How to interpret categorical variables in an OLS when only one category is statistically significant? I am running a simple OLS.


*

*Dependent Variable: Population Change In A Congressional District After An Election

*Independent Variable: Who won the election: Democrat, Republican, or, Independent (Categorical)


I've created dummies variables for the different categories and included two of them in two regression models:


*

*PopulationChange = DemocratWin + RepublicanWin + e

*PopulationChange = DemocratWin + IndependentWin + e
However, when I run the model, only DemocratWin is statistically significant. Substantively, how do I interpret that?
 A: 1st I would recommend including an intercept term. 
2nd I would suggest doing an F-test of joint significance for the set of categorical variables. If that is significant, proceed to looking at interpreting individual coefficients. The F-test is an indicator of the significance of the category. 
3rd if the F-test indicates joint significance, then as Charlie said, the interpretation of individual coefficients is done with respect to the intercept.
A: If you allow a constant term in your model it suggests that the average change when an Independent wins is close (in some sense) in your data to the average change when a Republican wins.  If this is modern United States with few Independent wins then it may be coincidence. 
If you do not allow a constant term in your model, it suggests that the average population change with Independent wins or Republican wins is close (again in some sense) to zero.
Perhaps you could tell us what the average population changes are in the three cases, and how many there are of each case.
A: Let's consider a simplified model that extends easily to your situation.
Let $x_i$ be an indicator variable, a variable that is 1 if $i$ has a particular quality or 0 otherwise.
We have
$$\begin{align*}
\text{E}[y_i \mid x_i] &= x_i \times \text{E}[y_i \mid x_i =1] + (1 - x_i)\times \text{E}[y_i \mid x_i = 0] \\
&= \text{E}[y_i \mid x_i = 0] + \big[\text{E}[y_i \mid x_i=1] - \text{E}[y_i \mid x_i=0]\big] x_i \\
&= \beta_0 + \beta_1 x_i.
\end{align*}$$
The coefficient on the dummy variable is the difference between the expected outcome for that group and the expected outcome when that dummy variable is 0---the expected outcome for the reference group.
In your case, the reference group is independent-wins districts. So the coefficient on the Democrat-win indicator is the difference in population growth between Democrat- and independent-win districts, with an analogous interpretation for the Republican-win coefficient.
If you only include the Democrat-win indicator, then your reference group is a district where an independent or a Republican won (whatever could happen when the Democrat-win dummy is 0).
My guess is that you have so few independent-win districts that you are getting big standard errors, giving you a lack of significance on your D-win and R-win coefficients. I might exclude independent-win districts and focus only on districts where one of the two major parties won. Then, I'd compare growth in D-win districts relative to R-win districts by looking at the coefficient on a D-win indicator variable, as given by the simple example above. But, maybe you care about I-win districts and want to include them in your analysis, despite a lack of power (the ability to detect the difference between your estimate and other possible values of the coefficients).
