Section 13.1 Analyzing Voting for Obama
Let’s start by analyzing voting for Obama (vote_1) as the dependent variable:
. reg vote_1 race_2 race_3 race_4 race_5 race_6
Source | SS df MS Number of obs = 3,036
-------------+---------------------------------- F(5, 3030) = 76.29
Model | 83.3981974 5 16.6796395 Prob > F = 0.0000
Residual | 662.426572 3,030 .218622631 R-squared = 0.1118
-------------+---------------------------------- Adj R-squared = 0.1104
Total | 745.824769 3,035 .245741275 Root MSE = .46757
------------------------------------------------------------------------------
vote_1 | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
race_2 | .4972868 .0281049 17.69 0.000 .4421802 .5523934
race_3 | .2078207 .0541766 3.84 0.000 .1015941 .3140472
race_4 | .1028423 .1353307 0.76 0.447 -.162507 .3681916
race_5 | .3135004 .032158 9.75 0.000 .2504466 .3765542
race_6 | .1042547 .0441427 2.36 0.018 .017702 .1908075
_cons | .480491 .0097901 49.08 0.000 .4612952 .4996868
Since our independent variable is qualitative, we have an omitted category. In this case, we’ve left category 1 (race_1) out of our regression, which indicates non-Hispanic White respondents. Our constant (or y-intercept) indicates the predicted value of the dependent variable when all independent variables are equal to zero. We can see this by writing out the regression equation:
\begin{equation}
\widehat{vote\_1}=.48+.50race\_2+.21race\_3 +.10race\_4+.31race\_5+.10race\_6\tag{13.1.1}
\end{equation}
For non-Hispanic White respondents (the omitted category), all the included race dummy variables (race_2 through race_6) equal zero, so we get:
\begin{equation*}
\widehat{vote\_1}=.48+.50(0)+.21(0)+.10(0)+.31(0)+.10(0)= .48
\end{equation*}
Remember, vote_1 is equal to zero if the respondent didn’t vote for Obama, and it is equal to one if the respondent did vote for Obama. Our predicted value is neither zero nor one; instead, we get .48. This can be interpreted as indicating the probability of a one. In other words, a non-Hispanic White respondent has a .48 probability of voting for Obama. We can also convert this probability to a percentage by moving the decimal place two spots to the right: a non-Hispanic White person is estimated to have a 48% chance of voting for Obama, according to this model.
Now, let’s look at the slope coefficients. The coefficient for Black (race_2) equals .50. Thus, a one-unit increase in race_2 is associated with a .50-unit increase in vote_1. Let’s break that down a bit to see if we can create a clearer interpretation. Since race_2 is a dummy variable and non-Hispanic White is the omitted category, a one-unit increase in race_2 correspondents to having a Black respondent instead of a White respondent. And since our dependent variable is binary, we should think in terms of probabilities, which can be converted to percentages: a .50-unit increase in vote_1 means a 50 percentage-point increase in the probability of voting for Obama. So putting this altogether, we’d say: (non-Hispanic) Black voters are 50 percentage points more likely to vote for Obama than (non-Hispanic) White voters, according to this model.
Similarly, Asian voters are 21 percentage points more likely to vote for Obama than (non-Hispanic) White voters. Native American voters are 10 percentage points more likely to vote for Obama than (non-Hispanic) White voters. Hispanic voters are 31 percentage points more likely to vote for Obama than non-Hispanic White voters. And voters identifying as multiracial or other race are 10 percentage points more likely to vote for Obama than (non-Hispanic) White voters. All of these differences are statistically significant, except for Native American versus White voters (probably because there are only 27 Native Americans in the sample, making the estimate of this difference very imprecise).
