Introduction To Linear Regression Analysis 5th Edition Douglas C. Montgomery, Elizabeth A. Peck, G. Geoffrey Vining - Solutions

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Suppose xt = Pp j=1 φjxt−j + wt, where φp 6= 0 and wt is white noise such that wt is uncorrelated with {xk; k < t}. Use the Projection Theorem to show that, for n > p, the BLP of xn+1 on sp{xk, k ≤ n} is xbn+1 = Xp j=1φjxn+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.
Fit a seasonal ARIMA model of your choice to the U.S. Live Birth Series(birth). Use the estimated model to forecast the next 12 months.
Fit a seasonal ARIMA model of your choice to the unemployment data(unemp) displayed in Figure 3.21. Use the estimated model to forecast the next 12 months.
Plot (or sketch) the ACF of the seasonal ARIMA(0, 1) × (1, 0)12 model with Φ = .8 and θ = .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 = X∞k=0πkxt−k.(c) Develop equations for the m-step ahead forecast, xen+m,
One of the series collected along with particulates, temperature, and mortality described in Example 2.2 is the sulfur dioxide series, so2. Fit an ARIMA(p,d, q) model to the data, performing all of the necessary diagnostics.After deciding on an appropriate model, forecast the data into the future
Fit an ARIMA(p,d, q) model to the global temperature data gtemp 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; see Appendix R for more details. Fit an ARIMA(p,d, q) model to the growth rate performing all necessary diagnostics. Comment.
In Example 3.39, 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 3.32, use the first 100 observations and calculate the EWMA, xet t+1, given in(3.150) 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, yn n+j = δ [1 + φ + · · · + φj−1] + φj yn.(b) Use part (a) to show that, for m = 1, 2, . . . , xn n+m = xn +δ1 − φh m − φ(1
Verify that the IMA(1,1) model given in (3.147) can be inverted and written as (3.148).
Suppose yt = β0 + β1t + · · · + βqt q + xt, βq 6= 0, where xt is stationary. First, show that ∇kxt is stationary for any k = 1, 2, . . . , and then show that ∇kyt is not stationary for k < q, but is stationary for k ≥ q.
Forecasting with estimated parameters: Let x1, x2, . . . , xn be a sample of size n from a causal AR(1) process, xt = φxt−1+wt. Let φb be the Yule–Walker estimator of φ.(a) Show φb − φ = Op(n−1/2). See Appendix A for the definition of Op(·).(b) Let xn n+1 be the one-step-ahead
A problem of interest in the analysis of geophysical time series involves a simple model for observed data containing a signal and a reflected version of the signal with unknown amplification factor a and unknown time delay δ. For example, the depth of an earthquake is proportional to the time
Consider the stationary series generated by xt = α + φxt−1 + wt + θwt−1, where E(xt) = µ, |θ| < 1, |φ| < 1 and the wt are iid random variables with zero mean and variance σ2 w.(a) Determine the mean as a function of α for the above model. Find the autocovariance and ACF of the process
as your guide, find the Gauss–Newton procedure for estimating the autoregressive parameter, φ, from the AR(1) model, xt = φxt−1 + wt, given data x1, . . . , xn. Does this procedure produce the unconditional or the conditional estimator? Hint: Write the model as wt(φ) =xt − φxt−1; your
Using Example
Generate n = 50 observations from a Gaussian AR(1) model with φ =.99 and σw = 1. Using an estimation technique of your choice, compare the approximate asymptotic distribution of your estimate (the one you would use for inference) with the results of a bootstrap experiment (use B = 200).
Generate 10 realizations of length n = 200 each of an ARMA(1,1) process with φ = .9, θ = .5 and σ2 = 1. Find the MLEs of the three parameters in each case and compare the estimators to the true values.
