Chapter08 - Indicator Variables STAT 563 Spring 2007...

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Indicator Variables STAT 563 Spring 2007
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General Concept Generally multiple regression accommodates only quantitative variables Dummy (or indicator) variables are useful in incorporating qualitative (or categorical) variables in the model
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Simplest Case Inclusion of one dichotomous and one quantitative predictor variable Assumptions Relationships are additive: the partial effect of each predictor is the same regardless of the specific value at which the other predictor variable is held constant The other assumptions of the regression model hold The motivation of including a dummy is same as including a continuous variable: To account more fully for the response variable by making the errors smaller To avoid biased assessment of the impact of predictor, as a consequence of omitting another one that is related to it
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Hypothetical Example
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Explanation In both cases, the within-gender regressors of income on education are parallel. Parallel regressions imply additive effects of education and gender on income In (a), gender and education are unrelated to each other; if we ignore gender and regress income on education alone, we obtain the same slope as is produced by the separate within-gender regressions; ignoring gender inflates the size of the errors, however.
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Explanation In (b), gender and education are unrelated to each other; if we regress income on education alone, we arrive at a biased assessment of the effect of education on income. The overall regression of income on education has a negative slope even though the within-gender regressions have positive slopes.
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Proposal Perform separate regressions for women and men. This is reasonable but has limitations Fitting separate regressions makes it difficult to estimate and test for gender differences in income Furthermore, if we can assume parallel regressions, then we can more efficiently estimate the common education slope by pooling sample data from both groups
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Common-slope model Formulate the common-slope model as – Where D i is called the dummy or indicator variable defined as For women the model can be written as i i i i D X Y ε γ β α + + + = = women for men for D i 0 1 i i i i i X X Y ε β α ε γ β α + + = + + + = ) 0 (
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Common Slope Model For men the model becomes These regression equations are graphed as: i i i i i X X Y ε β γ α ε γ β α + + + = + + + = ) ( ) 1 (
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Parameters in the additive dummy regression model
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Regressors and Explanatory Variables Difference between explanatory variables and regressors Gender is a qualitative explanatory variable with values male and female The dummy variable D is a regressor representing the explanatory variable gender The quantitative explanatory variable income and the regressor X are one and the same
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Interpretation Interpretation of the parameters: Here γ is the difference between intercepts for the two regression lines Because these regression lines are parallel, γ also represents
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