1. A two category dummy variable without interactions.

Transcription

1 Vartanian: Data Analysis 541 Dummy Variables in Regression Models 1. A two category dummy variable without interactions. You want to test to see if the mean values for males and females are different from one another. Since this is an OLS model, we need some interval level dependent variable. We'll use income. What you will do is create a single dummy variable that will indicate whether or not the individual is a male or a female. Z 1 is a (0,1) variable. If Z 1 =1 then the observation is a male. If Z 1 =0 then the observation is a female. (1) Y=a+c 1 Z 1 Where c 1 is the parameter estimate for the variable Z 1. c 1 will tell you the difference in income levels between the two groups. If c 1 is a positive number, this will indicate that males have a mean income level that is c 1 larger than females. If c 1 is negative, males will have a mean income level that is c 1 less than females. Why? What you have to do is substitute the different values of Z 1 in the equation to determine the overall mean income levels for the two groups. For males, you will substitute in Z 1 =1 in the equation 1 above. For females, you will substitute in Z 1 =0. You will get a mean income level for both groups by doing this. Why not include variables for both males and females? Because if you did, you would have perfect multicollinearity within your model. Why? If you know that the person is a male, you have perfect knowledge that the person is not a female. Thus, if you're coding your variables as 0,1 for male and female, you would have something that looks like the following: obs male female In other words, when you know the value of either male or female, you can perfectly predict the d:\wp61\probsets\lect2.phd\dummyvar.phd Page 1

2 value of the other variable. Thus, the correlation coefficient between these two variables is perfect, or r=1. We will not be able to determine a SE for these regression coefficients because SE by1.2 ' S 2 y.12 [ j X 2& ( j x 1 )2 ](1&r n ). Also, we cannot determine b coefficients since we cannot distinguish between the two variables. With perfect collinearity, SAS and other computer programs will reject one of the variables but will determine the b s and SEs for the other perfectly correlated variable. To determine the a (or the intercept) and c coefficients for males, (2) Y=a+c 1 *1 Y=a+c 1 For Females, (3) Y=a+c 1 *0 Y=a The difference between the two groups is c 1. Example: Let's say that we only have 6 observations in a sample so that it will be easy to see how dummy variables work. Sex Income ($) From this, you can see that the mean income levels are: Females: 300 Males: 200 In other words, there is a $100 difference in income levels between the two groups. If Z 1 =1 for Page 2

3 males and Z 1 =0 for females, then the equation will have a c 1 value = -100, since the difference in mean income levels is 100 and male mean income is lower than female mean income. What will a (or the intercept) be? We know the value for the mean income levels, so a, the intercept, must make the mean income levels come out to the 300 and 200 levels above. Thus, for females, Y= a (Z 1 ) We know that Z 1 =0 for females, therefore in order to make y=300, a=300. For males, Y= a (Z 1 ) We know that Z 1 =1 for males, therefore in order to make y=200, a=300. Of course, a will always turn out to be the same value in both equations (since they're the same equation but have different values for Z 1 ). This is the logic of dummy variables. When you actually use dummy variables, you will often use them with other variables within the model so that determining the values of a and c 1 will not always be so evident. Using dummy variables in an OLS regression model will give you similar information to using group t-tests or anova models with two groups. The significance levels will be exactly the same using any of these three different statistics (t-test, anova, or dummy variables). 2. Using categorical variables with more than two categories as dummy variables. In this second situation, you are examining the differences in income level among people of different races. Let's assume that there are people of 4 different races you're examining: Whites, African Americans, Hispanics and Asian Americans. What you will do is create 4 separate dummy variables -- one for each of the people of the different races. Let's call these variables z 1, z 2, z 3 and z 4. Let's say that if you're white, z 1 =1, otherwise z 1 =0; if you're African American, z 2 =1, otherwise z 2 =0; if you're Hispanic, z 3 =1, otherwise z 3 =0; and if you're Asian American, z 4 =1, otherwise z 4 =0. What the regression equation will do is examine the differences in income between each of the groups and an excluded group. Thus, what you will be doing is excluding one of the groups above from the regression analysis and the regression results, or the parameters you find will be d:\wp61\probsets\lect2.phd\dummyvar.phd Page 3

4 the difference in income between the group you're examining and the excluded group. Let's say that we exclude Asian Americans from the regression analysis. Our regression equation would be, Y=a+c 1 z 1 +c 2 z 2 +c 3 z 3 The value for c 1 will be the difference in mean income between whites and Asian Americans. c 2 will be the difference in income between African Americans and Asian Americans, and c 3 will be the difference in income between Hispanics and Asian Americans. If the values for the c coefficients are positive, this will indicate that the group being examined has a higher level of income than the excluded group, Asian Americans. Example: Race Income The mean income ($) levels for Whites 200 Af Am 700 Hispanics 400 As A 500 Thus, value for the coefficients will be: c 1 =-300 c 2 =200 c 3 =-100 Page 4