Repeat the following numerical exercise three times. Generate n = 500 observations from the ARMA model given by xt = .9xt−1 + wt − .9wt−1, with wt ∼ iid N(0, 1). Plot the simulated data, compute the sample ACF and PACF of the simulated data, and fit an ARMA(1, 1) model to the data. What
Suppose x1, . . . , xn are observations from an AR(1) process with µ = 0.(a) Show the backcasts can be written as xn t = φ1−t x1, for t ≤ 1.(b) In turn, show, for t ≤ 1, the backcasted errors are wbt(φ) = xn t − φxn t−1 = φ1−t(1 − φ2)x1.(c) Use the result of (b) to show P1
Fit an AR(2) model to the cardiovascular mortality series (cmort) discussed in Chapter 2, Example 2.2. using linear regression and using Yule–Walker.(a) Compare the parameter estimates obtained by the two methods.(b) Compare the estimated standard errors of the coefficients obtained by linear
Verify statement (3.87), that for a fixed sample size, the ARMA prediction errors are correlated.Section 3.6
Consider the ARMA(1,1) model discussed in Example 3.7, equation(3.27); that is, xt = .9xt−1 + .5wt−1 + wt. Show that truncated prediction as defined in (3.91) is equivalent to truncated prediction using the recursive formula (3.92).
For an AR(1) model, determine the general form of the m-step-ahead forecast xt t+m and show E[(xt+m − xt t+m)2] = σ2 w1 − φ2m 1 − φ2 .
Suppose xt is stationary with zero mean and recall the definition of the PACF given by (3.55) and (3.56). That is, lett = xt −h X−1 i=1 aixt−i andδt−h = xt−h −h X−1 j=1 bjxt−j be the two residuals where {a1, . . . , ah−1} and {b1, . . . , bh−1} are chosen so that they minimize
In the context of equation (3.63), show that, if γ(0) > 0 and γ(h) → 0 as h → ∞, then Γn is positive definite.
Consider the MA(1) series xt = wt + θwt−1, where wt is white noise with variance σ2 w.(a) Derive the minimum mean-square error one-step forecast based on the infinite past, and determine the mean-square error of this forecast.(b) Let xen n+1 be the truncated one-step-ahead forecast as given in
Let xt represent the cardiovascular mortality series (cmort) discussed in Chapter 2, Example 2.2.(a) Fit an AR(2) to xt using linear regression as in Example 3.17.(b) Assuming the fitted model in (a) is the true model, find the forecasts over a four-week horizon, xn n+m, for m = 1, 2, 3, 4, and the
Generate n = 100 observations from each of the three models discussed in Problem 3.8. Compute the sample ACF for each model and compare it to the theoretical values. Compute the sample PACF for each of the generated series and compare the sample ACFs and PACFs with the general results given in
Verify the calculations for the autocorrelation function of an ARMA(1, 1)process given in Example 3.13. Compare the form with that of the ACF for the ARMA(1, 0) and the ARMA(0, 1) series. Plot (or sketch) the ACFs of the three series on the same graph for φ = .6, θ = .9, and comment on the
For the AR(2) series shown below, use the results of Example 3.9 to determine a set of difference equations that can be used to find the ACFρ(h), h = 0, 1, . . .; solve for the constants in the ACF using the initial conditions. Then plot the ACF values to lag 10 (use ARMAacf as a check on your
For the AR(2) model given by xt = −.9xt−2 + wt, find the roots of the autoregressive polynomial, and then sketch the ACF, ρ(h).
Verify the causal conditions for an AR(2) model given in (3.28). That is, show that an AR(2) is causal if and only if (3.28) holds.Section 3.3
Identify the following models as ARMA(p, q) models (watch out for parameter redundancy), and determine whether they are causal and/or invertible:(a) xt = .80xt−1 − .15xt−2 + wt − .30wt−1.(b) xt = xt−1 − .50xt−2 + wt − wt−1.
Verify the calculations made in Example 3.3:(a) Let xt = φxt−1 +wt where |φ| > 1 and wt ∼ iid N(0, σ2 w). Show E(xt) = 0 and γx(h) = σ2 wφ−2 φ−h/(1 − φ−2).(b) Let yt = φ−1yt−1 + vt where vt ∼ iid N(0, σ2 wφ−2) and φ and σw are as in part (a). Argue that yt is causal
Let wt be white noise with variance σ2 w and let |φ| < 1 be a constant.Consider the process x1 = w1, and xt = φxt−1 + wt, t = 2, 3, . .
