Interpretation of multiple factor variables in linear regression

Question

I couldn't find a similar forum post, where multiple factor variables are included in a regression, which all have multiple base groups. I would like to make my question clear with a short example with a regression equation.

Let's assume we would like to predict console prices (Y) while using the age of the consoles as an independent variable, as well as two color dummies. There are in total 15 colors and only dummy variables of two of them are included, which are 1 when the console is blue (gold) and 0 otherwise.

${\rm price}_i\ =\ \beta_0\ +\ \beta_1{\rm age}_i+\beta_2{\rm blue}_i+\beta_3{\rm gold}_i\ +\upsilon_i$

Let's assume blue as well as gold are significant, what is the correct interpretation of blue?:

1. "Consoles which have the color blue have on average a $\beta_2$ higher/lower price than consoles with different colors, keeping everything else constant"
2. "Consoles which have the color blue have on average a $\beta_2$ higher/lower price than consoles with different colors, except gold, keeping everything else constant"

I am unsure if I need the "except gold" and couldn't find an answer elsewhere.

Please let me know, when you have suggestions how I can improve my question.

Thank you!

Answer 1

You need the except gold part. Notice you are comparing $$\mathbb{E}(price_i|age_i, blue_i = 1) - \mathbb{E}(price_i|age_i, blue_i = 0) .$$ $blue_i = 1 \implies gold_i = 0$. So the "everything else equal" has to fix $gold_i = 0$.