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principles of managerial statistics

Questions and Answers of Principles Of Managerial Statistics

2. Suppose that (21.1) holds with μ(r)=0.1(0.035 − r) and σ(r)=2.3r.(a) What is the expected value of rt given that rt−1 = 0.04?(b) What is the variance of rt given that rt−1 = 0.02?
1. A linear spline s(t) has knots at 1, 2, and 3. Also, s(0) = 1, s(1) = 1.3, s(2) = 5.5, s(4) = 6, and s(5) = 6.(a) What is s(0.5)?(b) What is s(3)?(c) What is 4 2 s(t) dt?
7. So far we have treated the sample mean vector and covariance matrix as fixed when considering the risk of a portfolio. Stated differently, estimation risk has been ignored. A methodology for
6. Expand the model in Exercise 5 so that 1,... is a GARCH(1,1) process.Revise the BUGS and R code of Exercise 5 to fit this expanded model.
5. One of the strength of fitting models by MCMC using BUGS is that a very wide range of models can be fit. As an example, in this exercise a regression model with MA(1) errors will be used. For
4. Continue the analysis in Example 20.16. Divide the data into 20 blocks of 250 days each, except that the last block will have only 212 days. Use each of the first 19 blocks as training data with
3. In the derivation of (20.51), it was stated that “{Y − E(μ|Y )} and{E(μ|Y ) − μ} are conditionally uncorrelated given Y .” Verify this statement.
2. Verify (20.26).
1. Show in Example 20.2 that the MAP estimator is 6/7.
6. This problem uses daily stock price data in the file Stock_Bond.csv on the book’s website. In this exercise, use only the first 500 prices on each stock.The following R code reads the data and
5. Suppose the risk measure R is VaR(α) for some α. Let P1 and P2 be two portfolios whose returns have a joint normal distribution with means μ1 and μ2, standard deviations σ1 and σ2, and
4. This exercise uses daily Microsoft price data in the msft.dat data set on the book’s website. Use the closing prices to compute daily returns.Assume that the returns are i.i.d., even though
3. Find a source of stock price data on the Internet and obtain daily prices for a stock of your choice over the last 1,000 days.(a) Assuming that the loss distribution is t, find the parametric
2. Assume that the loss distribution has a polynomial tail and an estimate of a is 3.1. If VaR(0.05) = $252, what is VaR(0.005)?
1. This exercise uses daily BMW returns in the bmwRet data set on the book’s website. For this exercise, assume that the returns are i.i.d., even though there may be some autocorrelation and
4. Compute the eigenvectors in Example 18.3 and offer an interpretation of the first two eigenvectors.
3. Verify equation (18.6).
2. Perform a statistical factor analysis of the returns in the data set midcapD.ts on the book’s website. How many factors did you select? Use(18.20) to estimate the covariance matrix of the
1. The file yields2009.csv on this book’s website contains daily Treasury yields for 2009. Perform a principal components analysis on changes in the yields. Describe your findings. How many
12. As an analyst, you have constructed 2 possible portfolios. Both portfolios have the same beta and expected return, but portfolio 1 was constructed with only technology companies whereas portfolio
11. Suppose there are three risky assets with the following betas and σ2 j when regressed on the market portfolio.
10. Suppose that the risk-free rate of interest is 0.07 and the expected rate of return on the market portfolio is 0.14. The standard deviation of the market portfolio is 0.12.(a) According to the
9. What is the beta of a portfolio if the expected return on the portfolio is E(RP ) = 15 %, the risk-free rate is μf = 6 %, and the expected return on the market is E(RM) = 12 %? Make the usual
8. Suppose there are two risky assets, call them C and D. The tangency portfolio is 60 % C and 40 % D. The expected yearly returns are 4 % and 6 % for assets C and D. The standard deviations of the
7. Suppose there are three risky assets with the following betas and σ2 j .j βj σ2 j1 0.9 0.010 2 1.1 0.015 3 0.6 0.011 Suppose also that the variance of RMt − μf t is 0.014.(a) What is the
6. Suppose that the riskless rate of return is 4 % and the expected market return is 12 %. The standard deviation of the market return is 11 %. Suppose as well that the covariance of the return on
5. True or false: The CAPM implies that investors demand a higher return to hold more volatile securities. Explain your answer.
