Chapter 15, Other Topics in the Use of Regression Analysis Video Solutions, Introduction to Linear Regression Analysis

Problem 1

The data below give the percentage share of market of a particular brand of canned peaches for the past 15 months and the relative selling price.
$$\begin{array}{ccc|rcc}\hline t & x_{t} & y_{t} & t & x_{t} & y_{t} \\\hline 1 & 100 & 15.93 & 9 & 85 & 16.60 \\2 & 98 & 16.26 & 10 & 83 & 17.16 \\3 & 100 & 15.94 & 11 & 81 & 17.77 \\4 & 89 & 16.81 & 12 & 79 & 18.05 \\5 & 95 & 15.67 & 13 & 90 & 16.78 \\6 & 87 & 16.47 & 14 & 77 & 18.17 \\7 & 93 & 15.66 & 15 & 78 & 17.25 \\8 & 82 & 16.94 & & & \\\hline\end{array}$$
a. Fit a simple linear regression model to these data. Plot the residuals versus time. Is there any. indication of autocorrelation?
b. Use the Durbin-Watson test to determine if there is positive autocorrelation in the errors. What are your conclusions?
c. Use one iteration of the Cochrane-Orcutt procedure to estimate the regression coefficients. Find the standard errors of these regression coefficients.
d. Is there positive autocórrelation remaining after the first iteration? Would you conclude that the iterative parameter estimation technique has been successful?

Sheryl Ezze

Numerade Educator

04:46

Problem 2

The data below give the monthly sales for a cosmetics manufacturer $\left(y_{t}\right)$ and the corresponding monthly sales for the entire industry $\left(x_{t}\right)$. The units of $x_{t}$ and $y_{t}$ are millions of dollars
a. Build a simple linear regression model relating company sales to industry sales. Plot the residuals against time. Is there any indication of autocorrelation?
b. Use the Durbin-Watson test to determine if there is positive autocorrelation in the errors. What are your conclusions?
c. Use one iteration of the Cochrane-Orcutt procedure to estimate the model parameters. Compare the standard error of these regression coefficients with the standard error of the least-squares estimates.
d. Test for positive autocorrelation following the first iteration. Has the iterative procedure -been successful?

Heather Duong

Numerade Educator

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Problem 3

Consider the simple linear regression model $y_{t}=\beta_{0}+\beta_{1} x+\varepsilon_{t},$ where the errors are generated by the second-order autoregressive process
$$\varepsilon_{t}=\rho_{1} \varepsilon_{t-1}+\rho_{2} \varepsilon_{t-2}+a_{t}$$
Discuss how the Cochrane-Orcutt iterative procedure could be used in this situation. What transformations would be used on the variables $y_{i}$ and $x_{i} ?$ How would you estimate the parameters $\rho_{1}$ and $\rho_{2} ?$

Oluwadamilola Ameobi

Numerade Educator

04:14

Problem 4

Consider the regression model with first-order autoregressive errors defined in Eq. (15.2). Derive the mean, variance, and covariance of the errors given in Eqs. (15.3a) to (15.3c).

Victor Salazar

Numerade Educator

06:53

Problem 5

Using the data in Problem 15.1 , define a new set of transformed variables as the first difference of the original variables, $x_{i}^{\prime}=x_{t}-x_{t-1}$ and $y_{t}^{\prime}=y_{t}-$ $y_{t-1}$. Regress $y_{t}^{\prime}$ on $x_{t}^{\prime}$ through the origin. Compare the estimate of the slope from this first-difference approach with the estimate obtained from the iterative method in Problem 15.1

Melvin Adkins

Numerade Educator

01:54

Problem 6

Consider the regression model in Problem 2.12 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 $15.3 ?$ In this particular application, how would you respond to the suggestion of building a regression model relating weight to systolic blood pressure?

James Kiss

Numerade Educator

03:12

Problem 7

Consider the regression model in Problem 2.4 relating gasoline mileage to engine displacement.
a. If a particuilar car has an observed gasoline mileage of 17 miles per gallon, find a point estimate of the corresponding engine displacement.
b. Find a $95 \%$ confidence interval on engine displacement.

Shu Naito

Numerade Educator

03:44

Problem 8

Consider a regression model relating total heat flux to radial deflection for the solar energy data in Table B.2.
a. Suppose that the observed total heat flux is $250 \mathrm{kW}$. Find a point estimate of the corresponding radial deflection.
b. Construct a $90 \%$ confidence interval on radial deflection.

