson data.
Verify the distributional statements made in Example 6.25.
We consider the stochastic volatility model (6.197).(a) Show that for any integer m, E[r 2m t] = β2mE[r 2m t] exp(m 2σ2 x/2), where σ2 x = σ2/(1 − φ2).(b) Show (6.198).(c) Show that for any positive integer h, var(Xt + Xt+h) = 2σ2 X(1 + φh).(d) Show that cov(r 2m t,r 2m t+h) = β4mE[v 2m
Fit a stochastic volatility model to the residuals of the GNP (gnp) returns analyzed in Example 3.39.
Argue that a switching model is reasonable in explaining the behavior of the number of sunspots (see Figure 4.22) and then fit a switching model to the sunspot data.
Fit a Poisson-HMM to the dataset polio from the gamlss.data package. The data are reported polio cases in the U.S. for the years 1970 to 1983. To get started, install the package and then type library(gamlss.data) # load package plot(polio, type='s') # view the data
Prove Property 6.7 and verify (6.143)
Verify (6.132), (6.133), and (6.134).
Let yt represent the global temperature series (globtemp) shown in Figure 1.2.(a) Fit a smoothing spline using gcv (the default) to yt and plot the result superimposed on the data. Repeat the fit using spar=.7; the gcv method yields spar=.5 approximately. (Example 2.14 on page 70 may help. Also in
Verify Property 6.5. 6.18 Verify Property 6.6.
The data set ar1miss is n = 100 observations generated from an AR(1) process, xt = φxt−1 +wt, with φ = .9 and σw = 1, where 10% of the data have been deleted at random (replaced with NA). Use the results of Problem 6.11 to estimate the parameters of the model, φ and σw, using the EM
As an example of the way the state-space model handles the missing data problem, suppose the first-order autoregressive process xt = φxt−1 + wt has an observation missing at t = m, leading to the observations yt = At xt, where At = 1 for all t, except t = m wherein At = 0. Assume x0 = 0 with
To explore the stability of the filter, consider a univariate state-space model.That is, for t = 1, 2, . . ., the observations are yt = xt + vt and the state equation is xt = φxt−1 + wt, where σw = σv = 1 and |φ| < 1. The initial state, x0, has zero mean and variance one.(a) Exhibit the
Consider the univariate state-space model given by state conditions x0 = w0, xt = xt−1 + wt and observations yt = xt + vt, t = 1, 2, . . ., where wt and vt are independent, Gaussian, white noise processes with var(wt) = σ2 w and var(vt) = σ2 v.(a) Show that yt follows an IMA(1,1) model, that
Projection Theorem Derivation of Property 6.2. Throughout this problem, we use the notation of Property 6.2 and of the Projection Theorem given in Appendix B, where H is L 2. If Lk+1 = sp{y1, . . ., yk+1}, and Vk+1 = sp{yk+1 − y kk+1}, for k =0, 1, . . ., n − 1, where y kk+1 is the projection
Simulate n = 100 observations from the following state-space model:xt = .8xt−1 + wt and yt = xt + vt where x0 ∼ N(0, 2.78), wt ∼ iid N(0, 1), and vt ∼ iid N(0, 1) are all mutually independent. Compute and plot the data, yt, the one-step-ahead predictors, y t−1 talong with the root mean
Consider the state-space model presented in Example 6.3. Let x t−1 t = E(xt|yt−1, . . ., y1) and let P t−1 t = E(xt − x t−1 t)2. The innovation sequence or residuals aret = yt − y t−1 t, where y t−1 t = E(yt| yt−1, . . ., y1). Find cov(s, t) in terms of x t−1 tand P t−1
Consider the data set econ5 containing quarterly U.S. unemployment, GNP, consumption, and government and private investment from 1948-III to 1988-II. The seasonal component has been removed from the data. Concentrating on unemployment (Ut), GNP (Gt), and consumption (Ct), fit a vector ARMA model to
The sunspot data (sunspotz) are plotted in Chapter 4, Figure 4.22. From a time plot of the data, discuss why it is reasonable to fit a threshold model to the data, and then fit a threshold model.
The stats package of R contains the daily closing prices of four major European stock indices; type help(EuStockMarkets) for details. Fit a GARCH model to the returns of one of these series and discuss your findings. (Note: The data set contains actual values, and not returns. Hence, the data must
Weekly crude oil spot prices in dollars per barrel are in oil; see Problem Problem 2.10 and Appendix R for more details. Investigate whether the growth rate of the weekly oil price exhibits GARCH behavior. If so, fit an appropriate model to the growth rate.
Plot the global temperature series, globtemp, and then test whether there is a unit root versus the alternative that the process is stationary using the three tests, DF, ADF, and PP, discussed in Example 5.3. Comment.
