# Dummy variables interpretation in multiple regression

If I have the following maximum model in an analysis: $y = b_0 + b_1 x_1 + b_2 x_2 + b_3 x_3 + b_4 x_4 + b_5 x_5 + b_6 x_6$

where

• $y$ = home price
• $x_1$ = number of rooms
• $x_2$ = number of bathrooms
• $x_3$ = sq.ft. and the following dummy variables which represent a category for state (where California is the reference)
• $x_3$ = Florida
• $x_4$ = Delaware
• $x_5$ = New York
• $x_6$ = Texas

and after analysis, my regression equation is: $y = b_0 + b_2 x_2 + b_3 x_3 + b_4 x_5$

How would I interpret that last variable since it's a dummy? Would I still have to compare it to the reference in my explanation? Or could I just refer to the coefficient in terms of New York?

• What happened/what did you do during analysis to arrive at the second equation? Apr 12, 2018 at 15:44
• I've never yet seen a dataset on such a subject in which working on log scale didn't produce a more reasonable model. Apr 12, 2018 at 16:01
• $b_4 x_5$ is a typo for $b_5 x_5$ but substantive errors should always be for the OP to fix. Apr 12, 2018 at 16:03
• Wouldn't it make more sense to keep ALL of your state dummy variables in the final model? Together, the states you consider cover the entire geographic area you are interested in. The dummy variables are used to encode the effect of the categorical "state" variable. Apr 13, 2018 at 16:40

Assuming you have 5 levels in the location variable, $\beta_5$ is the mean difference of New York compared to the reference group (the one that was not included in this model) adjusting for room, bat, and square footage, so it should be interpreted as such. $\beta$s and their associated p-value within a single categorical variable can change if you change your reference group, so it's crucial to mention "compared to what."
Since $x_4$ and $x_6$ are ignored in this model, for a fixed value of $x_2$ and $x_3$ homes in California, Delaware and Texas have the same mean.
So, $b_4$ is the marginal change in price for a house in New York relative to a house in the reference group (any of CA, DE, TX).