Cost Benefit Analysis Concepts And Practice 5th Edition Anthony E. Boardman, David H. Greenberg, Aidan R. Vining, David L. Weimer - Solutions

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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 φˆ be the Yule–Walker estimator of φ.(a) Show φˆ − φ = Op(n−1/2). See Appendix A for the definition of Op(·).(b) Let x nn+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
Using Example
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 x nt = φ1−t x1, for t ≤ 1.(b) In turn, show, for t ≤ 1, the backcasted errors are w˜t(φ) = x nt − φx nt−1 = φ1−t(1 − φ2)x1.(c) Use the result of (b) to show Í1
Fit an AR(2) model to the cardiovascular mortality series (cmort) discussed in 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 regression
Verify statement (3.87), that for a fixed sample size, the ARMA prediction errors are correlated.Section 3.5
Consider the ARMA(1,1) model discussed in Example 3.8, 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).
Suppose we wish to find a prediction function g(x) that minimizes MSE = E[(y − g(x))2], where x and y are jointly distributed random variables with density function f (x, y).(a) Show that MSE is minimized by the choice g(x) = E(y x).Hint:MSE = EE[(y − g(x))2| x].(b) Apply the above result to
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 −Õh−1 i=1 ai xt−i and δt−h = xt−h −Õh−1 j=1 bj xt−j be the two residuals where {a1, . . ., ah−1} and {b1, . . ., bh−1} are chosen so that they
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 x˜n n+1 be the truncated one-step-ahead forecast as given in
Let xt represent the cardiovascular mortality series (cmort) discussed in Example 2.2.(a) Fit an AR(2) to xt using linear regression as in Example 3.18.(b) Assuming the fitted model in (a) is the true model, find the forecasts over a fourweek horizon, x nn+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 3.14. Compare the form with that of the ACF for the ARMA(1, 0)and the ARMA(0, 1) series. Plot the ACFs of the three series on the same graph forφ = .6, θ = .9, and comment on the diagnostic
For the AR(2) series shown below, use the results of Example 3.10 to determine a set of difference equations that can be used to find the ACF ρ(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 plot 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.
Verify the calculations made in Example 3.4 as follows.(a) Let xt = φxt−1 + wt where |φ| > 1 and wt ∼ iid N(0, σ2 w). Show E(xt) = 0 andγx(h) = σ2 wφ−2 φ−h/(1 − φ−2) for h ≥ 0.(b) Let yt = φ−1 yt−1 + vt where vt ∼ iid N(0, σ2 wφ−2) and φ and σw are as in part
Let {wt; t = 0, 1, . . . } be a white noise process with variance σ2 w and let |φ| < 1 be a constant. Consider the process x0 = w0, and xt = φxt−1 + wt, t = 1, 2, . . . .We might use this method to simulate an AR(1) process from simulated white noise.(a) Show that xt =Ít j=0φjwt−j for any
Use two different smoothing techniques described in Section 2.3 to estimate the trend in the global temperature series globtemp. 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.(a) Plot the data on the same graph. Which of the simulated series displayed in Section 1.2 do these series most resemble? Do you believe the series are
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 2.7 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
Consider a process consisting of a linear trend with an additive noise term consisting of independent random variables wt with zero means and variances σ2 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 (2.15) and (2.16) 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 n × 1 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) = n−1 E1 log f (y; θ1)f (y; θ2), (2.41)where E1 denotes expectation with
In this problem, we explore the difference between a random walk and a trend stationary process.(a) Generate four series that are random walk with drift, (1.4), of length n = 100 with δ = .01 and σw = 1. Call the data xt for t = 1, . . ., 100. Fit the regression xt = βt + wt using least squares.