For an MA(1), xt = wt + θwt−1, show that |ρx(1)| ≤ 1/2 for any numberθ. For which values of θ does ρx(1) attain its maximum and minimum?
Use two different smoothing techniques described in §2.4 to estimate the trend in the global temperature series displayed in Figure 1.2. Comment.
Consider the two weekly time series oil and gas. The oil series is in dollars per barrel, while the gas series is in cents per gallon; see Appendix R for details.(a) Plot the data on the same graph. Which of the simulated series displayed in§1.3 do these series most resemble? Do you believe the
Consider the model (2.46) used in Example 2.9, xt = Xn j=0β1(j/n) cos(2πtj/n) + β2(j/n) sin(2πtj/n).(a) Display the model design matrix Z [see (2.5)] for n = 4.(b) Show numerically that the columns of Z in part (a) satisfy part (d) and then display (Z0 Z)−1 for this case.(c) If x1, x2, x3, x4
In this problem, we will explore the periodic nature of St, the SOI series displayed in Figure 1.5.(a) Detrend the series by fitting a regression of St on time t. Is there a significant trend in the sea surface temperature? Comment.(b) Calculate the periodogram for the detrended series obtained in
The glacial varve record plotted in Figure 2.6 exhibits some nonstationarity that can be improved by transforming to logarithms and some additional nonstationarity that can be corrected by differencing the logarithms.(a) Argue that the glacial varves series, say xt, exhibits heteroscedasticity by
Show (2.31) is stationary.
Consider a process consisting of a linear trend with an additive noise term consisting of independent random variables wt with zero means and variancesσ2 w, that is, xt = β0 + β1t + wt, where β0, β1 are fixed constants.(a) Prove xt is nonstationary.(b) Prove that the first difference series
Model Selection. Both selection criteria (2.19) and (2.20) are derived from information theoretic arguments, based on the well-known Kullback-Leibler discrimination information numbers (see Kullback and Leibler, 1951, Kullback, 1958). We give an argument due to Hurvich and Tsai (1989). We think of
Kullback-Leibler Information. Given the random vector y, we define the information for discriminating between two densities in the same family, indexed by a parameter θ, say f(y; θ1) and f(y; θ2), as I(θ1; θ2) = 1 nE1 log f(y; θ1)f(y; θ2), (2.57)where E1 denotes expectation with respect to
Repeat the following exercise six times and then discuss the results. Generate a random walk with drift, (1.4), of length n = 100 with δ = .01 andσw = 1. Call the data xt for t = 1, . . . , 100. Fit the regression xt = βt + wt using least squares. Plot the data, the mean function (i.e., µt =
For the mortality data examined in Example 2.2:(a) Add another component to the regression in (2.25) that accounts for the particulate count four weeks prior; that is, add Pt−4 to the regression in(2.25). State your conclusion.(b) Draw a scatterplot matrix of Mt, Tt, Pt and Pt−4 and then
For the Johnson & Johnson data, say yt, shown in Figure 1.1, let xt =log(yt).(a) Fit the regression model xt = βt + α1Q1(t) + α2Q2(t) + α3Q3(t) + α4Q4(t) + wt where Qi(t) = 1 if time t corresponds to quarter i = 1, 2, 3, 4, and zero otherwise. The Qi(t)’s are called indicator variables. We
Let {xt; t = 0, ±1, ±2, . . .} be iid(0, σ2).(a) For h ≥ 1 and k ≥ 1, show that xtxt+h and xsxs+k are uncorrelated for all s 6= t.(b) For fixed h ≥ 1, show that the h × 1 vectorσ−2n−1/2Xn t=1(xtxt+1, . . . , xtxt+h)0 d→ (z1, . . . , zh)0 where z1, . . . , zh are iid N(0, 1) random
For a linear process of the form xt = X∞j=0φjwt−j where {wt} satisfies the conditions of Theorem A.7 and |φ| < 1, show that √n(ρbx(1) − ρx(1))p1 − ρ2 x(1)d → N(0, 1), and construct a 95% confidence interval for φ when ρbx(1) = .64 and n = 100.