4. Show that equation (17.16) follows from equation (7.8).
3. Suppose that the risk-free interest rate is 0.023, that the expected return on the market portfolio is μM = 0.10, and that the volatility of the market portfolio is σM = 0.12.(a) What is the
2. Suppose that the risk-free rate of interest is 0.03 and the expected rate of return on the market portfolio is 0.14. The standard deviation of the market portfolio is 0.12.(a) According to the
1. What is the beta of a portfolio if E(RP ) = 16 %, μf = 5.5 %, and E(RM) = 11 %?17.10 Exercises 513
6. Stocks 1 and 2 are selling for $100 and $125, respectively. You own 200 shares of stock 1 and 100 shares of stock 2. The weekly returns on these stocks have means of 0.001 and 0.0015,
5. Suppose one has a sample of monthly log returns on two stocks with sample means of 0.0032 and 0.0074, sample variances of 0.017 and 0.025, and a sample covariance of 0.0059. For purposes of
4. Let RP be a return of some type on a portfolio and let R1,..., RN be the same type of returns on the assets in this portfolio. Is RP = w1R1 + ··· + wN RN true if RP is a net return? Is this
3. (a) Suppose that stock A shares sell at $75 and stock B shares at $115.A portfolio has 300 shares of stock A and 100 of stock B. What are the weights w and 1 − w of stocks A and B in this
2. Suppose there are two risky assets, C and D, the tangency portfolio is 65 % C and 35 % D, and the expected return and standard deviation of the return on the tangency portfolio are 5 % and 7 %,
1. Suppose that there are two risky assets, A and B, with expected returns equal to 2.3 % and 4.5 %, respectively. Suppose that the standard deviations of the returns are √6 % and √11 % and that
4. Suppose that Y t = (Y1,t, Y2,t) is the bivariate AR(1) process in Example 15.2. Is Y t stationary? (Hint: See Sect. 13.4.4.)
3. Verify that in Example 15.2 Y1,t − λY2,t is stationary.
2. In (15.2) and (15.3) there are no constants, so that Y1,t − λY2,t is a stationary process with mean zero. Introduce constants into (15.2) and (15.3)and show how they determine the mean of Y1,t
1. Show that (15.4) implies that Y1,t−1 − λY2,t−1 is an AR(1) process with coefficient 1 + φ1 − λφ2.References 463
10. Consider the daily log returns on the S&P 500 index (GSPC). Begin by running the following commands in R, then answer the questions below for the series y.25 library(rugarch)26
9. This problem uses monthly observations of the two-month yield, that is, YT with T equal to two months, in the data set Irates in the Ecdat package. The rates are log-transformed to stabilize the
8. On Black Monday, the return on the S&P 500 was −22.8 %. Ouch! This exercise attempts to answer the question, “what was the conditional probability of a return this small or smaller on Black
7. Suppose that 1, 2,... is a Gaussian white noise process with mean 0 and variance 1, and at and yt are stationary processes such that at = σtt where σ2 t =2+0.3a2 t−1, and yt =2+0.6yt−1 +
6. Let Yt be a stock’s return in time period t and let Xt be the inflation rate during this time period. Assume the model Yt = β0 + β1Xt + δσt + at, (14.30)where at = t#1+0.5a2 t−1.
5. Suppose that# t is white noise with mean 0 and variance 1, that at = t 7 + a2 t−1/2, and that Yt =2+0.67Yt−1 + at.(a) What is the mean of Yt?(b) What is the ACF of Yt?(c) What is the ACF of
4. Let yt be the AR(1)+ARCH(1) model at = t#ω + αa2 t−1,(yt − μ) = φ(yt−1 − μ) + at, where t is i.i.d. WN(0,1). Suppose that μ = 0.4, φ = 0.45, ω = 1, andα1 = 0.3.
3. Suppose that t is an i.i.d. WN(0, 1) process, that at = t#1+0.35a2 t−1, and that yt =3+0.72yt−1 + at.(a) Find the mean of yt.(b) Find the variance of yt.(c) Find the autocorrelation function
2. Suppose that fX(x)=1/4 if |x| < 1 and fX(x)=1/(4x2) if |x| ≥ 1. Show that ∞−∞fX(x)dx = 1, so that fX really is a density, but that 0−∞xfX(x)dx = −∞and ∞0 xfX(x)dx = ∞, so
1. Let Z have an N(0, 1) distribution. Show that E(|Z|) = ∞−∞1√2π|z|e−z2/2dz = 2 ∞0 1√2πze−z2/2dz = 2π .Hint: d dz e−z2/2 = −ze−z2/2.