Compute the sample ACF of the absolute values of the NYSE returns displayed in Figure 1.4 up to lag 200, and comment on whether the ACF indicates long memory.Fit an ARFIMA model to the absolute values and comment.
For the zero-mean complex random vector z = xc − ixs, with cov(z) = Σ =C − iQ, with Σ = Σ∗, define w = 2Re(a∗z), where a = ac − ias is an arbitrary non-zero complex vector. Prove cov(w) = 2a∗Σa.Recall ∗ denotes the complex conjugate transpose.
Prove Lemma C.4.
Show that condition (4.48) implies (C.19) by showing n−1/2 Õh≥0 h |γ(h)| ≤ σ2 wÕk≥0|ψk |Õj≥0 pj |ψj|.
Let wt be a Gaussian white noise series with variance σ2 w. Prove that the results of Theorem C.4 hold without error for the DFT of wt.
Consider the two-dimensional linear filter given as the output (4.143).(a) Express the two-dimensional autocovariance function of the output, say, γy(h1, h2), in terms of an infinite sum involving the autocovariance function of xs and the filter coefficients as1,s2.(b) Use the expression derived
Consider the same model as in the preceding problem. (a) Prove the optimal smoothed estimator of the form xˆt =Õ∞s=−∞as yt−s has as =σ2 wσ2θ|s |1 − θ2.(b) Show the mean square error is given by E{(xt − xˆt)2} =σ2 vσ2 wσ2(1 − θ2).(c) Compare mean square error of the
Consider the model yt = xt + vt, where xt = φxt−1 + wt, such that vt is Gaussian white noise and independent of xt with var(vt) = σ2 v, and wt is Gaussian white noise and independent of vt, with var(wt) = σ2 w, and |φ| < 1 and Ex0 = 0. Prove that the spectrum of the observed series yt is
Consider the signal plus noise model yt =Õ∞r=−∞βr xt−r + vt, where the signal and noise series, xt and vt are both stationary with spectra fx(ω)and fv(ω), respectively. Assuming that xt and vt are independent of each other for all t, verify (4.131) and (4.132).
The data set climhyd, contains 454 months of measured values for six climatic variables: (i) air temperature [Temp], (ii) dew point [DewPt], (iii) cloud cover [CldCvr],(iv) wind speed [WndSpd], (v) precipitation [Precip], and (vi) inflow [Inflow], at Lake Shasta in California; the data are
Prove the squared coherence ρ2 y·x(ω) = 1 for all ω when yt =Õ∞r=−∞ar xt−r, that is, when xt and yt can be related exactly by a linear filter.
Consider the problem of approximating the filter output yt =Õ∞k=−∞ak xt−k,Õ∞−∞|ak | < ∞, by yM t =Õ|k |
Suppose we are given a stationary zero-mean series xt with spectrum fx(ω) and then construct the derived series yt = ayt−1 + xt, t = ±1, ±2, ... .(a) Show how the theoretical fy(ω) is related to fx(ω).(b) Plot the function that multiplies fx(ω) in part (a) for a = .1 and for a = .8. This
Suppose xt is a stationary series, and we apply two filtering operations in succession, say, yt =Õr ar xt−r then zt =Õs bs yt−s .(a) Show the spectrum of the output is fz (ω) = |A(ω)|2|B(ω)|2 fx(ω), where A(ω) and B(ω) are the Fourier transforms of the filter sequences at and bt,
Let xt = cos(2πωt), and consider the output yt =Õ∞k=−∞ak xt−k, where Ík|ak | < ∞. Show yt = |A(ω)| cos(2πωt + φ(ω)), where |A(ω)| and φ(ω) are the amplitude and phase of the filter, respectively. Interpret the result in terms of the relationship between the input series, xt,
Consider the bivariate time series records containing monthly U.S. production(prod) as measured by the Federal Reserve Board Production Index and the monthly unemployment series (unemp).(a) Compute the spectrum and the log spectrum for each series, and identify statistically significant peaks.
For the processes in Problem 4.25:(a) Compute the phase between xt and yt.(b) Simulate n = 1024 observations from xt and yt for φ = .9, σ2 = 1, and D = 1.Then estimate and plot the phase between the simulated series for the following values of L and comment:(i) L = 1, (ii) L = 3, (iii) L = 41,
Consider two processes xt = wt and yt = φxt−D + vt where wt and vt are independent white noise processes with common variance σ2, φis a constant, and D is a fixed integer delay.(a) Compute the coherency between xt and yt.(b) Simulate n = 1024 normal observations from xt and yt for φ = .9, σ2
Analyze the coherency between the temperature and salt data discussed in Problem 4.10. Discuss your findings.