For the mortality data examined in Example 2.2:(a) Add another component to the regression in (2.21) that accounts for the particulate count four weeks prior; that is, add Pt−4 to the regression in (2.21). State your conclusion.(b) Draw a scatterplot matrix of Mt, Tt, Pt and Pt−4 and then
A Structural Model For the Johnson & Johnson data, say yt, shown in Figure 1.1, let xt = log(yt). In this problem, we are going to fit a special type of structural model, xt = Tt + St + Nt where Tt is a trend component, St is a seasonal component, and Nt is noise. In our case, time t is in quarters
Let {xt; t = 0, ±1, ±2, . . .} be iid(0, σ2).(a) For h ≥ 1 and k ≥ 1, show that xt xt+h and xs xs+k are uncorrelated for all s , t.(b) For fixed h ≥ 1, show that the h × 1 vectorσ−2 n−1/2Õn t=1(xt xt+1, . . ., xt xt+h)0 d→ (z1, . . ., zh)0 where z1, . . ., zh are iid N(0, 1)
For a linear process of the form xt =Õ∞j=0φjwt−j, where {wt } satisfies the conditions of Theorem A.7 and |φ| < 1, show that√n(ρˆx(1) − ρx(1))p 1 − ρ2 x(1)d→ N(0, 1), and construct a 95% confidence interval for φ when ρˆx(1) = .64 and n = 100.
Let xt be a linear process of the form (A.43)–(A.44). If we defineγ˜(h) = n−1Õn t=1(xt+h − µx)(xt − µx), show that n1/2γ˜(h) − γˆ(h)= op(1).Hint: The Markov Inequality Pr{|x| ≥ }
(a) Suppose xt is a weakly stationary time series with mean zero and with absolutely summable autocovariance function, γ(h), such thatÕ∞h=−∞γ(h) = 0.Prove that √n x¯p→ 0, where x¯ is the sample mean (1.34).(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 → ∞,ρˆx(h) → 1 for fixed h, where ρˆx(h) is the ACF (1.37).
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 stationary
Consider a collection of time series x1t, x2t, . . ., xN t that are observing some common signal µt observed in noise processes e1t, e2t, . . ., eN t, 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 ifÕn i=1Õn 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}, this
For the time series yt described in Example 1.26, verify the stated result thatρy(1) = −.47 and ρy(h) = 0 for h > 1.
(a) Simulate a series of n = 500 moving average observations as in Example 1.9 and compute the sample ACF, ρˆ(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, ρˆ(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 xt = µ + wt + θwt−1, where wt ∼ wn(0, σ2 w).(a) Show that mean function is E(xt) = µ.(b) Show that the autocovariance function of xt is given by γx(0) = σ2 w(1 + θ2),γx(±1) = σ2 wθ, and γx(h) = 0 otherwise.(c) Show that xt is stationary for all values of θ ∈ R.(d) Use
Suppose that xt is a linear process of the form (1.31). ProveÕ∞h=−∞|γ(h)| < ∞.Section 1.5
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 instructed
Consider the series xt = sin(2πUt), 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.
For two weakly stationary series xt and yt, verify (1.30).
Consider the linear process defined in (1.31).(a) Verify that the autocovariance function of the process is given by (1.32). Use the result to verify your answer to Problem 1.7. Hint: For h ≥ 0, cov(xt+h, xt) =cov(Ík ψkwt+h−k,Íj ψjwt−j). For each j ∈ Z, the only “survivor” will be
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 +Ít k=1 wk .(b) Find the mean function and the autocovariance function of xt.(c) Argue that xt is not
Consider the time series xt = β1 + β2t + wt, where β1 and β2 are known constants and wt is a white noise process with varianceσ2 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 + 1Õq
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.