7. This exercise uses the TbGdpPi.csv data set. In Sect. 12.15.1, nonseasonal models were fit. Now use auto.arima() to find a seasonal model. Which seasonal model is selected by AIC and by BIC? Do
6. (a) Find an ARIMA model that provides a good fit to the variable unemp in the USMacroG data set in the AER package.(b) Now perform a small model-based bootstrap to see how well auto.arima() can
5. Fit an ARIMA model to income, which is in the first column of the IncomeUK data set in the Ecdat package. Explain why you selected the model you did. Does your model exhibit any residual
4. In Example 13.10, a bivariate AR(1) model was fit to (Δcpi, Δip) andΦ = 0.767 0.0112−0.330 0.3014 .
3. Figure 13.22 contains ACF plots of 40 years of quarterly data, with all possible combinations of first-order seasonal and nonseasonal differencing.Which combination do you recommend in order to
2. Figure 13.21 contains ACF plots of 40 years of quarterly data, with all possible combinations of first-order seasonal and nonseasonal differencing.Which combination do you recommend in order to
1. Figure 13.20 contains ACF plots of 40 years of quarterly data, with all possible combinations of first-order seasonal and nonseasonal differencing.Which combination do you recommend in order to
17. In Sect. 12.9.1, it was stated that “if E(Yt) has an mth-degree polynomial trend, then the mean of E(ΔdYt) has an (m−d)th-degree trend for d ≤ m.For d>m, E(ΔdYt) = 0.” Prove these
16. Suppose you fit an AR(2) model to a time series Yt, t = 1,...,n, and the estimates were μ = 100.1, φ1 = 0.5, and φ2 = 0.1. The last three observations were Yn−2 = 101.0, Yn−1 = 99.5, and
15. This problem fits an ARIMA model to the logarithms monthly one-month T-bill rates in the data set Mishkin in the Ecdat package. Run the following code to get the variable:15 library(Ecdat)16
14. To decide the value of d for an ARIMA(p,d, q) model for a time series y, plots were created using the R program:8 par(mfrow=c(3,2))9 plot(y,type="l")10 acf(y)11 plot(diff(y),type="l")12
13. The ARMA(1,2) model Yt = μ + φ1Yt−1 + t + θ1t−1 + θ2t−2 was fit to data and the estimates are Parameter Estimateμ 103φ1 0.2θ1 0.4θ2 −0.25 The last two values of the observed
12. The MA(2) model Yt = μ + t + θ1t−1 + θ2t−2 was fit to data and the estimates are
11. The time series in the middle and bottom panels of Fig. 12.14 are both nonstationary, but they clearly behave in different manners. The time series in the bottom panel exhibits “momentum” in
10. For a univariate, discrete time process, what is the difference between a strictly stationary process and a weakly stationary process?
9. Show that if wt is defined by (12.34) then (12.35) is true.
8. Use (12.11) to verify Eq. (12.12).
7. Let Yt be a stationary AR(2) process,(Yt − μ) = φ1(Yt−1 − μ) + φ2(Yt−2 − μ) + t.(a) Show that the ACF of Yt satisfies the equationρ(k) = φ1ρ(k − 1) + φ2ρ(k − 2)for all
6. Let Yt be an MA(2) process, Yt = μ + t + θ1t−1 + θ2t−2.Find formulas for the autocovariance and autocorrelation functions of Yt.
5. An AR(3) model has been fit to a time series. The estimates are ˆμ = 104,φˆ1 = 0.4, φˆ2 = 0.25, and φˆ3 = 0.1. The last four observations were Yn−3 = 105, Yn−2 = 102, Yn−1 = 103, and
4. Suppose that Y1, Y2,... is an AR(1) process with μ = 0.5, φ = 0.4, andσ2= 1.2.(a) What is the variance of Y1?(b) What are the covariances between Y1 and Y2 and between Y1 and Y3?(c) What is the
3. Consider the AR(1) model Yt = 5 − 0.55Yt−1 + t and assume that σ2= 1.2.(a) Is this process stationary? Why or why not?(b) What is the mean of this process?(c) What is the variance of this
2. Next, fit AR(1) and AR(2) models to the CRSP returns:6 arima(crsp,order=c(1,0,0))7 arima(crsp,order=c(2,0,0))(a) Would you prefer an AR(1) or an AR(2) model for this time series?Explain your
1. This problem and the next use CRSP daily returns. First, get the data and plot the ACF in two ways:1 library(Ecdat)2 data(CRSPday)3 crsp=CRSPday[,7]4 acf(crsp)5 acf(as.numeric(crsp))(a) Explain
4. Least-squares estimators are unbiased in linear models, but in nonlinear models they can be biased. Simulation studies (including bootstrap resampling) can be used to estimate the amount of bias.
