12-6 ASPECTS OF MULTIPLE REGRESSION MODELING 467MIND-EXPANDING EXERCISESIMPORTANT TERMS AND CONCEPTS
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All possible regressions
Analysis of variance
test in multiple
regression
Categorical variables as
regressorsConfidence interval on
the mean response
Extra sum of squares
method
Inference (test and
intervals) on individ-
ual model parameters
Influential observations
Model parameters and
their interpretationin multiple
regression
Outliers
Polynomial terms in a
regression model
Prediction interval on a
future observation
Residual analysis and
model adequacy
checkingSignificance of
regression
Stepwise regression and
related methods
CD MATERIAL
Ridge regression
Nonlinear regression
models12-74. Consider a multiple regression model with kre-
gressors. Show that the test statistic for significance of
regression can be written asSuppose that n20, k4, and R^2 0.90. If 0.05,
what conclusion would you draw about the relationship
between yand the four regressors?
12-75. A regression model is used to relate a response
yto k4 regressors with n20. What is the smallest
value of R^2 that will result in a significant regression if
0.05? Use the results of the previous exercise. Are
you surprised by how small the value of R^2 is?
12-76. Show that we can express the residuals from a
multiple regression model as e(IH)y, where H
X(XX)^1 X.
12-77. Show that the variance of the ith residual eiin
a multiple regression model is and that the
covariance between eiand ejis ^2 hij, where the h’s are
the elements of HX(XX)^1 X.
12-78. Consider the multiple linear regression model
yX .If denotes the least squares estimator of
,show that where.
12-79. Constrained Least Squares.Suppose we wish
to find the least squares estimator of in the model y
Xsubject to a set of equality constraints, say,
Tc.(a) Show that the estimator isT[T(XX)–^1 T]–^1 (cT )where (XX)–^1 Xy.
(b) Discuss situations where this model might be appro-
priate.
12-80. Piecewise Linear Regression (I).Suppose
that yis piecewise linearly related to x.That is, different
linear relationships are appropriate over the intervals
and .Show how indicator
variables can be used to fit such a piecewise linear re-
gression model, assuming that the point is known.
12-81. Piecewise Linear Regression (II).Consider
the piecewise linear regression model described in
Exercise 12-79. Suppose that at the point a disconti-
nuity occurs in the regression function. Show how
indicator variables can be used to incorporate the dis-
continuity into the model.
12-82. Piecewise Linear Regression (III).Consider
the piecewise linear regression model described in
Exercise 12-79. Suppose that the point x* is not known
with certainty and must be estimated. Suggest an ap-
proach that could be used to fit the piecewise linear
regression model.x*x* xx* x* x ˆˆˆcˆ 1 X¿X 2 ^1ˆR, R 1 X¿X 2 ^1 X¿ˆ¿ 11 hii 2¿ ¿F 0 R^2 k
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