12-6 ASPECTS OF MULTIPLE REGRESSION MODELING

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MIND-EXPANDING EXERCISES

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12-74. Consider a multiple regression model with k regressors. Show that the test statistic for significance of regression can be written as 11 R k R 2 2 1n k

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(a) Show that the estimator is

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Suppose that n 20, k 4, and R2 0.90. If 0.05, what conclusion would you draw about the relationship between y and the four regressors 12-75. A regression model is used to relate a response y to k 4 regressors with n 20. What is the smallest value of R2 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 R2 is 12-76. Show that we can express the residuals from a multiple regression model as e (I H)y, where H X(X X) 1X . 12-77. Show that the variance of the ith residual ei in a multiple regression model is 11 hii 2 and that the 2 covariance between ei and ej is hij, where the h s are the elements of H X(X X) 1X . 12-78. Consider the multiple linear regression model y X . If denotes the least squares estimator of , show that R , where R 1X X2 1X . 12-79. Constrained Least Squares. Suppose we wish to find the least squares estimator of in the model y X subject to a set of equality constraints, say, T c.

where (X X) 1X y. (b) Discuss situations where this model might be appropriate. 12-80. Piecewise Linear Regression (I). Suppose that y is piecewise linearly related to x. That is, different linear relationships are appropriate over the intervals x x* and x* x . Show how indicator variables can be used to fit such a piecewise linear regression model, assuming that the point x* is known. 12-81. Piecewise Linear Regression (II). Consider the piecewise linear regression model described in Exercise 12-79. Suppose that at the point x* a discontinuity occurs in the regression function. Show how indicator variables can be used to incorporate the discontinuity 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 approach that could be used to fit the piecewise linear regression model.

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IMPORTANT TERMS AND CONCEPTS In the E-book, click on any term or concept below to go to that subject. All possible regressions Analysis of variance test in multiple regression Categorical variables as regressors Confidence interval on the mean response Extra sum of squares method Inference (test and intervals) on individual model parameters Influential observations Model parameters and their interpretation in multiple regression Outliers Polynomial terms in a regression model Prediction interval on a future observation Residual analysis and model adequacy checking Significance of regression Stepwise regression and related methods CD MATERIAL Ridge regression Nonlinear regression models

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12-2.3

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More about the Extra Sum of Squares Method (CD Only)

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The extra sum of squares method for evaluating the contribution of one or more terms to a model is a very useful technique. Basically, one considers how much the regression or model sum of squares increases upon adding terms to a basic model. The expanded model is called the full model, and the basic model is called the reduced model. Although the development in the text is quite general, Example 12-5 illustrates the simplest case, one in which there is only one additional parameter in the full model. In this case, the partial F-test based on the extra sum of squares is equivalent to a t-test. When there is more than one additional parameter in the full model, the partial F-test is not equivalent to a t-test. The extra sum of squares method is often used sequentially when tting a polynomial model, such as y

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