Let xt be a linear process of the form (A.43)–(A.44). If we defineγ˜(h) = n−1Xn t=1(xt+h − µx)(xt − µx), show that n1/2γ˜(h) − γb(h)= op(1).Hint: The Markov Inequality P{|x| ≥ }
(a) Suppose xt is a weakly stationary time series with mean zero and with absolutely summable autocovariance function, γ(h), such that X∞h=−∞γ(h) = 0.Prove that √n x¯ p→ 0, where ¯x is the sample mean (1.32).(b) Give an example of a process that satisfies the conditions of part (a).
Suppose xt = β0 +β1t, where β0 and β1 are constants. Prove as n → ∞,ρbx(h) → 1 for fixed h, where ρbx(h) is the ACF (1.35).
A concept used in geostatistics, see Journel and Huijbregts (1978) or Cressie (1993), is that of the variogram, defined for a spatial process xs, s = (s1, s2), for s1, s2 = 0, ±1, ±2, ..., as Vx(h) = 1 2E[(xs+h − xs)2], where h = (h1, h2), for h1, h2 = 0, ±1, ±2, ... Show that, for a
Consider a collection of time series x1t, x2t, . . . , xNt that are observing some common signal µt observed in noise processes e1t, e2t, . . . , eNt, with a model for the j-th observed series given by xjt = µt + ejt.Suppose the noise series have zero means and are uncorrelated for different j.
A real-valued function g(t), defined on the integers, is non-negative definite if and only if Xn i=1 Xn j=1 aig(ti − tj )aj ≥ 0 for all positive integers n and for all vectors a = (a1, a2, . . . , an)0 and t =(t1, t2, . . . , tn)0. For the matrix G = {g(ti − tj ); i, j = 1, 2, . . . , n},
For the time series yt described in Example 1.23, verify the stated result that ρy(1) = −.47 and ρy(h) = 0 for h > 1.
Simulate a series of n = 500 observations from the signal-plus-noise model presented in Example 1.12 with σ2 w = 1. Compute the sample ACF to lag 100 of the data you generated and comment.
Although the model in Problem 1.2(a) is not stationary (Why?), the sample ACF can be informative. For the data you generated in that problem, calculate and plot the sample ACF, and then comment.
(a) Simulate a series of n = 500 moving average observations as in Example 1.9 and compute the sample ACF, ρb(h), to lag 20. Compare the sample ACF you obtain to the actual ACF, ρ(h). [Recall Example 1.20.](b) Repeat part (a) using only n = 50. How does changing n affect the results?
(a) Simulate a series of n = 500 Gaussian white noise observations as in Example 1.8 and compute the sample ACF, ρb(h), to lag 20. Compare the sample ACF you obtain to the actual ACF, ρ(h). [Recall Example 1.19.](b) Repeat part (a) using only n = 50. How does changing n affect the results?
Suppose x1, . . . , xn is a sample from the process xt = µ + wt − .8wt−1, where wt ∼ wn(0, σ2 w).(a) Show that mean function is E(xt) = µ.(b) Use (1.33) to calculate the standard error of ¯x for estimating µ.(c) Compare (b) to the case where xt is white noise and show that (b) is
Suppose that xt is a linear process of the form (1.29). Prove X∞h=−∞|γ(h)| < ∞.Section 1.6
Suppose we have the linear process xt generated by xt = wt − θwt−1, t = 0, 1, 2, . . ., where {wt} is independent and identically distributed with characteristic function φw(·), and θ is a fixed constant. [Replace “characteristic function” with “moment generating function” if
Consider the series xt = sin(2πU t), t = 1, 2, . . ., where U has a uniform distribution on the interval (0, 1).(a) Prove xt is weakly stationary.(b) Prove xt is not strictly stationary. [Hint: consider the joint bivariate cdf(1.18) at the points t = 1, s = 2 with h = 1, and find values of ct, cs
Let wt, for t = 0, ±1, ±2, . . . be a normal white noise process, and consider the series xt = wtwt−1.Determine the mean and autocovariance function of xt, and state whether it is stationary.
