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regression analysis

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

to estimate the parameters of the model,φ and σw, using the EM algorithm, and then estimate the missing values.Section 6.5
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 has been zeroed out at random. Considering the zeroed out data to be missing data, use the results of Problem
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 = Atxt, where At = 1 for all t, except t = m wherein At = 0. Assume x0 = 0 with
Continuing with the previous problem, consider the evaluation of the Hessian matrix and the numerical evaluation of the asymptotic variance–covariance matrix of the parameter estimates. The information matrix satisfies E−∂2 lnLY (Θ)∂Θ ∂Θ0= E(∂ lnLY (Θ)∂Θ ∂ lnLY (Θ)∂Θ
In §6.3, we discussed that it is possible to obtain a recursion for the gradient vector, −∂ lnLY (Θ)/∂Θ. Assume the model is given by (6.1) and(6.2) and At is a known design matrix that does not depend on Θ, in which case Property 6.1 applies. For the gradient vector, show∂ lnLY
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 model yt = xt + vt, where vt is Gaussian white noise with variance σ2 v, xt are independent Gaussian random variables with mean zero and var(xt) = rtσ2 x with xt independent of vt, and r1, . . . , rn are known constants. Show that applying the EM algorithm to the problem of
Smoothing Splines and the Kalman Smoother. Consider the discrete time version of the smoothing spline argument given in (2.56); that is, suppose we observe yt = xt + vt and we wish to fit xt, for t = 1, . . . , n, constrained to be smooth, by minimizingShow that this problem is identical to
Let yt represent the global temperature series (gtemp) 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 75 may help. Also in R,
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 is,
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 L2. If Lk+1 = sp{y1, . . . , yk+1}, and Vk+1 =sp{yk+1 − yk k+1}, for k = 0, 1, . . . , n − 1, where yk k+1 is the projection
Suppose the vector z = (x0, y0)0, where x (p × 1) and y (q × 1) are jointly distributed with mean vectors µx and µy and with covariance matrix cov(z) = Σxx ΣxyΣyx Σyy.Consider projecting x on M = sp{1, y}, say, xb = b + By.(a) Show the orthogonality conditions can be written as E(x − b
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, yt−1 t along with the root mean
Consider the state-space model presented in Example 6.3. Let xt−1 t =E(xt|yt−1, . . . , y1) and let Pt−1 t = E(xt − xt−1 t )2. The innovation sequence or residuals are t = yt − yt−1 t , where yt−1 t = E(yt|yt−1, . . . , y1). Find cov(s, t)in terms of xt−1 t and Pt−1 t for
Consider a system process given by xt = −.9xt−2 + wt t = 1, . . . , n where x0 ∼ N(0, σ2 0), x−1 ∼ N(0, σ2 1), and wt is Gaussian white noise with variance σ2 w. The system process is observed with noise, say, yt = xt + vt, where vt is Gaussian white noise with variance σ2 v. Further,
Let y0,y1, . . . ,yn be independent Nμ ,σ 2random variables and compute ¯ y·, and s2 from observations 1 through n. Show that (y0 − ¯ y·)/σ 2+σ 2/n ∼ N(0,1) using results from Chapter 1 and the fact that linear combinations of independent normals are normal. Recalling that y0, ¯y·,
Shewhart (1931, p. 62) reproduces Millikan’s data on the charge of an election.These are repeated in Table 2.4. Check for outliers and nonnormality. Adjust the data appropriately if there are any problems. Give a 98% confidence interval for the population mean value. Give a 98% prediction
Give 99% confidence intervals for the population variances of all the variables in Exercise 2.8.3. Assume that the original data were normally distributed. Using α = 0.01, test whether the potassium variance could reasonably be 45,000. Could the formol number variance be 8?
Give a 95% confidence interval for the population variance associated with the data of Exercise 2.8.4. Remember that the inferences about variances require the assumption of normality.
Give a 95% confidence interval for the population variance associated with the data of Exercise 2.8.5. Remember that inferences about variances require the assumption of normality. Could the variance reasonably be 10?
Mosteller and Tukey (1977) extracted data from the Coleman Report. Among the variables considered was the percentage of sixth-graderswhose fathers were employed in whitecollar jobs. Data for 20 NewEngland schools are given in Table 2.3.Are the data reasonably normal?Do any of the standard
Jolicoeur and Mosimann (1960) gave data on female painted turtle shell lengths. The data are presented in Table 2.2. Give a 95% confidence interval for the population mean length. Give a 99% prediction interval for the shell length of a new female.