Suppose we wish to test the noise alone hypothesis H0 : xt = nt against the signal-plus-noise hypothesis H1 : xt = st +nt, where st and nt are uncorrelated zeromean stationary processes with spectra fs(ω) and fn(ω). Suppose that we want the test over a band of L = 2m+1 frequencies of the form
Suppose a sample time series with n = 256 points is available from the firstorder autoregressive model. Furthermore, suppose a sample spectrum computed with L = 3 yields the estimated value ¯fx(1/8) = 2.25. Is this sample value consistent withσ2 w = 1, φ = .5? Repeat using L = 11 if we just
Fit an autoregressive spectral estimator to the Recruitment series and compare it to the results of Example 4.16.
Analyze the chicken price data (chicken) using a parametric spectral estimation procedure. Compare the results to Problem 4.13.
Often, the periodicities in the sunspot series are investigated by fitting an autoregressive spectrum of sufficiently high order. The main periodicity is often stated to be in the neighborhood of 11 years. Fit an autoregressive spectral estimator to the sunspot data using a model selection method
Consider two time series xt = wt − wt−1, yt =1 2(wt + wt−1), formed from the white noise series wt with variance σ2 w = 1.(a) Are xt and yt jointly stationary? Recall the cross-covariance function must also be a function only of the lag h and cannot depend on time.(b) Compute the spectra
Use Property 4.2 to verify (4.71). Then verify (4.74) and (4.75).
Cepstral Analysis. The periodic behavior of a time series induced by echoes can also be observed in the spectrum of the series; this fact can be seen from the results stated in Problem 4.7. Using the notation of that problem, suppose we observe xt = st + Ast−D + nt, which implies the spectra
Repeat Problem 4.10 using a nonparametric spectral estimation procedure. In addition to discussing your findings in detail, comment on your choice of a spectral estimate with regard to smoothing and tapering.
Repeat Problem 4.9 using a nonparametric spectral estimation procedure. In addition to discussing your findings in detail, comment on your choice of a spectral estimate with regard to smoothing and tapering.
Analyze the chicken price data (chicken) using a nonparametric spectral estimation procedure. Aside from the obvious annual cycle discovered in Example 2.5, what other interesting cycles are revealed?
Prove the convolution property of the DFT, namely,Õn s=1 as xt−s =Õn−1 k=0 dA(ωk )dx(ωk ) exp{2πωk t}, for t = 1, 2, . . ., n, where dA(ωk ) and dx(ωk ) are the discrete Fourier transforms of at and xt, respectively, and we assume that xt = xt+n is periodic.
Let the observed series xt be composed of a periodic signal and noise so it can be written as xt = β1 cos(2πωk t) + β2 sin(2πωk t) + wt, where wt is a white noise process with variance σ2 w. The frequency ωk is assumed to be known and of the form k/n in this problem. Suppose we consider
The levels of salt concentration known to have occurred over rows, corresponding to the average temperature levels for the soil science data considered in Figure 1.18 and Figure 1.19, are in salt and saltemp. Plot the series and then identify the dominant frequencies by performing separate spectral
Figure 4.22 shows the biyearly smoothed (12-month moving average) number of sunspots from June 1749 to December 1978 with n = 459 points that were taken twice per year; the data are contained in sunspotz. With Example 4.13 as a guide, perform a periodogram analysis identifying the predominant
Suppose xt and yt are stationary zero-mean time series with xt independent of ys for all s and t. Consider the product series zt = xt yt.Prove the spectral density for zt can be written as fz (ω) =∫ 1 2−1 2fx(ω − ν) fy(ν) dν.
In applications, we will often observe series containing a signal that has been delayed by some unknown time D, i.e., xt = st + Ast−D + nt, where st and nt are stationary and independent with zero means and spectral densities fs(ω) and fn(ω), respectively. The delayed signal is multiplied by
A first-order autoregressive model is generated from the white noise series wt using the generating equations xt = φxt−1 + wt, where φ, for |φ| < 1, is a parameter and the wt are independent random variables with mean zero and variance σ2 w.(a) Show that the power spectrum of xt is given by
A time series was generated by first drawing the white noise series wt from a normal distribution with mean zero and variance one. The observed series xt was generated from xt = wt − θwt−1, t = 0, ±1, ±2, . . ., where θ is a parameter.(a) Derive the theoretical mean value and autocovariance
With reference to equations (4.1) and (4.2), let Z1 = U1 and Z2 = −U2 be independent, standard normal variables. Consider the polar coordinates of the point(Z1, Z2), that is, A 2 = Z 21 + Z 22 and φ = tan−1(Z2/Z1).(a) Find the joint density of A 2and φ, and from the result, conclude that A
Repeat the simulations and analyses in Example 4.1 and Example 4.2 with the following changes:(a) Change the sample size to n = 128 and generate and plot the same series as in Example 4.1:xt1 = 2 cos(2π .06 t) + 3 sin(2π .06 t), xt2 = 4 cos(2π .10 t) + 5 sin(2π .10 t), xt3 = 6 cos(2π .40 t) +
Verify that for any positive integer n and j, k = 0, 1, . . ., [[n/2]], where [[·]] denotes the greatest integer function:(a) Except for j = 0 or j = n/2, 4.12Õn t=1 cos2(2πt j/n) =Õn t=1 sin2(2πt j/n) = n/2.(b) When j = 0 or j = n/2,Õn t=1 cos2(2πt j/n) = n but Õn t=1 sin2(2πt j/n) =
Prove Property 3.2.