Let xt be a stationary normal process with mean µx and autocovariance function γ(h). Define the nonlinear time series yt = exp{xt}.(a) Express the mean function E(yt) in terms of µx and γ(0). The moment generating function of a normal random variable x with mean µ and variance σ2 is Mx(λ) =
Consider the two series xt = wt yt = wt − θwt−1 + ut, where wt and ut are independent white noise series with variances σ2 w and σ2 u, respectively, and θ is an unspecified constant.(a) Express the ACF, ρy(h), for h = 0, ±1, ±2, . . . of the series yt as a function of σ2 w, σ2 u, and
For two weakly stationary series xt and yt, verify (1.28).
Consider the linear process defined in (1.29).(a) Verify that the autocovariance function of the process is given by (1.30).Use the result to verify your answer to Problem 1.7.(b) Show that xt exists as a limit in mean square (see Appendix A).
Suppose we would like to predict a single stationary series xt with zero mean and autocorrelation function γ(h) at some time in the future, say, t + `, for ` > 0.(a) If we predict using only xt and some scale multiplier A, show that the mean-square prediction error MSE(A) = E[(xt+` − Axt)2]is
A time series with a periodic component can be constructed from xt = U1 sin(2πω0t) + U2 cos(2πω0t), where U1 and U2 are independent random variables with zero means and E(U2 1 ) = E(U2 2 ) = σ2. The constant ω0 determines the period or time it takes the process to make one complete cycle.
Consider the random walk with drift model xt = δ + xt−1 + wt, for t = 1, 2, . . . , with x0 = 0, where wt is white noise with variance σ2 w.(a) Show that the model can be written as xt = δt + Pt k=1 wk.(b) Find the mean function and the autocovariance function of xt.(c) Argue that xt is not
For a moving average process of the form xt = wt−1 + 2wt + wt+1, where wt are independent with zero means and variance σ2 w, determine the autocovariance and autocorrelation functions as a function of lag h = s − t and plot the ACF as a function of h.
Consider the time series xt = β1 + β2t + wt, where β1 and β2 are known constants and wt is a white noise process with variance σ2 w.(a) Determine whether xt is stationary.(b) Show that the process yt = xt − xt−1 is stationary(c) Show that the mean of the moving average vt = 1 2q + 1 Xq
For the two series, xt, in Problem 1.2 (a) and (b):(a) Compute and plot the mean functions µx(t), for t = 1, . . . , 200.(b) Calculate the autocovariance functions, γx(s, t), for s, t = 1, . . . , 200.Section 1.5.
Show that the autocovariance function can be written asγ(s, t) = E[(xs − µs)(xt − µt)] = E(xsxt) − µsµt, where E[xt] = µt.
(a) Generate n = 100 observations from the autoregression xt = −.9xt−2 + wt with σw = 1, using the method described in Example 1.10, page 13. Next, apply the moving average filter vt = (xt + xt−1 + xt−2 + xt−3)/4 to xt, the data you generated. Now plot xt as a line and superimpose vt as
Consider a signal-plus-noise model of the general form xt = st + wt, where wt is Gaussian white noise with σ2 w = 1. Simulate and plot n = 200 observations from each of the following two models (Save the data or your code for use in Problem 1.22 ):(a) xt = st + wt, for t = 1, ..., 200, where st
To compare the earthquake and explosion signals, plot the data displayed in Figure 1.7 on the same graph using different colors or different line types and comment on the results. (The R code in Example 1.11 may be of help on how to add lines to existing plots.)
