Chap15-SSM - 378 Chapter 15: Multiple Regression Model...

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378 Chapter 15: Multiple Regression Model Building CHAPTER 15 OBJECTIVES To be able to use quadratic terms in a regression model To be able to use transformed variables in a regression model To be able to measure the correlation among the independent variables To build a regression model using either the stepwise or best-subsets approach To understand the pitfalls involved in developing a multiple regression model OVERVIEW AND KEY CONCEPTS The Quadratic Regression Model In a quadratic regression model, the relationship between the dependent variable and one or more independent variable is a quadratic polynomial function. The quadratic regression model is useful when the residual plot reveals nonlinear relationship. The quadratic regression model: ± 2 01 1 2 1 ii i i YX X β ββε =+ + + ± The quadratic relationship can be between Y and more than one X variable as well. Testing for the overall significance of the quadratic regression model: ± The test for the overall significance of the quadratic regression model is exactly the same as testing for the overall significance of any multiple regression model. ± Test statistic: MSR F MSE = . Testing for quadratic effect: ± The hypotheses: 02 :0 H = (no quadratic effect) vs. 12 H (the quadratic term is needed) ± Test statistic: 2 22 b b t S = . ± This is a two-tail test with a left tail and a right-tail rejection region. Using Transformation in Regression Models The following three transformation models are often used to overcome violations of the homoscedasticity assumption, as well as to transform a model that is not linear in form into one that is linear. Square-root transformation: ± 011 i βε +
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Study Guide and Student’s Solutions Manual 379 ± The dependent variable is i Y and the independent variable is 1 i X . Transformed multiplicative model: ± 01 1 log log log log log ii k k i i YX X β ββ ε =+ + + + L ± The dependent variable is log i Y and the independent variables are 1 log i X , 2 log i X , etc. Transformed exponential model: ± 1 ln ln k k i i X + + + L ± The dependent variable is ln i Y and the independent variables are 1 i X , 2 i X , etc. Collinearity Some of the explanatory variables are highly correlated with each other. The collinear variables do not provide new information and it becomes difficult to separate the effect of such variables on the dependent variable. Y X 1 X 2 Large Overlap Overlap in variation of X 1 and X 2 is used in explaining the variation in Y but NOT in estimating and 1 2 Large Overlap Overlap reflects collinearity between X 1 and X 2 The values of the regression coefficient for the correlated variables may fluctuate drastically depending on which independent variables are included in the model.
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Chap15-SSM - 378 Chapter 15: Multiple Regression Model...

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