5 Y=a+c 1 z 1 +c 2 z 2 +c 3 z 3 If we want to examine the mean income level for whites, we would give z 1 a value of 1 and z2 and z3 values of 0. Y=a+c 1 We know that the mean income level for whites is =a-300 Therefore, a=500. We could figure out the value of a for the other groups as well. Y=a+c 2 700=a+200 Therefore, a=500. The equation for this, assuming that Asian Americans is the excluded category will be, Y= z z z The interaction between a two category dummy variable and an interval scale variable in a regression model. You re examining the effects of a two category race variable and age on income. X 1 is the variable for age and Z1 is the variable for being white (Z 1 =1 if you re white; Z 1 =0 otherwise). The equation will look like the following: Y p = a + b 1 X 1 +c 1 Z 1 + d 1 X 1 Z 1 b 1 is your coefficient estimate for age, c 1 is the difference in income between whites and people of other races, and d 1 is the difference in the effect of age between whites and people of a different races. If the coefficient estimate for the interaction variable is significant, this means that the interval scale variable, age, has a different effect on whites relative to those of other races. Let s say we re examining income (the DV) and have the following output. d:\wp61\probsets\lect2.phd\dummyvar.phd Page 5

6 Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > T INTERCEP WHITE AGE WHITEAGE Here, a=20830 b 1 = c 1 =27879 d 1 = The d1 coefficient is telling us that age has a more negative effect for whites than it does for people of other races. As the age of white people increases by 1 year, their income decreases by dollars more than it does for people of other races. Thus, the d 1 coefficient estimate is telling you the differential effect that the interval scale independent variable (age) has on the dependent variable (income) for the included group in the dummy variable (whites) versus the excluded group (non-whites). Since the estimate for Whiteage is negative, this is telling us that effect of age on whites is more negative than the effect of age on races other than white. If this coefficient estimate was positive and significant, it would mean that the effect of age on whites would be more positive than the effect of age on people of other races. In order to then determine the slope and intercept coefficients for members of the different races (whites and non-whites), we would do the following: For Whites: Yp = ( ) X + (27879) Z + ( )XZ We can set Z=1, so Yp=20830 ( )X ( )X We can now put like terms together, Yp= ( ) + ( )X Yp= ( )X So the slope coefficient is and the intercept is for whites. For Races other than White: Yp = ( ) X + (27879) Z + ( )XZ Page 6

7 Set Z=0. Yp = ( )X So the slope is and the intercept is for races other than white. 4. The interaction between a four category set of dummy variables and an interval variable in a regression model. Let's say that we're examining the effects of race and age on income. X 1 will be the variable for age. The model without any interactions would be Y=a+b 1 x 1 +c 1 z 1 +c 2 z 2 +c 3 z 3. If we believed that there was some interaction between race and age, we could interact these variables and the model would look like the following, Y=a+b 1 x 1 +c 1 z 1 +c 2 z 2 +c 3 z 3 + d 11 x 1 z 1 +d 12 x 1 z 2 +d 13 x 1 z 3 You will again be excluding one of the categorical variables as well as one of the interaction variables. d 11, d 12, and d 13 are the coefficient estimates the interactions between age and white, age and African American, and age and Hispanic, respectively. Each of the d coefficient estimates (the interaction coefficient estimates) will indicate if the effects of the interval scale variable are different for members of the different races relative to the excluded group. Thus, if d 11 is positive and significant, it indicates that the effects of age on whites is greater than the effects of age on Asian Americans (the excluded group) for the dependent variable (income). If d 12 is negative and significant, it indicates that the effects of age on African Americans is less than the effects of age on Asian Americans (the excluded group) for the dependent variable (income). You now want to determine the slope for the age variable and the intercept for each of the different races. You ll do this by setting the appropriate z variable equal to 1 and the other z variables equal to 0. For whites: Y=a+b 1 x 1 +c 1 +d 11 x 1 This is because the values of z 2 and z 3 are equal to 0 when we are examining whites. The value of z 1 =1 when we are examining whites. For African Americans: Set z2=1, z1=0, z3=0. Y=a+b 1 x 1 +c 2 +d 12 x 1 d:\wp61\probsets\lect2.phd\dummyvar.phd Page 7

8 For Hispanics: Set z1=0, z2=0 and z3=1. Y=a+b 1 x 1 +c 3 +d 13 x 1 For Asian Americans: Set z1=0, z2=0, and z3=0. Y=a+b 1 x 1 What we find is that once we set the z values equal to either 1 or 0, we are left with the c coefficient standing on their own. These c coefficients are simply constants. Since the c values are not being multiplied by an X variable, these c coefficients are not slopes. We can add these constants to the intercept term, a, and come up with a new intercept term. We can also group together the coefficients that go with x 1 to determine a new slope coefficient for each of the different races. Each race will have a different slope and intercept. For Whites: Y=(a+c 1 )+(b 1 +d 11 )x 1 The new intercept is a+c 1 and the new slope is (b 1 +d 11 ). For African Americans: Y=(a+c 2 )+(b 1 +d 12 )x 1 The new intercept is a+c 2 and the new slope is (b 1 +d 12 ). For Hispanics: Y=(a+c 3 )+(b 1 +d 13 )x 1 The new intercept is a+c 3 and the new slope is (b 1 +d 13 ). For Asian Americans: Y=a+b 1 x 1 The intercept is a and the slope is b 1. Page 8