Prove Theorem B.2.
Use Theorem B.2 and Theorem B.3 to verify (3.117).
Consider the series xt = wt −wt−1, where wt is a white noise process with mean zero and variance σ2 w. Suppose we consider the problem of predicting xn+1, based on only x1, . . ., xn. Use the Projection Theorem to answer the questions below.(a) Show the best linear predictor is xn n+1 = −1 n
Suppose xt =Íp j=1φj xt−j + wt, where φp , 0 and wt is white noise such that wt is uncorrelated with {xk ; k < t}. Use the Projection Theorem, Theorem B.1, to show that, for n > p, the BLP of xn+1 on sp{xk, k ≤ n} is xˆn+1 =Õp j=1φj xn+1−j
Fit an appropriate seasonal ARIMA model to the log-transformed Johnson and Johnson earnings series (jj) of Example 1.1. Use the estimated model to forecast the next 4 quarters.The following problems require supplemental material given in Appendix B.
Fit a seasonal ARIMA model of your choice to the unemployment data in UnempRate. Use the estimated model to forecast the next 12 months.
Fit a seasonal ARIMA model of your choice to the unemployment data in unemp.Use the estimated model to forecast the next 12 months.
Fit a seasonal ARIMA model of your choice to the chicken price data in chicken.Use the estimated model to forecast the next 12 months.
Plot the ACF of the seasonal ARIMA(0, 1) × (1, 0)12 model with Φ = .8 andθ = .5.
Consider the ARIMA model xt = wt + Θwt−2.(a) Identify the model using the notation ARIMA(p,d, q) × (P, D, Q)s.(b) Show that the series is invertible for |Θ| < 1, and find the coefficients in the representation wt =Õ∞k=0πk xt−k .(c) Develop equations for the m-step ahead forecast, x˜n+m,
Redo Problem 2.2 without assuming the error term is white noise.
One of the remarkable technological developments in the computer industry has been the ability to store information densely on a hard drive. In addition, the cost of storage has steadily declined causing problems of too much data as opposed to big data. The data set for this assignment is cpg,
Let St represent the monthly sales data in sales (n = 150), and let Lt be the leading indicator in lead.(a) Fit an ARIMA model to St, the monthly sales data. Discuss your model fitting in a step-by-step fashion, presenting your (A) initial examination of the data, (B)transformations, if necessary,
Fit an ARIMA(p,d, q) model to the sulfur dioxide series, so2, performing all of the necessary diagnostics. After deciding on an appropriate model, forecast the data into the future four time periods ahead (about one month) and calculate 95%prediction intervals for each of the four forecasts.
Fit an ARIMA(p,d, q) model to the global temperature data globtemp performing all of the necessary diagnostics. After deciding on an appropriate model, forecast(with limits) the next 10 years. Comment.
Crude oil prices in dollars per barrel are in oil. Fit an ARIMA(p,d, q) model to the growth rate performing all necessary diagnostics. Comment.
In Example 3.40, we presented the diagnostics for the MA(2) fit to the GNP growth rate series. Using that example as a guide, complete the diagnostics for the AR(1) fit.
For the logarithm of the glacial varve data, say, xt, presented in Example 3.33, use the first 100 observations and calculate the EWMA, x˜t t+1, given in (3.151) for t = 1, . . ., 100, using λ = .25, .50, and .75, and plot the EWMAs and the data superimposed on each other. Comment on the results.
For the ARIMA(1, 1, 0) model with drift, (1 − φB)(1 − B)xt = δ + wt, let yt = (1 − B)xt = ∇xt.(a) Noting that yt is AR(1), show that, for j ≥ 1, y nn+j = δ [1 + φ + · · · + φj−1] + φj yn.(b) Use part (a) to show that, for m = 1, 2, . . ., x nn+m = xn +δ1 − φh m −φ(1 −
Verify that the IMA(1,1) model given in (3.148) can be inverted and written as(3.149).
Suppose yt = β0 + β1t + · · · + βqt q + xt, βq , 0, where xt is stationary. First, show that ∇k xt is stationary for any k = 1, 2, . . ., and then show that ∇k yt is not stationary for k < q, but is stationary for k ≥ q.

Showing 1 - 100 of 172

Time Series Analysis And Its Applications 4th Edition Robert H. Shumway, David S. Stoffer - Solutions

Step by Step Answers