Consider the linear regression model y = β0 + β1x1 + β2x2 + ε , where the regressors have been coded so thata. Show that an orthogonal design ( X ′ X diagonal) minimizes the variance of ˆβ1 and ˆβ2 .b. Show that any design for fi tting this fi rst - order model that is orthogonal is also
An analyst is fi tting a simple linear regression model with the objective of obtaining a minimum - variance estimate of the intercept β0 . How should the data collection experiment be designed?
A Designed Experiment for Linear Regression. You wish to fi t a simple linear regression model over the region − 1 ≤ x ≤ 1 using n = 10 observations.Four experimental designs are under consideration: (i) 5 observations at x = − 1 and 5 observations at x = + 1, (ii) 4 observations at x = −
Consider the nonlinear regression model fi t to the data in Problem 12.11.Find approximate 95% bootstrap confi dence intervals for the regression coeffi cients ˆθ1, ˆθ2, and ˆθ3 using m = 1000 bootstrap samples. Compare these intervals to the ones based on the large - sample results. Based on
Consider the nonlinear regression model fi t to the data in Problem 12.11.Find the bootstrap standard errors for the regression coeffi cients ˆθ1, ˆθ2, andˆθ3 using m = 1000 bootstrap samples. Based on the results you obtain, comment on how the asymptotic theory seems to apply to this problem.
Describe how you would fi nd an approximate bootstrap confi dence interval on the mean response at a particular point, say x0 .
Describe how you would fi nd a bootstrap estimate of the standard deviation of the estimate of the mean response at a particular point, say x0 .
Consider the soft drink delivery time data in Example 3.1 . Find the bootstrap estimate of the standard deviation of ˆβ1 using the following numbers of bootstrap samples: m = 100, m = 200, m = 300, m = 400, and m = 500. Can you draw any conclusions about how many bootstrap samples are necessary
Consider the soft drink delivery time data in Example 3.1 . Find an approximate 95% bootstrap confi dence interval on the regression coeffi cient for distance using m = 1000 bootstrap samples. Compare this to the usual normal - theory confi dence interval.
Consider a regression model relating total heat fl ux to radial defl ection for the solar energy data in Table B.2 .a. Suppose that the observed total heat fl ux is 250 kW. Find a point estimate of the corresponding radial defl ection.b. Construct a 90% confi dence interval on radial defl ection.
Consider the regression model in Problem 2.4 relating gasoline mileage to engine displacement.a. If a particular car has an observed gasoline mileage of 17 miles per gallon, fi nd a point estimate of the corresponding engine displacement.b. Find a 95% confi dence interval on engine displacement.
Consider the regression model in Problem 2.10 relating systolic blood pressure to weight. Suppose that we wish to predict an individual ’ s weight given an observed value of systolic blood pressure. Can this be done using the procedure for predicting x given a value of y described in Section
Tukey ’ s Biwelght. A popular ψ function for robust regression is Tukey ’ s biweight, wherewith a = 5, 6 Sketch the ψ function for a = 5 and discuss its behavior. Do you think that Tukey ’ s biweight would give results similar to Andrews ’ wave function? y(z)= [z[1-(z/a)], |z|a |z|> a
Consider the continuous probability distribution f ( x ). Suppose that θ is an unknown location parameter and that the density may be written as f ( x − θ )for −∞ < θ < ∞ . Let x1 , x2 , . . . , x n be a random sample of size n from the density.a. Show that the maximum - likelihood
Consider the global surface air temperature anomaly data and the CO 2 concentration data in Table B.17. Fit a time series regression model to these data, using global surface air temperature anomaly as the response variable.Is there any indication of autocorrelation in the residuals? What
Consider the cosmetic sales data in Exercise 14.4. Fit the lagged variables regression models shown in Eq. (14.26) and (14.27) to these data. Compare these models with the results you obtained in Exercise 14.4 using the Cochrane – Orcutt procedure, and with the time series regression model from

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Introduction To Linear Regression Analysis 5th Edition Douglas C. Montgomery, Elizabeth A. Peck, G. Geoffrey Vining - Solutions

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