Ankit Gupta

Numerade Educator

02:42

Problem 9

Consider the soft drink delivery time data in Example 3.1 . Find an approximate $95 \%$ bootstrap confidence interval on the regression coefficient for distance using $m=1000$ bootstrap samples. Compare this to the usual normal-theory confidence interval.

Shu Naito

Numerade Educator

02:42

Problem 10

Consider the soft drink delivery time data in Example 3.1 . Find the bootstrap estimate of the standard deviation of $\hat{\beta}_{1}$ 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 to obtain a reliable estimate of the precision of estimation for $\hat{\beta}_{1} ?$

Shu Naito

Numerade Educator

02:39

Problem 11

Describe how you would find a bootstrap estimate of the standard deviation of the estimate of the mean response at a particular point, say $\mathbf{x}_{0}$

Andy Wong

Numerade Educator

02:21

Problem 12

Describe how you would find an approximate bootstrap confidence interval on the mean response at a particular point, say $\mathbf{x}_{0}$

Monique Whittaker

Numerade Educator

03:48

Problem 13

Consider the nonlinear regression model fit to the data in Problem 13.11 Find the bootstrap standard errors for the regression coefficients $\hat{\theta}_{1}, \hat{\theta}_{2}$ and $\hat{\theta}_{3}$ using $m=1000$ bootstrap samples. Based on the results you obtain, comment on how the asymptotic theory seems to apply to this problem.

Heather Duong

Numerade Educator

06:33

Problem 14

Consider the nonlinear regression model fit to the data in Problem 13.11 Find approximate $95 \%$ bootstrap confidence intervals for the regression coefficients $\hat{\theta}_{1}, \hat{\theta}_{2},$ and $\hat{\theta}_{3}$ using $m=1000$ bootstrap samples. Compare these intervals to the ones based on the large-sample results. Based on the comparison of these intervals, comment on how the asymptotic theory seems to apply to this problem.

Heather Duong

Numerade Educator

01:15

Problem 15

Consider the NFL, team performance data in Table B.1. Construct a regression tree for this data set.

Coach Rye

Numerade Educator

08:34

Problem 16

A Designed Experiment for Linear Regression. You wish to fit a simple linear regression model over the region $-1 \leq x \leq 1$ using $n=10$ observa tions. Four experimental designs are under consideration: (i) 5 observations at $x=-1$ and 5 observations at $x=+1,$ (ii) 4 observations at $x=-1,2$ observations at $x=0,$ and 4 observations at $x=+1,$ (iii) 2 observations at $x=-1,-\frac{1}{2}, 0,+\frac{1}{2},$ and $+1,$ and (iv) 1 observation at $x=-1,-0.8$ $-0.6,-0.4,-0.2,+0.2,+0.4,+0.6,+0.8,$ and $+1 .$ For each of these designs, find the number of degrees of freedom available for evaluating pure error and testing lack of fit, the standard error of the slope (up to a constant $\sigma$ ), and the value of the determinant of $\mathbf{X}^{\prime} \mathbf{X} .$ Based on these analyses, which design would you select?

Lucas Finney

Numerade Educator

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Problem 17

An analyst is fitting a simple linear regression model with the objective of obtaining a minimum-variance estimate of the intercept $\beta_{0} .$ How should the data collection experiment be designed?

Shu Naito

Numerade Educator

00:51

Problem 18

Suppose that you are fitting a simple linear regression model that will be used to predict the mean response at a particular point such as $x_{0} .$ How should the data collection experiment be designed so that a minimum-variance estimate of the mean of $y$ at $x_{0}$ is obtained?

Maxime Rossetti

Numerade Educator

02:16

Problem 19

Consider the linear regression model $y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\varepsilon,$ where the regressors have been coded so that $$\sum_{i=1}^{n} x_{i 1}=\sum_{i=1}^{n} x_{i 2}=0 \quad \text { and } \quad \sum_{i=1}^{n} x_{i 1}^{2}=\sum_{i=1}^{n} x_{i 2}^{2}=n$$
a. Show that an orthogonal design (X'X diagonal) minimizes the variance of $\hat{\beta}_{1}$ and $\hat{\beta}_{2}$
b. Show that any design for fitting this first-order model that is orthogonal is also rotatable.

Rashmi Sinha

Numerade Educator