(a) Generate n = 100 observations from the autoregression xt = −.9xt−2 + wt with σw = 1, using the method described in Example 1.10. 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 a dashed
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.(a) xt = st + wt, for t = 1, ..., 200, where st =0, t = 1, . . ., 100 10 exp{−(t−100)20 }
It may seem strange to partition in this way because one would normally think of existence as being a prerequisite for use. Some types of uses, however, could be provided without maintaining existence value. For example, through stocking it may be possible to provide game fishing without preserving
Depending on the slopes of the indifference curves, the consumption of good Y could either increase or decrease. As shown in Figure 3A.1(a), it slightly decreases in this particular example.Figure 3A.1(a) G 0 Price Po Pa 0 0 a Uo K U₁ H X Hicksian compensated variation demand curve Marshallian
In calculus notation, this decomposition can be represented as follows:This equation is known as the Slutsky equation. The first term to the right of the equal sign is the substitution effect, where utility is held constant. The second term is the income effect, where prices are held constant, and
One way of doing this first requires obtaining estimates of the relation between quantity purchased and prices and the relation between quantity purchased and income and then (as implied by the preceding note) using the Slutsky equation to derive the income-compensated (rather than the
For analyses of the size of the bias, see Ian J. Irvine and William A. Sims, “Measuring Consumer Surplus with Unknown Hicksian Demands.” American Economic Review, 88(1), 1998, 314–22; Robin W. Boadway and Neil Bruce Welfare Economics (Oxford: Basil Blackwell, 1984), 216–19; Julian M. Alston
Specifically, using compensating variation is only theoretically correct if consumers have homothetic preferences (i.e., the slopes of all indifferent curves are constant along any ray from the origin), which implies that each good in a consumer’s utility function has an income elasticity of one.
For a clear introduction to regression analysis, see Jeffrey M. Wooldridge, Introductory Econometrics: A Modern Approach (Boston, MA: South-Western Publishing, 2000).
See Greene, Econometric Analysis. For an application that presents both OLS and ridge regression estimates, see A. E. Boardman, S. Miller, and A. P. Schinnar, “Efficient Employment of Cohorts of Labor in the U.S. Economy: An Illustration of a Method.” Socio-Economic Planning Sciences, 13(6),
The OLS estimators are also efficient. Here, efficiency means that among all estimators whose formulas are linear in the dependent variable, the OLS estimator has the smallest variance. It therefore has the greatest “power” for rejecting the null hypothesis that a coefficient is zero.
Assuming all benefits arise at the end of each year. pv (B)= Σ. (1+1) B(1+g) Σ’ -Σ Setting i = (i - g)/(1 + g) = implies 1/(1 + i) = (1 + g)/(1 + i). Therefore, B₁ 1+g di=1(1+g){1+i " PV (B) = Σ t=1 1 B₁ *(1+g) (1+i) B₁ α (1+g)
The central limit theorem tells us that the distribution of the sum of independent random variables approaches the normal distribution as the number in the sum becomes large, The theorem applies for almost any starting distributions with finite variances. If we think of the error term as the sum of
Although we use compensating variation as a measure of the dollar value of utility here because we find it most intuitive (as discussed in the appendix to Chapter 3), equivalent variation, which measures welfare changes relative to the utility level after the price change, is generally considered a
Student is the pseudonym of William Gosset, a quality control engineer at the Guinness Brewery in Dublin, who originally derived the t distribution.
The probability we choose puts an upward bound on the probability of falsely rejecting the null hypothesis. Falsely rejecting the null hypothesis is referred to as Type I error. Failing to reject the null hypothesis when in fact the alternative hypothesis is true is referred to as Type II error. We
A more formal treatment can be found in Alan Randall, “Total and Nonuse Values,” in John B. Braden and Charles D. Kolstad (editors), Measuring the Demand for Environmental Quality (New York, NY: North-Holland, 1991), 303–21. The original analysis of the importance of the sequence of valuation
Of course, the value of TEV will depend on whether it is measured in terms of compensating variation or equivalent variation.
John P. Hoehn and Alan Randall, “Too Many Proposals Pass the Benefit Cost Test.” American Economic Review, 79(3), 1989, 544–51.
If E is a public good, then we would interpret pe as the tax per unit of E that the person pays. As everyone would have to consume the same quantity, the person would not be able to choose the value of E so that the optimization of utility would be over only X and Z.
The partial derivatives of L with respect to E, X, Z, and λ give four equations with four unknowns. Although these first-order conditions cannot be solved analytically, they can be rearranged so that E, X, and Z are each expressed as a function of λ and parameters. A solution can be found by
If we assume that E is fixed at its initial level, the more realistic case in evaluating an existing wilderness area, then the existence value equals only $5.47. It is smaller than in the example because the person does not have the opportunity to purchase more E when X is not available. This

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Cost Benefit Analysis Concepts And Practice 5th Edition Anthony E. Boardman, David H. Greenberg, Aidan R. Vining, David L. Weimer - Solutions

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