note07 - 1 Chapter 7 Qualitative Variables Previous...

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1 Chapter 7 Qualitative Variables Previous discussion and examples considered only continuous variables. We now consider discrete variables that represent categorical (qualitative) differences across individuals such as races (white vs nonwhite), gender (male vs female), etc. These differences are usually captured by binary variables , or dummy variables . Binary variable - Dummy variable . This is a variable that takes two values, typically either 0 or 1, as in the case of tossing a coin and recording the result by 1 for head and 0 for tail. Example. gender classification Example. race classification Example. marital status: 1 if married and 0 if not. Dummy Variable as an Explanatory Variable Consider a linear regression model Suppose our data includes both males and females and we are interested in whether there is any difference in the regression relationship between the two groups. There are three types of differences that can be considered. (a) Intercept term is different between the two groups, but the slope coefficient is the same between the two groups. (b) Intercept term is the same between the two groups, but the slope coefficient is different between the two groups. (c) All coefficients are different between the two groups.

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2 Case (a): Different Intercepts group 1: group 2: Let . Then, group 1: group 2: We can write these two equations in one equation by using a dummy variable: for group 1 and for group2: In this regression, the estimate of intercept for group 1 ( ) is , and the estimate of the intercept for group 2 ( ) is . The difference in intercept terms is of course . The estimate of the marginal effect of x is for both groups. We may define another dummy variable, for group 2 and for group 1. If we specify the regression equation as the estimate of intercept for group 2 ( ) is , and the estimate of the intercept for group 1 ( ) is .
3 Remark: What happens if we include both dummy variables in the same equation such as This cannot be estimated because of perfect multicollinearity , that is, for all observations. The intercept terms for group 1 and 2 are and , and there is no way to separate from . However, we can estimate the equation without the intercept term The intercept for each group is . Example 7.1. wage equations for female and male: for female and for male The estimate of male's intercept term is -1.57 and the estimate of female's intercept term is -1.57-1.81=-3.38. The difference between female and male is -1.81. This means that a female with the same education, job experience and tenure as a male receives \$1.81 less per hour. This estimation uses 1976 data, and the average wages for the male and female were \$7.10 and \$4.59, respectively. Is this difference of \$1.81 statistically significant? To test this, we use the estimate of the difference, the

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note07 - 1 Chapter 7 Qualitative Variables Previous...

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