Fuchs and Kenett (1987) presented data on citrus juice for fruits grown during a specific season at a specific location. The sample size was 80 but many variables were measured on each sample. Sample statistics for some of these variables are given below.Variable BX AC SUG K FORM PECT Mean 10.4 1.3
Box (1950) gave data on the weights of rats that were about to be used in an experiment. The data are repeated in Table 2.1. Assuming that these are a random sample from a broader population of rats, give a 95% confidence interval for the population mean weight. Test the null hypothesis that the
Mulrow et al. (1988) presented data on the melting temperature of biphenyl as measured on a differential scanning calorimeter. The data are given below; they are the observed melting temperatures in Kelvin less 340.3.02,2.36,3.35,3.13,3.33,3.67,3.54,3.11,3.31,3.41,3.84,3.27,3.28,3.30 Compute the
When I order a limo, 65% of the time the driver is clearly male, 30% of the time the driver is clearly female, and 5% of the time the gender of the driver is indeterminant.Assuming independence, what is the probability that among my next 8 drivers 5 are clearly male and 3 are clearly female? What
When I order a limo, 65% of the time the driver is male. Assuming independence, what is the probability that 6 of my next 8 drivers are male? What is the expected number of male drivers among my next eight? What is the variance of the number of male drivers among my next eight?
A pizza parlor makes small, medium, and large pizzas. Over the years they make 20% small pizzas, 35% medium pizzas, and 45% large pizzas. On a given Tuesday night they were asked to make only 10 pizzas. If the orders were independent and representative of the longterm percentages, what is the
Graph the function f (x) = 2x if 0 < x < 1 and f (x) = 0 otherwise. If we use this curve to define a probability function, what is the probability of getting an observation larger than 1/4? Smaller than 2/3? Between 1/3 and 7/9?
As of 1994,Duke University had been in the final four of the NCAA’s national basketball championship tournament seven times in nine years. Suppose their appearances were independent and that they had a probability of .25 for winning the tournament in each of those years.(a) What is the
Consider three independent random variables X, Y, and Z. Suppose E(X) =25, E(Y) = 40, and E(Z) = 55 with Var(X) = 4, Var(Y) = 9, and Var(Z) = 25.(a) Find E(2X +3Y +10) and Var(2X +3Y +10).(b) Find E(2X +3Y +Z+10) and Var(2X +3Y +Z+10).
Consider a random variable that takes on the values 25, 30, 45, and 50 with probabilities .15, .25, .35, and .25, respectively. Find the expected value, variance, and standard deviation of this random variable.
Appendix B.1 gives probabilities for a family of distributions that all look roughly like Figure 1.1. All members of the family are symmetric about zero and the members are distinguished by having different numbers of degrees of freedom (df ). They are called t distributions.For 0 ≤ α ≤ 1, the
LetW ∼ Bin(N, p) and for i = 1, . . . ,N take independent yis that are Bin(1, p).Argue thatW has the same distribution as y1+· · ·+yN. Use this fact, along with Exercise 1.6.1 and Proposition 1.2.11, to find E(W) and Var(W).
Arthritic ex-football players prefer their laudanum made with Old Pain-Killer Scotch by two to one. If we take a random sample of 5 arthritic ex-football players, what is the distribution of the number who will prefer Old Pain-Killer? What is the probability that only 2 of the ex-players will
Graph the function f (x)=1 if 0
Referring to Exercise 1.6.4, supposing I have a class of 40 students, what is the joint distribution for the numbers of students who get each of the five grades? Note that we are no longer looking at how many grade points an individual student might get, we are now counting how many occurrences we
Consider your letter grade for this course. Obviously, it is a random phenomenon.Define the ‘grade point’ random variable: y(A) = 4, y(B) = 3, y(C) = 2, y(D) = 1, y(F) = 0. If you were lucky enough to be taking the course from me, you would find that I am an easy grader. I give 5% As, 10% Bs,
Let y be the random variable consisting of the number of spots that face up upon rolling a die. Give the distribution of y. Find the expected value, variance, and standard deviation of y.
Let y be a random variable with E(y) =μ and Var(y) =σ 2. Show that Ey−μσ= 0 and Vary−μσ= 1.Let ¯ y· be the sample mean of n independent observations yi with E(yi) = μ and Var(yi) =σ 2.What is the expected value and variance of¯ y·−μσ /√n?Hint: For the first part, write
Use the definitions to find the expected value and variance of a Bin(1, p) distribution.
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
For the climhyd data set, consider predicting the transformed flows It =log it from transformed precipitation values Pt = √pt using a transfer function model of the form(1 − B12)It = α(B)(1 − B12)Pt + nt, where we assume that seasonal differencing is a reasonable thing to do. You may think
The data in climhyd have 454 months of measured values for the climatic variables air temperature, dew point, cloud cover, wind speed, precipitation(pt), and inflow (it), at Lake Shasta; the data are displayed in Figure 7.3. We would like to look at possible relations between the weather factors
Consider the correlated regression model, defined in the text by (5.58), say, y = Zβ + x, where x has mean zero and covariance matrix Γ. In this case, we know that the weighted least squares estimator is (5.59), namely,βbw = (Z0Γ −1Z)−1Z0Γ −1y.Now, a problem of interest in spatial series
Let St represent the monthly sales data in sales (n = 150), and let Lt be the leading indicator in lead. Fit the regression model ∇St = β0 +β1∇Lt−3 + xt, where xt is an ARMA process.
