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What is the proper test to determine whether the coefficients of variables in the same linear regression model are different from each other?

Specifically, the variables I am referring to are different levels of a categorical variable and the coefficients may be close enough to each other that there actually is no difference. If they were not correlated, I would merely obtain a z-score (difference between the estimates divided by the square root of the sum of the standard error, squared). However, not sure what to do in this situation.

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Fit the model where you constrain the coefficients to be equal and compare that to the unconstrained model. E.g. if you have two predictors and fit the model

$$ y_i = \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i} + \epsilon_i $$

as the unconstrained model. Then compare this to the model

$$ y_i = \beta_0 + \beta_1 (X_{1i} + X_{2i}) + \epsilon_i $$

And compare using the likelihood ratio test. Operationally, you can do this by by defining a new variable that is the sum of the two predictors and put that into the model.

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  • $\begingroup$ Are you sure X2 in your second equation? Why would you combine both X1 and X2? $\endgroup$
    – SmallChess
    Jan 30, 2017 at 3:17
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    $\begingroup$ @StudentT Because that's the same as forcing them to have the same coefficient. If $\beta_1 = \beta_2$ in the full model (the first model), you'd get the second model. $\endgroup$
    – gammer
    Jan 30, 2017 at 3:21
  • $\begingroup$ Ok. But does that make the two models non-nested? LR test should only work on nested models. $\endgroup$
    – SmallChess
    Jan 30, 2017 at 3:26
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    $\begingroup$ @StudentT, yes. The sub-model is the subspace where $\beta_1 = \beta_2$. Any possible model fitted in the submodel could also be captured in the larger model simply by estimating $\beta_1$ and $\beta_2$ to be the same. You can define nested models that aren't just a matter of deleting predictors. The model is nested as long as the span of the space covered by the sub-model is a subset of that spanned by the full model space, which is the case here. $\endgroup$
    – gammer
    Jan 30, 2017 at 3:27
  • $\begingroup$ @gammer, I understand the logic. However, I don't know how to do this without specifying it manually. I'm running this in R using the LM function, which creates the dummy variables automatically for me. How would I ask it to combine two of the categories? Is there a simpler way than just recoding the categories so that category 1 and category 2 both become category 1. $\endgroup$ Jan 30, 2017 at 14:53

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