3. The maturities (T) in years and prices in dollars of zero-coupon bonds are in file ZeroPrices.txt on the book’s website. The prices are expressed
2. Suppose one has a long position of F20 face value in 20-year Treasury bonds and wants to hedge this with short positions in both 10- and 30-year Treasury bonds. The prices and durations of 10-,
1. When we were finding the best linear predictor of Y given X, we derived the equations 0 = −E(Y ) + β0 + β1E(X)0 = −E(XY ) + β0E(X) + β1E(X2).Show that their solution isβ1 = σXYσ2
5. It was noticed that a certain observation had a large leverage (hat diagonal) but a small Cook’s D. How could this happen?
4. Residual plots and other diagnostics are shown in Fig. 10.13 for a regression of Y on X. Describe any problems that you see and possible remedies.
3. Residual plots and other diagnostics are shown in Fig. 10.12 for a regression of Y on X. Describe any problems that you see and possible remedies.
2. Residual plots and other diagnostics are shown in Fig. 10.11 for a regression of Y on X. Describe any problems that you see and possible remedies.
1. Residual plots and other diagnostics are shown in Fig. 10.10 for a regression of Y on X. Describe any problems that you see and possible remedies.
10. Pairs of random variables (Xi, Yi) were observed. They were assumed to follow a linear regression with E(Yi|Xi) = θ1+θ2Xi but with t-distributed noise, rather than the usual normally
9. Complete the following ANOVA table for the model Yi = β0 + β1Xi,1 +β2Xi,2 + i:Source df SS MS F P Regression ? ? ? ? 0.04 Error ? 5.66 ?Total 15 ?R-sq = ?
8. Sometimes it is believed that β0 is 0 because we think that E(Y |X = 0) =0. Then the appropriate model is yi = β1Xi + i.This model is usually called “regression through the origin” since
7. The quadratic polynomial regression model Yi = β0 + β1Xi + β2X2 i + i was fit to data. The p-value for β1 was 0.67 and for β2 was 0.84. Can we accept the hypothesis that β1 and β2 are both
6. A data set has 66 observations and five predictor variables. Three models are being considered. One has all five predictors and the others are smaller.Below is residual error SS for all three
5. A linear regression model with three predictor variables was fit to a data set with 40 observations. The correlation between Y and Y was 0.65. The total sum of squares was 100.(a) What is the
4. It was stated in Sect. 9.8 that centering reduces collinearity. As an illustration, consider the example of quadratic polynomial regression where X takes 30 equally spaced values between 1 and
3. Use (7.11), (9.3), and (9.2) to show that (9.8) holds.
2. Show that if 1,...,n are i.i.d. N(0, σ2), then in straight-line regression the least-squares estimates of β0 and β1 are also the maximum likelihood estimates.Hint: This problem is similar to
1. Suppose that Yi = β0 + β1Xi + i, where i is N(0, 0.3), β0 = 1.4, andβ1 = 1.7.(a) What are the conditional mean and standard deviation of Yi given that Xi = 1? What is P(Yi ≤ 3|Xi = 1)?(b)
10. Suppose Y = (Y1,...,Yd) has a meta-Gaussian distribution with continuous marginal distributions and copula CGauss(·|Ω). Show that ifρτ (Yi, Yj ) = 0 then Yi and Yj are independent.
9. A convex combination of k joint CDFs is itself a joint CDF (finite mixture), but is a convex combination of k copula functions a copula function itself?
8. Let ϕ(u|θ) = (1 − u)θ, for some θ ≥ 1, and show that for the twodimensional case this generates the copula C(u1, u2|θ) = max[0, 1 − {(1 − u1)θ + (1 − u2)θ}1/θ].Further, show that
7. Suppose that ϕ1,...,ϕk are k strict generator functions and define a new generator ϕ as a convex combination of these k generators, that isϕ(u) = a1ϕ1(u) + ··· + akϕk(u), in which
6. Show that as θ → ∞, CFr(u1, u2|θ) → min(u1, u2), the co-monotonicity copula C+.
5. Show that the generator of a Frank copulaϕFr(u|θ) = − loge−θu − 1 e−θ − 1, θ ∈ {(−∞, 0) ∪ (0,∞)}, satisfies assumptions 1–3 of a strict generator.

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