# Multiple linear regression and the betas

So I have a question about multiple linear regression.

$$Y^j=\beta_1 X_1^j+\beta_2 X_2^j +\cdots +\beta_p X_p^j + \epsilon \tag{*}$$

When I test the significance of the $$\beta$$ with student or Fisher statistic using for instance EXCEL or R some of the $$\beta$$ are non significant meaning according to the test some of the betas are equal to zero and some not.

My question is easy: do i have to take out the betas who are equal to zero in the model(*) and i have a new model( or is it the same ??) without the betas equal to zeros ?

thanks in advance i hope everyone understand me !).

• Yes, you omit the variables where the betas can be considered as 0. Then you can think about what you do with the omitted variables. You also should repeat the F-test for the whole model. Did the model get worse after reducing the variables? – callculus Mar 22 at 20:53
• In general you also make a reality check. It is plausible to omit a variable "height" when you want to predict the weight? – callculus Mar 22 at 20:59
• thanks for the answer. I've just read [link] math.stackexchange.com/questions/2269791/… and one of the answers say that you shouldn't omit the the variables where the betas can be considered as 0 – Minkowski Yaacov Mar 22 at 21:00
• I haven´t found what you said. But sure at the first step you omit the variable where t-test says that $\beta_i=0$. But only if it makes sense somhow, see my second comment. – callculus Mar 22 at 21:06
• the first answer by Yujie Zha – Minkowski Yaacov Mar 22 at 21:09

There are many other (and better) approaches for model selection, i.e., picking which covariates to include in the model. You can for example, use all-subset regression (\ell_0 penalty) together with any of these measures: AIC, BIC, Mallow's $$C_p$$, adjusted $$R^2$$, prediction error sum-of-squares (PRESS). You can also use the Lasso or concave penalties (like the MCP) to avoid going over all subsets.