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From the research I've done the common answer is that you can not remove insignificant dummy variables from a regression. I'm having a hard time finding academic papers or books that back up this thought. If anyone knows of any papers dealing with this issue or a good place to search through statistics papers that would be great!

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    $\begingroup$ This is contradictory. What "research" is this that doesn't involve reading? You seem to be saying "I've found some [unexplained] references arguing a point of view. Please suggest some more references arguing likewise." $\endgroup$ – Nick Cox Dec 3 '19 at 15:19
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    $\begingroup$ Pedantically or otherwise, the issue is not whether you can do this, but whether you should. In essence one or more indicator variables coding two or more categories are a team effort. The usual argument is that you accept the whole team, or not. But I can't see that some aggregation of infrequent categories might not make some sense, but you need to be open about why you do it. $\endgroup$ – Nick Cox Dec 3 '19 at 15:21
  • $\begingroup$ What I was trying to ask is if anyone knows of any stats papers that deal with this issue. The research I've done has been mostly web based forums like this one, but I was hoping to find a more concrete source. [relevant links: stats.stackexchange.com/questions/94441/…, stats.stackexchange.com/questions/154758/… $\endgroup$ – Peyton Dec 3 '19 at 15:27
  • $\begingroup$ See stats.stackexchange.com/questions/24298/… and search this site. $\endgroup$ – kjetil b halvorsen Dec 3 '19 at 18:14
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I fear that your question might be considered ambiguous. Let us say you have a regression of the form: $yY_i = a . X_i + b2 . D2_i + b3 . D3_i + \epsilon_i$ Where D2 and D3 are dummies variable respectively equal to 1 if the underlying variable $D_i==2$ (respectively $D_i==3$). Reference category is $D_i ==1$.

Do you ask whether you should withdraw: - case A: both D2 and D3 if (b2 and b3) prove jointly insignificant? - case B: D2 if for instance b2 proves insignificant?

Case A : I do not think there is something special with dummy variables. You should ask another question. (My own understanding would be: why not? But please note that it changes the interpretation of the coefficient a. More has to be known regarding your objectives, I imagine.)

Case B: why not? But this will change the interpretation of coefficient b3, as reference category is now: $D_i==$ 1 or 2.

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