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Unformatted text preview: Indicator Variables STAT 563 Spring 2007 General Concept • Generally multiple regression accommodates only quantitative variables • Dummy (or indicator) variables are useful in incorporating qualitative (or categorical) variables in the model 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 Hypothetical Example Explanation • In both cases, the withingender 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 withingender regressions; ignoring gender inflates the size of the errors, however. 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 withingender regressions have positive slopes. 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 Commonslope model • Formulate the commonslope 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 1 i i i i i X X Y ε β α ε γ β α + + = + + + = ) ( 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 ( Parameters in the additive dummy regression model 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 Interpretation • Interpretation of the parameters: – Here γ is the difference between intercepts for the two regression lines • Because these regression lines are parallel,...
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