The sunspot data (sunspotz) are plotted in Chapter 4, Figure 4.31.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.Section 5.6
The 2 × 1 gradient vector, l(1)(α0, α1), given for an ARCH(1) model was displayed in (5.47). Verify (5.47) and then use the result to calculate the 2 ×2 Hessian matrix l(2)(α0, α1) = ∂2l/∂α2 0 ∂2l/∂α0∂α1∂2l/∂α0∂α1 ∂2l/∂α2 1.Section 5.5
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 2.11 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.
Investigate whether the quarterly growth rate of GNP exhibit ARCH behavior. If so, fit an appropriate model to the growth rate. The actual values are in gnp; also, see Example 3.38.
Verify (5.33).Section 5.4
Plot the GNP series, gnp, and then test for a unit root against the alternative that the process is explosive. State your conclusion.
Plot the global temperature series, gtemp, 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.
The data set arf is 1000 simulated observations from an ARFIMA(1, 1, 0)model with φ = .75 and d = .4.(a) Plot the data and comment.(b) Plot the ACF and PACF of the data and comment.(c) Estimate the parameters and test for the significance of the estimates φb and d b.(d) Explain why, using the
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.
Finish the proof of Theorem C.5.
Prove Lemma C.4.
Show that condition (4.40) implies (C.19) by showing n−1/2 X h≥0 h |γ(h)| ≤ σ2 wX k≥0|ψk|X j≥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 (4.149).(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 xbt = X∞s=−∞asyt−s has as = σ2 wσ2θ|s|1 − θ2 .(b) Show the mean square error is given by E{(xt − xbt)2} = σ2 vσ2 wσ2(1 − θ2).(c) Compare mean square error of the estimator
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 = X∞r=−∞βrxt−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 (4.137) and (4.138).
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 = X∞r=−∞arxt−r, that is, when xt and yt can be related exactly by a linear filter.
Consider the problem of approximating the filter output yt = X∞k=−∞akxt−k, X∞−∞|ak| < ∞, by yM t = X|k|
Repeat the wavelet analyses of Examples 4.22 and 4.23 on all earthquake and explosion series in the data file eqexp. Do the conclusions about the difference between earthquakes and explosions stated in Examples 4.22 and 4.23 still seem valid?
Repeat the dynamic Fourier analysis of Example 4.21 on the remaining seven earthquakes and seven explosions in the data file eqexp. Do the conclusions about the difference between earthquakes and explosions stated in the example still seem valid?
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 zero-mean 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
Suppose a sample time series with n = 256 points is available from the first-order 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 with σ2 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 4.13.
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
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 = X rarxt−r then zt = X sbsyt−s.(a) Show the spectrum of the output is fz(ω) = |A(ω)|2|B(ω)|2fx(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 = X∞k=−∞akxt−k, where P k |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, and
Determine the theoretical power spectrum of the series formed by combining the white noise series wt to form yt = wt−2 + 4wt−1 + 6wt + 4wt+1 + wt+2.Determine which frequencies are present by plotting the power spectrum.
Consider the bivariate time series records containing monthly U.S. production as measured by the Federal Reserve Board Production Index and monthly unemployment as given in Figure 3.21.(a) Compute the spectrum and the log spectrum for each series, and identify statistically significant peaks.
For the processes in Problem 4.18:(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 varianceσ2, φ is 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 4.9. Discuss your findings.
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 (4.63). Then verify (4.66) and (4.67).
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 4.6(a). Using the notation of that problem, suppose we observe xt = st + Ast−D + nt, which implies the spectra satisfy fx(ω) =[1 +
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.
Repeat Problem 4.8 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.
Prove the convolution property of the DFT, namely, Xn s=1 asxt−s =n X−1 k=0 dA(ωk)dx(ωk) exp{2πωkt}, 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.Section 4.5
Let the observed series xt be composed of a periodic signal and noise so it can be written as xt = β1 cos(2πωkt) + β2 sin(2πωkt) + 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 Figures 1.15 and 1.16, are in salt and saltemp. Plot the series and then identify the dominant frequencies by performing separate spectral
Figure 4.31 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 4.10 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 = xtyt.Prove the spectral density for zt can be written as fz(ω) = Z 1/2−1/2 fx(ω − ν)fy(ν) dν.Section 4.4
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
Verify (4.4).Section 4.3
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, A2 = Z2 1 + Z2 2 and φ = tan−1(Z2/Z1).(a) Find the joint density of A2 and φ, and from the result, conclude that A2
Repeat the simulations and analyses in Examples 4.1 and 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) + 7
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
Use the Projection Theorem to derive the Innovations Algorithm, Property 3.6, equations (3.77)-(3.79). Then, use Theorem B.2 to derive the m-stepahead forecast results given in (3.80) and (3.81).

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